Symmetric Tensors And Geometry of {\mathbb{P}} ^N Subvarieties

This paper using a geometric approach produces vanishing and nonvanishing results concerning the spaces of twisted symmetric differentials on subvarieties , with k ≤ m. Emphasis is given to the case of k = m which is special and whose nonvanishing results on the dimensional range dim X > 2/3(N − 1) are related to the space of quadrics containing X and the variety of all tangent trisecant lines of X. The paper ends with an application showing that the twisted symmetric plurigenera, along smooth families of projective varieties X_t are not invariant even for α arbitrarily large. Received: September 2006, Revision: May 2007, Accepted: June 2007     

Intersecting subvarieties of $${G_m^n}$$ with algebraic subgroups

Let be an irreducible closed subvariety defined over . We bound the height of algebraic points on X that are in a certain sense close to the union of all algebraic subgroup of of dimension m < n/dim X. The bound we obtain is effective and will be expressed as a function of the height of X, the degree of X, and n. We then apply this bound to derive certain finiteness results if m is also strictly less than n − dim X.     

Amenability,tubularity, and embeddings into $$\mathcal{R}^{\omega}$$

Suppose M is a tracial von Neumann algebra embeddable into (the ultraproduct of the hyperfinite II₁-factor) and X is an n-tuple of selfadjoint generators for M. Denote by Γ(X; m, k, γ) the microstate space of X of order (m, k ,γ). We say that X is tubular if for any ε >  0 there exist and γ > 0 such that if then there exists a k × k unitary u satisfying for each 1 ≤  i ≤  n. We show that the following conditions are equivalent:

Holomorphic factorization of determinants of Laplacians on Riemann surfaces and a higher genus generalization of Kronecker’s first limit formula

For a family of compact Riemann surfaces X_t of genus g > 1, parameterized by the Schottky space we define a natural basis of which varies holomorphically with t and generalizes the basis of normalized abelian differentials of the first kind for n  =  1. We introduce a holomorphic function F(n) on which generalizes the classical product for n  =  1 and g  =  1. We prove the holomorphic factorization formula

The $${\overline{\partial}}$$-equation on homogeneous varieties with an isolated singularity

Let X be a regular irreducible variety in , Y the associated homogeneous variety in , and N the restriction of the universal bundle of to X. In the present paper, we compute the obstructions to solving the -equation in the L ^p -sense on Y for 1 ≤  p ≤  ∞ in terms of cohomology groups . That allows to identify obstructions explicitly if X is specified more precisely, for example if it is equivalent to or an elliptic curve.      

Error Correcting Sequence and Projective De Bruijn Graph

Let X be a finite set of q elements, and n, K, d be integers. A subset C ⊂ X ⁿ is an (n, K, d) error-correcting code, if #(C) = K and its minimum distance is d. We define an (n, K, d) error-correcting sequence over X as a periodic sequence {a _i } _i=0,1,... (a _i ∈ X) with period K, such that the set of all consecutive n-tuples of this sequence form an (n, K, d) error-correcting code over X. Under a moderate conjecture on the existence of some type of primitive polynomials, we prove that there is a error correcting sequence, such that its code-set is the q-ary Hamming code with 0 removed, for q > 2 being a prime power. For the case q = 2, under a similar conjecture, we prove that there is a error-correcting sequence, such that its code-set supplemented with 0 is the subset of the binary Hamming code [2 ^m  − 1, 2 ^m  − 1 − m, 3] obtained by requiring one specified coordinate being 0. Received: October 27, 2005. Final Version received: December 31, 2007     

Solutions of elliptic problems with nonlinearities of linear growth

In this paper, we study existence of nontrivial solutions to the elliptic equation

The <Emphasis Type="Italic">s</Emphasis>-energy of spherical designs on <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript>

This paper investigates the s-energy of (finite and infinite) well separated sequences of spherical designs on the unit sphere S ². A spherical n-design is a point set on S ² that gives rise to an equal weight cubature rule which is exact for all spherical polynomials of degree ≤n. The s-energy E _s (X) of a point set of m distinct points is the sum of the potential for all pairs of distinct points . A sequence Ξ = {X _m } of point sets X _m ⊂S ², where X _m has the cardinality card(X _m )=m, is well separated if for each pair of distinct points , where the constant λ is independent of m and X _m . For all s>0, we derive upper bounds in terms of orders of n and m(n) of the s-energy E _s (X _m(n)) for well separated sequences Ξ = {X _m(n)} of spherical n-designs X _m(n) with card(X _m(n))=m(n).      

