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

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<p>The Symmetric Meixner-Pollaczek polynomials are considered. We denote these polynomials in this thesis by <i>p</i><i>n</i>^(λ)(<i>x</i>) instead of the standard notation <i>p</i><i>n</i>^(λ) (<i>x</i>/2, <i>π</i>/2), where λ > 0. The limiting case of these sequences of polynomials <i>p</i><i>n</i>⁽⁰⁾ (<i>x</i>) =lim_λ→0 <i>p</i><i>n</i>^(λ)(<i>x</i>), is obtained, and is shown to be an orthogonal sequence in the strip, <i>S</i> = {<i>z</i> ∈ ℂ : −1≤ℭ (<i>z</i>)≤1}.</p><p>From the point of view of Umbral Calculus, this sequence has a special property that makes it unique in the Symmetric Meixner-Pollaczek class of polynomials: it is of convolution type. A convolution type sequence of polynomials has a unique associated operator called a delta operator. Such an operator is found for <i>p</i><i>n</i>⁽⁰⁾ (<i>x</i>), and its integral representation is developed. A convolution type sequence of polynomials may have associated Sheffer sequences of polynomials. The set of associated Sheffer sequences of the sequence <i>p</i><i>n</i>⁽⁰⁾(<i>x</i>) is obtained, and is found</p><p>to be ℙ = {{<i>p</i><i>n</i>^(λ) (<i>x</i>)} =0 : λ ∈ R}. The major properties of these sequences of polynomials are studied.</p><p>The polynomials {<i>p</i><i>n</i>^(λ) (<i>x</i>)}^∞<i>n</i>₌₀, λ < 0, are not orthogonal polynomials on the real line with respect to any positive real measure for failing to satisfy Favard’s three term recurrence relation condition. For every λ ≤ 0, an associated nonstandard inner product is defined with respect to which <i>p</i><i>n</i>^(λ)(x) is orthogonal. </p><p>Finally, the connection and linearization problems for the Symmetric Meixner-Pollaczek polynomials are solved. In solving the connection problem the convolution property of the polynomials is exploited, which in turn helps to solve the general linearization problem.</p>

The Symmetric Meixner-Pollaczek polynomials are considered. We denote these polynomials in this thesis by pn(λ)(x) instead of the standard notation pn(λ) (x/2, π/2), where λ &gt; 0. The limiting case of these sequences of polynomials pn(0) (x) =limλ→0 pn(λ)(x), is obtained, and is shown to be an orthogonal sequence in the strip, S = {z ∈ ℂ : −1≤ℭ (z)≤1}. From the point of view of Umbral Calculus, this sequence has a special property that makes it unique in the Symmetric Meixner-Pollaczek class of polynomials: it is of convolution type. A convolution type sequence of polynomials has a unique associated operator called a delta operator. Such an operator is found for pn(0) (x), and its integral representation is developed. A convolution type sequence of polynomials may have associated Sheffer sequences of polynomials. The set of associated Sheffer sequences of the sequence pn(0)(x) is obtained, and is found to be ℙ = {{pn(λ) (x)} =0 : λ ∈ R}. The major properties of these sequences of polynomials are studied. The polynomials {pn(λ) (x)}∞n=0, λ &lt; 0, are not orthogonal polynomials on the real line with respect to any positive real measure for failing to satisfy Favard’s three term recurrence relation condition. For every λ ≤ 0, an associated nonstandard inner product is defined with respect to which pn(λ)(x) is orthogonal. Finally, the connection and linearization problems for the Symmetric Meixner-Pollaczek polynomials are solved. In solving the connection problem the convolution property of the polynomials is exploited, which in turn helps to solve the general linearization problem.

Neste trabalho temos por objetivo apresentar alguns resultados relacionados aos temas abordados por Aron, Choi e Llavona (1995), Aron e Dimant (2002) e Aron e Rueda (1997). Primeiramente, vamos estudar as propriedades polinomiais (P) e (RP) para os espaços de Banach e a propriedade ACL para as funções definidas entre as bolas unitárias fechadas do espaço. Vamos apresentar novos exemplos de espaços de Banach que possuem a propriedade (P) onde é possível exibir funções que satisfazem a propriedade ACL. Vamos ainda estudar o conjunto de continuidade seqüencial fraca de um polinômio N-homogêneo contínuo com valores vetoriais. Apresentamos as suas propriedades básicas e algumas conexões com o caso dos polinômios escalares. No espaço dual faremos uma breve análise dos polinômios com certo tipo de continuidade com relação à topologia fraca-estrela. Numa outra direção, estudamos os zeros de polinômios N-homogêneos em várias variáveis complexas, mais especificamente, dados n, N números naturais existe um número natural m tal que para cada polinômio N-homogêneo complexo P definido no espaço vetorial C^ existe um subespaço vetorial X_ contido no conjunto dos zeros do polinômio P de dimensão n. Aqui, o principal objetivo é melhorar as limitações para m encontradas por Aron e Rueda (1997) como também generalizar os seus resultados. / Our purpose here is to study some results regarding the articles of Aron, Choi and Llavona (1995), Aron and Dimant (2002) and Aron and Rueda (1997). Firstly, we study properties (P) and (RP) for the Banach spaces and the ACL property for the functions defined between the closed unit balls. We give new examples of Banach spaces which have (P) property and some functions defined in those spaces satisfying the ACL property. We also study the set of weak sequential continuity of a vector-valued continuous Nhomogeneous polynomial. In the dual space we study the N-homogeneous polynomials which are weak-star continuous on bounded sets. Finally, we study the zeros of complex N-homogeneous polynomials. This means, given positive integers n and N, there is a positive integer m such that an complex N-homogeneous polynomial P defined in C^ has an ndimensional subspace contained in its zero set. We discuss the problem of finding a good bound on m as a function of n and N. We improve the results given by Aron and Rueda (1997) as also generalize their results.

