The Department does not offer a doctorate in statistics. However, students may choose statistics or mathematical statistics as a concentration in the doctor of philosophy programs in mathematics and applied mathematics. The Department is a part of the interdisciplinary program in Analytical Finance. To begin graduate work in mathematics a student must present evidence of adequate undergraduate preparation.
The undergraduate program should have included a year of advanced calculus, a semester of linear algebra, and a semester of abstract algebra. The master's program requires 30 credit hours of graduate courses with at least 18 hours at the level. With the permission of the chairperson, up to six hours of these courses can be replaced by a thesis.
All students in the master's program must also pass a comprehensive examination. The choice of courses must be approved by the graduate advisor, and up to six hours of coursework may be replaced with a thesis. All students in the program must also pass a comprehensive examination. The plan of work toward the doctor of philosophy degree will include a comprehensive examination, a qualifying examination, and an advanced topic examination. A language exam may be required at the discretion of the thesis committee. The content of the advanced topic examination is determined by a department committee.
A general examination, the doctoral dissertation and its defense complete the work for the Ph. Each candidate's plan of work must be approved by a special committee of the department. A language examination may be required at the discretion of the thesis committee. Intensive review of fundamental concepts in mathematics utilized in calculus, including functions and graphs, exponentials and logarithms, and trigonometry. This course is for students who need to take MATH 51 or 21, but who require remediation in precalculus.
The credits for this course do not count toward graduation, but do count toward GPA and current credit count. Consent of department required. Meaning, content, and methods of mathematical thought illustrated by topics that may be chosen from number theory, abstract algebra, combinatorics, finite or nonEuclidean geometries, game theory, mathematical logic, set theory, topology.
Systems of linear equations, matrices, introduction to linear programming. Sets, counting methods, probability, random variables, introduction to Markov chains. A first course in the basic concepts and methods of statistics with illustrations from the social, behavioral, and biological sciences. Descriptive statistics; frequency distributions, mean and standard deviation, two-way tables, correlation and regression; random sampling, rules of probability, probability distributions and parameters, parameter estimation, confidence intervals, hypothesis testing, statistical significance.
Functions and graphs; limits and continuity; derivative, differential, and applications; indefinite and definite integrals; trigonometric, logarithmic, exponential, and hyperbolic functions. Applications of integration; techniques of integration; separable differential equations; infinite sequences and series; Taylor's Theorem and other approximations; curves and vectors in the plane. Vectors in space; partial derivatives; Lagrange multipliers; multiple integrals; vector analysis; line integrals; Green's Theorem, Gauss's Theorem.
Matrices, vectors, vector spaces and mathematical systems, special kinds of matrices, elementary matrix transformations, systems of linear equations, convex sets, introduction to linear programming. The derivative and applications to extrema, approximation, and related rates. Exponential and logarithm functions, growth and decay. Trigonometric functions and related derivatives and integrals.
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Partial derivatives and extrema. Multiple integrals and applications. Meets three hours per week. Limits and continuity; exponential, logarithmic and trigonometric functions; derivatives; extrema; approximations; indefinite and definite integrals. Applications with emphasis on business and economics. A first course in logical theory, introducing the notions of logical consequence and proof, as well as related concepts such as consistency and contigency. Formal systems taught may include: Elements of statistics and probability with emphasis on biological applications.
Statistical analysis of experimental and observational data. An introduction to the discipline of mathematics for students considering a major in mathematics. Provides an introduction to rigorous mathematical reasoning, including basic proof techniques e.
Numerical Solution of Differential-Algebraic Equations—Wolfram Language Documentation
Study of a topic in mathematics under individual supervision. Intended for students with specific interests in areas not covered in the listed courses. Consent of department chair required. Practice in solving problems from mathematical contests using a variety of techniques. Permission of instructor required. Course may be repeated. Preparation for the first actuarial exam — probability.
Problems in calculus and probability with insurance applications. Preparation for the second actuarial exam - financial mathematics. Mathematics of interest and investments, interest rate measurement, present value, annuities, loan repayment schemes, bond valuation, introduction to derivative securities.
Practice in solving problems from past exams. Linear differential equations and applications; matrices and systems of linear equations; vector spaces; eigenvalues and application to linear systems of differential equations. Functions of a complex variable; calculus of residues; contour integration; applications to conformal mapping and Laplace transforms.
