Program in Applied Mathematics




Скачать 238.92 Kb.
НазваниеProgram in Applied Mathematics
страница7/7
Дата04.10.2012
Размер238.92 Kb.
ТипДокументы
1   2   3   4   5   6   7




Seminars

Fall Term

Winter Term

Spring Term

Departmental Seminar,
mandatory for second year MS students, 1 credit,

1 hour/week
(Gheorghe Morosanu)

Departmental Seminar,
mandatory for all MS students, 1 hour/week, no credit for first year students, 1 credit for second year students
(Gheorghe Morosanu)

Spring MS Seminar,
mandatory for first year students, 1 credit
(Pal Hegedus)




Winter MS Seminar,
mandatory for second year students, 2 credits,

2 hours/week
(Pal Hegedus)





Remarks:
1. Each course has 3 CEU credits (3 hours / week),  unless otherwise specified.
2. Spring Term course: Thesis Writing for Mathematics (12 hours = 6 lectures, no credit), mandatory for all MS students.


2009-2010 Syllabi


  • Name of Course: Basic Algebra 1 (MS)

  • Lecturer: Pál Hegedűs

  • No. of Credits: 3, and no. of ECTS credits: 6

  • Semester or Time Period of the course: Fall Semester of AY 2009-2010

  • Prerequisites: linear algebra, introductory abstract algebra

  • Course Level: introductory

  • Brief introduction to the course:


Basic concepts and theorems are presented. Emphasis is put on familiarising with the aims and methods of abstract algebra. Interconnectedness is underlined throughout. Applications are presented.


  • The goals of the course:


One of the main goals of the course is to introduce students to the most important concepts and fundamental results in abstract algebra. A second goal is to let them move confidently between abstract and concrete phenomena.



  • The learning outcomes of the course:


The students will learn some basic notions and results of abstract algebra. More importantly, they will gain expertise in using it in various areas of mathematics.


  • More detailed display of contents (week-by-week)



    1. Groups: permutations groups, orbit-stabilizer theorem, cycle notation, conjugation, conjugacy classes of S_n, odd/even permutations,

    2. commutator subgroup, free groups, geerators and relations, Dyck’s theorem,

    3. solvable and simple groups, simplicity of A_n, classical linear groups,

    4. Polynomials: Euclidean Algorithm, uniqueness of factorisatoin, Gauss Lemma, cyclotomic polynomials,

    5. polynomials in several variables, homogeneous polynomials, symmetric polynomials, formal power series, Newton's Formulas,

    6. Sturm’s Theorem on the number of real roots of a polynomial with real coefficients.

    7. Rings and modules: simplicity of matrix rings, quaternions, Frobenius Theorem, Wedderburn’s Theorem,

    8. submodules, homomorphisms, direct sums of modules, free modules,

    9. chain conditions, composition series.

    10. Partially ordered sets and lattices: Hasse-diagram, chain conditions, Zorn Lemma, lattices as posets and as algebraic structures,

    11. modular and distributive lattices, modularity of the lattice of normal subgroups, Boolean algebras, Stone Representation Theorem.

    12. Universal algebra: subalgebras, homomorphisms, direct products, varieties, Birkhoff Theorem.


Optional topics:


Resultants, polynomials in non-commuting variables, twisted polynomials, subdirect products, subdirectly irreducible algebras, subdirect representation. Categorical approach: products, coproducts, pullback, pushout, functor categories, natural transformations, Yoneda lemma, adjoint functors.


Books:

0. P J Cameron, Introduction to Algebra, Oxford University Press, Oxford, 2008.

1. N Jacobson, Basic Algebra I-II, WH Freeman and Co., San Francisco, 1974/1980.

2. I M Isaacs, Algebra, a graduate course, Brooks/Cole Publishing Company, Pacific Grove, 1994


Teaching format: lecture combined with classroom discussions, regular homework assignments


  • Assessment:


Attendance is mandatory.


Homework: will be assigned regularly. The tests will be based to a significant extent on homework assignments.


Tests and Grading: There will be a midterm test (worth 30%), a final exam (worth 40%) and homework (worth 30%).


  • Such further items as assessment deadlines, office hours, contact details etc are at the discretion of the department or the individual.


