2021–22–A

Discrete Mathematics
Pdf 201.1.2201 5.0 Credits

Dr. Yaar Solomon

  • יום א 14:00 - 12:00
  • יום ד 14:00 - 12:00

  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.

Infinitesimal Calculus 1
Pdf 201.1.1011 5.0 Credits

Dr. Gili Golan

  • 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.

Linear algebra 1
Pdf 201.1.1211 5.0 Credits

Prof. Ilya Tyomkin

  • יום א 16:00 - 14:00 בגולדברגר [28] חדר 204
  • יום ג 12:00 - 10:00 בביה”ס אילנות [97] חדר 201

  • Complex numbers. Fields: definition and properties. Examples.
  • Systems of Linear equations. Gauss elimination process.
  • Matrices and operations on them. Invertible matrices.
  • Determinant: definition and properties. Adjoint matrix. Cramer rule.
  • Vector spaces and subspaces. Linear spanning and linear dependence. Basis and dimension. Coordinates with respect to a given basis.
  • Linear transformations. Kernel and Image. Isomorphism of vector spaces. Matrix of a linear transformation with respect to given bases.
  • The space of linear transformations between two vector spaces. Dual space

Algebraic Structures
Pdf 201.1.7031 4.0 Credits

Prof. Ilya Tyomkin

  • יום א 11:00 - 09:00
  • יום ד 14:00 - 12:00

  • Groups as symmetries. Examples: cyclic, dihedral, symmetric, matrix groups.
  • Homomorphism. Subgroups and normal subgroups. Quotient groups. Lagrange’s theorem. The isomorphism theorems. Direct products of groups.
  • Actions of groups on sets. Cayley’s theorem.
  • Group automorphisms.
  • Sylow’s theorems. Application: classification of groups of small order.
  • Composition series and Jordan–Hoelder theorem. Solvable groups.
  • Classification of finite abelian groups, finitely-generated abelian groups.
  • Symmetric group and alternating group. The alternating group is simple.
  • Rings, maximal and prime ideals, integral domain, quotient ring. Homomorphism theorems.
  • Multilinear algebra: Quotient spaces. Tensor products of vector spaces. Action of $S_n$ on tensor powers. Exterior and symmetric algebras. Multilinear forms and determinant.
  • Optional topics: group of symmetries of platonic solids, free groups, semidirect products, representation theory of finite groups.

Fundamentals of Measure Theory(*)
Pdf 201.1.0081 4.0 Credits

Prof. Ilan Hirshberg

  • יום ג 14:00 - 12:00
  • יום ה 16:00 - 14:00

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. Assaf Hasson

  • יום ב 17:00 - 15:00
  • 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.

Probability
Pdf 201.1.8001 4.0 Credits

Prof. Ariel Yadin

  • יום א 16:00 - 14:00
  • יום ב 13:00 - 11:00

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

Prof. Fedor Pakovich

  • יום א 14:00 - 12:00
  • יום ג 10:00 - 08:00

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 Commutative Algebra(*)
Pdf 201.1.7071 4.0 Credits

Prof. Eitan Sayag

  • יום ד 14:00 - 12:00
  • יום ב 15:00 - 13:00

  1. Rings and ideals (revisited and expanded).
  2. Modules, exact sequences, tensor products.
  3. Noetherian rings and modules over them.
  4. Hilbert’s basis theorem.
  5. Finitely generated modules over PID.
  6. Hilbert’s Nullstellensatz.
  7. Affine varieties.
  8. Prime ideals and localization. Primary decomposition.
  9. Discrete valuation rings.

