Showing papers in "Israel Journal of Mathematics in 1998"
TL;DR: In this paper, the authors consider the case when the semigroup generated by A is only equicontinuous and obtain existence of integral solutions if, in particular, X* is uniformly convex and F satisfies β(F(t,B))≤k(t)β(B) for all boundedB⊂X where β denotes the Hausdorff-measure of noncompactness.
Abstract: Given anm-accretive operatorA in a Banach spaceX and an upper semicontinuous multivalued mapF: [0,a]×X→2
X
, we consider the initial value problemu′∈−Au+F(t,u) on [0,a],u(0)=x
0. We concentrate on the case when the semigroup generated by—A is only equicontinuous and obtain existence of integral solutions if, in particular,X* is uniformly convex andF satisfies β(F(t,B))≤k(t)β(B) for all boundedB⊂X wherek∈L
1([0,a]) and β denotes the Hausdorff-measure of noncompactness. Moreover, we show that the set of all solutions is a compactR
δ-set in this situation. In general, the extra condition onX* is essential as we show by an example in whichX is not uniformly smooth and the set of all solutions is not compact, but it can be omited ifA is single-valued and continuous or—A generates aC
o-semigroup of bounded linear operators. In the simpler case when—A generates a compact semigroup, we give a short proof of existence of solutions, again ifX* is uniformly (or strictly) convex. In this situation we also provide a counter-example in ℝ4 in which no integral solution exists.
253 citations
TL;DR: In this article, it was proved that every normalized weakly null sequence has a subsequence which is convexly unconditional and a hierarchy of summability methods was introduced. And with this, a complete classification of the complexity of weakly-null sequences was given.
Abstract: It is proved that every normalized weakly null sequence has a subsequence which is convexly unconditional. Further, a hierarchy of summability methods is introduced and with this we give a complete classification of the complexity of weakly null sequences.
111 citations
TL;DR: In this paper, the volume of central hyperplane sections of star bodies in R n ≥ 2 was analyzed in terms of the Fourier transform of a power of the radial function.
Abstract: We express the volume of central hyperplane sections of star bodies inR
n
in terms of the Fourier transform of a power of the radial function, and apply this result to confirm the conjecture of Meyer and Pajor on the minimal volume of central sections of the unit balls of the spacesl
with 0
85 citations
TL;DR: In this article, the authors investigated the structure of finitely generated shift-invariant spaces and solvability of linear operator equations, including linear partial difference equations and discrete convolution equations.
Abstract: In this paper we investigate the structure of finitely generated shift-invariant spaces and solvability of linear operator equations. Fourier trans-forms and semi-convolutions are used to characterize shift-invariant spaces. Criteria are provided for solvability of linear operator equations, including linear partial difference equations and discrete convolution equations. The results are then applied to the study of local shift-invariant spaces. Moreover, the approximation order of a local shift-invariant space is characterized under some mild conditions on the generators.
80 citations
TL;DR: For a hypergraph H with fractional cover number ρ*,N(G, H) = θ(lρ*) as mentioned in this paper, the upper bound is shown to be a simple consequence of an entropy lemma of J. Shearer.
Abstract: Given two hypergraphsH andG, letN(G, H) denote the number of subhypergraphs ofG isomorphic toH. LetN(l, H) denote the maximum ofN(G, H), taken over allG with exactlyl edges. In [1] Noga Alon analyzes the asymptotic behaviour ofN(l, H) forH a graph. We generalize this to hypergraphs: Theorem:For a hypergraph H with fractional cover number ρ*,N(G,H).=θ(lρ*) The interesting part of this is the upper bound, which is shown to be a simple consequence of an entropy lemma of J. Shearer. In a special case which includes graphs, we also provide a different proof using a hypercontractive estimate.
80 citations
TL;DR: In this paper, it was shown that every Frechet differentiable real function on C(K), K scattered with locally uniformly continuous derivative has locally compact derivative, and the existence of C 2-Frechet smooth surjections between various Banach spaces was investigated.
