Search

Let f (t 1 , ... , t n) be a nondegenerate integral quadratic form. We analyze the asymptotic behavior of the function D f (X) , the number of integers of absolute value up to X represented by f. When f is isotropic or n is at least 3, we show that there is a δ (f) ∈ Q ∩ (0 , 1) such that D f (X) ∼ δ (f) X and call δ (f) the density of f. We consider the inverse problem of which densities arise. Our main technical tool is a Near Hasse Principle: a quadratic form may fail to represent infinitely many integers that it locally represents, but this set of exceptions has density 0 within the set of locally represented integers. [ABSTRACT FROM AUTHOR]

Let a 15 (n) , a 20 (n) and a 24 (n) be defined by q ∏ k = 1 ∞ (1 - q k) (1 - q 3 k) (1 - q 5 k) (1 - q 15 k) = ∑ n = 1 ∞ a 15 (n) q n , q ∏ k = 1 ∞ (1 - q 2 k) 2 (1 - q 10 k) 2 = ∑ n = 1 ∞ a 20 (n) q n , q ∏ k = 1 ∞ (1 - q 2 k) (1 - q 4 k) (1 - q 6 k) (1 - q 12 k) = ∑ n = 1 ∞ a 24 (n) q n (| q | < 1) , and let p > 3 be a prime. In this paper, for p ≡ 3 (mod 4) we reveal the connection between a 20 (p) and residue-counts of x 4 - 4 x 2 + 4 x modulo p as x runs over 0 , 1 , … , p - 1 , and the connection between a 24 (p) and residue-counts of x 3 + c / x modulo p as x runs over 1 , 2 , … , p - 1 , where c is an integer not divisible by p. We also deduce the congruences for a 15 (p) , a 24 (p) modulo 16 and a 20 (p) modulo 4, and pose some analogous conjectures. [ABSTRACT FROM AUTHOR]

During the conceptual and preliminary ship design stages, designers must ensure that the selected principal dimensions and parameters are good enough to deliver a stable ship (statically and dynamically), besides deadweight and speed. To support this, the initial intact stability of the proposed ship must be calculated, and for that, the lightship weight and its detailed breakdown are necessary to be known. After hull (steel) weight, machinery weight, mainly marine diesel propulsion and power generation machinery, plays a vital role in the lightship weight estimate of a ship directly. It also affects the deadweight, displacement, draft, and trim, which finally influence stability. The correct estimation of respective weights improves calculating a ship's initial stability, typically designed and built. Hence, it would be advantageous for the ship designer to convince the shipowner. Weight control is not easy, especially since its consequences in terms of deadweight and stability could be disastrous. It should start right at the beginning and control throughout the design and construction phases. Marine diesel main engines (MEs) and auxiliary engines (AEs) of various power output (generators output for AEs), engine speed, and cylinder number of different engine manufacturers are collected. These are analyzed and presented in tabular and graphical forms to demonstrate the possible relationship between propulsion engine weight and power generation engine weight, and their respective power, speed, and power-speed ratio. In this article, the authors have attempted to investigate the appraisal behavior of ME weight and AE weight regarding engine power, generator power, engine speed, and power-speed ratio (independent variables). Further attempts have been made to identify those independent variables that influence the weight of the propulsion engine and power generation engine (dependent variables) and their interrelationships. A mathematical model has thus been developed and proposed, as a guiding tool, for the designer to estimate the weight of main and AEs more accurately during the conceptual and preliminary design stages. [ABSTRACT FROM AUTHOR]

A multiring is a ring-like structure where the sum is multivalued and a hyperring is a multiring with a strong distributive property. With every multiring we associate a structural presheaf, and when that presheaf is a sheaf, we say that the multiring is geometric. A characterization of geometric von Neumann hyperrings is presented. And we build a von Neumann regular hull for multirings which is used in applications to algebraic theory of quadratic forms. Namely, we describe the functor Q, introduced by M. Marshall in [J. Pure Appl. Alg., 205, No. 2, 452-468 (2006)], as a left adjoint functor for the natural inclusion of the category of real reduced multirings (similar to real semigroups) into the category of preordered multirings and explore some of its properties. Next, we employ sheaf-theoretic methods to characterize real reduced hyperrings as certain geometric von Neumann regular real hyperrings and construct the functor V , a geometric von Neumann regular hull for a multiring. Finally, we look at some interesting logical and algebraic interactions between the functors Q and V that are useful for describing hyperrings in the image of the functor Q and that will allow us to explore the theory of quadratic forms for (formally) real semigroups. [ABSTRACT FROM AUTHOR]

