Multilinear Forms Research Articles

Trilinear forms with Kloosterman fractions

We give new bounds for ∑a,m,nαmβnνae(am‾n) where αm, βn and νa are arbitrary coefficients, improving upon a result of Duke, Friedlander and Iwaniec [7]. We also apply these bounds to problems on representations by determinant equations and on the equidistribution of solutions to linear equations.

Exact Solutions of the Nonlocal Nonlinear Schrödinger Equation with a Perturbation Term

Abstract Analytical solutions of both the nonlinear Schrödinger equation (NLSE) and NLSE with a perturbation term have been attained. Besides, analytical solutions of nonlocal NLSE have also been obtained. In this paper, the nonlocal NLSE with a perturbation term is discussed. By virtue of the dependent variable substitution, trilinear forms of this equation is attained. Lax pairs and Darboux transformation of this equation are obtained. Via the Darboux transformation, two kinds solutions of this equation with the different seed solutions are attained.

Solidarity Value and Solidarity Share Functions for TU Fuzzy Games

TU games under both crisp and fuzzy environments describe situations where players make full (crisp) or partial (fuzzy) binding agreements and generate worth in return. The challenge is then to decide how to distribute the profit among them in a rational manner: we call this a solution. In this paper, we introduce the notion of solidarity value and the solidarity share function as a suitable solution to TU fuzzy games. Two special classes of TU fuzzy games, namely, TU fuzzy games in Choquet integral form and in multilinear extension form, are studied and the corresponding solidarity value and the solidarity share functions are characterized.

A note on the Hardy-Littlewood inequalities for multilinear forms

A note on the Hardy-Littlewood inequalities for multilinear forms

Numerical study of polynomial feedback laws for a bilinear control problem

An infinite-dimensional bilinear optimal control problem with infinite-time horizon is considered. The associated value function can be expanded in a Taylor series around the equilibrium, the Taylor series involving multilinear forms which are uniquely characterized by generalized Lyapunov equations. A numerical method for solving these equations is proposed. It is based on a generalization of the balanced truncation model reduction method and some techniques of tensor calculus, in order to attenuate the curse of dimensionality. Polynomial feedback laws are derived from the Taylor expansion and are numerically investigated for a control problem of the Fokker-Planck equation. Their efficiency is demonstrated for initial values which are sufficiently close to the equilibrium.

A novel hierarchy of two-family-parameter equations: Local, nonlocal, and mixed-local–nonlocal vector nonlinear Schrödinger equations

We use two families of parameters {(ϵxj,ϵtj)|ϵxj,tj=±1,j=1,2,…,n} to first introduce a unified novel hierarchy of two-family-parameter equations (simply called Qϵxn→,ϵtn→(n) hierarchy), connecting integrable local, nonlocal, novel mixed-local–nonlocal, and other nonlocal vector nonlinear Schrödinger (VNLS) equations. The Qϵxn→,ϵtn→(n) system with (ϵxj,ϵtj)=(±1,1),j=1,2,…,n is shown to possess Lax pairs and infinite number of conservation laws. Moreover, we also analyze the PT symmetry of the Hamiltonians with self-induced potentials. The multi-linear forms and some symmetry reductions are also studied. In fact, the used two families of parameters can also be extended to the general case {(ϵxj,ϵtj)|ϵxj=eiθxj,ϵtj=eiθtj,θxj,θtj∈[0,2π),j=1,2,…,n} to generate more types of nonlinear equations. The novel two-family-parameter (or multi-family-parameter for higher-dimensional cases) idea can also be applied to other local nonlinear evolution equations to find novel integrable and non-integrable nonlocal and mixed-local–nonlocal systems.

Strong stability of a class of difference equations of continuous time and structured singular value problem

This article studies the strong stability of scalar difference equations of continuous time in which the delays are sums of a number of independent parametersτi,i=1,2,…,K. The characteristic quasipolynomial of such an equation is a multilinear function ofe−τis. It is known that the characteristic quasipolynomial of any difference equation set in the form of one-delay-per-scalar-channel (ODPSC) model is also in such a multilinear form. However, it is shown in this article that some multilinear forms of quasipolynomials are not characteristic quasipolynomials of any ODPSC difference equation set. The equivalence between local strong stability, the exponential stability of a fixed set of rationally independent delays, and the stability for all positive delays is shown, and relations with the structured singular value problem are presented. A procedure to determine strong stability in the special case of up to three independent delay parameters in finite steps is developed. This procedure means that the structured singular value problem in the case of up to three scalar complex uncertain blocks can be solved in finite steps.

