Showing papers in "Israel Journal of Mathematics in 1992"
TL;DR: In this article, the influence of the k-th variable on the probability space is defined, denoted by If(k), as follows: Foru=(u1,u2,…,un−1) ∈ Xn−1 consider the setlk(u)={(u 1,u 2,...,uk−1,t,uk, etc, un−1):t ∈X}.
Abstract: LetX be a probability space and letf: Xn → {0, 1} be a measurable map. Define the influence of thek-th variable onf, denoted byIf(k), as follows: Foru=(u1,u2,…,un−1) ∈Xn−1 consider the setlk(u)={(u1,u2,...,uk−1,t,uk,…,un−1):t ∈X}.\(I_f (k) = \Pr (u \in X^{n - 1} :f is not constant on l_k (u)).\)
169 citations
TL;DR: In this paper, it was proved that a periodic pro-p-group is locally finite, and it is shown that the periodic pro p-group can not be locally finite.
Abstract: It is proved that a periodic pro-p-group is locally finite.
136 citations
TL;DR: In this paper, a strongly minimal expansion of an algebraically closed field of a given characteristic was constructed, implying that the existence of a set of strongly minimal sets with two different field structures of distinct characteristics.
Abstract: (1) We construct a strongly minimal expansion of an algebraically closed field of a given characteristic. Actually we show a much more general result, implying for example the existence of a strongly minimal set with two different field structures of distinct characteristics. (2) A strongly minimal expansion of an algebraically closed field that preserves the algebraic closure relation must be an expansion by (algebraic) constants.
109 citations
TL;DR: In this paper, the authors determined theSn×Sm-cocharacterXn,m of the algebra M1,1(E) and proved that the T2-ideal of its graded identities is generated by the polynomialsy1y2−y2y1 andz1z2z3+z3z2Z1.
Abstract: We determine theSn×Sm-cocharacterXn,m of the algebraM1,1(E) and prove that theT2-ideal of its graded identities is generated by the polynomialsy1y2−y2y1 andz1z2z3+z3z2z1.
91 citations
TL;DR: Using entropy arguments on a symbolic representation of the inverse limit of this action, this article showed that if any element inS has positive entropy with respect toμ, thenμ is Lebesgue.
Abstract: LetS be a nonlacunary subsemigroup of the natural numbers and letμ be anS-invariant and ergodic measure. Using entropy arguments on a symbolic representation of the inverse limit of this action, we show that if any element inS has positive entropy with respect toμ, thenμ is Lebesgue.
90 citations
TL;DR: In this article, it was shown that Tverberg's theorem follows easily from a theorem of which Barany has given a very short proof, which is called the Barany-Tverberg theorem.
Abstract: We show that Tverberg’s theorem follows easily from a theorem of which Barany [1] has given a very short proof.
80 citations
TL;DR: In this article, the authors give a complete description of reducibility, components and composition series of the degenerate principal series representations of the groupSp(n, F), F a non-archimedean local field, which are induced from a character of a maximal parabolic subgroupP = MN with Levi subgroupM ≊GL( n, F).
Abstract: In this paper we give a complete description of the points of reducibility, components and composition series of the degenerate principal series representations of the groupSp(n, F), F a non-archimedean local field, which are induced from a character of a maximal parabolic subgroupP = MN with Levi subgroupM ≊GL(n, F) We show that all of the reducibility is accounted for by submodules coming from the Weil representation associated to quadratic forms overF The local results of this paper play an essential role in our extension of the Siegel-Weil formula relating theta integrals and special values of Eisenstein series
80 citations
TL;DR: In this paper, the authors studied the convergence of the Schrodinger equation to the initial data and showed that f(x, t) →f(x) almost everywhere iff ∈H PsyNet p (R2), wherep may be chosen < 1/2.
Abstract: We study the almost everythere convergence to the initial dataf(x)=u(x, 0) of the solutionu(x, t) of the two-dimensional linear Schrodinger equation Δu=iϖ
t
u. The main result is thatu(x, t) →f(x) almost everywhere fort → 0 iff ∈H
p
(R2), wherep may be chosen <1/2. To get this result (improving on Vega’s work, see [6]), we devise a strategy to capture certain cancellations, which we believe has other applications in related problems.
79 citations
TL;DR: In this article, the joinings of simple systems with another simple system with another ergodic system are treated; then the pairwise independent joins of three simple systems one of which is simple.