Final exam: within two weeks after the final lecture

Office hours: by appointment


Pál Hegedűs

Zrinyi u. 14, Third Floor, Office # 312, hegedusp@ceu.hu


Name of Course: Introduction to Computer Science

  • Lecturer: Istvan Miklos

  • No. of Credits: 3 and no. of ECTS credits: 6

  • Semester or Time Period of the course: Fall Semester

  • Prerequisites: -

  • Course Level: introductory

  • Brief introduction to the course:

Greedy and dynamic programming algorithms. Famous tricks in computer science. the most important data structures in computer science. The Chomsky hierarchy of grammars, parsing of grammars, relationship to automaton theory. Computers, Turing machines, complexity classes P and NP, NP-complete. Stochastic Turing machines, important stochastic complexity classes. Counting classes, stochastic approximation with Markov chains.


  • The goals of the course:


To learn dynamic programming algorithms
To get an overview of standard tricks in algorithm design

To learn the most important data structures like chained lists, hashing, etc.
To learn the theoretical background of computer science (Turing machines, complexity classes)
 To get an introduction in stochastic computing



  • The learning outcomes of the course:


The students will be able to read and understand moderately involved scientific papers related to the topic.



  • More detailed display of contents.



Lecture 1.
Theory: The O,  and  notations. Greedy and dynamic programming algorithms. Kruskal’s algorithm for minimum spanning trees, the folklore algorithm for the longest common subsequence of two strings.
Practice: The money change problem and other famous dynamic programming algorithms

Lecture 2.
Theory: Dijstra’s algorithm and other algorithms for the shortest path problem.
Preactice: Further dynamic programming algorithms.

Lecture 3.
Theory: Divide-and-conqueror and checkpoint algorithms. The Hirshberg’s algorithm for aligning sequences in linear space
Practice: Checkpoint algorithms. Reduced memory algorithms.

Lecture 4.
Theory: Quick sorting. Sorting algorithms.
Practice: Recursive functions. Counting with inclusion-exclusion.

Lecture 5.
Theory: The Knuth-Morrison-Pratt algorithm. Suffix trees.
Practice: String processing algorithms. Exact matching and matching with errors.

Lecture 6.
Theory: Famous data structures. Chained lists, reference lists, hashing.
Practice: Searching in data structures.

Lecture 7.
Theory: The Chomsky-hierarchy of grammars. Parsing algorithms. Connections to the automaton theory.
Practice: Regular expressions, regular grammars. Parsing of some special grammars between regular and context-free and between context-free and context-dependent classes.

Lecture 8.
Theory: Introduction to algebraic dynamic programming and the object-oriented programming.
Practice: Algebraic dynamic programming algorithms.

Lecture 9.
Theory: Computers, Turing-machines, complexity and intractability, complexity of algorithms, the complexity classes P and NP. 3-satisfiability, and NP-complete problems.
Practice: Algorithm complexities. Famous NP-complete problems.

Lecture 10.
Theory: Stochastic Turing machines. The complexity class BPP. Counting problems, #P, #P-complete, FPRAS.
Practice: Stochastic algorithms.

Lecture 11.
Theory: Discrete time Markov chains. Reversible Markov chains, Frobenious theorem. Relationship between the second largest eigenvalue modulus and convergence of Markov chains. Upper and lower bounds on the second largest eigenvalue.
Practice: Upper and lower bounds on the second largest eigenvalue.

Lecture 12.
Theory: The Sinclair-Jerrum theorem: relationship between approximate counting and sampling.

Practice: Some classical almost uniform sampling (unrooted binary trees, spanning trees).

  • Assessment:

Mid-term test after the 7th week. Description of algorithms + exercises. (50%)

Colloquia at the end of the course. 1 exercise + 1 theorem covering the material of one lecture. (50%)

  • Office hours by appointment

Istvan Miklos


Rényi Institute of Mathematics

Reáltanoda utca 13-15, 1053 Budapest, miklosi@renyi.hu


Name of Course:  Probability and Statistics

  • Lecturer: Marianna Bolla

  • No. of Credits: 3,  no. of ECTS credits: 6

  • Semester or Time Period of the: Fall Semester

  • Prerequisites: Undergraduate Calculus and Real Analysis

  • Course Level: Introductory course

  • Brief introduction to the course:

Students will learn basic probability models with applications. Laws of large numbers, central limit and large deviation theorems will be introduced together with the notion of conditional expectation that plays a crucial role in statistics.

While probability theory describes random phenomena, mathematical statistics teaches us how to behave in the face of uncertainties, according to the famous mathematician Abraham Wald. Roughly speaking, we will learn strategies of treating randomness in everyday life.


  • The goals of the course:


The first part of the course gives an introduction to probability models and basic notion of conditional distributions, while the second part to the theory of estimation and hypothesis testing.

The main concept is that our inference is based on a randomly selected sample from a large population, and hence, our observations are treated as random variables. Through the course we intensively use laws of large numbers and the Central Limit Theorem. On this basis, applications are also discussed, mainly on a theoretical basis, but we make the students capable of solving numerical exercises.


