Oct 27, 2019-Jan 24, 2020 Exam Period Ends: March 6, 2020

Courses

  1. 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.
  2. Fields: Definition, properties, examples: Rationals, reals, complex numbers, integers mod p.
  3. 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.
  4. 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.
  5. Matrices multiplication, the algebra of square matrices, inverse determinants: properties, Cramer’s rule, adjoint and its use for finding the inverse.
  6. 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

Infinitesimal Calculus 1 Pdf 201.1.1011

5.0 Credits

יום ב 12:00 - 10:00 בקרייטמן-זלוטובסקי(חדש) [34] חדר 14
יום ד 12:00 - 10:00 בביה”ס אילנות [97] חדר 201

Axioms of the reals. Sequences: limits, monotone sequences, the Bolzano-Weierstrass theorem, Cauchy’s criterion, the number e. Limits of functions. Continuous functions: equivalent definitions of continuity, properties of the elementary functions, the exponential function, the Intermediate Value Theorem, existence of extrema in closed and bounded sets, uniform continuity and Cantor’s theorem. Introduction to derivatives: the definition of the derivative and differentiation rules, the derivative of an inverse function, derivatives of elementary functions, Fermat’s theorem, Rolle’s theorem and Lagrange’s Mean Value Theorem.

Discrete Mathematics Pdf 201.1.2201

Dr. Gili Golan 5.0 Credits

יום א 14:00 - 12:00 בצוקר, גולדשטיין-גורן [72] חדר 209
יום ג 16:00 - 14:00 בגולדברגר [28] חדר 301

  1. Introduction: Sets, subsets, permutations, functions, partitions. Indistinguishable elements, multisets, binary algebra of subsets. Rules of sum and product, convolutions, counting pairs. Binomial and multinomial coefficients. Stirling numbers of second kind, definition and a recurrenat formula.
  2. Graphs: General notions and examples. Isomorphism. Connectivity. Euler graphs. Trees. Cayley’s theorem. Bipartite graphs. Konig’s theorem, P. Hall’s theorem.
  3. The inclusion-exclusion method: The complete inclusion-exclusion theorem. An explicit formula for the Stirling numbers. Counting permutations under constraints, rook polynomials.
  4. Generating functions: General notion, combinatorial meaning of operations on generating functions. Theory of recurrence equations with constant coefficients: the general solution of the homogeneous equation, general and special cases of nonhomogeneity. Catalan numbers. Partitions of numbers, Ferrers diagrams. Exponential generating functions, counting words, set partitions, etc.

The goal of the workshop is to accompany first year mathematics majors, and to improve their skills in writing formal proofs. In the course of the workshop, the students will work in small groups on writing proofs, with an emphasis on topics related to the foundational first year courses.

  1. Ordered fields: Hilbert’s 17 problem, orderings and preorderings, sums of squares, real closed fields, Artin-Schreier theory
  2. Infinite Galois theory: profinite groups, the Galois correspondence in the infinite case
  3. Introduction to Galois cohomology

Algebraic Structures Pdf 201.1.7031

Prof. Nadya Gurevich 4.0 Credits

יום ג 11:00 - 09:00 in גולדברגר [28] חדר 203
יום ה 14:00 - 12:00 in קרייטמן-זלוטובסקי(חדש) [34] חדר 18

  • Groups, the factor group and the homomorphism theorems, Sylow’s theorems and permutation actions of groups.
  • Rings, Integral Domains and Fields. Ideals: maximal and prime. Unique Factorization Domains, Principle Ideal Domains, Euclidean Domains.
  • Modules, structure theorems for finitely generated modules over a PID, application to finitely generated abelian groups and to the Jordan Canonical Form.