Abstract: The purpose of this work is to study the joinings of simple systems. First the joinings of a simple system with another ergodic system are treated; then the pairwise independent joinings of three systems one of which is simple. The main results obtained are: (1) A weakly mixing simple system with no non-trivial factors with absolutely continuous spectral type is simple of all orders. (2) A weakly mixing system simple of order 3 is simple of all orders.
50 citations
TL;DR: In this paper, simple proofs of Raghunathan's conjectures for SL(2,R) were given in a simplified form, incorporating some of the ideas and methods used to prove the conjectures in general connected Lie groups.
Abstract: In this paper I give simple proofs of Raghunathan’s conjectures for SL(2,R). These proofs incorporate in a simplified form some of the ideas and methods I used to prove the Raghunathan’s conjectures for general connected Lie groups.
50 citations
TL;DR: In this article, the authors studied the approximation properties of the dilations of the integer translates of a smooth function, with some derivatives vanishing at infinity, and showed that polynomial reproduction is not sufficient for the derivation of exact order of convergence by dilated quasiinterpolants.
Abstract: Approximation properties of the dilations of the integer translates of a smooth function, with some derivatives vanishing at infinity, are studied. The results apply to fundamental solutions of homogeneous elliptic operators and to “shifted” fundamental solutions of the iterated Laplacian. Following the approach from spline theory, the question of polynomial reproduction by quasi-interpolation is addressed first. The analysis makes an essential use of the structure of the generalized Fourier transform of the basis function. In contrast with spline theory, polynomial reproduction is not sufficient for the derivation of exact order of convergence by dilated quasi-interpolants. These convergence orders are established by a careful and quite involved examination of the decay rates of the basis function. Furthermore, it is shown that the same approximation orders are obtained with quasi-interpolants defined on a bounded domain.
TL;DR: In this article, the authors give necessary and sufficient conditions for the existence of two random closed sets based on the same probability space and such that their distributions coincide with those of ψ and ϕ in a locally compact second countable topological space.
Abstract: Letψ andϕ be two given random closed sets in a locally compact second countable topological spaceS. (They need not be based on the same probability space.) The main result gives necessary and sufficient conditions on the distributions ofψ andϕ, for the existence of two random closed sets\(\hat \psi \) and\(\hat \varphi \), based on the same probability space and such that their distributions coincide with those ofψ andϕ, resp., and\(\hat \psi \subseteq \hat \varphi \) a.s.
TL;DR: In this paper, it was shown that the stableness property of the graded Lie algebra for certain derivations associated with pure sphere braid group onn strings is related to Grothendieck's conjecture.
Abstract: We shall prove some stability property of the graded Lie algebraD
n of certain derivations associated with pure sphere braid group onn strings; in fact, thatD
n forn≥6. These Lie algebrasD
n are connected with some bigl-adic Galois representations, and the stability property is related to some conjecture of Grothendieck.
TL;DR: By analyzing the distribution function of the convolution square of a convex and symmetric body, this paper obtained some affine invariants related to the body and these invariants have a geometric interpretation.
Abstract: By analyzing the distribution function of the convolution square of a convex and symmetric body we obtain some affine invariants related to the body. These invariants have a geometric interpretation.
TL;DR: In this article, a class of zero-sum, two-person stochastic games is shown to have a value which can be calculated by transfinite iteration of an operator, and games considered have a countable state space, finite action spaces for each player.
Abstract: A class of zero-sum, two-person stochastic games is shown to have a value which can be calculated by transfinite iteration of an operator. The games considered have a countable state space, finite action spaces for each player, and a payoff sufficiently general to include classical stochastic games as well as Blackwell’s infiniteGδ games of imperfect information.
TL;DR: In this article, an upper bound for the number of integer solutions of a system of homogeneous equations in a box is derived, assuming that the solution proportion is proportional to the volume of the box, provided the box is not very small.
Abstract: We estimate exponential sums with additive character along an affine variety given by a system of homogeneous equations, with a homogeneous function in the exponent. The proof uses the results of Deligne’s Weil Conjectures II and a generalization of Lefschetz hyperplane theorem to singular varieties. We apply our estimate to obtain an upperbound for the number of integer solutions of a system of homogeneous equations in a box. Another application is devoted to uniform distribution of solutions of a system of homogeneous congruences modulo a prime in the following sense: the portion of solutions in a box is proportional to the volume of the box, provided the box is not very small.
TL;DR: In this article, the authors constructed uniform embeddings of Cayley graphs of hyperbolic groups and cyclic extensions of torsion-free small cancellation groups in Hilbert spaces.
