Adjacency preservers on invertible hermitian matrices I

From the 19th century, the theory of permutation polynomial over finite fields, that are arose in the work of Hermite and Dickson, has drawn general attention. Permutation polynomials over finite fields are an active area of research due to their rising applications in mathematics and engineering. The last three decades has seen rapid progress on the research on permutation polynomials due to their diverse applications in cryptography, coding theory, finite geometry, combinatorics and many more areas of mathematics and engineering. For this reason, the study of permutation polynomials is important nowadays. In this chapter, we propose some new problems in connection to permutation polynomials over finite fields by the help of prime numbers.

This thesis focuses on the interplay of extremal combinatorics and finite geometry. Combinatorics is concerned with discrete (and usually finite) objects. Extremal combinatorics studies how large or how small a collection of finite objects can be under certain restrictions. Those objects can be sets, graphs, vectors, etc. These questions are often motivated by problems in information theory and computer science. Another branch of combinatorics is finite geometry over finite fields of order q. Although there is no field of order 1, certain substructures in finite geometry can be interpreted as versions of projective spaces for q = 1. A triangle in a projective plane is a good example of this phenomenon. Following an introductory chapter, Chapter 2 gives an overview of some classical problems and their q-analogues in extremal combinatorics. The most recent results for the q-analogues of Sperner's theorem, the Erdös-Ko-Rado theorem and several versions of Bollobás's theorem are described. For the q-analogue of the Hilton-Milner theorem we give some new results. We also give a new bound for the minimum size of the q-analogue of small maximal cliques. We conclude that sometimes results for a q-analogue can be obtained by using the same technique as in the original problem. In some cases the answer to the problem is even identical to that of the original problem. In other cases techniques used for the q-analogue could be used for improving bounds for the original problem. The third chapter describes the known results for the chromatic number of the Kneser graphs and gives new bounds for the chromatic number of the q-analogue. The vertices of a Kneser graph are subsets (of a fixed size) of a set, whereas two vertices are adjacent if they are disjoint in their subset representation. In 1955 M. Kneser conjectured the chromatic number of those graphs. In 1978 this was proven correct by L. Lóvasz. Two small cases in the projective space q-analogue were solved in 2001 by J. Eisfeld, L. Storme and P. Sziklai and in 2006 by A. Chowdhury, C. Godsil and G. Royle. We describe an asymptotic result for all cases (except for one parameter family, where we give a partial proof) using the bounds from the q-analogue of the Hilton-Milner theorem. Chapters 4, 5 and 6 describe other q-analogues of the Kneser graphs. In Chapter 4 we define a family of Kneser type graphs over pairs of incident points and hyperplanes of a projective space. We describe large maximal cocliques in these graphs and prove that for small dimensions these are the largest possible cocliques. We conjecture that this is the case for all dimensions. In Chapter 5 we extend the Kneser graphs to the case of finite polar spaces instead of finite projective spaces and in some cases give descriptions and chromatic numbers of these graphs. Chapter 6 describes the generalization of Kneser graphs over coset spaces of Chevalley groups (parameterized by q) with respect to parabolic subgroups. This encompasses all previous cases and extends to some interesting new cases. First we describe these graphs when q = 1 and give chromatic numbers for some families. Then we consider the graphs defined over coset spaces of Chevalley groups for general q with respect to parabolic subgroups and study what results of the q = 1 case can be translated to the general case. In this thesis we found different possibilities for the connection between the bounds for the q = 1 case and the case for general q. In some cases the bounds for the general case are identical to the bounds for q = 1. In other cases we obtain the bounds for q = 1 by taking the limit for q¿1 in the general bound. Other cases show a completely different bound in both cases.

Finite fields play important roles in many application areas such as coding theory, cryptography, Monte Carlo and quasi-Monte Carlo methods, pseudorandom number generation, quantum computing, and wireless communication. In this book we will focus on sequences, character sums, and polynomials over finite fields in view of the above mentioned application areas: Chapters 1 and 2 deal with sequences mainly constructed via characters and analyzed using bounds on character sums. Chapters 3, 5, and 6 deal with polynomials over finite fields. Chapters 4 and 9 consider problems related to coding theory studied via finite geometry and additive combinatorics, respectively. Chapter 7 deals with quasirandom points in view of applications to numerical integration using quasi-Monte Carlo methods and simulation. Chapter 8 studies aspects of iterations of rational functions from which pseudorandom numbers for Monte Carlo methods can be derived. The goal of this book is giving an overview of several recent research directions as well as stimulating research in sequences and polynomials under the unified framework of character theory.

