Electrical Networks

There are many applications of Linear Algebra in electrical engineering. In this project we shall study the application to electrical circuits, namely the linearity relationship between voltage and current. We shall analyze complicated circuits and learn how linear algebra is applied. The questions will be in the same spirit as the problems mentioned in the course text book.

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Theory of Games

This project involves studying a general game in which two competing players choose separate strategies to reach opposing objectives. Using matrix techniques, the optimal strategy of each player is found for a few special cases.

The problems will involve a two player zero sum game and calculating expected payoffs, optimal strategies and the value of the game. Guidelines of the what the report should contain and the grading scheme will be given as the semester progresses.


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Equilibrium Temperature Distributions

The flow of a temperature through a 2-dimensional object like a thin plate is governed by a partial differential equation called heat equation. If there are not external sources or sinks of heat then after a while the temperature will come into equilibrium and will no longer change in time. In this case the equilibrium temperature distribution satisfies a related partial differential equation called Laplace's equation. If one knows the temperature at the boundary of the plate, then the problem of calculating the equilibrium temperature distribution inside the plate is known as the Dirichlet problem. The temperature distribution function will be what is known as a harmonic function, and satisfies a particularly nice property called the mean value property which says its value at any point is the average of its values at nearby points. Physically this must be so for otherwise heat would flow to or from the central point if the average temperature nearby is warmer or closer.

Linear algebra can be used to approximately calculate the equilibrium temperature distribution in such a problem by replacing the partial derivatives by what is known as a finite difference approximation. There is a discrete version of the mean value property which reduces the Dirichlet problem to solving a system of linear equations for the values of the temperature at various mesh points in the plate. Often it is useful to calculate an approximation to the solution of the linear system by a procedure called Jacobi iteration, which can be used because of the special form of systems of equations which arise.


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Linear Programming

The study of linear programming theory has expanded greatly since the pioneer work of George Dantzig in the late nineteen-forties. Today, linear programming is applied to a wide variety of problems industry1 and science. Linear programming is concerned with the maximization or minimization of certain quantities depending on variables which themselves are subject to various linear inequalities, called the constraints. Such constraints arise in many different situations and consequently linear programming is applied to a wide variety of problems in industry and science.

In this project we learn the basic concepts of linear programming and how to solve simple linear programming problems which arise in different applications.


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Applications in Genetics

In this project, we shall examine the inheritance of traits in animals or plants. The inherited trait under consideration is assumed to be governed by a set of 2 genes, which we designate by the letters A and a. Under auto-somal inheritance each individual in the population, of either sex, possesses two of these genes: the possible pairings being designated by AA, Aa and aa. This pair of genes is called the individual's genotype, and it determines how the trait controlled by the genes is manifested in the individual. For example, in snapdragons a set of two genes determines the color of the flower. Genotype AA produces red flowers, genotype Aa produces pink flowers, and genotype aa produces white flowers. In humans, eye coloration is controlled through auto-somal inheritance. Genotypes AA and Aa have brown eyes, and genotype aa has blue eyes. In this case we say gene A dominates gene a, or that gene a is recessive to gene A, since genotype Aa has the same outward trait as genotype AA. In addition, as this project moves along we discuss X-linked inheritance. In this type of inheritance the male of the species possesses only one of the two possible genes (A or a), and the female possesses a pair of the two genes ( AA, Aa,aa). In humans, color blindness, hereditary baldness, hemophilia, and muscular dystrophy, to name a few, are traits controlled by X-linked inheritance.

We shall construct probability-matrix models which give the probable genotypes of the offspring in terms of the genotypes of the parents, and use these models to follow the genotype distribution of a population through successive generations.