Vector Valued Differentiation Theorems for Multiparameter Additive Processes in Lp Spaces

Let X be a Banach space and (,,µ) be a -finite measure space. We consider a strongly continuous d-dimensional semigroup T={T(u):u=(u₁,..., u_d, u_i >0, 1 i d} of linear contractions on L_p((,,µ); X), with 1 p<. In this paper differentiation theorems are proved for d-dimensional bounded processes in L_p((,,µ); X) which are additive with respect to T. In the theorems below we assume that each T(u) possesses a contraction majorant P(u) defined on L_p((,,µ); R), that is, P(u) is a positive linear contraction on L_p((,,µ); R) such that T(u)f(w) P(u)f(·)() almost everywhere on for all f L_p((,,µ); X).     

Existence and Construction of Simple B-Splines of Class C k on a Four-Directional Mesh of the Plane

In this paper we present a study of spaces of splines in C ^k (R ²) with supports the square ₁ and the lozenge ₁ formed respectively by four and eight triangles of the uniform four directional mesh of the plane. Such splines are called ₁ and ₁-splines. We first compute the dimension of the space of ₁-splines. Then we prove the existence of a unique ₁-spline of minimal degree for any fixed k0. By using this last result, we also prove the existence of a unique ₁-spline of minimal degree. Finally, we describe algorithms allowing to compute the Bernstein–Bézier coefficients of ₁-spline and ₁-spline of minimal degree.     

Fourier series of additive vector measures and their term-by-term differentiation

On a measurable space (T, , ) we choose an additive measure: Z (Z is a Banach space) with the following property: for alle , we have ; this measure defines an indefinite integral over the measure onL ² (T, ,). We prove that if { _n (t)} _n ^=1/ is an orthonormal basis inL ² and _n (e)=e _n (t) d, then any additive measure: Z whose Radon-Nikodým derivatived/d belongs toL ² is uniquely expandable in a series(e)= _n ^=1/ _n n(e) that converges to(e) uniformly with respect toe can be differentiated term-by-term, and satisfies _n ^=1/ _n ^/2 <. In the caseL ²[0,2],Z=, the Fourier series of a 2-periodic absolutely continuous functionF(t) such thatF'(t) L ²[0, 2] is superuniformly convergent toF(t).Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 180–184, August, 1998.     

Stochastic processes with Gaussian interaction of components

Summary LetU(x), x ^d-|0}, be a nonnegative even function such that _x 0U(x)1. In this paper, we consider an infinite system of stochastic process _t (x); x ^d with the following mechanism: at each sitex, after mean 1 exponential waiting time, _t(x) is replaced by a Gaussian random variable with mean _{yx t} (y) U(y-x) and variance 1. It is understood here that all the interactions are independent of one another. The behavior of this system will be investigated and some ergodic theorems will be derived. The results strongly depend whether _{x 0} U(x)<1 or =1.     

Estimation of the Multivariate Normal Precision Matrix under the Entropy Loss

Let X ₁, , X _n (n > p) be a random sample from multivariate normal distribution N _p (, ), where R ^p and is a positive definite matrix, both and being unknown. We consider the problem of estimating the precision matrix ^–1. In this paper it is shown that for the entropy loss, the best lower-triangular affine equivariant minimax estimator of ^–1 is inadmissible and an improved estimator is explicitly constructed. Note that our improved estimator is obtained from the class of lower-triangular scale equivariant estimators.     

Weighted Polynomial Approximation for Weights with Logarithmic Singularities in the Extremal Measure

Let w : [0, ) be a weight function on a set R. We assume that the associated extremal measure has density function v(t) with finitely many singularities of logarithmic type. We show that any continuous function f on which vanishes outside the set where v is positive or has a logarithmic singularity, is the uniform limit on of a sequence of weighted polynomials of the form wⁿ P_n, where P_n is of degree n. This extends previous results for continuous densities to densities having logarithmic singularities.     

C k -B-splines with Square Support on a Three-Direction Mesh of the Plane

Let be the uniform triangulation generated by the usual three-directional mesh of the plane and let ₁ be the unit square consisting of two triangles of . We study the space of piecewise polynomial functions in C ^k (R ²) with support ₁ having a sufficiently high degree n, which are symmetrical with respect to the first diagonal of ₁. Such splines are called ₁-splines. We first compute the dimension of this space in function of n and k. Then, for any fixed k0, we prove the existence of ₁-splines of class C ^k and minimal degree. These splines are not unique. Finally, we describe an algorithm computing the Bernstein–Bézier coefficients of these splines, and we give an example.     

