my finance lessons

2011-01-26T07:04:00.001-08:00

An online doctorate in financial engineering offers students the unique opportunity to combine financial theory, applied mathematics and computer science into a fast-growing interdisciplinary degree. Financial engineering is closely tied with quantitative finance and financial mathematics, although it presents a unique niche in the field. Due to the added experience of computer programming and science, graduates of an online PhD in financial engineering can pursue lucrative career paths in risk management, investment banking and the financial management industries. In addition, as an online doctoral program can be completed outside of standard employment hours, a student may continue obtaining work experience and advancement, giving them an additional advantage when pursuing employment.
Online PhD in Financial Engineering Requirements

Candidates pursuing an online doctorate in financial engineering can expect rigorous coursework. Successful students must have motivation, focus and a background in finance, applied mathematics and computer science. An online PhD program in financial engineering coursework assumes students have had previous experience in the discipline and most candidates apply to the program with a Master degree in hand.

Candidates for an online doctoral program in financial engineering typically have one of the following Master degrees:

* Master of Science in Computer Science
* Master of Science in Financial Engineering
* Master of Science in Quantitative Finance
* Master of Science in Financial Mathematics
* Master of Science in Applied Mathematics
* Master of Science in Economics
* MBA in Economics
* MBA in Finance

In addition, students may have prior experience in financial engineering as computer programmers, financial managers or investment bankers.
Online PhD in Financial Engineering Education

Online PhD programs in financial engineering take 4-5 years to complete. During the course, students have hands-on project experience, take advanced finance, computer science and mathematics courses and submit a lengthy dissertation or extensive project.
Information on Financial Engineering Courses

Some common courses found in a financial engineering program are:

* Financial Engineering I and II
These courses focus on modern theory and techniques in the following areas: financial instruments, pricing options, managing financial risks, applications for assets and complex trading strategies.
* Advanced Corporate Finance
Topics in Advanced Corporate Finance range from capital structure, mergers, acquisitions and buyouts, venture capital and pensions. The focus of this course is hands-on, applied case studies.
* International Finance
This course examines aspects of international corporate finance, including exchange rate risk management, hedging in foreign markets and competitive strategy in a global environment.
* Mergers and Acquisitions
The objective in this course is to give candidates experience in valuing firms. The primary focus is mergers and acquisitions, although students will also study initial public offerings, buyouts and divestitures.
* Economic Statistics
This course covers statistical interference in detail, using electronic data acquisition and computer modeling. Prior experience with statistics is required.
* Advanced Computer Applications
Students are introduced to some commonly used risk management and data structures and their applications using C++. Topics include abstract data types, object-oriented programming, stacks, queues, linked lists, sorting, binary search trees, heaps, and hashing. Prerequisite programming courses are required.
* Applications Programming for Financial Models
Options pricing and risk management are essential tools for a financial engineer. This course demonstrates techniques in implementation of computation-intensive option pricing models in evolving programming architectures.

Specializations and Concentrations

While a top PhD program in financial engineering provides a path into a niche market, there are several specializations on which candidates may focus their studies during their independent research and dissertation.

Some common concentrations in an online doctoral program in financial engineering include:

* Data Analysis
* Strategic Operations
* Systems Engineering
* Applied Operations
* Applied Probability
* Risk Management-Actuarial

Careers in Financial Engineering

Graduates who earn an online doctorate in financial engineering from an accredited institution enjoy a variety of employers seeking candidates with their unique skill set. Some common careers opportunities for graduates of the best online doctoral programs in financial engineering include:

* Investment Trader
Investment traders with a financial engineering degree are held in high esteem within investors for their extensive knowledge of trading systems, financial management and data analysis. Traders work for corporations and smaller banks or investment firms, and must have the knowledge and skills to make extremely quick decisions to increase profitability to the client. Investment traders earn an average of $118,000 per year.
* Risk Management
Risk managers oversee programs to minimize risks and losses that might arise from financial transactions and business operations at investment firms, banks and consultant firms. Risk managers control financial risk by using hedging and other techniques to limit a company’s exposure to damaging change. Risk managers earn between $73,530 and $138,010 per year.
* Computer Programming
Computer programmers convert project specifications into detailed logical flow charts for coding into computer language. Financial engineers with a specialty in programming develop and write computer programs to handle difficult risk management, trading or tracking elements of financial management. Computer programmers earn a median of $74,950 per year.
* Actuarial Risk Management
Actuaries handle risk management for the insurance sector of finance. They collect and analyze data to make predictions of risk factors for a specific population or problem. Actuarial risk managers earn an average of $95,980. This is the fastest growing sector of the finance industry.
* Research and Education
Financial engineering researchers typically work at a university or college that offers this degree. Professors are required to hold a PhD, and both teach courses in their subject area and conduct and publish research. A professor earns between $45,977 and $108,749 per year.

Financial engineering is a relatively new, exciting and fast-growing field. This innovative program combines extensive financial knowledge and skill with computer programming experience. Graduates with a doctoral degree in financial engineering have the benefit of an excellent variety of career options.

2011-01-26T06:47:00.000-08:00

The Pythagoreans

Formal Mathematics

Linear Polynomials

Systems of Linear Equations

Quadratic Polynomials

Polynomial Equations

More About Functions

Logarithms

Limits

The Number e

Interest Calculations

Pricing Fixed Income Instruments

Day Count Conventions

Physical Interpretation of Linear Polynomials

Average Rates

Instantaneous Rates

Forward Prices

Calculus Derivatives

Applications of Calculus Derivatives

Higher-Order Derivatives

More Calculus Derivatives

More Applications of Derivatives

Newton’s Method

Optimization

Duration and Convexity

Anti-Derivatives

Applications of Anti-derivatives
***************

The Pythagoreans

Formal Mathematics

Linear Polynomials

Systems of Linear Equations

Quadratic Polynomials

Polynomial Equations

More About Functions

Logarithms

Limits

The Number e

Interest Calculations

Pricing Fixed Income Instruments

Day Count Conventions

Day Two

Physical Interpretation of Linear Polynomials

Average Rates

Instantaneous Rates

Forward Prices

Calculus Derivatives

Applications of Calculus Derivatives

Higher-Order Derivatives

Day Three

More Calculus Derivatives

More Applications of Derivatives

Newton’s Method

Optimization

Duration and Convexity

Anti-Derivatives

Applications of Anti-derivatives
**********************
Matrix Computations

Partial Derivatives

Probability Distributions

Random Vectors

Lectures Syllabus

Day One

Statistics

Point Estimation

Determinants

Linear Independence

Singular Matrices

Eigenvalues and Eigenvectors

Optimizing Quadratic Forms

Method of Least Squares

Optimal Hedge Ratios

Dynamic Hedging

Day Two

Maximum-Likelihood Estimators

Time Series Analysis

White Noise, MA, AR and ARMA Processes

Efficient Markets

Capital Asset Pricing Model

Black-Scholes Theory

Fitting a Stochastic Process to Data

Differential Equations

Separable Equations, Exact Equations, and Integrating Factors

Day Three

Random Walks and Weiner Processes

Stochastic Differentiation

Ito's Lemma

Black-Scholes Option Pricing Formula

Stochastic Integration

Stochastic Differential Equations

Derivatives Pricing
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Random Walks and Weiner Processes

Stochastic Differentiation

Ito's Lemma

Stochastic Integration

Stochastic Differential Equations

Course Syllabus

Day One

Samuelson (1965) Model

Arbitrage Pricing vs. Expectations Pricing

Black-Scholes (1973) Model

Black-Scholes Partial Differential Equation

Partial Differential Equations

The Diffusion Equation

Initial Values and Boundary Conditions

Solutions for European Options

Day Two

American Options as Free Boundary Problems

American Options as Variational Inequalities

Dividends and Time-Dependent Parameters

Foreign Exchange

Risk Neutral Valuation

Examples

Fixed Income

Arbitrage-Free Pricing

Standard Term Structure Models
*********************

General areas of finance

2011-01-25T21:00:00.000-08:00

General areas of finance

Financial markets
Investment management
Financial institutions
Personal finance
Public finance
Mathematical finance
Quantitative behavioral finance
Financial economics
Experimental finance
Computational finance
Statistical finance
**********
Financial engineering

Artificial Intelligence and Investing (information science)

2011-01-15T03:45:00.000-08:00

Introduction

The techniques of artificial intelligence include knowledge- based, machine learning, and natural language processing techniques. The discipline of investing requires data identification, asset valuation, and risk management. Artificial intelligence techniques apply to many aspects of financial investing, and published work has shown an emphasis on the application of knowledge-based techniques for credit risk assessment and machine learning techniques for stock valuation. However, in the future, knowledge-based, machine learning, and natural language processing techniques will be integrated into systems that simultaneously address data identification, asset valuation, and risk management.
WHAT IS ARTIFICIAL INTELLIGENCE?

Computers play a role in many aspects of investing. For example, program trading is computer-driven, automatically executed trading of large volumes of shares, and has become increasingly prominent on stock exchanges. Artificial intelligence is a technique of computing that is perpetually on the cutting edge of what can be done with computers. Artificial intelligence could apply to program trading, but also other aspects of investing.

In the early days of computing, a typical task for a computer program was a numerical computation, such as computing the trajectory of a bullet. In modern days, a typical task for a computer program may involve supporting many people in important decisions, backed by a massive database across a global network. As the tasks that computers typically perform have become more complex and more closely intertwined with the daily decisions of people, the behavior of the computer programs increasingly assumes characteristics that people associate with intelligence. When, exactly, a program earns the label of "artificial intelligence" is unclear. The classic test for whether a program is intelligent is that a person would not be able to distinguish a response from an intelligent program from the response of a person. This famous Turing Test is dependent on factors not easily standardized, such as what person is making the assessment under what conditions.

A range of computer programming techniques that are currently, popularly considered artificial intelligence techniques includes (Rada 2008):

• Knowledge-based techniques, such as in expert systems.

• Machine learning techniques, such as genetic algorithms and neural networks.

• Sensory or motor techniques, such as natural language processing and image processing.

These methods may apply to investing. For instance, expert systems have been used to predict whether a company will go bankrupt. Neural networks have been used to generate buy and sell decisions on stock exchange indices. Natural language processing programs have been used to analyze corporate news releases, and to suggest a buy or sell signal for the corporate stock.

While artificial intelligence (AI) could apply to many areas of investing, much of what happens in computer-supported investing comes from non-AI areas. For instance, computational techniques not considered primarily AI techniques include numerical analyses, operations research, and probabilistic analyses. These non-AI techniques are routinely used in investing.
INVESTING AND DATA

The process of investing has three stages of:

• Data identification,

• Asset valuation, and

• Risk management.

AI has been most often applied to asset valuation, but is also applicable to data identification and risk management.

Two, high-level types of data used in financial investing are technical data and fundamental data. The price of an asset across time is technical data, and lends itself to various computations, such as the moving average or the standard deviation (volatility). Fundamental data should support cause-and-effect relationships between an asset and its price. For instance, the quality of management of a company should influence the profitability of a company and thus, the price of its stock.

The universe of fundamental data is infinite. Many streams of data that might be relevant, such as corporate earnings or corporate debt, might also be related to one another. Various non-AI tools, such as linear regression analysis and principal components analysis, might be used in identifying what sets of data are more likely to be useful than what other sets. Such non-AI, computational techniques can be combined with AI techniques in experimenting with various combinations of data and choosing what data to use in asset valuation.
ASSET VALUATION

Different computational techniques might be appropriate for different assets or for different types of data for a particular asset. For instance, both stocks and commodities have price histories that might be tracked by the same time series analysis methods. However, the knowledge bases that would apply to valuing these assets might be significantly different. For example, knowledge about corporate management is less germane to commodity valuation than to stock valuation, while knowledge about weather patterns is more germane to commodity valuation than stock valuation.

Many assets have derivatives, such as options, that are priced and exchanged on markets. The computational characteristic of the Black-Scholes option pricing equation means that option valuation is done with the support of computer programs. Solving the Black-Scholes equation is not an artificial intelligence operation, although adequately handling options valuation could well involve artificial intelligence techniques.

An index is a special kind of asset. For instance, Standard & Poor's 500 Index (S&P500) is a widely traded asset that represents, with a single number, the price of shares of 500 companies. Programs that would be appropriate for evaluating the fair price of the S&P500 would be different from programs designed to evaluate the fair price of a particular company's shares. Among other things, the S&P500 does not have corporate management, per se. Neural networks have been extensively applied to predicting prices of stock indices.

Atypical technical approach to a market problem (Chun & Park, 2006) took daily values over 5 years for five attributes of the Korean stock price index: daily high and low values, daily opening and closing values, and daily trading volume. On the other hand, the bond rating work of (Kim & Lee, 1995) looks at fundamental data with an expert system The input data for the bond rating work considers the quality of management and the quality of financial policies. The expert system's approach has a professional interactively answer questions from the system. Through this user interactivity, the system might collect subjective information, such as a company's management quality.

If a bank considers lending money to a company, the bank would be interested injudging the likelihood that the company would go bankrupt. More generally, financial institutions that lend money want to judge the credit worthiness of the entities to which they might lend money. These valuations of credit worthiness are a kind of asset valuation, but the techniques for doing this credit assessment would tend to be different from those for assessing the fair price of a company share. In particular, expert systems are more likely to be used for bankruptcy predictions, and neural networks are more likely to be used for stock price prediction. A bank may take its time in deciding what conditions, if any, to offer for a loan. Once the loan is made, its conditions are not subject to ready change. Investing in stocks or financial derivatives may be a fast-moving activity based on a history of prices. Those prices may be volatile, and entry and exit from the market may occur any time. The speculative financial investing problem is more of a time-series problem than the financial accounting problem and is thus, amenable to a different set of computational tools.
RISK MANAGEMENT

Risk or portfolio management involves choosing the asset classes in which to invest, and modifying the held assets across time, so as to suit the investment objectives. Various mathematical models, such as the Markowitz portfolio selection model, may be used by professional managers to guide the diversification of holdings so as to minimize risk for any specified rate of return. Implementing this kind of portfolio management may rely on numerical computation at one level, but can also benefit from various artificial intelligence techniques.

Lee et al. (Lee, Trippi, Chu, & Kim,1990) have described a knowledge-based system for supporting portfolio management. The system has different agents for different tasks. One agent elicits client goals, another agent implements dynamic hedging strategies, and another suggests market-timing decisions. Lee et al. note that the agents are only successful in narrow domains, and intervention of the human, portfolio manager is regularly necessary. In more recent work, Abdelazim and Wahba (2006) use genetic algorithms and neural networks to modify the parameters suggested by the Markowitz portfolio selection model, and obtain portfolios that earn higher returns at a specified risk level.
AI TRENDS

A multiagent architecture for an integrated system that considers data identification, asset valuation, and risk management has been proposed by researchers at Carnegie Mellon University. The system is called WARREN, which refers to the first name of the famous investor Warren Buffet (Sycara, Decker, Pannu, Williamson, & Zeng, 1996). The WARREN system design includes components for collecting large amounts of real-time data, both numeric and textual. The data would be preprocessed and then fed to various asset valuation agents that would, in turn, feed their assessments to a portfolio management agent. The portfolio management agent would interact with clients of WARREN. Systems with various features of WARREN are available from commercial vendors, and are developed in-house by large investing companies, but more research is needed on how to develop integrated, AI systems that support investing.

Natural language processing systems may include large bodies of domain knowledge and parse free text, so as to make inferences about the content of the text. However, such natural language processing systems do not seem as popular in investing applications as much simpler natural language processing techniques. The natural language processing work that has been applied to the investing seems to be largely of the sort in which the distribution of word frequencies in a document is used to characterize the document. In this word-frequency way, Thomas (2003) has shown a potential value to processing news stories to help anticipate stock price changes.

As one can see cycles in the value of financial assets, one can also see cycles in the frequency of publication of articles on certain topics. In the field of artificial intelligence, one might identify, roughly speaking, three phases, as follows (Rada, 2008):

1. machine learning, in what was then called perceptron and self-organizing systems research, was popular from 1955 to 1975,

2. knowledge-based, multiagent, or expert systems work was popular from 1975 to 1995, and

3. machine learning research, now called neural networks or genetic algorithms research, returned to dominate the AI research scene from 1995 to the date of this article.

When AI research has been applied to investing, the AI technique used has tended to be the technique popular at the time. This leaves, unaddressed, the question of whether investing is more appropriately addressed with one AI technique or another.

The recent literature is rich with neural network applications to investing, but a new trend is the combining of knowledge-based techniques with neural network and genetic algorithm techniques. For instance, Tsakonas et al. (Tsakonas, Dounias, Doumpos, & Zopounidis, 2006) use "logic" neural nets that can be directly understood by people (traditional neural nets are a "black box" to humans). Genetic programming modifies the architecture of the logic neural net by adding or deleting nodes of the network in a way that preserves the meaning of the neural net to people and to the net itself. Bhattacharyya et al. (Bhattacharyya, Pictet, & Zumbach,2002) have added knowledge-rich constraints to the genetic operators in their application for investing in foreign exchange markets.

A promising research direction is to combine the earlier knowledge-based work on financial accounting with the more recent work on machine learning for stock valuation. For instance, neural logic nets could represent some of the cause-effect knowledge from a bankruptcy system and become part of a learning system for predicting stock prices. Some of the bankruptcy variables are readily available online, such as a company's debt, cash flow, and capital assets.

The financial markets are human markets that evolve over time as opportunities to make profits in this zero-sum game depend on the changing strategies of the opponent. Thus, among other things, what is important in the input may change over time. An AI system should be able to evolve its data selection, asset valuation, and portfolio management components. The future direction for AI in investing is to integrate the three major tools of AI (knowledge-based systems, machine learning, and natural language processing) into a system that simultaneously handles the three stages of investing (data collection, asset valuation, and portfolio management). Such systems will interact with humans so that humans can specify their preferences and make difficult decisions, but in some arenas, such as program trading, these sophisticated AI systems could compete with one another.
key terms

Artificial Intelligence: The ability of a computer to perform activities normally considered to require human intelligence

Asset Valuation: The process of determining the worth of something

Expert System: A program that uses knowledge and inferences to solve problems in a way that experts might

Investing: The act of committing money to an endeavor with the expectation of obtaining profit.

Neural Networks: Programs that simulate a network of communicating nerve cells to achieve a machine learning objective

Risk Management: The process of managing the uncertainty in investment decision-making.
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Scope of Artificial Intelligence in Business

Introduction

Business applications utilize the specific technologies mentioned earlier to try and make better sense of potentially enormous variability (for example, unknown patterns/relationships in sales data, customer buying habits, and so on). However, within the corporate world, AI is widely used for complex problem-solving and decision-support techniques in real-time business applications. The business applicability of AI techniques is spread across functions ranging from finance management to forecasting and production.
In the fiercely competitive and dynamic market scenario, decision-making has become fairly complex and latency is inherent in many processes. In addition, the amount of data to be analyzed has increased substantially. AI technologies help enterprises reduce latency in making business decisions, minimize fraud and enhance revenue opportunities.

Definition of AI

AI is a broad discipline that promises to simulate numerous innate human skills such as automatic programming, case-based reasoning, neural networks, decision-making, expert systems, natural language processing, pattern recognition and speech recognition etc. AI technologies bring more complex data-analysis features to existing applications.
There are many definitions that attempt to explain what Artificial Intelligence (AI) is. I like to think of AI as a science that investigates knowledge and intelligence, possibly the intelligent application of knowledge. Knowledge and Intelligence are as fundamental as the universe within which they exist, it may turn out that they are more fundamental.
One of the aims of AI is said to be the investigation of human cognition and AI is part of Cognitive Science. AI is really an investigation into the creation of intelligence and that there is no reason for the intelligence that is created to be exactly the same as human intelligence.

Importance of AI

Enterprises that utilize AI-enhanced applications are expected to become more diverse, as the needs for the ability to analyze data across multiple variables, fraud detection and customer relationship management emerge as key business drivers to gain competitive advantage.
Artificial Intelligence is a branch of Science which deals with helping machines, finds solutions to complex problems in a more human-like fashion. This generally involves borrowing characteristics from human intelligence, and applying them as algorithms in a computer friendly way. A more or less flexible or efficient approach can be taken depending on the requirements established, which influences how artificial the intelligent behavior appears.
AI is generally associated with Computer Science, but it has many important links with other fields such as Maths, Psychology, Cognition, Biology and Philosophy, among many others. Our ability to combine knowledge from all these fields will ultimately benefit our progress in the quest of creating an intelligent artificial being.

Emergence of AI in business

Artificial Intelligence (AI) has been used in business applications since the early eighties. As with all technologies, AI initially generated much interest, but failed to live up to the hype. However, with the advent of web-enabled infrastructure and rapid strides made by the AI development community, the application of AI techniques in real-time business applications has picked up substantially in the recent past.
Computers are fundamentally well suited to performing mechanical computations, using fixed programmed rules. This allows artificial machines to perform simple monotonous tasks efficiently and reliably, which humans are ill-suited to. For more complex problems, things get more difficult... Unlike humans, computers have trouble understanding specific situations, and adapting to new situations. Artificial Intelligence aims to improve machine behavior in tackling such complex tasks.
Together with this, much of AI research is allowing us to understand our intelligent behavior. Humans have an interesting approach to problem-solving, based on abstract thought, high-level deliberative reasoning and pattern recognition. Artificial Intelligence can help us understand this process by recreating it, then potentially enabling us to enhance it beyond our current capabilities.

Applications of AI

The potential applications of Artificial Intelligence are abundant. They stretch from the military for autonomous control and target identification, to the entertainment industry for computer games and robotic pets, to the big establishments dealing with huge amounts of information such as hospitals, banks and insurances, we can also use AI to predict customer behavior and detect trends.
AI is a broad discipline that promises to simulate numerous innate human skills such as automatic programming, case-based reasoning, decision-making, expert systems, natural language processing, pattern recognition and speech recognition etc. AI technologies bring more complex data-analysis features to existing applications.
Business applications utilize the specific technologies mentioned earlier to try and make better sense of potentially enormous variability (for example, unknown patterns/relationships in sales data, customer buying habits, and so on). However, within the corporate world, AI is widely used for complex problem-solving and decision-support techniques in real-time business applications. The business applicability of AI techniques is spread across functions ranging from finance management to forecasting and product

Artificial Neural Networks

An artificial neural network (ANN), often just called a "neural network" (NN), is a mathematical model or computational model based on biological neural networks. It consists of an interconnected group of artificial neurons and processes information using a connectionist approach to computation. In most cases an ANN is an adaptive system that changes its structure based on external or internal information that flows through the network during the learning phase. In more practical terms neural networks are non-linear statistical data modeling tools. They can be used to model complex relationships between inputs and outputs or to find patterns in data.

Real life applications of ANN

The tasks to which artificial neural networks are applied tend to fall within the following broad categories:
• Function approximation, or regression analysis, including time series prediction and modeling.
• Classification, including pattern and sequence recognition, novelty detection and sequential decision making.
• Data processing, including filtering, clustering, blind source separation and compression.
Application areas include system identification and control (vehicle control, process control), game-playing and decision making (backgammon, chess, racing), pattern recognition (radar systems, face identification, object recognition and more), sequence recognition (gesture, speech, handwritten text recognition), medical diagnosis, financial applications (automated trading systems), data mining (or knowledge discovery in databases, "KDD"), visualization and e-mail spam filtering.
The proven success of Artificial Neural Networks (ANN) and expert systems has helped AI gain widespread adoption in enterprise business applications. In some instances, such as fraud detection, the use of AI has already become the most preferred method. In addition, neural networks have become a well-established technique for pattern recognition, particularly of images, data streams and complex data sources and, in turn, have emerged as a modeling backbone for a majority of data-mining tools available in the market. Some of the key business applications of AI/ANN include fraud detection, cross-selling, customer relationship management analytics, demand prediction, failure prediction, and non-linear control.
A majority of the enterprises adopt horizontal or vertical solutions that embed neural networks such as insurance risk assessment or fraud-detection tools, or data-mining tools that include neural networks (for instance, from SAS, IBM and SPSS) as one of the modeling options.
Artificial Intelligence in Manufacturing
As the manufacturing industry becomes increasingly competitive, sophisticated technology has emerged to improve productivity. Artificial Intelligence in manufacturing can be applied to a variety of systems. It can recognize patterns, plus perform time consuming and mentally challenging tasks. Artificial Intelligence can optimize your production schedule and production runs. In order for organizations to meet ever increasing customer demands, and to be able to survive in an environment where change is inevitable, it is crucial that they offer more reliable delivery dates and control their costs by analyzing them on a continual basis. For businesses, being capable of delivering high quality goods at low costs and short delivery times is akin to operating in a whirlpool environment like the Devil's Triangle, and this is no easy task for any organization. Managing so that production takes place at the right time, on the right equipment, and using the right tools will minimize any deviations in delivery dates promised to the customer. Utilizing equipment, personnel and tools to their maximal efficiency will no doubt improve any organization's competitive strength. In return, proper utilization of these capabilities will result in lower costs for the organization
Optimal scheduling of jobs on equipment, without the use of computer software, is a truly difficult undertaking. Performing planning using the "Deterministic Simulation Method" will provide you with schedules that will indicate job loads per equipment. Even in the case limited to a single piece of equipment, as the number of jobs to schedule on that equipment increases, finding the right solution in the "Possible Solutions Set" becomes next to impossible. And in the real world, the difficulties arising from the large size of the solutions set due to the recipes formed by jobs, equipment and products, and shaped by the technological restrictions, as well as the complexity in finding a close to ideal solution, are readily apparent.
Research and studies are being conducted worldwide on the subject of scheduling. Software vendors working in this area follow developments closely, and they are coming out with new products to better meet demands. "Genetic Algorithms", "Artificial Intelligence", and "Neural Networks" are some of the technologies being used for scheduling

Advantages

• View your best product runs and the corresponding settings.
• Increase efficiency and quality by using optimal settings from past production.
• Artificial Intelligence can optimize your schedule beyond normal human capabilities.
• Increase productivity by eliminating downtime due to unpredictable changes in the schedule.

Artificial Intelligence in Financial services

AI has found a home in financial services and is recognized as a valuable addition to numerous business applications. Sophisticated technologies encompassing neural networks and business rules along with AI-based techniques are yielding positive results in transaction-oriented scenarios for financial services. AI has been widely adopted in such areas of risk management, compliance, and securities trading and monitoring, with an extension into customer relationship management (CRM). Tangible benefits of AI adoption include reduced risk of fraud, increased revenues from existing customers due to newer opportunities, avoidance of fines stemming from non-compliance and averted securities trade exceptions that could result in delayed settlement, if not detected.
Warren Buffet is known as the ultimate investor in this age. So good is he, in fact, that artificial intelligence software developed in Carnegie Mellon that predicts stock movements was named after him by. But can machines really take the place of human traders, much less surpass them? When Deep Blue defeated Chess Grandmaster Kasparov in 1997, AI was propelled into the limelight. Indeed, if a machine can whiz through the intricacies of the ultimate game of strategy, why not beat man in other fields as well – thereby facilitating work, decreasing costs and errors and increasing productivity and quality. This study focuses on applying AI in Finance, particularly in stock trading. In the field of Finance, artificial intelligence has long been used. Some applications of Artificial Intelligence are
• Credit authorization screening
• Mortgage risk assessment
• Project management and bidding strategy
• Financial and economic forecasting
• Risk rating of exchange-traded, fixed income investments
• Detection of regularities in security price movements
• Prediction of default and bankruptcy
• Security/and or Asset Portfolio Management
Artificial intelligence types used in finance include neural networks, fuzzy logic, genetic algorithms, expert systems and intelligent agents. They are often used in combination with each other. When AI first appeared a decade ago, it generated mass media hype but delivered inconsistent results. A number of those who praised its ability were paralyzed in the end. One such case is Fidelity Investments. In this paper, we set the stage by describing how traditional stock trading differs from AI-powered stock trading. We define the various AI systems available and also explore the various solutions available in the market, their IT foundations and how salient they are. Then, we move into how AI systems for stock trading will affect traders, companies and individuals. Benefits, risks and competitive strategy will be defined and real-world examples cited, as grounding for our recommendations in the end. Recommendations include getting management buy-in, implementing the system and managing the whole structure to succeed.

Artificial Intelligence in Marketing

Advances in artificial intelligence (AI) eventually could turbo-boost customer analytics to give companies speedier insights into individual buying patterns and a host of other consumer habits.
Artificial intelligence functions are made possible by computerized neural networks that simulate the same types of connections that are made in the human brain to generate thought. Currently, the technology is used mostly to analyze data for genetics, pharmaceutical and other scientific research. It's seeing little use in CRM right now, though it has tremendous potential in the future
AI-enhanced analytics programs also provide survival modeling capabilities -- suggesting changes to products based on use. For example, customer patterns are analyzed to learn ways to extend the life of light bulbs or to help decide the correct dosage for medications.
High-tech data mining can give companies a precise view of how particular segments of the customer base react to a product or service and propose changes consistent with those findings. In addition to further exploring customers" buying patterns, analytics could help companies react much more quickly to the marketplace.
According to Meta Group vice president Liz Shahnam, intelligent agents could let companies make real-time changes to marketing campaigns. "New technologies would have the model refreshed on the fly based on each new incoming piece of customer information -- reaction to the campaign -- for a more targeted offer,"

Artificial Intelligence in HR

It is widely believed that the role of managers is becoming a key determinant for enterprises' competitiveness in today's knowledge economy era. Owing to fast development of information technologies (ITs), corporations are employed to enhance the capability of human resource management, which is called human resource information system (HRIS). Recently, due to promising results of artificial neural networks (ANNs) and fuzzy theory in engineering, they have also become candidates for HRIS. The artificial intelligence (AT) field can play a role in this, especially; in assuring that the fuzzy neural network has the characteristics and functions of training, learning, and simulation to make an optimal and accurate judgment according to the human thinking model. The main purposes of the study are to discuss the appointment of managers in enterprises through fuzzy neural network, to construct a new model for evaluation of managerial talent, and accordingly to develop a decision support system in human resource selection. Therefore, the research methods of reviewing literature, in-depth interview, questionnaire survey, and fuzzy neural network are used in the study. The fuzzy neural network is used to train the concrete database, based on 191 questionnaires from experts, for getting the best network model in different training conditions. In order to let decision-makers adjust weighted values and obtain decisive results of each phase's scores, we adopted the simple additive weighting (SAW) and fuzzy analytic hierarchy process (FAHP) methods in the study. Finally, the human resource selection system of Java user interface has been constructed by FNN in the study.

Conclusion

It is difficult for business to see general relevance from AI. This is probably one of the reasons for the compartmentalization of AI into things like Knowledge Based Systems, Neural Networks, and Genetic Algorithms etc. Some of these separate sub topics have been shown to be very useful in solving certain difficult business and industrial problems and consequently funding bodies influence research directions by encouraging work on these more application based areas. This can have a positive effect for business benefit and has lead to some very useful systems that have found their way into the heart of business activity. Business should not lose sight of where AI could go because there are many potential benefits to current and new businesses of future research. The idea of robotic domestic workers is still far fetched but companies are making progress even here. There is already a Robot Vacuum Cleaner marketed by Electrolux and doubtless improved systems with better functionality will follow. .
I would like to close by quoting from Tom Peters, a leading management guru: "When you think you've reached the top, tear down everything and do it all over again. If you don't, your competitor will." To this, I would like to add my own: "If your competitor won't, new investors will enter the market segment who will do the same job better."

Read more: http://www.articlesbase.com/management-articles/scope-of-artificial-intelligence-in-business-328608.html#ixzz1B6dJKDFL
Under Creative Commons License: Attribution

2011-01-14T22:44:00.000-08:00

Think and Grow Rich by Napoleon Hill
Think and Grow Rich is the single most popular personal development book of all time. It is a definite must-read for anyone who is interested in improving their finances or any other aspect of their life. It is the ultimate guide that teaches you how to harness the power of your spirit and connect to the infinite mind.

As a Man Thinketh by James Allen
This book is extremely short but potent. One Allen's most powerful ideas is that we attract the things and people that represent what we already are. Another notion, popularized by The Secret, is that we become precisely what we think about.

The Wealth of Nations by Adam Smith
Published in 1776, The Wealth of Nations is one of the most famous economic texts of all time. This influential book helped shape the thinking of some of the most well known economists in history.

Reminiscences of a Stock Operator by Edwin Lefevre
Reminiscences of a Stock Market Operator is one of the classic trading books of all time. The author was a successful stock trader during and after the Great Depression. This book will teach you that human emotion has not changed in the last 100 years. The technology has changed radically, but only to enhance the effect of the power of emotion on the markets.

Don't Just Survive! How to Thrive During Tough Economic Times by Matt Swayne and B7
Most people think it’s better to hunker down during a depression or recession. However, the times that everyone else is running in fear is actually the easiest time to make large sums of money. This report will show you how.

2011-01-13T00:23:00.000-08:00

http://www.statistics.com/course-catalog/

Introduction to Statistics for Beginners

Aim of Course:

To provide an easy introduction to statistics for those with little or no prior exposure to basic probability and descriptive statistics in preparation for the two course sequence Introduction to Statistics 1: Inference for a Single Variable and Introduction to Statistics 2: Working with Bivariate Data.
Prerequisite:

No prior study of statistics is required. The only mathematics you need is arithmetic.
Course Program:

SESSION 1: Designing a Statistical Study

* Gathering and organizing data
* Looking at your (measurement) data
* Simple numerical and graphical summaries
* Introduction to statistical inference via resampling

SESSION 2: Categorical Data and Probability

* Observational studies, polls and surveys
* Looking at your (categorical) data
* Letting the computer do the work
* Random variables and their distributions
* Expected value
* The normal distribution

SESSION 3: Relationships Between (Two Categorical) Variables

* Two-way Tables
* Conditional probabilities
* Bayes rule
* Independence
* Simpson's paradox
***********************
Interactive Data Visualization

Aim of Course:

This course is about the interactive exploration of data, and how it is achieved using state-of-the-art data visualization software. Participants will learn to explore a range of different data types and structures. They will learn about various interactive techniques for manipulating and examining the data and producing effective visualizations.

The course is very hands-on in terms of the learning process. Participants will be guided through an exploration of quantitative business data to discern meaningful patterns, trends, relationships, and exceptions that reveal business performance, potential problems and opportunities.
Prerequisite:

The equivalent of Introduction to Statistics 1: Inference for a Single Variable, and Introduction to Statistics 2: Working with Bivariate Data (and, if necessary before these courses, Introduction to Statistics for Beginners or Survey of Statistics for Beginners).
Course Program:

SESSION 1

* Information visualization characterization and history
* Elements of visual perception
* Software introduction and data preparation (merging data, getting started, export)

SESSION 2

* Interaction techniques
* Distribution analysis
* Hands-on visual exploration of business data

SESSION 3

* Time Series
* Multivariate views (scatterplots, parallel coordinate plots, trellising)
* Treemaps for hierarchical data

SESSION 4

* Specialized visualizations
* Video demonstrations of novel techniques
* From visualization to visual analytics
**********************
Text Mining

Aim of Course:

This course will introduce the essential techniques of text mining, understood here as the extension of data mining's standard predictive methods to unstructured text. This course will discuss these standard techniques, and will devote considerable attention to the data preparation and handling methods that are required to transform unstructured text into a form in which it can be mined.
Prerequisite:

Math beyond algebra is not required to learn what text mining methods do, and how they can be used, though there is some detail on algorithms that employs more advanced math, for those interested in pursuing it. You should have some familiarity with standard data mining supervised learning methods, such as those covered in Introduction to Data Mining and you should be comfortable with learning the software used in this course (see below).

Course Program:

SESSION 1: Introduction and data preparation

* Overview of text mining
* Tokenization
* Dictionary creation
* Vector generation for prediction
* Feature generation and selection
* Parsing

SESSION 2: Predictive models for text

* Document classification
* Document similarity and nearest-neighbor
* Decision rules
* Probabilistic models
* Linear models
* Performance evaluation
* Applications

SESSION 3: Retrieval and clustering of documents

* Measuring similarity for retrieval
* Web-based document search and link analysis
* Document matching
* Clustering by similarity
* k-means clustering
* Hierarchical clustering
* The EM algorithm for clustering
* Evaluation of clustering

SESSION 4: Information extraction

* Goals of information extraction
* Finding patterns and entities
* Entity Extraction: The Maximum Entropy method
* Template filling
* Applications
********************
Introduction to Support Vector Machines in R

Aim of Course:

Support vector machines (SVMs) have established themselves as one of the preeminent machine learning models for classification and regression over the past decade or so, frequently outperforming artificial neural networks in task such as text mining and bioinformatics.

The aim of this course is to give you an understanding on what is going on "under the hood" when using SVMs. After completing this course, you will be able to interpret the performance of SVM models and make appropriate choices for model parameters during the model evaluation and selection cycle. You will understand the difference between linear, polynomial, and gaussian kernels and know how to tune their parameters. In addition, you will have a deep understanding on how the cost constant "C" affects the quality of your models.

The course is based on the R statistical computing environment. However, the knowledge gained here is easily transferred to other knowledge discovery environments.
Prerequisite:

Familiarity with calculus and matrix algebra.

Introduction to R - Data Handling and Data Mining 1 will be helpful, but not required.

Introduction to Statistics 1: Inference for a Single Variable
Introduction to Statistics 2: Working with Bivariate Data
Course Program:

SESSION 1: The Foundations

* What is Knowledge Discovery?
* Describing Data Mathematically
* Linear Decision Surfaces and Functions
* Perceptron Learning
o Duality
* Maximum Margin Classifiers
o Quadratic Programming

SESSION 2: Support Vector Machines

* The Lagrangian Dual
* Dual Maximum Margin Optimization
* Linear/Non-Linear SVMs
o "The Kernel Trick"
* Soft-margin Classifiers

SESSION 3: Model Evaluation and Selection

* Performance metrics
o the Confusion Matrix
* Model Evaluation
o Hold-out
o Leave-one-out
o N-fold Cross-validation
* Confidence Intervals
* Elements of Statistical Learning Theory
o the VC-dimension
o Empirical Risk Minimization
o VC-confidence
o Structural Risk Minimization

SESSION 4: Extensions to the Basic Model

* Multi-class Classification
o One-versus-the-rest Classification
o Pairwise Classification
* Regression with SVMs
o Regression with Maximum Margin Machines
o Regression with Support Vector Machines
o Model Evaluation
*******************
Natural Language Processing

Aim of Course:

This course is designed to give you an introduction to the algorithms, techniques and software used in natural language processing (NLP). Their use will be illustrated by reference to existing applications, particularly speech understanding, information retrieval, machine translation and information extraction. The course will try to make clear both the capabilities and the limitations of these applications.

For real-world applications, NLP draws heavily on work in computational linguistics and artificial intelligence. The course textbook will provide the necessary background in linguistics and computer science for those students who need it. In this course only a portion of the textbook will be covered, however anyone going on to do further studies in NLP will find the textbook a very useful reference.

