Mar 5-Jun 22, 2018 Exam Period Ends:
- Real numbers (axiomatic theory). Supremum and Infimum of a set. Existence of an n-th root for any a > 0. 2. Convergent sequences, subsequences, Cauchy sequences. The Bolzano-Weierstrass theorem. Upper and lower limits. 3. Series. Partial sums, convergent and divergent series, Cauchy criterion. Series of non-negative terms. The root and the ratio tests. Series of arbitrary terms. Dirichlet, Leibnitz, and Abel theorems. Rearrangements of series. The Riemann Theorem. 4. The limit of a function. Continuous functions. Continuity of the elementary functions. Properties of functions continuous on a closed interval. Uniformly continuous functions. Cantor?s theorem. 5. The derivative of a function. Mean value theorems. Derivatives of higher order. L’Hospital’s rule. Taylor’s theorem.
- The Riemann integral: Riemann sums, the fundamental theorem of calculus. Methods for computing integrals (integration by parts, substitution, partial fractions). Improper integrals and application to series. Numerical integration. Stirling’s formula and additional applications time permitting.
- Uniform and pointwise convergence. Cauchy’s criterion and the Weierstrass M-test. Power series. Taylor series, analytic and non-analytic functions. Convolutions, approximate identities and the Weierstrass approximation theorem. Additional applications time permitting.
- A review of vectors in R^n and linear maps. The Euclidean norm and the Cauchy-Schwarz inequality. Basic topological notions in R^n. Continuous functions of several variables. Curves in R^n, arc-length. Partial and directional derivatives, differentiability and C^1 functions. The chain rule. The gradient. Implicit functions and Lagrange multipliers. Extrema in bounded domains.
- Partially ordered sets. Chains and antichains. Examples. Erdos-Szekeres’ theorem or a similar theorem. The construction of a poset over the quotient space of a quasi-ordered set.
- Comparison of sets. The definition of cardinality as as an equivalence class over equinumerousity. The Cantor-Bernstein theorem. Cantor’s theorem on the cardinality of the power-set.
- Countable sets. The square of the natural numbers. Finite sequences over a countable set. Construction of the ordered set of rational numbers. Uniqueness of the rational ordering.
- Ramsey’s theorem. Applications.
- The construction of the ordered real line as a quotient over Cauchy sequences of rationals.
- Konig’s lemma on countably infinite trees with finite levels. Applications. A countable graph is k-colorable iff every finite subgraph of it is k-colorable.
- Well ordering. Isomorphisms between well-ordered sets. The axiom of choice formulated as the well-ordering principle. Example. Applications. An arbitrary graph is k–colorable iff every finite subgraph is k-colorable.
- Zorn’s lemma. Applications. Existence of a basis in a vector space. Existence of a spanning tree in an arbitrary graph.
- Discussion of the axioms of set theory and the need for them. Russle’s paradox. Ordinals.
- Transfinite induction and recursion. Applications. Construction of a subset of the plane with exactly 2 point in every line.
- Infinite cardinals as initial ordinals. Basic cardinal arithmetic. Cardinalities of well known sets. Continuous real functions, all real runctions, the automorphisms of the real field (with and without order).
- General background: sets and operations on them, Complex numbers: definition (via ordered pairs), addition and multiplication, inverses, adjoint, absolute value. Real and complex polynomials and their roots.
- Fields: Definition, properties, examples: Rationals, reals, complex numbers, integers mod p.
- Linear equations over fields, matrices and elementary row operations, rank of a matrix, solutions of homogeneous and non homogeneous systems of linear equations and the connections between them.
- Vector spaces over fields, subspaces, bases and dimensions, coordinates change of coordinate matrix, row rank as rank of a subspace, sums, direct sums of subspaces and the dimension theorem.
- Matrices multiplication, the algebra of square matrices, inverse determinants: properties, Cramer’s rule, adjoint and its use for finding the inverse.
- Linerar transformationsbasic propertieskernel and image of a linear trasformationrepresentaion of linear transformations by matrices and the effect of change of bases.linear functionals, dual bases
- Polynomialsalgebras and idealsthe algebra of polynomials and its ideal sturctureLagrange interpolationthe prime factorization of a polynomial.2. Elementary canonical forms characteristic values and vectors of linear transformations and matrices.characteristic polynomials and annihilating polynomialsinvariant subspaces.direct sum decompostions .invariant direct sums. the primary decomposition theorem.diagonalization:necessary and sufficient conditions for diagonilaztion, computing diagonalizing matrices.3. Inner product spacesinner productsinner product spaces linear functionals and adjointsunitary operatorsHermitian operatorsnormal operators and the spectral decomposition theoremsingular value decomposition theorem and applications4. Jordan forms (optional)cyclic subspaces and annihilatorscyclic decompostionsthe Jordan form and its computation
Topological spaces and continuous functions (product topology, quotient topology, metric topology). Connectedness and Compactness. Countabilty Axioms and Separation Axioms (the Urysohn lemma, the Urysohn metrization theorem, Partition of unity). The Tychonoff theorem and the Stone-Cech compactification. Metrization theorems and paracompactness.
- Cesaro means: Convolutions, positive summability kernels and Fejer’s theorem.
- Applications of Fejer’s theorem: the Weierstrass approximation theorem for polynomials, Weyl’s equidistribution theorem, construction of a nowhere differentiable function (time permitting).