  • 9. The learning outcomes of the course:


Students will be able to identify probability models, further to find the best possible estimator for a given parameter by investigating the bias, efficiency, sufficiency, and consistency of an estimator on the basis of theorems and theoretical facts. Students will gain familiarity with basic methods of estimation and will be able to construct statistical tests for simple and composite hypotheses. They will become familiar with applications to real-world data and will be able to choose the most convenient method for given real-life problems.


  • More detailed display of contents (Week by week breakdown):




  1. Axioms of probability, conditional probability and independence, Lovász Local Lemma, Borel-Cantelli Lemma, famous problems.

  2. Random variables, distribution function, basic discrete probability distributions (Bernoulli, Binomial, Pascal, Poisson) and applications.

  3. Absolutely continuous distributions (Uniform, Exponential, Normal) with applications, density functions and moments. Conditional expectation.

  4. Laws of large numbers, Central Limit Theorem, large deviations.

  5. Statistical space, statistical sample. Basic statistics, empirical distribution function, Glivenko-Cantelli Theorem, histograms. Ordered sample, Kolmogorov-Smirnov Theorems.

  6. Sufficiency, Neyman-Fisher factorization. Completeness, exponential family.

  7. Theory of estimation: unbiased estimators, efficiency, consistency. Fisher information. Cramer-Rao inequality, Rao-Blackwell-Kolmogorov Theorem.

  8. Methods of point estimation: maximum likelihood estimation (asymptotic normality), method of moments, Bayes estimation. Interval estimation, confidence intervals.

  9. Theory of hypothesis testing, Neyman-Pearson Lemma for simple alternative and its extension to composite hypotheses. Parametric inference: u, t, F, chi-square, Welch, Bartlett tests.

  10. Nonparametric inference: chi-square, Kolmogorov-Smirnov tests.

  11. Sequential analysis, Wald-Wolfowitz Theorem, CO-curves.

  12. Two-variate normal distribution and common features of methods based on it. Theory of least squares, regression analysis, Gauss-Markov Theorem.


Literature:

S. Ross, A First Course in Probability. Fifth Edition, Prentice-Hall, 1998.

A. Rényi, Probability Theory, Acad. Press, 1978.

W. Feller, An Introduction to Probability. Theory and Its Applications, Wiley, 1966.

C.R. Rao, Linear statistical inference and its applications. Wiley, New York, 1973.

R. A. Johnson, G. K. Bhattacharyya, Statistics. Principles and methods. Wiley, New York, 1992.

M.G. Kendall, A. Stuart, The Theory of Advanced Statistics I-III. Griffin, London, 1966.


Handouts: tables of notable distributions and percentile values of basic test distributions.


  • Assessment, grading:


For passing grade, student must solve correctly at least 50% of homework exercises (there will be 4 homework assignments, each of them containing 5 exercises) and 50% of the test exercises (there will be 2 tests, each of them containing 10 exercises). Grading in the function of the collected (maximum 40) points:

A: 36-40

A-: 33-35

B+: 30-32

B: 26-29

B-: 23-25

C+: 20-22

However, the final grade can be contested by taking an oral exam at the end of the semester.


  • Office hours: after the classes in office 301.


Schedule of the tests: 6th and 12th weeks.


Contact details: Marianna Bolla, Budapest University of Technology and Economics,

Institute of Mathematics, 1111. Budapest, Egry József u. 1. Bldg. H5.2

Office phone: 06-1-4631111, ext. 5902.

E-mail: marib@math.bme.hu,

Homepage: www.math.bme.hu/~marib/ceu


Name of Course: Real Analysis


  • Lecturer: Laszlo Csirmaz

  • No. of Credits: 3, and no. of ECTS credits: 6

  • Semester or Time Period of the course: Fall Semester of AY 2009-2010

  • Prerequisites: Undergraduate calculus, Elementary Linear Algebra

  • Course level: introductory course

  • Class meeting time: once/week 9:00-12:00 pm, CEU Zrinyi 14, room 310/A


Textbooks: Online material is available at the following sites: http://www.indiana.edu/~mathwz/PRbook.pdf, http://compwiki.ceu.hu/mediawiki/index.php/Real_analysis


  • Objective and learning outcomes of the course:


Introduction to Lebesgue integration theory; measure, σ-algebra, σ-finite measures. Different notion of convergences; product spaces, signed measure, Radon-Nikodym derivative, Fubini and Riesz theorems; Weierstrass approximation theorem. Solid foundation in the Lebesgue integration theory, basic techniques in analysis. It also enhances student’s ability to make their own notes.