At the completion of the course, a student should be able to read the description of an NLP application and have an idea of how it is done, what the likely weaknesses are, and often which claims are probably exaggerated. The course also prepares students to do further work in NLP by giving them a good grasp of the basic concepts.
Prerequisite:

Students should be familiar with probability (e.g. the material covered in Introduction to Statistics for Beginners, and the succeeding two courses at statistics.com). Some familiarity with Bayesian statistics (such as that covered in "Introduction to Bayesian Statistics") is also helpful, although the text does cover the required Bayesian fundamentals to a limited degree. Keep in mind that this course is an introductory/survey course with a broad brush approach, and, as such, does not get into computational intensity on a comprehensive basis.
Course Program:

SESSION 1: Introduction and Word-level Analysis

* Overview of NLP
* Text Preprocessing
* Corpus Creation
* Fundamental Statistical Techniques in NLP (review)
* Lexical Analysis

SESSION 2: Sentence-level Processing

* Part-of-Speech Tagging
* Context-Free Grammars (CFG)
* Parsing of sentences with CFG
* Statistical parsing methods

SESSION 3: Semantics

* Representation of Meaning
* Semantic Analysis
* Word Sense Disambiguation

SESSION 4: Applications of NLP

* Information Retrieval
* Information Extraction
* Speech Recognition Systems
* Natural Language Generation
**********************
Statistical Analysis of Microarray Data with R

Aim of Course:

In this course, participants will learn the statistical tools required for the analysis of microarray data, how to apply them using R software and how to interpret the results meaningfully. We will review the biology relevant to microarray data, then cover microarray experiment set up, quantification of information generated from the experiment, preprocessing of data including statistical tools for between array and within array normalization, statistical inference procedures to identify differentially expressed genes under two different conditions, and its extension to situations involving more than two conditions. The course will also introduce multivariate statistical tools, such as principal component analysis & cluster analysis. These tools help to identify differentially expressed genes, sets of co-regulated genes, which in turn will help to assign functions to genes.
Prerequisite:

Some familiarity with statistical modeling will also be helpful. Any of the following statistics.com courses would provide useful background in modeling: Regression,Logistic Regression, Introduction to Data Mining. Participants should also be familiar with basic molecular biology and microarray experiments, including gene expression, transcription, splicing, and translation.

Please also read the note at the end of the course outline concerning the course's review materials in biology and statistics, and the time that you should budget for this course. Also, please note the use of R software, as described below.

Course Program:

SESSION 1: Introduction to R*

* Starting and stopping R, data types, using R as a simple calculator, methods of data input: c function, scan function and sequence function.
* Importing data from text file, using data editor for entering and editing data.
* Data frames and lists.
* Using resident data sets.
* Data accession or indexing from a vector and from a data frame, the functions attach and detach, transform function.
* Graphics with R: Histogram, box plot, scatter plot.
* Some functions such as mean, variance, standard deviation, coefficient of variation, mean absolute deviation, quantiles, sort, order.
* Using on-line help.
* Writing simple functions.

*If you are familiar with R and this material, you may join the course in its second week.

SESSION 2: Background of Microarrays and Normalization

* Microarray experimental set up and quantification of information available from microarray experiments.
* Data cleaning.
* Transformation of data.
* Between array & within array normalization.
* Concordance coefficients and their use in normalization.
* Numerical illustration for 4-6 with complete set of annotated R-commands.

SESSION 3: Statistical Inference procedures in comparative experiments

* Basics of statistical hypothesis testing.
* Two sample t- test.
* paired t-test.
* Tests for validating assumptions of t-test.
* Welch test.
* Wilcoxon rank sum test, signed rank test.
* Adjustments for Multiple hypotheses testing including false discovery rate.
* Numerical illustration for 2-8 with complete set of annotated R-commands.
* One way ANOVA.

SESSION 4: Multivariate Techniques

* Principal component analysis.

SESSION 5: Clustering.

* Cluster analysis.
***********************
Advanced Logistic Regression

Aim of Course:

After taking this course, participants will be able to specify, implement and interpret the output of a variety of advanced logistic regression models. This course moves beyond the topics covered in "Logistic Regression" and covers a number of situations that call for logistic-based modeling, including a variety of ordered-categorical response (both proportional and non-proportional) models, multinomial models, panel models with fixed and random effects, GEE and quasi-least-squares models, multi-level models, survey logistic models, discriminant logistic models, skewed and penalized logistic regression, median unbiased estimation, Monte Carlo sampling, and exact logistic regression.
Prerequisite:

Though it is not required for practical applications of material in this course, some familiarity with calculus (see statistics.com's brief Calculus Review course) is helpful for a complete understanding of model development.

Participants should also have taken the course Logistic Regression or have an equivalent level of statistical expertise as covered in that course.

Course Program:

SESSION 1

* Overview of binary logistic regression
* Overview of binomial logistic regression
* Proportional odds models

SESSION 2

* Ordered non-proportional models
* Multinomial logistic regression
* Multinomial probit regression
* Alternative categorical response models
* Marginal effects and discrete change

SESSION 3

* Panel models
* GEE/Quasi-least squares models
* Fixed- and random-effects models
* Multi-level models

SESSION 4

* Survey models
* Exact logistic regression
* Penalized logistic regression
* Monte Carlo sampling methods
* Median unbiased estimation
******************
Logistic Regression

Aim of Course:

Logistic regression is one of the most commonly-used statistical techniques. It is used with data in which there is a binary (success-failure) outcome (response) variable, or where the outcome takes the form of a binomial proportion. Like linear regression, one estimates the relationship between predictor variables and an outcome variable. In logistic regression, however, one estimates the probability that the outcome variable assumes a certain value, rather than estimating the value itself. This course will cover the functional form of the logistic model and how to interpret model coefficients. The concepts of "odds" and "odds ratio" are examined, as well as "risk ratio" and the difference between the two statistics. Our emphasis is on model construction, interpretation, and goodness of fit. Exercises include hands-on computer problems.
Prerequisite:

Though it is not required for practical applications of material in this course, some familiarity with calculus (see statistics.com's brief Calculus Review course) is helpful for a complete understanding of model development.

Familiarity with standard multiple linear regression is helpful, It is covered briefly in the above introductory courses; for a more complete treatment, see Regression Analysis. Some familiarity with statistical software that can do logistic regression is desirable (see below).

Course Program:

SESSION 1: BASIC TERMINOLOGY AND CONCEPTS

* Software for modeling logistic regression: Stata, R, SAS, SPSS, other
* History of the logistic model
* Concepts related to logistic regression
* 2x2, 2xn models of odds and risk ratios
* Fitting algorithms

SESSION 2: LOGISTIC MODEL CONSTRUCTION

* Derivation of the binary logistic model
* Model-building strategies
* Link tests, partial residual plots
* Standard errors: scaling, bootstrap, jackknife, robust
* Interpreting odds ratios as risk ratios - criteria
* Stepwise methods, missing values, constrained coefficients, etc Construction and interpretation of interactions

SESSION 3: ANALYSIS, FIT, AND INTERPRETATION OF THE LOGISTIC MODEL

* Goodness of fit tests
* Information criterion tests
* Residual analysis
* Validation models

SESSION 4: BINOMIAL LOGISTIC REGRESSION AND OVERDISPERSION

* The meaning and types of overdispersion
* Simulations: detecting apparent vs real overdispersion
* Methods of handling real overdispersion
*********************

Ito's Lemma for amateurs

2011-01-12T09:24:00.000-08:00

Over the years I have read several books purporting to make Ito's lemma accessible to the non-mathematician. While technically I see the mathematical explanation, I never felt like I intuitively understood what Ito's lemma was actually doing. I spoke with several supposed experts and they either never thought about it or clumsily referred to the mysteries of stochastic calculus. About a year ago, a picture formed in my mind that suddenly brought what was happening home. It can be explained as a thought experiment, but it isn't hard to test either. (I did it with the binomial model described below)

Imagine a binomial tree which goes out 50 steps where the price at each step is determined by drift +- vol. The average of returns at the end of the steps will be (drift - vol^2/2)*dtime. This is as Ito's Lemma would have it. But when you do this averaging to get that number, all of the outcomes(with their individual returns) have the same weighting. Its as though you weighted each outcome by its beginning value or price. Since the all the paths started at the same price, it turns out being a simple average. (actually a probability weighted average)

Here is my breakthrough in thinking. Try averaging each of the outcomes by their ending value - and lo and behold - the average mean turns out to be (drift + vol^2/2). Note the change in sign. So the formula has a minus if you use beginning weightings and a plus if you use ending value weightings. Conceivable, somewhere in the middle of the process (or maybe the average drift of the process) is just the beginning drift with no volatility adjustment. Just as a person would think, without the aid of stochastic calculus.

Why? Well when you weight by the beginning price, all of the paths share equal weightings - the bad performing paths carry as much weight as the good, in spite of the fact that they get smaller in size. So they are unreasonably bringing down the average return (thus the minus sigma^2/2). The opposite happens using ending values. The top paths get really big versus the bottom and accordingly seem to be dragging up the returns (kinda like what happens in some stock indexes). The "reality" is somewhere in the middle, where the number is the original drift.

In this context, Ito's lemma is just a weighted averaging fix. Once I came to this conclusion, most of the mystery disappeared. Maybe everyone else already knows this. If so, I wish someone had mentioned it to me.

Keynesian Economics vs. Supply Side Economics

2011-01-11T21:45:00.000-08:00

Supply side and demand side economics are philosophies designed to stimulate the economy by using divergent theories. In theory, supply side economics will cause an influx of investments by the wealthy, prompting new growth. Demand side economics, on the other hand, focuses on stimulating the average consumer to spend more money.

To supply-siders, not all tax cuts are equally good. They place their emphasis on the "marginal tax rate," the percentage taxed away from an extra dollar of income earned. A marginal tax rate of 40 percent, for example, would mean that 40 cents of each additional dollar of income would be taxed away, leaving an after-tax reward of 60 cents for the person who earned the dollar. High marginal tax rates kill economic initiative, they believe, and tax cuts that leave marginal tax rates high are useless. As an example, a fixed $500 tax credit to everyone would not affect the after-tax reward of earning an additional dollar of income. It would leave marginal tax rates unaffected and therefore would have no direct effect on promoting aggregate supply.

SUPPLY-SIDE ECONOMICS is based on the premise that high tax rates hurt the national economy by discouraging work, production, and innovation.Even as Keynesians and monetarists have debated how to increase aggregate demand, supply-side economists and their political allies have been insisting that demand is typically not the problem. They believe that conventional policies increasing spending will only give small upward bumps to the economy. Their cure, therefore, is tax cuts designed to increase productivity, entrepreneurship, and risk-taking. The resulting increase in aggregate supply, they believe, will lead to economic recovery.

The Simple Keynesian Model is important not so much for its ability to capture the details of recessions, but for its ability to demonstrate the possibility of a stable equilibrium at less than full employment. While the real wage rate adjusts in the Classical Model to move the economy to full employment, the real wage rate does not appear in the Simple Keynesian Model and equilibrium is achieved by adjustments in aggregate demand, which equals aggregate income. The equilibrium aggregate income need not imply full employment.

****************
Classical views:
1)Products are paid for with products

2)A glut can take place only when there are too many means of production applied to one kind of product and not enough to another

3)A rational businessman will never hoard money; he will promptly spend any money he gets "for the value of money is also perishable
*************
In the Keynesian interpretation, the assumptions of Say's law are:

* A barter model of money - "products are paid for with products;"
* Flexible prices - all prices can rapidly adjust upwards or downwards;
* No government intervention.

Under these assumptions, Say's law states that there cannot be a general glut, which, Keynesians conclude, means that there cannot be a persistent state where demand is generally less than productive capacity and high unemployment results.

Since there have been a great many persisting economic crises historically, one may either reject one or more of the assumptions of Say's law, its reasoning, or its conclusions.
************
Supply creates its own demand

"Supply creates its own demand" is the formulation of Say's law by John Maynard Keynes, and is considered by him one of the defining characteristics of classical economics. The rejection of this doctrine is a central component of The General Theory of Employment, Interest and Money (1936) and a central tenet of Keynesian economics.

Keynes's rejection of Say's law is on the whole accepted within mainstream economics since the 1940s and 50s, in the neoclassical synthesis, but debate continues between more Keynesian economists and more neoclassical economists – see saltwater and freshwater economics.

The exact phrase "supply creates its own demand" does not appear to be found in the writings of classical economists;[1] similar sentiments, though different wordings, appear in the work of John Stuart Mill (1848), whom Keynes credits and quotes, and his father, James Mill (1808), whom Keynes does not.

Keynes's interpretation is rejected as a misinterpretation or caricature of Say's law by proponents of same – see Say's law: Keynes vs. Say – and the advocacy of the phrase "supply creates its own demand" is today most associated with supply-side economics, which retorts that "Keynes turned Say on his head and instead stated that 'demand creates its own supply'".

Keynes's formulation

Keynes coined the phrase thusly (emphasis added):

From the time of Say and Ricardo the classical economists have taught that supply creates its own demand; —meaning by this in some significant, but not clearly defined, sense that the whole of the costs of production must necessarily be spent in the aggregate, directly or indirectly, on purchasing the product.

In J. S. Mill's Principles of Political Economy the doctrine is expressly set forth:

What constitutes the means of payment for commodities is simply commodities. Each person’s means of paying for the productions of other people consist of what he himself possesses. All sellers are inevitably, and by the meaning of the word, buyers. Could we suddenly double the productive powers of the country, we should double the supply of commodities in every market; but we should, by the same stroke, double the purchasing power. Everybody would bring a double demand as well as supply; everybody would be able to buy twice as much, because every one would have twice as much to offer in exchange. (Principles of Political Economy, Book III, Chap. xiv. § 2.)

—The General Theory of Employment, Interest and Money, John Maynard Keynes, Chapter 2, Section VI, p. 18

Keynes then restates this in the language of Keynesian economics as:

(3) [S]upply creates its own demand in the sense that the aggregate demand price is equal to the aggregate supply price for all levels of output and employment.
—The General Theory of Employment, Interest and Money, John Maynard Keynes, Chapter 2, Section VII

Other sources

Another source widely cited as a classical expression of the idea, and the original statement of Say's law in English, is by James Mill, in Commerce Defended (1808):

The production of commodities creates, and is the one and universal cause that creates a market for the commodities produced.
—James Mill, Commerce Defended (1808), Chapter VI: Consumption, p. 81

Keynes does not cite a specific source for the phrase, and, as it does not appear to be found in the pre-Keynesian literature,[1] some consider its ultimate origin a "mystery".[2] The phrase "supply creates its own demand" appears earlier, in quotes, in a 1934 letter of Keynes,[2] and has been suggested that the phrase was an oral tradition at Cambridge, in the circle of Joan Robinson,[2] and that it may have derived from the following 1844 formulation by John Stuart Mill:[3]

Nothing is more true than that it is produce which constitutes the market for produce, and that every increase of production, if distributed without miscalculation among all kinds of produce in the proportion which private interest would dictate, creates, or rather constitutes, its own demand.
—John Stuart Mill, Essays On Some Unsettled Questions of Political Economy (1844), "Of the Influence of Consumption On Production", p. 73
**************
Say, Jean Baptiste (zhäN bätēst' sā) , 1767–1832, French economist. In A Treatise on Political Economy (1803, tr. from the 4th ed. 1821) he effectively reorganized and popularized the theories of Adam Smith. Say also developed a noted theory of markets and the concept of the entrepreneur. Say's law of markets holds that supply creates its own demand. His works include Cours complet d'économie politique pratique (6 vol., 1828–29). CUPEn

Say's Law
He is well known for Say's Law (or Say's Law of Markets), often summarised as

*
"Aggregate supply creates its own aggregate demand",
*
“Supply creates its own demand”, or
*
“Supply constitutes its own demand”.

He argued that production and sale of goods in an economy automatically produces an income for the producers of the same value, which would then be reinjected into the economy and create enough demand to buy the goods. Thus production is determined by the supply of goods rather than demand. Unemployment of men, land or other resources would not be possible unless it were by choice, or due to some kind of restraint on trade.

He was also among the first to argue that money was neutral in its effect on the economy. Money is not desired for its own sake, but for what it can purchase. An increase in the amount of money in circulation would increase the price of other goods in terms of money (causing inflation), but would not change the relative prices of goods or the quantity produced. This idea was later developed by economists into the Quantity theory of money. Say's ideas helped to inspire neoclassical economics which arose later in the 19th century. He wrote Traite d’Economique Politique.
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Potential output
From Wikipedia, the free encyclopedia
Jump to: navigation, search

In economics, potential output (also referred to as "natural gross domestic product") refers to the highest level of real Gross Domestic Product output that can be sustained over the long term. The existence of a limit is due to natural and institutional constraints. If actual GDP rises and stays above potential output, then (in the absence of wage and price controls) inflation tends to increase as demand exceeds supply. This is because of the limited supply of workers and their time, capital equipment, and natural resources, along with the limits of our technology and our management skills. Graphically, the expansion of output beyond the natural limit can be seen as a shift of production volume above the optimum quantity on the average cost curve. Likewise, if GDP is below natural GDP, inflation will decelerate as suppliers lower prices to fill their excess production capacity.

Potential output in macroeconomics corresponds to one point on the production possibilities frontier (or curve) for a society as a whole seen in introductory economics, reflecting natural, technological, and institutional constraints.

Potential output has also been called the "natural gross domestic product." If the economy is at potential, the unemployment rate equals the NAIRU or the "natural rate of unemployment." There is great disagreement among economists as to what these rates actually are.

Generally speaking, most central banks and other economic planning agencies attempt to keep GDP at or around the natural GDP level. This can be done in a number of ways: the two most common strategies are expanding or contracting the government budget (fiscal policy), and altering the money supply to change consumption and investment levels (monetary policy).

The difference between potential output and actual output is referred to as the output or GDP gap
Natural gross domestic product
Natural gross domestic product (natural GDP) is defined as the optimal production capacity of a territory's economy. Once gross domestic product exceeds natural GDP, inflation accelerates as demand exceeds supply. Likewise, if GDP is below natural GDP, inflation will decelerate as suppliers lower prices to fill their excess production capacity.

Generally speaking, most central banks and other economic planning agencies attempt to keep GDP at or around natural GDP level. This can be done in a number of ways: the two most common strategies are expanding or contracting the government budget, and altering the money supply to change consumption and investment levels.
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SUPPLY DETERMINANT: One of five basic ceteris paribus factors that affect supply, but which are assumed constant when a supply curve is constructed. The five supply determinants are

1. resource prices,
2. technology,
3. other prices,
4. sellers' expectations,
5. number of sellers.

DEMAND DETERMINANT: One of five basic basic ceteris paribus factors that affect demand, but which are assumed constant when a demand curve is constructed. Changes in any one causes a shift of the demand curve. The five demand determinants are:
1. income,
2. preferences,
3. other prices,
4. buyers' expectations, and
5. number of buyers.

Microeconomic discussion generally ignores adjustment problems, at least at the introductory level. Microeconomics assumes that markets clear, that is, they are always in equilibrium. Its analysis begins with the assumption that equilibrium has been reached and then asks questions about that equilibrium. However, adjustment problems are very important in macroeconomics. Macroeconomics cannot assume there are no adjustment problems or else it assumes away one of the problems it wants to explain, unemployment. In fact, much of macroeconomics is about the forces that bump an economy away from equilibrium, and why, once it is away, it has problems reaching a new equilibrium.
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Say’s law
In economics, Say’s Law or Say’s Law of Markets is a principle attributed to French businessman and economist Jean-Baptiste Say (1767-1832) stating that there can be no demand without supply. A central element of Say's Law is that recession does not occur because of failure in demand or lack of money.

The more goods (for which there is demand) that are produced, the more those goods (supply) can constitute a demand for other goods. For this reason, prosperity should be increased by stimulating production, not consumption. In Say's view, creation of more money simply results in inflation; more money demanding the same quantity of goods does not represent an increase in real demand.

What Say said

James Mill was to restate Say's Law as "production of commodities creates, and is the one and universal cause which creates a market for the commodities produced". In Say's language, "products are paid for with products" (1803: p.153) or "a glut can take place only when there are too many means of production applied to one kind of product and not enough to another", (1803: p.178-9.) Explaining his point at length, he wrote that:

It is worthwhile to remark that a product is no sooner created than it, from that instant, affords a market for other products to the full extent of its own value. When the producer has put the finishing hand to his product, he is most anxious to sell it immediately, lest its value should diminish in his hands. Nor is he less anxious to dispose of the money he may get for it; for the value of money is also perishable. But the only way of getting rid of money is in the purchase of some product or other. Thus the mere circumstance of creation of one product immediately opens a vent for other products. (J.B. Say, 1803: p.138-9)

He also wrote:

It is not the abundance of money but the abundance of other products in general that facilitates sales... Money performs no more than the role of a conduit in this double exchange. When the exchanges have been completed, it will be found that one has paid for products with products.

Say argued against claims that business was suffering because people did not have enough money and more money should be printed. Say argued that the power to purchase could only be increased by more production. James Mill used Say's Law against those who sought to give economy a boost via unproductive consumption. Consumption destroys wealth, in contrast to production which is the source of economic growth. Production must be guided by demand, however, to prevent the creation of goods for which there is little demand. Thus, supply and demand interact in self-correcting feedback.

It is important to note that Say himself never used many of the later short definitions of Say's Law and that Say's Law actually developed due to the work of many of his contemporaries and those who came after him. The work of James Mill, David Ricardo, John Stuart Mill, and others evolved into what is sometimes called "law of markets" which was the framework of macroeconomics from mid 1800's until the 1930's.

Recession and unemployment

Keynes claimed that according to Say's Law, involuntary unemployment cannot exist due to inadequate aggregate demand. However, involuntary unemployment could be explained in a different way by the 19th century economists, and the neoclassical economists actually used Say's Law to understand and explain even long-term unemployment and recession.

Recession was explained as arising from production not meeting demand in quality. While in general, more is not produced than there could be demand for, some particular products are produced too much and consequently other products too little. This "disproportionality" would lead to a producer not being able to sell the products in cost-covering prices. Hence he will be lacking in the capability to buy and this will cause contraction in other parts of industry, too.
Such economic losses and unemployment were seen as an intrinsic property of the capitalistic system. Division of labour leads to a situation where one always has to anticipate what others will be willing to buy, and this will lead to miscalculations.

The kind of unemployment that results is what modern macroeconomics calls "structural unemployment". It differs from Keynesian "cyclical unemployment" that arises due to aggregate demand failure.

Role of money

It is not easy to say what exactly Say's Law says about the role of money apart from the claim that recession is not caused by lack of money. One can read the second long quotation above by Say quotation (see above) as stating simply that money is completely neutral, although Say did not concern himself about the question. The central notion that Say had concerning money can be seen in the first long quotation above. If one has money, it is irrational to hoard it.

To understand the role of this notion, restate Say's Law. To Say, as with other Classical economists, it is quite possible for there to be a glut (excess supply, market surplus) for one product, and it co-exists with a shortage (excess demand) for others. But there is no "general glut" in Say's view, since the gluts and shortages cancel out for the economy as a whole. But what if the excess demand is for money, because people are hoarding it? This creates an excess supply for all products, a general glut. Say's answer is simple: there is no reason to engage in hoarding. To quote Say from above:

Nor is [an individual] less anxious to dispose of the money he may get ... But the only way of getting rid of money is in the purchase of some product or other.

The only reason to have money, in Say's view, is to buy products. It would not be a mistake, in his view, to treat the economy as if it were a barter economy.

An alternative view is that all money that is held is done so in financial institutions (markets), so that any increase in the holding of money increases the supply of loanable funds. Then, with full adjustment of interest rates, the increased supply of loanable funds leads to an increase in borrowing and spending. So any negative effects on demand that results from the holding of money is canceled out and Say's Law still applies.

In Keynesian terms, followers of Say's Law would argue that on the aggregate level, there is only a transactions demand for money. That is, there is no precautionary, finance, or speculative demand for money. Money is held for spending and increases in money supplies lead to increased spending.

Classical economists did see that loss of confidence in business or collapse of credit will increase the demand for money which would cut down the demand for goods. This view was expressed both by Robert Torrens and John Stuart Mill. This would lead to demand and supply to move out of phase and lead to an economic downturn in the same way as miscalculation in productions, as described by William H. Beveridge in 1909.

However, in Classical economics, there was no reason for such a collapse to persist. Persistent depressions, such as that of the 1930s, were impossible according to laissez-faire principles. The flexibility of markets under laissez faire would allow prices, wages, and interest rates to adjust to abolish all excess supplies and demands.

Modern interpretations

A modern way of expressing Say's Law is that there can never be a general glut. Instead of there being an excess supply (glut or surplus) of goods in general, there may be an excess supply of one or more goods, but only when balanced by an excess demand (shortage) of yet other goods. Thus, there may be a glut of labor ("cyclical" unemployment), but that is balanced by an excess demand for produced goods. Modern advocates of Say's Law see market forces as working quickly -- via price adjustment -- to abolish both gluts and shortages. The exception would be the case where the government or other non-market forces prevent price changes.

According to Keynes, the implication of Say's "law" is that a free-market economy is always at what the Keynesian economists call full employment. Thus, Say's Law is part of the general world-view of laissez-faire economics, i.e., that free markets can solve the economy's problems automatically (here the problems are recessions, stagnation, depression, and involuntary unemployment). There is no need for any intervention by the government or the central bank (such as the U.S. Federal Reserve) to help the economy attain full employment. All that the central bank needs to be concerned with is the prevention of inflation.

In fact, some proponents of Say's Law argue that such intervention is always counterproductive. Consider Keynesian-type policies aimed at stimulating the economy. Increased government purchases of goods (or lowered taxes) merely "crowds out" the private sector's production and purchase of goods. To contradict this, Arthur Cecil Pigou, who, according to himself, followed Say's Law, wrote 1932 a letter signed by five other economists (among them Keynes) calling for more public spending to alleviate high levels of unemployment.

From a modern macroeconomic viewpoint Say's Law is subject to dispute. John Maynard Keynes and many other critics of Say's Law have paraphrased it as saying that "supply creates its own demand". Under this definition, once a producer has created a supply of a product, consumers will inevitably start to demand it. This interpretation allowed for Keynes to introduce his alternative perspective that "demand creates its own supply" (up to, but not beyond, full employment). Some call this "Keynes' law".

Keynes vs. Say

Keynesian economics places central importance on demand, believing that on the macroeconomic level, the amount supplied is primarily determined by effective demand or aggregate demand. For example, without sufficient demand for the products of labor, the availability of jobs will be low; without enough jobs, working people will receive inadequate income, implying insufficient demand for products. Thus, an aggregate demand failure involves a vicious circle: if I supply more of my labor-time (in order to buy more goods), I may be frustrated because no-one is hiring — because there is no increase in the demand for their products until after I get a job and earn an income. (Of course, most get paid after working, which occurs after some of the product is sold.) Note also that unlike the Say's law story above, there are interactions between different markets (and their gluts and shortages) that go beyond the simple price mechanism, to limit the quantity of jobs supplied and the quantity of products demanded.

Keynesian economists also stress the role of money in negating Say's Law. (Most would accept Say's Law as applying in a non-monetary or barter economy.) Suppose someone decides to sell a product without immediately buying another good. This would involve hoarding, increases in one's holdings of money (say, in a savings account). At the same time that it causes an increased demand for money, this would cause a fall in the demand for goods and services (an undesired increase in inventories (unsold goods) and thus a fall in production). This general glut would in turn cause a fall in the availability of jobs and the ability of working people to buy products. This recessionary process would be cancelled if at the same time there were dishoarding, in which someone uses money in his hoard to buy more products than he or she sells. (This would be a desired accumulation of inventories.)

Some classical economists suggested that hoarding would always be balanced by dishoarding. But Keynes and others argued that hoarding decisions are made by different people and for different reasons than decisions to dishoard, so that hoarding and dishoarding are unlikely to be equal at all times. (More generally, this is seen in terms of the equality of saving (abstention from purchase of goods) and investment in goods.)

Some have argued that financial markets and especially interest rates could adjust to keep hoarding and dishoarding equal, so that Say's Law could be maintained. (See the discussion of "excess saving" under "Keynesian economics".) But Keynes argued that in order to play this role, interest rates would have to fall rapidly and that there were limits on how quickly and how low they could fall (as in the liquidity trap). To Keynes, in the short run, interest rates were determined more by the supply and demand for money than by saving and investment. Before interest rates could adjust sufficiently, excessive hoarding would cause the vicious circle of falling aggregate production (recession). The recession itself would lower incomes so that hoarding (and saving) and dishoarding (and real investment) could attain balance below full employment.

Worse, a recession would hurt private real investment, by hurting profitability and business confidence, in what is called the accelerator effect. This means that the balance between hoarding and dishoarding would be even further below the full employment level of production.
Keynesians believe that this kind of vicious circle can be broken by stimulating the aggregate demand for products using various macroeconomic policies mentioned in the introduction above. Increases in the demand for products leads to increased supply (production) and an increased availability of jobs, and thus further increases in demand and in production. This cumulative causation is called the multiplier process.

Many modern advocates of laissez-faire economics have rejected Say's Law, except perhaps in the long run. Instead, the emphasis is on the automatic adjustment of the labor market to get to full employment: if wages are allowed to fall, this increases the availability of jobs and allows full employment. More interventionist economists (for example, those at the International Monetary Fund) are quite activist in their approach, advocating the use of state power to destroy unions, minimum wage laws, and the like in order to make labor markets more "flexible" so that this idealized vision of labor markets can be attained.

Modern Adherents of Say's Law

Economists such as Thomas Sowell of the Chicago school of economics have advocated Say's Law. Sowell also wrote his doctoral dissertation on the idea too. Arthur Laffer, who is now associated with the theory supply-side economics, also adhered to the law, as well as members of the Austrian school.
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Two controversial economic policies are Keynesian economics and Supply Side economics. They represent opposite sides of the economic policy spectrum and were introduced at opposite ends of the 20th century, yet still are the most famous for their effects on the economy of the United States when they were used.

The founder of Keynesian economic theory was John Maynard Keynes. He made many great accomplishments during his time and probably his greatest was what he did for America in its hour of need. During the 1920's, the U.S. experienced a stock market crash of enormous proportions which crippled the economy for years. Keynes knew that to recover as soon as possible, the government had to intervene and put a decrease on taxes along with an increase in spending. By putting more money into the economy and allowing more Americans to keep what they earned, the economy soon recovered and once again became prosperous. Keynes ideas were very radical at the time, and Keynes was called a socialist in disguise. Keynes was not a socialist, he just wanted to make sure that the people had enough money to invest and help the economy along.

As far as stressing extremes, Keynesian economics pushed for a "happy medium" where output and prices are constant, and there is no surplus in supply, but also no deficit. Supply Side economics emphasized the supply of goods and services. Supply Side economics supports higher taxes and less government spending to help economy. Unfortunately, the Supply Side theory was applied in excess during a period in which it was not completely necessary.

The Supply Side theory, also known as Reganomics, was initiated during the Regan administration. During the 1970's, the state and local governments increased sales and excise taxes. These taxes were passed from business to business and finally to the customer, resulting in higher prices. Along with raised taxes for the middle and lower classes, this effect was compounded because there was little incentive to work if even more was going to be taxed. People were also reluctant to put money into savings accounts or stocks because the interest dividends were highly taxed. There was also too much protection of business by the government which was inefficient and this also ran up costs, and one thing the Supply Side theory was quite good at was reinforcing inflation.

The two opposites of the Supply Side and Keynes' theories are well matched theories, but it was the time of use that made them good and bad. Keynes' theory was used during that aftermath of the Great Depression, a catastrophe America will never forget and will never be able to repay Keynes for the economic assistance in recovering from it. The Supply Side theory was used after a long period of prosperity, and although seeming to continue the practices of the past administration, was the cause of a fearful recession. The success of those or any economic theory is based on the time at which it is implemented.

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Supply-side economics is better known to some as "Reaganomics", or the "trickle-down" policy espoused by former U.S. president Ronald Reagan. He popularized the controversial idea that greater tax cuts for investors and entrepreneurs provide incentives to save and invest and produce economic benefits that trickle down into the overall economy. In this article, we summarize the basic theory behind supply-side economics.

Like most economic theories, supply-side economics tries to explain both macroeconomic phenomena and - based on these explanations - to offer policy prescriptions for stable economic growth. In general, supply-side theory has three pillars: tax policy, regulatory policy and monetary policy.

However, the single idea behind all three pillars is that production (i.e. the "supply" of goods and services) is the most important determinant of economic growth. The supply-side theory is typically held in stark contrast to Keynesian theory, which, among other facets, includes the idea that demand can falter, so if lagging consumer demand drags the economy into recession, the government should intervene with fiscal and monetary stimuli.

This is the single big distinction: a pure Keynesian believes that consumers and their demand for goods and services are key economic drivers, while a supply-sider believes that producers and their willingness to create goods and services set the pace of economic growth.

The Argument That Supply Creates Its Own Demand
In economics we review the supply and demand curves. The left-hand chart below illustrates a simplified macroeconomic equilibrium: aggregate demand and aggregate supply intersect to determine overall output and price levels. (In this example, output may be gross domestic product and the price level may be the Consumer Price Index.) The right-hand chart illustrates the supply-side premise: an increase in supply (i.e. production of goods and services) will increase output and lower prices.
Starting Point Increase in Supply
(Production)

Supply-side actually goes further and claims that demand is largely irrelevant. It says that over-production and under-production are not really sustainable phenomena. Supply-siders argue that when companies temporarily "over-produce", excess inventory will be created, prices will subsequently fall and consumers will increase their purchases to offset the excess supply. As put by the Fountainhead Capital Group, "After all, what would cause consumers and businesses to stop demanding goods and services and force the economy into a recession or a depression? Keynes had no idea, and said as much…."

This essentially amounts to the belief in a vertical (or almost vertical) supply curve, as shown below on the left-hand chart below. On the right-hand chart, we illustrate the impact of an increase in demand: prices rise but output doesn't change much.

Vertical Supply Curve
An Increase in Demand
→ Prices Go Up

Under such a dynamic - where the supply is vertical - the only thing that increases output (and therefore economic growth) is an increase in the production of the supply of goods and services. As illustrated below:
Supply-Side Theory
Only an Increase in Supply (Production) Raises Output
Three Pillars
The three supply-side pillars follow from this premise. On the question of tax policy, supply-siders argue for lower marginal tax rates. In regard to a lower marginal income tax, supply-siders believe that lower rates will induce workers to prefer work over leisure (at the margin). In regard to lower capital-gains tax rates, they believe that lower rates induce investors to deploy capital productively. At certain rates, a supply-sider would even argue that the government would not lose total tax revenue because lower rates would be more than offset by a higher tax revenue base - due to greater employment and productivity.

On the question of regulatory policy, supply-siders tend to ally with traditional political conservatives - those who would prefer a smaller government and less intervention in the free market. This is logical because supply-siders, although they may acknowledge that government can temporarily help by making purchases, they do not think this induced demand can either rescue a recession or have a sustainable impact on growth.

The third pillar, monetary policy, is especially controversial. By monetary policy, we are referring to the Federal Reserve's ability to increase or decrease the quantity of dollars in circulation (i.e. where more dollars means more purchases by consumers, thus creating liquidity). A Keynesian tends to think that monetary policy is an important tool for tweaking the economy and dealing with business cycles, whereas a supply-sider does not think that monetary policy can create economic value.

While both agree that the government has a printing press, the Keynesian believes this printing press can help solve economic problems. But the supply-sider thinks that the government (or the Fed) is likely to create only problems with its printing press by either (a) creating too much inflationary liquidity, or (b) not sufficiently "greasing the wheels" of commerce with enough liquidity. A strict supply-sider is therefore concerned that the Fed may inadvertently stifle growth by contributing to deflation and encouraging investors to horde dollars.

What’s Gold Got To Do with It?
Since supply-siders view monetary policy not as a tool that can create economic value, but rather a variable to be controlled, they advocate a stable monetary policy or a policy of gentle inflation tied to economic growth - for example, 3% to 4% growth in the money supply per year. This principle is the key to understanding why a supply-sider often advocates a return to the gold standard - which may seem strange at first glance. (And most economists probably do view this aspect as dubious.) The idea is not that gold is particularly special but rather that gold is the most obvious candidate as a stable "store of value". The supply-sider argues that if the U.S. were to peg the dollar to gold, the currency would be more stable, and fewer disruptive outcomes would result from currency fluctuations.

As an investment theme, supply-side theorists say that the price of gold - since it is a relatively stable store of value - provides investors with a "leading indicator", or signal for the direction of the dollar. Indeed, gold is typically viewed as an inflation hedge. And, although the historical record is hardly perfect, gold has often given early signals about the dollar. In the chart below, we compare the annual inflation rate in the United States (the year-to-year increase in the Consumer Price Index) with the high-low-average price of gold. An interesting example is 1997-98: gold started to descend ahead of deflationary pressures (lower CPI growth) in 1998.

Conclusion
Supply-side economics has a colorful history. Some economists view supply-side as a half-baked economic theory - economist and New York Times columnist Paul Krugman even called its founders "cranks" in a book dedicated to attacking the theory ("Peddling Prosperity"). Other economics are so utterly disagree with the theory that they dismiss it as offering nothing particularly new or controversial to an updated view of classical economics. We have discussed the three pillars, and, based on this, you can see how the supply side cannot be separated from the political realms: if true, it implies a reduced role for government and a less progressive tax policy.
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A REVIEW OF KEYNESIAN THEORY

Keynesian theory is central to understanding the Great Depression. We'll review just the theory here, and reserve for other sections the opportunity to see if the events of the 1930s bear out the theory.

Keynesianism is named after John Maynard Keynes, a British economist who lived from 1883 to 1946. He was a man of many contradictions: an elitist whose economic theories would be embraced by liberals the world over; a bisexual who enjoyed a happy and lifelong marriage to a Russian ballerina; a genius with an uncanny ability to predict the future, but whose works were often badly organized and sometimes very wrong. I mention this only because many of Keynes' critics try to refute his theories by pointing to the man himself. This is worse than irrelevant, of course; such criticisms are often prejudiced.

What is not in contention is that even Keynes' critics call him the greatest and most influential economist of the 20th century. For this reason, he is known as "the father of modern economics."

When the Great Depression hit worldwide, it fell on economists to explain it and devise a cure. Most economists were convinced that something as large and intractable as the Great Depression must have complicated causes. Keynes, however, came up with an explanation of economic slumps that was surprisingly simple. In fact, when he shared his theory and proposed solution with Franklin Roosevelt, the President is said to have dismissed them with the words: "Too easy."