Integral Transforms and Partial Differential Equations
Pdf 201.1.0291 4.0 Credits

Prof. Arkady Poliakovsky

  • יום א 18:00 - 16:00
  • יום ג 18:00 - 16:00

  1. The Fourier transform: convolutions, the inversion formula, Plancherel’s theorem, Hermite functions, tempered distributions. The Poisson summation formula. The Fourier transform in R^n.
  2. The Laplace transform. Connections with convolutions and the Fourier transform. Laguerre polynomials. Applications to ODE’s. Uniqueness, Lerch’s theorem.
  3. Classification of the second order PDE: elliptic, hyperbolic and parabolic equations, examples of Laplace, Wave and Heat equations.
  4. Elliptic equations: Laplace and Poisson equations, Dirichlet and Neumann boundary value problems, Poisson kernel, Green’s functions, properties of harmonic functions, Maximum principle
  5. 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.

Bibliography

  1. Stein E. and Shakarchi R., Fourier analysis, Princeton University Press, 2003.
  2. Korner T.W., Fourier analysis, Cambridge University Press, 1988.
  3. Katznelson Y., An Introduction to Harmonic Analysis, Dover publications. 4. John, Partial differential equations, Reprint of the fourth edition. Applied Mathematical Sciences, 1. Springer-Verlag, New York, 1991.
  4. Evans Lawrence C. Partial Differential Equations, Second Edition.
  5. 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.
  6. Zauderer E. Partial differential equations of applied mathematics, Second edition. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1989. xvi+891 pp. ISBN: 0-471-61298-7.

Introduction to Analysis
Pdf 201.1.1051 4.0 Credits

Dr. Izhar Oppenheim

  • יום ב 11:00 - 09:00
  • יום ד 16:00 - 14:00

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

Dr. Daniel Markiewicz

  • יום א 18:00 - 16:00
  • יום ג 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 Geometric Group Theory
Pdf 201.1.0311 4.0 Credits

Prof. Yair Glasner

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

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

Prof. Dmitry Kerner

  • יום א 16:00 - 14:00
  • יום ד 16:00 - 14:00

  • 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.

Homological Algebra

Prof. Amnon Yekutieli

Time: Wednesday, 12:00 – 14:00
Place: Math Building (no. 58), room 201
or Zoom – TBA

Course Topics: (as much as time permits)

  1. Review of prior material. On rings, ideals and modules (including noncommutative rings).

  2. Categories and functors. Emphasis on linear categories. (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.

  5. Exactness. Exact sequences and functors.

  6. Special modules. Projective, injective and flat modules.

  7. Complexes of modules. Operations on complexes, homotopies, the long exact cohomology sequence.

  8. Resolutions. Projective, flat and injective resolutions.

  9. Left and right derived functors. Applications to commutative algebra.

  10. Further applications of derived functors. Classification problems, extensions.

  11. Morita Theory.

(Some of the material might move to the subsequent course “Commutative Algebra”)

For an updated syllabus and course requirements see the course web page

Basic Concepts in Modern Analysis(#)

Prof. Victor Vinnikov

  • 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. Doubly periodic functions. Lattices, complex tori. The field of elliptic functions.
  2. Riemann surfaces: definitions, maps, the genus, Riemann–Hurwitz formula.
  3. Differential forms on a Riemann surface. The Abel–Jacobi map.
  4. Local study of holomorphic functions. Points of Riemann surfaces as valuations on the field of meromorphic functions. Classifications of the absolute values on the field of rational numbers. The field of p-adic numbers. The product formula.
  5. Algebraic curves over a field. Algebraic curves over C and relation to Riemann surfaces.
  6. Rational points. Arithmetic of curves according to the genus. The case of conics (genus 0). Hasse principle. Finite generation of the rational points on elliptic curves (genus 1): the case of Fermat quartic.
  7. Modular surfaces and modular forms. Analytic construction of rational points on elliptic curves.
  1. Expander graphs and their applications — Equivalent definitions, Mixing Lemma, Alon–Boppana Theorem, Applications, Constructions
  2. Kazhdan Property (T) - group representations (short intro.), Cayley and Schrier graphs, Property (T) — definition and properties, examples, construction of expander graphs via property (T)
  3. Additional topics (as time allows)
  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

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.

  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.
  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. Arkady Poliakovsky

  • 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

Prof. Alexander Ukhlov

  • 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.

