Compact Laws of the Iterated Logarithm for B-Valued Random Variables with Two-Dimensional Indices

Let be a real separable Banach space and {X, X _{n, m}; (n, m) N ²} B-valued i.i.d. random variables. Set . In this paper, the compact law of the iterated logarithm, CLIL(D), for B-valued random variables with two-dimensional indices ranging over a subset D of N ² is studied. There is a gap between the moment conditions for CLIL(N ¹) and those for CLIL(N ²). The main result of this paper fills this gap by presenting necessary and sufficient conditions for the sequence to be almost surely conditionally compact in B, where, for 0, 1 r 2, N ^r (, ) = {(n, m) N ²; n m n exp{(log n) ^r–1 (n)}} and (·) is any positive, continuous, nondecreasing function such that (t)/(log log t) is eventually decreasing as t , for some > 0.     

The lyapunov equation and the problem of stability for linear bounded discrete-time systems in Hilbert space

This paper is devoted to a study of the properties of the equationA ^*FA–F=–G, where FL() is unknown, AL(), GL() is positive and is a Hilbert space. It is shown that necessary and sufficient (in some sense) conditions for the existence of positive definite solutions of this equation are directly connected with the stability of infinite dimensional linear systemx _k+1=Ax _k . The relationships between stability of such a system and stability of a continuous-time system generated by a strongly continuous semigroup are given also. As an example the case of the delayed system in Rⁿ is considered.This work was supported in part by the Polish Academy of Sciences under the contract Problem Miedzyresortowy I.1, Grupa Tematyczna 3 This paper was written while the author was with the Instytut Automatyki, the same university.     

On the positive solutions of some nonlinear diffusion problems

We consider the nonlinear diffusion equationu _t –a(x, u _{x x} )+b(x, u)=g(x, u) with initial boundary conditions andu(t, 0)=u(t, 1)=0. Here,a, b, andg denote some real functions which are monotonically increasing with respect to the second variable. Then, the corresponding stationary problem has a positive solution if and only if(0, ^*) or(0, ^*]. The endpoint ^* can be estimated by , where ₁ u denotes the first eigenvalue of the stationary problem linearized at the pointu. The minimal positive steady state solutions are stable with respect to the nonlinear parabolic equation.

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Zusammenfassung Wir betrachten die nichtlineare Diffusionsgleichungu _t –a(x, u _x ) _x +b(x, u)=g(x, u) mit Randbedingungen undu (t, 0)=u (t, 1)=0. Dabei sinda, b, undg monoton wachsende Funktionen bzgl. des zweiten Argumentes. Das zugehörige stationäre Problem hat genau dann eine positive Lösung, falls (0, ^*) oder(0, ^*]. Der Endpunkt ^* kann durch abgeschätzt werden, wobei ₁ u den ersten Eigenwert des an der Stelleu linearisierten stationären Problems bezeichnet. Die minimale positive stationäre Lösung ist stabil bzgl. der obigen nichtlinearen parabolischen Gleichung.

     

Sharp Constants in Inequalities for Intermediate Derivatives (the Gabushin Case)

We solve Tikhomirov's problem on the explicit computation of sharp constants in the Kolmogorov type inequalities

On the Solvability and Asymptotics of Solutions of One Functional Differential Equation with Singularity

We prove the existence of continuously differentiable solutions with required asymptotic properties as t +0 and determine the number of solutions of the following Cauchy problem for a functional differential equation:

Interior Regularity for Free and Constrained Local Minimizers of Variational Integrals Under General Growth and Ellipticity Conditions

We consider strictly convex energy densities f: (x) under nonstandard growth conditions. More precisely, we assume that for some constants , and for all Z, Y ⁿ the inequality

On Kolmogorov-Type Inequalities Taking into Account the Number of Changes in the Sign of Derivatives

For 2-periodic functions and arbitrary q [1, ] and p (0, ], we obtain the new exact Kolmogorov-type inequality which takes into account the number of changes in the sign of the derivatives (x ^(k)) over the period. Here, = (r – k + 1/q)/(r + 1/p), _r is the Euler perfect spline of degree r, and . The inequality indicated turns into the equality for functions of the form x(t) = a _r (nt + b), a, b R, n N. We also obtain an analog of this inequality in the case where k = 0 and q = and prove new exact Bernstein-type inequalities for trigonometric polynomials and splines.     

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

A preconditioner for constrained and weighted least squares problems with Toeplitz structure

We study methods for solving the constrained and weighted least squares problem min _x by the preconditioned conjugate gradient (PCG) method. HereW = diag (₁, , _m ) with _{1 m} 0, andA ^T = [T ₁ ^T , ,T _k ^T ] with Toeplitz blocksT _l R ^{n × n} ,l = 1, ,k. It is well-known that this problem can be solved by solving anaugmented linear 2 × 2 block linear systemM +Ax =b, A ^T = 0, whereM =W ^–1. We will use the PCG method with circulant-like preconditioner for solving the system. We show that the spectrum of the preconditioned matrix is clustered around one. When the PCG method is applied to solve the system, we can expect a fast convergence rate.Research supported by HKRGC grants no. CUHK 178/93E and CUHK 316/94E.     

On Singular Functional Differential Inequalities

Classical theorems on differential inequalities [1, 2, 3] are generalized for initial value problems of the kind and where is a singular Volterra operator, is continuous and positive on ]a, b], is a norm in R ⁿ, and [u]₊ and [u]_– are respectively the positive and the negative part of the vector u R ⁿ.     

