Cycle of education: 2019/2020
The name of the faculty organization unit: The faculty Mathematics and Applied Physics
The name of the field of study: Mathematics
The area of study: sciences
The profile of studing:
The level of study: second degree study
Type of study: full time
discipline specialities : Applications of Mathematics in Economics
The degree after graduating from university: master
The name of the module department : Department of Mathematics
The code of the module: 1488
The module status: mandatory for teaching programme Applications of Mathematics in Economics
The position in the studies teaching programme: sem: 1 / W30 C45 / 5 ECTS / Z
The language of the lecture: Polish
The name of the coordinator: Prof. Józef Banaś, DSc, PhD
office hours of the coordinator: podane w harmonogramie pracy jednostki.
semester 1: Agnieszka Chlebowicz, PhD , office hours as in the work schedule of Department of Nonlinear Analysis.
The main aim of study: To familiarize students with the fundamentals of measure theory with particular emphasis on Lebesgue measure.
The general information about the module: The module is implemented in the first semester in the form of lectures (30 hours) and exercises (45 hours).
1 | E. DiBenedetto | Real Analysis | Birkhäuser, Springer, New York. | 2016 |
2 | S. Łojasiewicz | Wstęp do teorii funkcji rzeczywistych | PWN, Warszawa. | 1973 |
3 | W. Rudin | Analiza rzeczywista i zespolona | PWN, Warszawa. | 1986 |
4 | R. Sikorski | Funkcje rzeczywiste, tom I | PWN, Warszawa. | 1958 |
1 | W. Rudin | Analiza rzeczywista i zespolona | PWN, Warszawa. | 1986 |
2 | R. Sikorski | Funkcje rzeczywiste, tom I | PWN, Warszawa. | 1958 |
Formal requirements: The student satisfies the formal requirements set out in the study regulations
Basic requirements in category knowledge: A student has mathematical knowledge which allows him/her to understand the lectured material.
Basic requirements in category skills: Ability to use fundamental mathematical tools and the knowledge obtained during the first level studies.
Basic requirements in category social competences: A student is prepared to undertake substantiated mathematical operations in order to solve a task.
MEK | The student who completed the module | Types of classes / teaching methods leading to achieving a given outcome of teaching | Methods of verifying every mentioned outcome of teaching | Relationships with KEK | Relationships with PRK |
---|---|---|---|---|---|
01 | knows basic concepts and definitions given during the course of lectures i.e.: lower limit and upper limit of a sequence of sets, field of sets and sigma-field of sets, finitely additive measure and sigma additive measure, complete measure, Jordan measure, exterior measure, Caratheodory's condition | lecture, exercises | test |
K_W01++ K_W02+ K_W03+ K_W04+++ K_W05++ K_W07+ K_K04+ K_K07+ |
P7S_KK P7S_KO P7S_KR P7S_WG P7S_WK |
02 | knows how to perform fundamental operations on sets and is able to indicate lower limit and upper limit of a sequence of sets | lecture, exercises | test |
K_W01++ K_W02+ K_W03+ K_U01++ K_U02++ K_U04+ K_U08+ K_K01+ |
P7S_KK P7S_UK P7S_UO P7S_UW P7S_WG P7S_WK |
03 | knows how to check properties of family of sets | lecture, exercises | test |
K_W01++ K_W02+ K_W03+ K_U01++ K_U02++ K_U04+ K_U07+ K_U08+ K_K01+ |
P7S_KK P7S_UK P7S_UO P7S_UW P7S_WG P7S_WK |
04 | knows how to check if a given function is finitely additive measure and if it is sigma additive measure | lecture, exercises | test |
K_W01++ K_W02+ K_W03+ K_U01++ K_U02++ K_U03++ K_U04+ K_U05+ K_U07+ K_U08+ K_U09+ K_U13+ K_U14+ K_K01+ |
P7S_KK P7S_UK P7S_UO P7S_UW P7S_WG P7S_WK |
05 | knows how to calculate or estimate Jordan measure of a set contained in R or R^2 | lecture, exercises | test |
K_W01++ K_W02+ K_W03+ K_U01++ K_U02++ K_U03++ K_U04+ K_U07+ K_U15+ K_K01+ K_K02++ |
P7S_KK P7S_KO P7S_UK P7S_UO P7S_UU P7S_UW P7S_WG P7S_WK |
06 | knows how to check if a given function is exterior measure | lecture, exercises | test |
K_W01++ K_W02+ K_W03+ K_W07+ K_U01++ K_U03++ K_U05+ K_U09+ K_U13+ K_U14+ K_K01+ K_K02++ K_K07+ |
P7S_KK P7S_KO P7S_KR P7S_UK P7S_UW P7S_WG P7S_WK |
Attention: Depending on the epidemic situation, verification of the achieved learning outcomes specified in the study program, in particular credits and examinations at the end of specific classes, can be implemented remotely (real-time meetings).
