Showing papers in "Israel Journal of Mathematics in 1986"

A remark on regularization in Hilbert spaces

Abstract: We present here a simple method to approximate uniformly in Hilbert spaces uniformly continuous functions byC 1,1 functions. This method relies on explicit inf-sup-convolution formulas or equivalently on the solutions of Hamilton-Jacobi equations.

The metrical interpretation of superreflexivity in banach spaces

Abstract: The main result is a metrical characterization of superreflexivity in Banach spaces. A Banach spaceX is not superreflexive if and only ifX contains hyperbolic trees as a metric space. The notion of non-linear cotype in discussed.

Extensions of lipschitz maps into Banach spaces

Abstract: It is proved that ifY ⊂X are metric spaces withY havingn≧2 points then any mapf fromY into a Banach spaceZ can be extended to a map $$\hat f$$ fromX intoZ so that $$\left\| {\hat f} \right\|_{lip} \leqq c log n\left\| f \right\|_{lip} $$ wherec is an absolute constant. A related result is obtained for the case whereX is assumed to be a finite-dimensional normed space andY is an arbitrary subset ofX.

A szemerédi type theorem for sets of positive density inR k

Abstract: Letk≧2 andA a subset ofRk of positive upper density. LetV be the set of vertices of a (non-degenerate) (k−1)-dimensional simplex. It is shown that there existsl=l(A, V) such thatA contains an isometric image ofl′. V wheneverl′>l. The casek=2 yields a new proof of a result of Katznelson and Weiss [4]. Using related ideas, a proof is given of Roth’s theorem on the existence of arithmetic progressions of length 3 in sets of positive density.

Bi-convexity and bi-martingales

Abstract: A set in a product spaceX×Y isbi-convex if all itsx- andy-sections are convex. Abi-martingale is a martingale with values inX×Y whosex- andy-coordinates change only one at a time. This paper investigates the limiting behavior of bimartingales in terms of thebi-convex hull of a set — the smallest bi-convex set containing it — and of several related concepts generalizing the concept of separation to the bi-convex case.

Large time behaviour of solutions of the porous media equation with absorption

Abstract: We study the large time behaviour of nonnegative solutions of the Cauchy problemut=Δum −up,u(x, 0)=φ(x). Specifically we study the influence of the rate of decay ofφ(x) for large |x|, and the competition between the diffusion and the absorption term.

Changing cofinalities and the nonstationary ideal

Abstract: Aκ-c.c. iteration of a Prikry type forcing notion is defined. Applications to the nonstationary ideal are given.

Banach spaces with a weak cotype 2 property

Abstract: We study the Banach spacesX with the following property: there is a numberδ in ]0,1[ such that for some constantC, any finite dimensional subspaceE ⊂X contains a subspaceF ⊂E with dimF≧δ dimE which isC-isomorphic to a Euclidean space. We show that if this holds for someδ in ]0,1[ then it also holds for allδ in ]0,1[ and we estimate the functionC=C(δ). We show that this property holds iff the “volume ratio” of the finite dimensional subspaces ofX are uniformly bounded. We also show that (althoughX can have this property without being of cotype 2)L 2(X) possesses this property iffX if of cotype 2. In the last part of the paper, we study theK-convex spaces which have a dual with the above property and we relate it to a certain extension property.

Une inégalité du type de Slepian et Gordon sur les processus gaussiens

Abstract: A new proof and extension of the Slepian-Gordon inequality is given.

Une Caracterisation Volumique de Certains Espaces Normes de Dimension Finie

Abstract: We give a geometric characterization of finite dimensional normed spacesE, with a 1-unconditional basis, such that their volumetric product is minimal.

On theL p-bounds for maximal functions associated to convex bodies inR n

Abstract: The dimension-freeL 2-maximal inequality for convex symmetric bodies obtained in [2] is extended forp>3/2.

A Local Version of the Two-Circles Theorem.

Abstract: A necessary and sufficient condition is given so that in a domain Ω there are no functions whose average over all balls contained in Ω of radiir1,r2 vanish except the zero function.

