Zugl.: München, Techn. Univ., Diss., 2009

Universiẗat, Diss., 2003--Saarbrücken.

This thesis includes multinominals with one{\crq}s variables. Aim is create collection exercises hereto subject. Collection is divided on chaps with given to problems. In every chapter is totality theory buckthorn examples and straddle examples. All choice example are exemplary processed. Behind every chapter reader can try out problems for examles that are supplementeds record

Detta arbete tittar på rötter till de Eulerska polynomen. Dessa polynom har starka kopplingar till den enumerativa kombinatoriken. Polynomens koefficienter, de Eulerska talen, har kopplingar till många andra delar av matematiken. Arbetet jämför egenskaper med binomialkoefficienterna i både beteckning och beräkning. En beskrivning av de Eulerska polynomen ges, bland annat hur man räknar ut de Eulerska talen. Arbetet bevisar att samtliga rötter till de Eulerska polynomen är reella, negativa och enkla samt att dessa rötter ligger inflätade med rötterna för det Eulerska polynomet av en grad högre. Till sist visas att koefficienterna i de Eulerska polynomen är unimodala.

Universiẗat, Diss., 1999--Konstanz.

In the thesis we study two dimensional simplicial complexes and linear codes. We say that a linear code C over a field F is triangular representable if there exists a two dimensional simplicial complex ∆ such that C is a punctured code of the kernel ker ∆ of the incidence matrix of ∆ over F and dim C = dim ker ∆. We call this simplicial complex a geometric representation of C. We show that every linear code C over a primefield is triangular representable. In the case of finite primefields we construct a geometric representation such that the weight enumerator of C is obtained by a simple formula from the weight enumerator of the cycle space of ∆. Thus the geometric representation of C carries its weight enumerator. Our motivation comes from the theory of Pfaffian orientations of graphs which provides a polynomial algorithm for weight enumerator of the cut space of a graph of bounded genus. This algorithm uses geometric properties of an embedding of the graph into an orientable Riemann surface. Viewing the cut space of a graph as a linear code, the graph is thus a useful geometric representation of this linear code. We study embeddability of the geometric representations into Euclidean spaces. We show that every binary linear code has a geometric representation that can be embed- ded into R4 . We characterize...

Tato práce se zabývá použitím zobecněných Laguerrových funkcí pro modelování a identifikaci dynamických systémů. Po krátkém úvodu je uvedena definice zobecněných Laguerrových polynomů a funkcí a některé jejich vlastnosti. Je zavedena metoda pro identifikaci spojitých dynamických systémů z diskrétních nasamplovaných dat s použitím zobecněných Laguerrových funkcí a porovnána s metodou nejmenších čtverců. Je diskutována volba optimálních parametrů p a alpha pro konečné Fourierovy řady se zobecněnými Laguerrovými funkcemi. Na příkladech je porovnána navržená identifikační metoda s metodou nejmenších čtverců. Je zavedena diskrétní Laguerrova transformace a její rozšíření s použitím zobecněných Laguerrových funkcí. Na příkladech je předvedena aplikace diskrétní Laguerrovy transformace na kompresi dat. Je ukázáno, že použití diskrétní Laguerrovy transformace může vést k lepším výsledků při kompresi dat než tradiční postup s využitím diskrétní kosinové transformace.

This thesis deals with description of CRC codes, which is a type of polynomial error correction codes, and description of CAN and CAN FD protocols used in automobiles for data transmission between sensors. One of the security elements is usage of the CRC codes with the Hamming distance 6. Unfortunately, both protocols contain a design vulnerability which causes that some received messages with one wrong bit do not have to be detected by the protocol. The aim of the thesis was to describe this vulnerability and found out, if it was possible to eliminate it by using different CRC code. It managed to characterize all messages, which are not during this vulnerability detected by CRC code and based on that it was possible to prove, that the probability of error does not depend on a CRC code choice of a fixed length. 1