The interest in wireless communications among consumers has exploded since the introduction of the "3G" cell phone standards. One reason for their success is the increasingly higher data rates achievable through the networks. A further increase in data rates is possible through the use of multiple antennas at either or both sides of the wireless links. Precoding and receive filtering using matrices obtained from a singular value decomposition (SVD) of the channel matrix is a transmission strategy for achieving the channel capacity of a deterministic narrowband multiple-input multiple-output (MIMO) communications channel. When signalling over wideband channels using orthogonal frequency-division multiplexing (OFDM), an SVD must be performed for every sub-carrier. As the number of sub-carriers of this traditional approach grow large, so does the computational load. It is therefore interesting to study alternate means for obtaining the decomposition. A wideband MIMO channel can be modeled as a matrix filter with a finite impulse response, represented by a polynomial matrix. This thesis is concerned with investigating algorithms which decompose the polynomial channel matrix directly. The resulting decomposition factors can then be used to obtain the sub-carrier based precoding and receive filtering matrices. Existing approximative polynomial matrix QR and singular value decomposition algorithms were modified, and studied in terms of decomposition quality and computational complexity. The decomposition algorithms were shown to give decompositions of good quality, but if the goal is to obtain precoding and receive filtering matrices, the computational load is prohibitive for channels with long impulse responses. Two algorithms for performing exact rational decompositions (QRD/SVD) of polynomial matrices were proposed and analyzed. Although they for simple cases resulted in excellent decompositions, issues with numerical stability of a spectral factorization step renders the algorithms in their current form purposeless. For a MIMO channel with exponentially decaying power-delay profile, the sum rates achieved by employing the filters given from the approximative polynomial SVD algorithm were compared to the channel capacity. It was shown that if the symbol streams were decoded independently, as done in the traditional approach, the sum rates were sensitive to errors in the decomposition. A receiver with a spatially joint detector achieved sum rates close to the channel capacity, but with such a receiver the low complexity detector set-up of the traditional approach is lost. Summarizing, this thesis has shown that a wideband MIMO channel can be diagonalized in space and frequency using OFDM in conjunction with an approximative polynomial SVD algorithm. In order to reach sum rates close to the capacity of a simple channel, the computational load becomes restraining compared to the traditional approach, for channels with long impulse responses.

In this paper we investigate (d + 1)-point D-optimal designs for d-th degree polynomial regression with weight function w(x) > 0 on the interval [a, b]. Suppose that w'(x)/w(x) is a rational function and the information of whether the optimal support contains the boundary points a and b is available. Then the problem of constructing (d + 1)-point D-optimal designs can be transformed into a differential equation problem leading us to a certain matrix with k auxiliary unknown constants. We characterize the weight functions corresponding to the cases when k= 0 and k= 1. Then, we can solve (d + 1)-point D-optimal designs directly from differential equation (k = 0) or via eigenvalue problems (k = 1). The numerical results show us an interesting relationship between optimal designs and ordered eigenvalues.

This thesis examines graph polynomials and particularly their complexity. We give short proofs of two results from Gessel and Sagan (1996) which present new evaluations of the Tutte polynomial concerning orientations. A theorem of Massey et al (1997) gives an expression concerning the average size of a forest in a graph. We generalise this result to any simplicial complex. We answer a question posed by Kleinschmidt and Onn (1995) by showing that the language of partitionable simplicial complexes is in NP. We prove the following result concerning the complexity of the Tutte polynomial: Theorem 1. For any fixed k, there exists a polynomial time algorithm A, which will input any graph G, with tree-width at most k, and rational numbers x and y, and evaluate the Tutte polynomial, T(G;x,y). The rank generating function S of a graphic 2-polymatroid was introduced by Oxley and Whittle (1993). It has many similarities to the Tutte polynomial and we prove the following results. Theorem 2. Evaluating S at a fixed point (u,v) is #P-hard unless uv=1 when there is a polynomial time algorithm. Theorem 3. For any fixed k, there exists a polynomial time algorithm A, which will input any graph G, with tree-width at most k, and rational numbers u and v, and evaluate S(G;u,v). We consider a class of graphs $S$, which are those graphs which are obtainable from a graph with no edges using the unsigned version of Reidemeister moves. We examine the relationship between this class and other similarly defined classes such as the delta-wye graphs. There remain many open questions such as whether S contains every graph. However we have an invariant of the moves, based on the Tutte polynomial, which allows us to determine from which graph with no edges, if any, a particular graph can be obtained. Finally we consider a new polynomial on weighted graphs which is motivated by the study of weight systems on chord diagrams. We give three states model and a recipe theorem. An unweighted version of this polynomial is shown to contain as specialisations, a wide range of graph invariants, such as the Tutte polynomial, the polymatroid polynomial of Oxley and Whittle (1993) and the symmetric function generalisation of the chromatic polynomial introduced by Stanley (1995). We close with a discussion of complexity issues proving hardness results for very restricted classes of graphs.