Abstract: We show that every Frechet differentiable real function onC(K), K scattered with locally uniformly continuous derivative has locally compact derivative. Using this and similar results, we investigate the existence ofC2-Frechet smooth surjections between various Banach spaces.
44 citations
TL;DR: In this paper, the authors prove general results on complemented unconditional basic sequences in Banach lattices and apply it to give some new examples of spaces with unique unconditional bases, such as Tsirelson space and Nakano spaces.
Abstract: We prove a general results on complemented unconditional basic sequences in Banach lattices and apply it to give some new examples of spaces with unique unconditional basis We show that Tsirelson space and certain Nakano spaces have unique unconditional bases We also construct an example of a space with a unique unconditional basis with a complemented subspace failing to have a unique unconditional basis
43 citations
TL;DR: In this article, the exponential growth of the codimensions of T-ideals of matrix algebras was shown to be an integer, in particular, it is always an integer.
Abstract: By Kemer’s theory [9],T idealsJ
1 ∪…∪J
r andJ
1 …J
r, where eachJ
i is verbally prime, are of fundamental importance in the theory of P.I. algebras. We calculate, approximately and asymptotically, the codimensions of suchT-ideals, thereby extending the corresponding results about matrix algebras. In all such cases, the exponential growth of the codimensions is calculated; in particular, it is always an integer.
39 citations
TL;DR: In this article, the authors considered the stability of dissipative wave systems with nonlinear damping forces and strongly nonlinear potential energies, and showed that global asymptotic stability can then no longer be expected and should be replaced by local stability.
Abstract: We study the question of asymptotic stability, as time tends to infinity, of solutions of dissipative wave systems, governed by time-dependent nonlinear damping forces and by strongly nonlinear potential energies. This problem had been considered earlier for potential energies which arise from restoring forces, whereas here we allow as well for the effect of amplifying forces. Global asymptotic stability can then no longer be expected, and should be replaced by local stability. The conclusions are related to and supplement earlier work of Payne and Sattinger [7], who treated the nondissipative case, and of Hale [1], who showed the existence of connected global attractors.
39 citations
TL;DR: In this paper, it was shown that the l1-index of a separable Banach space not containing l1 must be of the form ωα for some countable ordinal α.
Abstract: A well known argument of James yields that if a Banach spaceX contains l
1
’s uniformly, thenX contains l
1
’s almost isometrically. In the first half of the paper we extend this idea to the ordinal l1-indices of Bourgain. In the second half we use our results to calculate the l1-index of certain Banach spaces. Furthermore we show that the l1-index of a separable Banach space not containing l1 must be of the form ωα for some countable ordinal α.
38 citations
TL;DR: In this article, a geometric realization of a harmonic function on a finite planar Markov chain was constructed as a tiling of a rectangle with trapezoids, each trapezoid having two horizontal edges.
Abstract: Letf be a harmonic function on a finite planar Markov chainM whose boundary consists of two vertices on the same face. We construct a geometric realization of (M, f) as a tiling of a rectangle with trapezoids, each trapezoid having two horizontal edges. Conversely, each such tiling arises in this way. Similar results hold for harmonic functions with more general boundary conditions. Certain prescriptions of transition probabilities on edges inM give rise to tilings with prescribed shapes. This allows us to give necessary conditions for the existence of a tiling of an arbitrary polygon with squares, equilateral triangles, and so on. Using this method, we classify all polygons with at most one non-convex vertex which can be tiled with squares. A similar classification holds for tiling with equilateral triangles. We determine the Euclidean tori which can be square-tiled.
TL;DR: For any set of characters on a compact group there exists a subset of proportional size such that, on the span of this subset, the l 1 and l 2 norm are equivalent up to a factorC logn log log logn)1/2 as mentioned in this paper.