On varieties parametrizing graded complex Lie algebras

We study spaces parametrizing graded complex Lie algebras from geometric as well as algebraic point of view. If R is a finite-dimensional complex Lie algebra, which is graded by a finite abelian group of order n, then a graded contraction of R, denoted by , is defined by a complex n × n-matrix , i, j = 1, . . . , n. In order for to be a Lie algebra, should satisfy certain homogeneous equations. In turn, these equations determine a projective variety X _R . We compute the first homology group of an irreducible component M of X _R , under some assumptions on M. We look into algebraic properties of graded Lie algebras where .      

The norm of products of free random variables

Let X _i denote free identically-distributed random variables. This paper investigates how the norm of products behaves as n approaches infinity. In addition, for positive X _i it studies the asymptotic behavior of the norm of where denotes the symmetric product of two positive operators: . It is proved that if EX _i = 1, then is between and c ₂ n for certain constant c ₁ and c ₂. For it is proved that the limit of exists and equals Finally, if π is a cyclic representation of the algebra generated by X _i , and if ξ is a cyclic vector, then for all n. These results are significantly different from analogous results for commuting random variables.     

On automorphisms of split metacyclic groups

Let D(m, n; k) be the semi-direct product of two finite cyclic groups and , where the action is given by yxy ⁻¹  =  x ^k . In particular, this includes the dihedral groups D _2m . We calculate the automorphism group Aut (D(m, n; k)).     

Positive bases in spaces of polynomials

For a nonempty compact set we determine the maximal possible dimension of a subspace of polynomial functions over Ω with degree at most m which possesses a positive basis. The exact value of this maximum depends on topological features of Ω, and we will see that in many of the cases m can be achieved. Whereas only for low m or finite sets Ω it is possible that we have a subspace X with positive basis and with dim X = m + 1. Hence there is no Ω for which a positive basis exists in for all . This work was accomplished during the 2^nd author’s stay in Paris under his Marie Curie fellowship, contract # MEIF-CT-2005-022927.     

Fundamental Solution of the Anisotropic Porous Medium Equation

We establish the existence of fundamental solutions for the anisotropic porous medium equation, ut = ∑n i=1（u^mi）xixi in R^n × （O,∞）, where m1,m2,..., and mn, are positive constants satisfying min1≤i≤n{mi}≤ 1, ∑i^n=1 mi 〉 n - 2, and max1≤i≤n{mi} ≤1/n（2 ＋ ∑i^n=1 mi）.     

Rates in the empirical bayes estimation problem with non-identical components

This paper considers empirical Bayes estimation of the mean θ of the univariate normal densityf ₀ with known variance where the sample sizesm(n) may vary with the component problems but remain bounded by <∞. Let {(θ _n ,X _n =(X _n,1,...,X _{n, m(n)} ))} be a sequence of independent random vectors where theθ _n are unobservable and iidG and, givenθ _n =θ has densityf _θ ^m(n) . The first part of the paper exhibits estimators for the density of and its derivative whose mean-squared errors go to zero with rates and respectively. LetR ^m(n+1)(G) denote the Bayes risk in the squared-error loss estimation ofθ _n+1 usingX _n+1. For given 0<a<1, we exhibitt _n (X₁,...,X _n ;X _n+1) such that . forn>1 under the assumption that the support ofG is in [0, 1]. Under the weaker condition that E[|θ|^2+γ]<∞ for some γ>0, we exhibitt _n ^* (X ₁,...,X _n ;X _n+1) such that forn>1.     

Zero Distribution of Composite Polynomials and Polynomials Biorthogonal to Exponentials

We analyze polynomials P _n that are biorthogonal to exponentials , in the sense that

Sums of random symmetric matrices and quadratic optimization under orthogonality constraints

Let B _i be deterministic real symmetric m × m matrices, and ξ _i be independent random scalars with zero mean and “of order of one” (e.g., ). We are interested to know under what conditions “typical norm” of the random matrix is of order of 1. An evident necessary condition is , which, essentially, translates to ; a natural conjecture is that the latter condition is sufficient as well. In the paper, we prove a relaxed version of this conjecture, specifically, that under the above condition the typical norm of S _N is : for all Ω > 0 We outline some applications of this result, primarily in investigating the quality of semidefinite relaxations of a general quadratic optimization problem with orthogonality constraints , where F is quadratic in X = (X ₁,... ,X _k ). We show that when F is convex in every one of X _j , a natural semidefinite relaxation of the problem is tight within a factor slowly growing with the size m of the matrices . Research was partly supported by the Binational Science Foundation grant #2002038.     

On the action of Lipschitz functions on vector-valued random sums

Let X be a Banach space and let (ξ_j)_j ≧ 1 be an i.i.d. sequence of symmetric random variables with finite moments of all orders. We prove that the following assertions are equivalent:

On some homotopical and homological properties of monoid presentations

If a monoid S is given by some finite complete presentation ℘, we construct inductively a chain of CW-complexes