Neste trabalho temos por objetivo apresentar alguns resultados relacionados aos temas abordados por Aron, Choi e Llavona (1995), Aron e Dimant (2002) e Aron e Rueda (1997). Primeiramente, vamos estudar as propriedades polinomiais (P) e (RP) para os espaços de Banach e a propriedade ACL para as funções definidas entre as bolas unitárias fechadas do espaço. Vamos apresentar novos exemplos de espaços de Banach que possuem a propriedade (P) onde é possível exibir funções que satisfazem a propriedade ACL. Vamos ainda estudar o conjunto de continuidade seqüencial fraca de um polinômio N-homogêneo contínuo com valores vetoriais. Apresentamos as suas propriedades básicas e algumas conexões com o caso dos polinômios escalares. No espaço dual faremos uma breve análise dos polinômios com certo tipo de continuidade com relação à topologia fraca-estrela. Numa outra direção, estudamos os zeros de polinômios N-homogêneos em várias variáveis complexas, mais especificamente, dados n, N números naturais existe um número natural m tal que para cada polinômio N-homogêneo complexo P definido no espaço vetorial C^ existe um subespaço vetorial X_ contido no conjunto dos zeros do polinômio P de dimensão n. Aqui, o principal objetivo é melhorar as limitações para m encontradas por Aron e Rueda (1997) como também generalizar os seus resultados. / Our purpose here is to study some results regarding the articles of Aron, Choi and Llavona (1995), Aron and Dimant (2002) and Aron and Rueda (1997). Firstly, we study properties (P) and (RP) for the Banach spaces and the ACL property for the functions defined between the closed unit balls. We give new examples of Banach spaces which have (P) property and some functions defined in those spaces satisfying the ACL property. We also study the set of weak sequential continuity of a vector-valued continuous Nhomogeneous polynomial. In the dual space we study the N-homogeneous polynomials which are weak-star continuous on bounded sets. Finally, we study the zeros of complex N-homogeneous polynomials. This means, given positive integers n and N, there is a positive integer m such that an complex N-homogeneous polynomial P defined in C^ has an ndimensional subspace contained in its zero set. We discuss the problem of finding a good bound on m as a function of n and N. We improve the results given by Aron and Rueda (1997) as also generalize their results.

Let K be a compact subset of Cn . The polynomially convex hull of K is defined as The compact set K is said to be polynomially convex if = K. A closed subset is said to be locally polynomially convex at if there exists a closed ball centred at z such that is polynomially convex. The aim of this thesis is to derive easily checkable conditions to detect polynomial convexity in certain classes of sets in This thesis begins with the basic question: Let S1 and S2 be two smooth, totally real surfaces in C2 that contain the origin. If the union of their tangent planes is locally polynomially convex at the origin, then is locally polynomially convex at the origin? If then it is a folk result that the answer is, “Yes.” We discuss an obstruction to the presumed proof, and use a different approach to provide a proof. When dimR it turns out that the positioning of the complexification of controls the outcome in many situations. In general, however, local polynomial convexity of also depends on the degeneracy of the contact of T0Sj with We establish a result showing this. Next, we consider a generalization of Weinstock’s theorem for more than two totally real planes in C2 . Using a characterization, recently found by Florentino, for simultaneous triangularizability over R of real matrices, we present a sufficient condition for local polynomial convexity at of union of finitely many totally real planes is C2 . The next result is motivated by an approximation theorem of Axler and Shields, which says that the uniform algebra on the closed unit disc generated by z and h — where h is a nowhereholomorphic harmonic function on D that is continuous up to ∂D — equals . The abstract tools used by Axler and Shields make harmonicity of h an essential condition for their result. We use the concepts of plurisubharmonicity and polynomial convexity to show that, in fact, the same conclusion is reached if h is replaced by h+ R, where R is a nonharmonic perturbation whose Laplacian is “small” in a certain sense. Ideas developed for the latter result, especially the role of plurisubharmonicity, lead us to our final result: a characterization for compact patches of smooth, totallyreal graphs in to be polynomially convex.