Mapping Properties of the Laplacian in Sobolev Spaces of Forms on Complete Hyperbolic Manifolds

For a complete manifold M with constant negative curvature, weprove that the rough Laplacian _R defines topological isomorphisms in the scale of Sobolev spaces H _p ^s (M) ofp-forms for all p, 0 < p< n. For the de Rham Laplacian and M= ⁿ , the Poincaréhyperbolic space, this is shown too for 0 pn,pn/2, p (n± 1)/2.     

Some functional inequalities and their Baire category properties

Summary In the paper we consider, from a topological point of view, the set of all continuous functionsf:I I for which the unique continuous solution:I – [0, ) of(f(x)) (x, (x)) and(x, (x)) (f(x)) (x, (x)), respectively, is the zero function. We obtain also some corollaries on the qualitative theory of the functional equation(f(x)) = g(x, (x)). No assumption on the iterative behaviour off is imposed.     

Approximation of the Distribution of the Supremum of a Centered Random Walk. Application to the Local Score

Let (X _n ) _{n 0} be a real random walk starting at 0, with centered increments bounded by a constant K. The main result of this study is: |P(S _n n x)–P( sup_{0 u 1} B _u x)| C(n,K) n/n, where x 0, ² is the variance of the increments, S _n is the supremum at time n of the random walk, (B _u ,u 0) is a standard linear Brownian motion and C(n,K) is an explicit constant. We also prove that in the previous inequality S _n can be replaced by the local score and sup₀ u 1 B _u by sup₀ u 1|B _u |.     

Balanced sets andQ-polynomial association schemes

The notion of a balanced set of vectors is defined, and the classification of such sets suggested. A slightly stronger condition is considered for association schemes, with the following result. LetX be ad-class symmetric association scheme with Bose-Mesner algebraM and Krein parametersq _ij ^h , and let E = E_t (0 t d) be any primitive idempotent ofM. For eachx X letx _E denote the diagonal matrix withy, y entryE _xy(y X). Define the representation diagramD _E on the nodes 0, 1, ...,d by drawing an undirected arc between any nodesh, j for whichq _tj ^h > 0. CallD _E an augmented tree if it has a single loop, whose removal yields a tree. Writexy = i ifx, y X are ith associates (0 i d).     

Eigenvalue Estimates for Submanifolds with Locally Bounded Mean Curvature

We present a method to obtain lower bounds for firstDirichlet eigenvalue in terms of vector fields with positivedivergence. Applying this to the gradient of a distance functionwe obtain estimates of eigenvalue of balls inside the cut locus and of domains M B _N (p, r) in submanifolds M Nwith locally bounded mean curvature. Forsubmanifolds of Hadamard manifolds with bounded mean curvaturethese lower bounds depend only on the dimension of the submanifold and the bound on its mean curvature.     

Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory

Let {X _k , 1 k n} be n independent and real-valued random variables with common subexponential distribution function, and let {k, 1 k n} be other n random variables independent of {X _k , 1 k n} and satisfying a _k b for some 0 < a b < for all 1 k n. This paper proves that the asymptotic relations P (max_{1 m n k=1} ^m _k X _k > x) P (sum _k=1 ⁿ _k X _k > x) sum _k=1 ⁿ P ( _k X _k > x) hold as x . In doing so, no any assumption is made on the dependence structure of the sequence { _k , 1 k n}. An application to ruin theory is proposed.     

Convex independent sets and 7-holes in restricted planar point sets

For a finite setA of points in the plane, letq(A) denote the ratio of the maximum distance of any pair of points ofA to the minimum distance of any pair of points ofA. Fork>0 letc (k) denote the largest integerc such that any setA ofk points in general position in the plane, satisfying for fixed , contains at leastc convex independent points. We determine the exact asymptotic behavior ofc (k), proving that there are two positive constants=(), such thatk ^1/3c (k)k ^1/3. To establish the upper bound ofc (k) we construct a set, which also solves (affirmatively) the problem of Alonet al. [1] about the existence of a setA ofk points in general position without a 7-hole (i.e., vertices of a convex 7-gon containing no other points fromA), satisfying . The construction uses Horton sets, which generalize sets without 7-holes constructed by Horton and which have some interesting properties.     

Graphs of Constant Mean Curvature in Hyperbolic Space

We study the problem of finding constant mean curvature graphsover a domain of a totally geodesic hyperplane andan equidistant hypersurface Q of hyperbolic space. We findthe existence of graphs of constant mean curvature H overmean convex domains Q and with boundary for –H < H |h|, where H > 0 is the mean curvature of the boundary . Here h is the mean curvature respectively of the geodesic hyperplane (h= 0) and of the equidistant hypersurface (0 < |h|< 1). The lower bound on H is optimal.     

Moser-type inequalities in Lorentz spaces

Moser-type estimates for functions whose gradient is in the Lorentz space L(n, q), 1q, are given. Similar results are obtained for solutions uH _inf0 ^sup1 of Au=(f _i ) _{x i} , where A is a linear elliptic second order differential operator and |f|L(n, q), 2q.Work partially supported by MURST (40%).     

Some remarks on compact constant mean curvature hypersurfaces in a halfspace of ℍ n+1

We consider embedded compact hypersurfacesM in a halfspace of hyperbolic space with boundaryM in the boundary geodesic hyperplaneP of the halfspace and with non-zero constant mean curvature. We prove the following. Let {M _n } be a sequence of such hypersurfaces withM _n contained in a disk of radiusr _n centered at a point P such thatr _n 0 and that eachM _n is a large. H-hypersurface,H > 1. Then there exists a subsequence of {M _n } converging to the sphere of mean curvatureH tangent toP at. In the case of smallH-hypersurfaces orH 1, if we add a condition on the curvature of the boundary, there exists a subsequence of {M _n } which are graphs. The convergence is smooth on compact subset of ³ .     

On the Minimum Size of Some Minihypers and Related Linear Codes

We denote by m_r,q(s) the minimum value of f for which an {f, _r-2+s ; r,q }-minihyper exists for r 3, 1 s q–1, where _j=(q^j+1–1)/(q–1). It is proved that m_3,q(s)=₁(₁+s) for many cases (e.g., for all q 4 when ) and that m_r,q(s) _r-1+s₁+q for 1 s q – 1,~q 3,~r 4. The nonexistence of some [n,k,n+s–q^k-2]_q codes attaining the Griesmer bound is given as an application.AMS classification: 94B27, 94B05, 51E22, 51E21     

Block LU factorization is stable for block matrices whose inverses are block diagonally dominant

Let A M_n (C) and let the inverse matrix B = A^–1 be block diagonally dominant by rows (columns) w.r.t. an m × m block partitioning and a matrix norm. We show that A possesses a block LU factorization w.r.t. the same block partitioning, and the growth factor for A in this factorization is bounded above by 1 + , where = max _{1im i} and _i, 0 _i 1, are the row (column) block dominance factors of B. Further, the off-diagonal blocks of A (and of its block Schur complements) satisfy the inequalities

On the distribution of square-full and cube-full integers

LetN _r (x) be the number ofr-full integers x and let _r (x) be the error term in the asymptotic formula forN _r (x). Under Riemann's hypothesis, we prove the estimates ₂(x)x^1/7+, ₃(x)x^97/804+(>0), which improve those of Cao and Nowak. We also investigate the distribution ofr-full andl-free numbers in short intervals (r=2,3). Our results sharpen Krätzel's estimates.     

Existence and uniqueness of solutions to two point boundary value problems for ordinary differential equations

In this paper, using the optimal control method, we deal with the following boundary value problemy + f(x, y)=0,y(0)=c,y(1)=d, under new nonresonance conditions of the form –A f _y ^/ (x, y) (x) B, where A > 0. We obtain the existence and uniqueness of solutions of the BVP (1).     

