All catalogue courses

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. 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.

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.

1. Complex numbers: Cartesian coordinates, polar coordinates. Functions of a complex variable. Basic properties of analytic functions, the exponential function, trigonometric functions. Definition of contour integral. The Cauchy Integral Formula. Residues and poles. Evaluation of impoper real integrals with the use of residues.
2. Inner product functional spaces. Orthogonal and orthonormal systems. Generalized Fourier series. Theorem on orthogonal projection. Bessel’s inequality and Parseval’s equality.
3. Trigonometric Fourier series. Complex form of Fourier series. Fourier series expansion defined over various intervals. Pointwise and uniform convergence of Fourier series. Completness of trigonometric system and Parseval’s equality. Differentiation and integration of Fourier series.
4. The Fourier integral as a limit of Fourier series. The Fourier transform: definition and basic properties. The inverse Fourier transform. The convolution theorem, Parseval’s theorem for the Fourier transform. A relation between Fourier and Laplace transforms. Application of Fourier transform to partial differential equations and image processing.
5. Distributions (generalized functions). The Heaviside step function, the impulse delta-function. Derivative of distribution. Convergence of sequences in the space of distributions. The Fourier transform of distributions.

1. Introduction to scalars and vectors: operations with scalars and vectors, scalar product, vector product, vector equations, indices, Einstein summation convention, kronecker delta, Levi-Civita symbol.
2. Introduction to functions: types, continuity, limits, derivatives, integrals, integration methods, taylor series, remainder.
3. Functions with scalar input and vector output: curves, tangent, normal, velocity, acceleration, curvature, torsion. Frenet-Serret basis and kinematics.
4. Functions with vector input and scalar output: extrema, contours, gradient, directional derivative, tangent space.
5. Multiple integrals
6. Functions with vector input and vector output: conservative fields, rotational fields, divergence, curl, line integral, surface integral, Stokes and Gauss
7. Differential equations: damped harmonic oscillator with driving
8. Rotations: scalars, vectors, tensors
9. Curvilinear coordinates

1. Real numbers and real line, elementary functions and graphs, some functions arising in economics. The limit of a function, calculating limits using the limit laws, continuity, the number e.2. The derivative of a function, differential rules, higher derivatives, L’Hospital rules.3. Extreme values of functions, monotonic functions, point of inflection, concavity, curve sketching, applications to economics.4. Indefinite integrals, techniques of integration, definite and improper integrals, areas between curves, applications to economics.5. Functions of several variables, economics examples, partial derivatives, linearization, the chain rile, implicit and homogeneous functions, maximum and minimum, Lagrange multipliers.6. Introduction to linear algebra, matrices, linear systems.

1. 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. 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. 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.

Prerequisites: 20119531 Linear Algebra

Brief syllabus

1. Operations over sets, logical notation, relations.

2. Enumeration of combinatorial objects: integer numbers, functions, main principles of combinatorics.

3. Elementary combinatorics: ordered and unordered sets and multisets, binomial and multinomial coefficients.

4. Principle of inclusion and exclusion, Euler function.

5. Graphs: representation and isomorphism of graphs, valency, paths and cycles.

6. Recursion and generating functions: recursive definitions, usual and exponential generating functions, linear recurrent relations with constant coefficients.

7. (Optional) Modular arithmetics: congruences of integer numbers, $\mathbb{Z}_m$, invertible elements in $\mathbb{Z}_m$.

1. 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.

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.

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.

