2020–21–A

Discrete Mathematics
Pdf 201.1.2201 5.0 Credits

  • יום א 14:00 - 12:00
  • יום ה 12:00 - 10:00

  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.

Algebra 1
Pdf 201.1.7011 5.0 Credits

  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

Prof. Tom Meyerovitch

  • יום ב 12:00 - 10:00
  • יום ד 12:00 - 10:00

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.

Algebraic Structures
Pdf 201.1.7031 4.0 Credits

Prof. Nadya Gurevich

  • יום ג 11:00 - 09:00
  • יום ה 14:00 - 12:00

  • 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 4.0 Credits

Prof. Victor Vinnikov

  • יום ב 16:00 - 14:00
  • יום ד 15:00 - 13:00

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 4.0 Credits

Prof. Menachem Kojman

  • יום ב 18:00 - 16:00
  • יום ד 12:00 - 10:00

  • 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.

תורת הגרפים
Pdf 201.1.6081 4.0 Credits

Prof. Shakhar Smorodinsky

  • יום א 12:00 - 10:00
  • יום ג 14:00 - 12:00

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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.

Probability
Pdf 201.1.8001 4.0 Credits

Prof. Ariel Yadin

  • יום א 16:00 - 14:00
  • יום ב 12:00 - 10:00

An introduction to the basic notions of probability theory:

sample spaces limits of events conditional probability independent events sigma algebras, continuous spaces, Lebesgue measure random variables and distributions independence expectation variance and covariance convergence of random variables: almost-sure, in Lp, in probability law of large numbers convergence in law central limit theorem

Theory of Numbers
Pdf 201.1.6031 4.0 Credits

Dr. Moshe Kamensky

  • יום ב 10:00 - 08:00
  • יום ה 12:00 - 10:00

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 4.0 Credits

Dr. Izhar Oppenheim

  • יום ב 14:00 - 12:00
  • יום ד 10:00 - 08:00

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 4.0 Credits

Prof. Fedor Pakovich

  • יום א 10:00 - 08:00
  • יום ד 18:00 - 16:00

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.

Introduction to Algebraic Geometry
Pdf 201.1.6171 4.0 Credits

Prof. Ilya Tyomkin

  • יום ג 12:00 - 10:00
  • יום ה 14:00 - 12:00

  1. Affine and projective spaces, affine and projective maps, Segre and Veronese embeddings, Desargues’s Theorem, Pappus’s Theorem, cross-ratio, projective duality
  2. Plane curves: rational curves, linear systems of curves, conics and the Butterfly Theorem, Pascal’s Theorem, Chasles’s Theorem, the group structure on a planar cubic, Bezout’s Theorem
  3. Affine algebraic varieties: Hilbert’s Basis Theorem, Zariski topology, irreducible components, Hilbert’s Nullstellensatz, the correspondence between the ideals and the algebraic sets, morphisms and rational maps between affine algebraic varieties
  4. Projective varieties: graded rings and homogeneous ideals, the projective correspondence, morphisms, blow-ups, birational equivalence and rational varieties, Grassmannians
  5. The basics of dimension theory
  6. The basics of smoothness
  7. Cubic surfaces and 27 lines. If time permits, other topics will be discussed such as abstract algebraic varieties, Chevaley’s Theorem, Riemann-Roch Theorem and its applications.

O-minimality: topology without pathologies
Pdf 201.1.6141 4.0 Credits

Prof. Assaf Hasson

  • יום א 14:00 - 12:00 בגולדברגר [28] חדר 201
  • יום ג 16:00 - 14:00 בגולדברגר [28] חדר 301

In the 1980s A. Grothendieck suggest a project for developing a tame topology that will not suffer from the many counter-examples and pathologies known in classical topology. Nowadays many view the notion of o-minimality as successful fulfillment of this program: in o-minimal fields all (unary) functions are piecewise differentiable (and therefore infinitely differentiable at almost every point); unary functions are piecewise monotone, connectedness is the same as path connectedness and the axiom of choice holds for definable sets. In the o-minimal setting most of the classical differential calculus can be developed, and so are large portion of the theory of Lie groups, algebraic topology and much more. O-minimality plays a key role in real geometry and in recent years had a crucial role in important breakthroughs in Diophantine geometry and in Hodge theory.

In the course we will define o-minimality and develop its basic theory. We will show that real closed fields are o-minimal and discuss – time permitting – some applications.

Workshop in Analysis
Pdf 201.1.1071 2.0 Credits

Dr. Yair Hartman

יום ג 10:00 - 08:00 בבניין כתות לימוד [35] חדר 115

The goal of the workshop is to examine and provide complimentary material for the course “Geometric Calculus 1” 201.1.1031 (as well as the course “Introduction to Analysis” 201.1.1051). The workshop is given in parallel with Geometric Calculus 1, and the workshop content follows the courses. Part of the main goals of the workshop, especially due to the Corona pandemic, is to improve the student’s teamwork skills. During the workshop, the students will work in small groups and will practice their “mathematical conversation” skills: how to think together, how to find the essence of an idea and how to present a mathematical idea to others.

Basic Concepts in Topology and Geometry(#)
Pdf 201.2.5221 4.0 Credits

Dr. Michael Brandenbursky

  • יום א 12:00 - 10:00
  • יום ג 14:00 - 12:00

  • 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.

Introduction to $C^*$-algebras

Prof. Ilan Hirshberg

  • יום ב 18:00 - 16:00
  • יום ד 12:00 - 10:00

  1. Background material in functional analysis and Hilbert space theory; basics of Banach algebras, and Gelfand theory.
  2. Basic results: commutative $C^*$-algebras, positivity, ideals and quotients, states and representations, the GNS constuction, the algebra of compact operators.
  3. A discussion of some basic examples in the theory (e.g. AF algebras, Toeplitz and Cuntz algebras, irrational rotation algebras, as time permits). Introduction to K-theory.

Further topics as time permits.

Functions of several complex variables and introduction to Singularities

Prof. Dmitry Kerner

  • יום א 16:00 - 14:00
  • יום ד 10:00 - 09:00
  • יום ד 18:00 - 17:00

  1. Riemann surfaces
  2. Holomorphic functions of several variables
  3. Isolated singularities of holomorphic functions
  4. Fundamentals of differential topology
  5. Topology of singularities.

See course web page for details

Please email me if you want to attend (even just for the first one or two lectures, to “taste” the course). I will send you the zoom link.

Basic Concepts in Modern Analysis(#)

Dr. Eli Shamovich

  • יום ב 12:00 - 10:00
  • יום ה 16:00 - 14:00

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.

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”.

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 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 B2
Pdf 201.1.9151 5.0 Credits

Dr. Natalia Karpivnik

יום ו 14:00 - 10:00

  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 4.5 Credits

Prof. Fedor Pakovich

  • יום א 16:00 - 14:00
  • יום ג 11:00 - 09:00

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.

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.

Introduction to Differential Equations C
Pdf 201.1.9271 4.5 Credits

Dr. Saak Gabrielyan

  • יום ג 17:00 - 15:00
  • יום ה 10:00 - 08:00

  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.
  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.

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.

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.

Linear Algebra for Engineering
Pdf 201.1.9531 4.5 Credits

Dr. Stewart Smith

  • יום א 12:00 - 10:00
  • יום ג 12:00 - 10:00

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.

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.

  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.
  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 $\mathbb{Z}_m$.

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..

  1. Fields: the definition of a field, complex numbers.

  2. Linear equations: elementary operations, row reduction, homogeneous and non-homogeneous equations, parametrization of solutions.

  3. Vector spaces: examplex, subspaces, linear independence, bases, dimension.

  4. Matrix algebra: matrix addition and multiplication, elementary operations, the inverse matrix, the determinant and Cramer’s law. Linear transformations: examples, kernel and image, matrix representation.

  1. Complex valued-functions and the complex exponential. Fourier coefficients of piecewise continuous periodic functions. Basic operations and their effects on Fourier coefficients: translation, modulation, convolutions, derivatives.
  2. Uniform convergence: Cesaro means, the Dirichlet and Fejer kernels, Fejer’s theorem. The Weierstrass approximation theorem for trigonometric polynomials and for polynomials. Uniqueness of Fourier coefficients. The Riemann-Lebesgue lemma. Hausdorff’s moment problem. Convergence of partial sums and Fourier series for $C^2$-functions.
  3. Pointwise convergence: Dini’s criterion. Convergence at jump discontinuities and Gibbs phenomenon.
  4. $L^2$-theory: orthonormal sequences and bases. Best approximations, Bessel’s inequality, Parseval’s identity and convergence in $L^2$.
  5. Applications to partial differential equations: the heat and wave equations on an interval with constant boundary conditions, the Dirichlet problem for the Laplace equation on the disk, the Poisson kernel.

Bibliography:

  • Korner, Fourier analysis
  • Stein and Shakarchi, Fourier analysis

2020–21–B

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.