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).

Probabilty Theory For EE
Pdf 201.1.9831 3.5 Credits

Dr. Dennis Gulko

יום ג 11:00 - 08:00

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.

  1. Integral calculus in one variable and its application: the integral, Riemann sums, integrability of bounded functions with countably many discontinuity points (the proofs only for continuous functions and monotone functions), antiderivatives and the Fundamental Theorem of Calculus, change of variables and integrations by parts, partial fractions (without proofs). Applications of integral calculus: computation of areas, volume of the solid of revolution, the length of a smooth curve. Improper integral, and convergence tests for positive functions, application to series.
  2. Functions of several variables: open, closed, and compact sets, level curves and surfaces, vector valued functions, paths and path-connectedness.
  3. Limits and continuity in several variables: arithmetic of limits, Weierstrass theorem, intermediate value theorem.
  4. Multivariable differential calculus: partial and directional derivatives, differentiability and the tangent plane, the chain rule, the orthogonality of the gradient to the level surfaces, implicit function theorem for a curve in the plane and a surface in the space (without a proof), the Hessian, Taylor approximation of order 2, critical points (classification only in dimension 2), Extremum problem, including Lagrange multipliers and gradient descent.
  5. Integration in dimension 2: Reimann integral in dimension 2, change of variables and Fubini theorem (without proofs), changing the order of integration, polar coordinates, computation of volumes. If time permits: integration in dimension 3.

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

2021–22–B

Ordered sets and well ordered sets. Ordinals. Linearly ordered sets. Uniqueness of countable linear orders without endpoints.

The set of finite ordinals, construction of the natural numbers, the induction principle and some of its equivalents.

Countable sets, construction of the rational numbers.

Construction of the real field.

Cardinality, cardinals, and the Cantor-Bernstein theorem.

Uncountable sets, Cantor’s theorem, applications.

The axiom of choice and its equivalents (the well ordering principle, Zorn’s lemma).

Applications of the axiom of choice. Transfinite induction.

Throughout the course we will see applications of the course’ material in algebra, logic, graph theory, Euclidean spaces and infinite combinatorics.

Infinitesimal Calculus 2
Pdf 201.1.1021 5.0 Credits

Prof. Eitan Sayag

  • יום ב 14:00 - 12:00 in בניין כתות לימוד [35] חדר 211
  • יום ד 14:00 - 12:00 in בניין כתות לימוד [35] חדר 211

The derivative as a function: continuously differentiable functions, Darboux’s theorem. Convex functions: definition, one-sided differentiability, connection to the second derivative. Cauchy’s generalized Mean Value Theorem and its applications: L’Hospital’s rule, Taylor polynomials with Lagrange remainder. The Newton-Raphson method. Series: Cauchy’s criterion, absolutely convergent series, the comparison, quotient and root tests, the Dirichlet test, change of the order of summation, the product formula for series, Taylor series, Taylor series of elementary functions. The definition of an analytic function, the radius of convergence of a power series. The Riemann integral. Riemann sums. The fundamental theorem of calculus (the Newton-Leibniz formula). Methods for computing integrals (the indefinite integral): integration by parts, change of variable, partial fractions. Improper integrals. Numerical integration: the midpoint, trapezoid and Simpson’s rules. Stirling’s formula. Introduction to convergence of functions, problems with pointwise convergence. Introduction to ordinary differential equations: the differential equation y’=ky, solution of first order ODE’s by separation of variables, initial value conditions.