Machines and mechanisms realize processes, from the shaping process of a milling machine and the motion process of an automotive system to the trajectory realization of a robot. The dynamics of a machine generated by a properly chosen set of constraints in combination with an appropriate drive system is designed to meet the prescribed requirements of the process, which is done by projecting the machine equations of motion on the process dynamics. We get a set of nonlinear relations, which represent the machine motion in terms of the required process motion. A well-known example is the projection of arbitrary many robot degrees of freedom (DOFs) on a given path resulting in a set of nonlinear equations with one DOF only, the path coordinate s. Application of this idea can be used to construct a mobility space $(\ddot{s}, \dot{s}, s)$ for any combination of coordinates and constraints. The paper presents a corresponding approach for n-link robots by applying multibody system theory. Method might be interesting for layout of machines and mechanisms. Practical aspects are discussed, and an example is given. [ABSTRACT FROM AUTHOR]

Theory predicts that the additive genetic covariance (G ) matrix determines a population’s short-term (in)ability to respond to directional selection—evolvability in the Hansen–Houle sense—which is typically quantified and compared via certain scalar indices called evolvability measures. Often, interest is in obtaining the averages of these measures across all possible selection gradients, but explicit formulae for most of these average measures have not been known. Previous authors relied either on approximations by the delta method, whose accuracy is generally unknown, or Monte Carlo evaluations (including the random skewers analysis), which necessarily involve random fluctuations. This study presents new, exact expressions for the average conditional evolvability, average autonomy, average respondability, average flexibility, average response difference, and average response correlation, utilizing their mathematical structures as ratios of quadratic forms. The new expressions are infinite series involving top-order zonal and invariant polynomials of matrix arguments, and can be numerically evaluated as their partial sums with, for some measures, known error bounds. Whenever these partial sums numerically converge within reasonable computational time and memory, they will replace the previous approximate methods. In addition, new expressions are derived for the average measures under a general normal distribution for the selection gradient, extending the applicability of these measures into a substantially broader class of selection regimes. [ABSTRACT FROM AUTHOR]

For certain types of quadratic forms lying in the n-th power of the fundamental ideal, we compute upper bounds and if possible exact values for the minimal number of general n-fold Pfister forms, that are needed to write the Witt class of that given form as the sum of the Witt classes of those n-fold Pfister forms. We restrict ourselves mostly to the case of so called rigid fields, i.e. fields in which anisotropic binary forms represent at most 2 square classes. [ABSTRACT FROM AUTHOR]

Let φ and ψ be quadratic forms over a field K of characteristic different from 2. In this paper, we give a criterion for isotropy of φ over the function field of ψ in terms of representations and we apply it to stably birational equivalence of φ and ψ. Then we use this criterion to investigate the case of stably birational equivalence of multiples of Pfister forms. Finally, we give a criterion of stably birational equivalence of quadratic forms in terms of isomorphisms of quotients of special Clifford groups by the kernel of the spinor norm. [ABSTRACT FROM AUTHOR]

Flat tori are among the only types of Riemannian manifolds for which the Laplace eigenvalues can be explicitly computed. In 1964, Milnor used a construction of Witt to find an example of isospectral nonisometric Riemannian manifolds, a striking and concise result that occupied one page in the Proceedings of the National Academy of Science of the USA. Milnor's example is a pair of 16-dimensional flat tori, whose set of Laplace eigenvalues are identical, in spite of the fact that these tori are not isometric. A natural question is, What is the lowest dimension in which such isospectral nonisometric pairs exist? This isospectral question for flat tori can be equivalently formulated in analytic, geometric, and number theoretic language. We explore this question from all three perspectives and describe its resolution by Schiemann in the 1990s. Moreover, we share a number of open problems. [ABSTRACT FROM AUTHOR]

