Dimension Gaps between Representability and Collapsibility

A simplicial complex K\mathsf{K} is called d -representable if it is the nerve of a collection of convex sets in ℝ ^d ; K\mathsf{K} is d -collapsible if it can be reduced to an empty complex by repeatedly removing a face of dimension at most d−1 that is contained in a unique maximal face; and K\mathsf{K} is d -Leray if every induced subcomplex of K\mathsf{K} has vanishing homology of dimension d and larger. It is known that d-representable implies d-collapsible implies d-Leray, and no two of these notions coincide for d≥2. The famous Helly theorem and other important results in discrete geometry can be regarded as results about d-representable complexes, and in many of these results, “d-representable” in the assumption can be replaced by “d-collapsible” or even “d-Leray.”     

Finite elements approximation of second order linear elliptic equations in divergence form with right-hand side in L 1

In this paper we consider, in dimension d≥ 2, the standard finite elements approximation of the second order linear elliptic equation in divergence form with coefficients in L ^∞(Ω) which generalizes Laplace’s equation. We assume that the family of triangulations is regular and that it satisfies an hypothesis close to the classical hypothesis which implies the discrete maximum principle. When the right-hand side belongs to L ¹(Ω), we prove that the unique solution of the discrete problem converges in (for every q with ) to the unique renormalized solution of the problem. We obtain a weaker result when the right-hand side is a bounded Radon measure. In the case where the dimension is d = 2 or d = 3 and where the coefficients are smooth, we give an error estimate in when the right-hand side belongs to L ^r (Ω) for some r > 1.     

Radial Fourier multipliers in high dimensions

Given a fixed p≠2, we prove a simple and effective characterization of all radial multipliers of FL^p( \mathbbR^d ) \mathcal{F}{L^p}\left( {{\mathbb{R}^d}} \right) , provided that the dimension d is sufficiently large. The method also yields new L ^q space-time regularity results for solutions of the wave equation in high dimensions.     

On the hot spots of a catalytic super-Brownian motion

Consider the catalytic super-Brownian motion X ^ϱ (reactant) in ℝ ^d , d≤3, which branching rates vary randomly in time and space and in fact are given by an ordinary super-Brownian motion ϱ (catalyst). Our main object of study is the collision local time L = L _[ϱ,Xϱ] (d(s,x) )of catalyst and reactant. It determines the covariance measure in themartingale problem for X ^ϱ and reflects the occurrence of “hot spots” of reactant which can be seen in simulations of X ^ϱ. In dimension 2, the collision local time is absolutely continuous in time, L(d(s,x) ) = ds K _s (dx). At fixed time s, the collision measures K _s (dx) of ϱ _s and X _s ^ϱ have carrying Hausdorff dimension 2. Spatial marginal densities of L exist, and, via self-similarity, enter in the long-term randomergodic limit of L (diffusiveness of the 2-dimensional model). We alsocompare some of our results with the case of super-Brownian motions withdeterministic time-independent catalysts. Received: 2 December 1998 / Revised version: 2 February 2001 / Published online: 9 October 2001     

Extension theorems for paraboloids in the finite field setting

In this paper we study the L ^p ? L ^r boundedness of the extension operators associated with paraboloids in ${{\mathbb F}_{q}^{d}}In this paper we study the L ^p − L ^r boundedness of the extension operators associated with paraboloids in \mathbb F_q^d{{\mathbb F}_{q}^{d}} , where \mathbbF_q{\mathbb{F}_{q}} is a finite field of q elements. In even dimensions d ≥ 4, we estimate the number of additive quadruples in the subset E of the paraboloids, that is the number of quadruples (x,y,z,w) ? E⁴{(x,y,z,w) \in E^4} with x + y = z+w. As a result, in higher even dimensions, we obtain the sharp range of exponents p for which the extension operator is bounded, independently of q, from L ^p to L ⁴ in the case when −1 is a square number in \mathbbF_q{\mathbb{F}_{q}} . Using the sharp L ^p −L ⁴ result, we improve upon the range of exponents r, for which the L ² − L ^r estimate holds, obtained by Mockenhaupt and Tao (Duke Math 121:35–74, 2004) in even dimensions d ≥ 4. In addition, assuming that −1 is not a square number in \mathbbF_q{\mathbb{F}_{q}}, we extend their work done in three dimension to specific odd dimensions d ≥ 7. The discrete Fourier analytic machinery and Gauss sum estimates make an important role in the proof.     

The Probability that a Convex Body Intersects the Integer Lattice in a <Emphasis Type="Italic">k</Emphasis>-dimensional Set

Let K be a convex body in ℝ ^d . It is known that there is a constant C ₀ depending only on d such that the probability that a random copy ρ(K) of K does not intersect ℤ ^d is smaller than \fracC₀|K|\frac{C_{0}}{|K|} and this is best possible. We show that for every k<d there is a constant C such that the probability that ρ(K) contains a subset of dimension k is smaller than \fracC|K|\frac{C}{|K|}. This is best possible if k=d−1. We conjecture that this is not best possible in the rest of the cases; if d=2 and k=0 then we can obtain better bounds. For d=2, we find the best possible value of C ₀ in the limit case when width(K)→0 and |K|→∞.     

Hardy spaces associated to the Schrödinger operator on strongly Lipschitz domains of {\mathbb{R}^d}

Let L = ?Δ + V be a Schrödinger operator and Ω be a strongly Lipschitz domain of ${\mathbb R^{d}}Let L = −Δ + V be a Schr?dinger operator and Ω be a strongly Lipschitz domain of \mathbb R^d{\mathbb R^{d}} , where Δ is the Laplacian on \mathbb R^d{\mathbb R^{d}} and the potential V is a nonnegative polynomial on \mathbb R^d{\mathbb R^{d}} . In this paper, we investigate the Hardy spaces on Ω associated to the Schr?dinger operator L.     

A bilinear estimate for biharmonic functions in Lipschitz domains

We show that a bilinear estimate for biharmonic functions in a Lipschitz domain Ω is equivalent to the solvability of the Dirichlet problem for the biharmonic equation in Ω. As a result, we prove that for any given bounded Lipschitz domain Ω in _boxclose^d{\mathbb{R}^{d}} and 1 < q < ∞, the solvability of the L ^q Dirichlet problem for Δ ² u = 0 in Ω with boundary data in WA ^1,q (∂Ω) is equivalent to that of the L ^p regularity problem for Δ ² u = 0 in Ω with boundary data in WA ^2,p (∂Ω), where \frac1p + \frac1q=1{\frac{1}{p} + \frac{1}{q}=1}. This duality relation, together with known results on the Dirichlet problem, allows us to solve the L ^p regularity problem for d ≥ 4 and p in certain ranges.     

On the fourier coefficients of functions concentrated on a subset of the circle

We consider ,mE > 0,G(E) is a certain subspace of L ¹ (E) consisting of functions concentrated on E and integrable, and {d_k}, (k ∈ ℤ) in a summable sequence of positive numbers. It is proved that if G(E)=L^p(E), p≥2, then there exists f∈G(E) such that |f(n)|≥d_n, (one of the questions involved in the majorization problem). Sufficient conditions are obtained for certain other function classes G(E). We study the question of partial majorization. Bibliography: 2 titles. Translated fromProblemy Matematicheskogo Analiza, No. 13, 1992, pp. 42–48.     

On the number of Galois points for a plane curve in positive characteristic,III

We consider the following problem: For a smooth plane curve C of degree d ≥ 4 in characteristic p > 0, determine the number δ(C) of inner Galois points with respect to C. This problem seems to be open in the case where d ≡ 1 mod p and C is not a Fermat curve F(p ^e  + 1) of degree p ^e  + 1. When p ≠ 2, we completely determine δ(C). If p = 2 (and C is in the open case), then we prove that δ(C) = 0, 1 or d and δ(C) = d only if d−1 is a power of 2, and give an example with δ(C) = d when d = 5. As an application, we characterize a smooth plane curve having both inner and outer Galois points. On the other hand, for Klein quartic curve with suitable coordinates in characteristic two, we prove that the set of outer Galois points coincides with the one of \mathbbF₂{\mathbb{F}_{2}} -rational points in \mathbbP²{\mathbb{P}^{2}}.     

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     

An exact Daugavet type inequality for small into isomorphisms in L 1

The well known Daugavet property for the space L ₁ means that || I  +  K || = 1+ || K || for any weakly compact operator K : L ₁ → L ₁, where I is the identity operator in L ₁. We generalize this theorem to the case when we consider an into isomorphism J : L ₁ → L ₁ instead of I and a narrow operator T. Our main result states that , where d  =  || J|| || J ⁻¹||. We also give an example which shows that this estimate is exact. Received: 21 August 2007     

Smoothness of convolution powers of orbital measures on the symmetric space SU(n)/SO(n)

We prove that if m_a = m_K*d_a*m_K{\mu _{a}\,{=}\,m_{K}*\delta _{a}*m_{K}} is the K-bi-invariant measure supported on the double coset KaK í SU(n){KaK\subseteq SU(n)} , for K = SO(n), then m_a^k{\mu _{a}^{k}} is absolutely continuous with respect to the Haar measure on SU(n) for all a not in the normalizer of K if and only if k ≥ n. The measure, μ _a , supported on the minimal dimension double coset has the property that m_a^n-1{\mu _{a}^{n-1}} is singular to the Haar measure.     

Curves with infinitely many points of fixed degree

Thed-th symmetric productC ^(d) of a curveC defined over a fieldK is closely related to the set of points ofC of degree ≤d. IfK is a number field, then a conjecture of Lang [Hi] proved by Faltings [Fa2] implies ifC ^(d) (K) is an infinite set, then there is aK-rational covering ofC → ℙ _|K ¹ of degree ≤2d. As an application one gets that for fixed fieldK and fixedd there are only finitely many primes ι such that the set of all elliptic curves defined over some extensionsL ofK with [L∶K]≤d and withL-rational isogeny of degree ι is infinite.     

Lyapunov exponents for products of matrices and multifractal analysis. Part I: Positive matrices

Let (Σ,σ) be a full shift space on an alphabet consisting ofm symbols and letM: Σ→L ⁺(ℝ ^d , ℝ ^d ) be a continuous function taking values in the set ofd×d positive matrices. Denote by λ _M (x) the upper Lyapunov exponent ofM atx. The set of possible Lyapunov exponents is just an interval. For any possible Lyapunov exponentα, we prove the following variational formula, , where dim is the Hausdorff dimension or the packing dimension,P _M(q) is the pressure function ofM, μ is aσ-invariant Borel probability measure on Σ,h(μ) is the entropy ofμ, and . The author was partially supported by a HK RGC grant in Hong Kong and the Special Funds for Major State Basic Research Projects in China.     

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.     

Geometric orbifolds with torsion free derived subgroup

A geometric orbifold of dimension d is the quotient space S = X/K, where (X,G) is a geometry of dimension d and K < G is a co-compact discrete subgroup. In this case {ie38-01} is called the orbifold fundamental group of S. In general, the derived subgroup K’ of K may have elements acting with fixed points; i.e., it may happen that the homology cover M_S = X/K’ of S is not a geometric manifold: it may have geometric singular points. We are concerned with the problem of deciding when K′ acts freely on X; i.e., when the homology cover M _S is a geometric manifold. In the case d = 2 a complete answer is due to C. Maclachlan. In this paper we provide necessary and sufficient conditions for the derived subgroup S to act freely in the case d = 3 under the assumption that the underlying topological space of the orbifold K is the 3-sphere S ³.     

Selmer groups of abelian varieties in extensions of function fields

Let k be a field of characteristic q, a smooth geometrically connected curve defined over k with function field . Let A/K be a non-constant abelian variety defined over K of dimension d. We assume that q = 0 or >  2d + 1. Let p ≠ q be a prime number and a finite geometrically Galois and étale cover defined over k with function field . Let (τ′, B′) be the K′/k-trace of A/K. We give an upper bound for the -corank of the Selmer group Sel _p (A × _K K′), defined in terms of the p-descent map. As a consequence, we get an upper bound for the -rank of the Lang–Néron group A(K′)/τ′B′(k). In the case of a geometric tower of curves whose Galois group is isomorphic to , we give sufficient conditions for the Lang–Néron group of A to be uniformly bounded along the tower. This work was partially supported by CNPq research grant 305731/2006-8.     

Existence and analyticity of eigenvalues of a two-channel molecular resonance model

We consider a family of operators H_γμ(k), k ∈ \mathbbT^d \mathbb{T}^d := (−π,π]^d, associated with the Hamiltonian of a system consisting of at most two particles on a d-dimensional lattice ℤ^d, interacting via both a pair contact potential (μ > 0) and creation and annihilation operators (γ > 0). We prove the existence of a unique eigenvalue of H_γμ(k), k ∈ \mathbbT^d \mathbb{T}^d , or its absence depending on both the interaction parameters γ,μ ≥ 0 and the system quasimomentum k ∈ \mathbbT^d \mathbb{T}^d . We show that the corresponding eigenvector is analytic. We establish that the eigenvalue and eigenvector are analytic functions of the quasimomentum k ∈ \mathbbT^d \mathbb{T}^d in the existence domain G ⊂ \mathbbT^d \mathbb{T}^d .