Non-auto Bäclund transformation, nonlocal symmetry and CRE solvability for the Bogoyavlenskii–Kadomtsev–Petviashvili equation

In this paper, we study the Bogoyavlenskii–Kadomtsev– Petviashvili (BKP) equation by using the truncated Painlevé method and consistent Riccati expansion (CRE). Through the truncated Painlevé method, its nonlocal symmetry and non-auto Bäcklund transformation are presented. The nonlocal symmetry is localized to a local Lie point group via a prolonged system. Moreover, with the help of the CRE method, we prove that the BKP equation is CRE solvable. Finally, the kink-lump wave interaction solution of BKP equation is explicitly given by using the trilinear form. The interaction between kink wave and lump wave is investigated and exhibited mathematically and graphically.

Approximation algorithms for optimization of real-valued general conjugate complex forms

Complex polynomial optimization has recently gained more attention in both theory and practice. In this paper, we study optimization of a real-valued general conjugate complex form over various popular constraint sets including the m-th roots of complex unity, the complex unit circle, and the complex unit sphere. A real-valued general conjugate complex form is a homogenous polynomial function of complex variables as well as their conjugates, and always takes real values. General conjugate form optimization is a wide class of complex polynomial optimization models, which include many homogenous polynomial optimization in the real domain with either discrete or continuous variables, and Hermitian quadratic form optimization as well as its higher degree extensions. All the problems under consideration are NP-hard in general and we focus on polynomial-time approximation algorithms with worst-case performance ratios. These approximation ratios improve previous results when restricting our problems to some special classes of complex polynomial optimization, and improve or equate previous results when restricting our problems to some special classes of polynomial optimization in the real domain. The algorithms are based on tensor relaxation and random sampling. Our novel technical contributions are to establish the first set of probability lower bounds for random sampling over the m-th root of unity, the complex unit circle, and the complex unit sphere, and to propose the first polarization formula linking general conjugate forms and complex multilinear forms. Some preliminary numerical experiments are conducted to show good performance of the proposed algorithms.

Summability of multilinear forms on classical sequence spaces

We present an extension of the Hardy-Littlewood inequality for multi-linear forms. This extension combines repetitions of indexes in the summands and perturbation of the exponent. In particular, we recover a recent result due to G. Araujo and D. Pellegrino.

Further results on the strong stability of difference equations of continuous time

This article studies the strong stability of scalar difference equations of continuous time in which the delays are sums of a number of independent parameters τi i = 1, 2,..., K. The characteristic quasipolynomial of such an equation is a multilinear function of e-Tis. It is known that the characteristic quasipolynomial of any difference equation set in the form of one-delay-per-scalar-channel (ODPSC) model is also in such a multilinear form. However, it is shown in this article that some multilinear forms of quasipolynomials are not characteristic quasipolynomials of any ODPSC difference equation set. The equivalence between local strong stability the exponential stability of a fixed set of rationally independent delays, and the stability for all positive delays is shown.

Partial-State Stabilization and Optimal Feedback Control for Stochastic Dynamical Systems

In this paper, we develop a unified framework to address the problem of optimal nonlinear analysis and feedback control for partial stability and partial-state stabilization of stochastic dynamical systems. Partial asymptotic stability in probability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that is positive definite and decrescent with respect to part of the system state which can clearly be seen to be the solution to the steady-state form of the stochastic Hamilton–Jacobi–Bellman equation, and hence, guaranteeing both partial stability in probability and optimality. The overall framework provides the foundation for extending optimal linear-quadratic stochastic controller synthesis to nonlinear-nonquadratic optimal partial-state stochastic stabilization. Connections to optimal linear and nonlinear regulation for linear and nonlinear time-varying stochastic systems with quadratic and nonlinear-nonquadratic cost functionals are also provided. Finally, we also develop optimal feedback controllers for affine stochastic nonlinear systems using an inverse optimality framework tailored to the partial-state stochastic stabilization problem and use this result to address polynomial and multilinear forms in the performance criterion.

Hom-Lie quadratic and Pinczon Algebras

ABSTRACTPresenting the structure equation of a hom-Lie algebra 𝔤, as the vanishing of the self commutator of a coderivation of some associative comultiplication, we define up to homotopy hom-Lie algebras, which yields the general hom-Lie algebra cohomology with value in a module. If the hom-Lie algebra is quadratic, using the Pinczon bracket on skew symmetric multilinear forms on 𝔤, we express this theory in the space of forms. If the hom-Lie algebra is symmetric, it is possible to associate to each module a quadratic hom-Lie algebra and describe the cohomology with value in the module.

Complexity of the satisfiability problem for multilinear forms over a finite field

Multilinear forms over finite fields are considered. Multilinear forms over a field are products in which each factor is the sum of variables or elements of this field. Each multilinear form defines a function over this field. A multilinear form is called satisfiable if it represents a nonzero function. We show the N P-completeness of the satisfiability recognition problem for multilinear forms over each finite field of q elements for q ≥ 3. A theorem is proved that distinguishes cases of polynomiality and NP-completeness of the satisfiability recognition problem for multilinear fields for each possible q ≥ 3.

