# Selected topics in applied mathematics

This course is part of the programme
Master in Engineering and Management (Second Level)

## Objectives and competences

The primary goal of this course is to teach students the basics of advanced mathematical topics which are indispensable in science and engineering. The focus is on probability theory and statistics, ordinary differential equations and integral transforms. The topics are mandatory for subjects like "Data mining" and "Introduction to control systems".
The competences gained by students encompass:
- the ability to formulate and solve some problems in natural sciences and engineering by using probabilistic setup
- the ability to solve the systems of ordinary differential equations by means of numerical methods
- sufficient skills to solve linear differential equations in analytical way by using integral transforms.

## Prerequisites

Basics of mathematical analysis and linear algebra.

## Content

1. Probability and statistics
a. Probability of an event
b. Basic concepts in statistics
c. Sampling
d. Confidence intervals
e. Analysis of variance, covariance and linear regression
2. Ordinary differential equations (ODE)
a. Linear differencial equations
b. Qualitative behaviours of systems of differencial equations
3. Integral transforms
a. Fourier series
b. Fourier transform
c. Laplace transform

## Intended learning outcomes

Knowledge and understanding:
- basic knowledge and understanding of how to deal with uncertainty in natural and engineering sciences and engineering,
- acquaintance with basic techniques of calculus with random variables
- distinguishing between linear and nonlinear as well as analytically and numerically solvable DE's
- basic skills for numerical solving of (systems) of nonlinear ordinary differential equations
- understanding the role of integral transforms and basic skills in analytic solving of linear ordinary differential equations

• R. Jamnik, Verjetnostni račun in statistika, DMFA Ljubljana, 1995.
• F. Križanič, Navadne diferencialne enačbe, DMFA Ljubljana, 1991.
• B. Zmazek: Diferencialna analiza, skripta, Maribor, 2006.
• A. Suhadolc, Integralske transformacije, Integralske enačbe, DMFA Ljubljana, 1994.
• D. Freedman, R. Pisani, R. Purves, Statistics, Norton, 1998.
• F.M. Dekking, et al., A modern introduction to Probability and Statistics, Springer 2005, https://cis.temple.edu/~latecki/Courses/CIS2033-Spring13/Modern_intro_probability_statistics_Dekking05.pdf
• C. R. Wylie, Differential equations, McGraw-Hill Singapore, 1985.
• L. Debnath, D. Bhatta, Integral transforms and their applications, CRC Press, 2015.

## Assessment

Written exam, Mandatory homework assignments 75/25

## Lecturer's references

Assoc. prof. dr. Irina Elena Cristea, hab. field mathematics

Principal education and research areas: Mathematics, algebraic hyperstructures and connections with fuzzy sets and their generalizations; mathematical models based on cluster analysis (hard and fuzzy version).
Professional career: Prof. Irina Elena Cristea received the PhD in Mathematics, with specialization in Algebra, in 2007, from the University of Constantza, Romania. She started the research career in 2003 at the University of Iasi, Romania, where she worked as Teaching Assistant for 4 years. After a postdoctoral position at Udine University, Italy, in January 2012, she was employed at the University of Nova Gorica, as Assistant Professor in Mathematics, conducting research in Algebraic Hyperstructure Theory. Since 2017, she is Associate Professor in Mathematics, being affiliated with the Centre for Information Technologies and Applied Mathematics. She is currently teaching Mathematics at the School of Engineering and Management, School of Environmental Sciences, School of Applied Sciences.

Selected bibliography

HEIDARI, Dariush, CRISTEA, Irina Elena. Breakable semihypergroups. Symmetry, ISSN 2073-8994, 2019, vol. 11, iss. 1, str. 1-10, [COBISS.SI-ID 5313787]

JANČIĆ-RAŠOVIĆ, Sanja, CRISTEA, Irina Elena. Hypernear-rings with a defect of distributivity. Filomat, ISSN 0354-5180, 2018, vol. 32, no. 4, str. 1133-1149, [COBISS.SI-ID 5175803]

NOROUZI, Morteza, CRISTEA, Irina Elena. Transitivity of the ϵm−relation on (m-idempotent) hyperrings. Open Mathematics, ISSN 2391-5455, 2018, vol. 16, iss. 1, str. 1012-1021, [COBISS.SI-ID 5201915]

BORDBAR, Hashem, CRISTEA, Irina Elena. Height of prime hyperideals in Krasner hyperrings. Filomat, ISSN 0354-5180, 2017, vol. 31, iss. 19, str. 6153-6163,[COBISS.SI-ID 4978939]

DAVVAZ, Bijan, CRISTEA, Irina Elena. Fuzzy algebraic hyperstructures : an introduction, (Studies in fuzziness and soft computing, vol. 321). Cham: Springer, cop. 2015. X, 242 str., ilustr. ISBN 978-3-319-14761-1 [COBISS.SI-ID 3704571]

DAVVAZ, Bijan, HASSANI SADRABADI, E., CRISTEA, Irina Elena. Atanassov's intuitionistic fuzzy grade of complete hypergroups of order less than or equal to 6. Hacettepe journal of mathematics and statistics, ISSN 1303-5010, 2015, vol. 44, no. 2, str. 295-315 [COBISS.SI-ID 3888123]

CRISTEA, Irina Elena, DAVVAZ, Bijan, HASSANI SADRABADI, E. Special intuitionistic fuzzy subhypergroups of complete hypergroups. Journal of intelligent & fuzzy systems, ISSN 1064-1246. 2015, vol. 28, no. 1, str. 237-245. [COBISS.SI-ID 3454203] 