On hilbertian subsets of finite metric spaces

Abstract: The following result is proved: For everye>0 there is aC(e)>0 such that every finite metric space (X, d) contains a subsetY such that |Y|≧C(e)log|X| and (Y, d Y) embeds (1 +e)-isomorphically into the Hilbert spacel 2

On existence of cone-maximal points in real topological linear spaces

Abstract: A sufficient condition for a convex coneC in a Hausdorff topological linear space is given in order to ensure the existence of cone-maximal points. The condition becomes a necessary one in a topological linear space with a countable local base, that is, if the space is pseudometrizable. The paper extends known results to infinite dimensions and we answer Corley’s question in the affirmative with the exception of a pathological case.

Renormages de Quelquesℒ(K)

Abstract: LetD([0, 1]) be the space of left continuous real valued functions on [0, 1] which have a right limit at each point. We show thatD([0, 1]) has no equivalent norm which is Gâteau differentiable. Hence the class of spaces which can be renormed by a Gâteau differentiable norm fails the three spaces property. We show that there is no norm onℒ([0, Ω]) such that its dual is strictly convex. However, there is an equivalent Frechet differentiable norm on this space.

Smash products and outer derivations

Abstract: LetR be a prime ring andL a Lie algebra acting onR as “Q-outer” derivations (if charR=p≠0, assume thatL is restricted). We study ideals and the center of the smash productR #U(L) (respectivelyR #u(L) ifL is restricted) and use these results to study the relationship betweenR and the ring of constantsR L . More generally, for any finite-dimensional Hopf algebraH acting onR such thatR #H satisfies the “ideal intersection property”, we useR #H to study the relationship betweenR and the invariant ringR H .

On the similarity problem for polynomially bounded operators on Hilbert space

Abstract: Partial solutions are obtained to Halmos’ problem, whether or not any polynomially bounded operator on a Hilbert spaceH is similar to a contraction. Central use is made of Paulsen’s necessary and sufficient condition, which permits one to obtain bounds on ‖S‖ ‖S −1‖, whereS is the similarity. A natural example of a polynomially bounded operator appears in the theory of Hankel matrices, defining $$R_f = \left( {\begin{array}{*{20}c} {S*} \\ 0 \\ \end{array} \begin{array}{*{20}c} {\Gamma _f } \\ S \\ \end{array} } \right)$$ onl 2 ⊕l 2, whereS is the shift and Γ f the Hankel operator determined byf withf′ ∈ BMOA. Using Paulsen’s condition, we prove thatR f is similar to a contraction. In the general case, combining Grothendieck’s theorem and techniques from complex function theory, we are able to get in the finite dimensional case the estimate $$\left\| S \right\|\left\| {S^{ - 1} } \right\| \leqq M^4 log(dim H)$$ whereSTS −1 is a contraction and assuming $$\left\| {p\left( T \right)} \right\| \leqq M\left\| p \right\|_\infty $$ wheneverp is an analytic polynomial on the disc.

Uniform convexity properties of norms on a super-reflexive Banach space

Abstract: We study uniform convexity and smoothness properties satisfied by all the equivalent norms of a super-reflexive Banach space We give some applications concerning quasi-transitive Banach spaces, and the uniform approximation property

Are there free groups in division rings

Abstract: The following conjecture is investigated: a noncentral subnormal subgroup of the multiplicative group of a division ring contains a noncyclic free subgroup. Special cases are proved, entailing several known commutativity theorems. Also a new framework is presented for some kinds of commutativity theorems, based on the existence of (group) words for which one can always find an appropriate substitution by elements of such a subnormal subgroup that yields a noncentral element. Several families of such words are given; one gets commutativity theorems imposing some restrictions (like periodicity) to the image of these words.

MORE ON p-GROUPS OF FROBENIUS TYPE

Abstract: We consider finitep-groups with the property that ifx ∈G − G′ andz ∈G′ thenx is conjugate toxz inG. In certain special casesG has class 2 or 3.

Ensembles Intersectifs et Récurrence de Poincaré

Abstract: We show that the intersective set and the set of Poincare are equal and we apply this fact to the repartition modulo 1 of a certain sequence of type (xθ n) n≧0.