Embeddings of anisotropic Sobolev spaces on unbounded domains

Summary Integral representations of junctions in the anisotropic Sobolev spaces on unbounded domains are used in the study of the embeddings of these spaces into Lebesgue spaces. Estimates of entropy numbers of the embedding are obtained, where k, p and q satisfy certain conditions and where is a certain type of quasibounded domain,     

A shift-invariant algebra of singular integral operators with oscillating coefficients

LetB be the Banach algebra of all bounded linear operators on the weighted Lebesgue spaceL ^p (T, ) with an arbitrary Muckenhoupt weight on the unit circleT, and the Banach subalgebra ofB generated by the operators of multiplication by piecewise continuous coefficients and the operatorse _h,S _T e _h, ^–1 I (hR, T) whereS _T is the Cauchy singular integral operator ande _h,(t)=exp(h(t+)/(t–)),tT. The paper is devoted to a symbol calculus, Fredholm criteria and an index formula for the operators in the algebra and its matrix analogue . These shift-invariant algebras arise naturally in studying the algebras of singular integral operators with coefficients admitting semi-almost periodic discontinuities and shifts being diffeomorphisms ofT onto itself with second Taylor derivatives.Partially supported by CONACYT grant, Cátedra Patrimonial, No. 990017-EX and by CONACYT project 32726-E, México.     

On the admissibility of function spaces of typeB p,q s andF p,q s,and boundary value problems for non-linear partial differential equations

u=f(x)+S(u), S — , u-G(u), G — . B _p,q ^s () -F _p,q ^s (). R _n — . — . _p,q ^s F _p,q ^s .     

A Delocalization Property for Assembly Maps and an Application in Algebraic K-Theory of Group Rings

Let be a group, F the free -module on the set of finite order elements in , with acting by conjugation, and the ring extension of by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaWaaiWaaeaada% WcaaqaaiaaigdaaeaatCvAUfKttLearyGqLXgBG0evaGqbciab-5ga% UbaaieaacaGFLbGaaGOmaiaabc8acqWFPbqAcaqGVaGae8NBa42aaq% qaaeaacqGHdicjcqaHZoWzcqGHiiIZcqqHtoWrcaqGGaGaae4Baiaa% bAgacaqGGaGaae4BaiaabkhacaqGKbGaaeyzaiaabkhacaqGGaGae8% NBa4gacaGLhWoaaiaawUhacaGL9baaaaa!563E!\[\left\{ {\frac{1}{n}e2{\text{\pi }}i{\text{/}}n\left| {\exists \gamma \in \Gamma {\text{ of order }}n} \right.} \right\}\]. For a ring R with , we build an injective assembly map , detected by the Dennis trace map. This is proved by establishing a delocalization property for the assembly map in Hochschild homology, namely providing a gluing of simpler assembly maps (i.e. localized at the identity of ) to build , and by delocalizing a known assembly map in K-theory to define . We also prove the delocalization property in cyclic homology and in related theories.     

Some Properties of Finite Morphisms on Double Points

For a finite morphism f : X Y of smooth varieties such that f maps X birationally onto X=f(X), the local equations of f are obtained at the double points which are not triple. If C is the conductor of X over X, and are the subschemes defined by C, then D and are shown to be complete intersections at these points, provided that C has the expected codimension. This leads one to determine the depth of local rings of X at these double points. On the other hand, when C is reduced in X, it is proved that X is weakly normal at these points, and some global results are given. For the case of affine spaces, the local equations of X at these points are computed.     

On a Family of Hyperplane Arrangements Related to the Affine Weyl Groups

Let be an irreducible crystallographic rootsystem in a Euclidean space V, with ⁺ theset of positive roots. For , , let be the hyperplane . We define a set of hyperplanes . This hyperplane arrangement is significant inthe study of the affine Weyl groups. In this paper it is shown that thePoincaré polynomial of is , where n is the rank of and h is the Coxeter number of the finiteCoxeter group corresponding to .     

QUADRATIC ESTIMATORS OF QUADRATIC FUNCTIONS WITH PARAMETERS IN NORMAL LINEAR MODELS

ＱＵＡＤＲＡＴＩＣＥＳＴＩＭＡＴＯＲＳＯＦＱＵＡＤＲＡＴＩＣＦＵＮＣＴＩＯＮＳＷＩＴＨＰＡＲＡＭＥＴＥＲＳＩＮＮＯＲＭＡＬＬＩＮＥＡＲＭＯＤＥＬＳ￥ＷＵＱＩＧＵＡＮＧ（吴启光）（ＩｎｓｔｉｔｕｔｅｏｆＳｙｓｔｅｍｅＳｃｉｅｎｃｅ,ｔｈｅＣｈｉｎｅｓｅ...     

Singular perturbations in foliations

Summary We define a constraint system , [0,₀), which is a kind of family of vector fields on a manifold. This is a generalized version of the family of the equations , [0,₀),x ^m ,y ⁿ . Finally, we prove a singular perturbation theorem for the system , [0,₀).Dedicated to Professor Kenichi Shiraiwa on his 60th birthday     

Imbedding theorems for the invariant subspaces of the backward shift operator

For subspaces K ^p of the form of the Hardy space H^p and for measures with support in the closed unit circleclos , one finds conditions that ensure the imbedding KL^p(). One considers measures with support inclos , satisfying the following condition: for some number >0 and for all circles with center on the circumference, intersecting the set , we have the inequality ()C(). Here C does not depend on , while () is the radius of the circle . For such measures one has the imbedding K ^p L^p(). From here one derives a criterion for the imbedding K _A ² L²(), found by B. Cohn for inner functions , such that the set is connected for some positive . In the paper one also proves that a condition on , necessary and sufficient for the imbedding of K ^p into L^p(), must depend on p.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 38–51, 1986.