Sem. | TK | The content | realized in | MEK |
---|---|---|---|---|
1 | TK01 | W01-W08, C01-C12 | MEK01 MEK02 MEK03 | |
1 | TK02 | W09-W16, C13-C24 | MEK01 MEK04 | |
1 | TK03 | W17-W22, C25-C33 | MEK01 MEK05 | |
1 | TK04 | W23-W30, C34-C45 | MEK01 MEK06 |
The type of classes | The work before classes | The participation in classes | The work after classes |
---|---|---|---|
Lecture (sem. 1) | contact hours:
30.00 hours/sem. |
complementing/reading through notes:
10.00 hours/sem. |
|
Class (sem. 1) | The preparation for a Class:
15.00 hours/sem. The preparation for a test: 15.00 hours/sem. |
contact hours:
45.00 hours/sem. |
Finishing/Studying tasks:
10.00 hours/sem. |
Advice (sem. 1) | The preparation for Advice:
3.00 hours/sem. |
The participation in Advice:
3.00 hours/sem. |
|
Credit (sem. 1) | The preparation for a Credit:
10.00 hours/sem. |
The written credit:
4.00 hours/sem. |
The type of classes | The way of giving the final grade |
---|---|
Lecture | A credit for the lecture is based on attendance at the lectures. |
Class | A credit for the exercises is based on the results of tests and oral answers. |
The final grade | The final grade is a credit for the exercises. |
Required during the exam/when receiving the credit
(-)
Realized during classes/laboratories/projects
(-)
Others
(-)
Can a student use any teaching aids during the exam/when receiving the credit : no
1 | J. Banaś; J. Madej | Asymptotically Stable Solutions of Infinite Systems of Quadratic Hammerstein Integral Equations | 2024 |
2 | J. Banaś; J. Madej | On solutions vanishing at infinity of infinite systems of quadratic Urysohn integral equations | 2024 |
3 | J. Banaś; J. Ochab; T. Zając | On the smoothness of normed spaces | 2024 |
4 | A. Ali; J. Banaś; . Mahfoudhi; B. Saadaoui | (P,Q)–ε-Pseudo Condition Spectrum for 2×2 Matrices. Linear Operator and Application | 2023 |
5 | J. Banaś; R. Taktak | Measures of noncompactness in the study of solutions of infinite systems of Volterra-Hammerstein-Stieltjes integral equations | 2023 |
6 | J. Banaś; V. Erturk; P. Kumar; A. Manickam; S. Tyagi | A generalized Caputo-type fractional-order neuron model under the electromagnetic field | 2023 |
7 | J. Banaś; A. Chlebowicz; M. Taoudi | On solutions of infinite systems of integral equations coordinatewise converging at infinity | 2022 |
8 | J. Banaś; R. Nalepa | The Space of Functions with Tempered Increments on a Locally Compact and Countable at Infinity Metric Space | 2022 |
9 | J. Banaś; R. Nalepa; B. Rzepka | The Study of the Solvability of Infinite Systems of Integral Equations via Measures of Noncompactness | 2022 |
10 | J. Banaś; W. Woś | Solvability of an infinite system of integral equations on the real half-axis | 2021 |
11 | J. Banaś; A. Chlebowicz; W. Woś | On measures of noncompactness in the space of functions defined on the half-axis with values in a Banach space | 2020 |
12 | J. Banaś; B. Krichen; B. Mefteh | Fixed point theorems in WC-Banach algebras and their applications to infinite systems of integral equations | 2020 |
13 | J. Banaś; L. Olszowy | Remarks on the space of functions of bounded Wiener-Young variation | 2020 |
14 | J. Banaś; A. Chlebowicz | On solutions of an infinite system of nonlinear integral equations on the real half-axis | 2019 |
15 | J. Banaś; B. Rzepka | Ocena efektywności inwestycji | 2019 |
16 | J. Banaś; B. Rzepka | Wykłady matematyki finansowej | 2019 |
17 | J. Banaś; L. Olszowy | On the equivalence of some concepts in the theory of Banach algebras | 2019 |
18 | J. Banaś; M. Krajewska | On solutions of semilinear upper diagonal infinite systems of differential equations | 2019 |
19 | J. Banaś; R. Nalepa | A measure of noncompactness in the space of functions with tempered increments on the half-axis and its applications | 2019 |
20 | J. Banaś; T. Zając | On a measure of noncompactness in the space of regulated functions and its applications | 2019 |
21 | L. Abadias; E. Alvarez; J. Banaś; C. Lizama | Solvability and uniform local attractivity for a Volterra equation of convolution type | 2019 |