At the end of the course students are expected to understand the difference between ”naive” and rigorous modern analysis. Should have a glimpse into the topics of functional analysis as well. They must know and recall the main results, proofs, definition. As a conclusion of the course, students take an oral exam where all acquired knowledge is checked and graded.


  • Detailed contents of the course:

    1. Outer measure, measure, σ-algebra, σ-finite measure

    2. lim inf and lim sup of sets; their measure. The Borel-Cantelli lemma. Complete measure

    3. Caratheodory outer measure on a metric space. Borel sets. Lebesgue measure. Connection between Lebesgue measurable sets and Borel sets

    4. Measurable functions. Measurable functions are closed under addition and multiplication. Continuous functions are measurable. Example where the composition of measurable functions is not measurable

    5. Limits of measurable functions, sup, inf, lim sup, lim inf

    6. Egoroff's theorem: if fi converges pointwise a.e to f then it converges uniformly with an exceptional set of measure <ε. Convergence in measure; pointwise convergence for a subsequence.

    7. Lusin's theorem: a Lebesgue measurable function is continuous with an exceptional set of measure <ε. Converging to a measurable function by simple functions.

    8. Definition of the integral; conditions on a measurable function to be integrable

    9. Fatou's lemma, Monotone Convergence Theorem; Lebesgue's Dominated Convergence Theorem. Counterexample: a sequence of functions tends to f, but the integrals do not converge to the integral of f.

    10. Hölder and Minkowsi inequalities; Lp is a normed space.

    11. Riesz-Fischer theorem: Lp is complete, conjugate spaces, basic properties

    12. Signed measure, absolute continuity, Jordan and Hahn decomposition

    13. Radon-Nikodym derivative

    14. Product measure, Fubini's theorem. Counterexample where the order of integration cannot be exchanged

    15. Example for a continuous, nowhere differentiable function

    16. Example for a strictly increasing function which has zero derivative a.e.

    17. An increasing function has derivative a.e.

    18. Weierstrass' approximation theorem

    19. Basic properties of convolution




  • Assessment:


At the end of the course students take an oral exam they must recall notions, definitions, theorem and proofs. Their grades are awarded on their performance on the final oral exam. During the exam students got the topics as outlined above, have about 20 minutes preparation time, then must present a short view of the topics including theorems, proofs, motivation, counterexamples as indicated by the topic.


  • Office hours: By appointment


László Csirmaz


Csirmaz@ceu.hu


Name of Course: Introduction to Discrete Mathematics

  • Lecturer: Ervin Gyori

  • No. of Credits: 3 and no. of ECTS credits: 6

  • Semester or Time Period of the course: Fall Semester

  • Prerequisite:-

  • Course Level: introductory

  • Brief introduction to the course:

Fundamental concepts and results of combinatorics and graph theory. Main topics: counting, recurrences, generating functions, sieve formula, pigeonhole principle, Ramsey theory, graphs, flows, trees, colorings.

  • The goals of the course:

The main goal is to study the basic methods of discrete mathematics via a lot of problems, to learn combinatorial approach of problems. Problem solving is more important than in other courses!

  • The learning outcomes of the course:

Knowledge of combinatorial techniques that can be applied not just in discrete mathematics but in many other areas of mathematics. Skills in solving combinatorial type problems.

  • More detailed display of contents:

Week 1. Basic counting problems, permutations, combinations, sum rule, product rule

Week 2. Occupancy problems, partitions of integers

Week 3. Solving recurrences, Fibonacci numbers

Week 4. Generating functions, applications to recurrences

Week 5. Exponential generating functions, Stirling numbers, derangements

Week 6. Advanced applications of generating functions (Catalan numbers, odd partitions)

Week 7. Principle of inclusion and exclusion (sieve formula), Euler function

Application of sieve formula to Stirling numbers, derangements, and other involved problems

Week 8. Pigeonhole principle, Ramsey theory, Erdos Szekeres theorem

Week 9. Basic definitions of graph theory, trees

Week 10. Special properties of trees, Cayley’s theorem on the number of labeled trees

Week 11. Flows in networks, connectivity

Week 12. Graph colorings, Brooks theorem, colorings of planar graphs


References:

Fred. S. Roberts, Applied Combinatorics, Prentice Hall, 1984

Fred. S. Roberts, Barry Tesman, Applied Combinatorics, Prentice Hall, 2004

Bela Bollobas, Modern Graph Theory, Springer, 1998


Teaching format: lectures combined with classroom exercises, problems and discussions, homework problems (not just easy exercises!), selected homework problems discussed in class

  • Assessment:

Written final exam within two weeks after the last lecture + graded homework problems.The final exam covers main results including connections between different areas, advanced problems.

