Multilinear differential operators on modular forms

We construct multilinear differential operators on modular forms and prove that they are essentially unique. We also discuss certain homogeneous polynomials associated to such differential operators as well as some related multilinear differential operators that do not produce modular forms.

     

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.     

Differential operators on Hermitian Jacobi forms

We study multilinear differential operators on a space of Hermitian Jacobi forms as well as on a space of Hermitian modular forms of degree 2. First we define a heat operator and construct multilinear differential operators on a space of Hermitian Jacobi forms of degree 2. As a special case of these operators, we also study Rankin-Cohen type differential operators on a space of Hermitian Jacobi forms. And we construct multilinear differential operators on a space of Hermitian modular forms of degree 2 as an application of multilinear differential operators on Hermitian Jacobi forms.     

New Sufficient Conditions for the Denseness of Norm Attaining Multilinear Forms

This paper gives new sufficient conditions for the density ofthe set of norm attaining multilinear forms in the space ofall continuous multilinear forms on a Banach space. The symmetriccase is also discussed.     

Hypercontraction principle and random multilinear forms

Summary We study a Banach space valued random multilinear forms in independent real random variables extensively using the concept of hypercontractive maps between L ^q-spaces. We show that multilinear forms share with linear forms a lot of properties, like comparability of L ^q-,L ⁰-and almost sure convergence.This author's contribution to a revision of this paper was supported by AFOSR Grant No. F49620 85C 0144     

Ideals of Multilinear Forms – a Limit Order Approach

A general theory of limit orders for ideals of multilinear forms is developed. We relate the limit order of an ideal to those of its maximal hull and its adjoint ideal. We study the limit orders of the ideals of dominated and multiple summing multilinear forms. Finally, estimates of the diagonal of a (non-necessarily diagonal) multilinear form are presented, in terms of the limit order of the ideals to which it belongs. The third author was supported by the MCYT and FEDER Project BFM2002-01423 and grant GV-GRUPOS04/45. The first and second authors were supported by CONICET-PIP 5272. The first author was also supported by UBACyT-X108 and ANPCyT-PICT 0315033.     

Approximation methods for complex polynomial optimization

Complex polynomial optimization problems arise from real-life applications including radar code design, MIMO beamforming, and quantum mechanics. In this paper, we study complex polynomial optimization models where the objective function takes one of the following three forms: (1) multilinear; (2) homogeneous polynomial; (3) symmetric conjugate form. On the constraint side, the decision variables belong to one of the following three sets: (1) the \(m\) -th roots of complex unity; (2) the complex unity; (3) the Euclidean sphere. We first discuss the multilinear objective function. Polynomial-time approximation algorithms are proposed for such problems with assured worst-case performance ratios, which depend only on the dimensions of the model. Then we introduce complex homogenous polynomial functions and establish key linkages between complex multilinear forms and the complex polynomial functions. Approximation algorithms for the above-mentioned complex polynomial optimization models with worst-case performance ratios are presented. Numerical results are reported to illustrate the effectiveness of the proposed approximation algorithms.     

On bounded multilinear forms on a class of lp spaces

In this paper we generalize an old result of Littlewood and Hardy about bilinear forms defined in a class of sequence spaces. Historically, Littlewood [Quart. J. Math.1 (1930)] first proved a result on bilinear forms on bounded sequences and this result was then generalized by Hardy and Littlewood in a joint paper [Quart. J. Math.5(1934)] to bilinear forms on a class of l^p spaces. Later Davie and Kaijser proved Littlewood's results for multilinear forms. In this paper, Theorems A and B generalize the results to multilinear forms on l^p spaces. All the results are stated at the end of Section 1. Theorems A and B are proved, respectively, in Sections 2 and 3.     

Spline approximation,Kronecker products and multilinear forms

Sums of Kronecker products occur naturally in high‐dimensional spline approximation problems, which arise, for example, in the numerical treatment of chemical reactions. In full matrix form, the resulting non‐sparse linear problems usually exceed the memory capacity of workstations. We present methods for the manipulation and numerical handling of Kronecker products in factorized form. Moreover, we analyze the problem of approximating a given matrix by sums of Kronecker products by making use of the equivalence to the problem of decomposing multilinear forms into sums of one‐forms. Greedy algorithms based on the maximization of multilinear forms over a torus are used to obtain such (finite and infinite) decompositions that can be used to solve the approximation problem. Moreover, we present numerical considerations for these algorithms. Copyright © 2016 John Wiley & Sons, Ltd.     

