On the ingham divisor problem

The main result of this paper is the following theorem. Suppose thatτ(n) = ∑ _d|n l and the arithmetical functionF satisfies the following conditions:

A Note on the KAM Theorem for Symplectic Mappings

The mapping version of Pöschel's theory on differentiable foliation structures of invariant tori is presented and the relevant estimates explicitly in terms of the diophantine constant and the nondegeneracy parameters of frequency maps are given. As a direct application of the main result, a generalization of Moser's small twist theorem to high dimensions is given.     

Mappings on Spaces of Continuous Functions

Let X and Y be locally compact Hausdorff spaces and T : C₀(X) C₀(Y) a ring homomorphism. We completely characterize such homomorphisms and show that if T is R-linear, then T is either C-linear or C-antilinear. In any case T is continuous and there is a continuous map : Y X such that Tf = f o , f C₀(X) (if T is C-linear) or (if T is C-antilinear). Thus, extending a result of Mólnar, we also derive the general form of an isometry T.AMS Subject Classification (2000): primary 46J05, 46E25(deceased) Passed away on 24 May 1999.     

NOTES ON GLAISHER＇S CONGRUENCES

Let p be an odd prime and let n≥1,k≥0 and r be integers,denote by Bk the kth Bernoulli number,It is proved that(i) If r≥1 is odd and suppose 1≥r+4,then ∑j=1^p-1 1/(np+j)^r=-(2n+1)r(r+1)/2(r+2)Bp-r-2p^2(mod p^3).(ii)If r≥2 is even and suppose p≥r+3, then p-1∑j=1 1/(np+j)^r=r/r+1Bv-r-1p(mod P^2).(iii) p-1∑j=1 1/(np+j)p-2=-(2n+1)p(mod P^2).This result generalizes the Glaisher‘s congruence. As a corollary, a generalization of the Wolsten-holme‘s theorem is obtained.     

Numerical analysis of a quadratic matrix equation

The quadratic matrix equation AX²+ BX + C = 0in n x nmatricesarises in applications and is of intrinsic interest as oneof the simplest nonlinear matrix equations. We give a completecharacterization of solutions in terms of the generalized Schurdecomposition and describe and compare various numerical solutiontechniques. In particular, we give a thorough treatment offunctional iteration methods based on Bernoulli’s method.Other methods considered include Newton’s method with exact line searches, symbolic solution and continued fractions.We show that functional iteration applied to the quadraticmatrix equation can provide an efficient way to solve the associated quadratic eigenvalue problem (²A + B + C)x = 0.     

Komlós Theorem for Unbounded Random Sets

We present the Komlós theorem for multivalued functions whose values are closed (possibly unbounded) convex subsets of a separable Banach space. Komlós theorem can be seen as a generalization of the SLLN for it deals with a sequence of integrable multivalued functions that do not have to be identically distributed nor independent. The Artstein–Hart SLLN for random sets with values in Euclidean spaces is derived from the main result. Finally, since the main theorem concerns multifunctions whose values are allowed to be unbounded, we can restate it in terms of normal integrands (random lower semicontinuous functions).     

Large volume growth and finite topological type

It is shown in this paper that a complete noncompact -dimensional Riemannian manifold with nonnegative Ricci curvature, sectional curvature bounded from below, and large volume growth is of finite topological type provided that the volume growth rate of the complement of the cone of rays from a fixed base point is less than .

     

Twin solutions to singular boundary value problems

In this paper we establish the existence of two nonnegative solutions to singular and singular focal boundary value problems. Our nonlinearity may be singular at , and/or .

     

HOW REYE'S CONFIGURATION HELPS IN PROVING THE BELL–KOCHEN–SPECKER THEOREM: A CURIOUS GEOMETRICAL TALE

It is shown that the 24 quantum states or rays used by Peres (J. Phys. A 24, 174-8 (1991)) to give a proof of the Bell–Kochen–Specker (BKS) theorem have a close connection with Reye's configuration, a system of twelve points and sixteen lines known to projective geometers for over a century. The interest of this observation stems from the fact that it provides a ready explanation for many of the regularities exhibited by the Peres rays and also permits a systematic construction of all possible non-coloring proofs of the BKS theorem based on these rays. An elementary exposition of the connection between the Peres rays and Reye's configuration is given, following which its applications to the BKS theorem are discussed.     

A one-dimensional lattice model for a quantum mechanical free particle

Two types of particles, A and B with their corresponding antiparticles, are defined in a onedimensional cyclic lattice with an odd number of sites. In each step of time evolution, each particle acts as a source for the polarization field of the other type of particle with nonlocal action but with an effect decreasing with the distance: . It is shown that the combined distribution of these particles obeys the time evolution of a free particle as given by quantum mechanics.