A Full Characterization of Multipliers for the Strong ϱ-Integral in the Euclidean Space

We study a generalization of the classical Henstock-Kurzweil integral, known as the strong -integral, introduced by Jarník and Kurzweil. Let be the space of all strongly -integrable functions on a multidimensional compact interval E, equipped with the Alexiewicz norm We show that each element in the dual space of can be represented as a strong -integral. Consequently, we prove that fg is strongly -integrable on E for each strongly -integrable function f if and only if g is almost everywhere equal to a function of bounded variation (in the sense of Hardy-Krause) on E.     

A Helly-type theorem for simple polygons

Let be a family of simple polygons in the plane. If every three (not necessarily distinct) members of have a simply connected union and every two members of have a nonempty intersection, then {P:P in } . Applying the result to a finite family of orthogonally convex polygons, the set {C:C in } will be another orthogonally convex polygon, and, in certain circumstances, the dimension of this intersection can be determined.Supported in part by NSF grant DMS-9207019.     

Strong solution for an elastic-plastic plate

Given a regular bounded open set R ²,, >0 andg L ^q () withq>2, we prove, under compatibility and safe load conditions ong, the existence of a minimizing pair for the functional, over closed setsK ² and functionsu C⁰( ) C²(/K); here ¦[Du]¦ denotes the jump ofDu acrossK and ¹ is the 1-dimensional Hausdorff measure.Dedicated to Enrico Magenes for his 70th birthday     

Some proximity and sensitivity results in quadratic integer programming

We show that for any optimal solution for a given separable quadratic integer programming problem there exist an optimal solution for its continuous relaxation such that wheren is the number of variables and(A) is the largest absolute subdeterminant of the integer constraint matrixA. Also for any feasible solutionz, which is not optimal for the separable quadratic integer programming problem, there exists a feasible solution having greater objective function value and with . We further prove, under some additional assumptions, that the distance between a pair of optimal solutions to an integer quadratic programming problem with right hand side vectorsb andb, respectively, depends linearly on b–b₁. Finally the validity of all the results for nonseparable mixed-integer quadratic programs is established. The proximity results obtained in this paper are extensions of some of the results described in Cook et al. (1986) for linear integer programming.This research was partially supported by Natural Sciences and Engineering Research Council of Canada Grant 5-83998.     

Cohomology of solvable lie algebras and solvmanifolds

The cohomology H^* (G/,) of the de Rham complex *(G/) of a compact solvmanifold G/ with deformed differential d = d + , where is a closed 1 -form, is studied. Such cohomologies naturally arise in Morse-Novikov theory. It is shown that, for any completely solvable Lie group G containing a cocompact lattice G, the cohomology H^*(G/, ) is isomorphic to the cohomology H^*( ) of the tangent Lie algebra of the group G with coefficients in the one-dimensional representation : defined by () = (). Moreover, the cohomology H ^*(G/,) is nontrivial if and only if -[] belongs to a finite subset of H ¹(G/,) defined in terms of the Lie algebra .Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 67–79.Original Russian Text Copyright © 2005 by D. V. Millionshchikov.This revised version was published online in April 2005 with a corrected issue number.     

Notes on Brown-McCoy-radicals for Γ-near-rings

The Brown-McCoy radical is known to be an ideal-hereditary Kurosh-Amitsur radical in the variety of zerosymmetric near-rings. We define the Brown-McCoy and simplical radicals, and , respectively, for zerosymmetric -near-rings. Both and are ideal-hereditary Kurosh-Amitsur radicals in that variety. IfM is a zerosymmetric -near-ring with left operator near-ringL, it is shown that , with equality ifM has a strong left unity. is extended to the variety of arbitrary near-rings, and and are extended to the variety of arbitrary -near-rings, in a way that they remain Kurosh-Amitsur radicals. IfN is a near-ring andA N, then , with equality ifA if left invariant.     

The Boundary Equivalence for Rings and Matrix Rings over Them

We study into the question of whether some rings and their associated matrix rings have equal decidability boundaries in the scheme and scheme-alternative hierarchies. Let be a decidability boundary for an algebraic system A; w.r.t. the hierarchy H. For a ring R, denote by an algebra with universe . On this algebra, define the operations + and in such a way as to extend, if necessary, the initial matrices by suitably many zero rows and columns added to the underside and to the right of each matrix, followed by ordinary addition and multiplication of the matrices obtained. The main results are collected in Theorems 1-3. Theorem 1 holds that if R is a division or an integral ring, and R has zero or odd characteristic, then the equalities hold for any n1. And if R is an arbitrary associative ring with identity then for any n 1 and i,j { 1,..., n}, where e _ij is a matrix identity. Theorem 2 maintains that if R is an associative ring with identity then . Theorem 3 proves that for any n 1.     

(C_k \oplus G,k,\lambda ) Difference Families

Let G be an additive group and C _k be the additive group of the ring Z _k of residues modulo k. If there exist a (G, k, ) difference family and a (G, k, ) perfect Mendelsohn difference family, then there also exists a difference family. If the (G, k, ) difference family and the (G, k, ) perfect Mendelsohn difference family are further compatible, then the resultant difference family is elementary resolvable. By first constructing several series of perfect Mendelsohn difference families, many difference families and elementary resolvable difference families are thus obtained.     

Quadrature formulas obtained by variable transformation

Quadrature formulas suitable for evaluation of improper integrals such as are obtained by means of variable transformations =tanhu and =erfu, and subsequent use of trapezoidal quadrature rule. Error analysis is carried out by the method of contour integral, and the results are confirmed on several concrete examples. Similar formulas are also obtained to accelerate the convergence of infinite integrals by means of variable transformations =sinhu and =tanu.     

