Showing papers in "Israel Journal of Mathematics in 1989"

Admissible observation operators for linear semigroups

Abstract: Consider a semigroupT on a Banach spaceX and a (possibly unbounded) operatorC densely defined inX, with values in another Banach space. We give some necessary as well as some sufficient conditions forC to be an admissible observation operator forT, i.e., any finite segment of the output functiony(t)=CTtx,t≧0, should be inLp and should depend continuously on the initial statex. Our approach is to start from a description of the map which takes initial states into output functions in terms of a functional equation. We also introduce an extension ofC which permits a pointwise interpretation ofy(t)=CTtx, even if the trajectory ofx is not in the domain ofC.

Homotopy spectral sequences and obstructions

Abstract: For a pointed cosimplicial spaceX •, the author and Kan developed a spectral sequence abutting to the homotopy of the total space TotX •. In this paper,X • is allowed to be unpointed and the spectral sequence is extended to include terms of negative total dimension. Improved convergence results are obtained, and a very general homotopy obstruction theory is developed with higher order obstructions belonging to spectral sequence terms. This applies, for example, to the classical homotopy spectral sequence and obstruction theory for an unpointed mapping space, as well as to the corresponding unstable Adams spectral sequence and associated obstruction theory, which are presented here.

On Λ( p )-subsets of squares

Abstract: This paper is a follow up of [B1]. It is shown that the sequence of squares {n 2|n=1, 2, ...} contains Λ(p)-subsets of “maximal density”, for any givenp>4. The proof is based on the probabilistic method developed in [B1] and a precise estimate of the Λ(p)-constant for the sequence of squares itself. Analogues of this phenomenon are obtained for other arithmetic sets, such as the sequence ofkth powers {n k |n=1, 2, ...} or the sequence of prime numbers. Sections 2 and 3 of the paper are of independent interest to orthogonal system theory.

Absolutely continuous invariant measures for piecewise expandingC 2 transformations inR N

Abstract: LetS be a bounded region inR N and let ℊ={S i} =1/ be a partition ofS into a finite number of subsets having piecewiseC 2 boundaries. We assume that whereC 2 segments of the boundaries meet, the angle subtended by tangents to these segments at the point of contact is bounded away from 0. Letτ:S →S be piecewiseC 2 on ℊ and expanding in the sense that there exists 0<σ< 1 such that for anyi=1, 2, ...,m, ‖Dτ −1 ‖<σ, whereDτ −1 is the derivative matrix ofτ −1 and ‖ ‖ is the euclidean matrix norm. The main result provides an upper bound onσ which guarantees the existence of an absolutely continuous invariant measure forτ.

Idempotents in compact semigroups and Ramsey theory

Abstract: We prove a theorem about idempotents in compact semigroups. This theorem gives a new proof of van der Waerden’s theorem on arithmetic progressions as well as the Hales-Jewett theorem. It also gives an infinitary version of the Hales-Jewett theorem which includes results of T. J. Carlson and S. G. Simpson.

All 2-manifolds have finitely many minimal triangulations

Abstract: A triangulation of a 2-manifoldM is said to be minimal provided one cannot produce a triangulation ofM with fewer vertices by shrinking an edge. In this paper we prove that all 2-manifolds have finitely many minimal triangulations. It follows that all triangulations of a given 2-manifold can be generated from the minimal triangulations by a process called vertex splitting.

The total reducibility order of a polynomial in two variables

Abstract: LetK be an algebraically closed field of characteristic zero. ForA ∈K[x, y] let σ(A) = {λ ∈K:A − λ is reducible}. For λ ∈ σ(A) letA − λ = ∏ (λ) A λ μ whereA iλ are distinct primes. Let ϱλ(A) =n(λ) − 1 and let ρ(A) = Σλɛσ(A)ϱλ(A). The main result is the following: Theorem.If A ∈ K[x, y] is not a composite polynomial, then ρ(A) < degA.

Ultrafilters with small generating sets

Abstract: It is consistent, relative to ZFC, that the minimum number of subsets ofω generating a non-principal ultrafilter is strictly smaller than the dominating number. In fact, these two numbers can be any two prescribed regular cardinals.

Boundaries of random walks on graphs and groups with infinitely many ends

Abstract: Consider an irreducible random walk {Z n} on a locally finite graphG with infinitely many ends, and assume that its transition probabilities are invariant under a closed group Γ of automorphisms ofG which acts transitively on the vertex set. We study the limiting behaviour of {Z n} on the spaceΩ of ends ofG. With the exception of a degenerate case,Ω always constitutes a boundary of Γ in the sense of Furstenberg, and {Z n} converges a.s. to a random end. In this case, the Dirichlet problem for harmonic functions is solvable with respect toΩ. The degenerate case may arise when Γ is amenable; it then fixes a unique end, and it may happen that {Z n} converges to this end. If {Z n} is symmetric and has finite range, this may be excluded. A decomposition theorem forΩ, which may also be of some purely graph-theoretical interest, is derived and applied to show thatΩ can be identified with the Poisson boundary, if the random walk has finite range. Under this assumption, the ends with finite diameter constitute a dense subset in the minimal Martin boundary. These results are then applied to random walks on discrete groups with infinitely many ends.