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 system of the real numbers: the natural numbers, well order, the inegers, the rational numbers, arithmetical operations and the field axioms, the order on the rational numbers, the Archimedian axiom, non completeness of the rationals, irrational numbers and the completeness of the reals, bounded sets, upper and lower bounds, maximum and minimum, supremum and infimum, rational and irrational powers, the basic inequalities (Bernoulli, Cauchy-Schwarz, the mean values, Holder, Minkowskii), the rational roots of a polynomial equation over the rational numbers.2) Sequences and their limits, the arithmetic of limits, divergence and tending to infinity, inequalities between sequences and between their limits, the Sandwitch Theorem, monotonic sequences, recursive sequences, Cantor’s Lemma, sub sequences, the Theorem of Bolzano-Weierstrass, the exponent, Cauchy’s criterion for the convergence of a sequence, upper limit, lower limit.3) Functions of a single variable, arithmetic operations on functions, monotone functions, the elementary functions.4) The limit of a function, the sequential definition and non sequential definition, the arithmetic of limits, one sided limits, Cauchy’s criterion, bounded functions, the order of magnitude of a function (big Oh and small Oh).5) Continuous functions, classification of discontinuities, the mean value theorem and its applications, continuity and being monotone.6) The derivative of a function, the graph of a function and the tangent line to the graph, the slope of the tangent line, the velocity, differentiability and continuity, the arithmetic of the differential operator in particular the Liebnitz rule, composition of functions, the chain rule, higher order derivatives, a theorem of Fermat, Roll’s Theorem, the maen value theorem of Lagrange, the mean value theorem of Cauchy, L’hospital rules, Taylor’s Theorem.7) Local and global maximum and minimum, inflection points, convex and concave functions, asymptotic lines, sketching

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The Lab will be based on a LINUX platform, and use the C programming language. The emphasis is on low-level programming. Goals of this lab are to introduce issues in low level programming, as well as techniques on how to learn needed information on demand

• Variable types: Numbers and strings, basic operations
• Input and output
• Conditionals and loops
• Lists, dictionaries and other data types
• Functions
• Recursions
• Object oriented programming
• Modules
• Introduction to computational complexity

1. Partially ordered sets. Chains and antichains. Examples. Erdos–Szekeres’ theorem or a similar theorem. The construction of a poset over the quotient space of a quasi-ordered set.
2. Comparison of sets. The definition of cardinality as as an equivalence class over equinumerousity. The Cantor-Bernstein theorem. Cantor’s theorem on the cardinality of the power-set.
3. Countable sets. The square of the natural numbers. Finite sequences over a countable set. Construction of the ordered set of rational numbers. Uniqueness of the rational ordering.
4. Ramsey’s theorem. Applications.
5. The construction of the ordered real line as a quotient over Cauchy sequences of rationals.
6. Konig’s lemma on countably infinite trees with finite levels. Applications. A countable graph is k-colorable iff every finite subgraph of it is k-colorable.
7. Well ordering. Isomorphisms between well-ordered sets. The axiom of choice formulated as the well-ordering principle. Example. Applications. An arbitrary graph is k–colorable iff every finite subgraph is k-colorable.
8. Zorn’s lemma. Applications. Existence of a basis in a vector space. Existence of a spanning tree in an arbitrary graph.
9. Discussion of the axioms of set theory and the need for them. Russel’s paradox. Ordinals.
10. Transfinite induction and recursion. Applications. Construction of a subset of the plane with exactly 2 point in every line.
11. Infinite cardinals as initial ordinals. Basic cardinal arithmetic. Cardinalities of well known sets. Continuous real functions, all real runctions, the automorphisms of the real field (with and without order).

סילבוס אינו מוגדר בתקופה המתבקשת

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.

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.

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.

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.

סילבוס אינו מוגדר בתקופה המתבקשת

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.

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

1. 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.

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. 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

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.

סילבוס:

1. קבוצות: שייכות, איחוד, חיתוך, הפרש.

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

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

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

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

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

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.

Free oscillations of simple systems (including linearity and superposition principle). Free oscillations of systems with many degrees of freedom (including vibrations of a string and Fourier analysis). Driven oscillations. Traveling waves. Reflection and Transmission. Modulations, pulses, and wave packets. Waves in two and three dimensions. Polarizations. Linear optics. Interference and diffraction. The particle properties of waves. The atom and the wave properties of particles. Elementary particles

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.

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.

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

1. Introduction: overall description of the multi-purpose digital computer operation, the central processing unit (CPU) organization, representation and processing of information (data and programs).
2. Machine language: representation of operations and commands, selecting the command set, designing the command architecture, addressing modes.
3. Control Unit: application methods by logic and micro-programming, interpretation and execution of command set.
4. Arithmetic Unit: numbers representation, basic arithmetic operations in digital systems, addition, subtraction, multiplication and division of fix and floating point numbers.
5. Memory Unit: memory types, addressing methods, memory organization, virtual memory, cache memory.
6. Input/Output Unit: I/O addressing modes, I/O data transferring control by serial polling, interrupt and direct memory access (DMA).

(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.

Hilbert space. The probability matrix. Unitary operations: Displacements, Rotations, Boosts, Gauge. Unitary representations of rotations and the spin concept. Actual procedure for building rotation matrices. Addition of angular momentum (elementary). Perturbation theory. Wigner decay. Fermi golden rule. The adiabatic equation. Particle in a few site system. Particle in 1D ring and AB effect. Particle in a uniform magnetic field (Landau, Hall). Spin a in magnetic field.

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.

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.

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

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

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

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

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

ביבליוגרפיה:

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

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

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

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

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.

סילבוס אינו מוגדר בתקופה המתבקשת

• 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

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.

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.

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”.

This course aims to provide algorithmic and numerical tools for solving common computational optimization problems in science and engineering. The course will include: linear algebra reminder: norms, least squares, eigenvalue and singular value decompositions, optimization of quadratic problems. Convexity, iterative methods for unconstrained optimization (Steepest Descent, Newton, Quasi-Newton, Conjugate Gradients, subspace methods, BFGS), and line-search methods. Constrained optimization with equality and inequality constraints (method of Lagrange multipliers and KKT optimality conditions), linear and quadratic programming, penalty, barrier and projection methods. Duality. Splitting methods and ADMM. Introduction to stochastic optimization (SGD). Introduction to non-smooth and sparse optimization. Robust statistics for dealing with outliers. The assignments in this course will include writing computer programs for practically implementing and demonstrating the algorithms taught in this course.

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.

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.

Vector analysis. electric field (including Coulomb law, Gauss law, and first Maxwell equation). Electric potential (including Laplace and Poisson equations, electric dipole). Conductors (including capacity and capacitors). Electric current (including Ohm’s law, direct current circuits, RC circuit). Magnetic field I (including special relativity connections, Lorenz force, Ampere force, magnetic moment). Magnetic field II (including vector potential, Bio-Savard’s law, Ampere’s law, absence of magnetic charge, second Maxwell equation, truncated third Maxwell equation, gauge, field of magnetic moment, solenoids). Electromagnetic induction (including fourth Maxwell equation, Lenz law, inductance, AC current circuits). Maxwell equations and electromagnetic waves. (Optional) Electric and magnetic properties of materials

Review of Newton laws. Generalized coordinates. The principle of least action, constraints. The Lagrangian for free particle and for a system of particles. Conservation laws (energy, momentum, angular momentum). The virial theorem. Motion in one dimension. The two body problem. Motion in a central field and Kepler’s problem. Scattering, total and differential cross sections. Rutherford’s formula. Small oscillations, forced and damped oscillations. Motion of a rigid body, the Inertia tensor. Space and body frames the Coriolis effect. 1The rotation group, infinitesimal and finite rotations. Euler’s angles, and Euler’s equations. Symmetric and asymmetric tops. The Hamiltonian and Hamilton’s equations. Canonical transformations and the Poisson brackets. Liouville’s theorem. The Hamilton-Jacobi equation.

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.

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.

Mathematical reminder: Vector calculus. Maxwell equations in vacuum, continuity and boundary conditions. Electrostatics in free space, Green’s function. Electrostatics with boundaries, Laplace equation for various symmetries. Magnetostatics. Electrodynamics, induction, conservation laws. Electromagnetic waves. Relativistic electrodynamics. Fields of moving charges. Radiation

Photoelectric effect, Frank-Hertz experiment, Compton effect, Two-slit experiment, Particle-wave duality, Bohr model, Periodic table Schrodinger eq, Operators, Harmonic oscillator, Stepwise potentials, Quantum mechanics in Hilbert space, Commutator, Central forces, Angular momentum, Hydrogen atom, Magnetic field and spin