Consent of instructor required. Discussion of geometry as an axiomatic system. History of and equivalent versions of Euclid's fifth postulate.
AMATH 568: Advanced Methods for Ordinary Differential Equations
NonEuclidean geometries based upon negation of the fifth postulate: Geometry on the sphere; Hyperbolic and elliptic geometries. Applications of the different geometries will be considered. Representation of numbers and rounding error; polynomial and spline interpolation; numerical differentiation and integration; numerical solution of nonlinear systems; Fast Fourier Transformation; numerical solution of initial and boundary value problems; Monte Carlo methods.
Probability and distribution of random variables; populations and random sampling; chi-square and t distributions; estimation and tests of hypotheses; correlation and regression theory of two variables. Metric spaces and iterated function systems; various types of fractal dimension; Julia and Mandelbrot sets. Other topics such as chaos may be included. Small amount of computer use. Solution of systems of linear equations, matrices, vector spaces, bases, linear transformations, eigenvalues, eigenvectors, additional topics as time permits.
Introduction to basic concepts of modern algebra: Topics in combinatorics and graph theory chosen to introduce the subjects and some of their common proof techniques. Graph theory topics include trees, connectivity, traversability, matching and coloring. Topics in discrete mathematical structures chosen for their applicability to computer science and engineering.
Sets, propositions, induction, recursion; combinatorics; binary relations and functions; ordering, lattices and Boolean algebra; graphs and trees; groups and homomorphisms. An introduction to the basics of Calculus-based theory of Probability. Includes combinatorial techniques, events, independence, and conditional probability; most important discrete and continuous probability distributions, expectation and variance; joint distributions and covariance; moment generating functions; basic form of the Laws of Large Numbers and the Central Limit Theorem.
Focuses on use of concepts to solve problems. Prior knowledge of Probability not required. Introduction to the basic concepts, logic and issues involved in statistical reasoning and statistical methods used to analyze data and evaluate studies. Topics include descriptive statistics and exploratory data analysis; elementary probability and statistical inference. Examples drawn from various areas of application. Use of computer software e. Three lectures and one computer laboratory. Demetrius at MIT based on the second law of thermodynamics. The essential idea is that life exists because if increases overall entropy quicker than other process es.
The course will include an introduction to thermodynamics as well as to the information theory difinition of entropy. Eventually I want to apply this to other systems. For instance, I think the theory explains why political candidates who create chaos, tweet and require more bits of information to describe each day are favored by the second law. Demetrius makes a distinction between robust high entropy environments and precarious low entropy environments which is a very interesting in many different systems.
Typically the problems considered will come from mathematics, chemistry, biology, and materials science but sometimes they will also come from economics, finance, and social sciences. The research problems in the course vary from year to year. Mathematics majors are advised that MATH cannot be taken to satisfy the major requirements. This course will introduce students to some of the most widely used algorithms and illustrate how they are actually used. Applications will include the use of matrix computations to computer graphics, use of the discrete Fourier transform and related techniques in digital signal processing, the analysis of systems of linear differential equations, and singular value deompositions with application to a principal component analysis.
The ideas and tools provided by this course will be useful to students who intend to tackle higher level courses in digital signal processing, computer vision, robotics, and computer graphics. MATH is a masters level version of this course. Linear algebra, eigenvalues and eigenvectors of matrices, groups, rings and fields.
After a review of vector calculus and a section on tensor algebra, we study manifolds and their intrinsic geometry, including metrics, connections, geodesics, and the Riemann curvature tensor.
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Introduction to the topology of metric spaces with an emphasis on higher dimensional Euclidean spaces. The contraction mapping principle. Inverse and implicit function theorems. Rigorous treatment of higher dimensional differential calculus. Introduction to Fourier analysis and asymptotic methods. The goal of this course will be to introduce students to some of the most important and widely used algorithms in matrix computation and to illustrate how they are actually used in various settings. Motivating applications will include: The course will cover the theoretical underpinnings of these problems and the numerical algorithms that are used to perform important matrixcomputations such as Gaussian Elimination, LU Decomposition and Singular Value Decomposition.