Variants of kazhdan's property for subgroups of semisimple groups

Abstract: Some variants of Kazhdan’s property (T) for discrete groups are presented. It is shown that some groups (e.g. SL n (Q),n≧3) which do not have property (T) still have some of these weaker properties. Applications to cohomology and infinitesimal rigidity for certain actions on manifolds are derived.

On the', existence of generalized complex space forms

Abstract: The existence of generalized complex space forms with nonconstant functionh is proved.

Propriétés de contraction d’un semi-groupe de matrices inversibles. Coefficients de Liapunoff d’un produit de matrices aléatoires indépendantes

Abstract: We study in this paper contraction properties of a matrix semi-groupT ⊂GL(d,R) acting on the flag space ofR d ; then we obtain properties of the Liapunoff exponents of theT-valued products of random matrices. The principal result is that, in this study, we can replaceT by its algebraic closureH inGL(d,R). This implies a “decomposition” of the action ofT in a proximal part and an isometric part; then we can write, modulo cohomology, the corresponding cocycle in a block-diagonal form, the blocks being similarities. In fact, we can express the multiplicities of the exponents in terms of the diagonal part of a conjugate of the groupH. So we obtain an extension of a recent result of Goldsheid and Margulis about the simplicity of Liapunoff’s spectrum [5]; this work uses their ideas as well as those of previous work [6].

Weak density and cupping in the d-r.e. degrees

Abstract: Consider the Turing degrees of differences of recursively enumerable sets (the d-r.e. degrees). We show that there is a properly d-r.e. degree (a d-r.e. degree that is not r.e.) between any two comparable r.e. degrees, and that given a high r.e. degreeh, every nonrecursive d-r.e. degree ≦h cups toh by a low d-r.e. degree.

Constructing modular classifying spaces

Abstract: Using the homotopy limit construction over a certain small category, we construct spaces whose modp cohomology algebras are the rings of invariants of some unitary reflection groups of order divisible byp.

Forcings with ideals and simple forcing notions

Abstract: Using generic ultrapower techniques we prove the following statements: (1) for every sequence 〈μn |n <ω〉 of 0–1σ-additive measures over the set of reals, there exists a set which is nonmeasurable in eachμn, (2) there is no nowhere primeσ-complete ℵ0-dense ideal, (3) ifI is a nowhere prime ideal over a setX then add (I) ≦d(I), (4) suppose thatμ is aσ-additive total nowhere prime probability measure over a setX, then add (μ)

Interpolation of compactness using Aronszajn-Gagliardo functors

Abstract: We prove that ifT: A 0 →B 0 andT: A 1 →B 1 both are compact, then $$T:F(\bar A) \to F(\bar B)$$ is also compact, whereF is the minimal or the maximal functor in the sense of Aronszajn-Gagliardo. We also derive some results for ordered couples.

The consistency strength of “every stationary set reflects”

Abstract: The consistency strength of a regular cardinal so that every stationary set reflects is the same as that of a regular cardinal with a normal idealI so that everyI-positive set reflects in aI-positive set. We call such a cardinal areflection cardinal and such an ideal areflection ideal. The consistency strength is also the same as the existence of a regular cardinal κ so that every κ-free (abelian) group is κ+-free. In L, the first reflection cardinal is greater than the first greatly Mahlo cardinal and less than the first weakly compact cardinal (if any).

Connectedness of Certain Random Graphs

Abstract: L. Dubins conjectured in 1984 that the graph on vertices {1, 2, 3, ...} where an edge is drawn between verticesi andj with probabilitypij=λ/max(i, j) independently for each pairi andj is a.s. connected forλ=1. S. Kalikow and B. Weiss proved that the graph is a.s. connected for anyλ>1. We prove Dubin’s conjecture and show that the graph is a.s. connected for anyλ>1/4. We give a proof based on a recent combinatorial result that forλ≦1/4 the graph is a.s. disconnected. This was already proved forλ<1/4 by Kalikow and Weiss. Thusλ=1/4 is the critical value for connectedness, which is surprising since it was believed that the critical value is atλ=1.

