The intersection property in the theory of radicals of topological algebras

LetK be a class of associative topological algebras that is closed under subalgebras with the induced topology, direct products, quotients, and semidirect topological products with respect to continuous homomorphisms. If is a radical of classK, then the following conditions are equivalent: 1) is a topological special radical; 2) the -semisimple algebras are topological subdirect products of prime -semisimple algebras ofK;This result is a corollary of a general result that establishes necessary and sufficient conditions for the radical to have the intersection property with respect to a class of prime algebras. Bibliography: 17 titles.Translated fromTrudy Seminara imeni I. G. Petrovskogo, No. 15, pp. 178–188.     

L-Splines

In this paper, we study the problem of unique interpolation and approximation by a class of spline functions,L-splines, containing as special cases the deficient and generalized spline functions ofAhlberg, Nilson, andWalsh [3, 5, 6], the Chebyshevian spline functions ofKarlin andZiegler [27], and the piecewise Hermite polynomial functions, as considered in [17]. We first give sufficient conditions for unique interpolation byL-spline functions in Section 2. Then, we obtain newL andL ² error estimates for interpolation byL-splines in Section 4, and show that these error estimates are, in a certain sense, sharp. In addition, we make a similar study for theg-splines ofSchoenberg, cf. [44, 3], in Section 5. In Section 6, an application of these new error estimates is made to the analysis of the error made in the use of finite dimensional subspaces ofL-splines andg-splines. in the Rayleigh-Ritz procedure for the class of nonlinear two-point boundary value problems studied in [17].Because of the rapid growth of the number of papers devoted to or connected with the topic of splines, we believe that a compilation of papers on splines for the reader's use is desirable, and such a list is found in the References at the end of this paper.This research was supported in part by NSF Grant GP-5553Papers not specifically concerned with splines are referred to in the text by [1, 2], etc.     

The parameterized local deduction theorem for quasivarieties of algebras and its application

Let be an algebraic type. To each classK of -algebras a consequence relation _K defined on the set of -equations is assigned. Some weak forms of the deduction theorem for _K and their algebraic counterparts are investigated. The (relative) congruence extension property (CEP) and its variants are discussed.CEP is shown to be equivalent to a parameter-free form of the deduction theorem for the consequence _K .CEP has a strong impact on the structure ofK: for many quasivarietiesK,CEP implies thatK is actually a variety. This phenomenon is thoroughly discussed in Section 5. We also discuss first-order definability of relative principal congruences. This property is equivalent to the fact that the deduction theorem for _K is determined by a finite family of finite sets of equations. The following quasivarietal generalization of McKenzie's [26] finite basis theorem is proved:LetK be quasivariety of algebras of finite type in which the principalK-congruences are definable. ThenK is finitely axiomatizable iff either the classK _FSI (of all relatively finitely subdirectly irreducible members ofK) or the class K_SI (of all relatively subdirectly irreducible members ofK) is strictly elementary.Applications of the theory to Heyting, interior, Sugihara, and ukasiewicz algebras are provided.Presented by R. McKenzie.The paper was presented in a talk given to the Conference on Algebraic Logic, Budapest, August 1988.     

Two equivalent definitions of a congruence on a finitary model in a quasivariety

LetL be a finitary language and letK be a subcategory of the category of allL-models andL-morphisms. For aK-objectA we consider two definitions of aK-congruence relation onA: that given by Rosenberg and Sturm [2], and that given by Adámek [1]. Both definitions are external definitions in the sense that they depend on the otherK-objects. IfK is a full subcategory, such that theK-objects form a quasivariety, then it is shown that the definitions ofK-congruence are equivalent and a purely internal characterisation is given.Presented by I. Rosenberg.I am indebted to Professor Teo Sturm as this paper originated from his seminar series on Algebraic Structures.     

