Universal lattices and property tau

We prove that the universal lattices – the groups G=SL_d(R) where R=ℤ[x₁,...,x_k], have property τ for d≥3. This provides the first example of linear groups with τ which do not come from arithmetic groups. We also give a lower bound for the τ-constant with respect to the natural generating set of G. Our methods are based on bounded elementary generation of the finite congruence images of G, a generalization of a result by Dennis and Stein on K₂ of some finite commutative rings and a relative property T of . Mathematics Subject Classification (2000) 20F69, 13M05, 19C20, 20G05, 20H05     

Quaternary universal forms over \mathbf{Q}(\sqrt{13})

Let be a real quadratic field over Q with m a square-free positive rational integer and be the integer ring in F. A totally positive definite integral n-ary quadratic form f=f(x ₁,…,x _n )=∑_1≤i,j≤n α _ij x _i x _j ( ) is called universal if f represents all totally positive integers in . Chan, Kim and Raghavan proved that ternary universal forms over F exist if and only if m=2,3,5 and determined all such forms. There exists no ternary universal form over real quadratic fields whose discriminants are greater than 12. In this paper we prove that there are only two quaternary universal forms (up to equivalence) over . For the proof of universality we apply the theory of quadratic lattices.      

Estimates for functionals by deviations of the Riesz sums in seminormed spaces

Suppose that (X, p) is a sermonized space, is a linearly independent system of elements in X, is a sequence of linear bounded functionals such that c _k (x _l ) = δ _kl ,

Convergence of solutions of stochastic differential equations to the Arratia flow

We consider the solution x _ε of the equation

INJECTIVE MAPS ON PRIMITIVE SEQUENCES OVER Z/（p^e）

Let Z/(pe) be the integer residue ring modulo pe with p an odd prime and integer e ≥ 3. For a sequence (a) over Z/(pe), there is a unique p-adic decomposition (a) = (a)0 (a)1·p … (a)e-1 ·pe-1, where each (a)i can be regarded as a sequence over Z/(p), 0 ≤ i ≤ e - 1. Let f(x) be a primitive polynomial over Z/(pe) and G' (f(x), pe) the set of all primitive sequences generated by f(x) over Z/(pe). For μ(x) ∈ Z/(p)[x] with deg(μ(x)) ≥ 2 and gcd(1 deg(μ(x)),p- 1) = 1,set ψe-1 (x0, x1,…, xe-1) = xe-1·[ μ(xe-2) ηe-3 (x0, x1,…, xe-3)] ηe-2 (x0, x1,…, xe-2),which is a function of e variables over Z/(p). Then the compressing map ψe-1: G'(f(x),pe) → (Z/(p))∞,(a) (→)ψe-1((a)0, (a)1,… ,(a)e-1) is injective. That is, for (a), (b) ∈ G' (f(x), pe), (a) = (b) if and only if ψe - 1 ((a)0, (a)1,… , (a)e - 1) =ψe - 1 ((b)0,(b)1,… ,(b)e-1). As for the case of e = 2, similar result is also given. Furthermore, if functions ψe-1 and ψe-1 over Z/(p) are both of the above form and satisfy ψe-1((a)0,(a)1,… ,(a)e-1) = ψe-1((b)0,(b)1,… ,(b)e-1) for (a),(b) ∈ G'(f(x),pe), the relations between (a) and (b), ψe-1 and ψe-1 are discussed.     

On the Posets ( \user1Wk2 , < ){\left( {{\user1{\mathcal{W}}}^{k}_{2} , < } \right)} and their Connections with Some Homogeneous Inequalities of Degree 2

A class of ranked posets {(D _h ^k , ≪)} has been recently defined in order to analyse, from a combinatorial viewpoint, particular systems of real homogeneous inequalities between monomials. In the present paper we focus on the posets D ₂ ^k , which are related to systems of the form {x _a x _b * _abcd x _c x _d : 0 ≤ a, b, c, d ≤ k, * _abcd ∈ {<, >}, 0 < x ₀ < x ₁ < ...< x _k}. As a consequence of the general theory, the logical dependency among inequalities is adequately captured by the so-defined posets . These structures, whose elements are all the D ₂ ^k 's incomparable pairs, are thoroughly surveyed in the following pages. In particular, their order ideals – crucially significant in connection with logical consequence – are characterised in a rather simple way. In the second part of the paper, a class of antichains is shown to enjoy some arithmetical properties which make it an efficient tool for detecting incompatible systems, as well as for posing some compatibility questions in a purely combinatorial fashion.     