Abstract: We construct uniform embeddings of the Cayley graphs of hyperbolic groups and cyclic extensions of torsion-free small cancellation groups in Hilbert spaces.
TL;DR: In this paper, a Banach space X such that the numerical radius attaining operators on X are not dense was shown to have no dense norm attaining operator on X. This result was conjectured by Cardassi in 1985.
Abstract: We answer a question posed by B. Sims in 1972, by exhibiting an example of a Banach spaceX such that the numerical radius attaining operators onX are not dense. Actually,X is an old example used by J. Lindenstrauss to solve the analogous problem for norm attaining operators, but the proof for the numerical radius seems to be much more difficult. Our result was conjectured by C. Cardassi in 1985.
TL;DR: For a family of dynamical properties, knowing that the condition holds for order 4 implies that it holds for all orders as discussed by the authors, which is the case for minimal self-joinings, simplicity and cartesiandisjointness.
Abstract: For a family of dynamical properties, knowing that the condition holds for order 4 implies that it holds for all orders. Here we establish this for the properties minimal self-joinings, simplicity and for cartesiandisjointness.
TL;DR: In this article, the existence of a scattered compact Hausdorff space with the following properties was shown: (i) for each continuous mapf on a Baire spaceB into (C(K), pointwise), the set of points of continuity of the mapf: B → (C (K), norm) is a denseG δ subset of B, and (ii) C(K) does not admit a Kadec norm that is equivalent to the supremum norm.
Abstract: Assuming that there exists in the unit interval [0, 1] a coanalytic set of continuum cardinality without any perfect subset, we show the existence of a scattered compact Hausdorff spaceK with the following properties: (i) For each continuous mapf on a Baire spaceB into (C(K), pointwise), the set of points of continuity of the mapf: B → (C(K), norm) is a denseG
δ subset ofB, and (ii)C(K) does not admit a Kadec norm that is equivalent to the supremum norm. This answers the question of Deville, Godefroy and Haydon under the set theoretic assumption stated above.
TL;DR: In this article, the authors identify the divisor of any fixed directional derivative of the Riemann theta function evaluated along the theta divisors with a subspace of holomorphic one-forms.
Abstract: In this note we will identify the divisor of any fixed directional derivative of the Riemann theta function evaluated along the theta divisor with the divisor formed from ag−1 dimensional subspace of holomorphic one-forms from the underlying Riemann surface.
TL;DR: In this article, a type of dual representation for martingale distributions has been proposed, where a convex order of p.m. distributions of the Hardy and Littlewood maximal functions of a probability measure is obtained from a set of martingales on ℝ with lim supx→∞xν[x,∞)=0xν [x, ∞] = 0 and finitex0=inf{z :ν(−∞,z]0}.
Abstract: Letμ be any probability measure on ℝ with ∫|x|dμ(x)<∞ and letμ* denote the associated Hardy and Littlewood maximal p.m., the p.m. of the Hardy and Littlewood maximal function obtained fromμ. Dubins and Gilat [6] showed thatμ* is the least upper bound, in the usual stochastic order, of the collection of p.m.’sν on ℝ for which there is a martingale (Xt)0≤t≤1 having distributions ofX1 and sup0≤t≤1Xt given byμ andν respectively. In this paper, a type of ‘dual representation’ is given. Specifically, letν be any p.m. on ℝ with lim supx→∞xν[x,∞)=0xν[x, ∞)=0 and finitex0=inf{z :ν(−∞,z]0}. Then there is a ‘minimal p.m.’νΔ which is the greatest lower bound, in the usual convex order, of the collection of p.m.’sμ on ℝ for which there is a martingale (Xt)0≤t≤1 having distributions ofX1 and sup0≤t≤1Xt given byμ andν respectively. To demonstrate existence and to obtain identification of these minimal p.m.’s, we use, in particular, a lattice structure on the set of p.m.’s with the convex order, and an equivalence between a convex order of p.m.’s and the stochastic order of their maximal p.m.’s. Consequences of these order results include sharp expectation-based inequalities for martingales. These martingale inequalities form a new class of ‘prophet inequalities’ in the context of optimal stopping theory.
TL;DR: In this article, the Banach-Mazur distance was extended to be a distance between non-symmetric sets by allowing affine transformations instead of linear transformations, and it was proved that for every convex body K is a simplex.