Intégration Symplectique des Variétés de Poisson Régulières

Asymplectic integration of a Poisson manifold (M, Λ) is a symplectic groupoid (Γ,η) whichrealizes the given Poisson manifold, i.e. such that the space of units Γ₀ with the induced Poisson structure Λ₀ is isomorphic to (M, Λ). This notion was introduced by A. Weinstein in [28] in order to quantize Poisson manifolds by quantizing their symplectic integration. Any Poisson manifold can be integrated by alocal symplectic groupoid ([4], [13]) but already for regular Poisson manifolds there are obstructions to global integrability ([2], [6], [11], [17], [28]). The aim of this paper is to summarize all the known obstructions and present a sufficient topological condition for integrability of regular Poisson manifolds; we will indeed describe a concrete procedure for this integration. Further our criterion will provide necessary and sufficient if we require Γ to be Hausdorff, which is a suitable condition to proceed to Weinstein’s program of quantization. These integrability results may be interpreted as a generalization of the Cartan-Smith proof of Lie’s third theorem in the infinite dimensional case.

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Recherche supportée par D.G.I.C.Y.T. Espagne (Proyecto PB90-0765) et Xunta de Galicia (Proxecto XUGA20704B90)     

A problem of P. Seymour on nonbinary matroids

The following statement fork=1, 2, 3 has been proved by Tutte [4], Bixby [1] and Seymour [3] respectively: IfM is ak-connected non-binary matroid andX a set ofk-1 elements ofM, thenX is contained in someU ₄ ² minor ofM. Seymour [3] asks whether this statement remains true fork=4; the purpose of this note is to show that it does not and to suggest some possible alternatives. Supported in part by the National Science Foundation     

On the uniqueness of solutions to linear complementarity problems

This paper characterizes the classU of all realn×n matricesM for which the linear complementarity problem (q, M) has a unique solution for all realn-vectorsq interior to the coneK(M) of vectors for which (q, M) has any solution at all. It is shown that restricting the uniqueness property to the interior ofK(M) is necessary because whenU, the problem (q, M) has infinitely many solutions ifq belongs to the boundary of intK(M). It is shown thatM must have nonnegative principal minors whenU andK(M) is convex. Finally, it is shown that whenM has nonnegative principal minors, only one of which is 0, andK(M)≠R ⁿ , thenU andK(M) is a closed half-space.     

Solutions ofLu=u α dominated byL-harmonic functions

LetL be a second order elliptic differential operator on a differentiable manifoldM and let 1 <α≤2. We investigate connections bewween classU of all positive solutions of the equationLu=u ^α and classH of all positiveL-harmonic functions (i.e., solutions of the equationsLh=0). Putu∈U ⁰ ifu∈U and ifu≤h for someh∈H. To everyu∈U ⁰ there corresponds the minimalL-harmonic functionh _u which dominatesu andu→h _u is a 1–1 mapping fromU ⁰ onto a subsetH ⁰ ofH. The inverse mapping associates with everyh∈H ⁰ the maximal element ofU dominated byh. Supposeg(x, dy) is Green's kernel,k(x, y) is the Martin kernel and ?M is the Martin boundary associated withL. A subset Γ of ?M is calledR-polar if it is not hit by the rangeR of the (L, α)-superdiffusion. It is calledM-polar if $\int\limits_M {g\left( {c,dx} \right)[\int\limits_\Gamma {k(x,y)v(dy)]^\alpha } } $ is equal to 0 or ∞ for everyc∈M and every measure ρ. Everyh∈H has a unique representation $h(x) = \int\limits_{\partial M} {k\left( {x,y} \right)v\left( {dy} \right)} $ where ρ is a measure concentrated on the minimal partM ^* of ?M. We show that the condition:

1. ρ(Γ)=0 for allR sets Γ is necessary and the condition:
2. ρ(Γ)=0 for allM-polar sets Γ is sufficient forh to belong toH ⁰. IfM is a bounded domain of classC ^{2, λ} in ? ^d , then conditions (a) and (b) are equivalent and therefore each of them characterizesH ⁰. This was conjectured by Dynkin a few years ago and proved in a recent paper of Le Gall forL=Δ, α=2 and domains of classC ⁵.