Main concepts of Statistical Physics: separation of microscopic and macroscopic description of a system; microstates vs. configurations; sharpness of the distribution of microstates over configuration space; basic idea of statistical ensembles and of probabilities. Main concepts of Statistical Physics as applied to a simple modelsystem of two-state spins: numbering of microscopic states; multiplicity function; continuous approximation of the multiplicity function; probabilities of micro- and macro- states; characteristics of probability function, mean, variance and standard deviation; sharpness of the probability distribution of a macroscopic variable; correlation of two random variables. Microcanonical ensemble: probabilities of microscopic states in an isolated system at equilibrium; example of equilibration in the system of spins; thermal equilibrium of two arbitrary systems; definition of temperature; entropy in microcanonical ensemble; additivity of entropy; the postulate of entropy increase in an isolated system (second Law of Thermodynamics); the direction of heat flow. Canonical ensemble: Boltzmann Distribution; partition function; microscopic state probabilities; average energy of a system in thermal equilibrium; fluctuations in energy and their relation to heat capacity; Helmoholtz Free energy and its relation to the partition function. Applications of the canonical ensemble: the system of spins; Schottky anomaly; partition function, energy and free energy of a classical ideal monoatomic gas. Thermodynamic equilibrium and thermodynamic processes; characteristic time scales; reversible and irreversible processes; quasi-stationary process; examples of quasi-stationary heat transfer and of work. Heat and Work in thermodynamics; First Law of Thermodynamics; differential relations between thermodynamic quantities; Maxwell relations; enthalpy; heat capacity; intensive and extensive quantities; entropy, pressure and heat capacity of a classical ideal

Introduction to computer science 201-1-1011 1. Basic Programming 2. Some order and arrays 3. Arrays and functions 4. Strings, Binary search and Selection sort 5. Insertion sort, handling matrices 6. Introduction to recursive programming 7. Recursive programming : binary search, selection sort, Merge sort 8. Last examples : recursive programming 9. Introduction to object oriented programming 10. Object oriented programming 11. Lecture canceled 12. Interfaces 13. Inheritance 14. Inheritance and abstract inheritance 15. Exceptions 16. Data structures 17. Stack application 18. Queue and linked lists 19. Lists - the recursice perspective 20. Trees

Number representation in different bases, binary codes and binary arithmetic. Combinational systems: Boolean algebra, switching function representations and minimization. Karnaugh maps, prime implicate table. Hazards. Combinational circuit design. Switching devices: logic gates (NAND, AND, NOR, OR, NOT, XOR); modules: HA, FA, HS, FS, Multipliers, Decoders, Multiplexers, Demultiplexers, PROM, PLA. Using modules to the implementation of combinational circuits. Sequential systems: Basic models, synchronous and asynchronous system structure. Bistable memory devices (Latches, Flip-Flops), transition table and state table. Master-Slave Flip-Flops, Edge-Triggered Flip-Flops. Design and implementation of sequential systems, synchronous and asynchronous. Race problem solution. Special modules: Registers, Shift Registers, Counters.

Thermodynamic potentials: Gibbs free energy; per molecule and per unit volume thermodynamic potentials. First order phase transitions: experimental observations; thermodynamic phases; free energy per unit volume; conditions for phase separation; local and global stability of a thermodynamic phase; graphical representation. Molecular interactions: hard-core repulsion; Van der Waals attraction; induced dipole; electronic and dipole polarizability; estimations of Van der Waals attraction for simple molecules; Lennard-Jones potential. Incorporation of molecular interactions into free energy: phase coexistence; common tangent construction; phase diagrams; spinodal and binodal lines; critical temperature; Clausius-Clapeyron equation. Van der Waals gas: free energy; pressure; spinodal; critical point; estimations of critical temperature; approximation of binodal far from the critical point. Van der Waals gas near the critical point: free energy; binodal line; first and second-order phase transition; critical opalescence; nucleation and growth; surface energy; critical nucleous. Ferromagnet-paramagnet second-order phase transition; Landau theory of phase transitions. Kinetic theory of gases: characteristic velocities; kinetic derivation of pressure of ideal gas; Maxwell distribution of velocities. Diffusion and random walks on a lattice model: mean-square displacement; diffusion coefficient; the diffusion equation for random walks; diffusion as transport phenomenon; derivation of Fick’s laws from microscopic considerations; self- diffusion and collective diffusion coefficients. Diffusion in gas: random walks; distribution of free paths; mean free path; mean square displacement of a molecule; self-diffusion coefficient; diffusion in concentration gradient; derivation of Fick’s laws; equivalence of self-diffusion and collective diffusion coefficients for gases. Other linear transport phenomena in gases: heat conductivity; Fourier law; thermalization;

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.

