Quadratic Forms and Graphs

Chapter 6 - Quadratic forms and Eigenvalue problems

There are close relations between Bernoulli numbers and quadratic fields or quadratic forms. In order to explain those, we give a survey of the ideal theory of quadratic fields and quadratic forms. Since Gauss , it is well known that there is a deep relation between the ideal theory of quadratic fields (i.e. quadratic extensions of the rational number field) and integral quadratic forms. This is obvious for specialists, but textbooks which explain this in detail are rare. In most textbooks, they treat the correspondence of ideals only in the case of maximal orders (the ring of all integers) for which the description is simple and easy. This is very disappointing. We cannot see the whole picture of the theory of quadratic forms by such a restricted treatment and sometimes it causes misunderstanding. We need the full description of this kind of relation when we explain later the relation between L-functions of prehomogeneous vector spaces and the Bernoulli numbers. So it would be a good chance to try to explain the full relation here.

Introduction Classical theory of symmetric bilinear forms and quadratic forms: Bilinear forms Quadratic forms Forms over rational function fields Function fields of quadrics Bilinear and quadratic forms and algebraic extensions $u$-invariants Applications of the Milnor conjecture On the norm residue homomorphism of degree two Algebraic cycles: Homology and cohomology Chow groups Steenrod operations Category of Chow motives Quadratic forms and algebraic cycles: Cycles on powers of quadrics The Izhboldin dimension Application of Steenrod operations The variety of maximal totally isotropic subspaces Motives of quadrics Appendices Bibliography Notation Terminology.

This dissertation presents computational tools for quadratic forms over global function fields of characteristic different from 2. The majority of the algorithms we develop in this study rely on our ability to find all places dividing any coefficient of the given quadratic form. This problem is equivalent to the factorization of fractional ideals in the ring of polynomial functions of a global function field. As a result, we begin by presenting alternative approaches for factoring fractional ideals that do not rely on determining the maximum order of the global function field in question. We then propose techniques for tackling the following quadratic form theory computational problems: how to detect whether a quadratic form is isotropic or not, how to detect whether a quadratic form is hyperbolic or not, how to compute the anisotropic dimension (or equivalently the Witt index) of a quadratic form, how to construct an anisotropic part of a quadratic form, how to determine if two forms are Witt-similar or are Ono-similar or not. We further explore algorithms for computing some important field invariants that are linked to quadratic forms. Those are: the length of a sum of squares, the level of a field, the Pythagoras number of a field, as well as a Pythagoras element of a field.

13 - Matrices and related results

Historically the theory of quadratic forms has its origins in number-theoretic questions of the following type: Which integers can be written in the form x 2 + 2y 2 , which are sums of three squares, or more generally, which integers can be represented by an arbitrary quadratic form Σ a ij x i x j integral coefficients? This general question is exceptionally difficult and we are still quite far from a complete solution. It is natural and considerably simpler to first investigate these questions over the field of rational numbers, that is, to ask for rational instead of integral solutions to the equation Σ a ij x i x j = a. This leads to the problem of classification of quadratic forms over \( \mathbb{Q} \), which was first solved by Minkowski. His solution appears in this chapter basically unaltered, except for a few simplifications and the use of modern terminology. The Gaussian sums of Gauss and Dirich-let play a significant role in the more formal algebraic part of the theory.

We consider a class of singular, zero-range perturbations of the quantum Hamiltonian of a system consisting of a test particle and N harmonic oscillators (Rayleigh gas). Using the theory of quadratic forms we construct the self-adjoint and bounded from below perturbed Hamiltonian and we give representation formulas for the resolvent and the unitary group. The one-dimensional, two-dimensional and three-dimensional cases are discussed.

Let the column vectors of \(\mathop {\text{X}}\limits_ \sim :\) pxn be distributed as independent normals with common covariance matrix \(\mathop \sum \limits_ \sim \). Then, the quadratic form in normal vectors is denoted by \(\mathop {\text{X}}\limits_ \sim \mathop {\text{A}}\limits_ \sim \mathop {\text{X}}\limits_ \sim '\, = \,\mathop {\text{S}}\limits_ \sim \) where \(\mathop {\text{A}}\limits_ \sim :\,{\text{nxn}}\) is a symmetric matrix which is assumed to be positive definite. This paper deals with a series representation of the density function of \(\mathop {\text{S}}\limits_ \sim \) when \({\text{E(}}\mathop {\text{X}}\limits_ \sim {\text{)}}\, \ne \,\mathop 0\limits_ \sim \), extending the idea of the author (1971) and Kotz et al. (1967b) to the multivariate non- central case. It is pointed out that this method gives a result which is not easy to obtain directly by integrating over an orthogonal space in the sense of James (1964).

Series representations of distributions of quadratic form in the normal vectors and generalised variance

We show that the theorem of Ellenberg and Venkatesh on representation of integral quadratic forms by integral positive definite quadratic forms is valid under weaker conditions on the represented form.

It seems that any deeper investigation of the algebraic theory of quadratic forms requires a thorough knowledge of the theory of Pfister forms. They are the subject of this chapter. The investigation of Pfister forms is based essentially on the use of transcendental extensions of the ground field: One considers quadratic forms Σ a ij X i X j . with indeterminates X i . This approach leads naturally to the important concept of the function field of a quadratic form which is the function field of the quadratic Σ a ij X i X j =0.

The noncommuting graph $nabla (G)$ of a group $G$ is a simple graph whose vertex set is the set of noncentral elements of $G$ and the edges of which are the ones connecting two noncommuting elements. We determine here, up to isomorphism, the structure of any finite nonabeilan group $G$ whose noncommuting graph is a split graph, that is, a graph whose vertex set can be partitioned into two sets such that the induced subgraph on one of them is a complete graph and the induced subgraph on the other is an independent set.

A finite and simple graph is said to be equitably colorable if its vertices can be partitioned into classes such that each is an independent set and holds for every . The smallest integer for which is equitable chromatic number of and denoted by . The equitable chromatic threshold of a graph , denoted by , is the minimum such that is equitably colorable for all . This paper focuses on the equitable colorability of rooted product of graphs, in particular, exact values or upper bounds of and when and are cycles, paths, complete graphs and complete partite graphs have been found.

Let [Formula: see text] be a simple graph with vertex set [Formula: see text] and edge set [Formula: see text]. A Roman dominating function on a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that every vertex [Formula: see text] for which [Formula: see text] is adjacent to at least one vertex [Formula: see text] for which [Formula: see text]. A Roman dominating function [Formula: see text] is called an independent Roman dominating function if the set of all vertices with positive weights is an independent set. The weight of an independent Roman dominating function [Formula: see text] is the value [Formula: see text]. The independent Roman domination number of [Formula: see text], denoted by [Formula: see text], is the minimum weight of an independent Roman dominating function on [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is a 2-independent set of [Formula: see text] if every vertex of [Formula: see text] has at most one neighbor in [Formula: see text]. The maximum cardinality of a 2-independent set of [Formula: see text] is the 2-independence number [Formula: see text]. These two parameters are incomparable in general, however, we show that for any tree [Formula: see text], [Formula: see text] and we characterize all trees attaining the equality.

Critical and maximum independent sets of a graph