- Pointwise and uniform convergence and divergence of partial sums: the Dirichlet kernel and its properties, construction of a continuous function with divergent Fourier series, the Dini test.
- approximations. Parseval’s formula. Absolute convergence of Fourier series of functions. Time permitting, the isoperimetric problem or other applications.
- Applications to partial differential equations. The heat and wave equation on the circle and on the interval. The Poisson kernel and the Laplace equation on the disk.
- Fourier series of linear functionals on . The notion of a distribution on the circle.
- Time permitting: positive definite sequences and Herglotz’s theorem.
- The Fourier transform: convolutions, the inversion formula, Plancherel’s theorem, Hermite functions. Time permitting: tempered distributions, further applications to differential equations.
- Fourier analysis on finite cyclic groups, and the Fast Fourier Transform algorithm.
Introduction to partial differential equations ? The first order equations: a linear equation, a quasilinear equation, resolving the initial value problem by the method of characteristic curves. ? Classification of the second order equations: elliptic, hyperbolic and parabolic equations, exam- ples of Laplace, Wave and Heat equations. ? Elliptic equations: Laplace and Poisson?s equations, Dirichlet and Neumann boundary value problems, Poisson?s kernel, Green?s functions, properties of harmonic functions, Maximum prin- ciple. ? Analytical methods for resolving partial differential equations: Sturm-Liouville problem and the method of separation of variables for bounded domains, applications for Laplace, Wave and Heat equations including non-homogenous problems. Applications of Fourier and Laplace transforms for resolving problems in unbounded domains. ? Heat equation: initial value problem in unbounded domain, basic formula for the solution, initial-boundary value problems in bounded domains, Maximum principle. ? Wave equation: Dalamber formula, non-homogenous equation, Wave equation in higher dimen- sions. ? If time permits: Legendre polynomials and spherical functions. Literature: ? Pinchover Y.; Rubinstein J. Introduction to partial differential equations (in Hebrew), Depart- ment of mathematics, Technion, 2011, ? John F. Partial differential equations, Reprin t of the fourth edition. Applied Mathematical Scien ces, 1. Springer-Verlag, New York, 1991, ? Evans Lawrence C. Partial Differential Equations, Second Edition, ? Gilbarg D.; Trudinger N. S. Elliptic partial differential equations of second order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Ver lag, Berlin, 2001, ? Zauderer E. Partial differential equations of applied mathematics, Second edition. Pure and Appli ed Mathematics (New York). A Wiley-Interscience Public ation. John Wiley & Sons, Inc., New York, 1989. xvi+891 pp. ISBN: 0-471-61298-7. 1
Coding Theory investigates error-detection and error-correction. Such errors can occur in various communication channels: satellite communication, cellular telephones, CDs and DVDs, barcode reading at the supermarket, etc. A mathematical analysis of the notions of error detection and correction leads to deep combinatorial problems, which can be sometimes solved using techniques ranging from linear algebra and ring theory to algebraic geometry and number theory. These techniques are in fact used in the above-mentioned communication technologies.
- The main problem of Coding Theory
- Bounds on codes
- Finite fields
- Linear codes
- Perfect codes
- Cyclic codes
- Sphere packing
- Asymptotic bounds
R. Hill, A First Course in Coding Theory, Clarendon Press, Oxford 1986
Graphs and sub-graphs, trees, connectivity, Euler tours, Hamilton cycles, matching, vertex and edge colorings, planar graphs, introduction to Ramsey theory, directed graphs, probabilistic methods and linear algebra tools in Graph Theory.
- Fields: basic properties and examples, the characteristic, prime fields
- Polynomials: irreducibility, the Eisenstein criterion, Gauss’s lemma
- Extensions of fields: the tower property, algebraic and transcendental extensions, adjoining an element to a field
- Ruler and compass constructions
- Algebraic closures: existence and uniqueness
- Splitting fields
- Galois extensions: automorphisms, normality, separability, fixed fields, Galois groups, the fundamental theorem of Galois theory.
- Cyclic extensions
- Solving polynomial equations by radicals: the Galois group of a polynomial, the discriminant, the Cardano-Tartaglia method, solvable groups, Galois theorem
- Roots of unity: cyclotomic fields, the cyclotomic polynomials and their irreducibility
- Finite fields: existence and uniqueness, Galois groups over finite fields, primitive elements
- Categories and functors: natural transformations, equivalence, adjoint functors, additive functors, exactness.
- Derived functors: projective, injective and flat modules; resolutions, the functors and ; examples and applications.
- Nonabelian cohomology and its applications.
Permutation representation and the Sylow theorems.
Representations of groups on groups, solvable groups, nilpotent groups, semidirect and central products.
Permutation groups, the symmetric and alternating groups.
The generalized Fitting subgroup of a finite group.
Extension of groups: The first and second cohomology and applications.
- Affine algebraic sets and varieties.
- Local properties of plane curves.
- Projective varieties and projective plane curves.
- Riemann–Roch theorem.
Prof. Eitan Sayag 2.0 Credits
Wed 17:00-19:00, Math 201
- vector bundles and K-groups of topological spaces
- Bott’s Periodicity theorem and applications to division algebras
- Index of Fredholm operators and K-theory
- If time permits: smooth manifolds, DeRham cohomology, Chern classes, Elliptic operators, formulation of Atiyah-Singer index theorem, relations to Gauss-Bonnet theorem
This course deals with random walks, harmonic functions, the relations between these notions, and their applications to geometry and algebra (mainly to finitely generated groups).
