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In the article the authors present improved Dijkstra's method for the military communication system by using the incidence matrix of nodes and communication lines instead of the incidence matrix of nodes. The improved method was adapted to the model of the military communication system, which is formalized by a block matrix of the incidence of nodes and communication lines with the division of matrix blocks by types and components of the military communication system. The improvement implies in the introduction of additional blocks to the standard Dijkstra algorithm for preliminary, more accurate calculation of the weighting factors of communication lines. At the same time, the authors propose a number of new analytical expressions based on the use of the block matrix of secondary incidence, as well as the quadratic form of the incidence matrix. Mathematical modeling based on a simplified model of the military communication system confirmed the efficiency of the improved method. The improved method synthesized by the authors is more relevant to the real conditions of managing the military communication system and can be used for the needs of the system management process automation as a basis for the development of information and analytical tasks.

Predmet istraživanja ove doktorske disertacije je sistematsko povezivanje dinamičkih svojstava kompleksnih sistema (kao što su stabilnost, robusnost, reaktivnost, tranziciono ponašanje, amplifikacioni okvir) sa teorijom H-matrica, koje je motivisalo nove rezultate iz oblasti lokalizacije blok i frakcionog pseudospektra, kao i ocene norme inverzne matrice, kako u tačkastoj, tako i u blok formi, ali i primenu ovih rezultata u dinamičkoj analizi kompleksnih sistema., This thesis provides a systematic link between the dynamical properties of complex systems (such as stability, robustness, reactivity, transient behavior, amplification envelope) and the theory of H-matrices, which has motivated new results as far as block and fractional pseudospectrum localizations are concerned, as well as some new upper bounds for the inverse of certain subclasses of H-matrices, both in the point-wise and block case, and the overall collective application of these results in the dynamical analysis of complex systems.

Effective arranging of numerical data and design of associated computational algorithms are important for any area of mathematics for teaching, learning and research purposes. Usage of various algorithms for the same area makes mathematics teaching goal-oriented and diverse. Matrices and linear-algebraic ideas can be used to make algorithms visual, two dimensional (2D) and easy to use. It may contribute to the planned educational reforms by teaching school and university students deeper mathematical thinking. In this article we give novel data arranging techniques (2D and 3D) for matrix multiplication. Our 2D method differs from the standard, formal approach by using block matrices. We find this method a helpful alternative for introducing matrix multiplication. We also give a new innovative 3D visualisation technique for matrix multiplication. In this method, matrices are positioned on the faces of a rectangular cuboid. Computerized implementations of this method may be considered as student project proposals.

Effective arranging of numerical data and design of associated computational algorithms are important for any area of mathematics for teaching, learning and research purposes. Usage of various algorithms for the same area makes mathematics teaching goal-oriented and diverse. Matrices and linear-algebraic ideas can be used to make algorithms visual, two dimensional (2D) and easy to use. It may contribute to the planned educational reforms by teaching school and university students deeper mathematical thinking. In this article we give novel data arranging techniques (2D and 3D) for matrix multiplication. Our 2D method differs from the standard, formal approach by using block matrices. We find this method a helpful alternative for introducing matrix multiplication. We also give a new innovative 3D visualisation technique for matrix multiplication. In this method, matrices are positioned on the faces of a rectangular cuboid. Computerized implementations of this method may be considered as student project proposals.

The focus of this thesis is on two functions: the exponential function and Euler’s factorial series. By constructing explicit Padé approximations, we are able to improve lower bounds for linear forms in the values of these functions. In particular, the dependence on the height of the coefficients of the linear form will be sharpened in the lower bound. The first chapter contains some necessary definitions and auxiliary results needed in later chapters. We give precise definitions for a transcendence measure and Padé approximations of the second type. Siegel’s lemma will be introduced as a fundamental tool in Diophantine approximation. A brief excursion to exterior algebras shows how they can be used to prove determinant expansion formulas. The reader will also be familiarised with valuations of number fields. In Chapter 2, a new transcendence measure for e is proved using type II Hermite-Padé approximations to the exponential function. An improvement to the previous transcendence measures is achieved by estimating the common factors of the coefficients of the auxiliary polynomials. The exponential function is the underlying topic of the third chapter as well. Now we study the common factors of the maximal minors of some large block matrices that appear when constructing Padé-type approximations to the exponential function. The factorisation of these minors is of interest both because of Bombieri and Vaaler’s improved version of Siegel’s lemma and because they are connected to finding explicit expressions for the approximation polynomials. In the beginning of Chapter 3, two general theorems concerning factors of Vandermonde-type block determinants are proved. In the final chapter, we concentrate on Euler’s factorial series which has a positive radius of convergence in p-adic fields. We establish some non-vanishing results for a linear form in the values of Euler’s series at algebraic integer points. A lower bound for this linear form is derived as well.

For the first time, with the help of mathematical modelling, clarified the properties of the received analytical relations and set the limits possibilities of OFDM (N-OFDM) signal demodulation method in the background of impulse signals in the integrated radar and telecommunication systems. It was found out that at angular spacing’s of sources of pulsed and OFDM signals at a value that not less than 0,75 width of a secondary beam. The same applies to the effects of active interference.

Matrices find many applications in various research areas and reallifeproblems. In spite of the availability of many innovative tools instatistics, the main tool of the applied statistician remains the linearmodel. Patterned matrices are often used to model dependence structureof longitudinal or repeated measures data. In this paper, someproperties of symmetric circular Toeplitz matrices will be outlined whichare useful for inference in linear models. Special focus is on block matriceshaving Kronecker structure since they arise in many applicationsfor modelling spatio-temporal data.

In this paper, we investigate additive results of the Drazin inverse of elements in a ring R. Under the condition ab = ba, we show that a + b is Drazin invertible if and only if aa^D(a+b) is Drazin invertible, where the superscript D means the Drazin inverse. Furthermore we find an expression of (a + b)^D. As an application we give some new representations for the Drazin inverse of a 2 × 2 block matrix.

In this article, we derive explicit expressions for the entries of the inverse of a patterned matrix that is a sum of Kronecker products. This matrix keeps the Kronecker structure under matrix inversion, and it is used, for example, in statistics, in particular in the linear mixed model analysis. The obtained results present new and extended existing algorithms for the inversion of the considered patterned matrices. We also obtain a closed-form inverse in terms of block matrices.

In this paper we derive explicit expressions for the entries of the inverse of a patternedmatrix that is a sum of Kronecker products. This matrix keeps the Kronecker structure under matrixinversion. We also obtain a closed form inverse in terms of block matrices.