An alternative proof of lower bounds for the first eigenvalue on manifolds

Recently, Andrews and Clutterbuck [1] gave a new proof of the optimal lower eigenvalue bound on manifolds via modulus of continuity for solutions of the heat equation. In this short note, we give an alternative proof of Theorem 2 in [1]. More precisely, following Ni's method (Section 6 of [5]), we give an elliptic proof of this theorem.     

Translation representation for automorphic solutions of the wave equation in non-euclidean spaces,IV

In Part I of this series of papers we have defined the incoming and outgoing translation representations for automorphic solutions of the hyperbolic wave equations; in Part II we have proved the completeness of these representations when the fundamental polyhedron F has a finite number of sides with a finite or infinite volume, but is not compact. In Part IV we present a proof of completeness which is simpler than our original proof contained in Section 7 of Part II for the case when F has cusps of less than maximal rank; and we supply a proof for the case, not covered in Section 7, when the parabolic subgroup associated with such cusps contains twists.     

Weighted norm inequalities for convolution operators and links with the Weiss Conjecture

We study convolution operators on weighted Lebesgue spaces and obtain weight characterisations for boundedness of these operators with certain kernels. Our main result is Theorem 3 which enables us to obtain results for certain kernel functions supported on bounded intervals; in particular we get a direct proof of the known characterisations for Steklov operators in Section 3 by using the weighted Hardy inequality. Our methods also enable us to obtain new results for other kernel functions in Section 4. In Section 5 we demonstrate that these convolution operators are related to operators arising from the Weiss Conjecture (for scalar-valued observation functionals) in linear systems theory, so that results on convolution operators provide elementary examples of nearly bounded semigroups not satisfying the Weiss Conjecture. Also we apply results on the Weiss Conjecture for contraction semigroups to obtain boundedness results for certain convolution operators.     

Backward Uniqueness for Thermoelastic Plates with Rotational Forces

This paper provides a backward uniqueness theorem for thermoelastic plate models which account for rotational forces, under all sets of canonical boundary conditions, including the most challenging case of so-called free boundary conditions. The proof is abstract and accomodates space variable coefficients in the model. This result is derived in two steps. First, in Section 3, a new backward uniqueness theorem for strongly continuous semigroups is given, which is of interest in itself. It is based on the assumption that the resolvent operator of the generator be bounded on suitable rays of the complex plane. Its proof uses the Phragmen-Lindelof Theorem. Next, the paper verifies a fortiori the required resolvent conditions, under all sets of canonical boundary conditions. The explicit proof (in Section 4) considers the most demanding case of free boundary conditions. An abstract version of this proof, and a corresponding backward uniqueness result, are then noted in Section 5, which gives the most general result of this paper. It covers thermoelastic wave equations as well. The results here presented were motivated by, and hence have important implications in, continuous observability/exact controllability problems for thermoelastic plates, and boundary observations/controls, see [8]. March 12, 1999     

Ideal Submodules Versus Ternary Ideals Versus Linking Ideals

We show that ideal submodules and closed ternary ideals in Hilbert modules are the same. We use this insight as a little peg on which to hang a little note about interrelations with other notions regarding Hilbert modules. In Section 3, we show that the ternary ideals (and equivalent notions) merit fully, in terms of homomorphisms and quotients, to be called ideals of (not necessarily full) Hilbert modules. The properties to be checked are intrinsically formulated for the modules (without any reference to the algebra over which they are modules) in terms of their ternary structure. The proofs, instead, are motivated from a third equivalent notion, linking ideals (Section 2), and a Theorem (Section 3) that all extends nicely to (reduced) linking algebras. As an application, in Section 4, we introduce ternary extensions of Hilbert modules and prove most of the basic properties (some new even for the known notion of extensions of Hilbert modules), by reducing their proof to the well-known analogue theorems about extensions of C^?–algebras. Finally, in Section 5, we propose several open problems that our method naturally suggests.

     

An Existence Theorem of Smooth Nonlocal Center Manifolds for Systems Close to a System with a Homoclinic Loop

Summary. In this paper we give a proof of the existence of smooth nonlocal center manifolds for systems close to a system with a homoclinic orbit to a saddle-type equilibrium point. Our proof is based on a consideration of some class of the boundary value problems (see Section 3). We obtain estimates for solutions of the boundary value problems that allow us to prove the theorem on the center manifolds at the C ¹ -assumptions for the smoothness of systems. Received June 4, 1997; final revision received April 24, 1998     

A new result on difference sets with -1 as multiplier

In this paper we study finite abelian groups admitting a difference set with multiplier -1. In these groups we have that each integer, which is relatively prime to the group order, is a multiplier (see [1] and Section 1 of this paper).About the arithmetical structure, there is an interesting result of Jungnickel [3] on primes dividing the order n of the corresponding design. Here we prove (see Theorem 2.1) that each odd prime divisor of the order v of the group divides n. The proof of Theorem 2.1 rests on character computations and is motivated by [5].     