Vector spaces, Basis and dimension, quotients; Linear maps and matrices; Determinants, Dual spaces and maps; Invariant subspaces, Cononical forms; Scalar products; Euclidean, unitary and symplectic spaces; Orthogonal and Unitary operators; Tensor products and polylinear maps; Symmetric and skew-symmetric tensors and exterior algebra. No background in finance is assumed. Martingales and optimal stopping. Stable laws and processes with independent increments. Brownian motion and the theory of weak convergence.
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Basics of homology and cohomology, singular and cellular; isomorphism with de Rham cohomology. The minors and cofactors of a matrix are found by computing the determinant of certain submatrices. A principal submatrix is a square submatrix obtained by removing certain rows and columns. The definition varies from author to author.
According to some authors, a principal submatrix is a submatrix in which the set of row indices that remain is the same as the set of column indices that remain. Matrices can be used to compactly write and work with multiple linear equations, that is, systems of linear equations.
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Using matrices, this can be solved more compactly than would be possible by writing out all the equations separately. If A has no inverse, solutions if any can be found using its generalized inverse. Matrices and matrix multiplication reveal their essential features when related to linear transformations , also known as linear maps. Conversely, each linear transformation f: The matrix A is said to represent the linear map f , and A is called the transformation matrix of f. These vectors define the vertices of the unit square.
The following table shows a number of 2-by-2 matrices with the associated linear maps of R 2. The blue original is mapped to the green grid and shapes. The origin 0,0 is marked with a black point. Under the 1-to-1 correspondence between matrices and linear maps, matrix multiplication corresponds to composition of maps: The rank of a matrix A is the maximum number of linearly independent row vectors of the matrix, which is the same as the maximum number of linearly independent column vectors. A square matrix is a matrix with the same number of rows and columns.
An n -by- n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied. The entries a ii form the main diagonal of a square matrix. They lie on the imaginary line that runs from the top left corner to the bottom right corner of the matrix. If all entries of A below the main diagonal are zero, A is called an upper triangular matrix. Similarly if all entries of A above the main diagonal are zero, A is called a lower triangular matrix.
If all entries outside the main diagonal are zero, A is called a diagonal matrix.
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The identity matrix I n of size n is the n -by- n matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, for example,. It is a square matrix of order n , and also a special kind of diagonal matrix. It is called an identity matrix because multiplication with it leaves a matrix unchanged:. A nonzero scalar multiple of an identity matrix is called a scalar matrix.
If the matrix entries come from a field, the scalar matrices form a group, under matrix multiplication, that is isomorphic to the multiplicative group of nonzero elements of the field. By the spectral theorem , real symmetric matrices and complex Hermitian matrices have an eigenbasis ; that is, every vector is expressible as a linear combination of eigenvectors.
In both cases, all eigenvalues are real. A square matrix A is called invertible or non-singular if there exists a matrix B such that. A symmetric matrix is positive-definite if and only if all its eigenvalues are positive, that is, the matrix is positive-semidefinite and it is invertible. Allowing as input two different vectors instead yields the bilinear form associated to A:. An orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors that is, orthonormal vectors.
Equivalently, a matrix A is orthogonal if its transpose is equal to its inverse:. The identity matrices have determinant 1 , and are pure rotations by an angle zero. The complex analogue of an orthogonal matrix is a unitary matrix. The trace , tr A of a square matrix A is the sum of its diagonal entries. While matrix multiplication is not commutative as mentioned above , the trace of the product of two matrices is independent of the order of the factors:.
Also, the trace of a matrix is equal to that of its transpose, that is,. The determinant det A or A of a square matrix A is a number encoding certain properties of the matrix. A matrix is invertible if and only if its determinant is nonzero. Its absolute value equals the area in R 2 or volume in R 3 of the image of the unit square or cube , while its sign corresponds to the orientation of the corresponding linear map: The determinant of 3-by-3 matrices involves 6 terms rule of Sarrus.
The more lengthy Leibniz formula generalises these two formulae to all dimensions. Adding a multiple of any row to another row, or a multiple of any column to another column, does not change the determinant. Finally, the Laplace expansion expresses the determinant in terms of minors , that is, determinants of smaller matrices. Determinants can be used to solve linear systems using Cramer's rule , where the division of the determinants of two related square matrices equates to the value of each of the system's variables. It is a monic polynomial of degree n.