The modern point of view will be presented, following recent texts by: Gromov, Kleiner, Ozawa, Shalom & Tao, among others.
The course is intended for 3rd year undergraduate as well as M.Sc and Ph.D. students both in computer science and mathematics. We will touch main topics in the area of discrete geometry. Some of the topics are motivated by the analysis of algorithms in computational geometry, wireless and sensor networks. Some other beautiful and elegant tools are proved to be powerful in seemingly non-related areas such as additive number theory or hard Erdos problems. The course does not require any special background except for basic linear algebra, and a little of probability and combinatorics. During the course many open research problems will be presented.
- Fundamental theorems and basic definitions: Convex sets, convex combinations, separation , Helly’s theorem, fractional Helly, Radon’s theorem, Caratheodory’s theorem, centerpoint theorem. Tverberg’s theorem. Planar graphs. Koebe’s Theorem. A geometric proof of the Lipton-Tarjan separator theorem for planar graphs.
- Geometric graphs: the crossing lemma. Application of crossing lemma to Erdos problems: Incidences between points and lines and points and unit circles. Repeated distance problem, distinct distances problem. Selection lemmas for points inside discs, points inside simplexes. Counting k-sets. An application of incidences to additive number theory.
- Coloring and hiting problems for geometric hypergraphs : VC-dimension, Transversals and Epsilon-nets. Weak eps-nets for convex sets. Conflict-free colorings .
- Arrangements : Davenport Schinzel sequences and sub structures in arrangements. Geometric permutations.
- Geometric Ramsey and Turan type theorems: Application of Dilworth theorem, Erdos-Szekeres theorem for convex sets, quasi-planar graphs.
We will present some basic notions and constructions from model theory, motivated by concrete questions about structures and their theories. Notions we expect to cover include:
- Types and spaces of types
- Homogeneous and saturated models
- Quantifier elimination and model companions
- Elimination of imaginaries
- Definable groups and fields
Students should be familiar with the following concepts from logic: Languages, structures, formulas, theories, the compactness theorem. In addition, some familiarity with field theory, topology and probability will be beneficial.
- Normed spaces and spaces with inner products. The projection theorem for finite dimensional subspaces. Orthogonal systems in infinite dimensional spaces. The Bessel inequality and the Parseval equality, closed orthogonal systems. The Haar system.
- The Fourier series (in real and complex form). Approximate identities, closedness of the trigonometric / exponential system. Uniform convergence of the Fourier series of piecewise continuously differentiable functions on closed intervals of continuity; the Gibbs phenomenon. Integrability and differentiability term by term.
- The Fourier transform. The convolution theorem. The Plancherel equality. The inversion theorem. Applications: low pass filters and Shannon’s theorem.
- The Laplace transform. Basic relations and connection with the Fourier transform. A table of the Laplace transforms. The convolution integral. Application of the Laplace transform for solution of ODEs.
- Introduction to the theory of distributions. Differentiation of distributions, the delta function and its derivatives. Fourier series, Fourier transforms, and Laplace transforms of distributions.
- Complex numbers, open sets in the plane.
- Continuity of functions of a complex variable
- Derivative at a point and Cauchy–Riemann equations
- Analytic functions; example of power series and elementary functions
- Cauchy’s theorem and applications.
- Cauchy’s formula and power series expansions
- Morera’s theorem
- Existence of a logarithm and of a square root
- Liouville’s theorem and the fundamental theorem of algebra
- Laurent series and classification of isolated singular points. The residue theorem
- Harmonic functions
- Schwarz’ lemma and applications
- Some ideas on conformal mappings
- Computations of integrals
- Second order linear equations with two variables: classification of the equations in the case of constant and variable coefficients, characteristics, canonical forms.
- Sturm-Liouville theory.
- String or wave equation. Initial and boundary value conditions (fixed and free boundary conditions). The d’Alembert method for an infinitely long string. Characteristics. Wave problems for half-infinite and finite strings. A solution of a problem for a finite string with fixed and free boundary conditions by the method of separation of variables. The uniqueness proof by the energy method. Well-posedness of the vibrating string problem.
- Laplace and Poisson equations. Maximum principle. Well-posedness of the Dirichlet problem. Laplace equation in a rectangle. Laplace equation in a circle and Poisson formula. An ill-posed problem - the Cauchy problem. Uniqueness of a solution of the Dirichlet problem. Green formula in the plane and its application to Neumann problems.
- Heat equation. The method of separation of variables for the one-dimensional heat equation. Maximum principle. Uniqueness for the one-dimensional heat equation. The Cauchy problem for heat equations. Green?s function in one dimension. If time permits: Green?s function in the two dimensional case.
- Non-homogeneous heat equations, Poisson equations in a circle and non-homogeneous wave equations.
- If time permits: free vibrations in circular membranes. Bessel equations.
- Complex numbers: Cartesian coordinates, polar coordinates. Functions of a complex variable. Basic properties of analytic functions, the exponential function, trigonometric functions. Definition of contour integral. The Cauchy Integral Formula. Residues and poles. Evaluation of impoper real integrals with the use of residues.
- Inner product functional spaces. Orthogonal and orthonormal systems. Generalized Fourier series. Theorem on orthogonal projection. Bessel’s inequality and Parseval’s equality.