Automorphisms on a Lie group contracting modulo a compact subgroup and applications to semistable convolution semigroups

Let ( _t ) _t0 be a -semistable convolution semigroup of probability measures on a Lie groupG whose idempotent ₀ is the Haar measure on some compact subgroupK. Then all the measures ₁ are supported by theK-contraction groupC _K() of the topological automorphism ofG. We prove here the structure theoremC _K()=C()K, whereC() is the contraction group of . Then it turns out that it is sufficient to study semistable convolution semigroups on simply connected nilpotent Lie groups that have Lie algebras with a positive graduation.     

Indices of Maximal Subgroups of Finite Soluble Groups

We look at the structure of a soluble group G depending on the value of a function m(G)= max m _p G), where m _p(G)=max{log_p|G:M| | M< G, |G:M|=p ^a}, p (G). Theorem 1 states that for a soluble group G, (1) r(G/ (G))= m(G); (2) d(G/ (G)) 1+ (m(G)) 3+m(G); (3) l _p(G) 1+t, where 2^t-1<m _p(G) 2^t. Here, (G) is the Frattini subgroup of G, and r(G), d(G), and l _p(G) are, respectively, the principal rank, the derived length, and the p-length of G. The maximum of derived lengths of completely reducible soluble subgroups of a general linear group GL(n,F) of degree n, where F is a field, is denoted by (n). The function m(G) allows us to establish the existence of a new class of conjugate subgroups in soluble groups. Namely, Theorem 2 maintains that for any natural k, every soluble group G contains a subgroup K possessing the following properties: (1) m(K); k; (2) if T and H are subgroups of G such that K T <_max <_max H G then |H:T|=p ^t for some prime p and for t>k. Moreover, every two subgroups of G enjoying (1) and (2) are mutually conjugate.     

Two-parameter Vilenkin multipliers and a square function

Two-parameter Vilenkin systems will be investigated. First we give a general sufficient condition for multipliers to be bounded between two-dimensional Hardy spaces H ^q(0<q1). By means of interpolation and duality argument, this theorem can be extended to other spaces. As a consequence, we can prove the (H ^q , L ^q)-boundedness of the Sunouchi operator U with respect to two-parameter Vilenkin systems for all 0 <q 1. Moreover, the equivalence f{H_q} ~ Uf_q (f H^q)follows for 1/2<q 1.     

On a theorem of lovász on covers inr-partite hypergraphs

A theorem of Lovász asserts that (H)/*(H)r/2 for everyr-partite hypergraphH (where and * denote the covering number and fractional covering number respectively). Here it is shown that the same upper bound is valid for a more general class of hypergraphs: those which admit a partition (V ₁, ...,V _k ) of the vertex set and a partitionp ₁+...+p _k ofr such that |eV _i |p _i r/2 for every edgee and every 1ik. Moreover, strict inequality holds whenr>2, and in this form the bound is tight. The investigation of the ratio /* is extended to some other classes of hypergraphs, defined by conditions of similar flavour. Upper bounds on this ratio are obtained fork-colourable, stronglyk-colourable and (what we call)k-partitionable hypergraphs.Supported by grant HL28438 at MIPG, University of Pennsylvania, and by the fund for the promotion of research at the Technion.This author's research was supported by the fund for the promotion of research at the Technion.     

Ebene algebraische Kurven vom Typ p,q

Let p,q be relatively prime integers with 2pr ^p,q be the numerical semigroup generated by p,q,{(p–1) (q–1)–1–(ip+jq)¦i+jr–2}. Then there exists a smooth projective curve X and a point x on X, such that H _r ^p,q is the set of orders of poles of the rational functions on X, which are regular on X\{x}; in other words: H _r ^p,q is a Weierstraß semigroup.