2022–23–A

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) 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.

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.

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. 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).

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. 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.

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

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. 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.

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