Multiple summing operators on Banach spaces

In this paper, we improve some previous results about multiple p-summing multilinear operators by showing that every multilinear form from spaces is multiple p-summing for 1?p?2. The proof is based on the existence of a predual for the Banach space of multiple p-summing multilinear forms. We also show the failure of the inclusion theorem in this class of operators and improve some results of Y. Meléndez and A. Tonge about dominated multilinear operators.     

Extendibility of bilinear forms on banach sequence spaces

We study Hahn-Banach extensions of multilinear forms defined on Banach sequence spaces. We characterize c ₀ in terms of extension of bilinear forms, and describe the Banach sequence spaces in which every bilinear form admits extensions to any superspace.     

Identical relations for the forms with a Young symmetry

Let there be given a Young symmetrizere . Consider the space of multilinear forms obtained by actione . In the paper the characteristic property in terms of identities is found for a multilinear form to belong to this space. This property is analogous to the well-known identities for the cases of the spaces of the symmetric and skew-symmetric forms. In addition, a generalization of Garnir identities is obtained.     

On integral formulas for submanifolds of spaces of constant curvature and some applications

Integral formulas of Minkowski type, involving the higher mean curvatures as multilinear forms on the normal bundle, are proved for compact oriented immersed submanifolds with arbitrary codimension in a Riemannian manifold of constant curvature, and as application a generalization of the Liebmann-Süss theorem as well as upper bounds for the first positive eigenvalue of the Laplace operator are given.     

On the convergence of random polynomials and multilinear forms

We consider different kinds of convergence of homogeneous polynomials and multilinear forms in random variables. We show that for a variety of complex random variables, the almost sure convergence of the polynomial is equivalent to that of the multilinear form, and to the square summability of the coefficients. Also, we present polynomial Khintchine inequalities for complex gaussian and Steinhaus variables. All these results have no analogues in the real case. Moreover, we study the Lp-convergence of random polynomials and derive certain decoupling inequalities without the usual tetrahedral hypothesis. We also consider convergence on “full subspaces” in the sense of Sjögren, both for real and complex random variables, and relate it to domination properties of the polynomial or the multilinear form, establishing a link with the theory of homogeneous polynomials on Banach spaces.     

Singular hypersurfaces and multilinear algebra

An algorithm for computing the discriminants of multilinear forms is given. A technique for computing the degeneracy conditions and other invariants ofn-ary forms is developed. An integral transform with respect to forms of degree 3 and higher (analog of the Gauss transform for quadratic forms) is introduced.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 45. Algebraic Geometry-8, 1997.     

The Gale transform and multi-graded determinantal schemes

Eisenbud and Popescu showed that certain finite determinantal subschemes of projective spaces defined by maximal minors of adjoint matrices of homogeneous linear forms are related by Veronese embeddings and a Gale transform. We extend this result to adjoint matrices of multihomogeneous multilinear forms. The subschemes now lie in products of projective spaces and the Veronese embeddings are replaced with Segre embeddings.     

Homogeneous and multilinear forms related to a theorem of Jordan and von Neumann

For vector spaces over certain algebraic fields, homogeneous functionals which are derivable from multilinear forms are characterized by an identity involving two-dimensional subspaces. The quadratic case is the “parallelogram law” which distinguishes Hilbert spaces among Banach spaces.     

Approximation numbers of nuclear and Hilbert-Schmidt multilinear forms defined on Hilbert spaces

We deal with multilinear forms t defined on Hilbert spaces and we investigate the relationships between nuclearity of t (respectively to be of Hilbert-Schmidt type) and summability properties of certain approximation numbers of t.     

Polynomials,Symmetric Multilinear Forms and Weak Compactness

In this paper we obtain some versions of weak compactness James theorem, replacing bounded linear functionals by polynomials and symmetric multilinear forms.