- Trigonometric Fourier series. Complex form of Fourier series. Fourier series expansion defined over various intervals. Pointwise and uniform convergence of Fourier series. Completness of trigonometric system and Parseval’s equality. Differentiation and integration of Fourier series.
- The Fourier integral as a limit of Fourier series. The Fourier transform: definition and basic properties. The inverse Fourier transform. The convolution theorem, Parseval’s theorem for the Fourier transform. A relation between Fourier and Laplace transforms. Application of Fourier transform to partial differential equations and image processing.
- Distributions (generalized functions). The Heaviside step function, the impulse delta-function. Derivative of distribution. Convergence of sequences in the space of distributions. The Fourier transform of distributions.
The aim of the course is to train students in creative problem solving in various areas of mathematics, in a manner simulating, to some extent, mathematical research (rather than ordinary courses, where homework assignments are, usually closely ? and quite obviously - related to the topics reviewed in class). The course will focus on problems whose solutions call for tools and ideas from several fields of mathematics, and will reveal ? on a miniature scale - the beauty of mathematics as an integral field of knowledge, where between seemingly unrelated subjects deep and surprising connections may arise. The problems will focus on topics in classical and modern mathematics that, due mostly to their interdisciplinary nature, are not ? as a rule ? covered in the core classes offered by the department (such topics as the axiom of choice and its applications, the Banach-Tarski paradox, transcendental number theory etc.). Classes will be divided between lectures, given by the instructor, filling in background material required to address the problems in question, and between students’ presentation of their solutions to the work sheets distributed previously. In addition to all of the above, the course will help students improve their skills in searching the mathematical literature and in the art of writing and presenting proofs.
The system of the real numbers (without Dedekind cuts). The supremum axiom. Convergent sequences, subsequences, monotonic sequences, upper and lower limits. Series: partial sums, convergent and divergent series, examples, nonnegative series, the root test, the quotient test, general series, Dirichlet, Leibnitz, absolute convergence implies convergence (without a proof). Limits of functions, continuity, the continuity of the elementary functions, extrema in compact intervals. The derivative of a function, Lagrange’s Mean Value Theorem, high order derivatives, L’hospital’s rules, Taylor’s Theorem, error estimates, lots of examples. The Riemann integral: only for piecewise continuous functions (finitely many points of discontinuity). Riemann sums and the definition of the integral, The Fundamental Theorem of Calculus, the existence of primitive functions (anti-derivatives). Integration techniques: integration by parts, substitutions, partial fractions (without proofs), improper integrals, applications of integrals, estimation of series with the aid of integrals, Hardy’s symbols O, o and Omega, approximation of momenta and the Stirling formula.
Prof. Michael Levin 5.0 Credits
Pointwise convergence of sequences and of series of functions. Functions of several variables: limits, continuity, directional derivatives, the gradient, the orthogonality of the gradient to level surfaces, the chain rule, critical points (the necessity of the vanishing of the first order derivatives and examples of saddle points). Integration in 2 variables, repeated integrals and changing the integration order, the dependence of the boundaries of the integrals on the order of integration. Optimization with Lagrange multipliers, examples and less proofs. Depending on time: the Euler Lagrange equation (in Variational Calculus).
Prerequisites: 20119531 Linear Algebra Brief syllabus 1. Operations over sets, logical notation, relations. 2. Enumeration of combinatorial objects: integer numbers, functions, main principles of combinatorics. 3. Elementary combinatorics: ordered and unordered sets and multisets, binomial and multinomial coefficients. 4. Principle of inclusion and exclusion, Euler function. 5. Graphs: representation and isomorphism of graphs, valency, paths and cycles. 6. Recursion and generating functions: recursive definitions, usual and exponential generating functions, linear recurrent relations with constant coefficients. 7. (Optional) Modular arithmetics: congruences of integer numbers, Zm, invertible elements in Zm.
Probability theory: discrete and continuous variables, independent vs dependent variables, six basic discrete distributions: Bernoulli, binomial, uniform, geometric, negative binomial, Poissonian. Mean, variance, moments, probability generating function. Five basic continuous distributions: uniform, normal, exponential, gamma, beta. Moment generating function. Events, conditional probability, aging of molecules, entropy and related concepts, scores and support. Generating various probabilities. Many random variables. EST library. Covariance and correlation, iid, minimum and maximum of many random variables.Theoretical statistics; random sampling. Classical vs Bayesian approach. Distributions of the mean and variance of the sample; methods of calculating point estimates; point estimator for the mean; point estimator for the variance; biased vs unbiased, MSE, confidence intervals for the parameters of distribution. Basic ideas and definitions for the test of the hypothesis; errors of type I and II; P-values, tests for mean values, variances and proportions; test for the goodness of fit; test of independence; correlation coefficient; and its tests; linear regression; Likelihood ratios, information and support; maximum value as test statistic. Nonparametric: Mann-Whitney and permutation tests.Bayesian approach to hypothesis testing and estimation.ANOVA - analysis of variance: one-way and two-way.More theory on classical estimation: optimality aspects.BLAST.
- Review of probability: a. Basic notions. b. Random variables, Transformation of random variables, Independence. c. Expectation, Variance, Co-variance. Conditional Expectation.
- Probability inequalities: Mean estimation, Hoeffding?s inequality.