Let ℓ ⩾ 2 be a fixed positive integer and Q(y) be a positive definite quadratic form in ℓ variables with integral coefficients. The aim of this paper is to count rational points of bounded height on the cubic hypersurface defined by u3 = Q(y)z. We can get a power-saving result for a class of special quadratic forms and improve on some previous work. [ABSTRACT FROM AUTHOR]

Let d = k² + 4 for some integer k ≥ 2. In this work, we first determined the set of automorphisms of the Pell form FΔ(x,y)=x² - dy² of discriminant Δ = 4d. Later, we deduced the set of all integer solutions of the Pell equations FΔ(x, y) = ±1 and FΔ(x,y) = ±k². [ABSTRACT FROM AUTHOR]

In this paper, we extend work from [M. Vancliff and P. P. Veerapen, Generalizing the notion of rank to noncommutative quadratic forms, in Noncommutative Birational Geometry, Representations and Combinatorics, eds. A. Berenstein and V. Retakh, Contemporary Mathematics, Vol. 592 (2013), pp. 241–250], where a notion of rank, called μ -rank, was proposed for noncommutative quadratic forms on two and three generators. In particular, we provide a definition of μ -rank one and two for noncommutative quadratic forms on four generators. We apply this definition to determine the number of point modules over certain quadratic AS-regular algebras of global dimension four. [ABSTRACT FROM AUTHOR]

We study formally real, non-pythagorean fields which have an anisotropic torsion form that contains every anisotropic torsion form as a subform. We obtain consequences for certain invariants and the Witt ring of such fields and construct examples. We obtain a theory analogous to the theory of supreme Pfister forms introduced by Karim Becher and see examples in which the Pythagoras number for formally real fields behaves like the level for nonreal fields. [ABSTRACT FROM AUTHOR]

For a field E of characteristic different from 2 and cohomological 2-dimension one, quadratic forms over the rational function field E (X) are studied. A characterisation in terms of polynomials in E [ X ] is obtained for having that quadratic forms over E (X) satisfy a local-global principle with respect to discrete valuations that are trivial on E. In this way new elementary proofs for the local-global principle are achieved in the cases where E is finite or pseudo-algebraically closed. The study is complemented by various examples. [ABSTRACT FROM AUTHOR]

This article explores additive codes with one-rank hull, offering key insights and constructions. The article introduces a novel approach to finding one-rank hull codes over finite fields by establishing a connection between self-orthogonal elements and solutions of quadratic forms. It also provides a precise count of self-orthogonal elements for any duality over the finite field F q , particularly odd primes. Additionally, construction methods for small rank hull codes are introduced. The highest possible minimum distance among additive one-rank hull codes is denoted by d 1 [ n , k ] p e , M. The value of d 1 [ n , k ] p e , M for k = 1 , 2 and n ≥ 2 with respect to any duality M over any finite field F p e is determined. Furthermore, the new quaternary one-rank hull codes are identified over non-symmetric dualities with better parameters than symmetric ones. [ABSTRACT FROM AUTHOR]

Given a positive integer n , a sufficient condition on a field is given for bounding its Pythagoras number by 2 n + 1. The condition is satisfied for n = 1 by function fields of curves over iterated formal power series fields over R , as well as by finite field extensions of R ((t 0 , t 1)). In both cases, one retrieves the upper bound 3 on the Pythagoras number. The new method presented here might help to establish more generally 2 n + 1 as an upper bound for the Pythagoras number of function fields of curves over R ((t 1 , ... , t n)) and for finite field extensions of R ((t 0 , ... , t n)). [ABSTRACT FROM AUTHOR]

Statistical surveillance for covariance matrices has attracted increasing attention recently. Many approaches have been developed for monitoring general shifts that are arbitrary deviations, as well as sparse shifts occurring in only a few elements. This paper considers directional shifts that occur in only one independent parameter, which is common if the process is relatively stable. A directional covariance matrix control chart is proposed, which fully exploits directional shift information and borrows the strong power of likelihood ratio test. Therefore, this chart provides a powerful tool for monitoring covariance matrices. In addition, the proposed chart does not require specifying the regularisation parameter, and it enjoys a concise quadratic form, thereby easy to implement. Furthermore, this chart naturally leads to a diagnostic prescription for identifying the shifting element in the covariance matrix. Simulation results have demonstrated the efficiency of the suggested control chart and its accompanying diagnostic scheme. [ABSTRACT FROM AUTHOR]