Matrix calculations can be often performed with different techniques. Many problems can be solved by both direct algorithms or iterative approaches. For example, the eigenvectors of a square matrix can be obtained by finding a sequence of vectors x n converging to an eigenvector when n tends to infinity.
To choose the most appropriate algorithm for each specific problem, it is important to determine both the effectiveness and precision of all the available algorithms. The domain studying these matters is called numerical linear algebra. Determining the complexity of an algorithm means finding upper bounds or estimates of how many elementary operations such as additions and multiplications of scalars are necessary to perform some algorithm, for example, multiplication of matrices.
For example, calculating the matrix product of two n -by- n matrix using the definition given above needs n 3 multiplications, since for any of the n 2 entries of the product, n multiplications are necessary. The Strassen algorithm outperforms this "naive" algorithm; it needs only n 2. In many practical situations additional information about the matrices involved is known.
An important case are sparse matrices , that is, matrices most of whose entries are zero. An algorithm is, roughly speaking, numerically stable, if little deviations in the input values do not lead to big deviations in the result. For example, calculating the inverse of a matrix via Laplace expansion adj A denotes the adjugate matrix of A. The norm of a matrix can be used to capture the conditioning of linear algebraic problems, such as computing a matrix's inverse.
Although most computer languages are not designed with commands or libraries for matrices, as early as the s, some engineering desktop computers such as the HP had ROM cartridges to add BASIC commands for matrices. Some computer languages such as APL were designed to manipulate matrices, and various mathematical programs can be used to aid computing with matrices. There are several methods to render matrices into a more easily accessible form. They are generally referred to as matrix decomposition or matrix factorization techniques.
The interest of all these techniques is that they preserve certain properties of the matrices in question, such as determinant, rank or inverse, so that these quantities can be calculated after applying the transformation, or that certain matrix operations are algorithmically easier to carry out for some types of matrices. The LU decomposition factors matrices as a product of lower L and an upper triangular matrices U.
Likewise, inverses of triangular matrices are algorithmically easier to calculate. The Gaussian elimination is a similar algorithm; it transforms any matrix to row echelon form. This can be used to compute the matrix exponential e A , a need frequently arising in solving linear differential equations , matrix logarithms and square roots of matrices.
Matrices can be generalized in different ways. Abstract algebra uses matrices with entries in more general fields or even rings , while linear algebra codifies properties of matrices in the notion of linear maps. It is possible to consider matrices with infinitely many columns and rows. Another extension are tensors , which can be seen as higher-dimensional arrays of numbers, as opposed to vectors, which can often be realised as sequences of numbers, while matrices are rectangular or two-dimensional arrays of numbers. Similarly under certain conditions matrices form rings known as matrix rings.
Though the product of matrices is not in general commutative yet certain matrices form fields known as matrix fields. This article focuses on matrices whose entries are real or complex numbers. However, matrices can be considered with much more general types of entries than real or complex numbers. As a first step of generalization, any field , that is, a set where addition , subtraction , multiplication and division operations are defined and well-behaved, may be used instead of R or C , for example rational numbers or finite fields.
For example, coding theory makes use of matrices over finite fields. Wherever eigenvalues are considered, as these are roots of a polynomial they may exist only in a larger field than that of the entries of the matrix; for instance they may be complex in case of a matrix with real entries. The possibility to reinterpret the entries of a matrix as elements of a larger field for example, to view a real matrix as a complex matrix whose entries happen to be all real then allows considering each square matrix to possess a full set of eigenvalues.
Alternatively one can consider only matrices with entries in an algebraically closed field , such as C , from the outset. More generally, matrices with entries in a ring R are widely used in mathematics. The very same addition and multiplication operations of matrices extend to this setting, too. The set M n , R of all square n -by- n matrices over R is a ring called matrix ring , isomorphic to the endomorphism ring of the left R - module R n.
The determinant of square matrices over a commutative ring R can still be defined using the Leibniz formula ; such a matrix is invertible if and only if its determinant is invertible in R , generalising the situation over a field F , where every nonzero element is invertible. One special but common case is block matrices , which may be considered as matrices whose entries themselves are matrices. The entries need not be square matrices, and thus need not be members of any ring ; but their sizes must fulfil certain compatibility conditions. More generally, any linear map f: