On Projections of Semi-Algebraic Sets Defined by Few Quadratic Inequalities

Let S⊂ℝ ^k+m be a compact semi-algebraic set defined by P ₁≥0,…,P _ℓ ≥0, where P _i ∈ℝ[X ₁,…,X _k ,Y ₁,…,Y _m ], and deg (P _i )≤2, 1≤i≤ℓ. Let π denote the standard projection from ℝ ^k+m onto ℝ ^m . We prove that for any q>0, the sum of the first q Betti numbers of π(S) is bounded by (k+m) ^{O(q ℓ)}. We also present an algorithm for computing the first q Betti numbers of π(S), whose complexity is . For fixed q and ℓ, both the bounds are polynomial in k+m. The author was supported in part by an NSF Career Award 0133597 and a Sloan Foundation Fellowship.     

Discrete Entropies of Orthogonal Polynomials

Given a nontrivial Borel measure on ℝ, let p _n be the corresponding orthonormal polynomial of degree n whose zeros are λ _j ⁽ⁿ⁾, j=1,…,n. Then for each j=1,…,n,

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→∞.      

On the Number of Topological Types Occurring in a Parameterized Family of Arrangements

Let be an o-minimal structure over ℝ, a closed definable set, and

Optimisation of Measures on a Hyperfinite Adapted Probability Space

The minimisation problem for a functional is considered, where is an ℝ ⁿ -valued stochastic process, defined on some filtered probability space , and P is an admissible probability measure in the sense that it obeys (1) some uniform equivalence condition with respect to the given measure ℙ on Γ, and (2) a finite number (possibly zero) of arbitrarily given other conditions that require the expectation (with respect to P) of some continuous bounded function φ of , for t ₁,…,t _k ∈[0,1], to lie within some closed set. We assume that u can be formulated through finite compositions of conditional expectations and bounded continuous functions. Under the assumption of |φ| being uniformly bounded from below and some condition on the dimension of , the existence of a solution on hyperfinite adapted probability spaces, as well as its minimality among admissible measures on any other adapted probability space, is proven. Also, a coarseness result for the Loeb operation is established. The main result of this paper, however, is a “standard result”: It does not include any reference to nonstandard analysis and can be perfectly understood without any familiarity with nonstandard analysis.      

Iterated Logarithm Law for Anticipating Stochastic Differential Equations

We prove a functional law of iterated logarithm for the following kind of anticipating stochastic differential equations

Point Counting in Families of Hyperelliptic Curves

Let E _Γ be a family of hyperelliptic curves defined by , where is defined over a small finite field of odd characteristic. Then with in an extension degree n field over this small field, we present a deterministic algorithm for computing the zeta function of the curve by using Dwork deformation in rigid cohomology. The time complexity of the algorithm is and it needs bits of memory. A slight adaptation requires only space, but costs time . An implementation of this last result turns out to be quite efficient for n big enough. H. Hubrechts is a Research Assistant of the Research Foundation–Flanders (FWO–Vlaanderen).     

Freeknot splines approximation of Sobolev-type classes of <Emphasis Type="Italic">s</Emphasis>-monotone functions

Let I be a finite interval, s ∈ ℕ₀, and r,ν,n ∈ ℕ. Given a set M, of functions defined on I, denote by M the subset of all functions y ∈ M such that the s-difference is nonnegative on I, ∀τ > 0. Further, denote by the Sobolev class of functions x on I with the seminorm . Also denote by Σ _ν,n , the manifold of all piecewise polynomials of order ν and with n – 1 knots in I. If ν ≥ max {r,s}, 1 ≤ p,q ≤ ∞, and (r,p,q) ≠ (1,1,∞), then we give exact orders of the best unconstrained approximation and of the best s-monotonicity preserving approximation . Part of this work was done while the first author visited Tel Aviv University in May 2003 and in March 2004.     

Generic initial ideals and exterior algebraic shifting of the join of simplicial complexes

In this paper, the relation between algebraic shifting and join which was conjectured by Eran Nevo will be proved. Let σ and τ be simplicial complexes and σ*τ be their join. Let J _σ be the exterior face ideal of σ and Δ(σ) the exterior algebraic shifted complex of σ. Assume that σ*τ is a simplicial complex on [n]={1,2,...,n}. For any d-subset S⊂[n], let denote the number of d-subsets R∈σ which are equal to or smaller than S with respect to the reverse lexicographic order. We will prove that for all S⊂[n]. To prove this fact, we also prove that for all S⊂[n] and for all nonsingular matrices ϕ, where Δ_ϕ(σ) is the simplicial complex defined by .     

Functoriality of automorphic <Emphasis Type="Italic">L</Emphasis>-functions through their zeros

Let E be a Galois extension of ℚ of degree l, not necessarily solvable. In this paper we first prove that the L-function L(s, π) attached to an automorphic cuspidal representation π of cannot be factored nontrivially into a product of L-functions over E. Next, we compare the n-level correlation of normalized nontrivial zeros of L(s, π₁)…L(s, π _k ), where π _j , j = 1,…, k, are automorphic cuspidal representations of , with that of L(s,π). We prove a necessary condition for L(s, π) having a factorization into a product of L-functions attached to automorphic cuspidal representations of specific , j = 1,…,k. In particular, if π is not invariant under the action of any nontrivial σ ∈ Gal _E/ℚ, then L(s, π) must equal a single L-function attached to a cuspidal representation of and π has an automorphic induction, provided L(s, π) can factored into a product of L-functions over ℚ. As E is not assumed to be solvable over ℚ, our results are beyond the scope of the current theory of base change and automorphic induction. Our results are unconditional when m,m ₁,…,m _k are small, but are under Hypothesis H and a bound toward the Ramanujan conjecture in other cases. The first author was supported by the National Basic Research Program of China, the National Natural Science Foundation of China (Grant No. 10531060), and Ministry of Education of China (Grant No. 305009). The second author was supported by the National Security Agency (Grant No. H98230-06-1-0075). The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein     

The Inverse Resonance Problem for Hermite Operators

In this paper the inverse resonance problem for the Hermite operator is investigated. The Hermite operator with the creation operator , the annihilation operator , and a finitely supported multiplication operator b, is an unbounded operator on ℓ ²(ℕ₀) having finitely many eigenvalues and infinitely many resonances (except for b=0, when there are no eigenvalues or resonances). It is shown that knowing the location of eigenvalues and resonances determines the potential b uniquely.      

Dual Wavelet Frames and Riesz Bases in Sobolev Spaces

This paper generalizes the mixed extension principle in L ₂(ℝ ^d ) of (Ron and Shen in J. Fourier Anal. Appl. 3:617–637, 1997) to a pair of dual Sobolev spaces H ^s (ℝ ^d ) and H ^−s (ℝ ^d ). In terms of masks for φ,ψ ¹,…,ψ ^L ∈H ^s (ℝ ^d ) and , simple sufficient conditions are given to ensure that (X ^s (φ;ψ ¹,…,ψ ^L ), forms a pair of dual wavelet frames in (H ^s (ℝ ^d ),H ^−s (ℝ ^d )), where

Invariant subspaces for operators in a general II<Subscript>1</Subscript>-factor

Let ℳ be a von Neumann factor of type II₁ with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μ _T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace affiliated with ℳ, such that the Brown measure of is concentrated on B and the Brown measure of is concentrated on ℂ∖B. Moreover, is T-hyperinvariant and the trace of is equal to μ _T(B). In particular, if T∈ℳ has a Brown measure which is not concentrated on a singleton, then there exists a non-trivial, closed, T-hyperinvariant subspace. Furthermore, it is shown that for every T∈ℳ the limit exists in the strong operator topology, and the projection onto is equal to 1_[0,r](A), for every r>0. Supported by The Danish National Research Foundation.     

Central polynomials and matrix invariants

LetK be a field, charK=0 andM _n (K) the algebra ofn×n matrices overK. If λ=(λ₁,…,λ _m ) andμ=(μ ₁,…,μ _m ) are partitions ofn ² let wherex ₁,…,x _n ²,y ₁,…,y _n ² are noncommuting indeterminates andS _n ² is the symmetric group of degreen ². The polynomialsF ^{λ, μ} , when evaluated inM _n (K), take central values and we study the problem of classifying those partitions λ,μ for whichF ^{λ, μ} is a central polynomial (not a polynomial identity) forM _n (K). We give a formula that allows us to evaluateF ^{λ, μ} inM(K) in general and we prove that if λ andμ are not both derived in a suitable way from the partition δ=(1, 3,…, 2n−3, 2n−1), thenF ^{λ, μ} is a polynomial identity forM _n (K). As an application, we exhibit a new class of central polynomials forM _n (K). In memory of Shimshon Amitsur Research supported by a grant from MURST of Italy.     

A Characterization of Quadric Constant Mean Curvature Hypersurfaces of Spheres

Let be an immersion of a complete n-dimensional oriented manifold. For any v∈ℝ ⁿ⁺², let us denote by ℓ _v :M→ℝ the function given by ℓ _v (x)=〈φ(x),v〉 and by f _v :M→ℝ, the function given by f _v (x)=〈ν(x),v〉, where is a Gauss map. We will prove that if M has constant mean curvature, and, for some v≠0 and some real number λ, we have that ℓ _v =λ f _v , then, φ(M) is either a totally umbilical sphere or a Clifford hypersurface. As an application, we will use this result to prove that the weak stability index of any compact constant mean curvature hypersurface M ⁿ in which is neither totally umbilical nor a Clifford hypersurface and has constant scalar curvature is greater than or equal to 2n+4. A. Brasil Jr. was partially supported by CNPq, Brazil, 306626/2007-1.     

On the Solution Existence of Generalized Quasivariational Inequalities with Discontinuous Multifunctions

We study the following generalized quasivariational inequality problem: given a closed convex set X in a normed space E with the dual E ^*, a multifunction and a multifunction Γ:X→2 ^X , find a point such that , . We prove some existence theorems in which Φ may be discontinuous, X may be unbounded, and Γ is not assumed to be Hausdorff lower semicontinuous. The authors express their sincere gratitude to the referees for helpful suggestions and comments. This research was partially supported by a grant from the National Science Council of Taiwan, ROC. B.T. Kien was on leave from National University of Civil Engineering, Hanoi, Vietnam.     

Wavelet Bases for Confinements of the Infrared Divergence

We propose the construction of wavelet bases with pseudo-polynomials adapted to the homogeneous Sobolev spaces , s−n/2∈ℕ. They provide a confinement of the infrared divergence by decomposing as a direct sum X ⊕ Y where X is a “small” space which carries the divergence and Y can be embedded in . In the case of we also construct such an orthonormal basis, which provides a confinement of the Mumford process.     

Module Arens regularity for semigroup algebras

We extend the concept of Arens regularity of a Banach algebra to the case that there is an -module structure on , and show that when S is an inverse semigroup with totally ordered subsemigroup E of idempotents, then A=ℓ ¹(S) is module Arens regular if and only if an appropriate group homomorphic image of S is finite. When S is a discrete group, this is just Young’s theorem which asserts that ℓ ¹(S) is Arens regular if and only if S is finite. An erratum to this article can be found at     

Map Preserving Local Properties of a Digital Image

In order to study digital topological properties of a k-surface in Z ⁿ , we generalize the topological number in Bertrand (Pattern Recogn. Lett. 15:1003–1011, 1994). Furthermore, we show that a local (k ₀,k ₁)-isomorphism preserves some digital-topological properties, such as a generalized topological number and a simple k ₀-point, and prove that a local (k ₀,k ₁)-isomorphism takes a simple k ₀-surface in into a simple k ₁-surface in .      

The extension of the operation in the Stone-Čech compactification of semigroups

For a large class of locally compact semitopological semigroups S, the Stone-Čech compactification β S is a semigroup compactification if and only if S is either discrete or countably compact. Furthermore, for this class of semigroups which are neither discrete nor countably compact, the quotient contains a linear isometric copy of ℓ _∞. These results improve theorems by Baker and Butcher and by Dzinotyiweyi.