Linear algebra 2
Pdf 201.1.1221 5.0 Credits

Dr. Yair Hartman

  • יום א 12:00 - 10:00 in בניין כתות לימוד [35] חדר 310
  • יום ג 14:00 - 12:00 in בניין כתות לימוד [35] חדר 213

  • Rings. Ring of polynomials and its ideal structure. The prime factorization of a polynomial. Lagrange interpolation.
  • Eigenvalues and eigenvectors of linear operators.
  • Characteristic polynomial and Cayley-Hamilton theorem. The primary decomposition theorem. Diagonalization. Nilpotent operators. Jordan decomposition in small dimension Jordan decomposition in general dimension- time permitted
  • Linear forms. Dual basis. Bilinear forms.
  • Inner product spaces. Orthogonal bases. Projections. Adjoint linear transformation. Unitary and Hermitian operators.
  • Normal operators and the spectral decomposition theorem. Singular value decomposition theorem and applications.

Graph Theory
Pdf 201.1.6081 4.0 Credits

Prof. Shakhar Smorodinsky

  • יום א 14:00 - 12:00 in גולדברגר [28] חדר 101
  • יום ד 17:00 - 15:00 in גוטמן [32] חדר 309

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.

Introduction to Topology
Pdf 201.1.0091 4.0 Credits

Prof. Michael Levin

  • יום ב 18:00 - 16:00 in קרייטמן-זלוטובסקי(חדש) [34] חדר 3
  • יום ה 16:00 - 14:00 in קרייטמן-זלוטובסקי(חדש) [34] חדר 18

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.

Theory of Functions of a Complex Variable
Pdf 201.1.0251 4.0 Credits

Dr. Eli Shamovich

  • יום ג 14:00 - 12:00 in גולדברגר [28] חדר 106
  • יום ה 10:00 - 08:00 in גולדברגר [28] חדר 104

  • Complex numbers. Analytic functions, Cauchy–Riemann equations.
  • Conformal mappings, Mobius transformations.
  • Integration. Cauchy Theorem. Cauchy integral formula. Zeroes, poles, Taylor series, Laurent series. Residue calculus.
  • The theorems of Weierstrass and of Mittag-Leffler. Entire functions. Normal families.
  • Riemann Mapping Theorem. Harmonic functions, Dirichlet problem.

Field Theory and Galois Theory(*)
Pdf 201.1.7041 4.0 Credits

Dr. David Corwin

  • יום ב 16:00 - 14:00 in גוטמן [32] חדר 108
  • יום ה 18:00 - 16:00 in גולדברגר [28] חדר 101

  • 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

Coding Theory

Prof. Ido Efrat

  • יום ג 10:00 - 08:00 in גוטמן [32] חדר 111
  • יום ה 12:00 - 10:00 in גולדברגר [28] חדר 107

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.

Topics
  1. The main problem of Coding Theory
  2. Bounds on codes
  3. Finite fields
  4. Linear codes
  5. Perfect codes
  6. Cyclic codes
  7. Sphere packing
  8. Asymptotic bounds
Bibliography:

R. Hill, A First Course in Coding Theory, Clarendon Press, Oxford 1986

Geometric infinitesimal calculus 2
Pdf 201.1.1041 4.0 Credits

Prof. Dmitry Kerner

  • יום א 18:00 - 16:00 in גולדברגר [28] חדר 202
  • יום ד 15:00 - 13:00 in קרייטמן-זלוטובסקי(חדש) [34] חדר 7

Embedded differentiable manifolds with boundary in Euclidean space. The tangent space, normal, vector fields. Orientable manifolds, the outer normal orientation. Smooth partitions of unity. Differential forms on embedded manifolds, the exterior derivative. Integration of differential forms and the generalized Stokes theorem. Classical formulations (gradient, curl and divergence and the theorems of Green, Stokes and Gauss). Closed and exact forms. Conservative vector fields and existence of potentials. Application to exact ordinary differential equations. Introduction to differential geometry: curvature of curves and surfaces in 3 dimensional space, the Gauss map, the Gauss-Bonnet theorem (time permitting).