- Convergence of random variables: a. Types of convergence. b. The law of large numbers. c. The central limit theorem.
- Statistical inference: a. Introduction. b. Parametric and non-parametric models. c. Point estimation, confidence interval and hypothesis testing.
- Parametric point estimation: a. Methods for finding estimators: method of moments; maximum likelihood; other methods. b. Properties of point estimators: bias; mean square error; consistency c. Properties of maximum likelihood estimators. d. Computing of maximum likelihood estimate
- Parametric interval estimation a. Introduction. b. Pivotal Quantity. c. Sampling from the normal distribution: confidence interval for mean, variance. d. Large-sample confidence intervals.
- Hypothesis testing concepts: parametric vs. nonparametric a. Introduction and main definitions. b. Sampling from the Normal distribution. c. p-values. d. Chi-square distribution and tests. e. Goodness-of-fit tests. f. Tests of independence. g. Empirical cumulative distribution function. Kolmogorov-Smirnov Goodness-of fit test.
- Regression. a. Simple linear regression. b. Least Squares and Maximum Likelihood. c. Properties of least Squares estimators. d. Prediction.
- Handling noisy data, outliers.
- Introduction to number theory. Intervals and segments. Concept of a function. Elementary functions. 2. Limit of a function.3. Continuity and discontinuity of functions.4. Derivative and differential. Basic derivatives. Differentiability and continuity. Linear approximation by differentials. High-order derivatives. The fundamental theorems of differentiation and their applications. L’Hopital’s theorem and its application to calculation of limits.5. Taylor’s polynom. Expansion of functions into Taylor’s and McLoran’s series. Expansions of some usage functions. Application of Taylor’s and McLoran’s polynoms a) to approximate calculations, and b) to calculation of limits.6. Investigation of a function. Extremal points. Necessary and sufficient conditions for extrema. Max. and min. of a function within a segment. Convexity and concavity, inflection point. Asymptotes. Graph construction.7. Primitive function and indefinite integral. Table integrals. Calculation of indefinite integrals by decomposition, by parts, by substitution. Integration of rational and trigonometric functions.8. Definite integrals. Reimann’s sum. The fundamental theorem. Formula of Newton-Leibnitz. Calculation of definite integrals. Integration by decomposition, by parts, by substitution.9. Use in definite integrals to calculation of areas, volumes and curve lengthes. Rectungular and polar coordinate systems.10. First-order ordinary differential equations. General definitions. Cauchy problem. Separated variables.
- Infinite series. Tests for convergence. Taylor series and Taylor polynomials. Absolute convergence. Alternating series. Conditional convergence. Power series for functions. Convergence of power series; differentiation and integration. 2. Vectors and parametric equations. Parametric equation in analytic geometry. Space coordinates. Vectors in space. The scalar product of two vectors. The vector product of two vectors in space. Equations of lines and planes. product of three vectors and more. Catalog of the quadratic surfaces. Cylindres.3. Vector functions and their derivatives. Vector functions. differentiation formulas. Velocity and acceleration. Tangential vectors. Curvature and normal vectors. Polar coordinates.4. Partial differentiation. Functions of two and more variables. The directional derivative. limits and continuity. Tangent plane and normal lines. The gradient. The chain rule for partial derivatives. The total differentiation. Maxima and minima of functions of several independent variables. Higher order derivatives.5. Multiple integrals. Double integrals. Area and volume by double integrals. Double integrals in polar coordinates. Physical applications. triple integrals. Integration in cylindrical and spherical coordinates. Surface area. Change of variable in multiple integrals.6. Vector analysis. Vector fields. Line integrals. Independence of path. Green’s theorem. Surface integrals. The divergence theorem. Stokes’ theorem.
- Real numbers and real line, elementary functions and graphs, some functions arising in economics. The limit of a function, calculating limits using the limit laws, continuity, the number e.2. The derivative of a function, differential rules, higher derivatives, L’Hospital rules.3. Extreme values of functions, monotonic functions, point of inflection, concavity, curve sketching, applications to economics.4. Indefinite integrals, techniques of integration, definite and improper integrals, areas between curves, applications to economics.5. Functions of several variables, economics examples, partial derivatives, linearization, the chain rile, implicit and homogeneous functions, maximum and minimum, Lagrange multipliers.6. Introduction to linear algebra, matrices, linear systems.
- Introduction: the real and complex numbers, polynomials.
- Systems of linear equations and Gauss elimination.
- Vector spaces: examples (Euclidean 2-space and 3-space, function spaces, matrix spaces), basic concepts, basis and dimension of a vector space. Application to systems of linear equations.
- Inverse matrices, the determinant, scalar products.
- Linear transformations: kernel and image, the matrix representation of a transformation, change of basis.
- Eigenvalues, eigenvectors and diagonalization.
- The real numbers, inequalities in real numbers, the complex numbers, the Cartesian representation, the polar representation, the exponential representation, the Theorem of de Moivre, root computations.
- Systems of linear equations over the real or complex numbers, the solution set and its parametric representation, echelon form and the reduced echelon form of a matrix, backwards substitution, forward substitution and their complexity, the Gauss elimination algorithm and its complexity, the reduction algorithm and its complexity.
- Vector spaces, sub-spaces of vector spaces, linear combinations of vectors, the span of a set of vectors, linear dependence and linear independence, the dimension of a vector space, row spaces and column spaces of matrices, the rank of a matrix.