Entropy loss and risk of improved estimators for the generalized variance and precision

Let the distributions of X(p×r) and S(p×p) be N(, I _r) and W _p(n, ) respectively and let them be independent. The risk of the improved estimator for || or {ei329-1} based on X and S under entropy loss (=d/|| –log(d/||)–1 or d||–log(d||)–1) is evaluated in terms of incomplete beta function of matrix argument and its derivative. Numerical comparison for the reduction of risk over the best affine equivariant estimator is given.Dedicated to Professor Yukihiro Kodama on his 60th birthday.     

Theorem of Bartle,Dunford, and Schwartz concerning vector measures

We show the existence, for an arbitrary vector measure: x (where X is a Banach space and gs is a-algebra of subsets of a set S) of a functional x X (X is the conjugate space of X) such that is absolutely continuous with respect to _x, _x (E)=(E)>, E gs.Translated from Matematicheskie Zametki, Vol. 7, No. 2, pp. 247–254, February, 1970.     

Spectral Properties of Hamiltonians of Charged Systems in a Homogeneous Magnetic Field: I. General Characteristic of the Spectrum

We study the spectrum of Hamiltonians of charged multiparticle systems in a homogeneous magnetic field with a fixed sum P of the pseudomomentum components and without it. We prove that if P is fixed, then the spectrum of Hamiltonians is independent of the value of P , while the spectrum without fixation of P coincides with the spectrum with fixation and differs from the latter only by some additional infinite degeneration (this is a principal difference between problems with a homogeneous magnetic field and problems without any field in which the absence of any fixation of the total angular momentum results in covering the spectrum of the relative motion by a continuous spectrum). We find the continuous spectrum of the Hamiltonians and characterize the spectrum of Hamiltonians of two-cluster mutually noninteracting systems obtained by decomposing the original system in the state with a fixed value of P . The last result is necessary for the study of the purely point spectrum.     

On Norden Metrics which Are Locally Conformal to Anti-Kählerian Metrics

We build examples of Norden metrics on × S ¹, where R ^{2 n} _n is either a pseudosphere or a pseudohyperbolic space. These turn out to be locally conformal to flat anti-Kählerian metrics, strongly non anti-Kählerian, and with a parallel Lee form. Conversely, any connected complete anti-Hermitian manifold possessing these properties is shown to be locally analytically homothetic to × S ¹.     

On some formalized conservation results in arithmetic

I _n andB _n are well known fragments of first-order arithmetic with induction and collection for _n formulas respectively;I _n ⁰ andB _n ⁰ are their second-order counterparts. RCA₀ is the well known fragment of second-order arithmetic with recursive comprehension;WKL ₀ isRCA ₀ plus weak König's lemma. We first strengthen Harrington's conservation result by showing thatWKL ₀ +B _n ⁰ is ₁ ¹ -conservative overRCA ₀ +B _n ⁰ . Then we develop some model theory inWKL ₀ and illustrate the use of formalized model theory by giving a relatively simple proof of the fact thatI ₁ provesB _n+1 to be _n+2-conservative overI _n . Finally, we present a proof-theoretic proof of the stronger fact that the _n+2 conservation result is provable already inI ₀ + superexp. ThusI _n+1 proves 1-Con (B _n+1) andI ₀ +superexp proves Con (I _n )Con(B _n+1).The first author was partially supported by NSF Grant #DCR-860615     

Uniform boundedness of a family of set functions

Let be a ring of sets, X a normed space, : X ( ) a bounded family of triangular functions. The following generalized Nikodym theorem is established: the family {} is uniformly bounded on if and only if it is bounded on every sequence of pairwise disjoint sets of which the union is a part of some set in . An analogous criterion is established also for semiadditive functions. In addition, it is shown that uniform boundedness of a family of triangular functions is preserved in passing from a ring to the -ring it generates.Translated from Matematicheskie Zametki, Vol. 23, No. 6, pp. 855–861, June, 1978.     

On the projected subgradient method for nonsmooth convex optimization in a Hilbert space

We consider the method for constrained convex optimization in a Hilbert space, consisting of a step in the direction opposite to an _k -subgradient of the objective at a current iterate, followed by an orthogonal projection onto the feasible set. The normalized stepsizes _k are exogenously given, satisfying _{k=0 k} = , _{k=0 k} ² < , and _k is chosen so that _{k k} for some > 0. We prove that the sequence generated in this way is weakly convergent to a minimizer if the problem has solutions, and is unbounded otherwise. Among the features of our convergence analysis, we mention that it covers the nonsmooth case, in the sense that we make no assumption of differentiability off, and much less of Lipschitz continuity of its gradient. Also, we prove weak convergence of the whole sequence, rather than just boundedness of the sequence and optimality of its weak accumulation points, thus improving over all previously known convergence results. We present also convergence rate results. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Research of this author was partially supported by CNPq grant nos. 301280/86 and 300734/95-6.     

