Berger-Shaw's theorem forp-hyponormal operators

For an-multicyclicp-hyponormal operatorT, we shall show that |T|^2p –|T ^*|^2p belongs to the Schatten and that tr Area ((T)).     

On the almost sure (a.s.) central limit theorem for random variables with infinite variance

We give criteria for a sequence (X _n ) of i.i.d.r.v.'s to satisfy the a.s. central limit theorem, i.e.,

Almost everywhere convergence of the spherical partial sums for radial functions

LetfL ^p( ⁿ ),n2, be a radial function and letS _Rf be the spherical partial sums operator. We prove that if thenS _Rf(x)f(x) a.e. asR. The result is false for and \frac{{2n}}{{n + 1}}$$ " align="middle" border="0"> .Partially supported by M.P.I.     

On Plane Maximal Curves

The number N of rational points on an algebraic curve of genus g over a finite field satisfies the Hasse–Weil bound . A curve that attains this bound is called maximal. With and , it is known that maximalcurves have . Maximal curves with have been characterized up to isomorphism. A natural genus to be studied is and for this genus there are two non-isomorphic maximal curves known when . Here, a maximal curve with genus g ₂ and a non-singular plane model is characterized as a Fermat curve of degree .     

When all trajectories in the Liénard plane cross the vertical isocline?

In this paper we study the problem whether all trajectories of the system =y–F(x) and =–g(x) cross the vertical isocline which is very important for the existence of periodic solutions and oscillation theory. The problem has not been solved for the critical case:

Hybrid triple systems and cubic feedback sets

Ac-hybrid triple system of orderv is a decomposition of the completev-vertex digraph intoc cyclic tournaments of order 3 and transitive tournaments of order 3. Hybrid triple systems generalize directed triple systems (c = 0) and Mendelsohn triple systems (c = v(v – 1)/3); omitting directions yields an underlying twofold triple system. The spectrum ofv andc for which ac-hybrid triple system of orderv exists is completely determined in this paper. Using (cubic) block intersection graphs, we then show that every twofold triple system of order underlies ac-hybrid triple system with . Examples are constructed for all sufficiently largev, for which this maximum is at most . The lower bound here is proved by establishing bounds onF _i (n, r), the size of minimum cardinality vertex feedback sets inn-vertexi-connected cubic multigraphs havingr repeated edges. We establish that , 8$$ " align="middle" border="0"> . These bounds are all tight, and the latter is used to derive the lower bound in the design theoretic problem.     

On the tail behaviour of the distribution function of multiple stochastic integrals

Summary Let F _n (u) denote the empirical distribution function of a sample of i.i.d. random variables with uniform distribution on [0, 1]. Define , and consider the integrals where f is a bounded measurable function. We give a good upper bound on the probability . An analogous estimate is given for multiple integrals with respect to a Poisson process.     

On the number of faces of centrally-symmetric simplicial polytopes

I. Bárány and L. Lovász [Acta Math. Acad. Sci. Hung.40, 323–329 (1982)] showed that ad-dimensional centrally-symmetric simplicial polytopeP has at least 2 ^d facets, and conjectured a lower bound for the numberf _i ofi-dimensional faces ofP in terms ofd and the numberf ₀ =2n of vertices. Define integers A. Björner conjectured (unpublished) that (which generalizes the result of Bárány-Lovász sincef _d–1 = h _i ), and more strongly that , which is easily seen to imply the conjecture of Bárány-Lovász. In this paper the conjectures of Björner are proved.Partially supported by NSF grant MCS-8104855. The research was performed when the author was a Sherman Fairchild Distinguished Scholar at Caltech.     

Almost sure invariance principles when EX 1 2 =∞

Summary Let {Y _i} be iid with EY ₁=0, EY ₁ ² =1. Let {X_i} be iid normal mean zero, variance one random variables. According to Strassen's first almost sure invariance principle {X _i} and {Y _i} can be reconstructed on a new probability space without changing the distribution of each sequence such that a.s., thus improving on the trivial bound obtainable from the law of the iterated logarithm: a.s. In this work we establish analogous improvements for symmetric {Y _i} in the domain of normal attraction to a symmetric stable law with index 0<<2. (We make this assumption of symmetry in order to avoid messy details concerning centering constants.) Let {X _i} be iid symmetric stable random variables with index 0<<2. Then, for example, hypotheses are stated which imply for a given satisfying 2> that a.s., thus improving on the trivial bound: a.s., >0.This research was supported in part by a National Science Foundation grant, USA     

Eigenvalue estimate on a compact Riemann manifold

LetM be a compact Riemann manifold with the Ricci curvature ≽ - R(R = const. > 0) . Denote by d the diameter ofM. Then the first eigenvalue λ₁ ofM satisfies . Moreover if , then      