- Linear mappings between vector spaces, invertible mappings and isomorphisms, the matrix representation of finite dimensional linear mappings, inversion of a square matrix, composition of mappings, multiplication of matrices, the algebra of matrices, the kernel and the image of a linear mapping and the computation of bases, changing of a basis, the dimension theorem for linear mappings.
- Inner product spaces, orthogonality, the norm of a vector, orthonormal sets of vectors, the Cauchy-Schwarz inequality, the orthogonal complement of a sub-space, orthogonal sequences of vectors, the Gram-Schmidt algorithm, orthogonal transformations and orthogonal matrices.
- The determinant of a square matrix, minors and cofactors, Laplace expansions of the determinant, the adjoint matrix and Laplace theorem, conjugation of a square matrix, similarity transformations and their invariants (the determinant and the trace).
- Eigenvalues, eigenvectors, eigenspaces, diagonalization and similarity, the characteristic polynomial, the algebraic and the geometric multiplicities of an eigenvalue, the spectral theorem for Hermitian matrices.
- Descriptive statistics: organizing, processing and displaying data. 2. Sampling distributions: Normal distribution, the student t-distribution, Chi-Square distribution and Fisher’s F-distribution 3. Estimation: A point estimate and Confidence Interval of population parameters: Mean variance and proportion. Tolerance interval. 4. Testing hypothesis about a population’s parameters: Mean, variance and proportion. 5. Evaluating the properties of a statistical test: errors, significance level and power of a test. 6. Testing hypothesis about equality of variances, equality of means and equality of proportions of two populations. 7. Testing for independence of factors: Normal and Chi-Square methods. 8. Testing for goodness of fit of data to a probabilistic model: Chi-Square test. 9. Linear regression: Inference about the utility of a linear regression model. Covariance and correlation coefficient. Confidence and prediction intervals. 10. Weibull distribution: estimating the distribution’s parameters
First-Order PDE Cauchy Problem Method of Characteristics The Wave Equation: Vibrations of an Elastic String D’Alembert’s Solution Fourier Series Fourier Sine Series Initial-Boundary Value Problems The Wave Equation .Separation of Variables Fourier Series Solution of the Heat Equation The Heat Equation. Duhamel’s Principle. Laplace’s Equation. Dirichlet Problem for a Disc
Ordinary Differential EquationsBasic concepts: ordinary differential equations, differential equations of the first order, general solution, initial value problems, partial solutions. Linear differential equations with separable variables, exact equations, integration factor, homogeneous equations. Existence and Uniqueness theorem (without proof). System of differential equation of first order, solution by matrixes. Linear differential equations of second order, non- homogeneous equations, Wronskian. Linear differential equations of n-th order.Integral TransformsLaplace transform, properties of the Laplace transform. Convolution of two functions and convolution theorem. Heavyside (unit step) function, ?-function (Dirac), particularly continuous functions, their Laplace transform. Solution of non-homogeneous differential equations by Laplace transform.Fourier transform, properties of the Fourier transform. Convolution of two functions and convolution theorem. Cosines and Sine Fourier transform. Solution of integral equations by Fourier transform..
Fields. Fields of rational, real and complex numbers. Finite fields. Calculations with complex numbers. Systems of linear equations. Gauss elimination method. Matrices. Canonical form of a matrix. Vector spaces . Homogeneous and non homogeneous systems. Vector spaces. Vector spaces. Vector subspace generated by a system of vectors. Vector subspace of solutions of a system of linear homogeneous equations. Linear dependence. Mutual disposition of subspaces. Basis and dimension of a vector space. Rank of a matrix. Intersection and sum of vector subspaces. Matrices and determinants. Operations with matrices. Invertible matrices. Change of a basis. Determinants. Polynomials over fields. Divisibility. Decomposition into prime polynomials over R and over C. Linear transformations and matrices. Linear transformations and matrices. Kernel and image. Linear operators and matrices. Algebra of linear operators. Invertible linear operators. Eigenvectors and eigenvalues of matrices and linear operators. Diagonalization of matrices and linear operators. Scalar multiplication. Orthogonalization process of Gram-Shmidt. Orthogonal diagonalization of symmetric matrices.
Analytic Geometry: planes and lines, quadric surfaces, cylinders.Vector functions: derivatives and integrals.Partial derivatives: functions of two or more arguments, chain rules, gradient, directional derivatives, tangent planes, higher order derivatives, linear approximation, differential of the first and higher order, maxima, minima and saddle points, Lagrange multipliers.Multiple integrals: double integrals, area, changing to polar coordinates, triple integrals in rectangular coordinates, physical applications.Vector analysis: vector and scalar fields, surface integrals, line integrals and work, Green’s theorem, the divergence theorem, Stokes’s theorem.Infinite series: tests for convergence of series with nonnegative terms, absolute convergence, Alternating series, conditional convergence, arbitrary series.Power series: power series for functions, Taylor’s theorem with remainder: sine, cosine and e , logarithm, arctangent, convergence of power series, integration, differentiation.