Abstract: We can extend the Banach-Mazur distance to be a distance between non-symmetric sets by allowing affine transformations instead of linear transformations. It was proved in [J] that for every convex bodyK we haved(K, D)≤n. It is proved that ifK is a convex body in ℝ
n
such thatd(K, D)=n, thenK is a simplex.
TL;DR: In this paper, the authors considered a bounded geometry non-compact Riemannian manifold and the graph obtained by discretizing this manifold, and showed that the uniform decay for large time of the heat kernel and the decay of the standard random walk on the graph are the same, in the polynomial scale.
Abstract: One considers a bounded geometry non-compact Riemannian manifold, and the graph obtained by discretizing this manifold One shows that the uniform decay for large time of the heat kernel on the manifold and the decay of the standard random walk on the graph are the same, in the polynomial scale As a consequence, such a large time behaviour of the heat kernel is invariant under rough isometries
TL;DR: In this article, the influence of small perturbations of a piecewise monotonic map on the Hausdorff dimension HD(R(T)) of R(T) is investigated.
Abstract: In this paper a piecewise monotonic mapT:X→ℝ, whereX is a finite union of intervals, is considered. DefineR(T)=
$$\mathop \cap \limits_{n = 0}^\infty \overline {T^{ - n} X} $$
. The influence of small perturbations ofT on the Hausdorff dimension HD(R(T)) ofR(T) is investigated. It is shown, that HD(R(T)) is lower semi-continuous, and an upper bound of the jumps up is given. Furthermore a similar result is shown for the topological pressure.
TL;DR: In this paper, it was shown that for every bounded linear operator T:C2p→H(1≤p<∞, H is a Hilbert space, C2pp is the Schatten space) there exists a continuous linear formf onCp such thatf≥0, ‖f‖(CCp)*=1 and==================\/\/\/\/\/\/
Abstract: We prove that for every bounded linear operatorT:C2p→H(1≤p<∞,H is a Hilbert space,C2pp is the Schatten space) there exists a continuous linear formf onCp such thatf≥0, ‖f‖(CCp)*=1 and
$$\forall x \in C^{2p} , \left\| {T(x)} \right\| \leqslant 2\sqrt 2 \left\| T \right\| 1/2$$
. Forp=∞ this non-commutative analogue of Grothendieck’s theorem was first proved by G. Pisier. In the above statement the Schatten spaceC2p can be replaced byEE2 whereE(2) is the 2-convexification of the symmetric sequence spaceE, andf is a continuous linear form onCE. The statement can also be extended toLE{(su2)}(M, τ) whereM is a Von Neumann algebra,τ a trace onM, E a symmetric function space.
TL;DR: In this paper, it was shown that if every integer is covered by a 1+n 1−n 1/n 2/n 3/n 4/n 6/1/n 9/1, 9/2/n 1.
Abstract: In this paper it is shown that if every integer is covered bya
1+n
1ℤ,…,a
k
+n
k
ℤ exactlym times then for eachn=1,…,m there exist at least (
) subsetsI of {1,…k} such that ∑
i
∈
I
1/n
i
equalsn. The bound (
) is best possible.
TL;DR: In this paper, it was shown that the Borsuk-Ulam theorem is a Rado type statement about (k + 1) measures in Rn where B(n, n − 1) turns out to be the ham sandwich theorem.
Abstract: This paper continues the search, started in [10], for relatives of the ham sandwich theorem. We prove among other results, the following implications {fx21-1} whereK(n, k) is an important instance of the Knaster’s conjecture so thatK(n, n − 1) reduces to the Borsuk-Ulam theorem,B(n, k) is a R. Rado type statement about (k + 1) measures inRn whereB(n, n − 1) turns out to be the ham sandwich theorem andC(n, k) is a topological statement, established in this paper in the caseC(n, n − 2),n = 3 orn ≥ 5.
TL;DR: In this article, the authors studied five domain constants related to the hyperbolic geometry in regions which are uniformly perfect and derived bounds for the variance ratio of each constant under conformal mappings of the regions.
Abstract: Five domain constants are studied in our paper, all related to the hyperbolic geometry in hyperbolic plane regions which are uniformly perfect (in Pommerenke’s terminology). Relations among these domain constants are obtained, from which bounds are derived for the variance ratio of each constant under conformal mappings of the regions, and we also show that each constant may be used to characterize uniformly perfect regions.
TL;DR: In this paper, it was shown that every division algebra of degreep t has a prime-to-p extension which is a crossed product, if and only if the crossing product is at most 2.
Abstract: Every division algebra of degreep t has a prime-to-p extension which is a crossed product, ifft ≤ 2.