Ordinary Differential Equations
Pdf 201.1.0061 4.0 Credits

Dr. Eli Shamovich

  • יום א 16:00 - 14:00 in קרייטמן-זלוטובסקי(חדש) [34] חדר 116
  • יום ה 14:00 - 12:00 in קרייטמן-זלוטובסקי(חדש) [34] חדר 18

Ordinary differential equations of first order, existence and uniqueness theorems, linear equations of order $n$ and the Wronskian, vector fields and autonomous equations, systems of linear differential equations, nonlinear systems of differential equations and stability near equilibrium

Approximation Theory
Pdf 201.1.0121 4.0 Credits

Prof. Amnon Besser

  • יום ב 12:00 - 10:00 in קרייטמן-זלוטובסקי(חדש) [34] חדר 5
  • יום ד 13:00 - 11:00 in קרייטמן-זלוטובסקי(חדש) [34] חדר 7

  1. Preliminaries: floating point arithmetic, round-off errors and stability. Matrix norms and the condition number of a matrix.
  2. Introduction to numerical solutions for ODE’s:initial value problems, Euler’s method, introduction to multistep methods. Boundary value problems.
  3. Numerical solution of linear equations: Gauss elimination with pivoting, LU decomposition. Iterative techniques: Jacobi, Gauss-Seidel, conjugate gradient. Least squares approximation.
  4. Numerical methods for finding eigenvalues: Gershgorin circles. The power method. Stability considerations in Gram-Schmidt: Hausholder reflections and Givens rotations. Hessenberg and tridiagonal forms. QR decomposition and the QR algorithm.

Introduction to representation theory of groups
Pdf 201.1.0511 4.0 Credits

Dr. Inna Entova-Aizenbud

  • יום א 12:00 - 10:00 בגולדברגר [28] חדר 107
  • יום ד 11:00 - 09:00 בגוטמן [32] חדר 207

  1. Introduction: Actions of groups on sets. Induced linear actions. Multilinear algebra.
  2. Representations of groups, direct sum. Irreducible representations, semi-simple representations. Schur’s lemma. Irreducible representations of finite abelian groups. Complete reducibility, Machke’s theorem.
  3. Equivalent representations. Morphisms between representations. The category of representations of a finite group. A description using the group ring. Multilinear algebra of representations: dual representation, tensor product (inner and outer).
  4. Decomposition of the regular representation into irreducible representations. The number of irreducibles is equal to the number of conjugacy classes. Matrix coefficients, characters, orthogonality.
  5. Harmonic analysis: Fourier transform on finite groups and the non-commutative Fourier transform.
  6. Frobenius divisibility and Burnside $p^aq^b$ theorem.
  7. Constructions of representations: induced representations. Frobenius reciprocity. The character of induced representation. Mackey’s formula. Mackey’s method for representations of semi-direct products.
  8. Induction functor: as adjoint to restrictions, relation to tensor product. Restriction problems, multiplicity problems, Gelfand pairs and relative representation theory.
  9. Examples of representations of specific groups: $SL_2$ over finite fields, Icosahedron group, Symmetric groups.
  10. Artin and Brauer Theorems on monomial representations

Model theory and applications

Dr. Moshe Kamensky

  • יום ב 18:00 - 16:00 in צוקר, גולדשטיין-גורן [72] חדר 119
  • יום ג 12:00 - 10:00

The aim of the course is to present applications of model theory (a branch of Mathematical Logic) in one or more area of mathematics. The particular direction will be determined by coordination with the students, but might include the following:

  • Algebraic theory of differential equations (Galois theory, dimension, classification in dimension 1)
  • Model theory of valued fields (imaginaries, integration theory, analytic spaces)
  • o-minimality, with applications to arithmetic
  • Difference fields and difference equations, applications to dynamics, asymptotic theory of the Frobenius
  • Continuous model theory, applications to operator algebras, probability etc. Background from logic and other relevant areas will be covered as necessary.

Introduction to Ergodic Theory

Prof. Tom Meyerovitch

  • יום ב 14:00 - 12:00 in גולדברגר [28] חדר 304
  • יום ג 14:00 - 12:00

  1. Measure preserving systems for Borel group actions
  2. Ergodicity, ergodic decomposition, mixing and weak mixing
  3. Minimality and unique ergodicity
  4. Mean and pointwise ergodic theorems for a single transformation
  5. (*) Joinings
  6. (*) Decomposition of a measure relative to a partition and conditional measures
  7. (*) Entropy
  8. (*) Ergodic theorems for general groups and amenability.