A positive quadratic form is (k , ℓ) -universal if it represents all the numbers k x + ℓ where x is a non-negative integer, and almost (k , ℓ) -universal if it represents all but finitely many of them. We prove that for any k , ℓ such that k ∤ ℓ there exists an almost (k , ℓ) -universal diagonal ternary form. We also conjecture that there are only finitely many primes p for which a (p , ℓ) -universal diagonal ternary form exists (for any ℓ < p) and we show the results of computer experiments that speak in favor of the conjecture. [ABSTRACT FROM AUTHOR]

We give examples of anisotropic n -fold quadratic Pfister forms (n ≥ 2) over a suitably chosen field F of characteristic 2 that split over a purely inseparable extension E of F of exponent ≥2 but that do not split over any subextension of F inside E of lower exponent. For n ≥ 3 , we also give examples of such Pfister forms that split over a purely inseparable extension E of F of degree ≥4 but that do not split over any simple subextension of F inside E. As a consequence, we get a negative answer to a question posed by Bernhard Mühlherr and Richard Weiss who asked whether an octonion division algebra over a field of characteristic 2 that splits over a purely inseparable extension will already split over some quadratic subextension. We use the fact that the splitting of such an algebra is controlled by the splitting of the associated norm form which in turn is given by a 3-fold Pfister form. [ABSTRACT FROM AUTHOR]

In this paper, we present an analysis of some cases where a positive integer cannot be represented by a diagonal quadratic form with four integer variables. [ABSTRACT FROM AUTHOR]

Let f(x_1,\ldots,x_s) be a translation invariant indefinite quadratic form of integer coefficients with s{\,\geqslant \,} 10. Let \mathcal {A}\subseteq \mathcal {P}\cap \{1,2,\ldots,X\}. Let X be sufficiently large. Subject to a rank condition, we prove that there exist distinct primes p_1,\ldots,p_s\in \mathcal {A} such that f(p_1,\ldots,p_s)=0 as soon as |\mathcal {A}|{\,\geqslant \,} \frac {X}{\log X} (\log \log X)^{-\frac {1}{80}}. [ABSTRACT FROM AUTHOR]

The weight hierarchy of a linear code have been an important research topic in coding theory since Wei’s original work in 1991. In this paper, choosing D={(x,y)∈(Fps1×Fps2)\{(0,0)}:f(x)+Tr1s2(αy)=0}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$D=\Big \{(x,y)\in \Big ({\mathbb {F}}_{p^{s_1}}\times {\mathbb {F}}_{p^{s_2}}\Big )\Big \backslash \{(0,0)\}: f(x)+\text {Tr}_1^{s_2}(\alpha y)=0\Big \}$$\end{document} as a defining set, where α∈Fps2∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha \in {\mathbb {F}}_{p^{s_2}}^*$$\end{document}, f(x) is a quadratic form over Fps1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {F}}_{p^{s_1}}$$\end{document} with values in Fp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {F}}_p$$\end{document} and f(x) can be non-degenerate or not, we construct a family of three-weight p-ary linear codes and determine their weight distributions and weight hierarchies completely. Most of the codes can be used in secret sharing schemes. [ABSTRACT FROM AUTHOR]

In this article we study weighted sums of $n$ i.i.d. Gamma($\alpha $) random variables with nonnegative weights. We show that for $n \geq 1/\alpha $ the sum with equal coefficients maximizes differential entropy when variance is fixed. As a consequence, we prove that among nonnegative quadratic forms in $n$ independent standard Gaussian random variables, a diagonal form with equal coefficients maximizes differential entropy, under a fixed variance. This provides a sharp lower bound for the relative entropy between a nonnegative quadratic form and a Gaussian random variable. Bounds on capacities of transmission channels subjects to $n$ independent additive gamma noises are also derived. [ABSTRACT FROM AUTHOR]