On a Geometric Inequality Related to Fractional Integration

In this paper we consider a new kind of inequality related to fractional integration, motivated by Gressman’s paper. Based on it we investigate its multilinear analogue inequalities. Combining with Gressman’s work on multilinear integral, we establish this new kind of geometric inequalities with bilinear form and multilinear form in more general settings. Moreover, in some cases we also find the best constants and optimisers for these geometric inequalities on Euclidean spaces with Lebesgue measure settings with \(L^{p}\) bounds.

Partially massless higher-spin theory

We study a generalization of the D-dimensional Vasiliev theory to include a tower of partially massless fields. This theory is obtained by replacing the usual higher-spin algebra of Killing tensors on (A)dS with a generalization that includes “third-order” Killing tensors. Gauging this algebra with the Vasiliev formalism leads to a fully non-linear theory which is expected to be UV complete, includes gravity, and can live on dS as well as AdS. The linearized spectrum includes three massive particles and an infinite tower of partially massless particles, in addition to the usual spectrum of particles present in the Vasiliev theory, in agreement with predictions from a putative dual CFT with the same symmetry algebra. We compute the masses of the particles which are not fixed by the massless or partially massless gauge symmetry, finding precise agreement with the CFT predictions. This involves computing several dozen of the lowest-lying terms in the expansion of the trilinear form of the enlarged higher-spin algebra. We also discuss nuances in the theory that occur in specific dimensions; in particular, the theory dramatically truncates in bulk dimensions D = 3, 5 and has non-diagonalizable mixings which occur in D = 4, 7.

Spaceability in norm-attaining sets

We study the existence of infinite-dimensional vector spaces in the sets of norm-attaining operators, multilinear forms, and polynomials. Our main result is that, for every set of permutations P of the set {1,…,n}, there exists a closed infinite-dimensional Banach subspace of the space of n-linear forms on ℓ1 such that, for all nonzero elements B of such a subspace, the Arens extension associated to the permutation σ of B is norm-attaining if and only if σ is an element of P. We also study the structure of the set of norm-attaining n-linear forms on c0.

The Study on the Differential and Integral Calculus in Pseudoeuclidean Space

In the vector space of real vectors, comparison was executed of the multilinear forms, covariant derivatives, the total differentials and derivatives in the direction which are calculated with different metrics—with Euclidean metric and with pseudoeuclidean metric of a zero index. Comparison was executed of the Taylor’s formulas to different metrics. What is established by us is that multilinear forms of different metrics have different values; covariant derivatives have identical values; the total differentials and derivatives in the direction have different values. In Euclidean space, Taylor’s formula with any order of accuracy assigned in advance is equal, but in pseudoeuclidean space Taylor’s formula is not equal with any order of accuracy. It is concluded that in space with a pseudoeuclidean metric, the computing sense of the differential and integral calculus created in Euclidean space is lost and the possibility of mathematical model operation of real physical processes in vector space with pseudoeuclidean metric is called into question.

Cyclic constitutive model for strain-hardening cementitious composites

A uniaxial cyclic constitutive model for strain-hardening cementitious composites (SHCCs) is proposed based on the response of the material at the stress–strain level under different loading regimes. The ascending branch of the compressive envelope curve was developed using the form of a parabolic curve, while the descending branch adopted the bilinear form. The full tensile stress–strain curve was approximately simulated by a trilinear form comprising ascending, strain-hardening and descending branches. Using the method of setting a reference point, the characteristics of the unloading and reloading process were determined, with consideration of the partial unloading–reloading scheme. Furthermore, the transition between tension and compression was taken into account in the model development. The proposed cyclic constitutive model was coded into the OpenSees platform and applied to simulate the responses of specimens at material and member levels. The simulation results indicate that the proposed model is reasonably accurate in simulating the mechanical properties of SHCC material and the cyclic behaviour of SHCC flexural elements.

Numerical simulation of large elastoplastic deformation of shells of revolution in the “spacesuit” under the influence of pulse overload

An axisymmetric problem of high strains in a spherical lead shell enclosed into an aluminum “spacesuit” under the influence of pulse overload is considered. The shell straining is described with the use of equations of mechanics of elastoviscoplastic media in Lagrangian variables. Kinematic relations are determined in the current state metrics. Equations of state are taken in the form of equations of the flow theory with isotropic hardening. Contact interaction of a shell and a “spacesuit” is modeled by conditions of non-penetration with friction. Numerical solution under the given boundary and initial conditions is based on the moment schema of the finite element method and the explicit time integration scheme of the "cross". For sampling the space variables 4-node isoparametric finite elements with multilinear forms features are used. As it was shown by the results of calculations, spherical shell suffers significant local forming, characterized by large displacements and rotation angles of finite elements as a rigid body in the process of loading. The calculation results of the residual form are in good agreement with the experimental data.