Isometric Group Actions on Banach Spaces and Representations Vanishing at Infinity

Our main result is that the simple Lie group G = Sp(n, 1) acts metrically properly isometrically on L ^p (G) if p > 4n + 2. To prove this, we introduce Property , with V being a Banach space: a locally compact group G has Property if every affine isometric action of G on V, such that the linear part is a C ₀-representation of G, either has a fixed point or is metrically proper. We prove that solvable groups, connected Lie groups, and linear algebraic groups over a local field of characteristic zero, have Property . As a consequence, for unitary representations, we characterize those groups in the latter classes for which the first cohomology with respect to the left regular representation on L ²(G) is nonzero; and we characterize uniform lattices in those groups for which the first L2-Betti number is nonzero.      

On Some Properties of Randomly Indexed Sequences of Random Elements

Let be a probability space with a nonatomic measure P and let (S,ρ) be a separable complete metric space. Let {N _n ,n≥1} be an arbitrary sequence of positive-integer valued random variables. Let {F _k ,k≥1} be a family of probability laws and let X be some random element defined on and taking values in (S,ρ). In this paper we present necessary and sufficient conditions under which one can construct an array of random elements {X _n,k ,n,k≥1} defined on the same probability space and taking values in (S,ρ), and such that , and moreover as  n→∞. Furthermore, we consider the speed of convergence to X as n→∞.      

Bounded universal functions for sequences of holomorphic self-maps of the disk

We give several characterizations of those sequences of holomorphic self-maps {φ _n } _n≥1 of the unit disk for which there exists a function F in the unit ball of H ^∞ such that the orbit {F∘φ _n :n∈ℕ} is locally uniformly dense in . Such a function F is said to be a -universal function. One of our conditions is stated in terms of the hyperbolic derivatives of the functions φ _n . As a consequence we will see that if φ _n is the nth iterate of a map φ of into , then {φ _n } _n≥1 admits a -universal function if and only if φ is a parabolic or hyperbolic automorphism of . We show that whenever there exists a -universal function, then this function can be chosen to be a Blaschke product. Further, if there is a -universal function, we show that there exist uniformly closed subspaces consisting entirely of universal functions.     

Equivariant vector bundles on Drinfeld’s upper half space

Let be Drinfeld’s upper half space over a finite extension K of ℚ _p . We construct for every GL _d+1-equivariant vector bundle on ℙ ^d _K , a GL _d+1(K)-equivariant filtration by closed subspaces on the K-Fréchet . This gives rise by duality to a filtration by locally analytic GL _d+1(K)-representations on the strong dual . The graded pieces of this filtration are locally analytic induced representations from locally algebraic ones with respect to maximal parabolic subgroups. This paper generalizes the cases of the canonical bundle due to Schneider and Teitelbaum [ST1] and that of the structure sheaf by Pohlkamp [P].     

On the minimum length of some linear codes

We determine the minimum length n _q (k, d) for some linear codes with k ≥ 5 and q ≥ 3. We prove that n _q (k, d) = g _q (k, d) + 1 for when k is odd, for when k is even, and for . This work was supported by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD). (KRF-2005-214-C00175). This research has been partially supported by Grant-in-Aid for Scientific Research of Japan Society for the Promotion of Science under Contract Number 17540129.     

( m,n)-Equidista nt Sets i n $\ mathbb{R}^{k},\ mathbb{S}^{k}$ , a nd ℙk

We call a metric space X (m,n)-equidistant if, when A⊆X has exactly m points, there are exactly n points in X each of which is equidistant from (the points of) A. We prove that, for k≥2, the Euclidean space ℝ ^k contains an (m,1)-equidistant set if and only if k≥m. Although the sphere is (3,2)-equidistant, and ℝ⁴ contain no (4,2)-equidistant sets. We discuss related results about projective spaces, and state a conjecture about analogous to the Double Midset Conjecture.     