Sharp Estimates for Deviations of Linear Approximation Methods for Periodic Functions by Linear Combinations of Moduli of Continuity of Different Order

Estimates for deviations are established for a large class of linear methods of approximation of periodic functions by linear combinations of moduli of continuity of different orders. These estimates are sharp in the sense of constants in the uniform and integral metrics. In particular, the following assertion concerning approximation by splines is proved: Suppose that is odd, . Then

Exponential Stability of Semigroups Related to Operator Models in Mechanics

In this paper, we consider equations of the form , where is a function with values in the Hilbert space , the operator B is symmetric, and the operator A is uniformly positive and self-adjoint in . The linear operator generating the C ₀-semigroup in the energy space is associated with this equation. We prove that this semigroup is exponentially stable if the operator B is uniformly positive and the operator A dominates B in the sense of quadratic forms.     

An Ω-theorem for an error term related to the sum-of-divisors function

Let denote the sum-of-divisors function, and set . Gronwall and Wigert proved (independently) in 1913 and 1914, respectively, thatE ₁ (x)= (x log logx). In this paper we obtain the more preciseE ₁ (x)=_–(x log logx). The method consists in averaging over suitable arithmetic progressions, and was suggested by the work ofP. Erdös andH. N. Shapiro [Canad. J. Math. 3–4, 375–385 (1951)] on the error term corresponding to Euler's functions, .     

The Gaussian measure of shifted balls

Summary Let be a centered Gaussian measure on a Hilbert spaceH and let be the centered ball of radiusR>0. ForaH and , we give the exact asymptotics of (B _R(t)+t·a) ast. Also, upper and lower bounds are given when is defined on an arbitrary separable Banach space. Our results range from small deviation estimates to large deviation estimates.Supported in part by NSF grant number DMS-9024961     

On the Lax Solution to the Hamilton–Jacobi Equation and Its Generalizations. Part II

This article is a continuation of [J. Math. Sci., 99, No.5, 1541–1547 (2000)] devoted to the validity of the Lax formula (cited in the article of Crandall, Ishii, and Lions [Bull. AMS, 27, No.1, 1–67 (2000)])

Existence,uniqueness and convergence as vanishing viscosity for a reaction-diffusion-convection system

A simple qualitative model of dynamic combustion

Light paths with an odd number of vertices in polyhedral maps

Let P _k be a path on k vertices. In an earlier paper we have proved that each polyhedral map G on any compact 2-manifold with Euler characteristic contains a path P _k such that each vertex of this path has, in G, degree . Moreover, this bound is attained for k = 1 or k 2, k even. In this paper we prove that for each odd , this bound is the best possible on infinitely many compact 2-manifolds, but on infinitely many other compact 2-manifolds the upper bound can be lowered to .     

Laplace approximations for sums of independent random vectors

Summary LetX _i,iN, be i.i.d.B-valued random variables whereB is a real separable Banach space, and a mappingB R. Under some conditions an asymptotic evaluation of is possible, up to a factor (1+o(1)). This also leads to a limit theorem for the appropriately normalized sums under the law transformed by the density exp .     

On Sikkema-Kantorovič polynomials of order K

Let a_n≥0 and F(u)∈C [0,1], Sikkema constructed polynomials: , ifα _n ≡0, then B_n (0, F, x) are Bernstein polynomials. Let , we constructe new polynomials in this paper: Q _n ^(k) (α _n ,f(t))=d ^k /dx ^k B _n+k (α _n ,F _k (u),x), which are called Sikkema-Kantorovic polynomials of order k. Ifα _n ≡0, k=1, then Q_n ⁽¹⁾ (0, f(t), x) are Kantorovič polynomials P_n(f). Ifα _n =0, k=2, then Q_n ⁽²⁾, (0, f(t), x) are Kantorovič polynomials of second order (see Nagel). The main result is: Theorem 2. Let 1≤p≤∞, in order that for every f∈L^P [0, 1], , it is sufficient and necessary that , § 1. Let f(t) de a continuous function on [a, b], i. e., f∈C [a, b], we define^{[1–2],[8–10]}: . As usual, for the space L^p [a,b](1≤p<∞), we have and L[a, b]=l¹[a, b]. Letα _n ⩾0and F(u)∈C[0,1],Sikkema-Bernstein polynomials ^{[3] [4]}. The author expresses his thanks to Professor M. W. Müller of Dortmund University at West Germany for his supports.     

Kolmogorov-type inequalities for mixed derivatives of functions of many variables

Let = (₁,...,_d) be a vector with positive components and let D be the corresponding mixed derivative (of order _j with respect to the jth variable). In the case where d > 1 and 0 < k < r are arbitrary, we prove that