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Some imitative words are more surprising than others. How to use a word that literally drives some people nuts. The awkward case of 'his or her'. Which of these things doesn't belong? Can you spell these 10 commonly misspelled words? Would you like to tell us about a lower price? If you are a seller for this product, would you like to suggest updates through seller support? This text introduces students to basic techniques of writing proofs and acquaints them with some fundamental ideas. The authors assume that students using this text have already taken courses in which they developed the skill of using results and arguments that others have conceived.
This text picks up where the others left off -- it develops the students' ability to think mathematically and to distinguish mathematical thinking from wishful thinking.
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An Accompaniment to Higher Mathematics (Undergraduate Texts in Mathematics)
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This is the text for a foundations course that is required at the college near me for math majors or others going into upper division math courses. This semester Spring I'm a senior citizen auditor of this course, having retired from a career in applied mathematics. Half-way through the course, having examined much of the material in this clear and well-illustrated text, I can say that I wish there had been a course based on a book like this decades ago, when I first got into higher math. This excellent text covers much of the abstract material that underpins higher math courses, which introduce said material only as needed.
Having to understand the abstract foundational material at the same time as trying to learn the main points of the advanced math course can be overwhelming to some students. Poisson approximation of the binomial distribution, Poisson distribution, Poisson process, applications. General theory of random variables: cumulative distribution function and its properties, singular continuous distributions, absolutely continuous distributions and probability density functions. Important continuous distributions: uniform, exponential, normal Gauss , Cauchy.
Distribution of a function of a random variable, transformation of probability densities. Quantities associated to distributions: expected value, moments, median, variance and their properties. Computation for the important distributions. Steiner formula. Joint distributions: joint distibution, mass and density functions, marginal and conditional distributions.
Important joint distributions: polynomial, polyhypergoemetric, uniform and mutlidimensional normal distribution. Conditional distribution and density functions. Conditional expectation and prediction, conditional variance. Vector of expected values, Covariance matrix, Cauchy-Schwartz inequality, correlation. Indicator random variables. Weak Law of Large numbers in full generality. Application: Weierstrass approximation theorem. Normal approximation of binomial distribution: Stirling formula, de Moivre-Laplace theorem.
Normal fluctuations. Central Limit Theorem. Ferenc Wettl Descripton: The aim of the course is to maintain the students' programming skills through programming problems associated with the topics of Probability Theory course helping the understanding of the basic concepts of probability simulations of random events at the same time.
Katalin Nagy Descripton: Ordinary differential equations. Explicitly solvable equations, exact and linear equations. Well-posedness of the initial value problem, existence, uniqueness, continuous dependence on initial values. Approximate solution methods. Linear systems of equations, variational system. Elements of stability theory, stability, asymptotic stability, Lyapunov functions, stability by the linear approximation.
Phase portraits of planar autonomous equations. Laplace transform, application to solve differential equations. Discrete-time dynamical systems. Boyce, R. Built-in data types: int, double, char, bool, complex. Control commands: if, switch, for, while, do.
Exception handling recall Python. Pointers, relationship with arrays. Basic algorithms: search, sort, etc. Dynamic memory management, new, delete. Header files. Literature: — E. Marianna Bolla Descripton: Statistical sample, descriptive statistics, empirical distributions. Most frequently used probabilictic models, likelihood function, sufficiency, maximum likelihood principle.
Methods of point estimation: maximum likelihood, method of moments, Bayes principle. Interval estimation, confidence intervals. Theory of hypothesis testing, likelihood ratios. Parametric inference: u, t, F tests, comparing two treatments. Two-way classified data, contingency tables, chi-square test. Nonparametric inference: Wilcoxon and sign tests, Spearman correlation.
Regression analysis. Linear regression, method of least squares, Pearson correlation. Multivariate regression, multiple correlation. Linear models, analysis of variance for one- and two-way classified data. Practical considerations: selecting the sample size, test for normality, resampling methods. Johnson, G. Set functions. Concept of Lebesgue measure. Outer measure. Measurable sets. Measure generated by an outer measure. Example for not Lebesgue-measurable set. Measure space, measurable functions. Null sets. The concept of convergence in measure and almost everywhere ae and relations between them.
Integral of measurable functions. Beppo-Levi theorem, Fatou's lemma, Lebesgue's dominated convergence theorem. Absolute continuity of the integral. Riemann sphere. Limits and properties of complex valued sequences. Limit and continuity of complex functions. Power series of elementary functions. Euler's formula. Complex logarithm function. Differentiability of complex functions. Cauchy-Riemann equations. Regularity of complex functions and elementary properties of regular functions. Harmonic functions, harmonic conjugate. Complex integral, integration by substitution. Newton-Leibniz formula.
Goursat lemma. Cauchy's integral theorem and integral formula on convex domain. Index of a curve. Simply-connected subsets. Cauchy integral theorem and integral formula. Primitive functions. Morera's theorem. Power series of regular functions. Liouville theorem and fundamental theorem of algebra. Multiplicity of roots. Laurent series. Isolated, removable and essential singularities of complex functions. Concept of residue and the residue theorem. Residue theorem with logarithmic functions.