  • Office hours by appointment

Ervin Gyori, Renyi Institute of Mathematics,1053 Budapest,

Realtanoda utca 13-15.Email: gyori@renyi.hu


Name of Course: Optimization in Economics

  • Lecturer: Alexandru Kristály

  • No. of Credits: 3, and no. of ECTS credits: 6 

  • Semester or Time Period of the course: Fall Semester of AY 2009-2010 

  • Course Level: introductory course for MS students 

  • Brief introduction to the course: 

In the last decades mathematical methods have become indispensable in the study of many economical problems, in particular, in the optimization of certain real-life phenomena. For instance, J. F. Nash received the Nobel Prize in Economics (1994) for his outstanding contributions in the field of Economics via mathematical tools. Our aim here is to emphasize the importance of Mathematics in the study of a broad range of economical problems. Many applications/examples will be discussed in detail.

  • The goals of the course:  

The main goal of the present course is to introduce Students into the most important concepts and fundamental results of Economics by using various tools from Mathematics as calculus of variations, critical points, matrix-algebra, or even Riemannian-Finsler geometry. Starting with basic economical problems, our final purpose is to describe some recent research directions concerning certain optimization problems in Economics.

  • The learning outcomes of the course: 

The Students will learn how to use well-known mathematical tools to treat both theoretical and practical economical problems.

  • More detailed display of contents

Lecture 1. Introduction and motivation: some basic problems from Economics via optimization.


Lecture 2. Economic applications of one-variable calculus (demand and marginal revenue, elasticity of price, cost functions, profit-maximizing output).

Lecture 3. Economic applications of multivariate calculus (consumer choice theory, production theory, the equation of exchange in Macroeconomics, Pareto-efficiency, application of the least square method).

Lectures 4. Linear programming (application of the geometric, simplex and dual simplex method).


Lecture 5. Linear economical problems (diet problem, Ricardian model of international trade).

Lecture 6. Comparative statics I (equilibrium comparative statics in one and two dimensions; comparative statics with optimization, perfectly competitive firms, Cournot duopoly model).


Lecture 7. Comparative statics II: n variables with and without optimization (equilibrium comparative statics in n dimensions, Gross-substitute system, perfectly competitive firms).


Lecture 8. Comparative statics III: Optimization under constraints (Lagrange-multipliers, specific utility functions, expenditure minimization problems).

Lecture 10. Nash equilibrium points (existence, location, dynamics, and stability).


Lecture 11. Optimal placement of a deposit between markets: a Riemann-Finsler geometrical approach.


Lecture 12. Economical problems via best approximations.


References:

  1. J.-P. Aubin, Optima and Equilibria, An Introduction to Nonlinear Analysis, Springer-Verlag, Berlin, Heidelberg, 1993.

  2. J.-P. Aubin, Analyse non lineaire et ses motivations economiques, Masson, 1984.

  3. D. W. Hands, Introductory Mathematical Economics, D.C. Heath and Company, Toronto, 1991.

  4. I.V. Konnov, Equilibrium models and Variational Inequalities, Math. in Science and Engineering, Elsevier, Amsterdam, 2007.

  5. A. Kristály, G. Moroşanu, A. Róth, Optimal placement of a deposit between markets: Riemann-Finsler geometrical approach. J. Optim. Theory Appl. 139 (2008), no. 2, 263—276.

  6. A. Kristály, Location of Nash equilibria: a Riemannian geometrical approach, Proc. Amer. Math. Soc., in press (2009).

  7. R. Wild, Essential of Production and Operations Management, Cassel, London, 1995.




    • Teaching format: lecture combined with classroom discussions 



    • Assessment:  

Attendance is mandatory.   

Homework: will be assigned regularly. The tests will be based to a significant extent on homework assignments. 

Tests and Grading: There will be two partial exams (a. In October: Lectures 1-6, 50%; b. In December: Lectures 7-12, 50%). Extra bonus points may be earned by presentations as well as by participation in class discussions.  

Final grade: from the two partial exams.

  • Office hours: by appointment  

Alexandru Kristaly, alexandrukristaly@yahoo.com 


Name of Course: Stochastic Processes and Applications


  • Lecturer: Balázs Székely

  • No. of Credits: 3, and no. of ECTS credits: 6

  • Semester or Time Period of the course: Fall Semester of AY 2009-2010

  • Prerequisites: Probability and Statistics

  • Course Level: advanced

  • Brief introduction to the course:


The most common classes of stochastic processes are presented that are important in applications an stochastic modeling. Several real word applications are shown. Emphasis is put on learning the methods and the tricks of stochastic modeling.