Keynes explanations of slumps ran something like this: in a normal economy, there is a high level of employment, and everyone is spending their earnings as usual. This means there is a circular flow of money in the economy, as my spending becomes part of your earnings, and your spending becomes part of my earnings. But suppose something happens to shake consumer confidence in the economy. (There are many possible reasons for this, which we'll cover in a moment.) Worried consumers may then try to weather the coming economic hardship by saving their money. But because my spending is part of your earnings, my decision to hoard money makes things worse for you. And you, responding to your own difficult times, will start hoarding money too, making things even worse for me. So there's a vicious circle at work here: people hoard money in difficult times, but times become more difficult when people hoard money.

The cure for this, Keynes said, was for the central bank to expand the money supply. By putting more bills in people's hands, consumer confidence would return, people would spend, and the circular flow of money would be reestablished. Just that simple! Too simple, in fact, for the policy-makers of that time.

If this is the proposed definition and cure for recessions, then what about depressions? Keynes believed that depressions were recessions that had fallen into a "liquidity trap." A liquidity trap is when people hoard money and refuse to spend no matter how much the government tries to expand the money supply. In these dire circumstances, Keynes believed that the government should do what individuals were not, namely, spend. In his memorable phrase, Keynes called this "priming the pump" of the economy, a final government effort to reestablish the circular flow of money.

Let's return now to the reasons why people start hoarding money in the first place. There are many possible explanations, all of which are open to argument. It may be a consumer loss of confidence in the economy, perhaps triggered by a visible event like a stock market crash. It may be a natural disaster, such as a drought, earthquake or hurricane. It may be a sudden loss of jobs, or a weak sector of the economy. It may be inequality of wealth, which results in the rich producing a surplus of goods, but leaving the poor too poor to buy them. It may be something intrinsic within the economy which causes it to go through a natural cycle of recessions and recoveries. Or the Federal Reserve may tighten the money supply too much, compelling people to hang on to their disappearing dollars. This last point is especially important, since many critics of activist government believe that is how the Great Depression started.

As mentioned above, Keynes' advice on ending the Great Depression was rejected. President Roosevelt tried countless other approaches, all of which failed. Almost all economists agree that World War II cured the Great Depression; Keynesians believe this was so because the U.S. finally began massive public spending on defense. This is a large part of the reason why "wars are good for the economy." Although no one knows the full secret to economic growth (the world's top economists are still working on this mystery), wars are an economic boon in part because governments always resort to Keynesian spending during them. Of course, such spending need not be directed only towards war -- social programs are much more preferable.

In seven short years, under massive Keynesian spending, the U.S. went from the greatest depression it has ever known to the greatest economic boom it has ever known. The success of Keynesian economics was so resounding that almost all capitalist governments around the world adopted its policies. And the result seems to be nothing less than the extinction of the economic depression! Before World War II, eight U.S. recessions worsened into depressions (as happened in 1807, 1837, 1873, 1882, 1893, 1920, 1933, and 1937). Since World War II, under Keynesian policies, there have been nine recessions (1945-46, 1949, 1954, 1956, 1960-61, 1970, 1973-75, 1980-83, 1990-92 ), and not one has turned into a depression. The success of Keynesian economics was such that even Richard Nixon once declared, "We are all Keynesians now."

Keynesianism in the Postwar Era

After the war, economists found Keynesianism a useful tool in controlling unemployment and inflation. And this set up a theoretical war between liberals and conservatives that continues to this day, although it appears that Keynesianism has survived the conservatives' attacks and has emerged the predominant theory among economists. Before describing this battle, however, we should take a look at how the money supply is expanded or contracted.

In the U.S., there are several ways to expand the money supply. The most common is for Federal Reserve banks to buy U.S. debt from commercial banks. The money that commercial banks collect from the sale of these government securities increases the amount they can lend. A second way is to loosen credit requirements, thereby increasing the amount of money generated by the banking system. A third way is to cut the prime lending rate, which is the rate the Federal Reserve loans to commercial banks. To reduce money in the economy, the Fed commits all the opposite actions.

To fight unemployment, the Fed traditionally expands the money supply. This creates more spending in the economy, which creates more jobs.

But what would happen if the Fed expanded the money supply too much? For example, let's suppose the Treasury printed so much money that it made every American a millionaire. After everyone retired, they would notice there would be no more workers or servants left to do their bidding… so they would attract them by raising their wages, sky-high if necessary. This, of course, is the essence of inflation. Eventually, prices would rise so much that it would no longer mean anything to be a millionaire. Soon, everyone would be back working at their same old jobs.

To fight inflation, then, the Fed contracts the money supply.

The Federal Reserve thus has an important role in balancing the economy. Too little money in the economy means crushing unemployment; too much money means runaway inflation. Finding the right balance is the job of the Federal Reserve Board, a job which calls for considerable discretion -- hence the term discretionary monetary policy. Making the correct decisions depends on reading the economy correctly, and some Boards have been better at it than others. In the early days especially, the Fed had a tendency to overreact to developments, sometimes causing more harm than good. But the art of discretionary policy has improved over time. And the effects of monetary policy, even when handled poorly, are immediate, profound and easily measurable. No serious economist claims otherwise -- supply-siders aside.

Milton Friedman's attack on Keynesianism

Of course, Keynesianism has its critics, most of them conservatives who loathe the idea that government could ever play a beneficial role in the economy. One of the first major critics was Milton Friedman. Although he accepted Keynes' definition of recessions, he rejected the cure. Government should butt out of the business of expanding or contracting the money supply, he argued. It should keep the money supply steady, expanding it slightly each year only to allow for the growth of the economy and a few other basic factors. Inflation, unemployment and output would adjust themselves according to market demands. This policy he named monetarism.

During the 70s, monetarism reached the peak of its popularity among conservative economists. Today, however, Friedman stands virtually alone among top economists in his belief that it contains any merit. Monetarism was tried in Great Britain during the 80s and it proved to be a disaster. For almost seven years, the Bank of England tried its best to make it work. According to monetarist theory, the British economy should have enjoyed low inflation and high stability. But in fact, it went berserk. The economy sank into a deep recession, while the lead economic indicators zigged and zagged. Although inflation came down, this was at the price of rising unemployment, which soared from 5.4 to 11.8 percent. Between 1979 and 1984, manufacturing output fell 10 percent, and manufacturing investment fell 30 percent. Eventually, the Bank of England came under overwhelming pressure to abandon monetarism, which it did in 1986. The experiment was such a failure that not even conservatives abroad wish to repeat it.

Along with Great Britain, President Reagan announced that the U.S. would also follow a monetarist policy. However, this was simply a cover story, meant for public consumption only. In reality, the government's policies were thoroughly Keynesian. Government borrowing and spending exploded under Reagan, with the national debt climbing to $3 trillion by the time he left office. Paul Volcker, Chairman of the Federal Reserve Board, battled inflation during the severe recession of 1980-82 through the Keynesian method of raising interest rates and tightening the money supply. When inflation looked defeated in 1982, he abruptly slashed the prime rate and flooded the economy with money. A few months later, the economy roared to life, in a recovery that would last over seven years. The American experience was in direct contrast to Great Britain's. As a result, most economists abandoned monetarist theory.

Friedman is also famous for a second theory, this one containing much more merit. It's called the natural rate of unemployment, and it goes something like this:

Imagine an economy where the cost of everything doubles. You have to pay twice as much for your groceries, but you don't mind, because your paycheck is also twice as large. Economists call this the neutrality of money. If inflation worked this way, then it would be harmless. Indeed, most presidents after World War II decided to accept high inflation if it meant low unemployment, and therefore urged the Federal Reserve to conduct an expansionary monetary policy. But why is it that when the Fed expands money by, say, 5 percent, that all prices and wages everywhere do not go up by 5 percent as well? Why is it that the neutrality of money does not make this expansion meaningless? Friedman argued that it was because the public was unaware of the expansion, or what it meant, or by how much if it did. In other words, they didn't know that they should raise their prices by 5 percent. When the extra money was pumped into the economy, therefore, it was unwittingly translated into more economic activity, not higher prices.

Of course, if businessmen knew that a 5 percent increase was coming, it would be in their best interest to just raise their prices 5 percent. That way, they would make the same increased profits without having to work for them. If everyone did this, then the Fed's monetary increases would become meaningless -- instead of resulting in more jobs, it would just create higher inflation. Friedman and others argued that as businessmen became savvier and learned to follow the Fed's actions, they would build their inflationary expectations into their prices. Not only would this make inflation worse, but the nation would be left with no tool to fight unemployment, which would eventually rise as well. The twin dragons of inflation and unemployment would therefore grow together, forming "stagflation."

Friedman showed that monetary policy could not be used to wipe out unemployment, one of the optimistic goals of the Keynesians shortly after World War II. Instead, the most monetary policy could do was keep unemployment at about 6 percent, which is the rate normally achieved when the inflation rate is what the market expects it to be. Friedman called this the "natural rate of unemployment," and it secured his fame. But Keynesian policies are still useful in keeping the unemployment rate as close to 6 percent as possible.

Robert Lucas' attack on Keynesianism

An even bigger attack on Keynesianism came from Robert Lucas, the founder of a theory called rational expectations. Although one aspect of this theory won Lucas the Nobel Prize in 1995, history has not been kind to the rest of it. Lucas himself has abandoned work on rational expectations, devoting himself nowadays to other economic problems, and his once broad following has almost completely dissipated.

There are two main parts to rational expectations. First, Lucas believed that recessions are self-correcting. Once people start hoarding money, it may take several quarters before everyone notices that a recession is occurring. That's because individual businessmen may know that they are making less money, but it may take awhile to realize that the same thing is happening to everyone else. Once they do recognize the recession, however, the market quickly takes steps to recover. Producers will cut their prices to attract business, and workers will cut their wage demands to attract work. As prices fall, the purchasing power of the dollar is strengthened, which has the same effect as increasing the money supply. Therefore, government should do nothing but wait the correction out.

Second, government intervention ranges from ineffectualness to harm. Suppose the Fed, looking at the leading economic indicators, learns that a recession has hit. But this information is also available to any businessman in any good newspaper. Therefore, any government attempt to expand the money supply cannot happen before a businessman's decision to cut prices anyway. Keynesians are therefore robbed of the argument that perhaps the Fed might be useful in hastening a recovery, since Lucas showed that the Fed is not much faster than anyone else in discovering the problem.

Lucas then gave a slightly fuller version of the Milton Friedman argument outlined above. Suppose the Fed established a predictable anti-recession policy: for every point the unemployment rate climbs, it increases the money supply by a certain percent. Businesses would come to expect these increases -- hence the term, rational expectations -- and would simply raise their prices by the anticipated amount. In order to be effective, monetary policy would have to surprise businesses with random increases. But true randomness would make the economy less stable, not more so. The only logical conclusion is that the government's efforts to control the economy can actually be harmful.

Lucas' work enjoyed incredible prestige in the 70s. But today we know there are at least two major flaws in the theory.

First, it is not reasonable to believe that business owners determine their prices by following macroeconomic trends. Can you cite the Federal Reserve's rates and policies at the moment? The inflation and unemployment rates? Growth in the GDP? Even more improbably, do you set your prices and wage demands by these indicators? Only an economist (who knows all these statistics anyway) would think this is natural behavior.

Second, recessions last for years, which is far longer than people's ignorance of their onset. Lucas and his followers searched for every model imaginable that would keep businessmen aware of the leading economic indicators and yet ignorant of the fact that they were in a recession. Needless to say, they failed.

The recessions of 80-82 and 90-92 were clear refutations of Lucas' theory. Jimmy Carter was explicitly voted out of office for a misery index (unemployment plus inflation) that crested 20 percent. Yet it was not until 1987 that the unemployment rate fell back to 1979 levels. It is ludicrous to believe that it took the public eight years to figure out that they were in a recession and that they needed to cut prices back to the required level. And voters were highly aware that they were in a slump for most of the 90-92 recession; James Carville found a resonating campaign slogan for an entire election season with "It's the economy, stupid." Yet the economy did not even start to recover until the summer of 92, with employment taking even longer to rebound.

By the mid-80s, it was already apparent that neither monetarism nor rational expectations were adequate theories, and neo-Keynesianism started making a comeback. (Lucas won the Nobel Prize for that part of his theory which states that businessmen can compensate for expected monetary increases by raising their prices accordingly. Which is true in principle, but not often in practice.) One of the basic problems of conservative theories is that they place an almost religious faith in the belief that leaving markets alone always results in the best. How, in that case, does one explain recessions and depressions? Or the fact that depressions have disappeared since government started taking an active role? Besides, the belief that we should let national disasters like the Great Depression run unchecked for years while waiting for the economy to correct itself borders on the immoral.

The rise of the New Keynesians

Today, neo-Keynesianism has returned to prominence. At the heart of this updated version is the theory that people are not perfectly rational, but nearly rational. That is, they do not carefully weigh the unemployment rate, inflation rate and monetary policy before deciding to cut their monthly prices by, say, $24.13. Instead, people have only a fuzzy idea of where their prices should be, and make their best guesses. But because people are self-interested animals, they tend to err in their own favor, underestimating how much they really need to cut. This results in a long lag between the recognition of a recession and the decision to cut prices in earnest. In fact, the lag is so long that discretionary monetary policy is warranted in cutting the recession short.

But won't a businessman's rational expectations negate the Fed's actions? The answer, it turns out, is not completely. The Fed's decision to expand the money supply in 1982 was widely debated and highly publicized. Yet businessmen generally did not compensate for the Fed's announced moves by raising their prices. There are many reasons: a large percentage of businessmen could still be expected to remain unaware of the Fed's actions, or what they mean. For many, raising prices incurs certain costs (reprinting, recalculating, reprogramming, etc., not to mention a dip in business) that eat into the increases and may not make them worth it. And even if they do deem the price hikes worth it, it takes many companies quite some time to put them into effect. (Sears, for example, has to reprint and remail all its catalogues.) Also, remember that the impulse to raise prices cancels out the impulse to lower them, which is also how Lucas believed markets cured recessions. Others may be engaged in price wars with their competitors. So, for these and other reasons, expanding the money supply still results in job-creation, despite the counter-effect of rational expectations.

The re-emergence of Keynesianism is testimony of its staying power. Almost certainly, future economic theories will incorporate its findings.

Must Read Books for Pre-Ph.D Economics Students

2011-01-11T07:24:00.000-08:00

Q:If I want to achieve a Ph.D. in economics what steps would you advise me to take and what books and courses would I need to study to gain the knowledge that is absolutely needed to be able to do and understand the research that is needed for a Ph.D.

A:Thank you for your question. It's a question that I'm frequently asked, so it's about time that I created a page that I could point people toward.

It's really difficult to give you a general answer, because a lot of it depends on where you'd like to get your Ph.D. from. Ph.D programs in economics vary widely in both quality and scope of what is taught. The approach taken by European schools tends to be different than that of Canadian and American schools. The advice in this article will mainly apply to those who are interested in entering a Ph.D. program in the United States or Canada, but much of the advice should also apply to European programs as well. There are four key subject areas that you'll need to be very familiar with to succeed in a Ph.D. program in economics.

1. Microeconomics / Economic Theory
Even if you plan to study a subject which is closer to Macroeconomics or Econometrics, it is important to have a good grounding in Microeconomic Theory. A lot of work in subjects such as Political Economy and Public Finance are rooted in "micro foundations" so you'll help yourself immensely in these courses if you've already familiar with high level microeconomics. Most schools also require you to take at least two courses in microeconomics, and often these courses are the most difficult you'll encounter as a graduate student.

Microeconomics Material You Must Know as a Bare Minimum
I would recommend reviewing the book Intermediate Microeconomics: A Modern Approach by Hal R. Varian. The newest edition is the sixth one, bu if you can find an older used edition costing less you may want to do that.

Advanced Microeconomics Material that would be Helpful to Know
Hal Varian has a more advanced book called simply Microeconomic Analysis. Most economics students are familiar with both books and refer to this book as simply "Varian" and the Intermediate book as "Baby Varian". A lot of the material in here is stuff you wouldn't be expected to know entering a program as it's often taught for the first time in Masters and Ph.D. programs. The more you can learn before you enter the Ph.D. program, the better you will do.

What Microeconomics Book You'll Use When You Get There
From what I can tell, Microeconomic Theory by Mas-Colell, Whinston, and Green is standard in many Ph.D. programs. It's what I used when I took Ph.D. courses in Microeconomics at both Queen's University at Kingston and the University of Rochester. It's an absolutely massive book, with hundreds and hundreds of practice questions. The book is quite difficult in parts so you'll want to have a good background in microeconomic theory before you tackle this one.
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2. Macroeconomics
Giving advice on Macroeconomics books is a lot more difficult because Macroeconomics is taught so differently from school to school. Your best bet is to see what books are used in the school that you would like to attend. The books will be completely different depending on whether your school teaches more Keynesian style Macroeconomics or "Freshwater Macro" which is taught at places like "The Five Good Guys" which includes the University of Chicago, the University of Minnesota, Northwestern University, University of Rochester, and University of Pennsylvania.

If anyone can recommend a good set of Keynesian textbooks, can you please contact me using the feedback form. The advice I'm going to give is for students who are going to a school that teaches more of a "Chicago" style approach.

Macroeconomics Material You Must Know as a Bare Minimum
I would recommend reviewing the book Advanced Macroeconomics by David Romer. Although it does have the word "Advanced" in the title, it's more suited for high level undergraduate study. It does have some Keynesian material as well. If you understand the material in this book, you should do well as a graduate student in Macroeconomics.

Advanced Macroeconomics Material that would be Helpful to Know
Instead of learning more Macroeconomics, it would be more helpful to learn more on dynamic optimization. See my section on Math Economics books for more detail.

What Macroeconomics Book You'll Use When You Get There
When I took Ph.D courses in Macroeconomics a few years ago we didn't really use any textbooks, instead we discussed journal articles. This is the case in most courses at the Ph.D. level. I was fortunate enough to have macroeconomics courses taught by Per Krusell and Jeremy Greenwood and you could spend an entire course or two just studying their work. One book that is used quite often is Recursive Methods in Economic Dynamics by Nancy L. Stokey and Robert E. Lucas Jr. Although the book is almost 15 years old, it's still quite useful for understanding the methodology behind many macroeconomics articles. I've also found Numerical Methods in Economics by Kenneth L. Judd to be quite helpful when you're trying to obtain estimates from a model which does not have a closed-form solution.

3. Econometrics
Econometrics Material You Must Know as a Bare Minimum
There's quite a few good undergraduate texts on Econometrics out there. When I taught tutorials in undergraduate Econometrics last year, we used Essentials of Econometrics by Damodar N. Gujarati. It's as useful as any other undergraduate text I've seen on Econometrics. You can usually pick up a good Econometrics text for very little money at a large second-hand book shop. A lot of undergraduate students can't seem to wait to discard their old econometrics materials.

Advanced Econometrics Material that would be Helpful to Know
I've found two books rather useful: Econometrics Analysis by William H. Greene and A Course in Econometrics by Arthur S. Goldberger. As in the Microeconomics section, these books cover a lot of material which is introduced for the first time at the graduate level. The more you know going in, though, the better chance you'll have of succeeding.

What Econometrics Book You'll Use When You Get There
Chances are you'll encounter the king of all Econometrics books Estimation and Inference in Econometrics by Russell Davidson and James G. MacKinnon. This is a terrific text, because it explains why things work like they do, and does not treat the matter as a "black box" like many econometrics books does. The book is quite advanced, though the material can be picked up fairly quickly if you have a basic knowledge of geometry.
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4. Mathematics
Having a good understanding of mathematics is crucial to success in economics. Most undergraduate students, particularly those coming from North America, are often shocked by how mathematical graduate programs in economics are. The math goes beyond basic algebra and calculus, as it tends to be more proofs, such as "Let (x_n) be a Cauchy sequence. Show that if (X_n) has a convergent subsequence then the sequence is itself convergent". I've found that the most successful students in the first year of a Ph.D. program tend to be ones with mathematics backgrounds, not economics ones. That being said, there's no reason why someone with an economics background can not succeed.
Mathematical Economics Material You Must Know as a Bare Minimum
You'll certainly want to read a good undergraduate "Mathematics for Economists" type book. The best one that I've seen happens to be called Mathematics for Economists written by Carl P. Simon and Lawrence Blume. It has a quite diverse set of topics, all of which are useful tools for economic analysis.

If you're rusty on basic calculus, make sure you pick up a 1st year undergraduate calculus book. There are hundreds and hundreds of different ones available, so I'd suggest looking for one in a second hand shop. You may also want to review a good higher level calculus book such as Multivariable Calculus by James Stewart.

You should have at least a basic knowledge of differential equations, but you do not have to be an expert in them by any means. Reviewing the first few chapters of a book such as Elementary Differential Equations and Boundary Value Problems by William E. Boyce and Richard C. DiPrima would be quite useful. You do not need to have any knowledge of partial differential equations before entering graduate school, as they are generally only used in very specialized models.

If you're uncomfortable with proofs, you may want to pick up The Art and Craft of Problem Solving by Paul Zeitz. The material in the book has almost nothing to do with economics, but it will help you greatly when working on proofs. As an added bonus a lot of the problems in the book are surprisingly fun.

The more knowledge you have of pure mathematics subjects such as Real Analysis and Topology, the better. I would recommend working on as much of Introduction to Analysis by Maxwell Rosenlicht as you possibly can. The book costs less than $10 US but it is worth it's weight in gold. There are other analysis books that are slightly better, but you cannot beat the price. You may also want to look at the Schaum's Outlines - Topology and Schaum's Outlines - Real Analysis. They're also quite inexpensive and have hundreds of useful problems. Complex analysis, while quite an interesting subject, will be of little use to a graduate student in economics, so you need not worry about it.
Advanced Mathematical Economics that would be Helpful to Know
The more real analysis you know, the better you will do. You may want to see one of the more canonical texts such as The Elements of Real Analysis by Robert G. Bartle. You may also want to look at the book I recommend in the next paragraph.
What Advanced Mathematical Economics Book You'll Use When You Get There
At the University of Rochester we used a book called A First Course in Optimization Theory by Rangarajan K. Sundaram, though I don't know how widely this is used. If you have a good understanding of real analysis, you will have no trouble with this book, and you'll do quite well in the obligatory Mathematical Economics course they have in most Ph.D. programs.
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You do not need to study up on more esoteric topics such as Game Theory or International Trade before you enter a Ph.D. program, although it never hurts to do so. You are not usually require to have a background in those subject areas when you take a Ph.D. course in them. I will recommend a couple of books I greatly enjoy, as they may convince you to study these subjects. If you're at all interested in Public Choice Theory or Virginia style Political Economy, first you should read my article "The Logic of Collective Action". After doing so, you may want to read the book Public Choice II by Dennis C. Mueller. It is very academic in nature, but it is probably the book that has influenced me most as an economist. If the movie A Beautiful Mind didn't make you frightened of the work of John Nash you may be interested in A Course in Game Theory by Martin Osborne and Ariel Rubinstein. It is an absolutely fabulous resource and, unlike most books in economics, it's well written.

If I haven't scared you off completely from studying economics, there's one last thing you'll want to look into. Most schools require you to take one or two tests as part of your application requirements. Here's a few resources on those tests:

Get familiar with the GRE General and GRE Economics Tests
The Graduate Record Examination or GRE General test is one of the application requirements at most North American schools. The GRE General test covers three areas: Verbal, Analytical, and Math. I've created a page called "Test aids for the GRE and GRE Economics" that has quite a few useful links on the GRE General Test. The Graduate School Guide also has some useful links on the GRE. I would suggest buying one of the books on taking the GRE. I can't really recommend any one of them as they all seem equally good.

It is absolutely vital that you score at least 750 (out of 800) on the math section of the GRE in order to get into a quality Ph.D. program. The analytical section is important as well, but the verbal not as much. A great GRE score will also help you get into schools if you have only a modest academic record.

There are a lot fewer online resources for the GRE Economics test. There are a couple of books that have practice questions that you may want to look at. I thought the book The Best Test Preparation for the GRE Economics was quite useful, but it's gotten absolutely horrid reviews. You may want to see if you can borrow it before committing to buying it. There is also a book called Practicing to Take the GRE Economics Test but I've never used it so I'm not sure how good it is. It is important to study for the test, as it may cover some material that you did not study as an undergraduate. The test is very heavily Keynesian, so if you did your undergraduate work at a school heavily influenced by the University of Chicago such as the University of Western Ontario, there will be quite a bit of "new" macroeconomics you'll need to learn.

Conclusion
Economics can be a great field in which to do your Ph.D., but you need to be properly prepared before you enter into a graduate program. I haven't even discussed all the great books available in subjects such as Public Finance and Industrial Organization. I'm sure I've also left out more great resources than I've included

Recommended Books on Quantitative Finance

2011-01-08T06:55:00.000-08:00

"The Concepts and Practice of Mathematical Finance"

C++: "C++ design patterns and derivatives pricing."
Its objective is to teach the reader C++ design using examples from quantitative finance. The target reader is the wannabe quant who knows how to program procedurally, and knows basic C++ syntax, but doesn't really get all this object-oriented stuff.

"Quant Job Interview Questions and Answers"

Encyclopedia

"the Princeton Companion to Mathematics," editor Timothy Gowers, associate editors June Barrow--Green and Imre Leader.

"Encyclopedia of Quantitative Finance" editor Rama Cont.

Introductory analysis

"Yet Another Introduction to Analysis " by Victor Bryant

"Mathematical Analysis: A Straightforward Approach" by K.G. Binmore

"Principles of Mathematical Analysis" by Walter Rudin.
This a great second book on analysis. It starts from first principles but is drier that Bryant. So first read Bryant to get some idea of what is going on, and then work through Rudin to get all the details and to learn enough to prepare you for measure theory.

Complex analysis

Complex analysis is not essential to learn probability theory and stochastic processes. However, contour integration and Fourier transforms are indispensable tools for the working quant. It is also one of the most beautiful and useful areas of mathematics.

"Introduction to Complex Analysis" by Hilary Priestley

Probability Theory and Stochastic Processes

Modern financial mathematics relies heavily on probability theory, if you want to do it well, you really need to learn to think probablistically and to study the theory. To really understand stochastic processes, you need to work through a program of

* basic probability theory
* basis analysis
* discrete-time martingales
* continuous-time martingales
* stochastic integration

"Elementary Probability Theory" by Kai Lai Chung

"Probability and Random Processes" by Geoffrey Grimmett and David Stirzaker

"Probability with Martingales" by David Williams

"Diffusions, Markov Processes and Martingales: by Chris Rogers and David Williams.
This is a two volume set. It is a natural sequel to Williams' "probability with martingales," although the authors quickly repeat much material from that book. This is a very good choice for getting the basics of Brownian motions and continuous time martingales in a rigorous fashion. The second volume then goes on to discuss stochastic calculus.

"Introduction to Stochastic Integration" by K. L. Chung, R.J. Williams.
This is the same Chung but a different Williams! I found this to be the most readable account of stochastic integration theory. It assumes knowledge of continuous time martingales, however, so you must learn those elsewhere first. The authors do all the details, and focus on trying to present the most important case in careful and clear detail rather than trying to work in absurd generality.

Basic mathematical

"Arbitrage Theory in Continuous Time" by Tomas Bjork

"Financial Calculus: An Introduction to Derivative Pricing" by Martin W. Baxter, Andrew J.O. Rennie.

"The Mathematics of Financial Derivatives: A Student Introduction" by Paul Wilmott, Sam Howison and Jeff Dewynne

"Stochastic Calculus for Finance" volumers I and II by Steven Shreve

Medium mathematical finance

"Martingale Methods in Financial Modelling" by Marek Musiela and Marek Rutkowski

"Financial Modelling with Jump Processes" by Rama Cont and Peter Tankov. Financial markets crash and are inherently jump. There has therefore been much effort devoted in recent years to derivatives pricing using jumpy processes. Cont and Tankov is a nice exposition of this theory covering both jump-diffusion processes and more general Levy processes. The point of view is quite applied with proofs deemphasized.

Interest rate modelling

"Interest Rate Modeling" Vols 1, 2 and 3. "Foundations and Vanilla Models, Term Structure Models, Products and Risk Management" 3 volume set by Leif Andersen and Vladimir Piterbarg.

Two of the world's leading interest rate quants have teamed up to give a comprehensive state of the art treatment of the pricing and Greeking of exotic interest rate derivatives. This is by far the best treatment of the topic available. It is not introductory so read them once you are comfortable with financial mathematics. They strike a reasonable middle ground between hand-waving and technical obscurities. It inevitably does not cover everything since that would require another three volumes, but it covers a lot.

"Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit" by Damiano Brigo, Fabio Mercurio.
This is a comprehensive book on the theory and implementation of interest rate models with an emphasis on the LIBOR market model. It has the great virtue that the authors do all the details.

"Engineering BGM" by Alan Brace
The writer is the "B" of BGM. This is the closest thing to a definitive text on the LIBOR market model also known as BGM. It's hard going at points and the writer believes that "less is more" when explaining material. However, it addresses many points not considered elsewhere and is a must for anyone working seriously in the area.

Credit derivatives

"Credit Derivatives Pricing Models: Models, Pricing and Implementation" by P.J. Schonbucher
Credit derivatives were a booming area. Schonbucher introduces and discusses many of the standard models with a reasonable level of detail.

"Modelling single-name and multi-name credit derivatives" by Dominic O'Kane
The author was an executive at Lehmans who left before the meltdown. He has much coverage of models for pricing portfolio credit derivatives than Schonbucher does and even includes a few pages on the Joshi-Stacey Intensity Gamma model!

Numerical Techniques

"Monte Carlo Methods in Financial Engineering" by Paul Glasserman.
Monte Carlo is the most effective technique for high-dimensional integration. This book is comprehensive and lucid, it's definitely indispensable if you are implementing Monte Carlo pricing models.

"Monte Carlo Methods in Finance" by Peter Jackel
Whilst Glasserman's book is the definitive reference for Monte Carlo pricing in finance, Peter's book is the best guide available on the use of low-discrepancy numbers particularly Sobol numbers for high dimensional quasi-Monte-Carlo. Since their use can improve convergence rates from O(n^(-1/2)) to O(n^-1), they are an important tool, and it's essential to get all the details right, Peter's book teaches you how to do this.

"Numerical Mathematics and Computing" by Cheney and Kincaid.
This is an undergraduate textbook designed to teach someone with a smattering of numerical analysis how to program models for numerical computation.

"Numerical Recipes in C++: The Art of Scientific Computing" by William H. Press, Saul A. Teukolsky, William Vetterling, Brian P. Flannery.
This is a compendious collection of C++ source code and discussion of numerical techniques that form an indispensable resource for the working quant. The code suffers a bit from being translated from FORTRAN but is very useful.

C++

C++ is a standard tool for implementing pricing models in banks. Some day, something better will supercede it, but for now you have to learn it if you want to get a job as a quantitative analyst. There are also very many books on this topic. The great virtue of C++ books is that they take a lot less time to read than mathematical finance books so you can get through a lot more of them.

"C++ How to Program" by Harvey M. Deitel, Paul J. Deitel.
This is an introductory textbook for American undergraduates. This means it goes slow, is comprehensive, uses lots of colour and is easy to read. I would recommend this book if you haven't done much computing in other languages.

"C++ Primer" by Stanley B. Lippman, Josee Lajoie, Barbara Moo.
This is an introduction to C++ but it is really suited to someone who is very competent in other programming languages. So if you are au fait with programming and want something that will get you going quickly buy this. It's a classic and has sold over half a million copies

"Thinking in C++' by Bruce Eckel.
This two volume set is about how to use the C++ language properly and aim to teach you the right way to think about C++. In this it succeeds. It is, however, hard going for those who do not know C, and the author assumes some knowledge of that language. It's long and a lot of hard work but if you work through it, you will really know how to program in C++.

Topic books

Once you've got the basics, there are a number of books that aim to get you from the novice level to the intermediate level.

"Effective C++", "More effective C++" and "Effective STL" by Scott Meyers.
Effective C++ was one of the first books to really discuss how to use C++ as a language rather than focussing on the syntax. Meyers' style is to give you lots of informal advice about the right way to do things and in my experience, if Scott gives you a guideline you really ought to follow it

"Exceptional C++", "More exceptional C++", "Exceptional C++ style" by Herb Sutter.
The author presents problems, invites the reader to solve them, and then generally demonstrates that the reader doesn't understand C++ nearly as well as he thought. There is a particular focus on writing exception-safe code -- hence the title. Whilst the presentation can be irritating at times, and I don't buy some of his advice, Sutter will definitely improve your understanding of C++.

"Large-Scale C++ Software Design" by John Lakos.
Ever had a large project that turned into spaghetti, or had a project where you were afraid to change certain files because of the time it would take to rebuild the project. This book is on how to avoid such problems by organizing your code correctly from the start. Whilst the book is a little-dated and there's a certain amount of overlap wth Sutter's books, it's still a good read.

"The C++ Standard Library: A Tutorial and Reference" by Nicolai M. Josuttis.
C++ ships with a lot of classes and algorithms; these are called the Standard Library. Learning to use them properly will make your code quicker to develop, more robust and more efficient. Reading Josutti is a great way to do the learning.

"C++ Templates: The Complete Guide" by David Vandevoorde, Nicolai M. Josuttis.
Everything you ever wanted to know about templates and quite a few things you didn't. Templates in C++ have gone way beyond their designers' original intention of providing a way of doing generic programming to being a method of doing computations at compile time. This book is the definitive book on the topic.

Reference books

"The C++ Programming Language" by Bjarne Stroustrup.
This is the definitive guide to the language from the guy who invented it. Very useful but paedagogy is not Bjarne's strength.

"The C++ Standard: Incorporating Technical Corrigendum No. 1" by British Standards Institute.
This has to be one of the driest books ever written, but sometimes you really want to know what the "legal" rule is for some piece of C++ and this book is then great.

Background

None of these books are essential reading, but they will all give you some idea of what goes on in banks and stop you being appalling ignorant of the background before you go for interviews. They can also help solve the problem of what to say when your parents ask what you do for a living...

"My life as a quant: reflections on physics and finance" by Emanuel Derman.

The author was one of the first quants and was fortunate enough to work under Fischer Black at Goldman Sachs. He takes us through his career in both physics and finance. Whilst he is a not natural writer, he lived through interesting times and this book is a natural read for the wannabe quant.

"Liar's poker" by Michael Lewis.
The author's account of life at Salamon Brothers in the 1980s. If you want to understand the excesses of Wall Street in boom years, this is the book to read. Most bankers have read this book at one time or another and the terminology and stories are legends.

"When Genius Failed: The Rise and Fall of Long Term Capital Management" by Roger Lowenstein.
How one hedge fund almost brought on the global financial crisis ten years early by taking their mathematical models far too seriously. The cast of characters overlaps with that of "Liar's poker."

"Wriston: Walter Wriston, Citibank, and the Rise and Fall of American Financial Supremacy" by Phillip L. Zweig

"Fooled by randomness" and "the Black Swan" by Nassim Taleb

"The poker face of Wall Street" by Aaron Brown

"The Partnership: The Making of Goldman Sachs" by Charles D. Ellis

"How I Became a Quant: Insights from 25 of Wall Street's Elite" edited by Richard Lindsey and Barry Schachter
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General finance

1. Michael Lewis- Liar's Poker: Rising Through the Wreckage on Wall Street (Strongly Recommended)

Michael Lewis has written a very funny book. He describes the life of investment bank in middle 1980's on wall street. For anyone who wants to know what is the life about trading and selling in the dynamic 1980's, this book will be invaluable for them. You will also learn how Soloman Brothers have earned tons of million by selling mortgage bonds and how it failed later. It's an interesting history in finance, a legend of a wall street bank and funny stories of the bankers.

2. John Rolfe, Peter Troob - Monkey Business : Swinging Through the Wall Street Jungle (Recommended)

Another funny book in investment banking. The authors talked about their miserable life as analyst and associate in an investment bank. How they thought there will be an oasis in the desert of investment bank and how they finally found they were inside a jungle of dangers without boundary and how they act as monkeys swinging from one tree to another without having any personal life. If you want to know what is the life of an investment bank associate, you'd better open this book.

3. Emanuel Derman - My Life as a Quant : Reflections on Physics and Finance(Recommended)

The book is Derman's self-portray of his journey from academia to wall street. Emanuel Derman has a Ph.D. in physics from Columbia. Shortly after his Post doc at University of Pennsylvania, he moved to AT&T Bell Lab where he began to program. In 1985, he finally moved to Goldman, Sachs & Co., where he became a managing director in 1997. He is the SunGard/IAFE Financial Engineer of the Year in 2000 and was appointed to the Risk Hall of Fame in 2002. He is now the Director of the Program in Financial Engineering at Columbia University and lives in New York City. It is of future quants' interest to learn how he converted from a physicist, an engineer to a quant and finally achieved high fame in the whole new world. He began the book from his Ph.D life at Columbia University and started to describe his academic life which I found no interest. In the second part of the book, he then talks about the life at Goldman Sachs, which is much interesting for quant-minds. You can also learn about the so-called 'implied volatility', which is the 'smile' intriguing many quants in equity and still an active research area in finance. He has a personal website with his publications listed.

4. Zvi Bodie, Alex Kane, Alan J. Marcus -- Investments (Strongly Recommended)

This so-called "BKM" Investments book is widely used in U.S. MBA schools. It is must-have for anyone wants to know the financial maket, portfolio theory, Capital Asset Pricing Model, Fixed Income Securities, Options, Portfolio Management. It is so comprehensive that you can find almost every aspect of finance in the book. As many quants don't have adequate background in finance, especially in background of portfolio theory, this is a good book for them to stand at the same line with MBAs. The portfolio theory is so important as well as the CAPM theory, they are another evidence of financial applications of theoretical research in academic. Their inventors won Nobel Prize as well. You don't need to finish all the book since it's so thick. But one should at least learn well the portfolio theory and CAPM and put them on your resume. The book requires no advanced mathmatics and therefore financial sense is strengthened through the reading.
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Quantitative finance

1. Steven E. Shreve --- Stochastic Calculus for Finance II : Continuous-Time Models (Springer Finance) (Strongly Recommended)
I personally believe this is one of the best books on stochastic calculus for finance applications. It is accessible for anyone who has a background in calculus, probability, algebra and finance. If you want to go beyond the Neftci level, you should use this textbook. This book is the volume II of Shreve's two volume series of stochastic calculus for finance. Volume I is Stochastic Calculus for Finance I: The Binomial Asset Pricing Model. The difference betwen the two volumes is that first discusses discrete-time models while the second discusses continuous-time models. Shreve has been writing his book for many years dated back to 1980s when option pricing theory was getting popular. He cofounded Carnegie Mellon University's Master’s program in Computational Finance and this book evolves from the class he taught there over ten years. Over the time, he distributed the draft freely to many quantitative finance students who have benefited a lot from his notes. The volume II is good enough for a quant to learn suffice stochastic calculus. Chapter 1, 2 3 present basic background in probability theory such as probability measure, conditional expectation and sigma-algebra, abstract concept for information. Chapter 4 discusses stochastic calculus, including Brownian motion, quadractic variation, Ito Integral, Ito's lemma, stochastic differential equation (SDE) and derivation of famous Black-Scholes-Merton equation etc. Chapter 5 discusses risk neutral measure, Girsanov theorem and chapter 6 discusses Feyman-Kac theorem which relates risk-neutral measure to partial differential equations. Chapter 7-11 discusses exotic options, American options, change of numeriaire, term structure of interest rate and jump diffusion process. A quant should finish the first six chapters to have a solid background in stochastic calculus. A quant would better finish the exercises in the book as well.