Geometrical implications of the existence of very smooth bump functions in Banach spaces

Abstract: We show that ifX is a Banach space and if there is a non-zero real-valuedC∞-smooth function onX with bounded support, then eitherX contains an isomorphic copy ofc0(N), or there is an integerk greater than or equal to 1 such thatX is of exact cotype 2k and, in this case,X contains an isomorphic copy ofl2k(N). We also show that ifX is a Banach space such that there is onX a non-zero real-valuedC4-smooth function with bounded support and ifX is of cotypeq forq<4, thenX is isomorphic to a Hilbert space.

Simplicial group models for Ωn S n X

Abstract: LetX be a pointed simplicial set. The free group functorsF [10] and Γ [1] provide simplicial models of ΩS |X| and Ω∞S∞ |X|. The simplicial groupFX is a simplicial subgroup of ΓX, and this corresponds to the inclusion ΩS |X| ⊂ ⊂Ω∞S∞X. In this paper we define free group functors Γ(n) such that Γ(n)X is a model of ΩnSn |X|. Moreover, there is natural filtration $$FX = \Gamma ^{(2)} X \subset \Gamma ^{(2)} X \subset \cdots \subset \Gamma ^{(n)} X \subset \cdots \subset \Gamma X,$$ (1) corresponding to the filtration $$\Omega S|X| \subset \Omega ^2 S^2 |X| \subset \cdots \subset \Omega ^2 S^2 |X| \subset \cdots \subset \Omega ^\infty S^\infty |X|.$$ (1) .

A blowup result for the critical exponent in cones

Abstract: We consider positive solutions of the initial value problem foru t=Δu+u p in conesD=R +×Ω⊆R N (Ω⊆S N−1). In an earlier paper, we determined a critical exponentp *(Ω) with the following properties: (a) if 1p *, then there are nontrivial global solutions (global existence case). Here we show thatp * belongs to the blowup case. This generalizes a well-known result for the critical exponentp *=1+2/N inD=R N .

ℏ-Normalizers and local definitions of saturated formations of finite groups

Abstract: We define, in each finite groupG, h-normalizers associated with a Schunck class ℏ of the formEΦ f with f a formation. We use these normalizers in order to give some sufficient conditions for a saturated formation of finite groups to have a maximal local definition.

Remarks on bifurcation for elliptic operators with odd nonlinearity

Abstract: On estimating the eigenvalues for a class of semilinear elliptic operators, we obtain bifurcation and comparison results concerning the eigenvalues of some related linear problem.

Fibrewise completion and unstable Adams spectral sequences

Abstract: We describe a tower of spaces whose inverse limit is a “fiberwise completion” of a fibrationE →B, and study the resulting spectral sequence converging to the homotopy groups of the space of lifts of a mapX →B. This is used to give a proof of the “generalized Sullivan conjecture”.

Onc 0 sequences in Banach spaces

Abstract: A Banach space has property (S) if every normalized weakly null sequence contains a subsequences equivalent to the unit vector basis ofc0. We show that the equivalence constant can be chosen “uniformly”, i.e., independent of the choice of the normalized weakly null sequence. Furthermore we show that a Banach space with property (S) has property (u). This solves in the negative the conjecture that a separable Banach space with property (u) not containingl1 has a separable dual.

On almost 1-1 extensions

Abstract: We show that a broad class of extensions of measure preserving systems in the context of ergodic theory can be realized by topological models for which the extension is “almost one-one”.

The covering numbers of the sporadic simple groups

Abstract: The covering numbers (cn) and the extended covering number (ecn) of all the sporadic simple groups and some other large groups are computed, using “economical” methods. An example is found, namelyD4(3), for which $$ecn(D_4 (3)) - cn(D_4 (3)) = 2$$

A conjecture on convolution operators, and a non-Dunford-Pettis operator onL1

Abstract: There exists a non-Dunford-Pettis operator fromL1 into a Banach latticeE that does not contain a copy ofc0 orL1. This problem is related to regularisation properties of convolution operators onL1.

Moduli of non-dentability and the radon-nikodým property in banach spaces

Abstract: We show that for a separable Banach spaceX failing the Radon-Nikodým property (RNP), ande > 0, there is a symmetric closed convex subsetC of the unit ball ofX such that every extreme point of the weak-star closure ofC in the bidualX** has distance fromX bigger than 1 −e. An example is given showing that the full strength of this theorem does not carry over to the non-separable case. However, admitting a renorming, we get an analogous result for this theorem in the non-separable case too. We also show that in a Banach space failing RNP there is, fore > 0, a convex setC of diameter equal to 1 such that each slice ofC has diameter bigger than 1 −e. Some more related results about the geometry of Banach spaces failing RNP are given.

Bousfield localization as an algebraic closure of groups

Abstract: We show that the notion ofHZ-local groups due to A. Bousfield which is based on considerations from algebraic topology can be defined and understood in terms of solubility of certain systems of equations overG.