(*) denotes optional topics to be covered according to class level and teacher discretion.

Basic notions in geometry and topology 2

Dr. Michael Brandenbursky

  • יום ב 11:00 - 09:00 in גולדברגר [28] חדר 304
  • יום ד 11:00 - 09:00 in צוקר, גולדשטיין-גורן [72] חדר 124

  1. Cohomology: definitions, Universal coefficient theorem, Orientation, Poincare duality, cap and cup products, cohomology ring, Kunneth formula
  2. Review of (Smooth manifolds, differential forms, orientability, Stokes theorem), degree of the map, Sard theorem, De-Rham cohomology.
  3. Isomorphism between De-Rham cohomology and singular cohomology.
  4. (If time permits) additional topics according to the instructor preferences

Fundamentals of Analysis for EE - part I

Prof. Arkady Leiderman

  • יום ג 14:00 - 12:00 in קרייטמן-זלוטובסקי(חדש) [34] חדר 205
  • יום ה 14:00 - 12:00 in גולדברגר [28] חדר 107

Metric spaces:

closed sets, open sets, Cauchy sequences, completeness, compactness, Theorem of Heine–Borel, continuity and uniform continuity of functions, uniform convergence of sequences of functions.

Measure theory:

algebras, measures and outer measures, measurable sets, discrete measure spaces, Lebesgue measure on the real line, measurable functions, Lebesgue integral, dominated convergence theorem, $L_p$-spaces as complete normed spaces. Time-permitting: signed measures and absolute continuity of measures and the Radon-Nykodim theorem.

Commutative Algebra
Pdf 201.2.2011 2.0 Credits

Prof. Amnon Yekutieli

יום ד 15:00 - 13:00

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.

Prodability and Statistics For Software Engineering
Pdf 201.1.2381 2.5 Credits

Dr. Guy Landsman

יום ד 12:00 - 10:00 in בניין אולמות להרצאות [92] חדר 002

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”.

  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
  • Rings. Ring of polynomials and its ideal structure. The prime factorization of a polynomial. Lagrange interpolation.
  • Eigenvalues and eigenvectors of linear operators. Characteristic polynomial and Cayley–Hamilton theorem. The primary decomposition theorem. Diagonalization. Nilpotent operators. Jordan decomposition in small dimension. Jordan decomposition in general dimension- time permitting.
  • Linear forms. Dual basis. Bilinear forms. Inner product spaces. Orthogonal bases. Projections. Adjoint linear transformation. Unitary and Hermitian operators. Normal operators and the spectral decomposition theorem. Singular value decomposition theorem and applications.

Optional topics:

  • Quadratic forms.
  • Sylvester theorem.
  • Classification of quadrics in two-dimensional spaces.

Introduction to Probability and Statistics A
Pdf 201.1.9091 2.5 Credits

Dr. Luba Sapir

יום ב 14:00 - 12:00 in בניין כתות לימוד [35] חדר 312

(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.

  1. Review of probability: a. Basic notions. b. Random variables, Transformation of random variables, Independence. c. Expectation, Variance, Co-variance. Conditional Expectation.
  2. Probability inequalities: Mean estimation, Hoeffding?s inequality.
  3. Convergence of random variables: a. Types of convergence. b. The law of large numbers. c. The central limit theorem.
  4. Statistical inference: a. Introduction. b. Parametric and non-parametric models. c. Point estimation, confidence interval and hypothesis testing.
  5. 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
  6. Parametric interval estimation a. Introduction. b. Pivotal Quantity. c. Sampling from the normal distribution: confidence interval for mean, variance. d. Large-sample confidence intervals.
  7. 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.
  8. Regression. a. Simple linear regression. b. Least Squares and Maximum Likelihood. c. Properties of least Squares estimators. d. Prediction.
  9. Handling noisy data, outliers.