Transformations with highly nonhomogeneous spectrum of finite multiplicity

Abstract: This paper studies a spectral invariant ℳ T for ergodic measure preserving transformationsT called theessential spectral multiplicities. It is defined as the essential range of the multiplicity function for the induced unitary operatorU T. Examples are constructed where ℳ T is subject only to the following conditions: (i) 1∈ℳ T , (ii) lcm(n, m)∈ℳ T wherevern, m ∈ ℳ T , and (iii) sup ℳ T <+∞. This shows thatD T, definedD T=card ℳ T , may be an arbitrary positive integer. The results are obtained by an algebraic construction together with approximation arguments.

Constraints on the degree of a sofic homomorphism and the induced multiplication of measures on unstable sets

Abstract: Let f be an endomorphism of an irreducible sofic system S, where S has entropy Iog)t. The degree of f is the number d such that f is d to 1 almost everywhere. Then d divides a power of the greatest common divisor of the nonleading coefficients of the minimal polynomial of )t. Also, f multiplies the natural measure on unstable sets of generic points by a positive unit of the ring generated by 1/A and the algebraic integers of Q[A]. Related results hold for bounded to one homomorphisms of sofic systems.

Exposed and denting points in duals of operator spaces

Abstract: We study the extremal structure of the dual unit balls of various operator spaces. Mainly, we show that the classes of [w*-] strongly exposed, [w*-] exposed, and denting points in the dual unit balls of spaces of compact operators between Banach spacesX andY are completely — and in a canonical way — determined by the corresponding classes of points in the unit balls of the (bi-)duals of the factor spacesX andY. Applications to the duality of operator spaces and differentiability properties of the norm in operator spaces are given.

Initial segments of the degrees of size ℵ1

Abstract: We settle a series of questions first raised by Yates at the Jerusalem (1968) Colloquium on Mathematical Logic by characterizing the initial segments of the degrees of unsolvability of size ℵ1: Every upper semi-lattice of size ℵ1 with zero, in which every element has at most countably many predecessors, is isomorphic to an initial segment of the Turing degrees.

On the number of certain subgraphs contained in graphs with a given number of edges

Abstract: All graphs considered are finite, undirected, with no loops, no multiple edges and no isolated vertices. For two graphsG, H, letN(G, H) denote the number of subgraphs ofG isomorphic toH. Define also, forl≧0,N(l, H)=maxN(G, H), where the maximum is taken over all graphsG withl edges. We determineN(l, H) precisely for alll≧0 whenH is a disjoint union of two stars, and also whenH is a disjoint union ofr≧3 stars, each of sizes ors+1, wheres≧r. We also determineN(l, H) for sufficiently largel whenH is a disjoint union ofr stars, of sizess 1≧s 2≧…≧s r>r, provided (s 1−s r)2

Theorem about the conjugacy representation ofS n

Abstract: Every group has two natural representations on itself, the regular representation and the conjugacy representation. We know everything about the construction of the regular representation, but we know very little about the conjugacy representation (for uncommutative groups). In this paper we will see that every irreducible complex character ofS n (n>2) is a constituent of conjugacy character ofS n .

Markov classes in certain finite quotients of artin's braid group

Abstract: This paper studies three finite quotients of the sequence of braid groups {B n;n = 1,2,…}. Each has the property that Markov classes in {ie160-1} = ∐B n pass to well-defined equivalence classes in the quotient. We are able to solve the Markov problem in two of the quotients, obtaining canonical representatives for Markov classes and giving a procedure for reducing an arbitrary representative to the canonical one. The results are interpreted geometrically, and related to link invariants of the associated links and the value of the Jones polynomial on the corresponding classes.

Periodicity and absolute regularity

Abstract: For a stationary ergodic process it is proved that the dependence coefficient associated with absolute regularity has a limit connected with a periodicity concept. Similar results can then be obtained for stronger dependence coefficients. The periodicity concept is studied separately and it is seen that the double tailσ-field can be trivial while the period is 2. The paper imbeds renewal theory in ergodic theory. The total variation metric is used.

The linear search problem rides again

Abstract: The Linear Search Problem concerns a search for a point in the real line by continuous motion starting at 0. The optimal turning points for such a search under the hypothesis that the location of the target is distributed normally about 0 have been approximated by mechanical calculation, but no proof has been given that there is only a single minimizing strategy or that the numbers calculated do indeed approximate that strategy. Plausible arguments have been made before, both by these authors and others. In this paper, the plausible arguments are supplanted by mathematical proofs.