Fundamentals of Measure Theory(*) Pdf 201.1.0081

Prof. Tom Meyerovitch 4.0 Credits

יום ב 16:00 - 14:00 in גוטמן [32] חדר 209
יום ד 16:00 - 14:00 in צוקר, גולדשטיין-גורן [72] חדר 123

Algebras and sigma-algebras of subsets, the extension theorem and construction of Lebesgue’s measure on the line, general measure spaces, measurable functions and their distribution functions, integration theory, convergence theorems (Egorov’s, relations between convergence in measure and a.e. convergence), the spaces $L_1$ and $L_2$ and their completeness, signed measures, the Radon-Nikodym theorem, measures in product spaces and Fubini’s theorem.

Logic Pdf 201.1.6061

Dr. Assaf Hasson 4.0 Credits

יום ב 14:00 - 12:00 in בנין 90 (מקיף ז’) [90] חדר 236
יום ד 12:00 - 10:00 in גולדברגר [28] חדר 304

  • An axiom system for predicate calculus and the completeness theorem.
  • Introduction to model theory: The compactness Theorem, Skolem–Löwenheim Theorems, elementary substructures.
  • Decidability and undecidability of theories, Gödel first Incompleteness Theorem.

Graph Theory Pdf 201.1.6081

Dr. Yaar Solomon 4.0 Credits

יום א 12:00 - 10:00 in גולדברגר [28] חדר 202
יום ה 16:00 - 14:00 in קרייטמן-זלוטובסקי(חדש) [34] חדר 116

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.

Theory of Numbers Pdf 201.1.6031

Prof. Fedor Pakovich 4.0 Credits

יום ב 18:00 - 16:00 in צוקר, גולדשטיין-גורן [72] חדר 211
יום ה 12:00 - 10:00 in גולדברגר [28] חדר 102

Number Theory studies the structure of the integers and the natural numbers. In addition to classical topics (prime numbers, congruences, quadratic residues, etc.) there is an emphasis on algorithmic questions and in particular on applications to cryptography.

  • Divisibility and prime numbers
  • Congruences
  • The multiplicative group of $\mathbb{Z}/m$
  • Quadratic residues
  • Continued fractions
  • Algebraic numbers and algebraic integers

Introduction to Analysis Pdf 201.1.1051

Dr. Izhar Oppenheim 4.0 Credits

יום ב 10:00 - 08:00 in גולדברגר [28] חדר 202
יום ד 10:00 - 08:00 in גולדברגר [28] חדר 202

Metric and normed spaces. Equivalence of norms in finite dimensional spaces, the Heine-Borel theorem. Convergence of sequences and series of functions: pointwise, uniform, in other norms. Term-by-term differentiation and integration of series of functions, application to power series. Completeness: completeness of the space of continuous functions on a closed interval and a compact metric space. The Weierstrass M-test. The Baire category theorem and applications, bounded linear functionals and the Banach-Steinhaus theorem. Compactness in function spaces and the Arzela-Ascoli theorem. Introduction to Fourier series: Cesaro means, convolutions and Fejer’s theorem. The Weierstrass approximation theorem. L^2 convergence. Pointwise convergence, the Dirichlet kernel and Dini’s criterion.

Geometric infinitesimal calculus 1 Pdf 201.1.1031

Prof. Dmitry Kerner 4.0 Credits

יום א 10:00 - 08:00 in גוטמן [32] חדר 208
יום ד 18:00 - 16:00 in קרייטמן-זלוטובסקי(חדש) [34] חדר 110

Open, closed and compact sets in Euclidean space. Matrix norms and equivalence of norms. Limits and continuity in several variables. Curves and path connectedness. Partial and directional derivatives, the gradient and differentiability. The implicit, open and inverse function theorems. Largange multipliers. Optimization: the Hessian matrix and critical points. Multivariable Riemann integration: Fubini’s theorem and the change of variables formula.

Representations of $SL_2$ Pdf 201.2.0051

Prof. Eitan Sayag

יום א 14:00 - 12:00 in גוטמן [32] חדר 207

The course will examine the role of the representation theory of the group G=SL(2) in the mathematics of 20th century. We plan to achieve this by surveying its connections to geometry, algebra, analysis and number theory.

