Real Quadratic Form Research Articles

Positive definiteness of real quadratic forms resulting from the variable-step L1-type approximations of convolution operators

Positive definiteness of real quadratic forms resulting from the variable-step L1-type approximations of convolution operators

An Upper Bound for the Size of $s$-Distance Sets in Real Algebraic Sets

In a recent paper, Petrov and Pohoata developed a new algebraic method which combines the Croot-Lev-Pach Lemma from additive combinatorics and Sylvester’s Law of Inertia for real quadratic forms. As an application, they gave a simple proof of the Bannai-Bannai-Stanton bound on the size of $s$-distance sets (subsets $\mathcal{A}\subseteq \mathbb{R}^n$ which determine at most $s$ different distances). In this paper we extend their work and prove upper bounds for the size of $s$-distance sets in various real algebraic sets. This way we obtain a novel and short proof for the bound of Delsarte-Goethals-Seidel on spherical s-distance sets and a generalization of a bound by Bannai-Kawasaki-Nitamizu-Sato on $s$-distance sets on unions of spheres. In our arguments we use the method of Petrov and Pohoata together with some Gröbner basis techniques.

A Crank–Nicolson-type finite-difference scheme and its algorithm implementation for a nonlinear partial integro-differential equation arising from viscoelasticity

A Crank–Nicolson-type finite-difference scheme is proposed and analyzed for a nonlinear partial integro-differential equation arising from viscoelasticity. The time derivative is approximated by the Crank–Nicolson scheme and the Riemann–Liouville fractional integral term is treated by means of the trapezoidal convolution quadrature rule. To construct a fully discrete difference scheme, the standard centered difference formula is utilized to approximate the second-order spatial derivative and the Galerkin method based on piecewise linear test functions is used to discrete the nonlinear convection term. Theoretical results of stability and convergence are derived using the non-negative character of the real quadratic form associated with the convolution quadrature, and combining with the discrete Gronwall inequality. Besides, a fixed point iterative numerical algorithm, which fills the gap that the existed numerical schemes have only theoretical analysis but no numerical results, is presented. Numerical results show the efficiency and feasibility of our scheme, and the orders of convergence are in good agreement with the theoretical results.

An automorphic generalization of the Hermite–Minkowski theorem

We show that for any integer $N$, there are only finitely many cuspidal algebraic automorphic representations of ${\rm GL}_n$ over $\mathbb{Q}$, with $n$ varying, whose conductor is $N$ and whose weights are in the interval $\{0,1,...,23\}$. More generally, we define a simple sequence $(r(w))_{w \geq 0}$ such that for any integer $w$, any number field $E$ whose root-discriminant is less than $r(w)$, and any ideal $N$ in the ring of integers of $E$, there are only finitely many cuspidal algebraic automorphic representations of general linear groups over $E$ whose conductor is $N$ and whose weights are in the interval $\{0,1,...,w\}$. Assuming a version of GRH, we also show that we may replace $r(w)$ with $8 \pi e^{\gamma-H_w}$ in this statement, where $\gamma$ is Euler's constant and $H_w$ the $w$-th harmonic number. The proofs are based on some new positivity properties of certain real quadratic forms which occur in the study of the Weil explicit formula for Rankin-Selberg $L$-functions. Both the effectiveness and the optimality of the methods are discussed.

Spaces of Type S as Topological Algebras under Twisted Convolution and Star Product

The properties of the generalized Gelfand-Shilov spaces $$S_{{b_n}}^{{a_k}}$$ are studied from the viewpoint of deformation quantization. We specify the conditions on the defining sequences (ak) and (bn) under which $$S_{{b_n}}^{{a_k}}$$ is an algebra with respect to the twisted convolution and, as a consequence, its Fourier transformed space $$S_{{a_k}}^{{b_n}}$$ is an algebra with respect to the Moyal star product. We also consider a general family of translation-invariant star products. We define and characterize the corresponding algebras of multipliers and prove the basic inclusion relations between these algebras and the duals of the spaces of ordinary pointwise and convolution multipliers. Analogous relations are proved for the projective counterpart of the Gelfand-Shilov spaces. A key role in our analysis is played by a theorem characterizing those spaces of type S for which the function exp(iQ(x)) is a pointwise multiplier for any real quadratic form Q.