2. Mark Joshi - The Concepts and Practice of Mathematical Finance (Mathematics, Finance and Risk) (Recommended)

This book is between introductory and intermediate level. You will get a better understanding of the text if you have read through John Hull's book before. The author writes clearly in his view of quantitative finance. He starts from risk, arbitrage and presents the simple binomial tree to price option. The good thing is that the author then points out the practical shortcomings of the theoretically perfect pricing method. Later, the author begins to discuss stochastic calculus and risk-neutral and uses them to price exotic options. Some advanced quantitative finance have been discussed such as volatility smile, jump-diffusion etc. I like the chapter on static replication, which is seldom discussed on other quantitative finance books. Since replication is the bedrock for the option pricing, it should desire more attentions than other pricing methods although many other books ignore the issue intentionally or unintentionally. At some points, where beginners normally ignore, the author gives great presentations of what things should look like. It's rather a practical book than conceptual. Many examples are very practical and away from ideal theoretical world. One more worthmention is that at the end of the book there are some programming projects which are useful for anyone who wants to practice their knowledge and programm skills in finance applications. You should try to finish the projects and put them on your resume if you don't have better words to brag yourself. There are also exercises at the end of each chapter, which can be used to examine your knowledge of finance.
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Programming

Programming should occupy 30%~40% of your time as a quant. If you spend more time on programming, you'd better improve your programming skills. It is benefical that you do coding as fast as possible because it saves you time on debugging and frees you from dirty works. The books mentioned in section will help you understand C++ better and program robust, reliable, resusable codes. These books ( #1-#6 ) are frequently referred to in many other C++ books and proved to be germs in so many C++ books. As a quant, you need not buy any other C++ books anymore. These books are good enough for both learning the language as well as serve as references. There are also other C++ programming books written mainly by quants and for quants, you definitely can try them out.

Mark Joshi -- C++ Design Patterns and Derivatives Pricing (Mathematics, Finance and Risk)(Strongly Recommended) quants-reading-list

This 200-page little book does its job very well. Mark starts the book with a simple Monte Carlo program to price a call option and he goes out adding other flavors around the center piece. How should the quant design the option pricing class such that he or other quants can later derive child class for other options, how should he encapsulate the data and operations, how to use wrapper pattern, how to add more functions through virtual function, etc. He also discusses random number class, template pattern for option class, binomial tree method, implied volatilities and finally all the previous knowledge comes into one big project. It is good for anyone who knows C++ language but does not have object-oriented programming experience. He also integrates finance into the codes very well. After reading his book, future quants will know why C++ programming is so important such days and how one can do in real life. The valuable thing is the exercises. Readers should try to follow the book and implement the exercises at the end of each chapter. Once you finish the book, you should be able to write the book into your resume and claim that you have implemented Black-Scholes model using Monte Carlo method in C++, that you have solved implied volatilities using Newton-Raphson method. You should remember the singleton and the virtual constructor patterns in case some interviewers ask if you know any design patterns. He also wrote another book, which is reviewed in 'quantitative finace book review' section.

1. Bjarne Stroustrup-The C++ Programming Language (Special 3rd Edition)

Bjarne Stroustrup is the creator of C++ and he writes a great book about the language he created himself. The author not only describes about almost every aspect of the C++ language, he also gives detail explaination and the underneath mechanism for the features of the language. Reading this book combined with the following books with give you sufficient background on internal mechanicsm of c++ and better understanding of the DOs and DONTs in practical programming. It can also serve as the reference for your daily programming needs.

2. Bjarne Stroustrup-The Design and Evolution of C++

In this book, Bjarne Stroustrup describes how he designed the language and how the design changes with the evolution of the language itself. You will learn why there will be public, private members, why we need friend functions, how virtual functions got introduced and many other features that make c++ easy to use as well as robust. Although this book is a little old, it definitely deserves your attentation if you want to know the ins and outs of c++ programming.

3. David Vandevoorde- C++ Templates: The Complete Guide

Templates have been used widely in c++, especially in standard template library (STL), which provides a very flexible plaform of programming. David Vandevoorde explains templates very well and the book is well suited for desk reference. For the reasons to read other people's codes as well as write your own codes in temples, this book is a must.

4. Erich Gamma, Richard Helm, Ralph Johnson, John Vlissides- Design Patterns: Elements of Reusable Object-Oriented Software (Addison-Wesley Professional Computing Series)

Also called the "Gang of Four". The design patterns can be applied to any object-oriented language, not c++ itself. The authors summarize the common programming design patterns that can be applied 'universally' to any predefined situations, which may be difficult to deal with without deep thinking. For quantatitative finance job interviews, one may need have a good understanding of singleton pattern, virtual constructor pattern since they are used widely in modern programming.

5. Nicolai M. Josuttis - The C++ Standard Library : A Tutorial and Reference

Modern c++ has been greatly expanded by the introducing of Standard Template Library (STL). STL is not only a library, it widely uses generic programing concept and makes daily programming much easier, reliable and reusable. This book introduces you the components of STL and explains the container, algorithm, iterator and functor. This is the only book you need for using STL.

6. Scott Meyers
--- Effective C++ : 55 Specific Ways to Improve Your Programs and Designs (3rd Edition) (Required)
--- More Effective C++: 35 New Ways to Improve Your Programs and Designs (Required)
--- Effective STL: 50 Specific Ways to Improve Your Use of the Standard Template Library (Required)
These books have been introduced in the books for review section. They are listed here for the completeness. Scott Meyers has written something classic. His effective series on C++ language have proved to be extremely valuable for both new and experienced C++ programmers. He teaches you the ins and outs of C++ language. He points out the common pitfalls encountered during object-oriented programming. The value of these books for quants are in the interviews. This has been proved by many quants. Many interviewers will ask you C++ questions whose answers can be found in the books. If you are not familiar with C++ and want to prepare for an interview of programming related position, you definitely need the books. Not only you learn from Scott Meyers, he also prepares you for the interviews.
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These books are mathematical books that have important applications in quantitative finance. Quant is supposed to have a solid background in mathematics. They include calculus, linear algebra, stochastic, statistics, probability. Good mathematical books are essential to master those subjects and refer to when necessary. As there are so many mathematical books, we try to list those most useful for quants and may ignore many other good books out there.

1. William H. Press etc. - Numerical Recipes in C++: The Art of Scientific Computing (strongly recommended)
This is not a pure mathematical book, but it is very important for quants to finish their algorithms for financial modeling quickly and correctly. This book has all the numerical algorithms you need for quantitative finance applications. Even though the writings of C++ programming are not perfect, the essence of the book, the algorithms, are always proved to be extremely useful for quants as well as for scientists and engineers. You only need one algorithm book, Numerical Recipes in C++, and NO MORE
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This reading list for quants is a combination of:
1. The well-know 'path to enlightment' in wilmott's website, which is a collection of book-reading suggestions from experienced quants' for newbies. The basic frame of the list is kept.
2. The financial engineering book recommendation from Quantlib.org.
3. My personal reading experience: not only in quantitative finance but also in general finance and some other interview-related interests for future quants.
It is really not enough for a quant to finish just the 'interview books' and states that he can do every job in front of him. Quants need to learn a lot of things: stochastic, finance, programming. At each stage, you will find more new things are waiting for you to learn. It is a journey both exciting and hardworking. New quants, take a look at the book list and get an idea of what is lying ahead of you.

First steps - general
A. Black-Scholes and Beyond: Option Pricing Models, N A Chriss
Many people like this book simply because it treats Black-Scholes so well that you have no difficulty learning the ins-and-outs of the Nobel-Prize winning formula.

B. Derivative Securities, R Jarrow, S Turnbull

C. Introduction to Mathematical Finance: Discrete Time Models, S R Pliska

First steps - Interest rates

A. Fixed Income Analytics, K Garbade

First steps - Stochastic Calculus

A.---Introduction to the Mathematics of Financial DerivativesIntroducation to financial calculus S N Neftci
As introduced in the 'interview books for quants' section, this book is rather superfacial. It skims through the stochastic calculus quickly, gives you a feel about it and then leave you there without a solid holding of what really stochastic calculus means. However, it DOES do a good job in leading a newbie into the world of stochastic calculus and finance. If you don't feel comfortable reading other stochastic calculus books, read this one first.

First steps - Honourable mention

A. Option Market Making : Trading and Risk Analysis for the Financial and Commodity Option Markets (Wiley Finance), A J Baird

1.0. Introductory - General

A. Options Markets, J C Cox, M Rubinstein (classic!!!)
This is a classic book written by two great researchers in finance. Once you started working as quant, you will find how many times their works have been cited in the literature you are reading. This book is out of print now, but you can always order second hand book from amazon.com

B. Options, Futures and Other Derivatives (6th Edition), J C Hull
Everyone on wall street has a copy, be it US version or international, hardcopy or soft.

C. An Introduction to Mathematical Finance : Options and Other Topics, S M Ross

D. Paul Wilmott Introduces Quantitative Finance, P Wilmott
Dr. Wilmott's simple, plain introduction to quantitative finance. If you don't want to buy his expensive two-volume book (listed below), this is also a good one for you to start thinking what are quants doing.

E. The Mathematics of Financial Derivatives : A Student Introduction, P Wilmott, S Howison, J Dewynne

1.1 Introductory - Interest rates

A. Modelling Fixed Income Securities and Interest Rate Options (2nd Edition), R A Jarrow

1.2 Introductory - Exotics

A.Structured Equity Derivatives: The Definitive Guide to Exotic Options and Structured Notes, H M Kat
1.3 Introductory - Stochastic Calculus

A. Elementary Stochastic Calculus With Finance in View (Advanced Series on Statistical Science & Applied Probability, Vol 6), T Mikosch

1.4 Introductory - Computational
A. Pricing Derivative Securities: An Interactive, Dynamic Environment with Maple V and Matlab, E Z Prisman

1.5 Introductory - Honourable mention

A. Investment under Uncertainty

Quantitative Finance Reading List

2011-01-08T00:23:00.000-08:00

Theoretical Foundations

Not everybody wants to become a theoretical physicist. Some consider the academic environment too relaxed, others are not keen on the politics or the necessity to continually hunt for funding early in their career. A job in Quantitative Finance offers an attractive alternative.

Financial engineering has both strong theoretical and applied components, is immensely intellectually stimulating and fast-paced. A significant degree of background knowledge and an exceptional academic record are required even to achieve an interview. If you have recently decided that academia is not where your career path lies and you possess strong technical skills then the reading list outlined below will get you started towards becoming a quant.

This is the first part in a multi-part series on textbooks suitable for becoming a quantitative analyst. The remaining parts will focus on implementation, further mathematical excursions, interview skills and numerical methods. This article will concentrate on the theory of financial engineering for those who have not had an exposure to finance before.

Mathematical Finance

A great place to start learning about the world of derivatives is with the classic text Options, Futures and Other Derivatives by John Hull. It is light on the mathematics, but covers a lot of ground. Specifically, it is a good introduction to derivative markets for those who haven't had prior exposure to finance.

Once you're comfortable with the concepts used in the financial markets the next step is to begin learning about arbitrage and the Black-Scholes model in a more mathematical manner. Dan Stefanica's A Primer for the Mathematics of Financial Engineering will provide all of the calculus (differentiation, integration, taylor expansion etc) needed to tackle the Black-Scholes equation. It will also cover "the Greeks" and basic risk neutral pricing. This is a great book for somebody who doesn't have the required undergraduate mathematical background needed for later texts.

At this stage you will be ready to tackle the intermediate works such as Mark Joshi's Concepts and Practice of Mathematical Finance (an excellent book, highly recommended), Paul Wilmott on Quantitative Finance (extremely comprehensive and humourous explanations!), Baxter and Rennie's Financial Calculus and Salih Neftci's Introduction to the Mathematics of Financial Derivatives. A good working knowledge of the contents of these books is sufficient theory for any front office desk quant interviews.

If you wish to delve deeper into the mathematical theory underpinning derivatives pricing then Bernt Oksendal's Stochastic Differential Equations is a great start, as it has plenty of SDE exercises to work through.

A rather heavy going text for desk work, but an essential book for researching financial engineering, is the two volume masterpiece by Steven Shreve - Stochastic Calculus for Finance (Vol I and Vol II). Vol I concentrates on the discrete pricing models while Vol II focuses on continuous models. Be warned that for the Vol II, a strong background in undergraduate mathematics is required - particularly in Real Analysis, Probability Theory and Measure Theory.

Summary and Suggested Reading Chronology

1. Options, Futures and Other Derivatives - John Hull
2. A Primer for the Mathematics of Financial Engineering - Dan Stefanica
3. The Concepts and Practice of Mathematical Finance - Mark Joshi
4. Financial Calculus: An Introduction to Derivative Pricing - Martin Baxter, Andrew Rennie
5. Stochastic Calculus for Finance II: Continuous-Time Models - Steven Shreve

In the next article, texts on implementation will be presented which will give you the knowledge you need to begin creating your own quant models.
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Numerical Methods

In the previous article the core C++ books required for a good grounding in quantitative programming were outlined. Now it is time to discuss the books useful for learning numerical methods, in particular Finite Difference Methods (FDM) and Monte Carlo Methods (MCM).

Finite Difference Methods

Finite Difference Methods are a class of numerical methods used to provide an approximate, discrete solution to various partial differential equations, in particular the Black-Scholes PDE. Finite Difference Methods work by discretising the derivative terms in the PDE, such that they can be implemented algorithmically. An explicit FDM has the quantities at the next time step calculated in terms of the values at the previous step. An implicit FDM has the quantities at the next time step calculated in terms of both the values of the next time step and the previous time step. Stability of the scheme is an important concept.

The following are some of the more well known (and recommended!) text books on Finite Difference Methods:

1. Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach - Duffy
2. Financial Instrument Pricing Using C++ - Duffy
3. Numerical Solution of Partial Differential Equations: Finite Difference Methods - Smith
4. Pricing Financial Instruments: The Finite Difference Method - Tavella and Randall
5. Option Pricing: Mathematical Models and Computation -Wilmott et al.

Monte Carlo Methods

Monte Carlo Methods rely on the concept of risk neutral valuation in order to price derivatives. In essence, many underlying random asset price paths are calculated and the associated derivative payoff is calculated for each path. The mean of the payoffs are taken and then the price is discounted to today's price. This will give an approximation of the the option price. Further accuracy can be obtained by increasing the number of random trials.

Here are some of the top financial modelling MCM books:

1. C++ Design Patterns and Derivatives Pricing - Joshi
2. Monte Carlo Methods in Financial Engineering - Glasserman
3. Monte Carlo Frameworks: Building Customisable High-performance C++ Applications - Duffy et al.
4. Monte Carlo Methods in Finance - Jaeckel
5. Monte Carlo Methodologies and Applications for Pricing and Risk Management - Dupire

Suggested Reading

The best books to start with from a C++/numerical point of view are Duffy's "Financial Instrument Pricing Using C++" and Joshi's "C++ Design Patterns and Derivatives Pricing" books. In fact, Joshi's can be read in conjunction with his "Concepts and Practice of Mathematical Finance". They will get you up to speed on intermediate usage of C++ as well as give you an insight into both FDM and MCM. Depending on which way you lean (FDM or MCM), you may wish to continue with Wilmott's "Option Pricing" or with Glasserman's "Monte Carlo Methods in Financial Engineering" and Duffy's "Monte Carlo Frameworks.
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Programming

Learning how to implement financial models is a three-stage process. The first stage requires a deep understanding of the theory, which provides necessary mathematical tricks which can be exploited to optimise the code. The second stage involves understanding the computational language of the implementation and how to apply it in a software engineering setting. Finally, the third aspect is the marriage of the two first stages. This is why PhD candidates in a technical discipline are highly sought for financial engineering, as they already possess the ability to independently model technical phenomena.

In the modern financial world C++ is by far the most prevalent programming language. A good understanding of the language will be a necessary prerequisite to gaining an interview. It is far easier to work through programming textbooks than mathematical texts, hence there are more listed here than in the theoretical foundations article. Since the aforementioned first stage of "implementation" has been discussed in the previous article, the second stage will be considered here, in particular the C++ language.
C++

The first consideration is where you will program your code. You will need to obtain an Integrated Development Environment (IDE) which is where you will enter your syntax and run your programs. Depending upon your operating system choice, you may wish to download the free version of Microsoft's Visual Studio C++ or use the GCC compiler that is part of most Linux distributions. In particular if you use Ubuntu Linux you will need to run "apt-get install build-essential" in order to obtain the tools. As for a Linux development environment the author prefers Emacs, but vi or Eclipse are equally appropriate.

There are many beginner guides to learning C++. The author has experience with Jesse Liberty's Teach Yourself C++ in One Hour a Day which is now in its 6th Edition. This book will give you a good foundation in the C++ language and syntax. It will teach you all of the basics of programming, including functions, program flow, memory management and object-orientation. It even touches on the Standard Template Library (STL). It is highly recommended.

The next stage in learning how to be a good C++ programmer is to consider style, software design principles, gain a deeper level of object orientation and generic programming. The author has personally found Solter and Kleper's Professional C++ Programming (Programmer to Programmer) to be highly useful in this regard. It has good chapters on memory management, style and C++ quirks. It is a little out of date regarding software design principles, but the remainder of the book is sound.

Scott Meyers has a well deserved reputation as a C++ expert and his two books on how to improve C++ coding will be useful even to seasoned developers. Most expert C++ developers will not even consider hiring you unless you have read these two books. The first book, Effective C++: 55 Specific Ways to Improve Your Programs and Designs is in its 3rd Edition and concentrates on memory management and object orientation. The second book More Effective C++: 35 New Ways to Improve Your Programs and Designs, spends more time on exception handling and efficiency. Herb Sutter's Exceptional C++ is also a noteworthy read, concentrating on exception safety and object orientation.

Learning C++ to the level of Meyers will be sufficient for desk quant job interviews. However, if mastery of C++ is your goal then learning about Design Patterns and the STL are the next logical steps. The "Gang Of Four" book Design patterns: Elements of Reusable Object-Oriented Software is the standard text on Design Patterns. Josuttis' text on the STL, The C++ Standard Library: A Tutorial and Reference is highly recommended but is quite a heavy read. It is only worth looking into once you are very comfortable with C++ syntax and idioms. Meyers also has a book on best practices for STL use - Effective STL: 50 Specific Ways to Improve the Use of the Standard Template Library - which is worth picking up.

Summary and Suggested Reading Chronology

1. Teach Yourself C++ in One Hour a Day - Liberty, et al.
2. Professional C++ Programming (Programmer to Programmer) - Solter, Kleper
3. Effective C++: 55 Specific Ways to Improve Your Programs and Designs - Meyers
4. More Effective C++: 35 New Ways to Improve Your Programs and Designs - Meyers
5. The C++ Standard Library: A Tutorial and Reference - Josuttis

In the next article, texts on numerical methods will be considered which will give you the knowledge you need to finally implement the models and obtain useful results.

2011-01-07T08:39:00.001-08:00

Computational Finance : The Quants Heaven

If you have an aptitude in Computer Programming, Mathematics and Finance then you are looking for the term known as Computational Finance or Financial Engineering. Computational Finance helps in making investment, trading and hedging decisions using Computer Intelligence, mathematical-numerical analysis and simulations. Generally, individuals who fill positions in computational finance are known as “quants”, referring to the quantitative skills necessary to perform the job. The main aim of Computational Finance is to calculate the financial risk of any financial problem by modeling it using computer intelligence and mathematical analysis.

Computational Finance uses the mathematical tools like Probability Distribution, Calculus, Differential Equations, Numerical analysis etc. to set up a model for a financial problem. One of the very famous models is the Black-Scholes model which is used for Option Pricing model. The model develops partial differential equations whose solution, the Black–Scholes formula, is widely used in the pricing of European-style options.

Computational finance is a wide umbrella of disciplines — mathematical science, and the use of computer simulations to explore the potential risks as well as the probable outcomes of trading, hedging and investment decisions. Applications of Computational finance are used in the fields of investment banking, financial risk management, options pricing, strategic planning and so on. Concepts such as Monte-Carlo simulations, portfolio selection & optimization and high frequency data analysis are the basic topics in computational finance

Some of the current work going on in this field are :-

1) Agent-Based Artificial Markets

2) Derivatives trading

3) Investment Banking

4) Forecasting

5) Security Trading

The 10 Quant schools are Carnegie Mellon University, Columbia University, Cornell University, New York University, Princeton University, Rutgers University, Stanford University, University of California at Berkeley, University of Chicago and University of Michigan as per the report from advanced trading

quantitative analysis

2011-01-07T01:54:00.000-08:00

Harry Markowitz's 1952 Ph.D thesis "Portfolio Selection" was one of the first papers to formally adapt mathematical concepts to finance. Markowitz formalized a notion of mean return and covariances for common stocks which allowed him to quantify the concept of "diversification" in a market. He showed how to compute the mean return and variance for a given portfolio and argued that investors should hold only those portfolios whose variance is minimal among all portfolios with a given mean return. Although the language of finance now involves Itō calculus, management of risk in a quantifiable manner underlies much of the modern theory.

In 1969 Robert Merton introduced stochastic calculus into the study of finance. Merton was motivated by the desire to understand how prices are set in financial markets, which is the classical economics question of "equilibrium," and in later papers he used the machinery of stochastic calculus to begin investigation of this issue.

At the same time as Merton's work and with Merton's assistance, Fischer Black and Myron Scholes were developing their option pricing formula, which led to winning the 1997 Nobel Prize in Economics. It provided a solution for a practical problem, that of finding a fair price for a European call option, i.e., the right to buy one share of a given stock at a specified price and time. Such options are frequently purchased by investors as a risk-hedging device. In 1981, Harrison and Pliska used the general theory of continuous-time stochastic processes to put the Black-Scholes option pricing formula on a solid theoretical basis, and as a result, showed how to price numerous other "derivative" securities.

Quantitative behavioral financec

2011-01-07T01:42:00.000-08:00

Quantitative behavioral finance[1] is a new discipline that uses mathematical and statistical methodology to understand behavioral biases in conjunction with valuation. Some of this endeavor has been led by Gunduz Caginalp (Professor of Mathematics and Editor of Journal of Behavioral Finance during 2001–2004) and collaborators including Vernon Smith (2002 Nobel Laureate in Economics), David Porter, Don Balenovich, Vladimira Ilieva, Ahmet Duran). Studies by Jeff Madura [2], Ray Sturm [3] and others have demonstrated significant behavioral effects in stocks and exchange traded funds.

The research can be grouped into the following areas:

1. Empirical studies that demonstrate significant deviations from classical theories[4].
2. Modeling using the concepts of behavioral effects together with the non-classical assumption of the finiteness of assets.
3. Forecasting based on these methods.
4. Studies of experimental asset markets and use of models to forecast experiments.

History

The prevalent theory of financial markets during the second half of the 20th century has been the efficient market hypothesis (EMH) which states that all public information is incorporated into asset prices. Any deviation from this true price is quickly exploited by informed traders who attempt to optimize their returns and it restores the true equilibrium price. For all practical purposes, then, market prices behave as though all traders were pursuing their self-interest with complete information and rationality.

Toward the end of the 20th century, this theory was challenged in several ways. First, there were a number of large market events that cast doubt on the basic assumptions. On October 19, 1987 the Dow Jones average plunged over 20% in a single day, as many smaller stocks suffered deeper losses. The large oscillations on the ensuing days provided a graph that resembled the famous crash of 1929. The crash of 1987 provided a puzzle and challenge to most economists who had believed that such volatility should not exist in an age when information and capital flows are much more efficient than they were in the 1920s.

As the decade continued, the Japanese market soared to heights that were far from any realistic assessment of the valuations. Price-earnings ratios soared to triple digits, as Nippon Telephone and Telegraph achieved a market valuation (stock market price times the number of shares) that exceeded the entire market capitalization of West Germany. In early 1990 the Nikkei index stood at 40,000, having nearly doubled in two years. In less than a year the Nikkei dropped to nearly half its peak.

Meanwhile, in the US the growth of new technology, particularly the internet, spawned a new generation of high tech companies, some of which became publicly traded long before any profits. As in the Japanese stock market bubble a decade earlier these stocks soared to market valuations of billions of dollars sometimes before they even had revenue. The bubble continued into 2000 and the consequent bust reduced many of these stocks to a few percent of their prior market value. Even some large and profitable tech companies lost 80% of their value during the period 2000-2003.

These large bubbles and crashes in the absence of significant changes in valuation cast doubt on the assumption of efficient markets that incorporate all public information accurately. In his book, “Irrational Exuberance”, Robert Shiller discusses the excesses that have plagued markets, and concludes that stock prices move in excess of changes in valuation. This line of reasoning has also been confirmed in several studies (e.g., Jeffrey Pontiff [5]), of closed-end funds which trade like stocks, but have a precise valuation that is reported frequently. (See Seth Anderson and Jeffrey Born “Closed-end Fund Pricing” [6] for review of papers relating to these issues.)

In addition to these world developments, other challenges to classical economics and EMH came from the new field of experimental economics pioneered by Vernon Smith who won the 2002 Nobel Prize in Economics. These experiments (in collaboration with Gerry Suchanek, Arlington Williams and David Porter and others) featuring participants trading an asset defined by the experimenters on a network of computers. A series of experiments involved a single asset which pays a fixed dividend during each of 15 periods and then becomes worthless. Contrary to the expectations of classical economics, trading prices often soar to levels much higher than the expected payout. Similarly, other experiments showed that many of the expected results of classical economics and game theory are not borne out in experiments. A key part of these experiments is that participants earn real money as a consequence of their trading decisions, so that the experiment is an actual market rather than a survey of opinion.

Behavioral finance (BF) is a field that has grown during the past two decades in part as a reaction to the phenomena described above. Using a variety of methods researchers have documented systematic biases (e.g., underreaction, overreaction, etc.) that occur among professional investors as well as novices. Behavioral finance researchers generally do not subscribe to EMH as a consequence of these biases. However, EMH theorists counter that while EMH makes a precise prediction about a market based upon the data, BF usually does not go beyond saying that EMH is wrong.

Research in Quantitative Behavioral Finance
The attempt to quantify basic biases and to use them in mathematical models is the subject of Quantitative Behavioral Finance. Caginalp and collaborators have used both statistical and mathematical methods on both the world market data and experimental economics data in order to make quantitative predictions. In a series of papers dating back to 1989, Caginalp and collaborators have studied asset market dynamics using differential equations that incorporate strategies and biases of investors such as the price trend and valuation within a system that has finite cash and asset. This feature is distinct from classical finance in which there is the assumption of infinite arbitrage.

One of the predictions of this theory by Caginalp and Balenovich (1999) [7] was that a larger supply of cash per share would result in a larger bubble. Experiments by Caginalp, Porter and Smith (1998) [8] confirmed that doubling the level of cash, for example, while maintaining constant number of shares essentially doubles the magnitude of the bubble.

Using the differential equations to predict experimental markets as they evolved also proved successful, as the equations were approximately as accurate as human forecasters who had been selected as the best traders of previous experiments (Caginalp, Porter and Smith).

The challenge of using these ideas to forecast price dynamics in financial markets has been the focus of some of the recent work that has merged two different mathematical methods. The differential equations can be used in conjunction with statistical methods to provide short term forecasts.

One of the difficulties in understanding the dynamics of financial markets has been the presence of “noise” (Fischer Black). Random world events are always making changes in valuations that are difficult to extract from any deterministic forces that may be present. Consequently, many statistical studies have only shown a negligible non-random component. For example, Poterba and Summers demonstrate a tiny trend effect in stock prices. White showed that using neural networks with 500 days of IBM stock was unsuccessful in terms of short term forecasts.

In both of these examples, the level of “noise” or changes in valuation apparently exceeds any possible behavioral effects. A methodology that avoids this pitfall has been developed during the past decade. If one can subtract out the valuation as it varies in time, one can study the remaining behavioral effects, if any. An early study along these lines (Caginalp and Greg Consantine) studied the ratio of two clone closed-end funds. Since these funds had the same portfolio but traded independently, the ratio is independent of valuation. A statistical time series study showed that this ratio was highly non-random, and that the best predictor of tomorrow’s price is not today’s price (as suggested by EMH) but halfway between the price and the price trend.

The subject of overreactions has also been important in behavioral finance. In his 2006 PhD thesis [9], Duran examined 130,000 data points of daily prices for closed-end funds in terms of their deviation from the net asset value (NAV). Funds exhibiting a large deviation from NAV were likely to behave in the opposite direction of the subsequent day. Even more interesting was the statistical observation that a large deviation in the opposite direction preceded such large deviations. These precursors may suggest that an underlying cause of these large moves—in the absence of significant change in valuation—may be due to the positioning of traders in advance of anticipated news. For example, suppose many traders are anticipating positive news and buy the stock. If the positive news does not materialize they are inclined to sell in large numbers, thereby suppressing the price significantly below the previous levels. This interpretation is inconsistent with EMH but is consistent with asset flow differential equations that incorporate behavioral concepts with the finiteness of assets. Research continues on efforts to optimize the parameters of the asset flow equations in order to forecast near term prices.

Statistical finance

2011-01-07T01:38:00.000-08:00

Statistical finance,[1] sometimes called econophysics,[2] is an empirical attempt to shift finance from its normative roots to a positivist framework using exemplars from statistical physics with an emphasis on emergent or collective properties of financial markets. The starting point for this approach to understanding financial markets are the empirically observed stylized facts.
Stylized facts

1. Stock markets are characterised by bursts of price volatility.
2. Price changes are less volatile in bull markets and more volatile in bear markets.
3. Price change correlations are stronger with higher volatility, and their auto-correlations die out quickly.
4. Almost all real data have more extreme events than suspected.
5. Volatility correlations decay slowly.
6. Trading volumes have memory the same way that volatilities do.
7. Past price changes are negatively correlated with future volatilities.

Research objectives

Statistical finance is focused on three areas:

1. Empirical studies focused on the discovery of interesting statistical features of financial time-series data aimed at extending and consolidating the known stylized facts.
2. The use of these discoveries to build and implement models that better price derivatives and anticipate stock price movement with an emphasis on non-Gaussian methods and models.
3. The study of collective and emergent behaviour in simulated and real markets to uncover the mechanisms responsible for the observed stylized facts with an emphasis on agent based models (see agent-based model).

Behavioral finance and statistical finance

Behavioural finance attempts to explain price anomalies in terms of the biased behaviour of individuals, mostly concerned with the agents themselves and to a lesser degree aggregation of agent behaviour. Statistical finance is concerned with emergent properties arising from systems with many interacting agents and as such attempts to explain price anomalies in terms of the collective behaviour. Emergent properties are largely independent of the uniqueness of individual agents because they are dependent on the nature of the interactions of the agents rather that the agents themselves. This approach has drawn strongly on ideas arising from complex systems, phase transitions, criticality, self-organized criticality, non-extensivity (see Tsallis entropy), q-Gaussian models, and agents based models (see agent based model); as these are known to be able to recover some of phenomenology of financial market data, the stylized facts, in particular the long-range memory and scaling due to long-range interactions.

Criticism

Within the subject the description of financial markets blindly in terms of models of statistical physics has been argued as flawed because it has transpired these do not fully correspond to what we now know about real finance markets. First, traders create largely noise, not long range correlations among themselves. A market is not at an equilibrium critical point, the resulting non-equilibrium market must reflect details of traders' interactions (universality applies only to a limited very class of bifurcations, and the market does not sit at a bifurcation). Even if the notion of a thermodynamics equilibrium is considered not at the level of the agents but in terms of collections of instruments stable configurations are not observed. The market does not 'self-organize' into a stable statistical equilibrium, rather, markets are unstable. Although markets could be 'self-organizing' in the sense used by finite-time singularity models these models are difficult to falsify. Although Complex systems have never been defined in a broad sense financial markets do satisfy reasonable criterion of being considered complex adaptive systems.[3] The Tallis doctrine has been put into question as it is apparently a special case of markov dynamics so questioning the very notion of a "non-linear Fokker-Plank equation". In addition, the standard 'stylized facts' of financial markets, fat tails, scaling, and universality are not observed in real FX markets even if they are observed in equity markets.

From outside the subject the approach has been considered by many as a dangerous view of finance which has drawn criticism from some economists because of:[4]

1. "A lack of awareness of work which has been done within economics itself."
2. "Resistance to more rigorous and robust statistical methodology."
3. "The belief that universal empirical regularities can be found in many areas of economic activity."
4. "The theoretical models which are being used to explain empirical phenomena."

In response to these criticism there are claims of a general maturing of these non-traditional empirical approaches to Finance.[5] This defense of the subject does not flatter the use of physics metaphors but does defend the alternative empirical approach of "econophysics" itself.

Some of the key data claims have been questioned in terms of methods of data analysis.[6]

Some of the ideas arising from nonlinear sciences and statistical physics have been helpful in shifting our understanding financial markets, and may yet be found useful, but the particular requirements of stochastic analysis to the specific models useful in finance is apparently unique to finance as a subject. There is much lacking in this approach to finance yet it would appear that the canonical approach to finance based optimization of individual behaviour given information and preferences with assumptions to allow aggregation in equilibrium are even more problematic.

It has been suggested that what is required is a change in mindset within finance and economics that moves the field towards methods of natural science [7]. Perhaps finance needs to be thought of more as an observational science where markets are observed in the same way as the observable universe in cosmology, or the observable ecosystems in the environmental sciences. Here local principles can be uncovered by local experiments but meaningful global experiments are difficult to envision as feasible without reproducing the system being observed. The required science becomes that based largely on pluralism (see scientific pluralism), as in most sciences that deal with complexity, rather than a singled unified mathematical framework that is to be verified by experiment.

Mathematical finance tools

2011-01-07T01:35:00.000-08:00

* Asymptotic analysis
* Calculus
* Copulas
* Differential equations
* Expected value
* Ergodic theory
* Feynman–Kac formula
* Fourier transform
* Gaussian copulas
* Girsanov's theorem
* Itô's lemma
* Martingale representation theorem
* Mathematical models
* Monte Carlo method
* Numerical analysis
* Real analysis
* Partial differential equations
* Probability
* Probability distributions
o Binomial distribution
o Log-normal distribution
* Quantile functions
o Heat equation
* Radon–Nikodym derivative
* Risk-neutral measure
* Stochastic calculus
o Brownian motion
o Lévy process
* Stochastic differential equations
* Stochastic volatility
o Numerical partial differential equations
+ Crank–Nicolson method
+ Finite difference method
* Value at risk
* Volatility
o ARCH model
o GARCH model

2011-01-06T23:57:00.000-08:00

http://www.cs.odu.edu/~toida/nerzic/content/intro2discrete/intro2discrete.html

What is Discrete Mathematics ?

Discrete mathematics is mathematics that deals with discrete objects. Discrete objects are those which are separated from (not connected to/distinct from) each other. Integers (aka whole numbers), rational numbers (ones that can be expressed as the quotient of two integers), automobiles, houses, people etc. are all discrete objects.

Discrete mathematics is the branch of mathematics dealing with objects that can assume only distinct, separated values. The term "discrete mathematics" is therefore used in contrast with "continuous mathematics," which is the branch of mathematics dealing with objects that can vary smoothly (and which includes, for example, calculus). Whereas discrete objects can often be characterized by integers, continuous objects require real numbers.

The study of how discrete objects combine with one another and the probabilities of various outcomes is known as combinatorics. Other fields of mathematics that are considered to be part of discrete mathematics include graph theory and the theory of computation. Topics in number theory such as congruences and recurrence relations are also considered part of discrete mathematics.

The study of topics in discrete mathematics usually includes the study of algorithms, their implementations, and efficiencies. Discrete mathematics is the mathematical language of computer science, and as such, its importance has increased dramatically in recent decades.

Although there is no agreed-upon definition of discrete mathematics, there is a general agreement that discrete mathematics includes three important areas: combinatorics, iteration and recursion, and vertex-edge graphs.

On the other hand real numbers which include irrational as well as rational numbers are not discrete.

Between any two different real numbers there is another real number different from either of them. So they are packed without any gaps and can not be separated from their immediate neighbors. In that sense they are not discrete.

Objects such as integers, propositions, sets, relations and functions,are all discrete.

Why Discrete Mathematics ?

Let us first see why we want to be interested in the formal/theoretical approaches in computer science.
Some of the major reasons that we adopt formal approaches are 1) we can handle infinity or large quantity and indefiniteness with them, and 2) results from formal approaches are reusable.

As an example, let us consider a simple problem of investment. Suppose that we invest $1,000 every year with expected return of 10% a year. How much are we going to have after 3 years, 5 years, or 10 years ? The most naive way to find that out would be the brute force calculation.

Let us see what happens to $1,000 invested at the beginning of each year for three years.
First let us consider the $1,000 invested at the beginning of the first year. After one year it produces a return of $100. Thus at the beginning of the second year, $1,100, which is equal to $1,000 * ( 1 + 0.1 ), is invested. This $1,100 produces $110 at the end of the second year. Thus at the beginning of the third year we have $1,210, which is equal to $1,000 * ( 1 + 0.1 )( 1 + 0.1 ), or $1,000 * ( 1 + 0.1 )2. After the third year this gives us $1,000 * ( 1 + 0.1 )3.

Similarly we can see that the $1,000 invested at the beginning of the second year produces $1,000 * ( 1 + 0.1 )2 at the end of the third year, and the $1,000 invested at the beginning of the third year becomes $1,000 * ( 1 + 0.1 ).

Thus the total principal and return after three years is $1,000 * ( 1 + 0.1 ) + $1,000 * ( 1 + 0.1 )2 + $1,000 * ( 1 + 0.1 )3, which is equal to $3,641.

One can similarly calculate the principal and return for 5 years and for 10 years. It is, however, a long tedious calculation even with calculators. Further, what if you want to know the principal and return for some different returns than 10%, or different periods of time such as 15 years ? You would have to do all these calculations all over again.

We can avoid these tedious calculations considerably by noting the similarities in these problems and solving them in a more general way.