- Basic notions: equations of the first order, general solution, initial value problem, particular solution. Linear equations, separable equations, exact equations, homogeneous equations, integrating factor. The existence and uniqueness theorem (without proof). The Riccatti equations, the Bernoulli equations. Linear systems of the first order differential equations. Solution via the matrix calculus. The second order linear equations. Non-linear equations and the Wronskian. The Euler equations. Linear equations of the first order. 2. The Laplace transformation, properties of the Laplace transformation, solutions of the linear non-homogeneous equations via the Laplace transformation, the Heaviside functions, the delta functions.3. The Fourier transformation, properties of the Fourier transformation. Cosine and sine Fourier transformation. Solution of the integral equations via the Fourier transformation.
The aim of the course is to learn basis of Calculus of functions of two and more variables. It includes: a) short study of vector algebra and analytic geometry in plane and space; b) differential calculus of two and more variables and its applications to local and global extremum problems, directional derivatives, Teylor’s formula, etc.;c) Integral calculus (line, double and triple integrals) and its applications; d) vector field theory and in particular its applications for studying potential vector fields.
- Lines and planes. Cross product. Vector valued functions of a single variable, curves in the plane, tangents, motion on a curve.
- Functions of several variables: open and closed sets, limits, continuity, differentiability, directional derivatives, partial derivatives, the gradient, scalar and vector fields, the chain rule, the Jacobian. Implicit differentiation and the implicit function theorem. Extremum problems in the plane and in space: the Hessian and the second derivatives test, Lagrange multipliers.
- Line integrals in the plane and in space, definition and basic properties, work, independence from the path, connection to the gradient, conservative vector field, construction of potential functions. Applications to ODEs: exact equations and integrating factors. Line integral of second kind and arclength.
- Double and triple integrals: definition and basic properties, Fubini theorem. Change of variable and the Jacobian, polar coordinates in the plane and cylindrical and spherical coordinates in space. Green’s theorem in the plane.
- Parametric representation of surfaces in space, normals, the area of a parametrized surface, surface integrals including reparametrizations
- Curl and divergence of vector fields. The theorems of Gauss and Stokes.
- Fields: definitions, the field of complex numbers.
- Linear equations: elementary operations, row reduction, homogeneous and inhomogeneous systems, representations of the solutions.
- Vector spaces: examples, subspaces, linear dependence, bases, dimension.
- Matrix algebra: matrix addition and multiplication, elemetary operations, the inverse of a matrix, the determinant, Cramer’s rule.
- Linear transformations: examples, kernel and image, matrix representation.
- Diagonalization: eigenvectors and eigenvalues, the characteristic polynomial, applications.
- Bilinear forms.
- Finite dimensional inner product spaces.
- Operators on finite dimensional inner product spaces: the adjoint, self adjoint operators, normal operators, diagonalization of normal operators.
Prof. Gregory Mashevitsky 5.0 Credits
The course provides a basic introduction to “naive” set theory, propositional logic and predicate logic. The course studies many important concepts which serve as the building blocks for any mathematical theory An emphasis is given to clear proof writing and correct usage of the mathematical language.A. Basics of Set Theory1. The notion of a set. Set operations: union, intersection, difference, complementation, and the power set.2. Cartesian products and binary relations. Operations on relations.3. Functions: domain and range, one to one and onto. Composition.4. Equivalence relations and set partitions.5. Order relations: partial, linear and well-founded.6. Induction theorems: mathematical, complete and well-founded.B. Set Cardinality1. The notion of cardinality. Finite, infinite and countable sets.2. Cantor’s theorem.3. The cardinality of the set of real numbers and other sets.C. Propositional Calculus1. The language of propositional calculus. Logical connectives.2. Logical implication and logical equivalence of propositional formulas.3. Disjunctive normal form.4. Complete sets of logical connectives.D. Predicate Calculus (First Order Logic)1. The language of predicate calculus: terms, formulas, sentences.2. Structures, assignments for a given structure.3. Logical implication and logical equivalence of first order formulas.4. Elementary equivalence of structures, definable sets in a given structure.
Sets. Set operations and the laws of set theory. Power set. Cartesian product of sets.The rules of sum and product. Permutations, combination, distributions. The Binomial Theorem. The well-ordering principle: mathematical induction. The principle of inclusion and exclusion. The pigeonhole principle. Recurrence relations. Generating functions.Relations and functions. Properties of relations. Equivalence relations and their properties. Partial order. Functions and their properties. Injective, surjective functions. Function composition and inverse functions.Graph, subgraph, complements. Graph isomorphism. Euler`s formula. Planar graph. Euler trails and circuits. Trees.Propositional logic. Syntax of propositional logic. Logical equivalence. The laws of logic. Logical implication. Equivalence and disjunctive normal form. Predicate logic. Syntax of predicate logic. Models. Equivalence of formulas. Normal form.Algebraic structures. Rings, groups, fields. The integer modulo n. Boolean algebra and its structure.
- Real numbers. Supremum and Infimum of a set. 2. Convergent sequences, subsequences, Cauchy sequences. The Bolzano-Weierstrass theorem. Limit superior and limit inferior. 3. Series. Partial sums, convergent and divergent series, Cauchy criterion. Series of non-negative terms. The root and the ratio tests. Conditional and absolute convergence. The Leibnitz test for series with alternating signs. Rearrangements of series (without proof) 4. The limit of a function. Continuous functions. Continuity of the elementary functions. Properties of functions continuous on a closed interval: boundedness and attainment of extrema. Uniform continuity, Cantor?s theorem. 5. The derivative of a function. Mean value theorems. Derivatives of higher order. L’Hospital’s rule. Taylor’s theorem. Lagrange remainder formula.