Abstract: We use majorizing measures to provide a simpler proof of the following unpublished result of J. Bourgain. For any set of characters on a compact group there exists a subset of proportional size such that, on the span of this subset, thel
1 andl
2 norm are equivalent up to a factorC logn log logn)1/2.
TL;DR: In this paper, the authors define HNN extensions for Lie algebras and prove some results analogous to results in group theory, and then use these results to prove some interesting results.
Abstract: In this paper, we define HNN extensions for Lie algebras. Given a Lie algebraL with a subalgebraA and a derivationd:A →L, the HNN extension containsL andd extends to an inner derivation of it. We then use it to prove some results analogous to results in group theory.
TL;DR: In this paper, the authors studied the connection between the dynamics of the subgroup action of H on G/Gℤ and the representation-theoretic properties of H being observable and epimorphic in G.
Abstract: LetH be an ℝ-subgroup of a ℚ-algebraic groupG. We study the connection between the dynamics of the subgroup action ofH onG/Gℤ and the representation-theoretic properties ofH being observable and epimorphic inG. We show that ifH is a ℚ-subgroup thenH is observable inG if and only if a certainH orbit is closed inG/Gℤ; that ifH is epimorphic inG then the action ofH onG/Gℤ is minimal, and that the converse holds whenH is a ℚ-subgroup ofG; and that ifH is a ℚ-subgroup ofG then the closure of the orbit underH of the identity coset image inG/Gℤ is the orbit of the same point under the observable envelope ofH inG. Thus in subgroup actions on homogeneous spaces, closures of ‘rational orbits’ (orbits in which everything which can be defined over ℚ, is defined over ℚ) are always submanifolds.
Abstract: We find a small weakly minimal theory with an isolated weakly minimal type ofM-rank ∞ and an isolated weakly minimal type of arbitrarily large finiteM-rank. These examples lead to the notion of an m-normal theory. We prove theM-gap conjecture for m-normalT. In superstable theories with few countable models we characterize traces of complete types as traces of some formulas. We prove that a 1-based theory with few countable models is m-normal. We investigate generic subgroups of small superstable groups. We compare the notions of independence induced by measure (μ-independence) and category (m-independence).
TL;DR: In this paper, the authors developed a theory for the asymptotic behavior of polynomials and polynomial maps over R and over C and applied it to the Jacobian conjecture.
Abstract: The aim of this paper is to develop a theory for the asymptotic behavior of polynomials and of polynomial maps overR and overC and to apply it to the Jacobian conjecture. This theory gives a unified frame for some results on polynomial maps that were not related before.
TL;DR: Theorem 3.1 in [K-T] has a gap in the proof of the fundamental criterium, Proposition 1.1 as discussed by the authors, which has been corrected in this paper.
Abstract: This note conta ins a corrected proof of the main resul t (which remains unchanged) from [K-T]. I t was recently observed t h a t an a rgument in a basic technical c r i te r ium has a gap. The first named author recently observed that our paper [K-T] contains a gap. Here we shall correct the formulation of a general result (Theorem 2.1 in [K-T]) and then we shall complete the proof of the main application to the homogeneous Banach space problem (Theorem 3.1 in [K-T]), that is now neccessary in view of modifications in the aforementioned general statement. Corollaries 4.3, 4.4 and 4.5 from [K-T] do not require any changes. This note is not self-contained. However, in order to make it possible for the readers a little familiar with [K-T] to read it without constantly referring to the original paper, we occasionally recall background information from [K-T]. The gap is in the proof of the fundamental criterium, Proposition 1.1 in [K-T]. One deals there with a Banach space Y which has a 2-dimensional Received July 11, 1996
TL;DR: This work considers the problem of embedding a certain finite metric space to the Euclidean space, trying to keep the bi-Lipschitz constant as small as possible, and introduces the notationc2(X, d) for the least distortion with which the metric space may be embedded in a Euclideans space.