  • The goals of the course:


The main goal of the course is to learn the basic tricks of stochastic modeling via studying many applications. It is also important to understand the theoretical background of the methods.


  • The learning outcomes of the course


The students will learn the most common methods in stochastic processes and their applications.


  • More detailed display of contents:




  1. Stochastic processes: Kolmogorov theorem, classes of stochastic processes, branching processes

  2. Poisson processes: properties, arrival times; compound, non-homogeneous and rarefied Poisson process; application to queuing

  3. Martingales: conditional expectation, martingales, stopping times, Wald's equation, convergence of martingales

  4. Applications of martingales: applications to risk processes, log-optimal portfolio

  5. Martingales and Barabási-Albert graph model: preferential attachment (BA model), degree distribution

  6. Renewal processes: renewal function, renewal equation, limit theorems, Elementary Renewal Theorem,

  7. Renewal processes: Blackwell's theorem, key renewal theorem, excess life and age distribution, delayed renewal processes

  8. Renewal processes: applications to queuing, renewal reward processes, age dependent branching process

  9. Markov chains: classification of states, limit theorems, stationary distribution

  10. Markov chains: transition among classes, absorption, applications

  11. Coupling: geometrically ergodic Markov chains, proof of renewal theorem

  12. Regenerative processes: limit theorems, application to queuing, Little's law


Books:

1. S. M. Ross, Applied Probability Models with Optimization Applications, Holden-Day, San Francisco, 1970.

2. S. Asmussen, Applied Probability and Queues, Wiley, 1987.


  • Teaching format: lecture combined with classroom discussions, regular homework assignments




  • Assessment:


Attendance is mandatory.

Homework: will be assigned regularly. The tests will be based to a significant extent on homework assignments.

Tests and Grading: There will be a final exam (worth 70%). The homeworks worth 30%.


  • Such further items as assessment deadlines, office hours, contact details etc are at the discretion of the department or the individual.


Final exam: within two weeks after the final lecture

Office hours: by appointment


Balázs Székely

szbalazs@math.bme.hu


Name of Course: Basic Algebra 2 (MS)

  • Lecturer: Pál Hegedűs

  • No. of Credits: 3, and no. of ECTS credits: 6

  • Semester or Time Period of the course: Winter Semester

  • Prerequisites: Basic Algebra 1

  • Course Level:  intermediate

  • Brief introduction to the course:

Further concepts and theorems are presented. Emphasis is put on difference of questions at different areas of abstract algebra and interconnectedness is underlined throughout. Applications are presented.

  • The goals of the course:

One of the main goals of the course is to introduce the main distinct areas of abstract algebra and the fundamental results therein. A second goal is to let them move confidently between abstract and concrete phenomena.

  • The learning outcomes of the course:

The students will learn in some depth the theories in the three main areas of abstract algebra: groups, rings and fields. More importantly, they will gain some expertise in using them in various areas of mathematics.

  • More detailed display of contents:

1. Groups: composition series, Jordan-Hölder Theorem,
2. conjugation, centralizer, normalizer, class equation, p-groups,
3. nilpotent groups, Frattini subgroup, Frattini argument,
4. direct product, Krull-Schmidt Theorem, semidirect product, groups of small order.
5. Commutative rings: unique factorization, principal ideal domains, Euclidean domains,
6. finitely generated modules over principal ideal domains, Fundamental Theorem of finite abelian groups, Jordan normal form of matrices,
7. Noetherian rings, Hilbert Basis Theorem, operations with ideals.
8. Fields: algebraic and transcendental extensions, transcendence degree,
9. splitting field, algebraic closure, the Fundamental Theorem of Algebra, normal extensions, finite fields, separable extensions,
10. Galois group, Fundamental Theorem of Galois Theory, cyclotomic fields,
11. radical expressions, insolvability of the quintic equation, traces and norms: Hilbert’s Theorem,
12. Artin-Schreier theorems, ordered and formally real fields.

Optional topics:

Categorical approach: products, coproducts, pullback, pushout, functor categories, natural transformations, Yoneda lemma, adjoint functors.

Books:
1. N Jacobson, Basic Algebra I-II, WH Freeman and Co., San Francisco, 1974/1980.
2. I M Isaacs, Algebra, a graduate course, Brooks/Cole Publishing Company, Pacific Grove, 1994

Teaching format: lecture combined with classroom discussions, regular homework assignments

  • Assessment:

Attendance is mandatory. 