Finding the singularity of a matrix is a basic problem in linear algebra. Chu and Schnitger [SC95] first considered this problem in the communication complexity model, in which Alice holds the first half of the matrix and Bob holds the other half. They proved that the deterministic communication complexity is Omega(n^2 log p) for an n by n matrix over the finite field F_p. Then, Clarkson and Woodruff [CW09] introduced the singularity problem to the streaming model. They proposed a randomized one pass streaming algorithm that uses O(k^2 log n) space to decide if the rank of a matrix is k, and proved an Omega(k^2) lower bound for randomized one-way protocols in the communication complexity model. We prove that the randomized/quantum communication complexity of the singularity problem over F_p is Omega(n^2 log p), which implies the same space lower bound for randomized streaming algorithms, even for a constant number of passes. The proof uses the framework by Lee and Shraibman [LS09], but we choose Fourier coefficients as the witness for the dual approximate norm of the communication matrix. Moreover, we use Fourier analysis to show the same randomized/quantum lower bound when deciding if the determinant of a non-singular matrix is a or b for non-zero a and b.

In 1940s, Hua established the fundamental theorems of geometry of rectangular matrices, symmetric matrices, skew-symmetric matrices, and hermitian matrices. In 1950s, Jacob generalized Hua's theorems to that of order-2 tensors and symmetric tensors. We extend Jacob's work to maps of order-3 symmetric tensors over C by proving that every surjective coherence invariant map on order-3 symmetric tensors over C is induced by a semilinear isomorphism apart from an additive constant.

Some partitions of a Hermitian matrix

13 - Matrices and related results

A large number of techniques are known for constructing orthogonal arrays. This chapter, the first of several describing these techniques, discusses some constructions due to Bush (1952b), Addelman and Kempthorne (1961a), Rao (1946a, 1947, 1949) and Bose and Bush (1952), that have the common theme of using Galois fields and finite geometries. We also describe a number of basic properties of orthogonal arrays, including the important concept of linearity.

The influence of Miroslav Fiedler on spectral graph theory

Characterizations and constructions of n-to-1 mappings over finite fields

In large-scale applications, parts of harmonic data are inevitably lost during transmission. This study presents an approach for the recovery of missing harmonic data based on the spectral graph theory. The proposed methodology involves graph theory for constructing a Laplacian matrix and a graph signal reconstruction function, the merging <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> -means algorithm for building a priori information model, and the accelerated segmentation Bregman iterative algorithm for solving the reconstruction function. Compared with existing methods on data recovery in power systems, the method maintains a good recovery accuracy when the data correlation of measurement units is low and the prior information is little. The proposed method has good anti-noise performances and low computational complexity. The feasibility and accuracy of the proposed method are verified through simulation and field recorded data.

This paper proposes lower bounds for the coupling strengths of oscillators in directed networks to guarantee global synchronization. The novel idea of graph comparison from spectral graph theory is employed so that the topological features of a given network can be fully utilized to simplify computations. For large networks that can be decomposed into a set of smaller strongly connected components, the comparison can be carried out at the local level as well.

The classical wavelet transform has been widely used in image and signal processing, where a signal is decomposed into a combination of basis signals. By analyzing the individual contribution of the basis signals, one can infer properties of the original signal. This dissertation presents an overview of the extension of the classical signal processing theory to graph domains. Specifically, we review the graph Fourier transform and graph wavelet transforms both of which based on the spectral graph theory, and explore their properties through illustrative examples. The main features of the spectral graph wavelet transforms are presented using synthetic and real-world data.

Using the technique of semantic mirroring a graph is obtained that represents words and their translations from a parallel corpus or a bilingual lexicon. The connectedness of the graph holds information about the semantic relations of words that occur in the translations. Spectral graph theory is used to partition the graph, which leads to a grouping of the words in different clusters. We illustrate the method using a small sample of seed words from a lexicon of Swedish and English adjectives and discuss its application to computational lexical semantics and lexicography.

We introduce the study of frames and equiangular lines in classical geometries over finite fields. After developing the basic theory, we give several examples and demonstrate finite field analogs of equiangular tight frames (ETFs) produced by modular difference sets, and by translation and modulation operators. Using the latter, we prove that Gerzon's bound is attained in each unitary geometry of dimension d=22l+1 over the field F32. We also investigate interactions between complex ETFs and those in finite unitary geometries, and we show that every complex ETF implies the existence of ETFs with the same size over infinitely many finite fields.