Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT
| MMA1016 | Topics of Mathematical Analysis I (for teachers education) |
| Teaching Staff in Charge |
Assoc.Prof. POPOVICI Nicolae, Ph.D., popovici math.ubbcluj.ro |
| Aims |
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This course aims to present some special topics of mathematical analysis concerning the sequences and series of real numbers, as well as several important classes of functions. |
| Content |
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- Sequences: the limit inferioar and limit superior of a sequence of extended real numbers and their role in the existence of the limit, the set of lmit points of a sequence, Kronecker and Dirichlet Theorems concerning the approximation of real numbers by rational numbers; sequences given by liniar or nonlinear reccurence relations; Toeplitz Theorem and some of its corollaries (Stolz-Cesaro and Cauchy Theorems).
- Series of real numbers: Cauchy and Riemann Theorems concerning the rearrangement of the terms of a series; Abel, Cauchy and Mertens Theorems concerning the Cauchy product of two series. - Semicontinuous functions: the characterization of semicontinuity by means of sequences, the limit inferioar and limit superioar of a function at a point and their relationship with semicontinuity. - Uniformly continuous functions between two normed spaces: the characterization of uniform continuity by means of sequences, the relationship between uniformly continuous functions and other important classes of functions (Lipschitz and Hölder continuous functions). - Functions having the Darboux property and functions admitting primitives. - Convex functions of one or several real variables: characterizations of convex functions; regularity properties of convex functions; generalized convexity; remarkable inequalities. |
| References |
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1. BORWEIN, J.M., LEWIS, A.S.: Convex Analysis and Nonlinear Optimization. Theory and Examples. CMS Books in Mathematics, Springer, 2000.
2. BRECKNER, B.E., POPOVICI, N.: Probleme de analiză convexă în R^n. Casa Cărţii de Ştiinţă, Cluj-Napoca, 2003. 3. COBZAŞ, ŞT.: Analiză matematică (Calcul diferenţial). Presa Universitară Clujeană, Cluj-Napoca, 1997. 4. MEGAN, M.: Bazele analizei matematice. Vol. I + Vol. II, Editura EUROBIT, Timişoara, 1997. Vol. III, Editura EUROBIT, Timişoara, 1998. 5. NICULESCU, C.P., PERSSON L.-E.: Convex Functions and Their Applications. A Contemporary Approach. Springer, 2006. 6. ROBERTS, A.W., VARBERG, D.E.: Convex Functions. Academic Press, 1973. 7. RUDIN, W.: Principles of Mathematical Analysis. 2nd Edition, McGraw-Hill, New York, 1964. 8. SIREŢCHI, GH.: Calcul diferenţial şi integral. Vol. 1: Noţiuni fundamentale. Vol. 2: Exerciţii, Editura Ştiinţifică şi Enciclopedică, Bucureşti, 1985. 9. TRIF, T.: Probleme de calcul diferenţial şi integral în R^n. Casa Cărţii de Ştiinţă, Cluj-Napoca, 2003. |
| Assessment |
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Continuous evaluation (contributes 20% to the assesment), written and oral exam (contributes 80% to the assesment). |
| Links: | Syllabus for all subjects Romanian version for this subject Rtf format for this subject |