  • Topological manifolds. The fundamental group and covering spaces. Applications.
  • Singular homology and applications.
  • Smooth manifolds. Differential forms and Stokes’ theorem, definition of de-Rham cohomology.
  • Additional topics as time permits.

Banach spaces and Hilbert spaces. Basic properties of Hilbert spaces. Topological vector spaces. Banach-Steinhaus theorem; open mapping theorem and closed graph theorem. Hahn-Banach theorem. Duality. Measures on locally compact spaces; the dual of $C(X)$. Weak and weak-$*$ topologies; Banach-Alaoglu theorem. Convexity and the Krein-Milman theorem. The Stone-Weierstrass theorem. Compact operators on Hilbert space. Introduction to Banach algebras and Gelfand theory. Additional topics as time permits.

  • Fundamental theorems and basic definitions: Convex sets, separation , Helly’s theorem, fractional Helly, Radon’s theorem, Caratheodory’s theorem, centerpoint theorem. Tverberg’s theorem. Planar graphs. Koebe’s Theorem.
  • Geometric graphs: the crossing lemma. Application of crossing lemma to Erdos problems: Geometric Incidences, Repeated distance problem, distinct distances problem. Selection lemmas. 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. $(p,q)$-Theorem, Conflict-free colorings.
  • Arrangements : Davenport Schinzel sequences and sub structures in arrangements.
  • Geometric Ramsey and Turan type theorems: Application of Dilworth theorem, Erdos-Szekeres theorem for convex sets, quasi-planar graphs.

Generally speaking, algebraic geometry deals with geometric objects defined by polynomial equations. We will cover topics 1-3 below, and maybe one other topic, depending on the level of the students.

Topics
  1. Topics from commutative algebra.
  2. Algebraic varieties over an algebraically closed field: definitions and main properties.
  3. Curves and their function fields.
  4. Intersection Theory in the projective plane.
  5. Vector bundles and coherent sheaves.
  6. Picard group and projective embeddings.
  7. Schemes: definitions, examples and basic properties.

This course concerns the physical notion of phase transition, specifically in the model known as “percolation”.

We will review the main mathematical results regarding percolation and its related counterparts the Ising model, Potts model and Fortuin-Kasteleyn cluster model, starting from works of Ising and Pierles in the beginning of the 20th century and culminating in modern work of Smirnov (for which he was awarded a Fields Medal).

The topics in the course are, time permitting:
  1. Percolation on graphs. Definitions and basic properties.
  2. Harris’ inequality
  3. van den Berg-Kesten inequality, Reimer’s inequality
  4. Russo’s formula
  5. Burton-Keane Theorem
  6. Exponential decay of correlations in sub-critical regime
  7. Planar percolation: Russo-Seymour-Welsh theory
  8. Planar percolation: the Harris-Kesten theorem.
  9. Conformal invariance: Cardy-Smirnov formula on the triangular lattice
  10. Percolation on groups
  11. Critical percolation on non-amenable groups: BLPS

Course Topics:

  1. Review of prior material. On rings, ideals and modules.

  2. Categories and functors. Emphasis on linear categories, linear functors and morphism between linear functors. Exactness. (This topic will be introduced gradually, as we go along.)

  3. Universal constructions. Free modules, products, direct sums. Polynomial rings.

  4. Tensor products. Definition, construction and properties. Flatness. Tensor products of rings, adjunction formulas, relations to Galois Theory. Symmetric and exterior tensor powers.

  5. Localization. Localizing rings and modules. Flatness of localization. Local rings and Nakayama’s Lemma.

  6. Prime spectrum. Definition, Zariski topology, dimension, support of modules, connectedness and idempotents, local properties of modules.

  7. Noetherian rings. Definition, basic properties, Hilbert Basis Theorem, Artin-Rees Lemma, completion, Cohen Structure Theorem.

  8. Dimension theory. Noether Normalization, transcendence degree, dimension theorems, Hilbert Nullstellensatz.

  9. Regularity. Regular sequences, grading, regular rings, normal rings, Cohen-Macaulay rings.

  10. Differential Algebra. Derivations, differential forms, smooth and étale homomorphisms, relations to Galois Theory and differential geometry.

Advanced Anlysis for Communication Engineering Pdf 201.1.0241

Prof. Arkady Leiderman 3.5 Credits

יום ב 12:00 - 09:00 in גוטמן [32] חדר 209

  1. 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.
  2. Inner product functional spaces. Orthogonal and orthonormal systems. Generalized Fourier series. Theorem on orthogonal projection. Bessel’s inequality and Parseval’s equality.
  3. 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.
  4. 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.
  5. 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.

Prodability and Statistics For Software Engineering Pdf 201.1.2381

Dr. Luba Sapir 2.5 Credits

יום ה 10:00 - 08:00 in בניין אולמות להרצאות [92] חדר 001

1) Probability space 2) Law of total probability 3) Conditional probability, independent events 4) Bayes’ law 5) Discrete random variables. Discrete distributions: uniform, Bernoulli, binomial, geometric, Poisson 6) Continuous random variable. Continuous distributions: uniform, exponential, normal 7) Discrete two-dimensional joint random variables 8) Independence of random variables 9) Expectation 10) Variance, covariance, correlation coefficient

Sample spaces and finite probability spaces with symmetric simple events, general probabilty spaces and the fields of events, the Borel filed and probabilities on it defined by densities, conditional probabilities and independent events, random variables and their distribution functions (discrete, absolutely continuous, mixed), the expectation of a random variable (for discrete, absolutely continuous and general distribution), the variance of a random variable, random vectors and the covariance, independent random variables, the central limit theorem for i.i.d. random variables, examples related to analysis of simple algorithms, joint densities (discrete or continuous) with computation of the covariance and the marginal distributions, the weak law of large numbers.1. A.M. Mood, F.A. Graybill And D.C.Boes. Introduction To The Theory Of Statistics 3rd Edition, Mcgraw-Hill, 1974. 2. A. Dvoretzky, Probability theory (in Hebrew), Academon, Jerusalem, 1968.3. B. Gnedenko, The theory of Probability, Chelsea 1967 (or Moscow 1982) in English; Russian origina titled ‘A course in probability theory”.

Introduction to Differential Equations A Pdf 201.1.9031

Dr. Alexander Ukhlov 3.5 Credits

יום א 18:00 - 15:00 in קרייטמן-זלוטובסקי(חדש) [34] חדר 110

First order differential equations.1. Separable equations.2. Exact equations. Integrating factors.3. Homogeneous equations.4. Linear equations. Equation Bernulli.5. The existence theoremSecond order equations.1. Reduction of order.2. Fundamental solutions of the homogeneous equations3. Linear independence. Liouville formula. Wronskian.4. Homogeneous equations with constant coefficients.5. The nonhomogeneous problem.6. The method of undetermined coefficients.7. The method of variation of parameters. 8. Euler equation.9. Series solutions of second order linear equatHigher order linear equations.1. The Laplace transform2. Definition of the Laplace transform3. Solution of differential equations by method of Laplase transform.4. Step functions.5. The convolution integral.Systems of first order equations.1. Solution of linear systems by elimination.2. Linear homogeneous systems with constant coefficients.3. The matrix method. Eigenvalues and eigenvectors.4. Nonhomogeneous linear systems.

(1) Probability space.(2) Conditional probability, independent events, Bayes’s theorem, complete probabilities.(3) Random discrete variable, discrete distributions: uniform, binomial, geometric, hypergeometric, negative binomial, Poisson.(4) Random continuos variable, continuos distributions: uniform, exponential, normal.(5) Random discrete two dimensional variable, independence of variables.(6) Mean, variance, correlation coefficient.(7) Chebyshev inequalitiy, large numbers law.(8) Central Limit Theorem, normal approximation.

Calculus B2 Pdf 201.1.9151

Dr. Natalia Karpivnik 5.0 Credits

יום ו 14:00 - 10:00 in קרייטמן-זלוטובסקי(חדש) [34] חדר 3

  1. 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.

Introduction to Differential Equations B Pdf 201.1.9171

Dr. Irena Lerman 4.5 Credits

יום א 16:00 - 14:00 in קרייטמן-זלוטובסקי(חדש) [34] חדר 3
יום ג 11:00 - 09:00 in קרייטמן-זלוטובסקי(חדש) [34] חדר 3

Ordinary differential equations: explicit solutions of first order equations. 2nd order equations. Higher order ordinary differential equations. Systems of ordinary differential equations. 2. Fourier series: Review of series of functions, Fourier expansions and properties of Fourier series, convergence of Fourier series, Gibbs phenomenon. Application to the heat conduction equation. 3. Additional applications as time permits.

Introduction to Differential Equations C Pdf 201.1.9271

Dr. Matan Ziv-Av 4.5 Credits

יום ג 17:00 - 15:00 in מנדל [14] חדר -102
יום ה 10:00 - 08:00 in מנדל [14] חדר -102

  1. Ordinary differential equations: explicit solutions of first -order equations. 2nd order equations. Higher order ordinary differential equations. Systems of ordinary differential equations.
  2. Fourier series: Review of series of functions, Fourier expansions and properties of Fourier series, convergence of Fourier series, Gibbs phenomenon. Application to periodic ODE’s.
  3. The Laplace transform and applications to ODE’s.

Ordinary Differential Equations for Chemistry Students Pdf 201.1.9341

Dr. Natalia Karpivnik 2.5 Credits

יום א 16:00 - 14:00 in קרייטמן-זלוטובסקי(חדש) [34] חדר 5

Basic concepts, direction fields. First order differential equations. Separable and exact equations, integrating factors. Methods for finding explicit solutions, Bernoulli equations. Euler approximations. Examples, polulation growth. Second order differential equations. Equations with constant coefficients, the solution space, the Wronskian. Nonhomogeneous equations. Variation of parameters. Systems of two first order equations with constant coefficients. Examples and applications.

Linear Algebra for Engineering Pdf 201.1.9531

Dr. Stewart Smith 4.5 Credits

יום א 12:00 - 10:00 in בניין אולמות להרצאות [92] חדר 001
יום ג 12:00 - 10:00 in בניין כתות לימוד [35] חדר 2

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.

Linear Algebra for Biotechnology Pdf 201.1.9551

Dr. Natalia Gulko 3.5 Credits

יום א 17:00 - 14:00 in צוקר, גולדשטיין-גורן [72] חדר 209

Complex numbers.Systems of linear equations. Solving linear systems: row reduction and echelon forms. Homogenous and inhomogenous systems.Rank of matrix.Vector spaces. Linearly independent and linearly dependent sets of vectors. Linear combinations of vectors.Inner (dot) product, length, and orthogonality. The Gram - Schmidt process.Matrices: vector space of matrices, linear matrix operations, matrix multiplication, inverse matrix. An algorithm for finding inverse matrix by means of elementary row operations.Rank of matrix and its invertibility. Solving systems of linear equations by means of inverse matrix.Determinants. Condition detA=0 and its meaning. Tranposed matrix.Eigenvectors and eigenvalues. The characteristic polynomial and characteristic equation. Finding of eigenvectors and eigenvalues.Diagonalization and diagonalizable matrices. Symmetric matrices.