Since all these problems ask for the result of invesing a certain amount every year for certain number of years with a certain expected annual return, we use variables, say A, R and n, to represent the principal newly invested every year, the return ratio, and the number of years invested, respectively. With these symbols, the principal and return after n years, denoted by S, can be expressed as
S = A(1 + R) + A(1 + R)2 + ... + A(1 + R)n .
As well known, this S can be put into a more compact form by first computing S - (1 + R)S as
S = A ( (1 + R)n + 1 - (1 + R) ) / R .
Once we have it in this compact form, it is fairly easy to compute S for different values of A, R and n, though one still has to compute (1 + R)n + 1 . This simple formula represents infinitely many cases involving all different values of A, R and n. The derivation of this formula, however, involves another problem. When computing the compact form for S, S - (1 + R)S was computed using S = A(1 + R) + A(1 + R)2 + ... + A(1 + R)n . While this argument seems rigorous enough, in fact practically it is a good enough argument, when one wishes to be very rigorous, the ellipsis ... in the sum for S is not considered precise. You are expected to interpret it in a certain specific way. But it can be interpreted in a number of different ways. In fact it can mean anything. Thus if one wants to be rigorous, and absolutely sure about the correctness of the formula, one needs some other way of verifying it than using the ellipsis. Since one needs to verify it for infinitely many cases (infinitely many values of A, R and n), some kind of formal approach, abstracted away from actual numbers, is required.

Suppose now that somehow we have formally verified the formula successfully and we are absolutely sure that it is correct. It is a good idea to write a computer program to compute that S, especially with (1 + R)n + 1 to be computed. Suppose again that we have written a program to compute S. How can we know that the program is correct ? As we know, there are infinitely many possible input values (that is, values of A, R and n). Obviously we can not test it for infinitely many cases. Thus we must take some formal approach.

Related to the problem of correctness of computer programs, there is the well known "Halting Problem". This problem, if put into the context of program correctness, asks whether or not a given computer program stops on a given input after a finite amount of time. This problem is known to be unsolvable by computers. That is, no one can write a computer program to answer that question. It is known to be unsolvable. But, how can we tell it is unsolvable ?. How can we tell that such a program can not be written ? You can not try all possible solution methods and see they all fail. You can not think of all (candidate) methods to solve the Halting Problem. Thus you need some kind of formal approaches here to avoid dealing with a extremely large number (if not infinite) of possibilities.

Discrete mathematics is the foundation for the formal approaches. It discusses languages used in mathematical reasoning, basic concepts, and their properties and relationships among them.Discrete mathematics is also concerned with techniques to solve certain types of problems such as how to count or enumerate quantities. The kind of counting problems includes: How many routes exist from point A to point B in a computer network ? How much execution time is required to sort a list of integers in increasing order ? What is the probability of winning a lottery ? What is the shortest path from point A to point B in a computer network ? etc.

The subject includes propositional logic, predicate logic, sets, relations, and functions, in particular growth of function.
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Because many discrete math problems are simply stated and have few mathematical prerequisites, they can be introduced at all grade levels, even with children who are not yet fluent readers.
It's an excellent tool for improving reasoning and problem-solving skills, and is appropriate for students at all levels and of all abilities. Teachers have found that discrete mathematics offers a way of motivating unmotivated students while challenging talented students at the same time.

Because many discrete math problems are simply stated and have few mathematical prerequisites, they can be easily be introduced at the middle school grade level.
EXAMPLE: Linear Programming

Minimize C = 3x + 2y on the given feasible set.

Students spent a lot of time graphing lines without seeing how it can be useful. Linear programming is a powerful tool for finding the optimal value of a linear function on some feasible set. The feasible set is created by solving a system of linear inequalities. Solutions can be found graphically so even students who have not studied systems of equations can solve these problems.

EXAMPLE: Systematic Listing & Counting

There are 45 creatures here. How many of them are fish?

Systematic listing and counting are crucial analytical skills which play a fundamantal role in many areas of mathematics, in particular probability. There are many nuances of counting which are often missed in elementary courses. One of our goals is to shed light on this topic by exploring many examples and employing a variety of learning styles.

EXAMPLE: How Many Possibilities?

Combinations and permutations can range from simple to highly complex problems, and the concepts used are relevant to everyday life. Problems and solution methods can range so much that these mathematical ideas can be used with students from elementary school to high school.

Even young students with limited reading skills can solve problems with combinations of small numbers of items. For example, given that a classmate has two shirts and three pairs of pants, students can determine that there are six possible outfits. They can reason about this problem and even draw out the different options.

For older students, more advanced solution strategies can allow them to handle more complex problems, such as the following:

I have a 6-CD player in my car and I own 100 CD's.
How many different ways can I load 6 CD's into my player?

1st Slot 2nd Slot 3rd Slot 4th Slot 5th Slot 6th Slot

EXAMPLE: Which pizza place is closest?

Voronoi diagrams allow students and teachers to explore a technique that is used in a variety of applications, while at the same time employing critical thinking skills and geometric concepts.

These types of diagrams allow you to map out the areas in a given space that are closest to one specified point or another. For example, if there are 17 ice cream shops of equal quality in your town, a Voronoi diagram can show you which one is the closest for each region of town. This example is shown in the picture below.

This technique is used in biology, chemistry, geology, forestry, and more, as well as in resource planning and placement. This last topic is easily familiar to students as it can include determining placement for a new cell phone tower. or a new pizza place!

These diagrams are constructed using perpendicular bisectors, but students can approach these either strictly through geometric constructions, or through a more algebraic approach for more advanced students.

You are lucky enough to live in a town with 17 ice cream shops, each as good as the next. On the town map below, the ice cream shops are marked with letters. For each numbered point, which ice cream shop or shops should you frequent?
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Here is a list of simply-stated problems that fall under the rubric of discrete mathematics:

Vertex-edge graphs

* Which way of connecting a number of sites into a network involves the least cable?
* What's the best route for a robot to pick up items stored in an automated warehouse, or for a courier to collect deposits from all ATM machines in some assigned region?
* What is the smallest number of colors needed to color the 48 states in the continental United States if states that share a border must be colored with different colors (so that all borders can be clearly distinguished)?

Combinatorics

* How many different pizzas can you have if each pizza must have at most three of the eight available toppings?
* How many tickets do you have to buy to make sure that you have a winning ticket in the contest that involves correctly selecting six numbers from 1 to 36?

Iteration and recursion

* What should be the daily dose of medication if, to function effectively, the medication must be at a specified concentration and if a given percentage of the medication in the body is eliminated each day?
* If the population of deer increases by 10% each year, how long will it take the population of deer to double?

Social choice

* What is the best system for reapportioning the 465 seats in the United States Congress among the states after each census? What system is actually used?
* What is a good strategy for dividing up a pie among three people so that each is satisfied with the portion he or she receives?

Information

* What is the quickest way of alphabetizing a list of 1000 names -- on index cards, or in a database?
* How are transmission errors detected and corrected when coded versions of pictures are sent from space?

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Preliminaries
1.1 Sets
1.2 Summation
1.3 Mathematical Induction
1.4 Functions
1.5 The Division Algorithm
1.6 Exercises

2 Combinatorial Analysis
2.1 Introduction
2.2 The Basic Principle of Counting
2.3 Permutations
2.4 Combinations
2.5 Counting the Number of Solutions
2.6 The Inclusion–Exclusion Identity
2.7 Using Recursion Equations
2.8 The Pigeonhole Principle
2.9 Exercises

3 Probability
3.1 Probabilities and Events
3.2 Probability Experiments Having Equally Likely
Outcomes
3.3 Conditional Probability
3.4 Computing Probabilities by Conditioning
3.5 Random Variables and Expected Values
3.6 Exercises

4 Mathematics of Finance
4.1 Interest Rates
4.2 Present Value Analysis
4.3 Pricing Contracts via Arbitrage
4.3.1 An Example in Options Pricing
4.3.2 Other Examples of Pricing via Arbitrage
4.4 The Arbitrage Theorem
4.5 The Multiperiod Binomial Model
4.5.1 The Black–Scholes Option Pricing Formula
4.6 Exercises

5 Graphs and Trees
5.1 Graphs
5.2 Trees
5.3 The Minimum Spanning Tree Problem
5.4 Cliques and Independent Sets
5.5 Euler Graphs
5.6 Exercises

6 Directed Graphs
6.1 Directed Graphs
6.2 The Maximum Flow Problem
6.3 Applications of the Maximum Flow Problem
6.3.1 The Assignment Problem
6.3.2 The Tournament Win Problem
6.3.3 The Transshipment Problem
6.3.4 An Equipment Selection Problem
6.4 Shortest Path in Digraphs
6.5 Exercises

7 Linear Programming
7.1 The Standard Linear Programming Problem
7.2 Transforming to the Standard Form
7.2.1 Minimization and Wrong-Way Inequality
Constraints
7.2.2 Problems with Variables Unconstrained in Sign
7.3 The Dual Linear Programming Problem
7.4 Game Theory
7.5 Exercises

8 Sorting and Searching
8.1 Introduction to Sorting
8.2 The Bubble Sort
8.3 The Quicksort Algorithm
8.4 Merge Sorts
8.5 Sequential Searching
8.6 Binary Searches and Rooted Trees
8.7 Exercises

9 Statistics
9.1 Introduction
9.2 Frequency Tables and Graphs
9.3 Summarizing Data Sets
9.3.1 Sample Mean, Sample Median, and
Sample Mode
9.3.2 Sample Variance and Sample Standard
Deviation
9.4 Chebyshev’s Inequality
9.5 Paired Data Sets and the Sample Correlation
Coefﬁcient
9.6 Testing Statistical Hypotheses
9.7 Exercises

10 Groups and Permutations
10.1 Permutations and Groups
10.2 Permutation Graphs
10.3 Subgroups
10.4 Lagrange’s Theorem
10.5 The Alternating Subgroup
10.6 Exercises
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# Problems in discrete mathematics can be introduced without much preparation.

* It can be used to provide students with a different view of mathematics, one where big ideas are discussed and where creativity is exercised.
* It is an arena where mathematical modeling is easily understood.
* For those students who have not been successful in mathematics, discrete mathematics offers a "new start"
o It provides mathematical topics that do not have algebraic skills as a prerequisite, but that rely mainly on reasoning and problem solving.
o For these students, discrete mathematics offers an arena where they can be successful in mathematics without realizing that they are doing mathematics.
o Enrichment through discrete mathematics has perhaps more potential than repeated remediation.

# Problems in discrete mathematics are engaging

* They are engaging because they are more visual than computational, more geometric than algebraic, and they often have a puzzle-like quality.
* Discrete mathematics provides an opportunity for students to enjoy mathematics again.
* Students find discrete mathematics challenging because, while many problems in discrete mathematics are easily stated, they are often not easy to solve.
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Calculus is a very powerful tool, based upon rates of change, used to solve problems which cannot be solved in any other fashion. The power that calculus wields is due to it's foundation of continuous mathematics.

Up until calculus, mathematics generally dealt with discrete quantities, such as money and other easy countable values. Calculus takes mathematics into a more flexible and usable state by using what are called continuous quantities. Arguably the most important continuous quantities are time and mass measurements, such as length, width, volume, and even into deeper unvisualized dimensions. What makes them continuous is the ability to split them into smaller and smaller units of measurement, thereby giving them an infinite amount of points inside any one interval. Time and mass measurements are consequently what most of applicable calculus is based upon.

The limit is the first way that calculus demonstrates it's ability to deal with continuous mathematics. It also shows some of the other key differences between calculus and algebra. While algebra often asserts that certain values of x in a function f(x) are undefinable, with limits we're able to see what a lot of (but not all) of these values should be, based on the tendency of the graph. This is the first time we see the important concept of an infinite linear approximation in calculus.

The concept of the derivative is the next building block of calculus. A derivative is most simply, a rate of change. This can be in two or more dimensions, as long as the value changing, and another value being compared is made known. The limit definition of a derivative is actually very similar in any number of dimensions. It basically finds the slope of secant lines until the values are infinitely close to the value at which you would like to find the derivative at. The slope of each individual secant line gives the average rate of change between two points, and the tendency of the slope as the two points get infinitely close to the point in question gives the instantaneous rate of change at the point. This tendency can be simplified as a linear approximation for values within a certain range of a point. This approximation makes finding the derivative much simpler.

The concept of differentiation has also been simplified using a set of rules to generalize different cases. Among these are the power rule, the product rule, the quotient rule, and most importantly: the chain rule and L'Hospital's rule. The chain rule is important because it can be used to disassemble most complex equations into simpler derivatives, and of course, because of its uses in integration. L'Hospital's rule, put simply, a way to compare different sizes of infinity, infinity-tending functions, and functions dealing with an infinitely small interval as the variable goes to 0 or infinity.

One of the most important applications of derivatives is the finding of extrema. This is where a lot of the basic linear approximation claims I have made are easily exposed. The most intuitive way to go about explaining why the formula for optimization works so well, is by explaining what is not an extrema. If the derivative as x does not equal zero, than it is either positive or negative. Therefore, the function is either increasing or decreasing at that point, and there are an infinite amount of points that are greater than and less than the point in question. This is the nature of continuous mathematics. By contradiction, a value is an extrema of a function f, if and only if the f has a derivative of zero at that point. This maximum or minimum is where the slope of the linearization at x, approaches zero. Although this is easiest to visualize in two dimensions, the concept can be brought visually into three-dimensions, and applicably into any finite n-dimensions greater than three. For as long as we know what the change of a variable is in respect to, we can find the value at which it has extrema. The same is true for related rate problems, and a variety of other applications, due to the nature of them being essentially special-case optimization problems.

Integration is often thought of as a different concept than differentiation. In reality, it is dealing with the exact same usage of rates, with similar approximation methods. It is effectively, the reverse or compliment of differentiation. Integration displays, more or less, the accumulation of something. Geometrically in two or three dimensions, it is usually used to find area under a curve or volume under a shape. This is an accumulation of one type, but there are many more.

The technical definition behind integrating is very similar to differentiation. Geometrically in two-space, we take progressively smaller areas under the curve and add them all up. As the number of rectangles under the curve reaches infinity, our answer becomes precise. This literal definition gets harder to carry out we introduce more dimensions.

Often though, we use verifiable short cuts to calculate derivatives. The most influential way that scholars have connected differentiation to integration is called the Fundamental Theorem of Calculus. Syntactically, this is “The integral of F from a to b is equal to the anti-derivative of F at b minus the anti-derivative of F at a.” This is often the easiest method to solving most simple integrals. Using the reverse-chain rule, u-substitution, we can break down complex problems down to easily computable parts. Many other variants of u-substitution (trig substitution, partial fractions, root substitution, etc) have been evolved for much more complicated integrals, and generally, this works quite well.

However, there are times when this definition does not work, and the core approximations behind the syntax must come into play. Examples of this include when the limits of integration go to infinity, or when the limits of integration cause the function to go through an asymptote. Another famous example is the integral from zero to one of sin(x^2). There is no anti-derivative to this integrand, and many others. In these cases, the Trapezoidal rule, the Simpson's rule, and the Midpoint rule are all different ways to form approximations of the accumulation of these functions from one point or set of points to another. These are all prone to error of some sort, but this error is easily calculated, and they have bounds of accuracy just as the approximations for differentiation have. These approximations work in much the same way. You are essentially even taking linear approximations when you are using the simpler Fundamental Theorem of Calculus equations, as have been shown by the infinite series definition of integration.

Higher branches of the sciences have made use of calculus in many ways. We have found that all of the rules of calculus can be generalized onto vector fields and used to calculate various solutions to physics related problems, if given the proper frame of reference. Fundamentally, it is the exact same concepts used in a different manor. However, many of these methods can not be effectively utilized without realizing the aforementioned nature of continuous mathematics and approximations.
It is difficult to really generalize what calculus is about. But for the most part, it is safe to say that calculus is the study of rates of changes and accumulations, and the approximations thereof. While easiest to visualize geometrically, if one truly understands the principles behind the equations, it is only a matter of analysis to transcribe the exact same methods to more abstract concepts.
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Mathematical analysis includes the following subfields.

* Differential equations
* Real analysis, the rigorous study of derivatives and integrals of functions of real variables. This includes the study of sequences and their limits, series.
o Multivariable calculus
o Real analysis on time scales - a unification of real analysis with calculus of finite differences
* Measure theory - given a set, the study of how to assign to each suitable subset a number, intuitively interpreted as the size of the subset.
* Vector calculus
* Functional analysis studies spaces of functions and introduces concepts such as Banach spaces and Hilbert spaces.
* Calculus of variations deals with extremizing functionals, as opposed to ordinary calculus which deals with functions.
* Harmonic analysis deals with Fourier series and their abstractions.
* Geometric analysis involves the use of geometrical methods in the study of partial differential equations and the application of the theory of partial differential equations to geometry.
* Complex analysis, the study of functions from the complex plane to itself which are complex differentiable (that is, holomorphic).
o Several complex variables
* Hypercomplex analysis or Clifford analysis
* p-adic analysis, the study of analysis within the context of p-adic numbers, which differs in some interesting and surprising ways from its real and complex counterparts.
* Non-standard analysis, which investigates the hyperreal numbers and their functions and gives a rigorous treatment of infinitesimals and infinitely large numbers. It is normally classed as model theory.
* Numerical analysis, the study of algorithms for approximating the problems of continuous mathematics.
* Stochastic calculus - analytical notions developed for stochastic processes.
* Set-valued analysis - applies ideas from analysis and topology to set-valued functions.
* Tropical analysis (or idempotent analysis) - analysis in the context of the semiring of the max-plus algebra where the lack of an additive inverse is compensated somewhat by the idempotent rule A+A=A. When transferred to the tropical setting, many nonlinear problems become linear.[7]

Classical analysis would normally be understood as any work not using functional analysis techniques, and is sometimes also called hard analysis; it also naturally refers to the more traditional topics. The study of differential equations is now shared with other fields such as dynamical systems, though the overlap with conventional analysis is large.

Analysis in other areas:

* Analytic number theory
* Analytic combinatorics
* Continuous probability
* Differential entropy in information theory
* Differential games
* Differential geometry, the application of calculus to specific mathematical spaces known as manifolds that possess a complicated internal structure but behave in a simple manner locally.
* Differential topology

MARKOV CHAINS

2011-01-06T23:50:00.000-08:00

MARKOV CHAINS

(This material comes from a variety of sources, including a presentation by
Frank Pitonyak at the NCTM conference in San Diego, Ralston and Maurer's
Discrete Algorithmic Mathematics, and Kenneth Bogart's Discrete Mathematics.
Hardly anything is original to me. This activity integrates three areas of
discrete mathematics -- recurrence relations, matrices, and graph theory.)

A stochastic process is one that has a finite number of outcomes (or states)
and the outcome (or state) occurs with a specific probability. For example,
the following are examples of stochastic processes:
% tossing a coin four times,
% picking four marbles without replacement from a jar containing six
green and four blue marbles, and
% having two people toss a ball four times where they have the option of
tossing straight up or to the other person.

In the first example, the probability of getting a head on the first toss is
0.5. Likewise the probability of a head on each succeeding toss is 0.5 and is
independent of any of the tosses before it.

In the second example, the probability of getting a green marble on the first
draw is 6/10. The probability of a green on the second draw is 5/9 or 6/9,
depending on what the first draw was. The probability of a green on the third
draw is either 6/8, 5/8, or 4/8 depending on the first two draws. The
probability of a green on the fourth draw depends on all the preceding draws.

In the third example, the probability that the ball is in the hands of person A
or B depends only on the immediately previous trial and the behavior of the
ball handler.

If the outcome (or state) of a stochastic process depends at most on the
immediately preceding trial, then it is called a Markov Chain (named after
Andrei Markov, a Russian mathematician.) Of the experiments described above,
only the first and third describe Markov Chains.

These situations can be represented in various ways -- as a digraph (arrows
with associated weights), as a linked recurrence relation (f(n) depends on
f(n-1) and g(n-1)), and as a matrix (where entry ij is the probability of
going from state i to state j).

One way to find probabilities in a Markov Chain after several steps is by using
a probability tree. These, however, can become rather unwieldy when you get
beyond two or three steps. What we want to do is to create an n by n matrix
containing the probabilities:

Next State
A B C
A | |
Current State B | |
C | |

This is called the Transition Matrix. The value in position AA is the
probability of going from A to A. Likewise, the value in position AB is the
probability of going from A to B, and so on. Each row consists of non-negative
values whose sum is equal to one, which qualifies it as a stochastic matrix. A
second 1 by n matrix, called the Initial State Matrix, holds the starting
position. For example, an initial state matrix might be [0 0 1], which
indicates that the starting position is C.

The product of the initial state matrix times the transition matrix yields the
probability of the second step. Continued multiplications by the transition
matrix yield probabilities for subsequent steps. In addition, regular
stochastic matrices converge when raised to a sufficiently high power. They
reach a steady state or equilibrium. This state can also be calculated
algebraically by setting f(n) = f(n-1) = f and g(n) = g(n-1) = g. Solving the
system of equations for f and g gives the probabilities for equilibrium.

To determine the expected number of times something occurs before equilibrium,
eliminate the row(s) and column(s) from T that are the terminating conditions.
For example, if when you get to A you stay there, then delete row A and column
A from the above matrix. This matrix is Q. I is the corresponding identity
matrix. Calculate I-Q, then find the inverse of that matrix. The entry in row
i column j is the expected number of times you end up in state j before
reaching the absorbing state, given that you started in state i. If you start
at C and you want to know how many times you end up at B before you stop, the
value will be in row C column B. If you start at C and you want to know how
many times you change to anywhere before you stop, sum the entries in row C.

1. Business Inventory

Trusty Rent-A-Car has offices in New York City and Los Angeles. It allows
customers to make local rentals or one-way rentals to the other location. Each
month, it finds that half the cars that start the month in NYC end it in LA,
and one third of the cars that start the month in LA end it in NYC. If at the
start of operations Trusty has 1000 cars in each city, what can it expect n
months later?

Let N(n) be the number of cars in New York at the beginning of month n, with
the start of operations considered to be the beginning of month 0. Define L(n)
similarly. You will have a pair of linked recurrence relations because each
variable depends on previous values of the other variable as well as of itself.
a. Write the linked recurrence relation for N(n) based on N(n-1) and
L(n-1). Similarly write the linked recurrence relation for L(n). In this
case, L(0) = N(0) = 1000.

Although individual cars will keep moving from NYC to LA and back, the
number of cars in each place might stabilize, or the agency gains equilibrium.
If equilibrium is attained, then N(n-1) = N(n) = N and L(n-1) = L(n) = L.
Determine values for N and L when equilibrium is achieved.

b. Create a 2x2 matrix called B as follows:

Next Month
N L
This Month N | |
L | |

where the rows are N and L and the columns are N and L. (Note b(NN) =
fraction of cars starting and ending in NYC.) Create a 1x2 matrix A = [.5 .5]
which indicates the original distribution of half the cars in each location.
Multiply AB. Multiply the answer times B. Keep multiplying by B. Eventually
the matrix product will stabilize. Compare this answer with the answer to (c).

b. Suppose the initial distribution is different from 1000 cars in each
location. Modify the values in matrix A to indicate a different distribution
(remember the two numbers must add up to 1) and repeat the multiplications in
(d). Explain your results.

c. Forget about matrix A and just take B and raise it to some power, such
as 20. What do you see? What does that tell you about the significance of A?
For what value of n does B^n stabilize?

d. Suppose Los Angeles suffers an earthquake and becomes less popular.
Trusty Rent-a-Car now finds that only 1/4 the cars in NYC end up in LA the next
month, whereas 4/5 the cars rented in LA and end up in NYC. Revise the linked
recurrence relations in (a) and solve for the new values.

2. Distributing Populations

Developing countries have a problem with too many people moving to the cities.
Country X does a study and finds that, in a given year, 10% of the rural
population moves to a city, but only 1% of the urban population goes back to
the country.

Let U(n) be the number of people in an urban setting at the beginning of year
n, with the start of the study considered to be the beginning of year 0.
Define R(n) similarly. You will have a pair of linked recurrence relations
because each variable depends on previous values of the other variable as
well as of itself.

a. Write the linked recurrence relation for U(n) based on U(n-1) and
R(n-1). Similarly write the linked recurrence relation for R(n). Assume that
initially, U(0) = R(0) = 50%.

Although individuals will keep moving from urban to rural settings and
back, the number of people in each place might stabilize and the country gain
equilibrium. If equilibrium is attained, then U(n-1) = U(n) = U and R(n-1) =
R(n) = R. Determine values for U and R when equilibrium is achieved.

b. Create a 2x2 matrix B as follows:
Next Year
U R
This Year U | |
R | |

where the rows are U and R and the columns are U and R. (Note b(UU) =
fraction of population starting and ending in urban setting.)

Create a 1x2 matrix A = [.5 .5] which indicates the original
distribution of half the population in each location. Multiply AB. Multiply
the answer times B. Keep multiplying by B. Eventually the matrix product will
stabilize. Compare this answer with the answer to (c).

c. Suppose the initial distribution is different from 50% people in each
location. Modify the values in matrix A to indicate a different distribution
(remember the two numbers must add up to 1) and repeat the multiplications in
(d). Explain your results.

d. Suppose an epidemic breaks out and the risk of contagion is
significantly higher in the cities. Surveys now find that only 5% of the rural
population moves to a city, but 15% of the urban population moves to the
country. Revise the linked recurrence relations in (a) and solve for the new
values.

3. Tylenol's Comeback

Several years ago the makers of Tylenol were devastated when someone, tampering
with Tylenol bottles, create a nationwide scare. Sales of Tylenol, the
number-one seller among acetaminophens, declined to almost zero. The company
removed all supplies of Tylenol from the shelves nationwide and designed new
tamper proof packaging. When the scare was over, Tylenol soon became the
number-one seller again. How could this occur? Most analysts believe it was a
result of the intense loyalty of Tylenol users.

Assume that for any purchase 70% of Tylenol users are loyal and will not
switch.
Similarly, assume that 40% of those using Brand X will not switch. Further,
assume that when Tylenol returned to the market with new packaging their market
share was effectively 0%.

a. Write the linked recurrence relation for the percent of people
purchasing Tylenol after n months T(n) based on T(n-1) and X(n-1). Similarly
write the linked recurrence relation for X(n). Assume that initially, T(0) =
0% and X(0) = 100%.

Although individuals will keep changing to Tylenol or Brand X, the number of
people purchasing each product might stabilize and the market gain equilibrium.
If equilibrium is attained, then T(n-1) = T(n) = T and X(n-1) = X(n) = X.
Solve algebraically.

b. Create a 2x2 matrix B as follows:

Next Purchase
T X
Current Purchase T | |
X | |

where the rows are T and X and the columns are T and X. (Note b(TT) =
fraction of purchasers starting and ending with Tylenol.)

Create a 1x2 matrix A = [0 1] which indicates the original
distribution of market share. Multiply AB. Multiply the answer times B.
Keep multiplying by B. Eventually the matrix product will stabilize. What
does this show?

c. Suppose the initial market share is different from 0% for Tylenol.
(Perhaps some true believers hoarded their Tylenol supply.) Modify the values
in matrix A to indicate a different distribution (remember the two numbers must
add up to 1) and repeat the multiplications in (c). Explain your results.

d. Suppose a second scare occurs and several people suddenly die after
taking Tylenol. Surveys now find that only 60% of the Tylenol users will not
switch, but 55% of the Brand X users will not switch. Revise the linked
recurrence relations in (a) and solve for the new values.

4. Predicting the Weather

Suppose a weather forecaster has collected data to predict whether tomorrow
will be sunny (S), cloudy (C), or rainy (R), given todayUs weather conditions.

% If today is sunny, then tomorrow the probabilities are 70% for S, 20%
for C, and 10% for R.
% If today is cloudy, then the probabilities for tomorrow are 30% for S,
60% for C, and 10% for R.
% If today is cloudy, then the probabilities for tomorrow are 25% for S,
20% for C, and 55% for R.

a. Create a 3x3 matrix where the rows indicate the current situation and
the columns indicate tomorrow's weather.

S C R
S | |
B = C | |
R | |

For example the first row would be 0.7 0.2 0.1. Now create a 1x3
matrix A to indicate today's weather. Suppose today is sunny, then A = [1 0
0].

b. Multiply AB to determine the probabilities for tomorrowUs weather.
Multiply by B again to determine the weather for 2 days from now. Repeat this
so that you have a set of probabilities for the next 5 days.

c. Keep multiplying by B. Eventually the matrix product will stabilize.
What does this show?

d. Change today's weather to indicate a rainy day in matrix A. Repeat the
multiplication in (b) to formulate a 5-day forecast.

e. Forget about matrix A and just take B and raise it to some power, such
as 20. What do you see? What does that tell you about the significance of A?
For what value of n does B^n stabilize? Once it stabilizes it no longer
forecasts the weather, but rather tells you about the climate.

f. Create a matrix that would be appropriate for Buffalo, NY. You may
want to include snow as well as rain. Decide on today's weather and
illustrate a 5-day forecast.

5. Genetics

A given plant species has red, pink, or white flowers according to the
genotypes RR, RW, and WW respectively. If each genotype is crossed with a
pink flowering plant (genotype RW), the transition matrix is as follows:

Next generation
Red Pink White
Red 0.5 0.5 0
This generation Pink 0.25 0.5 0.25
White 0 0.5 0.5

(These numbers are determined as follows: RR x RW yields RR, RW, RR, and RW.
RW x RW yields RR, RW, WR, WW. WW x RW yields WR, WW, WR, and WW.) Assume the
plants of each generation are crossed only with pink plants to produce the next
generation.

a. This matrix is B. Now create a 1x3 matrix A to indicate the initial
distribution of genotypes. Assume the initial distribution is 70% Red, 10%
Pink, and 20% White.

b. Multiply AB to determine the distribution for the next generation.
Multiply the answer times B. Keep multiplying by B. Eventually the matrix
product will stabilize. What does this show? What will be the eventual
distribution of genotypes after many generations?

c. Suppose the initial distribution is different. Modify the values in
matrix A to indicate a different distribution (remember the three numbers must
add up to 1) and repeat the multiplications in (b). Explain your results.

d. Suppose we decide to cross plants of each generation with only white
plants. Create a transition matrix for this situation. (To figure the
probabilities, recall that RR x WW yields RW all the time. Calculate the other
probabilities.) Use the same initial distribution of genotypes and determine
the eventual distribution after many generations.

6. Ball Toss Experiment

Two people are tossing a ball to each other. A person may toss the ball up and
catch it themselves or throw it to the other person. The probability the ball
is in the hands of person A or B depends only on the immediately previous
trial.

a. Have two people toss the ball 50 times so you can get some empirical
probabilities. Choose one person to be A and one to be B. A third person
should tally where the ball is, while a fourth person calls out what is
happening. The tally sheet is as follows:

A to A _____

A to B _____

B to B _____

B to A _____

(Hint: it is very boring if each person does each type of toss half
the time.) Total each category. Total the first 2 and the second 2.
Now P(AA) = (total A to A)/(total of first 2 categories) .
Similarly calculate P(AB), P(BB), and P(BA).

b. Construct a probability tree showing the first 3 tosses. Assume you
start by giving the ball to A. What is the probability of the ball being in
A's hand after 3 tosses?
Construct a second tree with the ball starting with B.
Similarly find the probability of the ball being in A's hand after 3 tosses.

c. Construct a weighted digraph of the situation. On each arrow, write
the probabilities.

d. Express the probabilities as a transition matrix T. In location AA put
the probability A tosses the ball to herself. In AB put the probability A
tosses the ball to B.
Next Toss
A B
Now A | |
B | |

Next suppose the ball is in the hands of A at first. This can be
represented by the Initial State Matrix as S = [1 0]. Multiply ST to
determine the probability for the location of the ball after the first toss.
Multiply the answer times T two more times. This should match the results from
the probability tree after 3 tosses.

e. Keep multiplying by T. Eventually the matrix product will stabilize.
What does this show? What will be the eventual location of the ball after many
tosses?

f. Suppose the initial situation is different. Modify the values in
matrix S to indicate the ball starting with B and repeat the multiplications
in (d). Explain your results.

g. Forget about matrix S and just take T and raise it to some power, such
as 20. What do you see? What does that tell you about the significance of S?
For what value of n does T^n stabilize?

7. Barbecue Begging

A puppy smells a number of neighbors barbecuing. One unsupervised grill is two
houses downhill from his yard, and another unsupervised grill is three houses
uphill from his yard. Because so many people are barbecuing, he goes randomly
from house to house in search of food, going downhill with twice the
probability that he goes uphill. We record his progress from house to house,
using 0 to stand for one unsupervised grill, 2 to stand for his yard, and 5
to stand for the other unsupervised grill.

a. Construct a weighted digraph of the situation. On each arrow, write
the probabilities that the puppy will follow that path.

Assuming that the puppy will stop and eat if and when he finds
unsupervised food, you would put a loop at 5 with probability 1 and a loop at 0
with probability 1.

b. Write a matrix T of transition probabilities of going from one house to
another. The first row would be 1 0 0 0 0 0 to indicate that if the puppy gets
to house 0 he will stay there. In position 1, 0 put the probability the puppy
will go from house 1 to house 0.
Next House
0 1 2 3 4 5
0 | |
1 | |
Now 2 | |
3 | |
4 | |
5 | |

Next create the Initial State Matrix as S = [0 0 1 0 0 0]
indicating the puppy started at house 2. Multiply ST to determine the
probability for the location of the puppy after the first move.

c. Find the probability of the puppy going from his yard to the grill down
the hill in two, three, or four moves. To make reading the results easier, you
may want to set the mode to 2 digits.

d. Keep multiplying by T. Eventually the matrix product will stabilize.
What does this show? What will be the eventual location of the puppy after
many moves?

e. What is the expected number of houses the dog will visit before finding
food?

8. Gambling Game

In a gambling game (such as coin flipping for a dollar a flip), your total
fortune can go up one dollar or down one dollar (each with probability 1/2) at
each stage. Suppose you begin with 3 dollars and your opponent begins with 2.

a. Construct a weighted digraph of the situation. The states are your
total fortune, either 0, 1, 2, 3, 4, or 5 dollars. The probability of going
from state k to state k + 1 or k - 1 is 1/2 unless k = 0 or k = 5. On each
arrow, write the probabilities that you will gain a dollar or lose a dollar.

Since the game is over if you reach either 0 or 5, you would put a loop
at 5 with probability 1 and a loop at 0 with probability 1.

b. Write a matrix T of transition probabilities of going from one amount
of money to another. The first row would be 1 0 0 0 0 0 to indicate that if
you get to $0 the game will be over. In position 1, 0 put the probability you
will go from $1 to $0.
Next Amount
0 1 2 3 4 5
0 | |
1 | |
Current 2 | |
3 | |
4 | |
5 | |

Next create the Initial State Matrix as S = [0 0 0 1 0 0]
indicating you started with $3. Multiply ST to determine the probability for
the amount of money you have after the first flip.

c. Determine the probability that you win your opponent's entire fortune
by the time four coins have been flipped.

d. Keep multiplying by T. Eventually the matrix product will stabilize.
What does this show? What will be the eventual outcome of the game after many
flips?

e. How many coin flips should we expect before someone wins the game?

9. Mouse Maze

We place a mouse in the maze consisting of a square cut into quarters. The
upper left quarter is chamber 1, lower left is 2, lower right is 3, upper right
is 4. There is 1 passageway between chamber 1 and chamber 4, 2 passageways
between chamber 1 and chamber 2, 2 passageways between chamber 2 and chamber 3,
and 1 passageway between chamber 3 and chamber 4.

The maze has food in chamber 4. The maze is designed so that the odor of the
food pervades all the chambers. The hypothesis to be tested is that after some
practice the mouse will move directly to and remain in chamber 4. At the
opposite extreme is the possibility that no matter how many times the mouse is
placed in the maze, it moves randomly through the maze until it encounters the
food. If the mouse chooses an opening at random and moves through it, then we
expect that the mouse will be equally likely to pick any opening in the chamber
it is in. Thus, for example, we expect the probability of moving from chamber
1 to chamber 4 to be 1/3 because there are three passageways from chamber 1 and
one of these leads to chamber 4. We expect the probability of moving from
chamber 1 to chamber 3 to be 0, because there is no passageway between chambers
1 and 3.

a. From this diagram, draw a weighted digraph. You draw an edge of weight
1 from vertex 4 to itself to signify that (because of the food) the mouse stays
in chamber 4 if it gets there. (We observe the mouse when it changes chambers
or is eating.)

b. From this digraph, write a matrix T of transition probabilities of
going from one chamber to another. The entry in row 1 column 4 would be the
probability the mouse goes from chamber 1 to chamber 4, which is 1/3.
Next Chamber
1 2 3 4
1 | |
Current 2 | |
3 | |
4 | |

c. Use this matrix to determine the probability that the mouse moves from
chamber 2 to chamber 4 in one or two movements. To do this, you need to create
the Initial State Matrix as S = [0 1 0 0] indicating you started the mouse
in chamber 2. Multiply ST to determine the probability for the mouse's
location after the first observation. Keep multiplying by T to get subsequent
observations.

d. What is the probability that after placing the mouse in chamber 2 and
making two, four, or eight observations, we find the mouse in chamber 2 again?

e. What is the expected number of times the mouse in the maze will change
chambers before finding food, given that we start the mouse in chamber 2?

10. Amusement Park

An amusement park has five adult attractions -- Roller coaster, Flume, Wheel,
Smasher, and Crunch. Two of these, the exciting Roller coaster and Flume, are
usually the last attractions visited before people leave the park. There is a
path between the Roller coaster and Flume, Roller coaster and Wheel, Roller
coaster and Smasher, Flume and Wheel, Flume and Crunch, Smasher and Wheel,
Wheel and Crunch.

a. Assuming that people are equally likely to take any path leaving an
attraction and that 25% of the people who ride the Roller coaster or the Flume
then leave the park, draw the weighted graph that shows the five attractions
and the exit and shows the probability that someone at one vertex goes to any
other vertex. Assume that someone who has exited the park does not return.

b. From this digraph, write a matrix T of transition probabilities for
going from one attraction to another. E stands for exit, R for roller coaster,
etc. The entries in row E would be 1 0 0 0 0 0 to indicate that someone
who exited the park did not return.
Next Attraction
E R F C W S
E | |
R | |
Current F | |
C | |
W | |
S | |

c. Use this matrix to determine the probability that a person who rides
the roller coaster rides it once again after four changes of rides. To do this,
you need to create the Initial State Matrix as S = [0 1 0 0 0 0] indicating
you started on the roller coaster. Multiply ST to determine the probability for
the next ride. Keep multiplying by T to get subsequent observations.

d. What is the probability that someone who is now riding the roller
coaster rides the wheel after changing rides four times?

e. Find the expected number of times that someone who is now riding the
roller coaster rides the wheel before leaving the park.

f. Find the expected number of rides that a person who is now riding the
roller coaster rides on any ride whatsoever before leaving the park.