Let φ and ψ be regular quadratic forms over a field F of characteristic different from 2. We say that ψ is a quasisubform of φ if there is a ∈ F * such that a ψ is a subform of φ. Let L/F be an odd degree field extension. Assume that ψL is a quasisubform of φ L . A natural question is whether ψ is a quasisubform of φ. We give a positive answer to this question in any of the following cases: The 2-cohomological dimension of F equals 2. dim φ − dim ψ ≤ 1 , or dim φ − dim ψ = 2 and dim φ is even. dim ψ = 3 , and dim φ ≤ 6. dim ψ = 3 , and ⟪ a , b ⟫ I 2 (F) = 0 , where ⟪ a , b ⟫ is the Pfister form associated with ψ (the most difficult case). [ABSTRACT FROM AUTHOR]

We partially characterize criteria for the n-universality of positive-definite integer-matrix quadratic forms. We then obtain the uniqueness of Oh's 8-universality criterion [11] as a corollary. [ABSTRACT FROM AUTHOR]

We examine the variance of a linear statistic defined on the symmetric group endowed with the Ewens probability. Despite the dependence of the summands, it can be bounded from above by a constant multiple of the sum of variances of the summands. We find the exact value of this constant. The analysis of the appearing quadratic forms and eigenvalue search is built upon the exponential matrices and discrete Hahn's polynomials. [ABSTRACT FROM AUTHOR]

Let d k (n) denote the k -th divisor function. In this paper, we study the asymptotic formula of the sum ∑ 1 ≤ n 1 , n 2 , ... , n l ≤ x d k (n 1 2 + n 2 2 + ⋯ + n l 2) , where n 1 , n 2 , ... , n l ∈ Z + , k ≥ 4 and l ≥ 3 are integers. Previously only the cases of k = 2 , 3 are studied. [ABSTRACT FROM AUTHOR]

The aim of this paper is first to introduce and study quadratic Hom-Jacobi-Jordan algebras, which are Hom-Jacobi-Jordan algebras with symmetric invariant nondegenerate bilinear forms. We provide several constructions leading to examples. We reduce the case where the twist map is invertible to the study of involutive quadratic Jacobi-Jordan algebras. Also elements of a representation theory for Hom-Jacobi-Jordan algebras, including adjoint and coadjoint representations are supplied with application to quadratic Hom-Jacobi-Jordan algebras. [ABSTRACT FROM AUTHOR]

In this paper, we first study the (non)-degeneracy of a class of quadratic forms. Then with a proper defining set determined by the given quadratic form, we construct a class of linear codes, decide their weight enumerators and show that they are two- or three-weight codes by using the Weil sums. Some of the punctured codes of the discussed linear codes are optimal or distance optimal to the Griesmer bound. [ABSTRACT FROM AUTHOR]

Let $\mathbb{Z}$ and $\mathbb{Z}^{+}$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in \mathbb{Z}^{+}$ , let $t(a,b,c,d;n)$ be the number of representations of $n$ by $\frac{1}{2}ax(x+1)+\frac{1}{2}by(y+1)+\frac{1}{2}cz(z+1)+\frac{1}{2}dw(w+1)$ with $x,y,z,w\in \mathbb{Z}$. Using theta function identities we prove 13 transformation formulas for $t(a,b,c,d;n)$ and evaluate $t(2,3,3,8;n)$ , $t(1,1,6,24;n)$ and $t(1,1,6,8;n)$. [ABSTRACT FROM AUTHOR]

This study considers method of constructing tests for the hypothesis on parameter β of a family of lifetime distributions under type-II censored sample. We propose three test statistics Q (sd) , F and Q 1. The performance of these statistics, in terms of power, is studied through simulation for different values of parameter of Burr-XII distribution, log-bathtub distribution, Weibull distribution, and two-parameter exponential distribution. The percentile points and powers for testing the parameter β , in each of these distributions, are simulated. For these distributions, the test based on quadratic form (i. e. Q (sd) and Q 1) performs better than the test F in identifying β > β 0 or β < β 0 alternatives for some member of the family of distributions. One numerical example is also given. [ABSTRACT FROM AUTHOR]