Homework: will be assigned regularly. The tests will be based to a significant extent on homework assignments.

Tests and Grading: There will be a midterm test (worth 30%), a final exam (worth 40%) and homework (worth 30%).

Final exam: within two weeks after the final lecture

  • Office hours: by appointment

    Pál Hegedűs
    Zrinyi u. 14, Third Floor, Office # 312, hegedusp@ceu.hu



Name of Course: Complex Function Theory

  • Lecturer: Róbert Szőke

  • No. of Credits: 3 and no. of ECTS credits: 6

  • Semester or Time Period of the course: Winter Semester

  • Prerequisites: calculus

  • Course Level: introductory

  • Brief introduction to the course:

Fundamental concepts and themes of classic function theory in one complex variable are presented: complex derivative of  complex valued  functions, contour integration, Cauchy's  integral theorem,  Taylor and Laurent series, residues, applications, conformal maps.

  • The goals of the course:

The goal of the course is to acquaint the students with the fundamental concepts and results of classic complex function theory.

  • The learning outcomes of the course :

The students will learn some basic notion and theorems (with some applications) of classic complex function theory.

  • More detailed display of contents.

Week 1: Complex numbers
Week 2: Complex differentiable functions
Week 3: Complex line integral
Week 4: Cauchy theorem on star regions and its corollaries
Week 5: Winding number, the general form of Cauchy's theorem
Week 6: Harmonic functions
Week 7: Isolated singularities
Week 8: The residue theorem and its applications
Week 9: Further applications of the residue theorem
Week 10: Conformal maps
Week 11: Riemann mapping theorem
Week 12: Picard's theorems

Books:
1.  R. Remmert, Theory of complex functions, Springer-Verlag, 1991 
2. J. B. Conway: Functions of one complex variable, Springer-Verlag, 1978,
3. A. I. Markushevich: Theory of functions of a complex variable, Chelsea, 1977
4. L. V. Ahlfors: Complex analysis, McGraw-Hill, 1979,
 
Teaching format: lecture combined with classroom discussions

  • Assessment:

Attendance is mandatory.

Homework: will be assigned regularly. The final exam will be based to a significant extent on homework assignments.

Tests and grading: the homeworks worth 30% and the final exam 70%.

Final exam:  written exam in two weeks after the final lecture.

  • Office hours: by appointment
    Róbert Szőke,
    rszoke@cs.elte.hu



Name of Course: Functional Analysis and Differential Equations

  • (syllabus TBA)


Name of Course: Combinatorial Optimization

  • Lecturer: Ervin Győri

  • No. of Credits: 3 and no. of ECTS credits: 6

  • Semester or Time Period of the course: Winter Semester

  • Pre-requisites: discrete mathematics, graph theory, linear algebra

  • Course Level: introductory

  • Brief introduction to the course:

Basic concepts and theorems are presented. Some significant applications are analyzed to illustrate the power and the use of combinatorial optimization. Special attention is paid to algorithmic questions.

  • The goals of the course:

One of the main goals of the course is to introduce students to the most important results of combinatorial optimization. A further goal is to discuss the applications of these results to particular problems, including problems involving applications in other areas of mathematics and practice. Finally, computer science related problems are to be considered too.

  • The learning outcomes of the course:

The students will learn some basic notions and results of combinatorial optimization. They will learn how to use these tools in solving every day life problems as well as in software developing.

  • More detailed display of contents:

Week 1: Typical optimization problems, complexity of problems, graphs and digraphs

Week 2: Connectivity in graphs and digraphs, spanning trees, cycles and cuts, Eulerian and Hamiltonian graphs

Week 3: Planarity and duality, linear programming, simplex method and new methods

Week 4: Shortest paths, Dijkstra method, negative cycles

Week 5: Flows in networks

Week 6: Matchings in bipartite graphs, matching algorithms

Week 7: Matchings in general graphs, Edmonds’ algorithm

Week 8: Matroids, basic notions, system of axioms, special matroids

Week 9: Greedy algorithm, applications, matroid duality, versions of greedy algorithm

Week 10: Rank function, union of matroids, duality of matroids

Week 11: Intersection of matroids, algorithmic questions

Week 12: Graph theoretical applications: dedge disjoint and coverong spanning trees, directed cuts


Books:

E.L. Lawler, Combinatorial Optimization: Networks and Matroids, Courier Dover Publications, 2001 or earlier edition: Rinehart and Winston, 1976

Teaching format: lecture combined with classroom discussions and possible presentations.