11. Temperature Control

A computer is monitoring the temperature changes in a controlled process whose
temperature must remain in the range between 20{ C and 26{ C. If the
temperature reaches 20{ C or 26{ C, the computer stops the process. The
temperature-measuring device measures only integer temperatures. The heating
system has the property that, during any minute, the temperature is equally
likely to go up one degree, down one degree, or stay the same.

a. Draw a digraph of this situation. The states would be the range of
temperatures from 20 to 26. Any number between 21 and 25, inclusive, has two
arrows and a loop. The end temperatures would have only a loop with weight 1,
indicating that the process stops.

b. From this digraph, write a matrix T of transition probabilities for
going from one temperature to another. The entries in row 20 would be 1 0 0
0 0 0 0 to indicate that the process stopped.
Next Temperature
20 21 22 23 24 25 26
20 | |
21 | |
Current 22 | |
23 | |
24 | |
25 | |
26 | |

c. Use this matrix to determine the probability that the process stops at
26{ C within four minutes, given that the room is at 24{ C initially. To do
this, you need to create the Initial State Matrix as S = [0 0 0 0 1 0 0]
indicating you started at 24{ C. Multiply ST to determine the probability for
the temperature for the next minute. Keep multiplying by T to get subsequent
observations.

d. How many minutes should we expect the computer to allow the process to
continue, given an initial temperature of 24{ C.

e. Suppose we change the initial room temperature to 22{ C. Redo the
calculations in (c) and (d). Are the results the same? Why or why not?

f. Keep multiplying by T. Eventually the matrix product will stabilize.
What does this show? What will be the eventual outcome of the process after
many minutes?

2011-01-05T21:17:00.000-08:00

BiGSEM, Bielefeld University, Germany"Educating tomorrow’s leaders in economics and management for science, society, and the economy."

The Bielefeld Graduate School of Economics and Management (BiGSEM) welcomes applications for its Ph.D. program in Economics and Managementfor the academic year 2007. BiGSEM offers a modern 3-year Ph.D. program that emphasizes the importance of a comprehensive theoretical and interdisciplinary education. The major difference to traditional German Ph.D. studies is a compulsory course program guaranteeing a solid foundation in the field. An international network of partner universities enables a wide variety of courses taught by renowned experts.

The Department of Business Administration and Economics at Bielefeld University established BiGSEM in 2001 in close cooperation with the Institute of Mathematical Economics. The goal was to offer a Ph.D. program competitive with successful US-American graduate programs. To facilitate the international scientific cooperation as well as the integration of international students, English was chosen as language of instruction.

The Ph.D. program is open to exceptional and highly motivated graduates with a background in business administration, economics, mathematics, or related fields, such as engineering, operations research, or statistics. BiGSEM encourages especially applications from international graduates. A formal requirement for admission is a master’s degree or equivalent with a written thesis. Essential for a successful participation in the program is a sound knowledge of modern mathematical techniques.

Application forms and further information are available online at http://www.bigsem.de. Complete application packages should be received by the graduate school no later than December 31, 2006 for the academic year 2007. Questions can be directed to Prof. Volker Böhm, Ph.D., chairman of the program, phone: +49 - 521 – 106 5637, eMail: bigsem @ wiwi.uni-bielefeld.de.

Application has to be received by 31 December 2006

2011-01-05T21:13:00.000-08:00

Scholarships for the PhD Program in General Management, University of Bologna, Italy

Scholarships for the PhD Program in General Management, University of Bologna, Italy

The PhD program in General Management of the University of Bologna trains brilliant researchers in the fields of management, banking and finance by offering courses in two different tracks: Track in Management; Track in Banking and Finance. The program is created by the integration of two doctoral programs previously offered by the Department of Management: the PhD in Business Administration (Dottorato in Direzione Aziendale) and the PhD in Financial Markets and Intermediaries (Dottorato in Mercati e Intermediari Finanziari).

In addition to the attendance of core and elective courses, PhD students have the opportunity to attend seminars held by internationally renowned scholars, present their work at conferences, spend a research period of at least 6 months in international academic institutions. The courses are held in English. The PhD in General Management (3-year program) is organized by the Department of Management of the University of Bologna (www.sa.unibo.it).

In order to apply, a Master degree (laurea specialistica/magistrale) is required. The next edition of the PhD in General Management (26th cycle) will start in October 2010. 8 students will be admitted to the program and 4 scholarships will be offered. The number of scholarship-assisted places may increase thanks to the financial support of public or private institutions. The net amount of the scholarship is around 1000 Euro per month, for a 3-year period. The amount of the scholarship can be increased during the research period spent abroad (+50%) and through assistantships in teaching or contract research activities. Applications from foreign students are encouraged.

For further details on the PhD program in General Management, the admission requirements and the selection procedures, please visit the website.

or contact the PhD Coordinator, Professor Federico Munari (federico.munari[ at ]unibo.it).

The call for application will be available on this website.

Deadline to submit all the application documents: 21 June 2010

Ratios

2011-01-05T04:47:00.001-08:00

(1)Incremental Capital Output Ratio - ICOR

A metric that assesses the marginal amount of investment capital necessary for an entity to generate the next unit of production. Overall, a higher ICOR value is not preferred because it indicates that the entity's production is inefficient. The measure is used predominantly in determining a country's level of production efficiency.

For example, suppose that Country X has an ICOR of 10. This implies that $10 worth of capital investment is necessary to generate $1 of extra production. Furthermore, if country X's ICOR was 12 last year, this implies that Country X has become more efficient in its use of capital.

Some critics of ICOR have suggested that its uses are restricted as there is a limit to how efficient countries can become as their processes become increasingly advanced. For example, a developing country can theoretically increase its GDP by a greater margin with a set amount of resources than its developed counterpart can. This is because the developed country is already operating with the highest level of technology and infrastructure. Any further improvements would have to come from more costly research and development, whereas the developing country can implement existing technology to improve its situation.

(2) Finance Ratio = Ratio of Primary Issues (i.e., issues by all sectors other than banks and other financial institutions) to National Income.

(3) Financial Interrelations Ratio = Ratio of Total Issues to Net Domestic Capital Formation.

(4) New Issue Ratio = Ratio of Primary Issues to Net Domestic Capital Formation.

(5) Intermediation Ratio = Ratio of Secondary (i.e., issues by banks and other financial institutions) Issue to Primary Issues.

(6) National Income refers to Net National Product at factor cost at current prices

Neoclassical macromodel

2011-01-03T20:28:00.000-08:00

http://homepage.newschool.edu/het//essays/macro/neoclass.htm

summary of the essence of the Neoclassical macromodel:

(1) Factor supplies and factor demands determine factor returns and factor employment.

(2) Factor employment and technological possibilities determine aggregate supply.

(3) Aggregate supply and aggregate demand determine the equilibrium rate of interest.

(4) Money demand and money supply determine the price level.

The essential features of the Neoclassical macromodel are shown diagramatically in Figure 1, with causality running from Quadrant I (upper right) to Quadrant III (bottom left).

Figure 1 - The Neoclassical Macromodel

In order to work through this diagram, let us then begin with the first line of the catena, the factor markets. Let w/p be the real wage (where w is nominal wage and p is the price level). Let N be labor. Then we are allowed to write out the labor demand function as:

Nd = Nd(w/p)

where dNd/d(w/p) < 0 so that labor demand is a negative function of the real wage. This relationship arises from the substitution by profit-maximizing firms between labor and other factors. At high real wages, firms will opt for less labor-intensive techniques and at low wages, they will opt for more labor-intensive techniques. So, if labor is relatively more expensive, the demand for labor declines and thus, the labor demand curve is downward sloping, as shown in Quadrant I of Figure 1.

Let us now turn to labor supply. Here we have:

Ns = Ns(w/p)

where dNd/d(w/p) > 0 so that labor supply is a positive function of the real wage. The upward-sloping labor supply function in Quadrant I of Figure 1 also arises from substitution - this time between work and leisure on the part of the household. The greater the real wage, the more labor is supplied.

The equilibrium in the labor market and is given by:

Nd(w/p) = Ns(w/p)

which gives us the equilibrium real wage (w/p)* and the equilibrium level of employment (N*), a shown in Quadrant I of Figure 1. Given these conditions, then by total differentiation we know that:

dw/w = dp/p

i.e. nominal wages (w) are fully flexible and will accompany changes in the price level (p) by the same proportion in order to maintain (w/p)* and, by extension, N*. This flexibility need not be necessarily instantaneous - adjustment can take time - but it is certainly the "long-run" tendency. In sum, in the long run, employment does not change with a different price level since nominal wages will change proportionately.

Thus, we have obtained the first line of the catena and the first quadrant of Figure 1. We have also, by extension, obtained the second one as well. Proposing a short-run production function Y = � (N), then given N* from the labor-market clearing we just derived, we obtain, immediately, the level of aggregate supply, Y*. This is shown in Quadrant II of Figure 1.

Now, turning to the goods market in Quadrant III of Figure 1, we notice that the aggregate supply curve is horizontal (or vertical, in normal perspective). In other words, aggregate supply, Y*, is derived entirely from factor-market clearing - thus the output of goods in an economy is wholly "supply-determined". The greater the level of employment, N*, the greater the level of output, Y*. Thus, the supply of goods enters the goods market already determined and is invariant to anything that happens within the goods market alone.

Now we turn to the third line in the catena. The given output Y* will be factor income which is, in turn, consumed, saved or taxed away, thus:

Y = C + S + T

Aggregate demand, Yd, however, is composed of consumption, investment and government expenditures, thus:

Yd = C + I + G

Thus, for there to be an equilibrium in the goods market, for aggregate demand to be equal to aggregate supply of goods (Yd = Y), then it must be that C + I + G = C + S + T or, cancelling out C and flipping G over:

I = S + (T - G)

Thus the condition for equilibrium in the goods market is that investment be equal to savings -- where savings here is defined as private sector saving, S , and public sector saving, (the excess of government revenue over expenditures, or T-G). This is equivalent to the loanable funds theory which says that the supply of loanable funds (public plus private savings) be equal to the demand for loanable funds (investment) in equilibrium.

What brings this equilibrium about? Movements in the rate of interest. The interest rate adjustes to bring Yd in equality with Y (or, equivalently, I into equality with S). This is in stark contrast to the Keynesian multiplier where it is output itself that adjusts to equate investment and savings. Thus, we shall argue that interest in the Neoclassical model does not affect aggregate supply, but rather affects only aggregate demand.

2011-01-03T09:56:00.000-08:00

Courses & Programs of Study
Descriptions of the Main Courses

Included below are descriptions of the main courses in the MS QCF Program. Each of these courses will carry 3 semester hours of credit. The outlines are given to give a sense of the type of material in the courses, and should not be interpreted as the exact content of the courses.

Quick links:

* Core Courses
* First Category Elective Courses
* Second Category Elective Courses
* Additional QCF Elective Courses

Core Courses
Finance and Investments MGT 6078
Stochastic Processes in Finance I ISYE/MATH 6759
Numerical Methods in Finance MATH 6635
Derivative Securities MGT 6081
Design and Implementation of Systems
to Support Computational Finance ISYE/MATH 6767
Fixed Income Securities ISYE/MATH/MGT 6769

Finance and Investments (MGT 6078)

This course is an introduction to finance that contains the fundamental concepts of financial accounting, corporate finance and portfolio optimization. This includes financial statement analysis, time value of money, cash flow analysis, capital budgeting, risk and return, capital structure, mean-variance portfolio optimization, and risk management.

The prerequisite for the course is MATH 3215. The text is at the level of Corporate Finance by Ross, Westerfield, and Jaffe, published by Irwin/McGraw-Hill; course handouts are also used.

It is expected that this course will have a more quantitative emphasis than the MSM 'Financial Management' and 'Investments' Courses.

The specific course topics are the following:

* Introduction to Financial Markets
* Financial Accounting Concepts and Financial Statement Structure.
* Cash Flow and Financial Statement Analysis.
* Time Value of Money and Net Present Value.
* Bond Valuation and Term Structure of Interest Rates. Stock Valuation.
* Capital Budgeting Techniques
* Mean-Variance Portfolio Optimization
* The Capital Asset Pricing Model.
* The Arbitrage Pricing Model.
* Capital Structure.
* Risk, Return and Capital Budgeting.
* Introduction to Derivative Securities and Risk Management.
* Hedging strategies. Portfolio Insurance

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Stochastic Processes in Finance I (ISYE/MATH 6759)

This course introduces basic probability concepts and uses these to model underlying and derivative securities in financial markets. This includes the probabilistic concepts of conditional expectation, convergence in distribution and central limit theorems, martingale processes, and Markov processes, and the modeling concepts of pricing, hedging and trading in the Binomial market model, more general discrete time market models, and the continuous time Black-Scholes market model. Mathematical concepts are introduced as needed.

The prerequisites for the course are MATH 3215 and some knowledge of computer programming. Class notes are used for this course; these notes are at the level of Introduction to Mathematical Finance: Discrete Time Models by S.Pliska, published by Blackwell Publishers.

The specific course topics are the following:

* Discussion of prerequisites, including basic probability background and linear systems of equations
* Some probability background, including Riemann-Stieltjes integrals and conditional probabilities and conditional expectations. Definitions of some financial terms.
* The Binomial Market Model and its use in pricing and hedging claims. European style options, some exotic options.
* Model implementation, implied volatility. Probability background: convergence in distribution, and a central limit theorem
* Convergence of Binomial option prices to Black-Scholes option prices, and a sketch of the Black-Scholes Market Model
* Use of derivative securities, and strategies for trading. The greeks, and their use in option trading.
* Probability and mathematics background: martingales, separating hyperplanes, linear programming and duality
* The Discrete-time, Stochastic Market Model, conditions of no-arbitrage and completeness, and pricing and hedging claims
* Variations of the basic models: American style options, foreign exchange derivatives, derivatives on stocks paying dividends, and forward prices and futures prices
* Probability background: Markov chains. Stochastic volatility and implied trees.
* Stochastic interest rates, bonds and interest rate derivatives. Model implementation.
* Mathematical background: optimization techniques. Incomplete Market Models. Utility-based pricing in complete markets and in incomplete markets. Portfolio optimization

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Numerical Methods in Finance (MATH 6635)

This course contains the basic numerical and simulation techniques for the pricing of derivative securities.

The prerequisites for the course are MATH 2403 and MATH 3215 (or the equivalent), knowledge of computer programming, and MS QCF standing or some previous exposure to the topics of stocks, bonds and options. Class notes are used for this course; these notes are at the level of The Mathematics of Financial Derivatives: A Student Introduction by P. Wilmott, S. Howison and J. Dewynne, published by Cambridge University Press.

The specific course topics are the following:

* Solution of a single non-linear equation and its application in simulating geometric Brownian motion and computing implied volatility from the Black-Scholes formula.
* Smooth interpolation and approximation of data by splines.
* The heat equation and its solution, analytic properties and issues in its numerical solution.
* The Black-Scholes equation for European options: derivatives, boundary conditions, the Black-Scholes formula, the binomial method, and finite-difference methods.
* Solutions of the American option problem: boundary conditions implied by early exercise; numerical methods for the free boundary problem (finite differences with projection, the sweep method ??he method of lines); and barriers, jumps and transaction costs and their numerical treatment.
* Pricing bonds: derivation of the diffusion equation, boundary conditions, and numerical considerations.
* Path dependent options, stochastic volatility, options on multiple assets and other advanced models, and the need for Monte Carlo methods.

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Derivative Securities (MGT 6081)

This course provides an introduction to options, futures, and swaps. Concepts of arbitrage, index trading, and portfolio insurance are discussed.

The prerequisite for this course is MGT 6060, Financial Management, or MS QCF standing and MGT 6078. The course is at the level of the text Derivative Securities by R. Jarrow and S. Turnbull, published by South-Western College Publishing.

The specific course topics are the following:

* Introduction to derivative securities.
* Simple arbitrage relationships for forward and futures contracts.
* Hedging, basis risk, and speculation.
* Stock index futures.
* Short-term interest rate futures.
* Swaps.
* Option markets.
* Simple arbitrage relationships for options.
* Trading strategies involving options.
* Valuation of options using a binomial model.
* The Black-Scholes analysis.
* Options on stock indices, currencies, and futures contracts.
* Hedging positions in options, and portfolio insurance.
* Non-standard (exotic) options

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Design and Implementation of Systems to Support Computational Finance (ISYE/MATH 6767)

The prerequisites for this course are some knowledge of computer programming, and MS QCF standing or some previous exposure to the topics of stocks, bonds and options.

Course Description:
Introduction to large scale-system design to support computational finance for options, stocks, or other instruments. The course weaves together the tools to obtain Web-based financial data, store it in a data base, and use various mathematical toolkits to compute desired parameters. The course includes acquisition of functional literacy in Java and object-oriented programming, use of Java (1) to obtain data from the Web, and store and retrieve it from a data base, such as Oracle; (2) to interact with mathematical and computational finance tools as necessary, for example, MatLab; and (3) to design Java-based graphical user interfaces to manage individual applications. The course uses a sequence of examples, increasingly more challenging over time, to introduce various concepts. Students extend class examples and conclude with a final project that integrates the various concepts, principles, and skills the course contains.

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Fixed Income Securities (ISYE/MATH/MGT 6769)

The course will contain numerical work to implement the modeling; the prerequisites for this course are MGT 6060 or MGT 6078, MATH 3215 (or the equivalent), and some knowledge of programming.

Tentative Topics:

* Introduction to Fixed Income Securities
* Bond Calculations
* Quantifying Interest Rate Risk
* Floating Rate Notes and Interest Rate Swaps
* Risk Management, Accounting, and Control
* Stochastic Interest Rate Models
o Bonds, Forward and Futures Contracts: Discrete- and Continuous-Time Models
o Term Structure: Discrete- and Continuous-Time Models
o Factor Spot Rate Models: Discrete- and Continuous-Time
o Yield Curve Models and the Heath-Jarrow-Morton Model
* Forwards, Futures and Options, caps and caplets, swaps
* Credit Risk on Corporate Bonds
* Emerging Market Debt
* Mortgages and Mortgage Derivatives

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First Category Elective Courses
Stochastic Processes in Finance II MATH 6235
Financial Optimization Models ISYE 6673
Management of Financial Institutions MGT 6090

Stochastic Processes in Finance II (MATH 6235)

This is the second of a two-semester sequence that develops basic probability concepts and models for working with financial markets and derivative securities. Continuous-time parameter stochastic processes are emphasized in this course. Mathematical concepts are introduced as needed.

The prerequisites for this course are MATH 2403 and ISYE/MATH 6759.

The specific course topics are the following.

* Background on integration and on simulation
* Brownian Motion, and Continuous-Time Martingales and their Variation.
* The Ito Stochastic Integral and its Properties, and Ito's Change-of-Variable Formula.
* Stock Prices as Geometric Brownian Motions.
* Black-Scholes Option Pricing.
* Ito Processes and Stochastic Differential Equations.
* Continuous-Time Markov Processes and the Kolmogorov Equations.
* Additional Results on Black-Scholes Option Pricing
* Girsanov's Theorem for Change of Measure, and Martingale Representation Theorems
* Asset Pricing theory, Risk Neutral Measures (Equivalent Martingale Measures), and Hedging
* Pricing Specific Exotic Options
* Continuous-Time Optimal Stopping and Pricing American Style Options

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Financial Optimization Models (ISYE 6673)

Financial optimization models are indispensable tools for managing risk, structuring portfolios and customizing financial products in the banking, insurance, corporate, and financial service sectors of our economy. This course will introduce different applications of optimization models, with special emphasis on formulation, analysis and implementation obtained by hands-on experience with computer modeling languages and cutting-edge optimization software. The prerequisites are ISYE 6225 or the MGT 6078 Finance and Investments course. ISYE 6669 or its equivalent is strongly recommended.. The course uses a text at the level of Operations Research, 3rd ed., Wayne L. Winston, Duxdury Press, and course handouts.

The specific course topics are the following.

* Portfolio Selection Models
* Asset Allocation Models
* Index Construction Models for Equity and Bond Portfolios
* Immunization Models to Manage Interest-Rate Risk
* Cash Matching Models for Asset-Liability Management
* Models to Structure Collateralized Mortgage Obligations
* Firm Valuation Models
* Valuation Bound Models on Financial Options
* Dynamic Hedging Models for Risk Management

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Management of Financial Institutions [Risk Management] (MGT 6090)

This course provides an introduction to the various risks faced by financial institutions and a detailed analysis of the tools used to manage these risks.

The prerequisite for this course is MGT 6060, Financial Management, or MS QCF standing and the MGT 6078 Finance and Investments course . The course is at the level of the text Management of Financial Institutions by A. Saunders.

The specific course topics are the following.

* Introduction (depository institutions)
* Unique characteristics of financial institutions
* Management of interest rate risk
* Mortgage backed securities
* Option adjusted spread analysis
* Management of credit risk
* Management of off-balance-sheet risk.
* Management of foreign exchange risk.
* Management of liquidity risk.
* Deposit insurance.
* Security underwriting.
* Role of investment banks in treasury and municipal markets

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Second Category Elective Courses
Empirical Finance MGT 7061
Statistical Techniques of Financial Data Analysis ISYE/MATH 6783
The Practice of Quantitative and Computational Finance ISYE/MATH/MGT 6785

Empirical Finance (MGT 7061)

The material and level of this course will be similar to that of the book Campbell, J., Lo, A. and MacKinlay, C. (1997) The Econometrics of Financial Markets. Princeton University Press, Princeton, N.J.

The topics of the course will be as follows:

* Overview of Econometrics
* Testing of Models Related to the Following Topics and Areas
* Capital Asset Pricing Model
* Arbitrage Pricing
* Conditional Asset Pricing
* Market Efficiency
* Information and Volatility Issues
* Option Pricing
* The Course also covers some topics involving
* Time Series Analysis and Prediction
* Market Microstructure Issues
* Event Studies
* Investment Performance Evaluation

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Statistical Techniques of Financial Data Analysis (ISYE/MATH 6783)

Fundamentals of statistical inference are presented and developed for models used in the modern analysis of financial data. Techniques are motivated by examples and developed in the context of applications.

The prerequisites for the course are MATH 3215 (or the equivalent), some knowledge of programming, and MS QCF standing or some previous exposure to the topics of stocks, bonds and options.

The specific course topics are the following.

The following probability topics are covered in the models that are presented:

* Distributions such as the normal (Gaussian), lognormal, geometric, binomial, Poisson, Student's t, F, chi-square, gamma, and Pareto
* Characteristic functions, sums of independent random variables, a-stable random variables
* Limit Theorems for sums
* order statistics
* Limit Theorems for extremes
* Elementary stochastic processes such as Markov chains
* Dynamic linear models
* Time series models

The following topics in statistical inference are covered in the models that are presented:

* Likelihood functions
* Estimation
* Testing Hypotheses via Neyman-Pearson tests, likelihood ratio tests, and Wald tests
* Tests of fit
* Markov chain and time series inference
* Regression
* Principal components analysis
* Non-parametric analyses

Applications to financial data are made throughout and include the topics such as the following:

* Testing hypotheses of independence, normality, homoscedascticity, and symmetry for returns, and the Bachelier and Mandelbrot models
* Efficient frontier in portfolio analysis under short selling and riskless borrowing and lending, optimal portfolio under single index and multi-index models, principal components analysis, stability tests of betas from auxiliary data
* simulation and Monte-Carlo, estimation and assessment of accuracy of path integrals arising in option pricing
* Hill's estimator of the Pareto index, application to solvency analysis and ruin probabilities, connections with a-stability
* Analysis of ar, ma, arma, arima, arch, garch, and stochastic volatility time series models applied to exchange rates, indexes, interest rates, and returns.

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The Practice of Quantitative and Computational Finance (ISYE/MATH/MGT 6785)

This course is jointly listed with the College of Management, the School of Mathematics and the School of Industrial and Systems Engineering. The course will consist of case studies, visiting lecturers from financial institutions and student group projects of an advanced nature - all centered around quantitative and computational finance. The group projects deal with applicable problems in areas such as portfolio management and optimization, pricing of derivatives, and data analysis and testing of models. The groups will be required to formulate and analyze the project problem, and implement and present their solutions to the problems. The prerequisite for the course is MS QCF major, or consent of the instructor. Normally the course is taken during the student's third semester in the QCF program.

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Additional QCF Elective Courses

For the Third Category Elective Courses Requirement there are many possible Elective Choices. Here is one of the courses directly related to quantitative and computational finance.
Advanced Topics in QCF

ISYE/MATH/MGT 6793

This course is jointly listed with the College of Management, the School of Mathematics and the School of Industrial and Systems Engineering. The course will deal with advanced research material in quantitative and computational finance. The prerequisite for the course is graduate standing, and consent of the instructor. Normally the course is taken during the student's third semester in the QCF program. The course will also be suitable for students pursuing Ph.D. work in areas related to quantitative and computational finance.
**********************
The QCF Lab

The QCF Trading Floor gives students access to a considerable number of computer applications in mathematics, statistics, optimization, programming languages, and finance.

In the area of finance, there is the following:

* Bloomberg data feed, with data download
* SAS software
* OSFinancial System which includes Market Simulation applications (see events)
* Financial CAD
* Wharton Research Data Services (WRDS)
* Other financial investment applications such as the MATLAB Financial Toolboxes

Advanced C++ for Computational Finance

2011-01-03T08:37:00.001-08:00

The goal of this three-day intensive hands-on course is to learn those advanced features in C+ that are of direct relevance to writing and extending application for quantitative and computational finance.

The course uses the object-oriented and generic (templates) programming models (OOP, GP) in combination with design patterns and the STL and boost libraries to allow you to create robust and flexible applications. We develop the contents of the course by discussing important C++ language, using OOP and GP models to write clean and effective code. We also discuss how to improve the performance of your application. In all cases, the examples and test cases are based on finance
experience.
This is one of the few courses (in our opinion) that focuses on the application of C++ to quantitative and computational finance. It is a practical course for practitioners.To participate in this course, you need to bring your own laptop computer with a C++ compiler (ideally Microsoft's Visual Studio or GNU GCC for example)
In this course we introduce state-of-the-art design and programming techniques in C++ and their application to Computational Finance. In particular, the following topics are discussed in detail:
• Advanced C++ syntax and its application
• Template classes and the Standard Template Library (STL)
• Combining the object-oriented and generic programming paradigms
• The famous Gamma (GOF) design patterns applied to QF
• Interfacing to Excel: COM Add-ins
• Creating applications: Monte Carlo, Finite Difference and lattice methods

What do you receive?
As attendee you receive a full set of slides, C++ source code and a copy of Daniel Duffy's book
"Financial Instrument Pricing using C++" (Wiley 2004), including CD with C++ code. In short, you will receive what is needed to start developing your own applications.
Prerequisites
We assume that the student has experience of C++. This is not a beginners course and we assume you know what constructors, destructors and operator overloading are in C++ and how memory management works.
Who should attend?
This course has been developed for financial professionals who design and implement pricing and hedging models in C++ and Excel. The course introduces and elaborates on how to apply C++ to creating flexible and reliable applications in Quantitative Finance using the most modern software design techniques. There is ample room for questions on your own specific applications as well as hands-on programming sessions. It is assumed that the attendees have some working knowledge of
C++ and have developed applications or prototype applications in that language.
Course contents
Day 1: Advanced Object-Oriented and Generic C++
Memory Management, Issues
- Static, stack and heap memory management
- Wild pointers, dangling pointers, double free bugs, memory
leaks
- Using object factories to control object lifecycle
- Single object and object array allocation and deallocation
Memory Management, Solutions
- The pimpl idiom; STL auto_ptr
- Using Builder pattern to coordinate object lifetime
- Overview of Boost smart pointers
- Scoped and shared pointers
- Casting and Run-Time Type Information (RTTI)
- Static and dynamic casting overview
- dynamic_cast and static_cast
- Exception handling issues
- Boost lexical_cast
Polymorphism Functions
- Dynamic polymorphism: virtual and pure virtual functions
- Polymorphism and algorithms in computational finance
- Polymorphism, inheritance and composition, what’s best?
- Performance issues with dynamic polymorphism
Static Polymorphism
- Curiously Recurring Template Pattern (CRTP)
- Where does static_cast fit it?
- Where to use CRTP
- CRTP and performance improvements
Advanced Generic Programming (GP) in C++
- Template classes and template functions
- Partial specialization
- Default parameters, template template parameters
- Template member functions
- Nested templates and data structures
Combining OOP and GP
- Inheritance scenarios
- Composition; combining composition with inheritance
- Policy-based design and traits classes
- When OOP and when GP?
Day 2: STL, Boost Libraries and Design
STL Containers
- Sequential and associative containers
- Lists, vector and queues
- Maps, sets and multimaps
- Modelling option data with maps and Property Sets
STL Algorithms and Iterators
- Mutating and non-mutating algorithms
- Searching and sorting
- Inserting and removing data
Using STL in Applications
- Using STL with class adapters
- STL-compatible data structures
- Complexity analysis and performance tests
Function Pointers and Function Objects
- C function pointers: advantages and disadvantages
- Function objects and their applications
- Functions in boost; binding
- Performance issues
String Algorithms
- Regular expressions
- Trimming and conversion
- Find and replace
- Find iterator
- Join and split
Random Number Library
- Concepts
- Random number variate generators
- Random number library distributions
- Applications
uBLAS (Basic Linear Algebra System)
- Vectors, matrices and their operations
- Patterned matrices (sparse, triangular, ..)
- Expression templates to improve performance
- Applications of uBLAS classes
Essential Patterns: Creational
- Factory method and abstract factory
- Creating complex objects using Builder
- Singleton
Essential Patterns: Structural
- Composite and nested objects
- Bridge and implementation-independence
- Extending object structure with Decorator
Essential Patterns: Behavioural
- Strategy and algorithms
- Extending class functionality with Visitor
- Template method pattern and customisable frameworks
Day 3: Design and Applications to Computational
Finance
Multithreading: Theory
- What is a thread? Thread lifecycle
- Shared data models
- Thread synchronization and notification
- Speedup and race conditions
Multithreading: OpenMP and boost Thread
- Overview of functionality
- Creating threads
- Speedup, accuracy and robustness
Excel-C++ Integration
- An overview of Excel add-ins
- Creating Automation add-ins and worksheet functions
- Creating COM add-ins: The steps using ATL projects
The Monte Carlo Method
- Description of the problem
- Creating a software framework for a MC engine
- Using design patterns to create flexible MC systems
- Plain options, Asians and barriers
The Finite Difference Method
- A quick introduction to FDM
- C++ classes for a FDM solver
- Explicit and implicit schemes (Crank-Nicolson, Euler, ADE)
- Presentation in Excel
Improving Application Performance
- Call by reference versus call by value
- Appropriate use of virtual functions
- Function objects versus function pointers
- Preventing unnecessary temporary object creation
Loop Optimisation
- Loop interchange
- Loop fission and fusion
- Making loops multi-threaded (OpenMP)
- Application profiling; determining a program’s serial
fraction
Note:
To participate in this course, you need to bring your own
laptop computer with a C++ compiler (ideally Microsoft's
Visual Studio or GNU GCC for example)

2011-01-03T06:05:00.000-08:00

http://www.vias.org/tmdatanaleng/cc_deriv_regression.html

Univariate Regression
Derivation of Equations

The principle of this derivation is quite simple: the least squares regression curve is one that minimizes the sum of squared differences between the estimated and the actual y values for given x values. Therefore you first have to define the equation of the sum of squares, calculate the partial derivatives (with respect to each parameter), and equate them to zero. The rest is just plain algebra to obtain an expression for the parameters.

Let us conduct this procedure for a particular example:
y = ax + bx2

This formula is to be estimated from a series of data points [xi,yi], where the xi are the independent values, and the yi are to be estimated. By substituting the yi values with their estimates axi+bxi2 we obtain the following series of data points: [xi, axi+bxi2]. The actual values of the y values are, however, the yi. Thus the sum of squared errors S for n data points is defined by

S = (ax1+bx12-y1)2 + (ax2+bx22-y2)2 + (ax3+bx32-y3)2 + ...... + (axn+bxn2-yn)2

Now we have to calculate the partial derivatives with respect to the parameters a and b, and equate them to zero:

dS/da = 0 = 2(ax1+bx12-y1)x1 + 2(ax2+bx22-y2)x2 + 2(ax3+bx32-y3)x3 + ...... + 2(axn+bxn2-yn)xn
dS/db = 0 = 2(ax1+bx12-y1)x12 + 2(ax2+bx22-y2)x22 + 2(ax3+bx32-y3)x32 + ...... + 2(axn+bxn2-yn)xn2

These two equations can easily be reduced by introducing the sums of the individual terms:

Now, solve these equations for the coefficients a and b:

And then substitute the expressions for a and b into their counterparts, with the following final results:
***************
Vectors
Introduction

Vectors form the basis for many mathematical methods and are important for data analysis. Note that a vector may be seen as a 1 by n matrix.

Vector We define an ordered set of n equal objects written in a column vector of order n, and the row counterpart of m objects a row vector (of order m). Please keep in mind that these definitions are simplified and cover only part of the exact, mathematical definition. However, the definition given here is sufficient for our purposes concerning data analysis.

To denote a specific vector, we shall use a lowercase, bold letter, such as a, for example. Whether this vector a is a column or row vector, will usually be clear from the context in which the letter is used. When written explicitly, vectors are put in parenthesis.

Scalar The n elements that form a vector according to definition above are called scalars. All scalars are taken from the same basic set. For most purposes the basic set is the space of real numbers R.

To denote a certain element of a vector, we shall use a lowercase letter (to describe the vector from which we are taking an element) with an index (for the index within the vector), such as a2, for example.
Example: Vector
v:=(4, 3, 5.1, p, -e) is a row vector of order 5, which we also could denote by (vk). v5 is the fifth element of this vector, and thus -e.
It is an important aspect of vector and matrix algebra that we now have two sets of objects to handle: the vectors (or matrices, respectively) and their scalars, which also play a significant role.
***********************
Addition of Vectors

Addition of vectors is only defined concerning vectors with the same type of elements and the same form (row or column) and order (number of elements in a vector).

Definition
Let a, b, and c be vectors of the same order and form. The sum c := a+b is then calculated by adding up the corresponding elements of the vectors a and b: ck:=ak+bk

There are several rules for the addition of vectors:

* the commutative law: a+b = b+a
* the associative law: (a+b)+c = a+(b+c)
* and the law that the solution x of the equation (ak) + (xk) = (yk) is unique to the given vectors a and y
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Subtraction of Vectors

The subtraction of vectors is accomplished in the same way as the addition of vectors:

Definition

Let a, b, and c be vectors of the same order and form. The difference vector c := a - b is then calculated by subtracting the corresponding elements of the vectors a and b: ck := ak - bk.

Note that the subtraction can be formally seen as a summation: a-b:=a+(-b).

Subtracting two equal vector results in a vector with all elements being zero. This vector is also called a zero vector, usually depicted by o.
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GRE

2011-01-01T04:49:00.000-08:00

The GRE General Test measures verbal reasoning, quantitative reasoning, critical thinking and analytical writing skills that are not related to any specific field of study.

* Analytical Writing — Measures critical thinking and analytical writing skills, specifically the test taker's ability to articulate complex ideas clearly and effectively
* Verbal Reasoning — Measures reading comprehension skills and verbal and analogical reasoning skills, focusing on the test taker's ability to analyze and evaluate written material
* Quantitative Reasoning — Measures problem-solving ability, focusing on basic concepts of arithmetic, algebra, geometry and data analysis

*******************
Subject test
Mathematics
* The test consists of approximately 66 multiple-choice questions drawn from courses commonly offered at the undergraduate level.
* Approximately 50 percent of the questions involve calculus and its applications — subject matter that can be assumed to be common to the backgrounds of almost all mathematics majors.
* About 25 percent of the questions in the test are in elementary algebra, linear algebra, abstract algebra and number theory. The remaining questions deal with other areas of mathematics currently studied by undergraduates in many institutions.

The following content descriptions may assist students in preparing for the test. The percents given are estimates; actual percents will vary somewhat from one edition of the test to another.

CALCULUS — 50%

Material learned in the usual sequence of elementary calculus courses — differential and integral calculus of one and of several variables — includes calculus-based applications and connections with coordinate geometry, trigonometry, differential equations and other branches of mathematics.

ALGEBRA — 25%

* Elementary algebra: basic algebraic techniques and manipulations acquired in high school and used throughout mathematics
* Linear algebra: matrix algebra, systems of linear equations, vector spaces, linear transformations, characteristic polynomials and eigenvalues and eigenvectors
* Abstract algebra and number theory: elementary topics from group theory, theory of rings and modules, field theory and number theory

ADDITIONAL TOPICS — 25%

* Introductory real analysis: sequences and series of numbers and functions, continuity, differentiability and integrability, and elementary topology of R and Rn
* Discrete mathematics: logic, set theory, combinatorics, graph theory and algorithms
* Other topics: general topology, geometry, complex variables, probability and statistics, and numerical analysis

The above descriptions of topics covered in the test should not be considered exhaustive; it is necessary to understand many other related concepts. Prospective test takers should be aware that questions requiring no more than a good precalculus background may be quite challenging; such questions can be among the most difficult questions on the test. In general, the questions are intended not only to test recall of information but also to assess test takers' understanding of fundamental concepts and the ability to apply those concepts in various situations.

Finance PhD Programs

2011-01-01T04:16:00.000-08:00

Following are the top leading universities offering Finance PhD Programs:

University of Michigan (Stephen M. Ross School of Business)
- The Ross School of Business Ph.D. program in Finance is one of the top programs in the country.
- Students complete most of the course work for the Ph.D. during the first two years of the program.
http://www.bus.umich.edu/Academics/Phd/WhyRoss.htm

We prefer the GMAT, but we will accept a GRE score. We only will accept official GMAT or GRE test scores for exams less than five years old. We do not have required minimum scores. Average GMAT scores for admitted applicants range between 660-770, while average GRE scores range from 550-800 for the verbal component and 700-800 for the quantitative component.

The University of North Carolina at Chapel Hill (Kenan-Flagler Business School)
- UNC is consistently ranked among the top business schools in the world by The Wall Street Journal, Forbes, and U.S. News & World Report.
- The Finance PhD program at UNC is a four year program designed for students who wish to become Finance faculty members at research universities.
http://areas.kenan-flagler.unc.edu/finance/phd/Pages/default.aspx

The University of Oklahoma (Michael F. Price College of Business)
- The Price College Finance PhD program is a high-quality program.
- A study in the Journal of Finance Literature (winter 2005) ranked two of its finance faculty among the 50 most prolific authors in the top seven finance journals.
- A study in Financial Management in 2002 ranked it 56th among all finance departments and 26th among finance departments at public universities in terms of Journal of Finance equivalent pages published in top finance journals.