Traversing a Set of Points with a Minimum Number of Turns

Given a finite set of points S in ℝ ^d , consider visiting the points in S with a polygonal path which makes a minimum number of turns, or equivalently, has the minimum number of segments (links). We call this minimization problem the minimum link spanning path problem. This natural problem has appeared several times in the literature under different variants. The simplest one is that in which the allowed paths are axis-aligned. Let L(S) be the minimum number of links of an axis-aligned path for S, and let G _n ^d be an n×…×n grid in ℤ ^d . Kranakis et al. (Ars Comb. 38:177–192, 1994) showed that L(G _n ²)=2n−1 and and conjectured that, for all d≥3, We prove the conjecture for d=3 by showing the lower bound for L(G _n ³). For d=4, we prove that For general d, we give new estimates on L(G _n ^d ) that are very close to the conjectured value. The new lower bound of improves previous result by Collins and Moret (Inf. Process. Lett. 68:317–319, 1998), while the new upper bound of differs from the conjectured value only in the lower order terms. For arbitrary point sets, we include an exact bound on the minimum number of links needed in an axis-aligned path traversing any planar n-point set. We obtain similar tight estimates (within 1) in any number of dimensions d. For the general problem of traversing an arbitrary set of points in ℝ ^d with an axis-aligned spanning path having a minimum number of links, we present a constant ratio (depending on the dimension d) approximation algorithm. Work by A. Dumitrescu was partially supported by NSF CAREER grant CCF-0444188. Work by F. Hurtado was partially supported by projects MECMTM2006-01267 and Gen. Cat. 2005SGR00692. Work by P. Valtr was partially supported by the project 1M0545 of the Ministry of Education of the Czech Republic.     

On pair correlations and Hausdorff dimension

We consider a system of “generalised linear forms” defined at a point x = (x _{(i, j)}) in a subset of R ^d by

On Convergence and Convolutions of Random Signed Measures

Let μ _n be a sequence of random finite signed measures on the locally compact group G equal to either or ℝ ^d . We give weak conditions on the sequence μ _n and on functions K such that the convolution product μ _n *K, and its derivatives, converge in law, in probability, or almost surely in the Banach spaces or L ^p (G). Examples for sequences μ _n covered are the empirical process (possibly arising from dependent data) and also random signed measures where is some (nonparametric) estimator for the measure ℙ, including the usual kernel and wavelet based density estimators with MISE-optimal bandwidths. As a statistical application, we apply the results to study convolutions of density estimators.      

On Sums of Two Coprime k-th Powers(II)

For fixed k ≥ 3, let E_k(x) denote the error term of the sum , where 1. It is proved that if the Riemann hypothesis is true, then , . A short interval result is also obtained.     

A Centrally Symmetric Version of the Cyclic Polytope

We define a centrally symmetric analogue of the cyclic polytope and study its facial structure. We conjecture that our polytopes provide asymptotically the largest number of faces in all dimensions among all centrally symmetric polytopes with n vertices of a given even dimension d=2k when d is fixed and n grows. For a fixed even dimension d=2k and an integer 1≤j<k we prove that the maximum possible number of j-dimensional faces of a centrally symmetric d-dimensional polytope with n vertices is at least for some c _j (d)>0 and at most as n grows. We show that c ₁(d)≥1−(d−1)⁻¹ and conjecture that the bound is best possible. Research of A. Barvinok partially supported by NSF grant DMS 0400617. Research of I. Novik partially supported by Alfred P. Sloan Research Fellowship and NSF grant DMS-0500748.     

The Mixed Norm Spaces of Polyharmonic Functions

Let be a bounded domain with C ² boundary. And let H _k be the set of all polyharmonic functions f with order k on Ω. For 0<p, q≤∞ and ϕ a normal weight, the mixed-norm space consists of all function f in H _k for which the mixed-norm ||·|| _{p, q, ϕ} <∞. The main result of the paper is the norm equivalence: