Showing papers in "Israel Journal of Mathematics in 1973"

Acyclic colorings of planar graphs

Abstract: A coloring of the vertices of a graph byk colors is called acyclic provided that no circuit is bichromatic. We prove that every planar graph has an acyclic coloring with nine colors, and conjecture that five colors are sufficient. Other results on related types of colorings are also obtained; some of them generalize known facts about “point-arboricity”.

A note on spherical summation multipliers

Abstract: We give a new proof of a theorem of L. Carleson and P. Sjolin onL p -boundedness of spherical summation operators in two variables.

The central limit theorem for geodesic flows onn-dimensional manifolds of negative curvature

Abstract: In this paper we prove a central limit theorem for special flows built over shifts which satisfy a uniform mixing of type $$\gamma ^{n^\alpha } $$ , 0 0. The function defining the special flow is assumed to be continuous with modulus of continuity of type $$f(z) = \sum olimits_{n = 0}^\infty {a_n z^n } $$ , 0 0, andd is the natural metric on sequence space. Geodesic flows on compact manifolds of negative curvature are isomorphic to special flows of this kind.

Markov partitions for anosov flows onn-dimensional manifolds

Abstract: In this work we construct a Markov partition for transitive Anosov flows, such that the measure of the boundary of the partition is zero. Symbolic dynamics for these flows is also developed.

L p bounds for pseudo-differential operators

Abstract: SharpL p boundedness results are proven for pseudo-differential operators in the classS pδ m .

Geodesic flows are Bernoullian

Abstract: A geometric method is developed for proving that transformations are isomorphic to Bernoulli shifts. The method is applied to the geodesic flows on surfaces of negative curvature and it is shown that they are isomorphic to Bernoulli flows.

Graph theorems for manifolds

Abstract: Two basic theorems about the graphs of convex polytopes are that the graph of ad-polytope isd-connected and that it contains a refinement of the complete graph ond+1 vertices. We obtain generalizations of these theorems, and others, for manifolds. We also supply some details for a proof of the lower bound inequality for manifolds.

Mean values and differential equations

Abstract: We show that the well-known equivalence between the mean-value theorem and harmonicity extends to arbitrary measures of compact support: a continuous function satisfies the generalized mean-value condition (1) with respect to a given measure if and only if it is annihilated by a certain system of homogeneous linear partial differential operators with constant coefficients determined by the measure Extensions of this result are obtained, primarily in the direction of replacing systems of differential equations by a single equation

The asymptotic behaviour of the solution of the filtration equation

Abstract: It is proved that the self-similar solution of the nonlinear equation of filtration gives the asymptotic representation of the solution of the Cauchy problem for the same equation.

On symmetric basic sequences in lorentz sequence spaces II

Abstract: It is shown that if {y n} is a block of type I of a symmetric basis {x n} in a Banach spaceX, then {y n} is equivalent to {x n} if and only if the closed linear span [y n] of {y n} is complemented inX. The result is used to study the symmetric basic sequences of the dual space of a Lorentz sequence spaced(a, p). Let {x n,f n} be the unit vector basis ofd(a, p), for 1≤p<+∞. It is shown that every infinite-dimensional subspace ofd(a, p) (respectively, [f n] has a complemented subspace isomorphic tol p (respectively,l q, 1/p+1/q=1 when 1

Prime flows in topological dynamics

Abstract: We present some results in topological dynamics and number theory. The number-theoretical results are estimates of the rates of convergence of sequences {fx26-1}, wherena is irrational,a is taken mod 1, and 0<β<1. One of these results is used to construct a homorphismT of a compact metric spaceX such that the minimal flow (X, T) had no nontrivial homomorphic images, i.e. is a prime flow. We define an infinite family of such flows, and describe other interesting properties of these flows.

Subsequences of normal sequences

Abstract: In this paper, we characterize a set of indices τ={τ(0)<τ(1)<…} such that forany normal sequence (α(0), α(1),…) of a certain type, the subsequence (α(τ(0)), α(τ(1)),…) is a normal sequence of the same type. Assume thatn→∞. Then, we prove that τ has this property if and only if the 0–1 sequence (θ τ (0), whereθ τ (i)=1 or 0 according asi∈{τ(j);j=0, 1,…} or not, iscompletely deterministic in the sense of B. Weiss.

The ergodic infinite measure preserving transformation of boole

Abstract: G. Boole proved that the transformation φ of the real line, defined by φ(x)=x−1/x, preserves Lebesgue measure. A general method is applied to proving that φ is ergodic. Some further applications of the method are also indicated.

Notes on spline functions III: On the convergence of the interpolating cardinal splines as their degree tends to infinity

Abstract: It is shown that for entire functionsf(x) defined by a Fourier-Stieltjes integral (9) the cardinal splineS m (x) of the odd degree 2m−1, which interpolatesf(x) at all integers, converges tof(x) asm tends to infinity. Properties of the exponential Euler spline are used in the proof.

Multivariate majorization and rearrangement inequalities with some applications to probability and statistics

Abstract: Multivariate generalizations of the concept of a Schur convex-function are defined and characterized These characterizations are shown to be useful in obtaining majorization and rearrangement inequalities We give simple derivations of known results as well as new ones with applications in probability and statistics

Markov random fields and gibbs random fields

Abstract: Spitzer has shown that every Markov random field (MRF) is a Gibbs random field (GRF) and vice versa when (i) both are translation invariant, (ii) the MRF is of first order, and (iii) the GRF is defined by a binary, nearest neighbor potential. In both cases, the field (iv) is defined onZ v, and (v) at anyxeZv, takes on one of two states. The current paper shows that a MRF is a GRF and vice versa even when (i)−(v) are relaxed, i.e., even if one relaxes translation invariance, replaces first order bykth order, allows for many states and replaces finite domains of Zv by arbitrary finite sets. This is achieved at the expense of using a many body rather than a pair potential, which turns out to be natural even in the classical (nearest neighbor) case when Zv is replaced by a triangular lattice.

The return of the linear search problem

Abstract: The linear search problem concerns a search on the real line for a point selected at random according to a given probability distribution. The search begins at zero and is made by a continuous motion with constant speed, first in one direction and then the other. The problem is to determine when it is possible to devise a “best” search plan. In former papers the best plan has been selected according to the criterion of minimum expected path length. In this paper we consider a more general, nonlinear criterion for a “best” plan and show that the substantive requirements of the earlier results are not affected by these changes.

First order theory of permutation groups

Abstract: We solve the problem of the elementary equivalence (definability) of the permutation groups over cardinals ℵα. We show that it suffices to solve the problem of elementary equivalence (definability) for the ordinals α in certain second order logic, and this is reduced to the case of α < (2ℵ 0)+. We solve a problem of Mycielski and McKenzie on embedding of free groups in permutation groups, and discuss some weak second-order quantifiers.

Lower bounds at infinity for solutions of differential equations with constant coefficients

Abstract: Two extensions of a classical theorem of Rellich are proved: (1) LetP=P(−iϖ/ϖx) be a partial differential operator with constant coefficients in\(\mathbb{R}^n \), let the manifolds contained in\(\left\{ {\xi \in \mathbb{R}^n ;P(\xi ) = 0} \right\}\) have codimension ≧k>0, and denote by Γ an open cone in\(\mathbb{R}^n \) intersecting each normal plane of every such manifold. If Open image in new window ,Pu=0 and Open image in new window it follows thatu=0. (2) Assume in addition that each irreducibe lfactor ofP van shes on a real hypersurface and that Γ contains both normal directions at some such point. If Open image in new window andP(D) u has compact support, the same condition withk=1 implies thatu has compact support. In both results the hypotheses on the cone Γ and on the operatorP are minimal.

A limit theorem for random coverings of a circle

Abstract: LetN α, m equal the number of randomly placed arcs of length α (0<α<1) required to cover a circleC of unit circumferencem times. We prove that limα→0 P(Nα,m≦(1/α) (log (1/α)+mlog log(1/α)+x)=exp ((−1/(m−1)!) exp (−x)). Using this result for m=1, we obtain another derivation of Steutel's resultE(Nα,1)=(1/α) (log(1/α)+log log(1/α)+γ+o(1)) as α→0, γ denoting Euler's constant.

M-structure in dual Banach spaces

Abstract: A result previously known only for certain ordered Banach spaces is generalized to arbitrary real Banach spaces Let ℒ be the Banach algebra of operators generated by theL-projections of a real Banach spaceU, and let ℳ (U * be the bounded operators on the dual spaceU * with adjoint in ℒ(U ** Then the adjoint operation maps ℒ (U) onto ℳ (U *) In particular, anyM-projection ofU * is weak* continuous

On subsequences of the Haar system inL p [0, 1], (1<p<∞)

Abstract: We show that ifX is the closed linear span inL p [0,1] of a subsequence of the Haar system, thenX is isomorphic either tol p or toL p [0,1], [1

Upward closure and cohesive degrees

Abstract: It is shown that the degrees of unsolvability of sets having almost any sort of immunity or cohesiveness property studied in recursion theory are closed upwards. From this it follows that every degreea witha′≧0″ contains a cohesive set.

Some essentially self-adjoint Dirac operators with spherically symmetric potentials

Abstract: It is shown that the one electron Dirac operator in a stationary electric field is essentially self-adjoint, on the domain of infinitely differentiable functions of compact support, for a class of spherically symmetric potentials including the Coulomb potential, for atomic numbers less than or equal to 118. In addition, the domain of the closure of the perturbed operator is the same as the domain of the closure of the unperturbed operator. We also give an abstract theorem on domain-preserving essential self-adjointness for perturbed operators, which is perhaps of independent interest.

Some examples concerning strictly convex norms onC(K) spaces

Abstract: Some examples ofC(K) spaces which admit (respectively, do not admit) an equivalent strictly convex norm are given. These examples consist of ideals inl c ∞ (I) (the bounded, real-valued functions on the unit intervalI having a countable support) which containc 0(I).

Differentially closed fields

Abstract: We prove that even the prime, differentially closed field of characteristic zero, is not minimal; that over every differential radical field of characteristicp, there is a closed prime one, and that the theory of closed differential radical fields is stable.

Beth’s theorem in cardinality logics

Abstract: We prove that the Beth definability theorem fails for a comprehensive class of first-order logics with cardinality quantifiers. In particular, we give a counterexample to Beth’s theorem forL(Q), which is finitary first-order logic (with identity) augmented with the quantifier “there exists uncountably many”.

Notes on combinatorial set theory

Abstract: We shall prove some unconnected theorems: (1) (G.C.H.) \omega _{\alpha + 1} \to \left( {\omega _\alpha + \xi } \right)_2^2 when ℵα is regular, │ξ│+ ℵ0 is a strong limit cardinal, then among the graphs with ≦λ vertices each of valence <λ there is a universal one. (4)(G.C.H.) If f is a set mapping on \omega _{a + 1} (ℵα regular) │f(x)∩f(y│<ℵα, then there is a free subset of order-type ζ for every ζ<ωα+1.

Torsion-free covers

Abstract: This paper studies the existence and properties of a torsion-free cover with respect to a faithful hereditary torsion theory (T, F) of modules over a ring with unity. A direct sum of a finite number of torsion-free covers of modules is the torsion-free cover of the direct sum of the modules. The concept of aT-near homomorphism, which generalizes Enochs’ definition of a neat submodule, is introduced and studied. This allows the generalization of a result of Enochs on liftings of homomorphisms. Hereditary torsion theories for which every module has a torsion-free cover are called universally covering. If the inclusion map ofR into the appropriate quotient ringQ is a left localization in the sense of Silver, the problem of the existence of universally-covering torsion theories can be reduced to the caseR=Q. As a consequence, many sufficient conditions for a hereditary torsion theory to be universally covering are obtained. For a universally-covering hereditary torsion theory (T, F), the following conditions are equivalent: (1) the product ofF-neat homomorphisms is alwaysT-neat; (2) the product of torsion-free covers is alwaysT-neat; (3) every nonzero module inT has a nonzero socle.

Maximal sets in α-recursion theory

Abstract: Let α be an admissible ordinal, and leta * be the Σ1-projectum ofa. Call an α-r.e. setM maximal if α→M is unbounded and for every α→r.e. setA, eitherA∩(α-M) or (α-A)∩(α-M) is bounded. Call and α-r.e. setM amaximal subset of α* if α*−M is undounded and for any α-r.e. setA, eitherA∩(α*-M) or (⇌*-A)∩(α*-M) is unbounded in α*. Sufficient conditions are given both for the existence of maximal sets, and for the existence of maximal subset of α*. Necessary conditions for the existence of maximal sets are also given. In particular, if α ≧ ℵ L then it is shown that maximal sets do not exist.