  • Assessment:

Attendance is mandatory.Homework and quiz are to be assigned regularly. Worth of them in grading is 20 %.Midterm exam (worth 40 %) and Final exam (worth 40 %) are based on or related to homework and quizzes. Bonus might be earned by presentations.

  • Final exam is to be in at most two weeks after the final lecture.

Office hours on Monday at 11:30 am or by appointment.


Ervin Győri

Rényi Institute, Reáltanoda u.13-15. 1053 Budapest

gyori@renyi.hu


Name of Course: Cryptology
(Laszlo Csirmaz) (syllabus TBA)

Name of Course: Quantitative Financial Risk Analysis
(Balazs Janecsko and Imre Kondor) (syllabus TBA)

Name of Course: Topics in Financial Mathematics
(Alexandru Kristaly) (syllabus TBA)

Name of Course: Statistics of Stochastic Processes
(Balazs Szekely) (syllabus TBA)


Annex


Mathematics Entrance Examination

The exam takes 3 hours and consists of problems in algebra and analysis. Of course, problems may involve a mixture of analysis and algebra. Some problems are computational, some ask for proofs, and some ask for examples or counterexamples.

Here is a list of subjects required for the entrance exam:

Algebra


    Linear Algebra:

  • Vector spaces over R, C, and other fields: subspaces, linear independence, basis and dimension.

  • Linear transformations and matrices: constructing matrices of abstract linear transformations, similarity, change of basis, trace, determinants, kernel, image, dimension theorems, rank; application to systems of linear equations.

  • Eigenvalues and eigenvectors: computation, diagonalization, characteristic and minimal polynomials, invariance of trace and determinant.

  • Inner product spaces: real and Hermitian inner products, orthonormal bases, Gram-Schmidt orthogonalization, orthogonal and unitary transformations, symmetric and Hermitian matrices, quadratic forms.

    Abstract Algebra:

  • Groups: finite groups, matrix groups, symmetry groups, examples of groups (symmetric, alternating, dihedral), normal subgroups and quotient groups, homomorphisms, Sylow theorems.

  • Rings: ring of integers, induction and well ordering, polynomial rings, roots and irreducibility, unique factorization of integers and polynomials, homomorphisms, ideals, principal ideals, Euclidean domains, prime and maximal ideals, quotients, fraction fields, finite fields.



Analysis

  • Real numbers as a complete ordered field. Extended real number system. Topological concepts: neighbourhood, interior point, accumulation point, etc. 

  • Sequences of real numbers. Convergent sequences. Subsequences. Fundamental results. 

  • Numerical series. Standard tests for convergence and divergence. 

  • Real functions of one real variable. Limits, continuity, uniform continuity, differentiation, Riemann integration,  fundamental theorem of calculus, mean value theorem, L'Hopital's rule, Taylor's theorem, etc. 

  • Sequences and series of functions. Pointwise and uniform convergence. Fundamental results. Power series and radii of convergence. 

  • The topology of  Rk. Connected and convex subsets of  Rk.  

  • Functions of several real variables. Limits, continuity, uniform continuity. Continuous functions on compact or  connected sets. Partial derivatives. Differentiable functions. Taylor's theorem. Maxima and minima. Implicit and inverse function theorems. 

  • Multiple integrals. Integrals in various coodinate systems. Vector fields in Euclidean space (divergence, curl, conservative fields), line and surface integrals, vector calculus (Green's theorem in the plane, the divergence theorem in 3-space).  

  • Ordinary differential equations. Elementary techniques for solving special differential equations (separable, homogeneous, first order linear, Bernoulli's, exact, etc.). Existence and uniqueness of solutions to initial value problems (Picard's theorem). Linear differential equations and systems. Fundamental results. 


1   2   3   4   5   6   7

Похожие:

Program in Applied Mathematics iconProgram in Applied Mathematics

Program in Applied Mathematics iconMat-2 applied mathematics

Program in Applied Mathematics iconApplied Engineering Mathematics

Program in Applied Mathematics iconApplied Engineering Mathematics

Program in Applied Mathematics iconFoundations of Applied Mathematics (foam)

Program in Applied Mathematics iconComputer Science and applied mathematics

Program in Applied Mathematics iconApplied Mathematics for Engineers and Scientists

Program in Applied Mathematics iconOn Modern Problems in Applied Mathematics

Program in Applied Mathematics iconPure and Applied Mathematics, Volume 115

Program in Applied Mathematics iconMA1602 Applied Mathematics 3 1 0 100 sm1601

Разместите кнопку на своём сайте:
Библиотека


База данных защищена авторским правом ©lib.znate.ru 2014
обратиться к администрации
Библиотека
Главная страница