     

Kostant's problem,Goldie rank and the Gelfand-Kirillov conjecture

Let g be a complex semisimple Lie algebra andU(g) its enveloping algebra. GivenM a simpleU(g) module, letL(M, M) denote the subspace of ad g finite elements of Hom(M, M). Kostant has asked if the natural homomorphism ofU(g) intoL(M, M) is surjective. Here the question is analysed for simple modules with a highest weight vector. This has a negative answer if g admits roots of different length ([7], 6.5). Here general conditions are obtained under whichU(g)/AnnM andL(M, M) have the same ring of fractions—in particular this is shown to always hold if g has only typeA _n factors. Combined with [21], this provides a method for determining the Goldie ranks for the primitive quotients ofU(g). Their precise form is given in typeA _n (Cartan notation) for which the generalized Gelfand-Kirillov conjecture for primitive quotients is also established.This paper was written while the author was a guest of the Institute for Advanced Studies, the Hebrew University of Jerusalem and on leave of absence from the Centre Nationale de la Recherche Scientifique     

On three-dimensional conformally flat quasi-Sasakian manifolds

LetM be a 3-dimensional quasi-Sasakian manifold. On such a manifold, the so-called structure function is defined. With the help of this function, we find necessary and sufficient conditions forM to be conformally flat. Next it is proved that ifM is additionally conformally flat with = const., then (a)M is locally a product ofR and a 2-dimensional Kählerian space of constant Gauss curvature (the cosymplectic case), or (b)M is of constant positive curvature (the non cosymplectic case; here the quasi-Sasakian structure is homothetic to a Sasakian structure). An example of a 3-dimensional quasi-Sasakian structure being conformally flat with nonconstant structure function is also described. For conformally flat quasi-Sasakian manifolds of higher dimensions see [O₁]     

Approximate factorizations with S/P consistently ordered M-factors

The behaviour of PCG methods for solving a finite difference or finite element positive definite linear systemAx=b with a (pre)conditioning matrixB=U ^TP^–1 U (whereU is upper triangular andP=diag(U)) obtained from a modified incomplete factorization, isunpredictable in the present status of knowledge whenever the upper triangular factor is not strictly diagonally dominant and 2P –D, whereD=diag(A), is not symmetric positive definite. The origin of this rather surprising shortcoming of the theory is that all upper bounds on the associated spectral condition number (B ^–1 A) obtained so far require either the strict diagonal dominance of the upper triangular factor or the strict positive definiteness of 2P –D. It is our purpose here to improve the theory in this respect by showing that, when the triangular factors are S/P consistently orderedM-matrices, nonstrict diagonal dominance is generally a sufficient requirement, without additional condition on 2P –D. As a consequence, the new analysis does not require diagonal perturbations (otherwise needed to keep control of the diagonal dominance ofU or of the positive definiteness of 2P –D). Further, the bounds obtained here on (B ^–1 A) are independent of the lower spectral bound ofD ^–1 A meaning that quasi-singular problems can be solved at the same speed as regular ones, an unexpected result.     

Abschätzungen für\sum\limits_{n \leqslant N} {B_1 (n\alpha )}

Estimates for . Let and letB ₁(x)={x}–1/2. In this paper we shall give best possible estimates for . On the importance of this sum see the papers ofBehnke [2],Hardy andLittlewood [5],Hecke [6] andOstrowski [9].

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Meinem verehrten Lehrer, Herrn Professor E. Hlawka zum siebzigsten Geburtstag gewidmet     

Sur l'interpolation des opérateurs maximaux monotones

On the Interpolation of Maximal Monotone Operators. We study here one way to extend to the maximal monotone case the results of linear interpolation, exposed bybalakrishnan in [2]. We obtain a necessary and sufficient condition of convergence for sequences(A ⁿ ) _n of maximal monotone operators on a real Hilbert spaceH.     

