Restriction theorems for homogeneous bundles

We prove that for an irreducible representation , the associated homogeneous -vector bundle Wτ is strongly semistable when restricted to any smooth quadric or to any smooth cubic in , where k is an algebraically closed field of characteristic ≠2,3 respectively. In particular Wτ is semistable when restricted to general hypersurfaces of degree?2 and is strongly semistable when restricted to the generic hypersurface of degree?2.     

Limit theorems for polylinear forms

The limit theorems for polylinear forms are obtained. Conditions are found under which the distribution of the polylinear form of many random variables is essentially the same as if all the distributions of arguments were normal.     

Compactness theorems for differential forms

We prove compactness properties of various sets of differential forms with bounds on their exterior derivatives. This gives simple proofs of the Federer-Fleming result on normal currents and of ?compensated compactness”? lemmas.     

Nuttall-Pommerenke theorems for homogeneous Padé approximants

We investigate Nuttall-Pommerenke theorems for several variable homogeneous Padé approximants using ideas of Goncar, Karlsson and Wallin.     

Eigenvalue distribution theorems for certain homogeneous spaces

A classical theorem of Szegö describes the limiting behavior of the eigenvalues of P_nAP_n, where A is a multiplication operator on L₂(S¹) and P_n is the projection on the subspace spanned by e^ikθ (0 ? k ? n). A generalization of this is proved, whereby S¹ is replaced by an arbitrary rank one homogeneous space (e.g. S^m), A by a pseudodifferential operator of order zero, and P_n by a sequence of spectral projections for the Laplace-Beltrami operator. A by-product is an alternate proof of a theorem of A. Weinstein on the eigenvalue clusters of the Laplacian plus a potential.     

Structure theorems over polynomial rings

Given a polynomial ring R over a field k and a finite group G, we consider a finitely generated graded RG-module S. We regard S as a kG-module and show that various conditions on S are equivalent, such as only containing finitely many isomorphism classes of indecomposable summands or satisfying a structure theorem in the sense of [D. Karagueuzian, P. Symonds, The module structure of a group action on a polynomial ring: A finiteness theorem, preprint, http://www.ma.umist.ac.uk/pas/preprints].     

Distortion theorems for convex mappings on homogeneous balls

Let B be the open unit ball of a complex Banach space X and let B be homogeneous. We prove distortion results for normalized convex mappings f:B→X which generalize various finite dimensional distortion theorems and improve some infinite dimensional ones. In particular, our results are valid for the open unit balls of complex Hilbert spaces and the Cartan domains.     

Generalized incidence theorems, homogeneous forms and sum–product estimates in finite fields

In recent years, sum–product estimates in Euclidean space and finite fields have received great attention. They can often be interpreted in terms of Erdős type incidence problems involving the distribution of distances, dot products, areas, and so on, which have been studied quite extensively by way of combinatorial and Fourier analytic techniques. We use both kinds of techniques to obtain sharp or near-sharp results on the distribution of volumes (as examples of d-linear homogeneous forms) determined by sufficiently large subsets of vector spaces over finite fields and the associated arithmetic expressions. Arithmetic–combinatorial techniques turn out to be optimal for dimension d≥4 to this end, while for d=3 they have failed to provide us with a result that follows from the analysis of exponential sums. To obtain the latter result we prove a relatively straightforward function version of an incidence results for points and planes previously established in [D. Hart, A. Iosevich, Sums and products in finite fields: An integral geometric viewpoint, in: Radon Transforms, Geometry, and Wavelets, Contemp. Math. 464 (2008); D. Hart, A. Iosevich, D. Koh, M. Rudnev, Averages over hyperplanes, sum–product theory in vector spaces over finite fields and the Erdős–Falconer distance conjecture, arXiv:math/0711.4427, preprint 2007].More specifically, we prove that if E=A××A is a product set in , d≥4, the d-dimensional vector space over a finite field , such that the size |E| of E exceeds (i.e. the size of the generating set A exceeds ) then the set of volumes of d-dimensional parallelepipeds determined by E covers . This result is sharp as can be seen by taking , a prime sub-field of its quadratic extension , with q=p². For in three dimensions, however, we are able to establish the same result only if (i.e., , for some C; in fact, the bound can be justified for a slightly wider class of “Cartesian product-like” sets), and this uses Fourier methods. Yet we do prove a weaker near-optimal result in three dimensions: that the set of volumes generated by a product set E=A×A×A covers a positive proportion of if (so ). Besides, without any assumptions on the structure of E, we show that in three dimensions the set of volumes covers a positive proportion of if |E|≥Cq², which is again sharp up to the constant C, as taking E to be a 2-plane through the origin shows.     

Distributions of polynomial forms

Institute of Mathematics and Cybernetics of the Lithunian Academy of Sciences. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 31, No. 2, pp. 242–257, April–June, 1991.     

Approximation algorithms for homogeneous polynomial optimization with quadratic constraints

In this paper, we consider approximation algorithms for optimizing a generic multi-variate homogeneous polynomial function, subject to homogeneous quadratic constraints. Such optimization models have wide applications, e.g., in signal processing, magnetic resonance imaging (MRI), data training, approximation theory, and portfolio selection. Since polynomial functions are non-convex, the problems under consideration are all NP-hard in general. In this paper we shall focus on polynomial-time approximation algorithms. In particular, we first study optimization of a multi-linear tensor function over the Cartesian product of spheres. We shall propose approximation algorithms for such problem and derive worst-case performance ratios, which are shown to be dependent only on the dimensions of the model. The methods are then extended to optimize a generic multi-variate homogeneous polynomial function with spherical constraint. Likewise, approximation algorithms are proposed with provable approximation performance ratios. Furthermore, the constraint set is relaxed to be an intersection of co-centered ellipsoids; namely, we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin, and propose polynomial-time approximation algorithms with provable worst-case performance ratios. Numerical results are reported, illustrating the effectiveness of the approximation algorithms studied.     

Eigenvalue analysis of constrained minimization problem for homogeneous polynomial

In this paper, the concepts of Pareto H-eigenvalue and Pareto Z-eigenvalue are introduced for studying constrained minimization problem and the necessary and sufficient conditions of such eigenvalues are given. It is proved that a symmetric tensor has at least one Pareto H-eigenvalue (Pareto Z-eigenvalue). Furthermore, the minimum Pareto H-eigenvalue (or Pareto Z-eigenvalue) of a symmetric tensor is exactly equal to the minimum value of constrained minimization problem of homogeneous polynomial deduced by such a tensor, which gives an alternative methods for solving the minimum value of constrained minimization problem. In particular, a symmetric tensor \({\mathcal {A}}\) is strictly copositive if and only if every Pareto H-eigenvalue (Z-eigenvalue) of \({\mathcal {A}}\) is positive, and \({\mathcal {A}}\) is copositive if and only if every Pareto H-eigenvalue (Z-eigenvalue) of \({\mathcal {A}}\) is non-negative.