In an earlier work, the authors prove Stillman's conjecture in all characteristics and all degrees by showing that, independent of the algebraically closed field K or the number of variables, n forms of degree at most d in a polynomial ring R over K are contained in a polynomial subalgebra of R generated by a regular sequence consisting of at most ηB(n,d) forms of degree at most d; we refer to these informally as ''small'' subalgebras. Moreover, these forms can be chosen so that the ideal generated by any subset defines a ring satisfying the Serre condition Rη. A critical element in the proof is to show that there are functions ηA(n,d) with the following property: in a graded n-dimensional K-vector subspace V of R spanned by forms of degree at most d, if no nonzero form in V is in an ideal generated by ηA(n,d) forms of strictly lower degree (we call this a strength condition), then any homogeneous basis for V is an Rη sequence. The methods of our earlier work are not constructive. In this paper, we use related but different ideas that emphasize the notion of a key function to obtain the functions ηA(n,d) in degrees 2, 3, and 4 (in degree 4 we must restrict to characteristic not 2, 3). We give bounds in closed form for the key functions and the ηA functions, and explicit recursions that determine the functions ηB from the ηA functions. In degree 2, we obtain an explicit value for ηB(n,2) that gives the best known bound in Stillman's conjecture for quadrics when there is no restriction on n. In particular, for an ideal I generated by n quadrics, the projective dimension R/I is at most 2n+1(n − 2) + 4. [ABSTRACT FROM AUTHOR]

Testing for series correlation among error terms is a basic problem in linear regression model diagnostics. The famous Durbin–Watson test and Durbin's h-test rely on certain model assumptions about the response and regressor variables. The present paper proposes simple tests for series correlation that are applicable in both fixed and random design linear regression models. The test statistics are based on the regression residuals and design matrix. The test procedures are robust under different distributions of random errors. The asymptotic distributions of the proposed statistics are derived via a newly established joint central limit theorem for several general quadratic forms and the delta method. Good performance of the proposed tests is demonstrated by the simulation results. [ABSTRACT FROM AUTHOR]

We show that for all integers m ⩾ 2 , and all integers k ⩾ 2 , the orthogonal groups O ± (2 m , F 2 k) act on abstract regular polytopes of rank 2m, and the symplectic groups Sp (2 m , F 2 k) act on abstract regular polytopes of rank 2 m + 1 . [ABSTRACT FROM AUTHOR]

The method of sliding modes (relaxation) was originally invented in optimal control in order to give a transparent proof of the maximum principle (a first-order necessary condition for a strong local minimum) using the local maximum principle (a first-order necessary condition for a weak local minimum). In the present work, we use this method to derive second-order necessary conditions for a strong local minimum on the base of such conditions for a weak local minimum. For simplicity, we confine ourselves to the consideration of the Mayer problem with endpoint equality and inequality constraints and control inequality constraints given by a finite number of twice smooth functions. Assuming that the gradients of active control constraints are linearly independent, we provide a rather short proof of second-order necessary conditions for a strong local minimum. [ABSTRACT FROM AUTHOR]

By an n -dimensional quadratic form over a field F (F -form) we mean a homogeneous quadratic polynomial in n variables with coefficients in F. We call two pairs of n -dimensional quadratic F -forms (f 1 , g 1) , (f 2 , g 2) isomorphic if there exists a nondegenerate linear change of variables taking (f 1 , g 1) to (f 2 , g 2). We give a complete classification (up to isomorphism) of 3-dimensional pairs of forms (f , g) over an arbitrary field F of characteristic different from 2. We consider separately the cases of isotropic form f + t g (which is equivalent to existence of a common zero of f and g), and the anisotropic one. In the first case we make a classification depending on det (f + t g) , and in the second case depending on the even Clifford algebra C 0 (f + t g) and det (f + t g). [ABSTRACT FROM AUTHOR]

The structure of an undirected graph is completely determined by a symmetric matrix: its adjacency matrix with respect to an ordering of its vertices; and that matrix can be used to define an integral quadratic form. The main purpose of this paper is to raise this question: "What can quadratic forms tell us about graphs?" As an initial answer, the theory of quadratic forms will be applied to the graph isomorphism problem. The essential definitions and facts from the theory of quadratic forms will be sketched without proof. [ABSTRACT FROM AUTHOR]