Mathematical introduction: Coordinates, vector algebra, partial derivatives. Particle kinematics: general concepts, velocity, acceleration, rotational motion. Newton’s laws, Galilean relativity, inertial and non-inertial frames Particle dynamics: applications of Newton’s laws, work, kinetic energy, momentum, angular momentum. Potential energy, conservation laws. Motion in a potential I: 1D potential, central forces, Keplerian motion. Motion in a potential II: Oscillations. Mathematical addition: complex numbers. Many particle systems, conservation laws, collisions. Rigid body rotation I: Theory (angular velocity and acceleration, tensor of inertia, moment of inertia, torque, angular momentum, kinetic energy). Rigid body rotation II: Applications (rolling, precession, gyroscopes). Basics of special relativity (Optional) Gravitation.

Experiments list: Scanning Tunneling Microscope (STM), Ultra High Vacuum (UHV), Low Energy Electron Diffraction (LEED), X Rays, Magnetic Susceptibility, Mossbauer Effect, Scintillation, Superconductivity, Nuclear Magnetic Resonance (NMR), Hall Effect, Magneto Optical Trapping, Laser Frequency Locking and Atomic Clock. Academic requirements: All students must perform 2 experiments from the list of experiments and submit a report on both of them. There will be pre-exam prior to each experiment.

סילבוס אינו מוגדר בתקופה המתבקשת

Experiments list: The ratio between electron charge and its mass (e/m), Polarization of light, Frank-Hertz experiment, Michelson and Fabry-Perot Interferometers, Photo-Electric Effect, Lasers, Micro Waves, Nuclear Radiation White Noise. Academic requirements: In each semseter all the students must chose and perform 4 experiments from the list of the experiments and submit a report on each of them. There will be pre-exam prior to each experiment, as well as the final exam at the end of the semester.

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

• 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.

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).

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

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.

• 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.

Basic combinatorics

• Sample space and events. Probability mass function
• Conditional probability. Law of total probability. Bayes? formula
• Independence
• Useful discrete distributions: uniform, binomial, geometric, Poisson, hypergeometric, negative binomial.
• Expectation, variance, median, percentile
• Functions of a random variable
• Two dimensional distributions: joint distribution, marginal distribution, covariance, conditioning on marginals
• Markov, Chebychev and Jensen inequalities
• Normal distribution
• Sequences of random variables: law of large numbers and central limit theorem

1. Counting states. Combinatorics. Distinguishable and indistinguishable particles. Ordered and unordered arrangements. Permutations without replacement and with replacement. Combinations without and with replacement. Multisets. N 1/2 spins. Cell gas. Fermions, para-fermions, bosons.
2. Repeated experiments, outcomes. Frequentist, Bayesian, and Kolmogorov probability, interrelation. Reproducible and irreproducible experiments. Probability in a finite universe. Expanding universe. Time dependent probability. Laws of probability. Mutually exclusive events/outcomes. Conditional probability. Bayes rule. Geometric probability, Betrand paradox.
3. Random variables. Discrete random variables. Probability of a random variable. Functions of random variables. Mean, variance, moments in general. Spin 1/2. Paramagnetism. Binomial distribution. Large numbers. Most probable state. Rare events. Radioactive decay. Poisson distribution. Information entropy. Maximum entropy principle without constraints. Uniform distribution. Maximum entropy principle with energy constraint. Boltzmann distribution.
4. Gas of particles in the velocity space. Continuous random variable. Probability density. Mean, variance, moments. Delta-function. Maxwellian (normal, gaussian) distribution. Localized magnetic moment in a magnetic field. Classical paramagnetism. Fluctuations of magnetization. Other observed distributions: merging black holes. Line width: Breit? Wigner distribution. Entropy. Uniform distribution. Particle size distribution of aerosols and mass distribution in the Universe: log-normal distribution. Collisions in accelerators: from binomial distribution to Poisson.
5. Multivariate continuous distributions. Joint and marginal distribution. Gas in 3D. Most probable components and most probable magnitude of the velocity. Isotropic and anisotropic distributions. Plasma pressure tensor. Covariance, correlation. Transformation of variables in joint distributions. Beams in plasmas. Cosmic rays: energy spectrum and pitch-angle distribution. Covariance vs independence.
6. Laws of large numbers. Gaussian as the limiting distribution for the Binomial and and Poisson distributions. Chebyshev’s inequality. Independent random variables. Sum of random variables. Convolution. Convolution of Gaussians. Central limit theorem. Applications and limitations of the theorem: velocity component of air molecules, Coulomb scattering, energy loss of charged particle traversing thin gas layer (Landau distribution).
7. Statistics in physics: from data to hypothesis. Main sequence: assume theory, devise an experiment, measure relevant parameters, estimate uncertainties, quantify agreement with theory, accept of reject. Examples: search for Higgs in LHC, dark matter in the Universe. More examples: CP violation.
8. Measurements and errors. Propagation of errors. Measured distribution vs true distribution: convolution with the resolution function (detector). Assumption of normal distribution of measurement uncertainties. Distortion of measured distribution: line width.
9. Measurements, samples, population, sample statistics. Sample mean and variance. Central limit theorem in statistics. Parameter estimates: frequentist and Bayesian approach. Prior and posterior probabilities. Basic estimators. Maximum-likelihood method: distance from the Sun to the center of the galaxy, mass distribution from LIGO.
10. LHC experiments: application of Chi-squared. Degrees of freedom. Nuisance parameters. Role of uncertainties (overestimate vs underestimate). Unbiased estimators. Correlation functions.
11. Hypothesis testing. Simple and composite hypotheses; statistical tests; Neyman?Pearson; generalised likelihood-ratio; Student?s t; Fisher?s F; goodness of fit.
12. (Optional) Random walk. Diffusion processes.
13. (Optional) Monte-Carlo methods.