Calculus B1
Pdf 201.1.9141 5.0 Credits

יום ו 15:00 - 11:00 in גולדברגר [28] חדר 101

  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. Avi Goren

  • יום ב 17:00 - 15:00 in בניין אולמות להרצאות [92] חדר 001
  • יום ד 16:00 - 14:00 in בניין כתות לימוד [35] חדר 212

  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.

Calculus C
Pdf 201.1.9221 5.0 Credits

יום ב 14:00 - 11:00 in קרייטמן-זלוטובסקי(חדש) [34] חדר 110

  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 Linear Algebra C
Pdf 201.1.9281 3.5 Credits

Mr. Avrham Bourla

יום ד 18:00 - 15:00 in בנין 90 (מקיף ז’) [90] חדר 225

  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.

Vector calculus for Electric Engineering
Pdf 201.1.9631 5.0 Credits

  • יום ב 16:00 - 14:00 in גולדברגר [28] חדר 202
  • יום ה 11:00 - 09:00 in גולדברגר [28] חדר 203

  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.

Introduction to Discrete Mathematics
Pdf 201.1.9661 3.5 Credits

Prof. Gregory Mashevitsky

יום א 18:00 - 15:00 in בניין אולמות להרצאות [92] חדר 001

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.

Differential Calculus for EE
Pdf 201.1.9671 5.0 Credits

  • יום ב 10:00 - 08:00 in גולדברגר [28] חדר 105
  • יום ד 10:00 - 08:00 in גולדברגר [28] חדר 107

  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.

Calculus 1 for engineering
Pdf 201.1.9711 5.0 Credits

  • יום א 14:00 - 12:00 in גולדברגר [28] חדר 107
  • יום ד 12:00 - 10:00 in גוטמן [32] חדר 309

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).

Probabilty Theory For EE
Pdf 201.1.9831 3.5 Credits

Prof. Ariel Yadin

יום א 17:00 - 14:00 in קרייטמן(אוד.) [26] חדר 5

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.

  1. Complex numbers, open sets in the plane.
  2. Continuity of functions of a complex variable
  3. Derivative at a point and Cauchy–Riemann equations
  4. Analytic functions; example of power series and elementary functions
  5. Cauchy’s theorem and applications.
  6. Cauchy’s formula and power series expansions
  7. Morera’s theorem
  8. Existence of a logarithm and of a square root
  9. Liouville’s theorem and the fundamental theorem of algebra
  10. Laurent series and classification of isolated singular points. The residue theorem
  11. Harmonic functions
  12. Schwarz’ lemma and applications
  13. Some ideas on conformal mappings
  14. Computations of integrals

Partial Differential Equations
Pdf 201.1.0101 2.5 Credits

Prof. Boris Zaltzman

יום ה 14:00 - 12:00 in בניין כתות לימוד [35] חדר 211

  1. Second order linear equations with two variables: classification of the equations in the case of constant and variable coefficients, characteristics, canonical forms.
  2. Sturm-Liouville theory.
  3. 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.
  4. 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.
  5. 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.
  6. Non-homogeneous heat equations, Poisson equations in a circle and non-homogeneous wave equations.
  7. If time permits: free vibrations in circular membranes. Bessel equations.