Abstract: We consider the problem of embedding a certain finite metric space to the Euclidean space, trying to keep the bi-Lipschitz constant as small as possible. We introduce the notation c2 (X, d) for the least distortion with which the metric space (X, d) may be embedded in a Euclidean space. It is known that if (X,d) is a metric space with n points, then c2(X,d) 0(logn) and the bound is tight. Let T be a tree with n vertices, and d be the metric induced by it. We show that c2(T,d) < 0(loglogn), that is we provide an embedding f of its vertices to the Euclidean space, such that
TL;DR: In this paper, a family of infinite measure preserving odometers is presented which exhibit examples of p-recurrent but notp+1-recurrence ergodic transformations for everyp>1.
Abstract: A family of infinite measure preserving odometers is presented which exhibit examples ofp-recurrent but notp+1-recurrent ergodic transformations for everyp>1.
TL;DR: For a compact manifold of dimension 2n, this article showed that the canonical homology group is not isomorphic to the de Rham cohomology group for a Poisson manifold.
Abstract: For a compact symplectic manifoldM of dimension 2n, Brylinski proved that the canonical homology groupH
(M) is isomorphic to the de Rham cohomology groupH
2n-k
(M), and the first spectral sequence {E
r
(M)} degenerates atE
1(M). In this paper, we show that these isomorphisms do not exist for an arbitrary Poisson manifold. More precisely, we exhibit an example of a five-dimensional compact Poisson manifoldM
5 for whichH
1
(M
5) is not isomorphic toH
4(M
5), andE
1(M
5) is not isomorphic toE
2(M
5).
TL;DR: In this paper, a local ring R whose Jacobson radical is locally nilpotent, but M 2(R) is not strongly π-regular, has been shown to be locally nil-potent.
Abstract: We fill a gap in [4], and provide a rigorous example of a local ringR whose Jacobson radical is locally nilpotent, butM 2(R) is not strongly π-regular
Abstract: This paper is devoted to some local-global type questions about fields of definition of algebraic covers. Letf:X→B be a covera priori defined over
. Assume that the coverf can be defined over each completion ℚ{p} of ℚ. Does it follow that the cover can be defined over ℚ? This is thelocal-to-global principle. It was shown to hold for G-covers [DbDo], i.e., for Galois covers given with their automorphisms. Here we prove that, in the situation ofmere covers, the local-to-global principle holds under some additional assumptions on the groupG of the cover and the monodromy representationG→S
d (withd=deg(f)). This local-to-global problem is closely related to the obstruction to the field of moduli being a field of definition. This problem was studied in [DbDo], which is the main tool of the present paper.
TL;DR: In this article, it was shown that for every positive integer number n, a 2-dimensional continuum contains a 1-dimensional sub-contiuum with hyperspace of dimension ≥ n.
Abstract: In [8] Y. Sternfeld and this author gave a positive answer to the following longstanding open problem: Is the hyperspace (=the space of all subcontinua endowed with the Hausdorff metric) of a 2-dimensional continuum infinite dimensional? This result was improved in [9] where it was shown that for every positive integer numbern a 2-dimensional continuum contains a 1-dimensional subcontiuum with hyperspace of dimension ≥n. And it was asked there: Does a 2-dimensional continuum contain a 1-dimensional subcontinuum with infinite dimensional hyperspace? In this note we answer this question in the positive.
TL;DR: In this paper, it was shown that the coefficients of the right 2-characteristic polynomial of a supermatrix over a Grassmann algebra are integral of degreen 2 over G ≥ 0.
Abstract: We prove that the coefficients of the so-called right 2-characteristic polynomial of a supermatrix over a Grassmann algebraG=G
0⊕G
1 are in the even componentG
0 ofG. As a consequence, we obtain that the algebra ofn×n supermatrices is integral of degreen
2 overG
0.
TL;DR: For the classical Banach spaces X = lp,C(K) as discussed by the authors, every polynomial of degreen + 1 on X is uniformly approximable on the unit ball by elements of the algebra generated by all polynomials of degree up ton on X.