On Properties of a Graph that Depend on its Distance Function

If G is a connected graph with distance function d, then by a step in G is meant an ordered triple (u, x, v) of vertices of G such that d(u, x) = 1 and d(u, v) = d(x, v) + 1. A characterization of the set of all steps in a connected graph was published by the present author in 1997. In Section 1 of this paper, a new and shorter proof of that characterization is presented. A stronger result for a certain type of connected graphs is proved in Section 2.     

The undecidability of pseudo real closed fields

The aim of this note is to establish the following result: THEOREM: Let be a non-empty class of Boolean spaces and letPRC() be the class of pseudo real closed fields whose spaces of orderings belong to . Then the elementary theory ofPRC() is undecidable.Our proof appears to be an interesting application of the theory of Artin-Schreier structures, which has been initiated in [5] for the purpose of characterization of the absolute Galois groups of PRC fields. In Section 1 we define and investigate Frattini covers of Artin-Schreier structures, in analogy with [6], Section 2. In Section 2 we consider the analogues of proofs of [1] and [3], to attain the Theorem.This work corresponds to a part of the doctoral dissertation done under the supervision of Prof. Moshe Jarden at Tel-Aviv University     

Some Special Tauberian Theorems for Fourier Type Integrals and Eigenfunction Expansions of Sturm-Liouville Equations on the Whole Line

We offer a new proof of a special Tauberian theorem for Fourier type integrals. This Tauberian theorem was already considered by us in the papers [1] and [2]. The idea of our initial proof was simple, but the details were complicated because we used Bochner's definition of generalized Fourier transform for functions of polynomial growth. In the present paper we work with L. Schwartz's generalization. This leads to significant simplification. The paper consists of six sections. In Section 1 we establish an integral representation of functions of polynomial growth (subjected to some Tauberian conditions), in Section 2 we prove our main Tauberian theorems (Theorems 2.1 and 2.2.), using the integral representation of Section 1, in Section 3 we study the asymptotic behavior of M. Riesz's means of functions of polynomial growth, in Sections 4 and 5 we apply our Tauberian theorems to the problem of equiconvergence of eigenfunction expansions of Sturm-Liouville equations and expansion in ordinary Fourier integrals, and in Section 6 we compare our general equiconvergence theorems of Sections 4 and 5 with the well known theorems on eigenfunction expansions in classical orthogonal polynomials. In some sense this paper is a re-made survey of our results obtained during the period 1953-58. Another proof of our Tauberian theorem and some generalization can be found in the papers [3] and [4].     

Corrigendum: The cover time of the giant component of a random graph,Random Structures and Algorithms 32 (2008), 401–439

The above paper gives an asymptotically precise estimate of the cover time of the giant component of a sparse random graph. The proof as it stands is not tight enough, and in particular, Eq. (64) is not strong enough to prove (65). The o(1) term in (64) needs to be improved to o(1/(lnn)²) for (65) to follow. The following section, which replaces Section 3.6, amends this oversight. The notation and definitions are from the paper. A further correction is needed. Property P2 claims that the conductance of the giant is whp , Ω(1/lnn). The proof of P2 in the appendix of the paper is not quite complete. A complete proof of Property P2 can be found in 1 , 2 . © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Rigidity of the critical point equation

On a compact n ‐dimensional manifold M, it was shown that a critical point metric g of the total scalar curvature functional, restricted to the space of metrics with constant scalar curvature of volume 1, satisfies the critical point equation ([5], p. 3222). In 1987 Besse proposed a conjecture in his book [1], p. 128, that a solution of the critical point equation is Einstein (Conjecture A, hereafter). Since then, number of mathematicians have contributed for the proof of Conjecture A and obtained many geometric consequences as its partial proofs. However, none has given its complete proof yet. The purpose of the present paper is to prove Theorem 1, stating that a compact 3‐dimensional manifold M is isometric to the round 3‐sphere S³ if ker s^′*_g ≠ 0 and its second homology vanishes. Note that this theorem implies that M is Einstein and hence that Conjecture A holds on a 3‐dimensional compact manifold under certain topological conditions (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bifurcation of periodic orbits in coupled chemical reactors II

In this part of the paper we deal with computational aspects of the bifurcation problem introduced in Part I, cf. [1]. It follows from Section 3 that the bifurcation behaviour is essentially determined by the expression KP(Ω) – ∧β, where β is some constant which was assumed to be nonzero, while P(Ω) is some 2π-periodic function which was implicitly defined in Section 2. In Section 4 we first derive easily computable expressions for P(Ω) and β from the definitions of Section 2. Secondly we offer a direct, although formal, derivation for these quantities. In Section 5 we briefly show how P(Ω). β can actually be computed by means of a high speed computer and discuss a number of examples. In Section 6 finally we give some asymptotic results for systems depending on an additional parameter.     