Nourdin and Peccati (2009a) established a neat characterization of Gamma approximation on a fixed Wiener chaos in terms of convergence of only the third and fourth cumulants. In this paper, we provide an optimal rate of convergence in the d2-distance in terms of the maximum of the third and fourth cumulants analogous to the result for normal approximation in Nourdin and Peccati (2015). In order to achieve our goal, we introduce a novel operator theory approach to Stein's method. The recent development in Stein's method for the Gamma distribution of Döbler and Peccati (2018) plays a pivotal role in our analysis. Several examples in the context of quadratic forms are considered to illustrate our optimal bound. [ABSTRACT FROM AUTHOR]

The sphere packing problem is an old puzzle. We consider packings with m spheres in the unit cell (m-periodic packings). For the case m = 1 (lattice packings), Voronoi proved there are finitely many inequivalent local optima and presented an algorithm to enumerate them, and this computation has been implemented in up to d = 8 dimensions. We generalize Voronoi's method to m > 1 and present a procedure to enumerate all locally optimal 2-periodic sphere packings in any dimension, provided there are finitely many. We implement this computation in d = 3 , 4 , and 5 and show that no 2-periodic packing surpasses the density of the optimal lattices in these dimensions. A partial enumeration is performed in d = 6 . [ABSTRACT FROM AUTHOR]

Summary: Finite mixtures offer a rich class of distributions for modelling of a variety of random phenomena in numerous fields of study. Using the sample interpoint distances (IPDs), we propose the IPD‐test statistic for testing the hypothesis of homogeneity of mixture of multivariate power series distribution or multivariate normal distribution. We derive the distribution of the IPDs that are drawn from a finite mixture of the multivariate power series distribution and multivariate normal distribution. Based on the empirical distribution of the IPDs, we construct a bootstrap test of homogeneity for other multivariate finite mixture models. The IPD test is applied to mixture models for matrix‐valued distributions and a test of homogeneity for Wishart mixture is presented. Numerical comparisons show that IPD test has accurate type I errors and is more powerful in most multivariate cases than the expectation–maximization (EM) test and modified likelihood ratio test. [ABSTRACT FROM AUTHOR]

We introduce a generalization, called a skew Clifford algebra , of a Clifford algebra, and relate these new algebras to the notion of graded skew Clifford algebra that was defined in 2010. In particular, we examine homogenizations of skew Clifford algebras, and determine which skew Clifford algebras can be homogenized to create Artin-Schelter regular algebras. Just as (classical) Clifford algebras are the Poincaré-Birkhoff-Witt (PBW) deformations of exterior algebras, skew Clifford algebras are the Z 2 -graded PBW deformations of quantum exterior algebras. We also determine the possible dimensions of skew Clifford algebras and provide several examples. [ABSTRACT FROM AUTHOR]

We introduce the notion of a symmetric basis of a vector space equipped with a quadratic form, and provide a sufficient and necessary condition for the existence to such a basis. Symmetric bases are then used to study Cayley graphs of certain extraspecial 2-groups of order 2 2 r + 1 ( r ≥ 1 ), which are further shown to be normal Cayley graphs and 2-arc-transitive covers of 2r-dimensional hypercubes. [ABSTRACT FROM AUTHOR]

Abstract We propose an algorithm for finding zero divisors in quaternion algebras over quadratic number fields, or equivalently, solving homogeneous quadratic equations in three variables over Q (d) where d is a square-free integer. The algorithm is randomized and runs in polynomial time if one is allowed to call oracles for factoring integers. [ABSTRACT FROM AUTHOR]

We examine a pair of dynamical systems on the plane induced by a pair of spanning trees in the Cayley graph of the Super-Apollonian group of Graham, Lagarias, Mallows, Wilks, and Yan. The dynamical systems compute Gaussian rational approximations to complex numbers and are “reflective” versions of the complex continued fractions of A. L. Schmidt. They also describe a reduction algorithm for Lorentz quadruples, in analogy to work of Romik on Pythagorean triples. For these dynamical systems, we produce an invertible extension and an invariant measure, which we conjecture is ergodic. We consider some statistics of the related continued fraction expansions, and we also examine the restriction of these systems to the real line, which gives a reflective version of the usual continued fraction algorithm. Finally, we briefly consider an alternate setup corresponding to a tree of Lorentz quadruples ordered by arithmetic complexity. [ABSTRACT FROM AUTHOR]