On the Approximation of Denotational Mu-Semantics

A signature gives rise to a language L(Var) by extending with variables x Var and binding constructs x and x, corresponding to least and greatest fixed points respectively. The natural denotational models for such languages are bicomplete dcpos as monotone -algebras. We prove that several approximating denotational semantics have the usual compositional semantics as their limit. These results provide techniques for relating syntactic and semantic concepts such as in full abstraction or completeness proofs. In the presence of an involutive antitone map on a bicomplete dcpo D we may translate the language L(Var) into one with least fixed points only such that meanings are preserved. This allows an approximative semantics where least and greatest fixed points are simultaneously approximated by unwindings in the syntax, provided that the limit semantics is substitutive. We discuss the principal difficulties of simultaneous unwindings in the absence of such semantic negations.     

4-Dimensional compact projective planes with small nilradical

We consider 4-dimensional compact projective planes with a solvable 6-dimensional collineation group and with orbit type (2, 1), i.e. fixes a flagv W, acts transitively onL \{W} and fixes no point in the setW\{v}. We We prove a series of lemmas concerning the action of invariant subgroups of . These lemmas are applied to prove that the maximal connected nilpotent invariant subgroup of has dimension at least 4.Dedicated to Prof. H. Salzmann on the occasion of his 65th birthday     

Cofibrations of algebras

Summary The chain complex of a twisted free product A_*t, FK, is chain homotopy equivalent to a differential graded algebra, which is identified to be a confibration of algebras as defined by Quillen. Under certain connectivity conditions we obtain a long exact sequence connecting the homologies of A, K, and A_*t FK. In particular we derive a long exact sequence connecting the homologies of Y, X and (Y U_g CX) (, C, are the loop, the cone and the suspension constructions respectively). A chain complex equivalent to the chain complex of the Milnor free group FX is recognized, from which results a theorem of Bott and Samelson that H(X) is freely generated as a graded algebra by H(X).     

On an optimum test of the equality of two covariance matrices

Let X: p × 1, Y: p × 1 be independently and normally distributed p-vectors with unknown means ₁, ₂ and unknown covariance matrices ₁, ₂ (>0) respectively. We shall show that Pillai's test, which is locally best invariant, is locally minimax for testing H ₀: ₁=₂ against the alternative H ₁: % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaaeiDaiaabkhacaqGOaWaaabmaeaadaaeqaqaaiabgkHiTiaadMea% caGGPaGaaiiiaiabg2da9iaacccacqaHdpWCcaGGGaGaeyOpa4Jaai% iiaiaaicdaaSqaaiaaigdaaeqaniabggHiLdaaleaacaqGYaaabaGa% aeylaiaabgdaa0GaeyyeIuoaaaa!4E3F!\[{\rm{tr(}}\sum\nolimits_{\rm{2}}^{{\rm{ - 1}}} {\sum\nolimits_1 { - I) = \sigma > 0} }\]as 0. However this test is not of type D among G-invariant tests.Research supported by the Canadian N.S.E.R.C. Grant.     

Spreads admitting net generating regulizations

We say a spread S carries a regulization , if is a collection of reguli contained in S and if each element of S, except at most two lines, is contained either in exactly one regulus of or in all reguli of . Replacement of each regulus of by its complementary regulus (exceptional lines remain unchanged) yields the complementary congruence S^c of S with respect to . If S^c is a hyperbolic or parabolic or elliptic linear congruence of lines, then is called a net generating, in particular, a hyperbolic or parabolic or elliptic regulization, respectively. For hyperbolic and parabolic regulizations we also give other geometric characterizations.     

Combinatorial principles concerning approximations of functions

Two infinite combinatorial principlesP( _n ) andT( _n ) concerning the existence of approximations of functions are studied.T( _n ) is shown to be equivalent toI _n andP( _n ) is shown to be incomparable withB _n+1 . Finally Pudláks principle, which is a finite miniaturization of bothT andP, is studied and its instances are related to instances of other known combinatorial statements.This paper was completed in September 1985 when the second author was a guest of the Mathematical Institute of the Czechoslovak Academy of Science in Prague.