Certified lattice reduction

Quadratic form reduction and lattice reduction are fundamental tools in computational number theory and in computer science, especially in cryptography. The celebrated Lenstra–Lenstra–Lovasz reduction algorithm (so-called LLL) has been improved in many ways through the past decades and remains one of the central methods used for reducing integral lattice basis. In particular, its floating-point variants---where the rational arithmetic required by Gram–Schmidt orthogonalization is replaced by floating-point arithmetic---are now the fastest known. However, the systematic study of the reduction theory of real quadratic forms or, more generally, of real lattices is not widely represented in the literature. When the problem arises, the lattice is usually replaced by an integral approximation of (a multiple of) the original lattice, which is then reduced. While practically useful and proven in some special cases, this method doesn't offer any guarantee of success in general. In this work, we present an adaptive-precision version of a generalized LLL algorithm that covers this case in all generality. In particular, we replace floating-point arithmetic by Interval Arithmetic to certify the behavior of the algorithm. We conclude by giving a typical application of the result in algebraic number theory for the reduction of ideal lattices in number fields.

Copositive and Positive Quadratic Forms on Matrices

A real symmetric quadratic form $$f = f(Z_1,\ldots ,Z_n)$$ in the n non-commuting indeterminates $$Z_1,\ldots ,Z_n$$ is said to be d-positive (respectively, d-copositive) if for all real symmetric (respectively, positive semidefinite) $$(d \times d)$$-matrices $$A_1,\ldots ,A_n$$, the matrix $$f(A_1,\ldots ,A_n)$$ is positive semidefinite. When $$d=1$$, i.e., when $$Z_i$$ can take real numbers as values, simple characterizations of real positive and copositive symmetric quadratic forms are given, for example, by Hajja (Math Inequal Appl 6:581–593, 2003). In this paper, similar characterizations are obtained for all d.

New Approximation to Distribution of Positive RVs Applied to Gaussian Quadratic Forms

This letter introduces a new approach to the problem of approximating the probability density function (PDF) and the cumulative distribution function (CDF) of a positive random variable. The novel approximation strategy is based on the analysis of a suitably defined sequence of auxiliary variables which converges in distribution to the target variable. By leveraging such convergence, simple approximations for both the CDF and PDF of the target variable are given in terms of the derivatives of its moment generating function (MGF). In contrast to classical approximation methods based on truncated series of moments or cumulants, our approximations always represent a valid distribution and the relative error between variables is independent of the variable under analysis. The derived results are then used to approximate the statistics of positive-definite real Gaussian quadratic forms, comparing our proposed approach with other existing approximations in the literature.

A backward Euler difference scheme for the integro-differential equations with the multi-term kernels

ABSTRACTA backward Euler difference scheme is formulated and analyzed for the integro-differential equations with the multi-term kernels. The convolution quadrature is used to deal with the Riemann-Liouville fractional integral terms. A fully discrete difference scheme is constructed with time derivative by backward Euler scheme and space discretization by standard central difference approximation. We prove the stability and convergence based on the nonnegative character of the real quadratic form. The and -norms stability and convergence are given. Finally, numerical experiments validate the theoretical results.

Shubin type Fourier integral operators and evolution equations

We study the Cauchy problem for an evolution equation of Schrodinger type. The Hamiltonian is the Weyl quantization of a real homogeneous quadratic form with a pseudodifferential perturbation of negative order from Shubin’s class. We prove that the propagator is a Fourier integral operator of Shubin type of order zero. Using results for such operators and corresponding Lagrangian distributions, we study the propagator and the solution, and derive phase space estimates for them.

Bounds on real finite linear groups that preserve a symmetric or skew symmetric bilinear form

Abstract We extend the results of our earlier work on Jordan-type bounds for finite subgroups of complex classical groups to real groups. The bounds that we obtain are related to our previous results by means of structural results for finite linear groups that we can generalise here to compact groups. For the real analogue of orthogonal groups, we take into account the signature of a real quadratic form to determine bounds in every case.

Conway River and Arnold Sail

We establish a simple relation between two geometric constructions in number theory: the Conway river of a real indefinite binary quadratic form and the Arnold sail of the corresponding pair of lines.