For A ⊆ { 1 , 2 , ... } , we consider R (A) = { a / a ′ : a , a ′ ∈ A }. If A is the set of nonzero values assumed by a quadratic form, when is R (A) dense in the p -adic numbers? We show that for a binary quadratic form Q , R (A) is dense in Q p if and only if the discriminant of Q is a nonzero square in Q p , and for a quadratic form in at least three variables, R (A) is always dense in Q p. This answers a question posed by several authors in 2017. [ABSTRACT FROM AUTHOR]

There is a mapping of the set of primitive integral representations of a quadratic form in n − 1 variables by a quadratic form in n variables into the set of primitive representations of a rational number by a quadratic form in n variables. Such a correspondence was introduced by Shimura in 2004. The purpose of this paper is to study the former representations by describing the inverse image of the latter set under that map. As an application we discuss some specific Fourier coefficients of theta series of degree n − 1. [ABSTRACT FROM AUTHOR]

Let ℤ and ℕ be the set of integers and the set of positive integers, respectively. For a 1 , a 2 , ... , a k , n ∈ ℕ , let N (a 1 , a 2 , ... , a k ; n) be the number of representations of n by a 1 x 1 2 + a 2 x 2 2 + ⋯ + a k x k 2 , and let t (a 1 , a 2 , ... , a k ; n) be the number of representations of n by a 1 x 1 (x 1 − 1) 2 + a 2 x 2 (x 2 − 1) 2 + ⋯ + a k x k (x k − 1) 2 (x 1 , ... , x k ∈ ℤ). In this paper, by using Ramanujan's theta functions φ (q) and ψ (q) , we reveal many relations between t (a 1 , a 2 , ... , a k ; n) and N (a 1 , a 2 , ... , a k ; 8 n + a 1 + ⋯ + a k) for k = 3 , 4. [ABSTRACT FROM AUTHOR]

In this paper we prove that there is a direct relationship between Salem numbers and translation lengths of hyperbolic elements of arithmetic hyperbolic groups that are determined by a quadratic form over a totally real number field. As an application we determine a sharp lower bound for the length of a closed geodesic in a noncompact arithmetic hyperbolic n-orbifold for each dimension n. We also discuss a short geodesic conjecture, and prove its equivalence with Lehmer's conjecture for Salem numbers. [ABSTRACT FROM AUTHOR]

The sample mean and sample variance are commonly used statistics in our study. In this paper, distributions of the sample mean and sample variance from a skew normal population are derived under closed skew normal (CSN) settings. The noncentral closed skew chi-square distribution is defined, and the distribution of quadratic forms is discussed. Our results generalize the corresponding results given under skew normal settings. Several examples are given for illustration of our results. [ABSTRACT FROM AUTHOR]

Abstract Highly regular graphs for which not all regularities are explainable by symmetries are fascinating creatures. Some of them like, e.g., the line graph of W. Kantor's non-classical GQ (5 2 , 5) , are stumbling stones for existing implementations of graph isomorphism tests. They appear to be extremely rare and even once constructed it is difficult to prove their high regularity. Yet some of them, like the McLaughlin graph on 275 vertices and Ivanov's graph on 256 vertices are of profound beauty. This alone makes it an attractive goal to strive for their complete classification or, failing this, at least to get a deep understanding of them. Recently, one of the authors discovered new methods for proving high regularity of graphs. Using these techniques, in this paper we study a classical family of strongly regular graphs, originally discovered by A.E. Brouwer, A.V. Ivanov, and M.H. Klin in the late 80s. We analyse their symmetries and show that they are (3 , 5) -regular but not 2-homogeneous. Thus we promote these graphs to the distinguished club of highly regular graphs with few symmetries. [ABSTRACT FROM AUTHOR]