The University of Georgia (Terry College of Business)
- The PhD program in Banking and Finance is designed to prepare students for careers in teaching and research at a major university or as financial researchers.
- Terry is ranked 92nd among all business schools in the world and 45th in the US in the Economist Intelligence Unit Full Time MBA Rankings 2009.
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1 University of California–Berkeley

1 Stanford University (CA)

1 Massachusetts Institute of Technology

4 Princeton University (NJ)

4 Harvard University (MA)

6 Yale University (CT)

6 Uni of Michigan–Ann Arbor

8 Uni of Wisconsin–Madison

8 Uni of Chicago

8 Cornell Uni (NY)

8 Columbia Univ (NY)

12 Univy of California–Los Angeles

13 Univ of Texas–Austin

2010-12-30T22:12:00.000-08:00

Financial markets
Investment management
Financial institutions
Personal finance
Public finance
Mathematical finance
Quantitative behavioral finance
Financial economics
Experimental finance
Computational finance
Statistical finance
************************
RESEARCH INTERESTS
• Mathematical finance and economics
• Numerical/nonlinear optimization
• Financial optimization
• Financial engineering
• Risk management
• Investments and capital markets
• Behavioral finance
• Financial economics
• Algorithmic trading
• Mathematical modeling
• Stochastic volatility models
• Data analysis and programming
• Inverse problems
• Numerical analysis
• Differential equations (ODE, PDE, SDE)
• Dynamical systems
• Wireless networking algorithms and simulation
• Computer algebra

complex analysis

2010-12-30T05:35:00.000-08:00

http://pirate.shu.edu/~wachsmut/complex/numbers/index.html
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1.1. Complex Numbers
ICA

Natural numbers (N), integers (Z), rational numbers (Q), and real numbers (R) should be quite familiar by now. Each more advanced set includes the previous one, and indeed extends its properties:

N Z Q R

* N derives from simple counting numbers, but addition does not have an inverse
* Z adds inverse numbers with respect to addition, but multiplication does not have an inverse
* Q adds inverse numbers with respect to multiplication, but the limit process is not complete

Example 1.1.1: Properties of number systems
Find

* a natural number without an additive inverse
* an integer without a multiplicative inverse
* a sequence of rational numbers that converges but the limit is not rational

R, the set of real numbers, on the other hand, is (by definition) complete with respect to the limiting and has all properties that are useful in daily life. We can add, subtract, multiply, and divide (albeit not by zero), we can take limits of real numbers and get back other real numbers, and any two real numbers are comparable (i.e. either they are identical or one is smaller than the other). Such a system, incidentally, is called a field:

Definition 1.1.2: A Field
A field is a set F together with two operations commonly denoted as + and *, as well as two different special elements commonly denoted as 0 and 1, that satisfies the following axioms:

1. Both + and * are associative, i.e. a+(b+c)=(a+b)+c and a*(b*c)=(a*b)*c
2. Both + and * are commutative, i.e. a+b=b+a and a*b=b*a
3. The distributive law holds, i.e. a*(b+c)=(a*b)+(a*c)
4. 0 is the additive identity, and 1 is the multiplicative identity, i.e. for all x we have x+0=x and x*1=x
5. There are additive and multiplicative inverses, i.e. for all x exists y such that x+y=0 and for all non-zero a exists b such that a*b=1

In other words, R is a field where in addition the limit process works (the proper way of saying it is that R is a complete field). The above properties just formalize in general some of the properties of real numbers, addition, and multiplication that we use (seemingly) automatically.

Example 1.1.3: Is Q a field
Is Q a field?

So, what is wrong with R to - possibly - warrant yet another number system? One problem is that:

not every polynomial equation is solvable in R

In other words:

* (1) x - 1 = 0 has one solution
* (2) x2 - 1 = 0 has two solutions
* (3) x3 - 1 = 0 has one solution ?
* (4) x2 + 1 = 0 has no solution ??

We can certainly live with those facts, but they seem not logical (as Mr. Spock from the USS Enterprise would say): the third eqution should really have three solutions, not just 1, and the fourth equation should have two, not zero, solutions! The solution to the problem will come from a simple, yet ingenious definition:

Definition 1.1.4: The Imaginary Unit
Define a new symbol called the imaginary unit, or i, as:

i =

This may not make sense, since traditionally we are not allowed to take even roots of negative numbers. However, this is a definition so henceforth the symbol i simply means: square root of negative one. If you are worried, think about another, probably more familiar symbol: . It looks familar, but what does it really mean? Of course it simply means that it is that symbol that by definition solves the equation x2-2=0, just as i or will be that symbols that solves x2+1=0.

What matters are the consequences and reprecussions that the introduction, or definition, of a new symbol has: does it result in any contradictions to known results? Does it yield new results that are compatible - but perhaps extend - existing theorems? Is it 'useful'?

Just as the symbol provides a reason of extending the rational numbers to the real ones, the symbol i = will cause us to extend the real numbers to the complex numbers (a step that we will formally do below).
In particular, i is not a regular number (in the sense of R), but a new symbol whose properties we need to explore.

Example 1.1.5: Properties ofi
Use the definition of i to:

* find i2, i3, i4, i5, ...
* solve x2+1=0 and x4+1=0
* simplify 1/i

Using our new symbol we could define a new number system: informally we simply combine real numbers with our new symbol as:

z = x + i*y, where x, y R

We can use the definition of i to add, subtract, multiply, and divide such entities. For example:

The new number system could now consist of all symbols of the form x + i*y, but we will use a more formalized approach to avoid the somewhat strange symbol i (just in case someone still objects to it). Just as is not used in the definition of the real numbers, the symbol i should not be used in the formal definition of the complex numbers.

Definition 1.1.6: Complex Numbers
The set of complex numbers C is defined as the set of all pairs (x,y) , x, y R, where for z = (x,y) and w = (u, v) we define:

* z + w = (x,y) + (u,v) = (x+u, y+v)
* z * w = (x,y) * (u,v) = (x*u - y*v, x*v + y*u)

Two complex numbers z1 = (x1,y1) and z2 = (x2,y2) are called equal if both x1 = x2 and y1 = y2. We will frequently use the symbol 0 to mean (0,0).

This definition does not pull any strange symbols out of the hat (i is nowhere to be seen), but it does look perhaps a little contrived: addition of two tuples is defined as it should, but why is multiplication defined so strangely? Why not simply define (x,y) * (u,v) = (x u, y v)?

Please note that two complex numbers being equal results in two equations that need to be true simultaneously. You should also observe that we have defined equality of two complex numbers, but not inequality. In other words, we can not decide if one complex number is less or greater than another! But first we want to find out what this definition has to do with our new symbol i. Let's play with the definition to find out:

Example 1.1.7: Simple Complex Numbers

* Find the additive and multiplicative identities in C
* Define the complex numbers z = (2,3), w = (-2,0), v = (0,2). Find z*w+v, v2
* Find the additive and multiplicative inverses for the above numbers z, w, v
* Which complex number (a,b) should correspond to our symbol i?

Now we can formalize the properties of our new set of numbers:

Theorem 1.1.8: Complex Numbers are a Field
The set of complex numbers C with addition and multiplication as defined above is a field with additive and multiplicative identities (0,0) and (1,0). It extends the real numbers R via the isomorphism (x,0) = x.

We define the complex number i = (0,1). With that definition we can write every complex number interchangebly as

z = (x,y) = x + i*y = x + i y

Proof Proof

Complex numbers are an extension of the reals, but not all properties of real numbers extend to complex ones: the real numbers are ordered, i.e. two real numbers are either equal or one is great than the other. Complex numbers, however, are not ordered: two complex numbers are either equal or not, but if they are not equal we can not decide which one is greater.

We now have two representations of the field of complex numbers:

1. as pairs of real numbers (x,y) with a 'funny' multiplication
2. as a sum of two real numbers z = x + iy using the symbol i with the property that i2 = -1

The first definition is the mathematically proper one, since it does not need a new symbol. However, the second definition is the easier one to work with in almost all situations, and we will hence think of complex numbers as:

Complex Numbers
C = {z = x+iy, where x,y R and i2 = -1}

Let's put our new numbers to work:

Example 1.1.9: Algebra with complex numbers

* If z = 1-i and w = 2+3i, find z w2
* Simplify (1-2i)(1-2i)/(3i-4)
* Rewrite z/w in the form a + i b
* Solve the equations z2 = i and z2 = 1+2i

Now we can also remedy, at least partially, our original problem where polynomial equations of degree 2 sometimes have two, one, or no solution.

Proposition 1.1.10: Two complex roots
Show that every equation z2 = c has exactly two solutions in C (unless c = 0). In other words, every complex (non-zero) number c has exactly two complex square roots.

Proof Proof

Now we have an elementary understanding of complex numbers, but to better work with them we need a helpful visual representation. That will be the topic of the next section.
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1.2. The Complex Plane
ICA

Complex numbers are defined as tuples (x,y) or equivalently x+iy. Therefore is is natural to visualize them as objects in the two-dimensional plane.

Definition 1.2.1: The Complex Plane
The field of complex numbers is represented as points or vectors in the two-dimensional plane. If z = (x,y) = x+iy is a complex number, then x is represented on the horizonal, y on the vertical axis. The horizontal axis is called real axis while the vertical axis is the imaginary axis.

The complex plane

Visually, C looks like R2, and complex numbers are represented as "simple" 2-dimensional vectors. Even addition is defined just as addition in R2. The big difference between C and R2, though, is the definition of multiplication. In R2 no multiplication of vectors is defined. To be precise, there is the idea of dot-product and (if you think of R2 as embedded in R3) cross-product, but the definition of complex multiplication gives C its special structure.

Example 1.2.2: Multiplication in 2-D and C
Explore the differences between complex multiplication in C, the dot-product in R2, and the cross-product in R2 as embedded in R3.

Before we explore multiplication further, let's define a few easier operations in C and see their impact in the complex plane.

Definition 1.2.3: Re, Im, Conj, and Modulus
If z = x + iy is a complex number, then we define the following functions:

The Real Part
Re: C R, Re(z) = Re(x+iy) = x is the real part of z
The Imaginary Part
Im: C R, Im(z) = Im(x+iy) = y is the imaginary part of z
The Conjugate or z-bar
conj: C C, conj(z) = conj(x+iy) = x - iy is the conjugate of z. The conjugate of z is sometimes also denoted as (z-bar)
The Modulus, or Absolute Value
|.|: C R, |z| = |x+iy| = is the modulus, absolute value, or length of z

Geometrically, these operations are easy to visualize:

Re(z), Im(z)
conj(z), |z|

* Re(z) is the projection of z onto the real axis
* Im(z) is the projection of z onto the imaginary axis
* conj(z) is the vector z reflected around the real axis
* |z| is the standard Euclidean length of the vector z, using good old Pythagoras

Here are a few concrete (simple) examples:

Example 1.2.4: Complex numbers in the plane
Let u = 3 - 4i and v = i, and w = -2 Find algebraically and visually:

1. Re(z), Im(z), |z| and conj(z) for z = u, v, w, respectively.
2. Draw -u, -v, -w and then describe how -z looks in relation to z for arbitrary complex numbers.
3. Draw 1/u, 1/v, 1/w and then describe how 1/z looks in relation to z for arbitrary complex numbers.

Our new operations can also be composed with one another:

Example 1.2.5: Complex operations composed
Show that

* Re(Re(z)) = Re(z)
* Re(Im(z)) = Im(z)
* Im(Re(z)) = Im(Im(z)) = 0

Similar relations hold when composing the conjugate function, but since we will need these identities frequently we will state them in the form of a proposition:

Proposition 1.2.6: The conjugate operator
For every complex number z we have:

(1) |z| = |conj(z)| = ||

(2) |z|2 = z*conj(z) = z

(3) conj(conj(z)) = = z

Proof Proof

Addition and in particular multiplication are what really define the structure of the complex plane and we need to figure out how to visualize those operations. Addition (and subtraction) will be familiar from vectors in the plane:

Proposition 1.2.7: Adding complex numbers geometrically
For any complex numbers z and w you can find the sum z+w by attaching the vector w to the tip of the vector z and completing a paralellogram. The diagonal of that paralellogram is the vector z+w.

To find the difference z-w, reverse the vector w and find z+(-w).

Proof Proof

Adding vectors z and w
Subtracting vectors z and w

Multiplication is more difficult to figure out, but feel free to come up with a conjecture before reading on.

Example 1.2.8: Conjecture for multiplying geometrically
Let z=1+i, w=1-i, and v=i. Compute and visualize z*w, v*z, v*w, z2, w2, and v2. Try to come up with a conjecture how to visualize multiplication of two general complex numbers.

To see if your conjecture about multiplication is correct, we first need to introduce a more suitable coordinate system. Similar to vectors in the 2-D plane, we define polar coordinates as follows:

Definition 1.2.9: Polar coordinates and cis
A complex, non-zero number z=x+iy with rectangular coordinates (x,y) can also be represented in polar coordinates (r,t). Let

r = |z| =

and t be an angle such that

x = r cos(t) und y = r sin(t)

Then

z = x+iy = r cos(t) + i r sin(t) = r (cos(t)+i sin(t)) = r cis(t)

where we define cis(t) = cos(t) + i sin(t). r is called the radius or length, t the angle or argument of z.

(x,y) rectangular (r,t) polar
(1,0)=1 (1,0)=1
(0,1)=i (1,/2)=cis(/2)
(-1,0)=-1 (1,)=cis()
(0,-1)=-i (1,3/2)=cis(3/2)
(1,1)=1+i (,/4)=cis(/4)
(1,-1)=1-i (,-/4)=cis(-/4)
Polar coordinates of a complex number Sample points

It is easy to convert a number z from polar coordinates (r,t) to rectangular coordinates (x,y): simply use the fact that z = r cis(t) = r(cos(t) + i sin(t)), i.e. compute

* x = r cos(t)
* y = r sin(t)

You can verify, for example, that the complex number with polar coordinates r=2 and angle t=/4 can be written in rectangular coordinates as

z = 2 cis(/4) = 2 (cos(/4) + i sin(/4)) = 2 (1/ + i 1/) = + i

Going the other way, i.e. converting from rectangular to polar coordinates, is a little more confusing since the angle in the above definition is not unique. For example, the complex number 1+i clearly has a radius of but any angle /4 + 2k will work. In other words, z=1+i=(1,1) in rectangular coordinates is equal to

(,/4) = (,9/4) = (,17/4) = ...

in polar coordinates. To avoid confusion, we will often (somewhat arbitrarily) restrict the angle of a complex number in polar coordinates to be between - and and call it principle angle.

Definition 1.2.10: The Principle Argument Arg
For any complex number z 0 we define the principle argument or Arg(z) as the angle which the vector z makes with the positive (real) x-axis and for which

- < Arg(z)

In other words, a non-zero complex number has many arguments, but only one principle argument. To find a principle argument we use the fact that

tan(t) = sin(t)/cos(t) = r sin(t)/r cos(t) = y/x = Im(z)/Re(z)

so that we can use the tan-1 = arctan adjusted in such a way as to give us the principle angle: take any non-zero complex number z:

If Re(z) = 0 then

* if Im(z) > 0 then Arg(z) = /2
* if Im(z) < 0 then Arg(z) = -/2

If Re(z) 0 compute

and adjust the angle as follows:

* for z in 1st quadrant: Arg(z) =
* for z in 2nd quadrant: Arg(z) = -
* for z in 3rd quadrant: Arg(z) = - +
* for z in 4th quadrant: Arg(z) = -

Example 1.2.11: Polar coordinates examples
Convert the following numbers into the indicated coordinates and draw them in the complex plane:

* z=(2,0), w=(3,), v=(2,5/6), u=(2,-3/4) from polar to rectangular
* z=(-2,0), w=(0,-2), v=(3,4), u=(3,-4) from rectangular to polar
* Prove that if z = r cis(t) then = r cis(-t)

Using polar coordinates will finally allow us to understand how multiplication of complex numbers works geometrically:

Theorem 1.2.12: Multiplying complex numbers geometrically
If z = r1 cis(s) and w = r2 cis(t) in polar coordinates then

z*w = r1 r2 cis(s + t)

and

zn = rn cis(ns)

Proof Proof

You might wonder why this theorem is called "multiply complex numbers geometrically" when in fact it shows only formulas. But what the formulas mean is:

* two complex numbers are multiplied by multiplying their length and adding their angles
* a complex number is raised to a power by raising the length to the power and multiplying the angle

Multiplying vectors z and w

A similar trick works for division of two complex numbers:

Corollary 1.2.13: Dividing complex numbers geometrically
If z = r1 cis(s) and w = r2 cis(t) in polar coordinates then

z/w = r1 / r2 cis(s - t)

Proof Proof

Here are some examples that should convince you that in cases where the angle of a complex number is easy to determine, multiplication and division in polar coordinates can be quite simple.

Example 1.2.14: Multiplying geometrically

* Explain geometrically why (1+i)(1-i), i2, and (1+i)4 are all purely real numbers, i.e. the imaginary part of each answer will be zero.
* Draw the vectors zn, n=1, 2, ... 8 for z=cis(/4) and z=0.85 cis(/8)
* Find both geometrically and algebraically and confirm that both answers agree.
* Compute in your head, then confirm your answer.

If a complex number z lies on the unit circle it has length one so that z=sin(t)+i cos(t) for somes t (i.e. the radius r is 1. In that case the above theorem has another well-known corollary:

Corollary 1.2.15: DeMoivre's Formula
For any integer n and any real number t we have

(cos(t) + i sin(t))n = cos(nt) + i sin(nt)

Proof Proof

DeMoivre's Formula is quite something. It says that if you take a number on the unit circle (i.e. with lenght 1) with initial argument (angle) t and multiply it by itself, it simply rotates around the unit circle by that angle t. Each time you multiply the number by itself, the vector rotates another t degrees. In other words, in this case the power operator results in a simple rotation.

Powers of a vector z with |z|=1

Two interesting questions related to this rotation, taken from the field of Complex Dynamics, are: suppose z is a complex number with |z|=1. Then:

* find conditions for Arg(z) such that zn = z for some integer n. Such a point, incidentally, is called periodic of order n.
* if Arg(z)/ is irrational, what can you say about the sequence {z, z2, z3, z4, ...}? Does it, for example, converge? Such a sequence, incidentally, is called the orbit of z.

Polar coordinates can be especially helpful for finding roots, in particular for complex numbers of lenght 1.

Proposition 1.2.16: Finding Roots
For any positive integer n and any non-zero complex number a = r cis(t) the equation zn = a has exactly n distinct roots given by:

z =

where k = 0, 1, 2, ... n-1.

Proof Proof

In a previous example we found the two square roots of i, which turned out to be a fair amount of work. The above proposition allows us to dispense with such a question quickly. For example, the two solutions for

z2 = i = cis(/2)

are:

z1 = cis(/2/2) = cis(/4)
z2 = cis((/2 + 2)/2) = cis(5/4)

which you can quickly check using DeMoivre's Formula. Here is a geometric interpretation of this proposition: let's find, for example, the three third-roots of i, i.e. we want to find all solutions to z3 = i.

1: Draw the vector i 2: Divide angle by 3 for first root 3: Draw 3 equally spaced segments,
starting at the first root

Note, in particular, that the third root of i turned out to be -i, which indeed checks out:

(-i)(-i)(-i) = i*i*(-i) = (-1)*(-i) = i

This proposition is very satisfying: it says that at least every simple polynomial equation of degree n has n solutions. Later we will see that this is true in general: every n-th degree polynomial has n roots, no "if's" and "but's". This, in fact, is a sign of things to come: many theorems in complex analysis will turn out to be very "satisfying" and nicely structured, which is one reason that the study of complex analysis is a lot of fun (I think -:). But first a few more 'profane' examples.

Example 1.2.17: Finding roots geometrically

* Find the cube roots of 8 and i and draw them.
* Find all 4 fourth-roots of -1 and draw them geometrically
* Find all 5 fifth-roots of 1 and draw them geometrically
* Find both square-roots of 3i-2 by (a) using polar coordinates and (b) using rectangular coordinates and a formula from the previous section. Confirm that both methods result in the same answers.

Let's conclude this chapter with a result that illustrates what a 'nicely structured' theorem in complex analysis can look like.

Proposition 1.2.18: Roots of Unity
The n n-th roots of unity are given by wnk, where k = 0, 1, 2, ... n-1 and

wn = cis(2/n)

They form the vertices of a regular polygon and add up to zero, i.e. they satisfy the equation:

1 + wn + wn2 + ... + wnn-1 = 0

Proof Proof

This is neat: not only does the equation zn = 1 have exactly n solutions (one of which is, of course, z=1), but the solutions have this really pretty geometric structure of forming a regular polygon, which implies algebraically that they add up to zero as vectors. Here are, for example, the eight roots of z8=1:

You can see their regular structure. When you add them all as vectors, you indeed get the zero vector.

We will explore some more geometric structures in the next section.
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1.3. Basic Topology
ICA

Theorem 1.3.X: Triangle Inequality
For all complex numbers z and w we have:

|z + w| |z| + |w|

blah blah

Definition 1.3.x: Disks, Open, and Closed Sets
The set D(z0, r) is the open disk centered at z0 with radius r. In other words:

D(z0, r) = {z C : |z - z0| < r }

A set U is called open if for every z U there exists an r > 0 such that D(z, r) U. A set C is called closed if its complement comp(C) is open.

blah blah

Example 1.3.X: Simple sets inC
If c is a (complex) constant, then clearly the set {z = c} is a single point in the plane. Describe the sets

* { Re(z) = c}
* { Im(z) = c}
* { Arg(z) = c}
* { |z| = c}
* { z = }
* { r cis(t) = c}
* { z = r cis(t): r = c}
* { z = r cis(t): t = c}

blah blah

Example 1.3.X: Disks and friends
The set { |z - c| < r } is an open disk centered at c with radius r (explain). What are:

* { |z - c| > r }
* { |z - c| r }
* { r1 < |z - c| < r2 }
* { c1 < Arg(z) < c2 }
* { |z - 2z| < r}

circles, disks (inside/outside), annulus, lines, half-planes
****************
1.3. Basic Topology
ICA

Theorem 1.3.X: Triangle Inequality
For all complex numbers z and w we have:

|z + w| |z| + |w|

blah blah

Definition 1.3.x: Disks, Open, and Closed Sets
The set D(z0, r) is the open disk centered at z0 with radius r. In other words:

D(z0, r) = {z C : |z - z0| < r }

A set U is called open if for every z U there exists an r > 0 such that D(z, r) U. A set C is called closed if its complement comp(C) is open.

blah blah

Example 1.3.X: Simple sets inC
If c is a (complex) constant, then clearly the set {z = c} is a single point in the plane. Describe the sets

* { Re(z) = c}
* { Im(z) = c}
* { Arg(z) = c}
* { |z| = c}
* { z = }
* { r cis(t) = c}
* { z = r cis(t): r = c}
* { z = r cis(t): t = c}

blah blah

Example 1.3.X: Disks and friends
The set { |z - c| < r } is an open disk centered at c with radius r (explain). What are:

* { |z - c| > r }
* { |z - c| r }
* { r1 < |z - c| < r2 }
* { c1 < Arg(z) < c2 }
* { |z - 2z| < r}

circles, disks (inside/outside), annulus, lines, half-planes
**************

the history of real analysis

2010-12-30T05:08:00.000-08:00

9.1. Abel, Niels (1802-1829)
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Niels Abel was one of the innovators in the field of elliptic functions, discoverer of Abelian functions and one of the leaders in the use of rigor in mathematics. His work was so revolutionary that one mathematician stated: "He has left mathematicians something to keep them busy for five hundred years." However, his life did not mirror his mathematical success and his story is one of the most tragic in the sciences.

Niels Henrik Abel was born to a Lutheran minister in Finnoy, Norway, on August 5, 1802. His family, which moved to Gjerstad shortly after his birth, was poor, but somehow they managed to support seven children. His early education came at home and at the age of 13, he was admitted to the Cathedral School in Oslo. The school had recently lost most of its teachers to the new University of Oslo and was staffed with inexperienced and incompetent instructors. Under these circumstances, Abel's performance was rather unimpressive. However, this changed when his first mathematics professor was dismissed for beating a student to death while disciplining him. He was replaced by a young Bernt Michael Holmboe, an assistant to Christopher Hansteen at the university. Both of these men would become close friends and strong supporters of Abel. Holmboe saw Abel's ability in mathematics and encouraged him with books and problems. Soon, the student was teaching the teacher as Abel quickly outpaced his professor.

Abel's first main contribution to mathematics came before entering college. For hundreds of years, mathematicians had searched in vain to discover the general solution for the quintic equation a x5 + b x4 + c x3 + d x2 + e x + f = 0. Abel developed what he thought was the answer. Holmboe and Hansteen knew there was no one in Norway with the ability to understand if the answer was correct, so they sent the paper to the mathematician Ferdinand Degen in Denmark. Before receiving an answer, Abel discovered a mistake in his figures and questioned if there was an answer. Taking the tract that there was not, he eventually proved that an algebraic solution to the quintic equation was impossible. More important, however, Degen suggested that Abel take up the subject of elliptic integrals, which would become the focus of his work and the source of his fame.

Before entering the University of Oslo in 1821, Abel's father died, leaving his son to support his mother and six siblings. Unable to meet his financial needs, he relied on grants from the university, gifts from his math professors and tutoring positions to keep his family afloat. However, his mathematics flourished. After fulfilling the requirements for graduation in one year, he was left on his own to study. In 1823, he published his first important paper on definite integrals, which included the first ever solutions of an integral equation. He also produced another valuable work on the integration of functions. Both of these works would have brought him instant renown and a professorship -- if anyone would have read them. Unfortunately, the works were written in Norwegian while the leading mathematicians of Europe wrote in French and German. The papers were ignored.

Desperately trying to make a name for himself, Abel tried to get a royal grant to travel Europe, but was forced to wait two years to learn French and German. During this period, he was engaged to marry. Mathematically, he sent a French copy of his work on the quintic equation to Gauss, the time's leading mathematician who could have brought him instant fame. However, the German rejected it as "another of those monstrosities" without ever reading it. Finally, in 1825, he was given a grant to travel Europe for two years by the nearly bankrupt Norwegian government.

The trip would prove to be a near disaster. His first stop was Denmark to meet with Degen, only to discover that he had died. His next trip brought him to Berlin, the only successful leg of the journey. Here he met August Leopold Crelle, an influential engineer and amateur mathematician who was pondering the creation of a journal dedicated to new mathematical ideas. According to the story, Crelle originally thought Abel was a candidate for the trade school where he worked. After a long struggle to find a language both understood, Abel replied that he was interested in mathematics. They then discussed one of Crelle's papers and after praising it as interesting, Abel impolitely pointed out several mistakes. Fortunately, the German kept an open mind and listened. Despite not understanding most of what he was talking about, he understood he was in the presence of genius. Crelle started his journal with Abel's works as the features of his first few editions. The two would hold a close friendship and Crelle would use his influence to try to get Abel the recognition he deserved.

Together, the two planned to travel first to Gottingen to meet Gauss and then onto Paris, the hub of mathematics. Unfortunately, Crelle was unable to go and Abel went to Paris alone. He arrived just in time to catch almost every mathematician on vacation. When they returned, they were civil but uninterested in talking about anything but their own work. However, Abel did manage to have his "masterpiece," a paper on elliptic functions and integrals which included Abel's theorem, presented to the French Academy of Sciences. If the work was accepted, he would have been made. Unfortunately, the Academy picked Legendre and Cauchy as referees to judge it. The former, who was in his seventies, claimed that he could not read the handwriting and left all the work to the latter. The latter, who was much more interested in his own work and possibly just a bit jealous, brought the work home and promptly "misplaced" it. Not until 1830, a year after Abel's death, was the paper given the recognition it deserved and awarded the grand prize by the Academy. It was not published until 1841.

Abel returned to Norway in failure. Not only was he unable to get the recognition he deserved and the professorship he desperately needed, he was in debt and had contracted tuberculosis. To add insult to injury, he had been passed over to fill a vacancy in the mathematics department at the university. The position had instead been given to his friend Holmboe, who had accepted it only after they threatened to give the job to a foreigner if he did not agree to take it. Abel survived on grants and gifts from both the university and his friends.

However, his mathematics did not suffer. He produced several papers on the theory of equations, including sections that introduced a new class of equations, now known as the Abelian. Meanwhile, he gained a rival in his study of elliptic functions and integrals in Carl Jacobi. Spurred on by his competitor and also fearing his illness would soon finish him, Abel production on the subject increased at a blazing pace. His work laid the foundation of all further studies into the field. Finally, people began to notice. Legendre, who had failed to read his masterpiece, started a correspondence with both Abel and Jacobi, praising them as two of "the foremost analysts of our times." Slowly, mathematicians all across Europe were calling for a professorship for the Norwegian.

Unfortunately, this was all too late. In 1829, he suffered an attack from his tuberculosis that would slowly kill him. With his fiancee at his side, he lost his battle to the disease on April 6, 1829. Two days later, Crelle sent him notice that he had finally been able to secure a position for him at the University of Berlin.

This tragic life contributed much to the field of mathematics. His proof of no solution to the quintic equation, the solutions to definite integrals, Abel's theorem and Abelian functions and equations were all valuable additions to the science. His most important achievements were his discovery of elliptic functions and his use of rigor. On the first subject, mathematicians had been studying elliptic integrals with limited success over the past century. Abel inverted these integrals into elliptic functions which were much easier to manipulate. This is similar to inverting the complicated inverse trigonometric functions arcsin and arccos into the much simpler sin and cos. On the second subject, Abel quickly realized that much of the previous mathematical work was unproved. He saw it as his responsibility to fill these holes in mathematics and provide the proofs that had been left out. He most significant work was the first proof of the general binomial theorem, which had been stated by Newton and Euler.

For related information on Abel, see: Augustin Cauchy , Leonhard Euler , Abel's test for series.
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9.2. Archimedes (287? -212 B.C.)

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Archimedes is considered one of the three greatest mathematicians of all time along with Newton and Gauss. In his own time, he was known as "the wise one," "the master" and "the great geometer" and his works and inventions brought him fame that lasts to this very day. He was one of the last great Greek mathematicians.

Born in 287 B.C., in Syracuse, a Greek seaport colony in Sicily, Archimedes was the son of Phidias, an astronomer. Except for his studies at Euclid's school in Alexandria, he spent his entire life in his birthplace. Archimedes proved to be a master at mathematics and spent most of his time contemplating new problems to solve, becoming at times so involved in his work that he forgot to eat. Lacking the blackboards and paper of modern times, he used any available surface, from the dust on the ground to ashes from an extinguished fire, to draw his geometric figures. Never giving up an opportunity to ponder his work, after bathing and anointing himself with olive oil, he would trace figures in the oil on his own skin.

Much of Archimedes fame comes from his relationship with Hiero, the king of Syracuse, and Gelon, Hiero's son. The great geometer had a close friendship with and may have been related to the monarch. In any case, he seemed to make a hobby out of solving the king's most complicated problems to the utter amazement of the sovereign. At one time, the king ordered a gold crown and gave the goldsmith the exact amount of metal to make it. When Hiero received it, the crown had the correct weight but the monarch suspected that some silver had been used instead of the gold. Since he could not prove it, he brought the problem to Archimedes. One day while considering the question, "the wise one" entered his bathtub and recognized that the amount of water that overflowed the tub was proportional the amount of his body that was submerged. This observation is now known as Archimedes' Principle and gave him the means to solve the problem. He was so excited that he ran naked through the streets of Syracuse shouting "Eureka! eureka!" (I have found it!). The fraudulent goldsmith was brought to justice. Another time, Archimedes stated "Give me a place to stand on and I will move the earth." King Hiero, who was absolutely astonished by the statement, asked him to prove it. In the harbor was a ship that had proved impossible to launch even by the combined efforts of all the men of Syracuse. Archimedes, who had been examining the properties of levers and pulleys, built a machine that allowed him the single-handedly move the ship from a distance away. He also had many other inventions including the Archimedes' watering screw and a miniature planetarium.

Though he had many great inventions, Archimedes considered his purely theoretical work to be his true calling. His accomplishments are numerous. His approximation of between 3-1/2 and 3-10/71 was the most accurate of his time and he devised a new way to approximate square roots. Unhappy with the unwieldy Greek number system, he devised his own that could accommodate larger numbers more easily. He invented the entire field of hydrostatics with the discovery of the Archimedes' Principle. However, his greatest invention was integral calculus. To determine the area of sections bounded by geometric figures such as parabolas and ellipses, Archimedes broke the sections into an infinite number of rectangles and added the areas together. This is known as integration. He also anticipated the invention of differential calculus as he devised ways to approximate the slope of the tangent lines to his figures. In addition, he also made many other discoveries in geometry, mechanics and other fields.

The end of Archimedes life was anything but uneventful. King Hiero had been so impressed with his friend's inventions that he persuaded him to develop weapons to defend the city. These inventions would prove quite useful. In 212 B.C., Marcellus, a Roman general, decided to conquer Syracuse with a full frontal assault on both land and sea. The Roman legions were routed. Huge catapults hurled 500 pound boulders at the soldiers; large cranes with claws on the end lowered down on the enemy ships, lifted them in the air, and then threw them against the rocks; and systems of mirrors focused the sun rays to light enemy ships on fire. The Roman soldiers refused to continue the attack and fled at the mere sight of anything projecting from the walls of the city. Marcellus was forced to lay siege to the city, which fell after eight months. Archimedes was killed by a Roman soldier when the city was taken. The traditional story is that the mathematician was unaware of the taking of the city. While he was drawing figures in the dust, a Roman soldier stepped on them and demanded he come with him. Archimedes responded, "Don't disturb my circles!" The soldier was so enraged that he pulled out his sword and slew the great geometer. When Archimedes was buried, they placed on his tombstone the figure of a sphere inscribed inside a cylinder and the 2:3 ratio of the volumes between them, the solution to the problem he considered his greatest achievement.

For related information on Archimedes, see: Euclid , Archimedien property.
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9.3. Bernoulli, Johan (1667-1748)

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Johann Bernoulli was one of the pioneers in the field of calculus and helped apply the new tool to real problems. His life was one of the most controversial of any mathematician. He was a member of the world's most successful mathematical family, the Bernoullis.

Johann (also known as Johannes, Jean or John, depending on the translation) Bernoulli was born in Basel, Switzerland, on August 6, 1667. His family had originally been from Antwerp, Belgium, but had fled to avoid persecution by Catholics. After first settling down in Frankfurt, Johann's grandfather moved to Basel in 1622. Johann was the tenth son of a successful merchant and local official.

Originally, Johann's father had attempted to make a merchant out of his child but the son failed miserably as an apprentice. In 1683, he was given permission to enter the University of Basel, where his older brother Jacob, who was also a great mathematician, was already a professor. While pursuing a degree in medicine, Johann was tutored in mathematics by his older brother and soon developed a mastery of the new Leibnizian calculus. In 1694, he had committed himself to this new field and received his doctorate on a mathematical paper on muscular movement.

Unable to get the seat of mathematics at University of Basel because his brother held it, Johann accepted a position at the University of Groningen. In 1705, he returned to Basel after his brother's death to take Jacob's old position at the university. During his life he was awarded many honors including membership at the Academies of Science at Paris, Berlin, St. Petersburg and many others. He died in Basel on January 1, 1748.

Johann's life was always full of controversy. His first conflict occurred in 1691 on a visit to Paris. There he met France's leading mathematician L'Hospital, who persuaded him to teach him the calculus. For a fee, Bernoulli gladly complied and corresponded afterwards. However, the friendship died suddenly when L'Hospital published a textbook on differential calculus. Everything in the book was from Johann's notes and letters but the Frenchman claimed it as his own original work. Bernoulli was obviously enraged by the theft and ceased helping his freeloading friend.

Family troubles soon took over his attention. The relationship with his brother Jacob crumbled. After educating his brother in calculus, Jacob just could not accept his brother as his equal mathematically. After Johann solved a complicated problem before his brother, Jacob referred to him as "his pupil" and stated all of Johann achievements could be linked back to his teacher. Johann detested this belittling and the two engaged in public criticism of each other's work. At times, the letters became rather heated as insults seemed just as important as the mathematical problem being discussed. Curiously, however, the two seemed to develop ideas building off the other brother's work and in many cases it is likely the two at least collaborated on several occasions.

The family problems did not end there. In 1700, his son Daniel was born. Daniel, who would later become famous in physics for the founding of hydrodynamics, did not have a good relationship with his father. It appears Johann was jealous of his son. In one incident, Daniel was kicked out of the house for winning a prize that both of them had competed for. In an even more horrible, not to mention dishonest, display, Johann wanted credit for a discovery his son had made. In a flagrant case of plagiarism, he stole one of his son's papers, changed the name and date and claimed it as his own. These incidents were only two of many atrocious deeds committed by Johann against his son.

Finally, Bernoulli also became engaged in the mathematical dispute of his day. Having learned Leibnizian calculus, he began a correspondence with Leibniz himself. In 1713, with the dispute over who invented calculus between Leibniz and Newton in full swing, Bernoulli took the side of his friend. Because of his support, Leibniz's calculus gained preference in continental Europe.

When not engaged in some dispute or another, Johann had many discoveries in the field of calculus. He and his brother Jacob are credited with the beginnings of the calculus of variations, which was inspired over an argument on a geometric figure known as the brachistochrone. The two brothers also discovered many new applications for the new calculus which spread its popularity. Johann himself did important work on the study of the equation y = xx, discovered the Bernoulli series and made advances in theory of navigation and ship sailing. In addition, he is famous for his tremendous letter writing (over 2500 messages) and his tutoring of another great mathematician, Leonhard Euler.

As for the rest of the Bernoulli family, there were eight good mathematicians over three generations. The three most significant were Jacob, Johann and Daniel who have already been mentioned. However, also of importance were Johann's brother Nikolaus, Johann's sons Nikolaus and Johann II, and his grandsons Johann III and Jacob.

For related information on Bernoulli, see: Leonhard Euler , Bernoulli's inequality.
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9.4. Bolzano, Bernhard (1781-1848)

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Bernard Bolzano was a philosopher and mathematician whose contributions were not fully recognized until long after his death. He is especially important in the fields of logic, geometry and the theory of real numbers.

Bernardus Placidus Johann Nepomuk Bolzano was born in Prague, Bohemia (now part of the Czech Republic), on October 5, 1781. His father was an art dealer and both parents were very pious Christians. Coming from such a religious household, Bernard grew up with a high moral code and a belief in holding to his principles. It was this background that attracted him to the Church and the priestly life.

Bolzano entered the University of Prague in 1796, where he studied philosophy, mathematics and physics. After graduation, he joined the theology department at the university and was ordained a Catholic priest in 1804. Despite his dedication to the Church, he did not give up his mathematical interests and was at one time recommended for the chair of the mathematics department.

The year 1805 started a struggle that would dominate the rest of his life. In a political move, the Austrian-Hungarian Empire set up a chair in the philosophy of religion at each university. The empire was comprised of many different ethic groups that were prone to nationalistic movements for independence. Spurred by the "free thinking" of the recent French Revolution, these movements were becoming a serious problem to holding the empire together. The creation of the chair was part of a greater plan to support the Catholic Church. The authorities considered the Church to be conservative and hoped it would control the liberal thinking of the time. Bolzano was appointed to the position at the University of Prague. As far as the authorities were concerned, this was a bad idea. Bolzano, though a priest, was a "free thinker" himself and was not afraid to express his beliefs in Czech nationalism.