Partial Differential Equations For Biotechnology Pdf 201.1.9591

Ms. Tamar Pundik 3.5 Credits

יום א 12:00 - 09:00 in צוקר, גולדשטיין-גורן [72] חדר 209

  1. Classification of linear Partial Differential Equations of order 2, canonical form.
  2. Fourier series (definition, Fourier theorem, odd and even periodic extensions, derivative, uniform convergence).
  3. Examples: Heat equation (Dirichlet’s and Newman’s problems), Wave equation (mixed type problem), Potential equation on a rectangle.
  4. Superposition of solutions, non-homogeneous equation.
  5. Infinite and semi-infinite Heat equation: Fourier integral, Green’s function. Duhamel’s principle.
  6. Infinite and semi-infinite Wave equation: D’Alembert’s solution.
  7. Potential equation on the disc: Poisson’s formula and solution as series.

Fourier analysis and orthonormal systems for physics Pdf 201.1.2021

Dr. Yosef Strauss 3.5 Credits

יום ד 18:00 - 15:00 in גולדברגר [28] חדר 301

  1. מרחבים נורמיים ומרחבי מכפלה פנימית, הקירוב הטוב ביותר והטלות אורתוגונליות, מערכות אורתונורמליות. התכנסות במרחבים נורמיים. מערכות אורתונורמליות אינסופיות, שוויון פרסבל ומערכות אורתונורמליות שלמות.

  2. פולינומים אורתוגונליים. משפט הקירוב של ויירשטראס. שלמות של פולינומים אורתוגונליים בקטע סופי.

  3. טורי פורייה. שלמות, התכנסות נקודתית ותנאים להתכנסות במידה שווה.

  4. טרנספורם פורייה. משפט פלנשרל. נוסחת ההיפוך של פורייה. קונבולוציות. פולינומי הרמיט.

  5. משוואות שטורם-ליוביל בקטע סופי. אורתוגונליות של פונקציות עצמיות. קיום ושלמות של מערכת פונקציות עצמיות עבור בעיית שטורם-ליוביל רגולארית (עם הוכחה חלקית).

ביבליוגרפיה:
  1. Hartman, Philip. Ordinary differential equations. Corrected reprint of the second (1982) edition. With a foreword by Peter Bates. Classics in Applied Mathematics, 38. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002.

  2. Jackson, Dunham. Fourier series and orthogonal polynomials. Reprint of the 1941 original. Dover Publications, Inc., Mineola, NY, 2004.

  3. K?rner, T. W. Fourier analysis. Second edition. Cambridge University Press, Cambridge, 1989.

  1. 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.
  2. 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.
  3. The Fourier transform. The convolution theorem. The Plancherel equality. The inversion theorem. Applications: low pass filters and Shannon’s theorem.
  4. 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.
  5. Introduction to the theory of distributions. Differentiation of distributions, the delta function and its derivatives. Fourier series, Fourier transforms, and Laplace transforms of distributions.
סילבוס:
  1. קבוצות: שייכות, איחוד, חיתוך, הפרש.

  2. מכפלה קרטזית, מושג היחס, יחסי שקילות, יחס סדר חלקי, יחס סדר קווי. הגדרת פונקציה כקבוצת סדורים.

  3. תחשיב הפסוקים: ו/או גרירה, שקילות וטבלאות האמת שלהם, ערך האמת של פסוקים בהשמה, שקילות לוגית וגרירה לוגית, טאוטולוגיות ופסוקים שקריים, הטאוטולוגיות החשובות: למשל, חוקי הפילוג, ונוסחאות דה-מורגן.

  4. תחשיב הפרדיקטים: הגדרת שפת תחשיב הפרדיקטים ומשמעותה; הגדרת מבנים; נוסחאות ופסוקים; הסתפקות במבנה ובהשמה, אמיתיות לוגית, גרירה לוגית, שקילות לוגית; השקילויות החשובות, סדר הכמתים, הכנסת השלילה פנימה.