Argument principle. Rouche's theorem. Maximum and minimum principles. Rudin: Real and complex analysis. Theorema Egregium. Marianna Eisenberg-Nagy Descripton: Introduction to operations research; convex sets, polyhedron, polytope Krein-Milmann theorem. Separation, Farkas' lemma. Linear programming problem, basis, basic solution, optimal solution. Simplex algorithm. Two-phase simplex algorithm, degeneration, index selection rules.
Modified simplex algorithm. Sensitivity testing. Weak and strong duality theorem. Network flow problems, algorithms. Network simplex algorithm. Transportation problem, assignment problem, the Hungarian method. Integer programming: Branch and bound method, dynamic programming, cutting plane procedures. Game theory: matrix games. Literature: — K. Murty: Linear and combinatorial programming, John Wiley and Sons. Chvatal: Linear programming, W. Katalin Friedl Descripton: Pattern matching: naive algorithm, the fingerprinting method of Rabin and Karp, solution by finite automata.
Deterministic and non-deterministic finite automata and their equivalence. Regular expressions, regular languages, and their connections to finite automata. Finite automaton as lexical analyser. Context free grammars. Parse tree, left and right derivation. Ambiguous words, grammars, languages. The importance of unambiguous grammars for algorithms.
Pushdown automaton. The main task of a parser. The general automaton: Turing machine. Church-Turing thesis. Karp reduction and the notion of NP completeness. Theorem of Cook and Levin. Linear and integer programming. LP and IP as algorithmic tools, translation of combinatorial problems to integer programming. Another tool: branch and bound. Dynamical programming example: knapsack, longest common substring. The objective in approximation algorithms.
Fro the TSP even the approximation s hard in general but there is efficient 2-approximation in the euclidean case.
An Accompaniment to Higher Mathematics | George R. Exner | Springer
Comparison based sorting: bubble sort, insertion sort, merge sort, quick sort. Lower bound for the number of comparisons. Other sorting methods: counting sort, bin sort, radix sort. Linear and binary search. The binary search is optimal in the number of comparisons. Notion of search tree, their properties and analysis.
Red-black tree as a balanced search tree. The tree, and its generalization, the B tree. Comparisons of the different data structures. Corman, C. Leiserson, R. Rivest, C. Sipser: Introduction to the Theory of Computing, Thomson. Ferenc Wettl Descripton: The aim of the course is to maintain the students' programming skills through programming problems associated with the topics of Algorithm Theory course helping the understanding of the basic concepts of algorithms.
Normal extensions, splitting field, separable extension, finite fields, Wedderburn's theorem, Galois group, irreducibility of the cyclotomic polynomials, Galois groups of radical extensions, Galois correspondence, Fundamental theorem of Galois theory. Applications of Galois theory: Fundamental theorem of algebra, ruler and compass constructions, solvability of equations by radicals, Abel—Ruffini theorem.
Existence and uniqueness of algebraic closure, transcendental extensions, transcendence of e, Gelfand-Schneider theorem. Linear congruences and systems of congruences, binomial congruences of higher degree, discrete logarithm, congruences of prime power moduli. Quadratic congruences, Legendre and Jacobi symbol, quadratic reciprocity.
Diophantine equations: linear diophantine equations, Pythagorean triples, Fermat's two squares theorem, Gaussian integers. Literature: — I. Model reformulations: rewrite complex transportation problem to simple transportation problem, rewrite maximum flow problem to minimum cost maximal flow problem. Modeling problems in economy. Integer modeling tricks, set covering, set partitioning problems. Modeling Facility Location problems. Numerical errors. Dynamic programming. Scheduling problems, heuristics, approximations, online versions. Decision Theory. Inventory tasks. Fourer, D.
Gay, B. Winston: Operations Research: Applications and Algorithms.
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Literature: — Essentials of Stochastic Processes 2nd edition , Springer, Press, The spreading of knowledge and culture of applied mathematics. The development of the connections and cooperation of students and professors of the Mathematical Institute, on the one hand, and of personal, researchers of other departments of the university or of other firms, interested in the applications of mathematics.
The speakers talk about problems arising in their work. They are either applied mathematicians or non-mathematicians, during whose work the mathematical problems arise. An additional aim of this course to make it possible for interested students to get involved in the works presented for also promoting their long-range carrier by building contacts that can lead for finding appropriate jobs after finishing the university.
Cheney, D. Integral curve of a vector field. Vector bundles and related algebraic constructions direct sum, tensor product, dual, homomorphisms. Differential forms, pull-back, exterior product, exterior derivation. Integration on compact oriented manifolds, Stokes' theorem.
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Lie-derivative, Lie-Cartan formula. Riemannian metric, examples. Geodetics, exponential map. Lie groups and algebras. Hopf-Rinow theorem and its consequences. Connections on a vector bundle, parallel transport, integrability. Levi-Civita connection, the Riemann curvature tensor. Properties of the curvature tensor, Ricci curvature. First and second variation of arc length, Jacobi vector fields.
Petersen: Riemannian geometry, Graduate Texts in Mathematics, Gallot, D. Hulin, J.