A backward Euler alternating direction implicit difference scheme for the three‐dimensional fractional evolution equation

A backward Euler alternating direction implicit (ADI) difference scheme is formulated and analyzed for the three‐dimensional fractional evolution equation. In our method, the Riemann‐Liouville fractional integral term is treated by means of first order convolution quadrature suggested by Lubich. Meanwhile, an ADI technique is adopted to reduce the multidimensional problem to a series of one‐dimensional problems. A fully discrete difference scheme is constructed with space discretization by finite difference method. Two new inner products and corresponding norms are defined to analyze the scheme. The verification of stability and convergence is based on the nonnegative character of the real quadratic form associated with the convolution quadrature. Numerical experiments are reported to demonstrate the efficiency of our scheme.

What is the Probability that a Random Integral Quadratic Form innVariables has an Integral Zero?

We show that the density of quadratic forms in nn variables over ZpZp that are isotropic is a rational function of pp, where the rational function is independent of pp, and we determine this rational function explicitly. When real quadratic forms in nn variables are distributed according to the Gaussian Orthogonal Ensemble (GOE) of random matrix theory, we determine explicitly the probability that a random such real quadratic form is isotropic (i.e., indefinite). As a consequence, for each nn, we determine an exact expression for the probability that a random integral quadratic form in nn variables is isotropic (i.e., has a nontrivial zero over ZZ), when these integral quadratic forms are chosen according to the GOE distribution. In particular, we find an exact expression for the probability that a random integral quaternary quadratic form is isotropic; numerically, this probability of isotropy is approximately 98.3%.

Sums of squares and varieties of minimal degree

Let X ⊆ P n X \subseteq \mathbb {P}^n be a real nondegenerate subvariety such that the set X ( R ) X(\mathbb {R}) of real points is Zariski dense. We prove that every real quadratic form that is nonnegative on X ( R ) X(\mathbb {R}) is a sum of squares of linear forms if and only if X X is a variety of minimal degree. This substantially extends Hilbert’s celebrated characterization of equality between nonnegative forms and sums of squares. We obtain a complete list for the cases of equality and also a classification of the lattice polytopes Q Q for which every nonnegative Laurent polynomial with support contained in 2 Q 2Q is a sum of squares.

Gaussian variational ansatz in the problem of anomalous sea waves: Comparison with direct numerical simulation

The nonlinear dynamics of an obliquely oriented wave packet on a sea surface is analyzed analytically and numerically for various initial parameters of the packet in relation to the problem of the so-called rogue waves. Within the Gaussian variational ansatz applied to the corresponding (1+2)-dimensional hyperbolic nonlinear Schrodinger equation (NLSE), a simplified Lagrangian system of differential equations is derived that describes the evolution of the coefficients of the real and imaginary quadratic forms appearing in the Gaussian. This model provides a semi-quantitative description of the process of nonlinear spatiotemporal focusing, which is one of the most probable mechanisms of rogue wave formation in random wave fields. The system of equations is integrated in quadratures, which allows one to better understand the qualitative differences between linear and nonlinear focusing regimes of a wave packet. Predictions of the Gaussian model are compared with the results of direct numerical simulation of fully nonlinear long-crested waves.

Random matrices and the average topology of the intersection of two quadrics

Let XR be the zero locus in RP n of one or two independently and Weyl distributed random real quadratic forms. Denoting by XC the complex part in CP n of XR and by b(XR) and b(XC) the sums of their Betti numbers, we prove that: (1) lim n→∞ Eb(XR) n = 1. In particular for one quadric hypersurface asymptotically Smith’s inequality b(XR) ≤ b(XC) is expected to be sharp. The methods we use combine Random Matrix Theory, Integral Geometry and spectral sequences.

Quadratic Cohomology

We study homological invariants of smooth families of real quadratic forms as a step towards a multipliers rule in the large that intends to describe topology of smooth maps in terms of scalar Lagrange functions.

Monomials in quadratic forms

We obtain some constraints on the zero-nonzero pattern of entries in the matrix of a real quadratic form which attains a minimum on a large set of vertices in the multidimensional cube centered at the origin whose edges are parallel to the coordinate axes. In particular, if the graph of the matrix contains an articulation point then the set of the minima of the corresponding quadratic form is not maximal (with respect to set inclusion) among all such sets for various quadratic forms.

Real fields, valuations, and quadratic forms

We study several field invariants arising in quadratic form theory. Some of the invariants considered are of particular interest in the study of real fields, including the length, the u-invariant, and the (reduced) stability index. In this context we give a systematic account of valuation theoretic arguments that lead to lower bounds for these invariants.