Calculus 2 For Computer Science and Software Engineering
Pdf 201.1.2371 5.0 Credits

  • יום ד 10:00 - 08:00 in גולדברגר [28] חדר 105
  • יום ה 10:00 - 08:00 in גולדברגר [28] חדר 107

  1. Integral calculus in one variable and its application: the integral, Riemann sums, integrability of bounded functions with countably many discontinuity points (the proofs only for continuous functions and monotone functions), antiderivatives and the Fundamental Theorem of Calculus, change of variables and integrations by parts, partial fractions (without proofs). Applications of integral calculus: computation of areas, volume of the solid of revolution, the length of a smooth curve. Improper integral, and convergence tests for positive functions, application to series.
  2. Functions of several variables: open, closed, and compact sets, level curves and surfaces, vector valued functions, paths and path-connectedness.
  3. Limits and continuity in several variables: arithmetic of limits, Weierstrass theorem, intermediate value theorem.
  4. Multivariable differential calculus: partial and directional derivatives, differentiability and the tangent plane, the chain rule, the orthogonality of the gradient to the level surfaces, implicit function theorem for a curve in the plane and a surface in the space (without a proof), the Hessian, Taylor approximation of order 2, critical points (classification only in dimension 2), Extremum problem, including Lagrange multipliers and gradient descent.
  5. Integration in dimension 2: Reimann integral in dimension 2, change of variables and Fubini theorem (without proofs), changing the order of integration, polar coordinates, computation of volumes. If time permits: integration in dimension 3.

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..

Calculus 2 for Biotechnology
Pdf 201.1.9571 5.0 Credits

  • יום ב 18:00 - 16:00 in בנין 90 (מקיף ז’) [90] חדר 127
  • יום ד 11:00 - 09:00 in בנין 90 (מקיף ז’) [90] חדר 230

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.

Ordinary Differential Equations for BE
Pdf 201.1.9581 3.5 Credits

Dr. Natalia Gulko

יום ד 15:00 - 12:00 in קרייטמן-זלוטובסקי(חדש) [34] חדר 210

  1. 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.

  1. 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.
  1. Infinite series of nonnegative terms and general series. Absolute and conditional convergence. Power series.
  2. Vector algebra. Dot product, cross product and box product.
  3. Analytic geometry of a line and a plane. Parametric equations for a line. Canonic equations for a plane. Points, lines and planes in space.
  4. Vector-valued functions. Derivative. Parametrized curves. Tangent lines. Velocity and acceleration. Integration of the equation of motion.
  5. Surfaces in space. Quadric rotation surfaces. Cylindrical and spherical coordinates.
  6. 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.
  7. Vector-valued functions of several variables. Vector field. Field curves. Divergence and curl.
  8. Line and path integrals. Work, circulation. Conservative fields. Potential function.
  9. Double integral and its applications. Green’s theorem.
  10. Parametrized surfaces. Tangent plane and normal line. Surface integrals. Flux. Stokes’s theorem.
  11. Triple integral and its applications. Divergence theorem.
  1. 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.
  2. 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
  3. 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.
  4. 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.

Introduction to Statistics A
Pdf 201.1.9421 2.5 Credits

Dr. Matan Ziv-Av

יום א 16:00 - 14:00 in גוטמן [32] חדר 308

  1. 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

Linear Algebra for Electrical Engineering 1
Pdf 201.1.9511 3.5 Credits

יום א 14:00 - 11:00 in גוטמן [32] חדר 111

  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.

Part A: Logic and set-theory. Propositional calculus, Boolean operations. Truth tables, the truth-value of a propositional formula (without induction at this stage), logical implication and logical equivalence, tautologies and contradictions, the useful tautologies, distributivity and de-Morgan’s Law. Sets: the notion of a set, membership and equality, operations: union, intersection, set-difference and power-set. Ordered pairs and Cartesian products. Equivalence relations, quotient spaces and partitions.Partial orders. Functions, injective and surjective functions, invertibility of a function. The ordered set of natural numbers.The axiom of induction in different forms.

Part B: Finite and infinite sets. The notion of cardinality. Countable sets. Cantor’s theorem on the power set of a set.

Part C: Combinatorics. Basic counting formulas. Binomials. Inclusion-exclusion technique. Recursive definition and formulas.

Part D: Graph Theory. Graphs, examples, basic facts, vertex degrees, representing a graph, neighborhood matrices, connected components, Euler graphs, bipartite graphs, matching in bipartite graphs, Hall’s marriage theorem, graph colorings.

  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.

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.