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.

1. Series of numbers, both positive and general. Absolute and conditional convergence. Root and Ratio tests. Leibniz Alternating series test. 2. Power Series. 3. First order equations: separable equations, exact equations, linear equations, Bernoulli equations. Existence and uniqueness. 4. Second order equations. Reduction of order. Linear homogeneous equations, fundamental solutions and Wronskian. Inhomogeneous equations, variation of parameters. Equations with constant coefficients and the method of undetermined coefficients. Linear equations of higher order. Euler equations. 5. Systems of differential equations.

How does a search engine decide what search results to present? How can a car camera identify automatically whether a pedestrian is crossing? How can we find out which drug is suitable for which patient? How can any application use data from many users to automatically improve user experience? These problems and many more can be solved with modern methods based on machine learning. In this class we will survey diverse practical and successful methods, which can be implemented efficiently and simply, even with very large input sizes. We will understand the mathematical reasoning behind these methods, and apply them in practice on data from real problems. The following topics will be covered: - Supervised learning methods for automatic classification and prediction: Nearest neighbots, SVM, kernels, decision trees, random forests, neural networks, naive Bayes, linear regression. - Unsupervised learning: PCA dimensionality reduction, clustering with k-means and other methods. - Model fitting methods such as Gaussian mixtures, maximum likelihood, and EM.

Experiments on mechanics: measurement of density and calculation of errors, Impulse and momentum, moment inertia, angular momentum, harmonic oscillations. Experiments on electricity and magnetism: oscilloscope, conductivity, magnetic force. Experiments on waves: wave propagation, Doppler effect, geometrical optics. Heat and fluids: surface tension, thermocouple. All students must perform 10-12 experiments and submit report on 5 of them. There will be exam on all the experiments. Sources: Sears F.W., Zemansky M.W., Young H.D., College physics, 7 ed., Addison-Wesley,Reading MA 1990; QC 23.S369 1990 Sears F.W.: Optics, 3rd ed., Addison-Wesley, Reading MA 1949; QC 355.S45 1949 Halliday D, Resnick R, Krane K S, Physics, vol. 2, 4 ed., Wiley, New York 1992; QC21.2.H355 1992b

1. Real numbers (axiomatic theory). Supremum and Infimum of a set. Existence of an n-th root for any a > 0. 2. Convergent sequences, subsequences, Cauchy sequences. The Bolzano-Weierstrass theorem. Upper and lower limits. 3. Series. Partial sums, convergent and divergent series, Cauchy criterion. Series of non-negative terms. The root and the ratio tests. Series of arbitrary terms. Dirichlet, Leibnitz, and Abel theorems. Rearrangements of series. The Riemann Theorem. 4. The limit of a function. Continuous functions. Continuity of the elementary functions. Properties of functions continuous on a closed interval. Uniformly continuous functions. Cantor?s theorem. 5. The derivative of a function. Mean value theorems. Derivatives of higher order. L’Hospital’s rule. Taylor’s theorem.