Dr. Daniel Markiewicz 5.0 Credits
- The Riemann integral: Riemann sums, the fundamental theorem of calculus and the indefinite integral. Methods for computing integrals: integration by parts, substitution, partial fractions. Improper integrals and application to series. 2. Uniform and pointwise convergence. Cauchy criterion and the Weierstrass M-test. Power series. Taylor series. 3. First order ODE’s: initial value problem, local uniqueness and existence theorem. Explicit solutions: linear, separable and homogeneous equations, Bernoulli equations. 4. Systems of ODE’s. Uniqueness and existence (without proof). Homogeneous systems of linear ODE’s with constant coefficients. 5. Higher order ODE’s: uniqueness and existence theorem (without proof), basic theory. The method of undetermined coefficients for inhomogeneous second order linear equations with constant coefficients. The harmonic oscillator and/or RLC circuits. If time permits: variation of parameters, Wronskian theory.
In this course the basic concepts of one-dimensional analysis (a limit, a derivative, an integral) are introduced and explored in different applications: graphing functions, approximations, calculating areas etc. 1. Limit of a function, continuity. 2. Derivative, basic derivative formulas. 3. Derivative of an inverse function; derivative of a composite function, the chain rule; derivative of an implicit function. 4. Derivatives of high order. 5. The mean value problem theorem. Indeterminate forms and l’Hopital’s rule. 6. Rise and fall of a function; local minimal and maximal values of a function. 7. Concavity and points of inflection. Asymptotes. Graphing functions. 8. Linear approximations and differentials. Teylor’s theorem and approximations of an arbitrary order. 9. Indefinite integrals: definition and properties. 10. Integration methods: the substitution method, integration by parts. 11. Definite integrals. The fundamental theorem of integral calculus (Newton-Leibniz’s theorem). 12. Calculating areas. Bibliography Thomas & Finney, Calculus and Analytic Geometry, 8th Edition, Addison-Wesley(World Student Series).
- Infinite series of nonnegative terms and general series. Absolute and conditional convergence. Power series.
- Vector algebra. Dot product, cross product and box product.
- Analytic geometry of a line and a plane. Parametric equations for a line. Canonic equations for a plane. Points, lines and planes in space.
- Vector-valued functions. Derivative. Parametrized curves. Tangent lines. Velocity and acceleration. Integration of the equation of motion.
- Surfaces in space. Quadric rotation surfaces. Cylindrical and spherical coordinates.
- Scalar functions of several variables. Scalar field. Level surfaces. Limit and continuity. Partial derivatives. Directional derivative. Gradient vector. Differential. Tangent plane and normal line. Chain rules. Implicit function and its derivative. Taylor and MacLaurin formulas. Local extreme values. Absolute maxima and minima on closed bounded regions.
- Vector-valued functions of several variables. Vector field. Field curves. Divergence and curl.
- Line and path integrals. Work, circulation. Conservative fields. Potential function.
- Double integral and its applications. Green’s theorem.
- Parametrized surfaces. Tangent plane and normal line. Surface integrals. Flux. Stokes’s theorem.
- Triple integral and its applications. Divergence theorem.
- Analytic geometry in space. Vector algebra in R3. Scalar, cross and triple product and their geometric meaning. Lines, planes and quadric surfaces in space including the standard equations for cones, ellipsoids, paraboloids and hyperboloids.
- Functions of several variables.Graphs and level curves and surfaces. Limits and continuity. Properties of the continuous functions on a closed bounded domain. Partial derivatives. The plane tangent to graph of the function. Differentiability, the total differential and the linear approximation. Differentiability implies continuity. The chain rule. The gradient vector and the directional derivative. Tangent plane and the normal line to a surface at a point. 201.1.9761
- Maxima and minima for functions of several variables. Higher-order partial derivatives and differentials. Taylor’s formula. Local extrema and saddle points. Necessary conditions for local maxima and minima of a differentiable function. Sufficient conditions for local maxima and minima via the Hessian. Global extrema in closed bounded sets. Lagrange Multipliers.
- Double integrals . Double integrals on rectangles. Connection with the volume. Properties and evaluation of double integrals in non-rectangular domains. Iterated integrals and change of order of integration. Change of variables formula for the double integral and the Jacobian. Double integrals in polar coordinates. Applications of the change of variables formula to the computation of area.
Prof. Shakhar Smorodinsky 5.0 Credits
Logic and Set theory: propositional calculus, boolean operators and their truth tables, truth values of formulae, logical equivalence and logical inference, tautologies and contradictions, the important tautologies, e.g., the distributive laws and De Morgan formulae. Sets: inclusion, union, intersection, difference, cartesian product, relations, equivalence relations, partial orders, linear orders, and functions. Basic Combinatorics: induction, basic counting arguments, binomial coefficients, inclusion-exclusion, recursion and, generating functions. Graphs: general notions and examples, isomorphism, connectivity, Euler graphs, trees.
- Courses marked with (*) are required for admission to the M.Sc. program in Mathematics.
- The M.Sc. degree requires the successful completion of at least 2 courses marked (#). See the graduate program for details
- The graduate courses are open to strong undergraduate students who have a grade average of 85 or above and who have obtained permission from the instructors and the head of the teaching committee.
- Please see the detailed undergraduate and graduate programs for the for details on the requirments and possibilities for complete the degree.