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     

Approximation of convolutions of multi-dimensional symmetric distributions by accompanying laws

Let F be a symmetric k-dimensional probability distribution, whose characteristic function satisfies for allt R ^k the inequality –1 + , where 0 < < 2. Let n be an arbitrary natural number, let Fⁿ be the n-fold convolution of the distribution F with itself, and let e(nF) be the accompanying infinitely divisible distribution with characteristic function exp(n( –1)). It is proved that the uniform distance (·,·) between corresponding distribution functions admits estimate (F ⁿ ,e(nF))c₁(k)(n^–1+exp(–n+ckn ³ n)), where c₁ (k) depends only on the dimension k, while c₂ is an absolute constant.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 55–72, 1989.     

A set of orthogonal polynomials induced by a given orthogonal polynomial

Summary Given an integern 1, and the orthogonal polynomials _n (·; d) of degreen relative to some positive measured, the polynomial system induced by _n is the system of orthogonal polynomials corresponding to the modified measure . Our interest here is in the problem of determining the coefficients in the three-term recurrence relation for the polynomials from the recursion coefficients of the orthogonal polynomials belonging to the measured. A stable computational algorithm is proposed, which uses a sequence ofQR steps with shifts. For all four Chebyshev measuresd, the desired coefficients can be obtained analytically in closed form. For Chebyshev measures of the first two kinds this was shown by Al-Salam, Allaway and Askey, who used sieved orthogonal polynomials, and by Van Assche and Magnus via polynomial transformations. Here, analogous results are obtained by elementary methods for Chebyshev measures of the third and fourth kinds. (The same methods are also applicable to the other two Chebyshev measures.) Interlacing properties involving the zeros of _n and those of are studied for Gegenbauer measures, as well as the orthogonality—or lack thereof—of the polynomial sequence .Work supported in part by the National Science Foundation under grant DMS-9023403.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

QUADRATIC ESTIMATORS OF QUADRATIC FUNCTIONS WITH PARAMETERS IN NORMAL LINEAR MODELS

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

On the convergence rate of imperfect minimization algorithms in Broyden'sβ-class

This paper presents a local convergence analysis of Broyden's class of rank-2 algorithms for solving unconstrained minimization problems, ,h C¹(R ⁿ ), assuming that the step-size ai in each iterationx _i+1 =x _i - _i H _i h(x _i ) is determined by approximate line searches only. Many of these methods including the ones most often used in practice, converge locally at least with R-order, .     

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.     

An Optimal Way of Moving a Sequence of Points onto a Curve in Two Dimensions

Let (t), 0 t T, be a smooth curve and let _i , i = 1, 2, , n, be a sequence of points in two dimensions. An algorithm is given that calculates the parameters t_i, i = 1, 2, , n, that minimize the function max{ _i – (t_i) ₂ : i = 1, 2, , n } subject to the constraints 0 t₁ t₂ t_n T. Further, the final value of the objective function is best lexicographically, when the distances _i – (t_i)₂, i = 1, 2, , n, are sorted into decreasing order. The algorithm finds the global solution to this calculation. Usually the magnitude of the total work is only about n when the number of data points is large. The efficiency comes from techniques that use bounds on the final values of the parameters to split the original problem into calculations that have fewer variables. The splitting techniques are analysed, the algorithm is described, and some numerical results are presented and discussed.     

Generalization of some classical inequalities in the theory of orthogonal series

Let {X_i} _– be a sequence of random variables, E(X_i) 0. If 1, estimates for the -th moments can be derived from known estimates of the -th moment. Here we generalized the Men'shov-Rademacher inequality for =2 for orthonormal X_i, to the case 1 and dependent random variables. The Men'shov-Payley inequality >2 for orthonormal X_i) is generalized for >2 to general random variables. A theorem is also proved that contains both the Erdös -Stechkin theorem and Serfling's theorem withv > 2 for dependent random variables.Translated from Matematicheskie Zametki, Vol. 17, No. 2, pp. 219–230, February, 1975.This article was written while the author was working in the V. A. Steklov Mathematics Institute, Academy of Sciences of the USSR.     

Lattices of interpretability types of varieties

Let be the set of all primes, the field of all algebraic numbers, and Z the set of square-free natural numbers. We consider partially ordered sets of interpretability types such as , and , where AD is a variety of -divisible Abelian groups with unique taking of the pth root _p(x) for every p , is a variety of -modules over a normal field , contained in , and G_n is a variety of n-groupoids defined by a cyclic permutation (12 ...n). We prove that , and are distributive lattices, with and where ub and ub_f are lattices (w.r.t. inclusion) of all subsets of the set and of finite subsets of , respectively.Deceased.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 198–210, March–April, 2005.     

Stochastic integrals on general topological measurable spaces

Summary A general theory of stochastic integral in the abstract topological measurable space is established. The martingale measure is defined as a random set function having some martingale property. All square integrable martingale measures constitute a Hilbert space M ². For each M ², a real valued measure on the predictable -algebra is constructed. The stochastic integral of a random function with respect to is defined and investigated by means of Riesz's theorem and the theory of projections. The stochastic integral operator I is an isometry from L ²() to a stable subspace of M ², its inverse is defined as a random Radon-Nikodym derivative. Some basic formulas in stochastic calculus are obtained. The results are extended to the cases of local martingale and semimartingale measures as well.     

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