Lozenges are polygons constructed by gluing two equilateral triangles along an edge. We can fit lozenge pieces together to form larger polygons and given an appropriate polygon we can tile it with lozenges. Lozenge tilings of the semi-regular hexagon with sides A,B,C can be viewed as the 2D picture of a stack of cubes in a A x B x C box. In this project we investigate the typical shape of a tiling as the sides A,B,C of the box grow uniformly to infinity and we consider two cases: The uniform case where all tilings occur with equal probability and the q^Volume case where the probability of a tiling is proportional to the volume taken up by the corresponding stack of cubes. To investigate lozenge tilings we transform it into a question on families of non-intersecting paths on a corresponding graph representing the hexagon. Using the Lindström–Gessel–Viennot theorem we can define the probability of a non-intersecting path crossing a particular point in the hexagon both for the uniform and the $q$-Volume case. In each case this probability function is connected to either the Hahn or the $q$-Hahn orthogonal polynomials. The orthogonal polynomials depend on the sides of the hexagon and so we consider the asymptotic behaviour of the polynomials as the sides grow to infinity using a result due to Kuijlaars and Van Assche. This determines the density of non-intersecting paths through every point in the hexagon, which we calculate, and a ``Arctic curve" result which shows that the six corners of the hexagon are (with probability one) tiled with just one type of lozenge. / "Lozenger" är polygoner konstruerade genom att limma två liksidiga trianglar längs en kant. Vi kan montera lozengstycken ihop för att bilda större polygoner och med en lämplig polygon kan vi lozengplatta den. Lozengplattor av den semi-liksidiga hexagonen med sidorna A, B, C kan ses som 2D-bilden av en stapel kuber i en A x B x C-box. I det här projektet undersöker vi den typiska formen på en platta när sidorna A, B, C på rutan växer till oändlighet och vi tar an två fall: Det likformiga fallet där alla plattor sker med samma sannolikhet och q ^ Volymfallet då sannolikheten för en platta är proportionell mot volymen som tas upp av motsvarande kubstapel. För att undersöka plattor förvandlar vi det till en fråga om samlingar av icke-korsande vägar på en motsvarande graf som representerar hexagonen. Med hjälp av satsen Lindström – Gessel – Viennot kan vi definiera sannolikheten för att en icke-korsande väg går genom en viss punkt i hexagonen både för det enhetliga och $ q $ -volymfallet. I båda fallen är dessa sannolikhetsfunktioner relaterade till Hahn eller $ q $ -Hahn ortogonala polynomer. Dessa ortogonala polynom beror på hexagonens sidor så vi betraktar polynomens asymptotiska beteende när sidorna växer till oändlighet genom ett resultat från Kuijlaars och Van Assche. Detta bestämmer densiteten för de icke-korsande vägarna genom varje punkt i det hexagon vi beräknar. Detta bestämmer också också en '' arktisk kurva '' som visar att hexagonens sex hörn är (med sannolikhet ett) plattade med bara en typ av lozeng.

Nowadays high-precision computations are still more desired. Either for simulation on a level of atoms where every digit is important and inaccurary in computation can cause invalid result or numerical approximations in partial differential equations solving where a small deviation causes a result to be useless. The computations are carried over data types with precision of order hundred to thousand digits, or even more. This creates pressure on time complexity of problem solving and so it is essential to find very efficient methods for computation. Every complex physical problem is usually described by a system of equations frequently containing elementary functions like sinus, cosines or exponentials. The aim of the work is to design and implement methods that for a given precision, arbitrary elementary function and a point compute its value in the most efficent way. The core of the work is an application of methods based on AGM (arithmetic-geometric mean) with a time complexity of order $O(M(n)\log_2{n})$ 9(expresed for multiplication $M(n)$). The complexity can not be improved. There are many libraries supporting multi-precision atithmetic, one of which is GMP and is about to be used for efficent method implementation. In the end all implemented methods are compared with existing ones.