Abstract: For the classical Banach spacesX = lp,C(K) we identify alln such that every polynomial of degreen + 1 onX is uniformly approximable on the unit ball by elements of the algebra generated by all polynomials of degree up ton onX.
TL;DR: In this paper, it is shown that given any sequence of natural numbers satisfying a certain growth rate, all BernoulliG actions are equivalent to a certain natural action of G, which we call the translation action.
Abstract: In this paper we look atr equivalence, which is an equivalence relation that is implicit in Vershik's classification ofr-adic decreasing sequences of σ-algebras, and also in work of Stepin. this equivalence relation is used to classify group actions of the group\(G = \sum _{n = 1}^\infty Z/r_n Z,r = (r_1 ,r_2 ,...)\) It is shown that given any sequence of natural numbers satisfying a certain growth rate, all BernoulliG actions arer equivalent to a certain natural action ofG, which we call the translation action. Furthermore, these actions are zeror entropy andr finitely determined, where this notion arises canonically from the restricted orbit equivalence theory of Kammeyer and Rudolph.
TL;DR: In this paper, the authors constructed a C 1 expanding circle map with the property that it has no σ-finite invariant measure equivalent to Lebesgue measure, and extended the construction to interval maps and maps on higher dimensional tori and the Riemann sphere.
Abstract: In this paper we construct aC
1 expanding circle map with the property that it has no σ-finite invariant measure equivalent to Lebesgue measure. We extend the construction to interval maps and maps on higher dimensional tori and the Riemann sphere. We also discuss recurrence of Lebesgue measure for the family of tent maps.
TL;DR: In this article, a homogeneous cocycle with the scaling property was defined, which is a realization of fractal functions which admit the continuous scalings, and a self-similar process with strictly ergodic, stationary increments which has 0 entropy.
Abstract: We consider a compact space Θ on whichR acts additively andR
+ acts multiplicatively satisfying the distributive law. Moreover,R-action is strictly ergodic. Such Θ is constructed as a space of colored tilings corresponding to a weighted substitution, which is a kind of natural extension of thef-expansion for a piecewise linearf. We define a homogeneous cocycleF on Θ, which was called a cocycle with the scaling property in [2]. This is a realization of fractal functions which admit the continuous scalings. This also defines a self-similar process with strictly ergodic, stationary increments which has 0 entropy.
TL;DR: In this article, a lattice of sub-synchronizing subshifts which describes the synchronizing parts of a sofic shift of finite type is presented, which preserves their mutual relationship and the corresponding covers within the Krieger covers are topologically conjugate.
Abstract: We introduce a notion of magic words and, through them, we present a lattice of sub-synchronizing subshifts which describes the synchronizing parts of a sofic shiftS. We show that topological conjugacy maps subsynchronizing subshifts onto sub-synchronizing subshifts, it preserves their mutual relationship (i.e. the corresponding lattices are isomorphic) and the corresponding covers within the Krieger covers are topologically conjugate. Using the magic words, a full characterization of the syntactic monoid of a shift of finite type is given. We show that a synchronizing deterministic presentation of every sub-synchronizing subshift ofS can be seen within a two-sided ideal of the syntactic monoid ofS.
TL;DR: In this paper, the spectral multiplicity of unitary operators defined by cocycles over an irrational rotation α was studied and it was shown that the multiplicity is finite whenever the cocycle has bounded variation and gave explicit bounds.
Abstract: We study the spectral multiplicity of unitary operators of\(L^2 (\mathbb{T})\) defined by cocycles over an irrational rotation α. We prove that the multiplicity is finite whenever the cocycle has bounded variation and we give explicit bounds. For a cocycle given by an absolutely continuous function ϕ on [0,1], we show that the multiplicity is strictly less than max (2.•∫ϕ(x)dx•+1), which is optimal in the case ϕ(x)=nx (where the multiplicity is exactlyn). The proofs are based on the representation of the rotation as a “local rank one” transformation, which arises from the continued fraction expansion of α.