For the next 14 years, Bolzano taught at the university, lecturing mainly on ethics, social questions and the links between mathematics and philosophy. He was very popular with both the student body, who appreciated his straightforward expression of his beliefs, and his fellow professors, who recognized his intelligence. In 1818, he became Dean of the philosophy department. However, the Austro-Hungarian authorities became displeased with his liberal views. In 1819, he was suspended from his professorship, forbidden to publish and put under police surveillance. Bolzano refused to back down. However, despite the backing of the Church, he was unable to get his job back. In 1824, after refusing to sign an official "recantation" of his nationalistic views, he resigned his seat.

After leaving the university, he moved to the small village of Techobuz , where he stayed until 1842. He then returned to Prague to continue his philosophical and mathematical studies. He died on December 18, 1848. Bolzano had many new mathematical and logical ideas during his lifetime; however, because he was prohibited from publishing by the government, most of his writings existed only in manuscript. They were not published until 1962.

Being a philosopher, Bolzano attacked his mathematics philosophically. He believed that first clear concepts could only be obtained by using logic on basic principles and definitions. By finding the foundations, the user was guaranteed rigorous proof. Sometimes, this system gave him discoveries that were amazing. At other times, especially in mathematics, it gave him wrong answers.

Bolzano did contribute much to mathematics. His work attacked mainly three subjects: geometry, the theory of real numbers and logic. In geometry, he attempted to handle the problem of Euclid's parallel postulate. He found several problems in Euclid's reasoning but was unable to solve them because he lacked the proper mathematical tool of topology which had not yet been invented. He did establish definitions for basic geometric concepts and was the first person to state the Jordan curve theorem, that a simple closed curve divides a plane into two parts. In the theory of real numbers, he tried to find its foundation and reconcile infinite quantities, a concept that had stumped previous mathematicians. Although he did not succeed, he did come up with some important discoveries including the Bolzano-Weierstrass theorem, a modern definition of a continuous function and the non- differentiable Bolzano function. In addition, he recognized some of the paradoxical qualities of infinite sets, a breakthrough which he did not pursue and would be later stated by Cantor. In logic, his ideas were generally ignored until the modern day. Not just trying to place mathematics on a logical foundation, he went a step further and tried to place all the sciences and human thinking under its scope. In his works, he tackles basic ideas like abstract truth, human judgment and rules of science. Today, he is now considered one of the precursors to modern logic.

For related information on Bolzano, see: Georg Cantor , Euclid , Karl Weierstrass , Bolzano theorem , Bolzano-Weierstrass theorem.
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9.5. Cantor, Georg (1845-1918)
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Georg Cantor put forth the modern theory on infinite sets that revolutionized almost every mathematics field. However, his new ideas also created many dissenters and made him one of the most assailed mathematicians in history.

Georg Ferdinand Ludwig Philipp Cantor was born in St. Petersburg, Russia, on March 3, 1845. Georg's background was very diverse. His father was a Danish Jewish merchant that had converted to Protestantism while his mother was a Danish Roman Catholic. The family stayed in Russia for eleven years until the father's ailing health forced them to move to the more acceptable environment of Frankfurt, Germany, the country Georg would call home for the rest of his life.

All the Cantor children displayed an early artistic talent with Georg excelling in mathematics. His father, the eternal pragmatic, saw this gift and tried to push his son into the more profitable but less challenging field of engineering. In one of his letters, he pressed upon his son that his entire family and God Himself were expecting him to become a "shining star" as an engineer. Georg was not at all happy about this idea but he lacked the assertiveness to stand up to his father and relented. However, after several years of training, he became so fed up with the idea that he mustered up the courage to beg his father to become a mathematician. Finally, just before entering college, his father let Georg study mathematics. The son accepted his decision with the same submission that he had before, thanking his father for the fact that he would not "displease him."

In 1862, Georg Cantor entered the University of Zurich only to transfer the next year to the University of Berlin after his father's death. At Berlin he studied mathematics, philosophy and physics. There he studied under some of the greatest mathematicians of the day including Kronecker and Weierstrass. After receiving his doctorate in 1867, he was unable to find good employment and was forced to accept a position as an unpaid lecturer and later as an assistant professor at the backwater University of Halle. In 1874, he married and eventually had six children.

It was in that same year of 1874 that Cantor published his first paper on the theory of sets. While studying a problem in analysis, he had dug deeply into its "foundations," especially sets and infinite sets. What he found flabbergasted him so much that he wrote to a friend: "I see it but I don't believe it.". In a series of papers from 1874 to 1897, he was able to prove among other things that the set of integers had an equal number of members as the set of even numbers, squares, cubes, and roots to equations; that the number of points in a line segment is equal to the number of points in an infinite line, a plane and all mathematical space; and that the number of transcendental numbers, values such as and e that can never be the solution to any algebraic equation, were much larger than the number of integers. Interestingly, the Jesuits also used his theory to "prove" the existence of God and the Holy Trinity. However, Cantor, who was also an excellent theologian, quickly distanced himself away from such "proofs."

Before in mathematics, infinity had been a taboo subject. Previously, Gauss had stated that infinity should only be used as "a way of speaking" and not as a mathematical value. Most mathematicians followed his advice and stayed away. However, Cantor would not leave it alone. He considered infinite sets not as merely going on forever but as completed entities, that is having an actual though infinite number of members. He called these actual infinite numbers transfinite numbers. By considering the infinite sets with a transfinite number of members, Cantor was able to come up his amazing discoveries. For his work, he was promoted to full professorship in 1879.

However, his new ideas also gained him numerous enemies. Many mathematicians just would not accept his groundbreaking ideas that shattered their safe world of mathematics. One great mathematician, Henri Poincare expressed his disapproval, stating that Cantor's set theory would be considered by future generations as "a disease from which one has recovered." However, he was kinder than another critic, Leopold Kronecker. Kronecker was a firm believer that the only numbers were integers and that negatives, fractions, imaginary and especially irrational numbers had no business in mathematics. He simply could not handle "actual infinity." Using his prestige as a professor at the University of Berlin, he did all he could to suppress Cantor's ideas and ruin his life. Among other things, he delayed or suppressed completely Cantor's and his followers' publications, raged both written and verbal personal attacks against him, belittled his ideas in front of his students and blocked Cantor's life ambition of gaining a position at the prestigious University of Berlin.

Not all mathematicians were antagonistic to Cantor's ideas. Some greats such as Mittag-Leffler, Karl Weierstrass, and long-time friend Richard Dedekind supported his ideas and attacked Kronecker's actions. However, it was not enough. Like with his father before, Cantor simply could not handle it. Stuck in a third-rate institution, stripped of well-deserved recognition for his work and under constant attack by Kronecker, he suffered the first of many nervous breakdowns in 1884. The rest of his life was spent in and out of mental institutions and his work nearly ceased completely. Much too late for him to really enjoy it, his theory finally began to gain recognition by the turn of the century. In 1904, he was awarded a medal by the Royal Society of London and was made a member of both the London Mathematical Society and the Society of Sciences in Gottingen. He died in a mental institution on January 6, 1918.

Today, Cantor's work is widely accepted by the mathematical community. His theory on infinite sets reset the foundation of nearly every mathematical field and brought mathematics to its modern form. In addition, his work has helped to explain Zeno's paradoxes that plagued mathematics for 2500 years. However, his theory also has led to many new questions, especially about set theory, that should keep mathematicians busy for centuries.

For related information on Cantor, see: Karl Weierstrass , Zeno of Elea , Cantor function, Cantor set , Cantor-Bernstein theorem , Zeno's paradoxes.
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9.6. Cauchy, Augustin (1789-1857)

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Augustin Cauchy was the mathematician that set the foundation of rigor in modern analysis. A product of the revolutions in France during the eighteenth and nineteenth centuries, he provided the revolutionary ideas that set this branch of mathematics on its present course. He is also known as being one of the most prolific writers in the history of the science.

Augustin Louis Cauchy was born in Paris, France, on August 21, 1789, only a month after the storming of the Bastille. His father, a government official and staunch royalist, recognized the coming revolution and quickly moved his family to a country cottage in Arcueil. Having escaped the guillotine, the family was poor and the young boy was generally malnourished. For the rest of his life, this early poverty left the future mathematician in a state of ill-health. During his eleven year stay at the cottage, Augustin received a classical education and a strong disposition for the monarchy from his father, who wrote his own textbooks in verse, and strict Catholic religious training from his mother. This training would influence the rest of his life. His zealous political and religious beliefs would alienate this great mathematician from the majority of his countrymen.

In 1800, the Cauchy's family returned to Paris after the political situation stabilized. During this early period in his life, Augustin's talent was recognized by two great mathematicians, Marquis Laplace and Joseph Lagrange. Both, after seeing the young boy's work, encouraged him to continue in mathematics. As Lagrange once predicted, he would eventually outdo both of them. Cauchy's education, however, was in engineering. After attending the Ecole Polytechnique, a military engineering school taught by some of the country's greatest mathematicians, and the Ecole des Ponts et Chaussees, he took a position as an engineer in Napoleon's army at Cherbourg. Somehow during his busy schedule, he found time to dabble in mathematics. During his three years there, he produced several significant mathematical papers, including one on determinants that gave the term its modern meaning. All this mathematical output also accomplished to ruin his health and ended his career in military engineering.

Now with all his efforts focused on mathematics, Cauchy became a star on the mathematics scene. After a slew of achievements, he became a professor at the Polytechnique in 1816. Then at the unheard of age of 27, he was elected to the Academy of Sciences in Paris. To be selected for the Academy was one of the highest honors that could be given to a scientist. Unfortunately, his acceptance of the position was filled with controversy. At the time, Napoleon had just been overthrown and the Bourbons had been returned to the throne. The new king immediately went about removing the former emperor's supporters including Gaspard Monge, a member of the Academy. Despite his political views, Monge was one of the greatest mathematicians in France and his removal was considered an outrage. However, when Cauchy was offered his seat, he accepted without reserve. Being a staunch royalist, he saw nothing wrong with the removal of an enemy of the king and saw it as his duty to the monarchy to take the position. This action did not bode well with many of countrymen and made him many enemies. Nevertheless, the mathematician continued his work and somehow found time to be married two years later.

This would not be the last time his political views would get him into trouble. In 1830 after the overthrow of King Charles X, all members of the Academy were obligated to swear an oath of allegiance to the new king. Having already taken an oath to Charles, Cauchy refused. He was removed from his position and self-exiled to Switzerland without his family. There he became a professor at the University of Turin and planned to spend the rest of his life working on mathematics. That was not to be. Two years later, Charles X, now in exile, asked the professor to supervise the education of his heir Henri. Being a good royalist, he agreed and was joined with his family in Vienna. His new duties overwhelmed him and his mathematical work lessened to a trickle. He found his escape in 1838 when he returned to Paris. Before he left, the king had given him the impressive sounding but practically useless title of baron. He still refused to take the oath and constantly struggled to find and hold a position. Finally in 1848, the oath was abolished and he resumed his old posts. Recognizing his value to Academy, he was exempted when the oath was reestablished in 1852.

Augustin Cauchy died on May 23, 1857, after contracting a fever on a trip to the country to help restore his health. His last words were, "Men die but their works endure."

Cauchy's life was one as unusual and complicated as the times he lived in. Brought up as a devout Catholic in a time most Frenchmen were opposed to the Church, he suffered prejudice from many people. However, the discrimination did not discourage him from engaging in his life's favorite hobby, charity. When he was not involved in some math problem, he was often working on some new mercy mission for the less fortunate. On the other hand, he could be bigoted against those who did not hold his religious views. For example, part of the reason Cauchy delayed the publication of fellow mathematician Niels Abel's masterpiece was because the latter called him a "bigoted Catholic."

He also tended to be just as opinionated in matters of politics. A supporter of the monarchy, he came into direct conflict with the supporters of both the republic and Napoleon. Again, he was both discriminated against and prejudiced against others. On one hand, his life was put into constant turmoil because of the affair with the oath. On the other hand, he helped repress the mathematical work of Nicolas Galois because the latter was a radical republican. Certainly, Cauchy led a very complicated and intricate life.

Cauchy is famous in the field of mathematics for two main reasons: his numerous contributions to the science and his immense publishing. His works spanned every branch of mathematics and are simply too long to list. He is especially famous for his works with convergent series and rigor in analysis. Early in his career, Cauchy developed the criteria for determining if an infinite series is convergent or divergent. While attending a lecture on the subject, it is told that Laplace became panicked and rushed home. He had just finished his masterpiece that used infinite series as its backbone and desperately checked each one for convergence, which they did. Cauchy second great contribution was setting the groundwork for rigor in analysis and all of mathematics. Rigor is discovery of the logical foundations of a science. Over the previous centuries, mathematicians had tried in vain to discover what were the underlying principles of calculus and many had asserted that Newton's discovery was flawed. Cauchy took the first step toward unifying the science. First, he defined continuity and derivative in terms of the limit. Second, he gave the first good definition of the limit as : "When the values successively assigned to the same variable indefinitely approach a fixed value, so as to end by differing from it as little as desired, this fixed value is called the limit of all the others." Though this is not a mathematical definition, it is a good approximation of the idea, which would be further clarified by future mathematicians. Other important works include determinants, polygonal numbers, complex numbers and the theory of substitutions.

Cauchy is also famous for his writings. Simply put, he overwhelmed the mathematics world with the number and size of his works. All in all, his total output included 789 full length papers, one of the largest outputs ever. It was not uncommon for him to finish two such papers in one week. In addition, these works tended to be rather long, sometimes extending for over 300 pages. In fact, after submitting several large papers to be published in the weekly bulletin, the Academy was forced to limit submissions to four pages to save their small budget from Cauchy's pen. However, all this writing did get his work out into the public and spread his ideas. A lot of his fame can be assessed to the fact that he simply overwhelmed all his competitors on the bookshelves. Because of this fact, his name is prominent in almost any analysis textbook.

For related information on Cauchy, see: Niels Abel , Cauchy condensation test for series, Cauchy criteria, Cauchy criterion for series, Cauchy sequence, Cauchy sequence and convergence.

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9.7. De Morgan, Augustus (1806-1871)

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Augustus De Morgan was an important innovator in the field of logic. In addition, he had many contributions to the field of mathematics and the chronicling of the history of mathematics.

Augustus De Morgan was born in Mandura, India, on June 27, 1806. His father was a colonel in the Indian Army. His family soon moved to England where they lived first at Worcester and then at Taunton. His early education was in private schools where he learned Latin, Greek, Hebrew, mathematics and a dislike of exams. He entered Trinity College, Cambridge, in 1823 and graduated four years later.

After graduation, De Morgan reached the point of deciding what to do with the rest of his life. Dubious of competitive fellowships and master degrees, he refused to continue his education. Fearful of hypocrisy and religious bigotry, he also rejected his parents' wish of becoming a priest. After contemplating medicine and law, he finally decided to become a mathematician. In 1828, he was awarded the position of first Professor of Mathematics at University College in London.

His time at the university was far from quiet. In 1831, he resigned on principle after another professor was fired without explanation. He regained his job five years later when his replacement died in an accident. He would resign again in 1861. As a teacher, he was highly praised at making mathematics alive and interesting to his students. In addition, he wrote textbooks on numerous subjects in mathematics and logic.

He was married in 1837 to Sophia Frend, who would later write his biography. During his life, De Morgan was constantly involved in various activities. A member of the Astronomical Society and the Society for the Diffusion of Useful Knowledge, he founded the London Mathematical Society and was its first president. He wrote thousands of books and articles on mathematics, logic, philosophy and many other subjects. In addition, he assembled a large personal library of over 3000 books, a vast feat considering he was never wealthy. Unfortunately with all his work, he had little time for the rest of his life, but he was known as a kind and humorous individual.

Augustus De Morgan died on March 18, 1871, in London, England. His library was later donated to the London University library.

De Morgan contributed many accomplishments to the field of mathematics on many different subjects. He was the first person to define and name "mathematical induction" and developed De Morgan's rule to determine the convergence of a mathematical series. His definition of a limit was the first attempt to define the idea in precise mathematical terms. In addition, he also devised a decimal coinage system, an almanac of all full moons from 2000 B.C. to 2000 A.D. and a theory on the probability of life events which is used by insurance companies.

However, he biggest contribution was in the field of logic. His most important work, Formal Logic, included the concept of the quantification of the predicate, an idea that solved problems that were impossible under the classic Aristotelian logic. For example, the following is only workable using De Morgan's method:

* In a particular group of people,
o most people have shirts
o most people have shoes
o therefore, some people have both shirts and shoes.

He devised the idea around the same time as a Scottish philosopher, Sir William Hamilton, who accused him of stealing his ideas. However, it is clear that De Morgan's work is clearer, more developed and all around superior to Hamilton's version. With no evidence to back him up, the Scot's charge of plagiarism has been dismissed as sour grapes. De Morgan's other works include a system of notations for symbolic logic that could denote converses and contradictions and the famous De Morgan laws and .

Of special note to this author, De Morgan was also deeply interested in the history of mathematics. He wrote biographies of Newton and Halley and produced a dictionary of all the important mathematicians of the seventeenth century. In 1847, he published the book Arithmetical Books, in which he describes the work of over fifteen hundred mathematicians and discusses subjects such as the history of the length of a foot. This work is also considered the first scientific bibliography. De Morgan felt that it was important for students to know the history of mathematics to understanding the development of the field.

For related information on De Morgan, see: De Morgan's laws.

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9.8. Euclid (330?-275? B.C.)
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Euclid is one of the most influential and best read mathematician of all time. His prize work, Elements, was the textbook of elementary geometry and logic up to the early twentieth century. For his work in the field, he is known as the father of geometry and is considered one of the great Greek mathematicians.

Very little is known about the life of Euclid. Both the dates and places of his birth and death are unknown. It is believed that he was educated at Plato's academy in Athens and stayed there until he was invited by Ptolemy I to teach at his newly founded university in Alexandria. There, Euclid founded the school of mathematics and remained there for the rest of his life. As a teacher, he was probably one of the mentors to Archimedes.

Personally, all accounts of Euclid describe him as a kind, fair, patient man who quickly helped and praised the works of others. However, this did not stop him from engaging in sarcasm. One story relates that one of his students complained that he had no use for any of the mathematics he was learning. Euclid quickly called to his slave to give the boy a coin because "he must make gain out of what he learns." Another story relates that Ptolemy asked the mathematician if there was some easier way to learn geometry than by learning all the theorems. Euclid replied, "There is no royal road to geometry" and sent the king to study.

Euclid's fame comes from his writings, especially his masterpiece Elements. This 13 volume work is a compilation of Greek mathematics and geometry. It is unknown how much if any of the work included in Elements is Euclid's original work; many of the theorems found can be traced to previous thinkers including Euxodus, Thales, Hippocrates and Pythagoras. However, the format of Elements belongs to him alone. Each volume lists a number of definitions and postulates followed by theorems, which are followed by proofs using those definitions and postulates. Every statement was proven, no matter how obvious. Euclid chose his postulates carefully, picking only the most basic and self-evident propositions as the basis of his work. Before, rival schools each had a different set of postulates, some of which were very questionable. This format helped standardize Greek mathematics. As for the subject matter, it ran the gamut of ancient thought. The subjects include: the transitive property, the Pythagorean theorem, algebraic identities, circles, tangents, plane geometry, the theory of proportions, prime numbers, perfect numbers, properties of positive integers, irrational numbers, 3-D figures, inscribed and circumscribed figures, LCD, GCM and the construction of regular solids. Especially noteworthy subjects include the method of exhaustion, which would be used by Archimedes in the invention of integral calculus, and the proof that the set of all prime numbers is infinite.

Elements was translated into both Latin and Arabic and is the earliest similar work to survive, basically because it is far superior to anything previous. The first printed copy came out in 1482 and was the geometry textbook and logic primer by the 1700s. During this period Euclid was highly respected as a mathematician and Elements was considered one of the greatest mathematical works of all time. The publication was used in schools up to 1903. Euclid also wrote many other works including Data, On Division, Phaenomena, Optics and the lost books Conics and Porisms.

Today, Euclid has lost much of the godlike status he once held. In his time, many of his peers attacked him for being too thorough and including self-evident proofs, such as one side of a triangle cannot be longer than the sum of the other two sides. Today, most mathematicians attack Euclid for the exact opposite reason that he was not thorough enough. In Elements, there are missing areas which were forced to be filled in by following mathematicians. In addition, several errors and questionable ideas have been found. The most glaring one deals with his fifth postulate, also known as the parallel postulate. The proposition states that for a straight line and a point not on the line, there is exactly one line that passes through the point parallel to the original line. Euclid was unable to prove this statement and needing it for his proofs, so he assumed it as true. Future mathematicians could not accept such a statement was unproveable and spent centuries looking for an answer. Only with the onset of non- Euclidean geometry, that replaces the statement with postulates that assume different numbers of parallel lines, has the statement been generally accepted as necessary. However, despite these problems, Euclid holds the distinction of being one of the first persons to attempt to standardize mathematics and set it upon a foundation of proofs. His work acted as a springboard for future generations.

For related information on Euclid, see: Archimedes, Euclid's theorem.
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9.9. Euler, Leonard (1707-1783)

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Leonhard Euler was one of top mathematicians of the eighteenth century and the greatest mathematician to come out of Switzerland. He made numerous contributions to almost every mathematics field and was the most prolific mathematics writer of all time. It was said that "Euler calculated without apparent effort, as men breathe...." He was dubbed "Analysis Incarnate" by his peers for his incredible ability.

Leonhard Euler was born in Basel, Switzerland, on April 15, 1707. His father, a Calvinist pastor and former mathematician, planned the life of a clergyman for his son and originally Leonhard followed that path. He graduated from the University of Basel in 1724 where he studied theology and Hebrew. During his time at the school, however, he was privately tutored in mathematics by Johann Bernoulli. Johann was so impressed by his pupil's ability that he convinced Euler's father to allow Leonhard to become a mathematician.

Euler took up a position at the Academy of Sciences in St. Petersburg, Russia, in 1727 and became the professor of mathematics six years later. During his stay, he was married and would over his lifetime have thirteen children, five of which would survive to adulthood. While in Russia, he lost sight in one eye after working day and night for three days to solve a problem. The question, which was a public contest, took all the other mathematicians involved months to figure out. He also discovered that the Czar's government was far from democratic as he was followed by secret police. He looked for a way out.

He found it in 1741, when he moved his family to Berlin to take over as director of mathematics at the Academy of Sciences under Frederick the Great. While in Prussia, his home was destroyed by invading Russian armies, but he was held in such high esteem by both countries that he was compensated for more than he lost. He also frustrated Frederick's mother to no end by refusing to engage in conversation. When she finally asked him for a reason, he responded: "Madam, it is because I have just come from a country where every person who speaks in hanged." He also could not handle the intrigues and feuds that plagued the Academy. When the previous president died, Euler should have been the obvious successor except for the fact that Frederick disliked him. The monarch asked D'Alembert, a French mathematician, to take the position. D'Alembert, who saw the injustice, refused on the basis that no one could be placed above Euler. However, it became clear it was time for Leonhard to find a new home.

Meanwhile, Russia had come under the rule of the more liberal Catherine the Great. In 1766, he returned to St. Petersburg and became the director of the Academy. Soon afterwards, he went completely blind but continued his mathematical work by dictating to a secretary. His house burned down in 1771 and his life was saved only by the heroic efforts of a servant to carry him out of the flames. He died of a stroke on September 7, 1783. Appropriately to this simple mathematician, his final words were simply "I die."

Euler was especially famous from his writings. Simply put, he produced more scholarly work on mathematics than anyone. It was said that he could produce an entire new mathematical paper in about thirty minutes and had huge piles of his works lying on his desk. Even more impressive, Euler contemplated new problems not in quiet privacy but in the presence of his young children. It was not uncommon to find "Analysis Incarnate" ruminating over a new subject with a child on his lap.

Though Euler is best remembered for his contributions to mathematics, he was involved in some extent in almost all fields. Especially close to his heart was philosophy. While in Berlin, he would constantly get involved in philosophical debates, especially with Voltaire. Unfortunately, Euler's philosophical ability was limited and he often blundered to the amusement of all involved. However, when he returned to Russia, he got his revenge. Catherine the Great had invited to her court the famous French philosopher Diderot, who to the chagrin of the czarina, attempted to convert her subjects to atheism. She asked Euler to quiet him. One day in the court, the French philosopher, who had no mathematical knowledge, was informed that someone had a mathematical proof of the existence of God. He asked to hear it. Euler then stepped forward and stated: "Sir,,hence God exists; reply!" Diderot had no idea what Euler was talking about. However, he did understand the chorus of laughter that followed and soon after returned to France.

Euler's contributions to every mathematical field that existed at the time. He standardized modern mathematics notation when he used symbols such as f(x), e, , i and in his textbooks. He was the first person to represent trigonometric values as ratios and prove that e is an irrational number. His invention of the calculus of variations led to the general method to solve max and min value problems. In physics, he developed the general equations for hydrodynamics and for motion. He was also one of the first people to recognize that infinite series had to be convergent to be used safely. Possibly his most impressive work was his approximation of the three-body problem of the sun, earth and moon, which he solved while completely blind and performing all the computations in his head. Among his other endeavors were proofs of Fermat's final theorem for cubes and quads, the use of calculus in mechanics and the computation of logs for negative and imaginary numbers.

For related information on Euler, see: Johann Bernoulli , e (Euler's numbers), Euler's sequence, Euler's series.
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9.10. Peano, Guiseppe (1858-1932)

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Giuseppe Peano was one of the pioneers in mathematical logic and axiomatization of mathematics. He also had many important discoveries in the field of analysis and was one of the leading authorities on auxiliary languages.

Giuseppe Peano was born to a poor farming family in Spinetta, Italy, on August 27, 1858. Being born in such a poor village, he and his brother were forced to walk to the neighboring town of Cueno to attend school. However, this handicap did not stop him from excelling in his studies and he was sent to Turin with his uncle to finish his primary schooling. In 1876, he enrolled at the University of Turin to study engineering but later decided on mathematics. The university would be his home for the rest of his life. After graduating, he became a University Assistant in 1880, professor at the Royal Military Academy in 1886, extraordinary professor in 1890 and ordinary professor in 1895. In 1887, he was married but had no children.

For the first part of his life, mathematics dominated Peano's life. During this period, almost all of his mathematical discoveries were made. He proved that y' = f(x,y) on the sole condition that f is continuous and set the minimum conditions for second order partials of an equation to be equal. He was the first person to develop a space-filling curve, a one-dimensional curve that fills all the points in a two-dimensional space. He also developed independently a method of successive approximations for the solution of differential equations and developed the idea of the cluster point of a function. He was also an early supporter of the use of recursive functions and vectors.

Peano's greatest contributions, however, were in the fields of axiomatization of mathematics and mathematical logic . Axiomatization of mathematics is the development of the postulates (axioms) and definitions that are the basis of the mathematical system. His most famous set, known as Peano's postulates, put forth the axioms of natural numbers. In addition, he also developed a related set for geometry.

Mathematical logic is the use of symbols instead of words to write mathematical statements. The idea had be raised over a century earlier by Leibniz who had suggested a universal language. By using symbols, the equations are simplified and made easily understood to anyone, no matter what language they speak. For example, Peano introduced the symbols and to represent "belongs to the set of" and "there exists" respectively. Mathematical logic quickly became the focus of his work. In 1889, Peano published his first version of a system of mathematical logic in his work Arithmetics principia, which included his famous axioms of natural numbers. Two years later, he established a journal, the Rivista, in which he proposed the symbolizing of all mathematical propositions into his system. The project, which became known as the Formulario, became his focus for the next fifteen years. When it was finished in 1908, the book contained over 4200 symbolized formulas and theorems with proofs in only 516 pages. His work did not go unnoticed. He was elected to the Academy of Sciences in Turin in 1891 and was a speaker at several International Congress of Mathematics. In addition, he was honored by the Italian government with several knighthoods, including the Order of the Crown of Italy.

It was at this time that Peano became interested in the auxiliary language movement. This movement was one of the first efforts in attempting to develop a universal language understood by everyone, especially for the scientific community. Intrigued by the idea, he became active at the turn of the century and in 1908 he was elected director of the Academia pro Interlingua, a congress for the development of an auxiliary language. His proposal was called Latino sine flexione, a version of Latin stripped of all grammar. Because of his work in this field, his mathematical work ended almost completely.

It was at this time that Peano's career declined. Interlingua, as his universal language would eventually be named, failed to gain widespread acceptance because of the international popularity of English. The Formulario, though impressive in its accomplishment, found little interest in the mathematical community. This was partially because of the confusion over the symbols and partially because it was published in Latino sine flexione, which nobody really understood. Meanwhile, back at the university, opposition was mounting. Professors objected to his insistence that he teach all his students mathematical logic and the fact that he never gave examinations. Meanwhile, students resented having to learn "the symbols" when they never would use them in real life and labeled pupils that liked his classes as suffering from "Peanitis." In 1901, he was forced to resign from the military academy and was stripped of teaching all the engineering students at the university. Even after his death, there remained great opposition to the mathematician.

Despite all this, Peano was happy for the end of his life. He was active in organizations for primary and secondary education, developed a perpetual calendar and became politically involved in a cotton workers' strike, hosting a party for the strikers at his country villa. He died of a heart attack in Turin on April 20, 1932.

For related information on Peano, see: Peano axioms.
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9.11. Weierstrass, Karl (1815-1897)
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Karl Weierstrass was one of the leaders in rigor in analysis and was known as the "father of modern analysis." In addition, he is considered one of the greatest mathematics teachers of all-time.

Karl Wilhelm Theodor Weierstrass was born October 31, 1815, in Ostenfelde, Westphalia, Germany. He was the first of four children of a customs official under Napoleon. His father would later enter the Prussian taxation service and would move his family often. The father himself was the stereotypical overbearing father who attempted to dominate the lives of all his children. Karl was on the receiving end of lectures well past the age of forty. Curiously, none of his children ever married. Despite the many schools and inept parenting, Karl still managed to excel in school and held down a part-time job as a bookkeeper in his spare time.

Unfortunately, this was the start of his trouble. Karl's father saw that his son was intelligent because of all the prizes he brought home. He also deduced that his son must be good bookkeeper. From these two facts, he figured his son could be a great accountant and the best accounting jobs were in the government. Therefore, his son would study commerce and law to prepare for a government career. The only problem was that his son was much more interested in mathematics. However, Karl bowed to his father's wishes and entered the University of Bonn in 1834 to pursue a career as an accountant. That was about as far as he followed his father's advice. Sick of his lectures, he simply stopped attending them and spent most of his time fencing (with swords), drinking beer, partying and reading mathematics books. After four years, he returned home without a degree in anything.

His father was not pleased. However, a friend of the father suggested that Karl enroll at the Theological and Philosophical Academy at Munster, where he could obtain a secondary schools teaching degree. In 1839, Karl was admitted after promising to the school's authorities he would not follow in his old ways. There he met his mathematical inspiration Christof Gudermann. After scaring away all his students except for Karl after the first lecture, Gudermann was able to give personal training to his prize pupil. Especially important, the professor introduced to Weierstrass the idea of the power series, which he would use as the basis of his work. Throughout the rest of his life, Karl always expressed his gratitude to his teacher. As part of his written examination, he presented a revolutionary essay on elliptic functions. The paper, which was the starting point of Weierstrass's discoveries, was so well received by Gudermann that he compared his student to "discoverers who were crowned with glory." However, Karl did not see the praise or publish the paper. He received his teacher's certificate in 1841.

For the next fifteen years, Weierstrass was employed as a secondary school teacher, teaching subjects such as mathematics, physics, German, geography, gym and penmanship(?). He spent most of his free time split between socializing at the local beer hall and working on his mathematics. For this entire period, he was practically cut off from the mathematical community. With no fellow mathematicians to talk to, no mathematical libraries to visit and no money to sustain an exchange of ideas through the mail system, he was basically left alone to completely explore his ideas. One story relates one night, he was so caught up on working on a problem that he worked through the morning and refused to teach his class until he was finished. However, he still had time to have some fun. In 1848, revolution was in the air and the local censor had his hands full keeping the seditious stuff out of the paper. Since the censor hated poetry, he gave all the poems and songs to Weierstrass to examine. He, of course, made sure that all the most extreme items were published behind the censor's back. Not all was good though. In addition to being bored with his job, he began to suffer seizures that would plague him for over a decade.

However, Karl's life was about to change. In 1854, he sent a paper on Abelian functions to the famous Crelle mathematical journal for publication. He had published only once before in the school journal but no one had noticed it (or should have). When this paper was published in 1854, it startled the mathematical community who could only wonder how this genius had been stuck teaching children. Almost immediately, he received an honorary doctorate from the University of Konigsberg and the mathematical community scrambled to find him a proper position. In 1856, he accepted an associate professorship at the University of Berlin.

Weierstrass was understandably overwhelmed by the entire chain of events. Overloaded by the new responsibilities of being a professor, he suffered a nervous breakdown and collapsed in the middle of a lecture in 1861. He would not return for two years. After his return, he never trusted himself to write at the blackboard again, delegating a student to write for him. Despite this, in 1864 he was promoted to full professor, a position he would hold for the rest of his life.

As a professor, Weierstrass slowly gained the reputation as a master lecturer. Originally, despite the fact that his lectures were generally disorganized, students flocked to his classes because he was the only person offering his advanced subjects. As the years progressed, he acquired the famous teaching skills that have given him the label as one of the all-time great lecturers. In addition, he was also well known for his constant availability, his productive leads for future study and his insistence on paying for his students at the local tavern. Eventually, his classes swelled with over 250 pupils from around the world. One of his students, Sonja Kovalevsky, became one of his closest friends and the two met or corresponded for the rest of their lives. She was the closest any woman ever came to claiming the heart of this lifelong bachelor. He was fundamental in her attaining the position of professor at the University of Stockholm in 1883.

Unfortunately, the end of Weierstrass' life was filled with tragedy. First, his friend and colleague, Leopold Kronecker, publicly attacked him for his support of the revolutionary theories of Georg Cantor and generally made his life miserable until Kronecker's death in 1891. Second, after his critic's death, Sonja died at the age of 41. The death mentally crushed Karl and he burned everything he owned that reminded him of her. He spent the last few years of his life confined to a wheelchair and died from pneumonia on February 19, 1897.

Weierstrass is famous in mathematics for numerous accomplishments. He was the first person to create a continuous function that is not differentiable at any point. He developed a general theory of Abelian integrals and Abelian functions, which he considered his lifetime work. He used the power series as the basis of functions, an idea that was key in the development of much of mathematical physics, and created the sequential definition of irrational numbers based on convergent series. However, he is most famous for his insistence on rigor in all his works, especially analysis. Karl demanded that mathematics be based on clear and correct proofs. For this reason, he rarely published because he would not release his work until he was sure it was on a firm mathematical foundation. This was generally a monumental task considering that he started most of his work basically from scratch.

Ironically, he understood his own limitations. He once said: "It is true that a mathematician who is not also something of a poet will never be a perfect mathematician." He understood that mathematical perfection, just like poetic perfection, is impossible, though there is nothing wrong in trying.

For related information on Weierstrass, see: Niels Abel, Georg Cantor, Bolzano-Weierstrass theorem.
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9.12. Zeno of Elea (495?-435? B.C.)
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Zeno of Elea was the first great doubter in mathematics. His paradoxes stumped mathematicians for millennia and provided enough aggravation to lead to numerous discoveries in the attempt to solve them.

Zeno was born in the Greek colony of Elea in southern Italy around 495 B.C. Very little is known about him. He was a student of the philosopher Parmenides and accompanied his teacher on a trip to Athens in 449 B.C. There he met a young Socrates and made enough of an impression to be included as a character in one of Plato's books Parmenides. On his return to Elea he became active in politics and eventually was arrested for taking part in a plot against the city's tyrant Nearchus. For his role in the conspiracy, he was tortured to death. Many stories have arisen about his interrogation. One anecdote claims that when his captors tried to force him to reveal the other conspirators, he named the tyrant's friends. Other stories state that he bit off his tongue and spit it at the tyrant or that he bit off the Nearchus' ear or nose.

Zeno was a philosopher and logician, not a mathematician. He is credited by Aristotle with the invention of the dialectic, a form of debate in which one arguer supports a premise while another one attempts to reduce the idea to nonsense. This style relied heavily on the process of reductio ad absurdum, which is the reduction of an idea to absurdity by finding an inherent contradiction. Zeno wrote only one known work, Epicheiremata, in which he attacks the opponents of his teacher Parmenides.

Zeno's greatest fame is from his paradoxes. With only 200 words surviving from his only book, the only records we have of his brainchilds are from secondary sources, mainly Aristotle. Originally, there were about forty but only eight have survived. The purpose of the arguments was a defense of his teacher's ideas. Parmenides believed that reality was one, immutable and unchanging. Motion, change, time and plurality were all mere illusions. This, of course, attracted many critics. Zeno's paradoxes attempted to show that holding the opposite position, that reality was many, was contradictory and absurd. Therefore, "the one" must be the correct philosophy. Curiously, using Zeno's methods, his own position can also be shown to be contradictory as well.

The four most famous paradoxes are the Dichotomy, the Achilles, the Arrow, and the Stadium.

1. The Dichotomy: Motion cannot exist because before that which is in motion can reach its destination, it must reach the midpoint of its course, but before it can reach the middle, it must reach the quarterpoint, but before it reaches the quarterpoint, it first must reach the eigthpoint, etc. Hence, motion can never start.
2. The Achilles: The running Achilles can never catch a crawling tortoise ahead of him because he must first reach where the tortoise started. However, when he reaches there, the tortoise has moved ahead, and Achilles must now run to the new position, which by the time he reaches the tortoise has moved ahead, etc. Hence the tortoise will always be ahead.
3. The Arrow: Time is made up of instants, which are the smallest measure and indivisible. An arrow is either in motion or at rest. An arrow cannot move, because for motion to occur, the arrow would have to be in one position at the start of an instant and at another at the end of the instant. However, this means that the instant is divisible which is impossible because by definition, instants are indivisible. Hence, the arrow is always at rest.
4. The Stadium: Half the time is equal to twice the time. Take the three rows below. They start at the first position. Row A stays stationary while rows B & C move at equal speeds in opposite directions. When they have reached the second position, each B has passed twice as many C's as A's. Thus it takes row B twice as long to pass row A as it does to pass row C. However, the time for rows B & C to reach the position of row A is the same. So half the time is equal to twice the time.

Though all four arguments seem illogical, not to mention confusing, they are not that simple to explain away and lead to some very serious problems for mathematics. To the Greek mathematicians, who had no real concept of convergence or infinity, these reasonings were incomprehensible. Aristotle discarded them as "fallacies" without really showing why and Zeno's paradoxes were hidden away in the mathematical closet for the next 2500 years. For that time, they were reduced mainly as novelties of philosophy. However, they were revived mathematically in the twentieth century by the efforts of people like Bertrand Russell and Lewis Carroll. Today, armed with the tools of converging series and Cantor's theories on infinite sets, these paradoxes can be explained to some satisfaction. However, even today the debate continues on the validity of both the paradoxes and the rationalizations.

For related information of Zeno, see: Georg Cantor , Zeno's paradox.