Some topological properties of ω-covering sets

We prove the following theorems:1. There exists an -covering with the property s ₀.2. Under cov there exists X such that is not an -covering orX \ B is not an -covering].3. Also we characterize the property of being an -covering.     

Classifying countable Boolean terms

We deal with the Borel and difference hierarchies in the space P of all subsets of endowed with the Scott topology. (The spaces P and 2 coincide set-theoretically but differ topologically.) We look at the Wadge reducibility in P. The results obtained are applied to the problem of characterizing ₁-terms t which satisfy C = t( ₁ ⁰ ) for a given Borel-Wadge class C. We give its solution for some levels of the Wadge hierarchy, in particular, all levels of the Hausdorff difference hierarchy. Finally, we come up with a discussion of some relevant facts and open questions.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 173–197, March–April, 2005.     

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.     

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 local automorphisms of certain quadrics of codimension 2

The article considers nondegenerate quadrics in Cⁿ⁺¹ with codimension 2 that are of the form M={zCⁿ, C²: Im_j=z, z_j; j=1, 2}, where are Hermitian forms, and thje stability groups Aut_x M that preserve the point x. It is proved that if the matrix ¹ is stable and the matrix (¹)^–12 has more than two different eigenvalues, all automorphisms of Aut_x M are linear transformations.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 9–14, July, 1992.     

Equivalence Bimodule Between Non-Commutative Tori

The non-commutative torus C ^*(ⁿ,) is realized as the C*-algebra of sections of a locally trivial C*-algebra bundle over S with fibres isomorphic to C ^*n/S, ₁) for a totally skew multiplier ₁ on ⁿ/S. D. Poguntke [9] proved that A is stably isomorphic to C(S) C(^*( Zⁿ/S, ₁) C(S) A M_kl( C) for a simple non-commutative torus A and an integer kl. It is well-known that a stable isomorphism of two separable C*-algebras is equivalent to the existence of equivalence bimodule between them. We construct an A-C(S) A-equivalence bimodule.     

Quasimonotonicity,regularity and duality for nonlinear systems of partial differential equations

Summary We prove partial regularity for the vector-valued differential forms solving the system (A(x, ))=0, d=0, and for the gradient of the vector-valued functions solving the system div A(x, Du)=B(x, u, Du). Here the mapping A, with A(x, w) (1+ + ¦¦²)^{(p – 2)/2} (p2), satisfies a quasimonotonicity condition which, when applied to the gradient A(x, )=Df(x, ) of a real-valued functionf, is analogous to but stronger than quasiconvexity for f. The case 1相似文献   

Definability properties and the congruence closure

We introduce a natural class of quantifiersTh containing all monadic type quantifiers, all quantifiers for linear orders, quantifiers for isomorphism, Ramsey type quantifiers, and plenty more, showing that no sublogic ofL (Th) or countably compact regular sublogic ofL (Th), properly extendingL , satisfies the uniform reduction property for quotients. As a consequence, none of these logics satisfies either-interpolation or Beth's definability theorem when closed under relativizations. We also show the failure of both properties for any sublogic ofL (Th) in which Chang's quantifier or some cardinality quantifierQ , with 1, is definable.     

On the best approximation in the metric of L to certain classes of functions by Haar-system polynomials

Let H, H ^L be classes of functionsf(x) whose modulus of continuity (f; t) and, respectively, integral modulus of continuity(f; t)_L do not exceed a given modulus of continuity(t), while H_v is a class of functionsf(x) whose variation fdoes not exceed a given number V > 0. Bounds are obtained for the upper limit of the best approximations in the metric of L by Haar-system polynomials on the classes just introduced (on the class H ^L only when (t)=Kt). These bounds are exact for class H_V and, in case(t) is convex, also for the classes H and H ^L .Translated from Matematicheskie Zametki, Vol. 6, No. 1, pp. 47–54, July, 1969.The author wishes to thank N. P. Korneichuk for having posed the problem and for his constant attention to this work.     

Computation of efficient solutions of discretely distributed stochastic optimization problems

In engineering and economics often a certain vectorx of inputs or decisions must be chosen, subject to some constraints, such that the expected costs (or loss) arising from the deviation between the outputA() x of a stochastic linear systemxA()x and a desired stochastic target vectorb() are minimal. Hence, one has the following stochastic linear optimization problem minimizeF(x)=Eu(A()x b()) s.t.xD, (1) whereu is a convex loss function on ^m , (A(), b()) is a random (m,n + 1)-matrix, E denotes the expectation operator andD is a convex subset of ⁿ . Concrete problems of this type are e.g. stochastic linear programs with recourse, error minimization and optimal design problems, acid rain abatement methods, problems in scenario analysis and non-least square regression analysis.Solving (1), the loss functionu should be exactly known. However, in practice mostly there is some uncertainty in assigning appropriate penalty costs to the deviation between the outputA ()x and the targetb(). For finding in this situation solutions hedging against uncertainty a set of so-called efficient points of (1) is defined and a numerical procedure for determining these compromise solutions is derived. Several applications are discussed.