  5. תורת הקבוצות: התאמות חד-חד-ערכיות, הרכבת פונקציות והפונקציה ההפוכה; יחסי שקילות; הגדרת העוצמה, שיוויון עוצמות ואי-שיוויון עוצמות; משפט קנטור ברנשטיין (ללא הוכחה), המשפט שכל שתי עוצמות נתנות להשוואה (ללא הוכחה); משפט קנטור על עוצמת קבוצות החזקה $|\mathbb{R}|=|\mathcal{P}(\mathbb{N})|$, $|\mathbb{Q}|=|\mathbb{N}\times\mathbb{N}|=|\mathbb{N}|$.

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.

Calculus B1 Pdf 201.1.9141

5.0 Credits

  1. 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.

Calculus C Pdf 201.1.9221

5.0 Credits

  1. 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.
  1. Introduction: the real and complex numbers, polynomials.
  2. Systems of linear equations and Gauss elimination.
  3. 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.
  4. Inverse matrices, the determinant, scalar products.
  5. Linear transformations: kernel and image, the matrix representation of a transformation, change of basis.
  6. Eigenvalues, eigenvectors and diagonalization.
  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. 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).
  7. Eigenvalues, eigenvectors, eigenspaces, diagonalization and similarity, the characteristic polynomial, the algebraic and the geometric multiplicities of an eigenvalue, the spectral theorem for Hermitian matrices.

Topics: 1. Limits and Continuity of functions, applications 2. Differentiability of functions, applications 3. Differentiation techniques 4. Differentiation of Implicit functions, applications 5. Investigation of functions. 6. Multivariable functions, Partial derivatives, applications 7. The Definite Integral 8. The Indefinite Integral 9. Applications of Integrals 10. Integration techniques 11. Taylor polynomials 12. Simple Differential Equations.

  1. Lines and planes. Cross product. Vector valued functions of a single variable, curves in the plane, tangents, motion on a curve.
  2. 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.
  3. 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.
  4. 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.
  5. Parametric representation of surfaces in space, normals, the area of a parametrized surface, surface integrals including reparametrizations
  6. Curl and divergence of vector fields. The theorems of Gauss and Stokes.

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.

  1. 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.

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).

The aim of the course is to study main principles of probability theory. Such themes as probability spaces, random variables, probability distributions are given in details.Some applications are also considered.1. Probability space: sample space, probability function, finite symmetric probability space, combinatorial methods, and geometrical probabilities.2. Conditional probability, independent events, total probability formula, Bayes formula. 3. Discrete random variable, special distributions: uniform, binomial, geometric, negative binomial, hypergeometric and Poisson distribution. Poisson process.4. Continuous random variable, density function, cummulative distribution function. Special distributions: uniform, exponential, gamma and normal. Transformations of random variables. Distribution of maximum and minimum. Random variable of mixed type.5. Moments of random variable. Expectation and variance. Chebyshev inequality.6. Random vector, joint probability function, joint density function, marginal distributions. Conditional density, covariance and correlation coefficient.7. Central Limit Theorem. Normal approximation. Law of Large Numbers.

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, $\mathbb{Z}_m$, invertible elements in $\mathbb{Z}_m$.

  1. טורים מספריים חיוביים וכלליים. התכנסות בהחלט ובתנאי. מבחני שורש והמנה. מבחן ליבניץ

  2. טורי חזקות.

  3. משוואות דיפרנציאליות מסדר ראשון: משוואות ניתנות להפרדת משתנים, משוואות מדויקות, משוואות לינאריות ומשוואות ברנולי. קיום ויחידות.

  4. משוואות דיפרנציאליות מסדר שני: שיטות להורדת סדר, משוואות לינאריות, ורונסקיאן, וריאציה של פרמטרים, משוואות לינאריות עם מקדמים קבועים ושיטת השוואת מקדמים. משוואות דיפרנציאליות מסדר $n$. משוואות אוילר.
  5. מערכות של משוואות דיפרנציאליות: שיטת חילוץ, שימוש באלגברה לינארית.

Notes

  • 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.