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# SZZMI

**Course:** Mathematics and Informatics

**Department/Abbreviation:** SLO/SZZMI

**Year:** 2020

**Guarantee:** 'doc. RNDr. Michal Krupka, Ph.D.'

**Annotation:** Final exam for verification and evaluation of the level of knowledge.

**Course review:**

MATHEMATICS
- Polynomials, matrix calculus, determinants.
- Algebraic structures, vector spaces - groups, vector spaces, bases, subspaces, linear map.
- Solving systems of linear equations - Gauss elimination method, Cramer's rule.
- Scalar product - scalar product in vector spaces over R and C, orthogonal vectors, vector length, the angle between vectors, orthonormal base.
- Sequence - limit of a sequence, sequence theorems, relationship of limits and convergence.
- Function - the function definition and operations with functions, continuous function and properties of continuous functions, limit of a function, limit of a composite function, basic functions and their calculus.
- Derivative - relationship between continuity and limit, differentials, mean value theorems, graphs of functions, Taylor polynomial and L'Hospital rule.
- Integration - Newton's formula and connection with derivatives, primitive functions, integration by parts, the first and the second substitution, integration of rational functions and other numerical techniques Riemann integral and proof of Newton's formula.
- Applications of integral calculus - lengths, areas, volumes, center of gravity, moments of inertia, surfaces, methods of numerical integration.
- Differential equations - existence and uniqueness of solutions of ordinary differential equations.
Separable equations and numerical techniques. Linear differential equations. Linear differential equations of higher order and numerical techniques.
- Functions of several variables - definition of a function of several variables, continuity and limits. Derivatives of functions of several variables. Potentials, vector fields, gradients, divergence, rotation, and their applications. Implicit function. Extremes of functions of several variables: Lagrange multipliers.
- Series - series with non-negative members, absolute and relative convergence. Series of functions - Fourier analysis.
- Lebesgue measure and integral. Integrals dependent on parameters. Fubini's theorem and substitution theorem.Line integrals and potential. Surface integrals. Gauss-Ostrogradsky's, Green's and Stokes' theorems.
INFORMATICS
- The terms - problem and algorithm, algorithm complexity, asymptotic notation.
- Linear data structures: array, list, stack, queue. The complexity of operations with linear structures.
- The problem of sorting. Sorting algorithms and their complexity: insertsort, selectsort, quicksort, heapsort, mergesort. The lower limit of algorithms' complexity for sorting by comparison. Sorting methods without comparison: counting sort, bucket sort.
- The terms - graph and tree, their basic properties. Algorithms of passage through a graph.
- Binary search trees and operations with them. Other variants of search trees: red-black trees, AVL-trees.
- Hash table, hash function and its properties. Methods of conflict solution in the hash table: open addressing, chaining.
- Von Neumann architecture. CPU, registers, machine instructions and their execution. Assembler: controlling of calculation, function calls, interrupts handling.
- Operating system: the importance, architecture, monolithic and microkernel approach. The terms - memory, process, thread, input-output device.
- Memory management, direct and indirect addressing, cache memory and algorithms for managing it.
- Managing processes and threads, life cycle of a process, algorithms for allocating processor.
- Synchronization. Operating system resources for synchronization, passive and active waiting. Deadlock conditions for its emergence, detection and prevention, the banker's algorithm.
- File management, file systems: FAT, NTFS, Ext3. Journaling. Arrays of discs, RAID.
- Managing of I/O device, drivers.