1. The Riemann integral: Riemann sums, the fundamental theorem of calculus. Methods for computing integrals (integration by parts, substitution, partial fractions). Improper integrals and application to series. Numerical integration. Stirling’s formula and additional applications time permitting.
2. Uniform and pointwise convergence. Cauchy’s criterion and the Weierstrass M-test. Power series. Taylor series, analytic and non-analytic functions. Convolutions, approximate identities and the Weierstrass approximation theorem. Additional applications time permitting.
3. A review of vectors in R^n and linear maps. The Euclidean norm and the Cauchy-Schwarz inequality. Basic topological notions in R^n. Continuous functions of several variables. Curves in R^n, arc-length. Partial and directional derivatives, differentiability and C^1 functions. The chain rule. The gradient. Implicit functions and Lagrange multipliers. Extrema in bounded domains.

1. Normed spaces and spaces with inner products. The projection theorem for finite dimensional subspaces. Orthogonal systems in infinite dimensional spaces. The Bessel inequality and the Parseval equality, closed orthogonal systems. The Haar system.
2. The Fourier series (in real and complex form). Approximate identities, closedness of the trigonometric / exponential system. Uniform convergence of the Fourier series of piecewise continuously differentiable functions on closed intervals of continuity; the Gibbs phenomenon. Integrability and differentiability term by term.
3. The Fourier transform. The convolution theorem. The Plancherel equality. The inversion theorem. Applications: low pass filters and Shannon’s theorem.
4. The Laplace transform. Basic relations and connection with the Fourier transform. A table of the Laplace transforms. The convolution integral. Application of the Laplace transform for solution of ODEs.
5. Introduction to the theory of distributions. Differentiation of distributions, the delta function and its derivatives. Fourier series, Fourier transforms, and Laplace transforms of distributions.

State space and transfer functions, stability-state space and input output, observability and controllability, realization theory, Hankel Matrices, statefeedback ans stabilization, Ricatti equations.

The aim of the course is to train students in creative problem solving in various areas of mathematics, in a manner simulating, to some extent, mathematical research (rather than ordinary courses, where homework assignments are, usually closely ? and quite obviously - related to the topics reviewed in class). The course will focus on problems whose solutions call for tools and ideas from several fields of mathematics, and will reveal ? on a miniature scale - the beauty of mathematics as an integral field of knowledge, where between seemingly unrelated subjects deep and surprising connections may arise. The problems will focus on topics in classical and modern mathematics that, due mostly to their interdisciplinary nature, are not ? as a rule ? covered in the core classes offered by the department (such topics as the axiom of choice and its applications, the Banach-Tarski paradox, transcendental number theory etc.). Classes will be divided between lectures, given by the instructor, filling in background material required to address the problems in question, and between students’ presentation of their solutions to the work sheets distributed previously. In addition to all of the above, the course will help students improve their skills in searching the mathematical literature and in the art of writing and presenting proofs.

The process diffusion limited aggregation, or DLA, is a remarkable process stemming from physics. Although it is known for almost 40 years, it still is mysterious and difficult to mathematically understand. Classically it has been studied in the plane. For this project we will simulate the process in other geometries, Euclidean as well as non-Euclidean. The goal is to obtain large-scale simulations, which may lead to further understanding and new ideas. Further, we will use the simulations repeatedly to measure experimentally different quantities associated with DLA, such as growth rate, fractal dimension, etc.