The geometric and numerical properties of duality in projective algebraic geometry

In this paper we investigate some fundamental geometric and numerical properties ofduality for projective varieties inP _k ^N =P ^N . We take a point of view which in our opinion is somewhat moregeometric and lessalgebraic andnumerical than what has been customary in the literature, and find that this can some times yield simpler and more natural proofs, as well as yield additional insight into the situation. We first recall the standard definitions of thedual variety and theconormal scheme, introducing classical numerical invariants associated with duality. In section 2 we recall the well known duality properties these invariants have, and which was noted first byT. Urabe. In section 3 we investigate the connection between these invariants andChern classes in the singular case. In section 4 we give a treatment of the dual variety of a hyperplane section of X, and the dual procedure of taking the dual of a projection of X. This simplifies the proofs of some very interesting theorems due toR. Piene. Section 5 contains a new and simpler proof of a theorem ofA. Hefez and S. L. Kleiman. Section 6 contains some further results, geometric in nature.     

On a Characterization of the Gamma Distribution: The Independence of the Sample Mean and the Sample Coefficient of Variation

Let n 3 and let X ₁,...,X _n be positive i.i.d. random variables whose common distribution function f has a continuous p.d.f. Using earlier work of the present authors and a method due to Anosov for solving certain integro-functional equations, it is shown that the independence of the sample mean and the sample coefficient of variation is equivalent to that f is a gamma function. While the proof is of methodological interest, this conclusion can also be arrived at without any assumptions by appealing to the Laplace-Stieltjes transform, as in the Concluding Romark (Section 3).     

On Positive Sasakian Geometry

A Sasakian structure =(\xi,\eta,\Phi,g) on a manifold Mis called positiveif its basic first Chern class c₁( ) can be represented by a positive (1,1)-form with respect to its transverse holomorphic CR-structure. We prove a theorem that says that every positive Sasakian structure can be deformed to a Sasakian structure whose metric has positive Ricci curvature. This provides us with a new technique for proving the existence of positive Ricci curvature metrics on certain odd dimensional manifolds. As an example we give a completely independent proof of a result of Sha and Yang that for every nonnegative integer kthe 5-manifolds k#(S ²×S ³) admits metrics of positive Ricci curvature.     

Two operators on the lattice of completely regular semigroup varieties

In this paper, varieties of completely regular semigroups are studied. This paper is divided into six sections. Section 1 contains an introduction to varieties of completely regular semigroups and preliminaries. Most of the notation needed in this paper is given. In Section 2, the operators \La ( ) and \Ra ( ) on the lattice of subvarieties of varieties of completely regular semigroups are investigated. In Section 3, some further properties of the operators \La ( ) and \Ra ( ) are given. In Section 4, the semigroups generated by various subset of some operators are considered. In Section 5, the operators \La ( ) and \Ra ( ) are used in finding the join of two given varieties. The word problem for free objects in the variety OLBG is considered in Section 6 using the operator \La ( ) . June 1, 1999     

Obstructions to positive curvature and symmetry

We discuss new obstructions to positive sectional curvature and symmetry. The main result asserts that the index of the Dirac operator twisted with the tangent bundle vanishes on a 2-connected manifold of dimension ≠8 if the manifold admits a metric of positive sectional curvature and isometric effective S¹-action. The proof relies on the rigidity theorem for elliptic genera and properties of totally geodesic submanifolds.     

On Bayes estimates

In this paper the author tries to give general conditions for the existence of Bayes estimates and for the consistency of sequences of Bayes estimates.In Section 3 we prove existence theorems for Bayes estimates, which contain those of DeGroot and Rao [3], as a special case. The proof is based on a theorem of Landers [5].Section 4 gives a characterization of Bayes estimates with convex loss and linear decision space. This theorem is also a generalization of a similar theorem of DeGroot and Rao [3].In Section 5 we generalize the theory of minimum contrast estimates (the foundations of which were laid by Huber [4], cf. Pfanzagl [6]) in such a way that we can apply it to the theory of Bayes estimates.Section 6 tries to give a general theory of consistency for Bayes estimates using the martingale argument of Doob [1] and the theory of minimum contrast estimates. Confer in this connection the results of Schwartz [8].Section 7 contains some auxiliary results.     

New light on Hensel''s lemma

The historical development of Hensel's lemma is briefly discussed (Section 1). Using Newton polygons, a simple proof of a general Hensel's lemma for separable polynomials over Henselian fields is given (Section 3). For polynomials over algebraically closed, valued fields, best possible results on continuity of roots (Section 4) and continuity of factors (Section 6) are demonstrated. Using this and a general Krasner's lemma (Section 7), we give a short proof of a general Hensel's lemma and show that it is, in a certain sense, best possible (Section 8). All valuations here are non-Archimedean and of arbitrary rank. The article is practically self-contained.