Probability theory: discrete and continuous variables, independent vs dependent variables, six basic discrete distributions: Bernoulli, binomial, uniform, geometric, negative binomial, Poissonian. Mean, variance, moments, probability generating function. Five basic continuous distributions: uniform, normal, exponential, gamma, beta. Moment generating function. Events, conditional probability, aging of molecules, entropy and related concepts, scores and support. Generating various probabilities. Many random variables. EST library. Covariance and correlation, iid, minimum and maximum of many random variables.Theoretical statistics; random sampling. Classical vs Bayesian approach. Distributions of the mean and variance of the sample; methods of calculating point estimates; point estimator for the mean; point estimator for the variance; biased vs unbiased, MSE, confidence intervals for the parameters of distribution. Basic ideas and definitions for the test of the hypothesis; errors of type I and II; P-values, tests for mean values, variances and proportions; test for the goodness of fit; test of independence; correlation coefficient; and its tests; linear regression; Likelihood ratios, information and support; maximum value as test statistic. Nonparametric: Mann-Whitney and permutation tests.Bayesian approach to hypothesis testing and estimation.ANOVA - analysis of variance: one-way and two-way.More theory on classical estimation: optimality aspects.BLAST.

The course discusses elements of linear algebra over the real and complex numbers. 1. The complex numbers. Linear equations: elementary operations, row reduction, homogeneous and non-homogeneous linear equations. 2. Real and complex vector spaces. Examples, subspaces, linear dependence, bases, dimension. 3. Matrix algebra. Matrix addition and multiplication, elementary operations, LU decomposition, inverse matrix, the determinant, Cramer’s law. 4. Linear transformations: examples, kernel and image, matrix representation. 5. Inner products, orthonormal sequences, the Gram-Schmidt process. 6. Diagonalization: eigenvalues and eigenvectors, the characteristic polynomial. Time permitting: unitary and orthogonal matrices and diagonalization of Hermitian and symmetric matrices.

First order equations: separable equations, exact equations, linear equations, Bernulli equations. Existence and uniqueness. Second order equations. Reduction of order. Linear homogeneous equations, fundamental solutions and Wronskian. Inhomogeneous equations, variation of parameters. Equations with constant coefficients and the method of undetermined coefficients. Linear equations of higher order. Euler equations. Systems of differential equations. Laplace transform and its properties. Solution of initial value problems with discontinuous forcing functions. Impulse functions. Convolution.

• Fields: definitions, the field of complex numbers.
• Linear equations: elementary operations, row reduction, homogeneous and inhomogeneous systems, representations of the solutions.
• Vector spaces: examples, subspaces, linear dependence, bases, dimension.
• Matrix algebra: matrix addition and multiplication, elemetary operations, the inverse of a matrix, the determinant, Cramer’s rule.
• Linear transformations: examples, kernel and image, matrix representation.
• Diagonalization: eigenvectors and eigenvalues, the characteristic polynomial, applications.
• Bilinear forms.
• Finite dimensional inner product spaces.
• Operators on finite dimensional inner product spaces: the adjoint, self adjoint operators, normal operators, diagonalization of normal operators.

The course provides a basic introduction to “naive” set theory, propositional logic and predicate logic. The course studies many important concepts which serve as the building blocks for any mathematical theory An emphasis is given to clear proof writing and correct usage of the mathematical language.A. Basics of Set Theory1. The notion of a set. Set operations: union, intersection, difference, complementation, and the power set.2. Cartesian products and binary relations. Operations on relations.3. Functions: domain and range, one to one and onto. Composition.4. Equivalence relations and set partitions.5. Order relations: partial, linear and well-founded.6. Induction theorems: mathematical, complete and well-founded.B. Set Cardinality1. The notion of cardinality. Finite, infinite and countable sets.2. Cantor’s theorem.3. The cardinality of the set of real numbers and other sets.C. Propositional Calculus1. The language of propositional calculus. Logical connectives.2. Logical implication and logical equivalence of propositional formulas.3. Disjunctive normal form.4. Complete sets of logical connectives.D. Predicate Calculus (First Order Logic)1. The language of predicate calculus: terms, formulas, sentences.2. Structures, assignments for a given structure.3. Logical implication and logical equivalence of first order formulas.4. Elementary equivalence of structures, definable sets in a given structure.

Logic and Set theory: propositional calculus, boolean operators and their truth tables, truth values of formulae, logical equivalence and logical inference, tautologies and contradictions, the important tautologies, e.g., the distributive laws and De Morgan formulae. Sets: inclusion, union, intersection, difference, cartesian product, relations, equivalence relations, partial orders, linear orders, and functions. Basic Combinatorics: induction, basic counting arguments, binomial coefficients, inclusion-exclusion, recursion and, generating functions. Graphs: general notions and examples, isomorphism, connectivity, Euler graphs, trees.