On theLU factorization ofM-matrices

Summary In this paper, we give in Theorem 1 a characterization, based on graph theory, of when anM-matrixA admits anLU factorization intoM-matrices, whereL is a nonsingular lower triangularM-matrix andU is an upper triangularM-matrix. This result generalizes earlier factorization results of Fiedler and Pták (1962) and Kuo (1977). As a consequence of Theorem 1, we show in Theorem 3 that the conditionx ^T A0 ^T for somex>0, for anM-matrixA, is both necessary and sufficient forPAP ^T to admit such anLU factorization for everyn×n permutation matrixP. This latter result extends recent work of Funderlic and Plemmons (1981). Finally, Theorem 1 is extended in Theorem 5 to give a characterization, similarly based on graph theory, of when anM-matrixA admits anLU factorization intoM-matrices.Dedicated to Professor Ky Fan on his sixty-seventh birthday, September 19, 1981.Research supported in part by the Air Force Office of Scientific Research, and by the Department of Energy     

The torsion of a submanifold of a Riemannian manifold

For a submanifoldM ⁿ of a Riemannian manifoldM ^q, the concept of a torsion bivector at the point x M ⁿ for given one- and two-dimensional directions fromT _x M ⁿ is introduced using only the first and second fundamental forms ofM ⁿ. Its relation to the concept of Gaussian torsion is then established. It is proved that: 1) equality to zero of the torsion bivector is necessary and, whenM ⁿ is a nondevelopable surface of a space of constant curvature with nonzero second fundamental form, is also sufficient for the "flattening" ofM ⁿ into some totally geodesicM ⁿ⁺¹ inM ^q; 2) when n = 2, the independence of the nonzero torsion bivector of direction characterizes a minimalM ² inM ^q.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 39–42, 1991.     

Subdirectly irreducible generalized sums of upper semilattice ordered systems of algebras

In [15] the generalized sum of an upper (F ₁ , F ₂ )-semilattice ordered system of algebras was defined. In this paper we find necessary and sufficient conditions under which this construction yields subdirectly irreducible algebras. Received March 3, 2000; accepted in final form June 14, 2001.     

Densely two-valued metric projections in uniformly convex Banach spaces

For every uniformly convex Banach spaceX with dimX2 there is a residual setU in the Hausdorff metric spaceB(X) of bounded and closed sets inX such that the metric projection generated by a set fromU is two-valued and upper semicontinuous on a dense and everywhere continual subset ofX. For any two closed and separated subsetsM ₁ andM ₂ ofX the points on the equidistant hypersurface which have best approximations both inM ₁ andM ₂ form a dense G set in the induced topology.The author is partially supported by the National Fund for Scientific Research at the Bulgarian Ministry of Science and Education under contract MM 408/94.     

Remarks on the Cauchy problem for first order hyperbolic systems with constant coefficient principal part

Abstract In this article we shall introduce the results obtained in [16], i.e., we shall give a necessary and sufficient condition that the Cauchy problem for first order hyperbolic systems with constant coefficient principal part is C^∞ well-posed under the maximal rank condition (see the condition (R) below). We shall also give a simple sufficient condition without any assumptions on the rank. Keywords: Hyperbolic system, Cauchy problem, Constant coefficient principal part     

On the sampling theorem for wavelet subspaces

In [13], Walter extended the classical Shannon sampling theorem to some wavelet subspaces. For any closed subspace V₀/L² (R), we present a necessary and sufficient condition under which there is a sampling expansion for everyf V₀-Several examples are given.     

Counting extreme <Emphasis Type="Italic">U</Emphasis><Subscript>1</Subscript> matrices and characterizing quadratic doubly stochastic operators

The U ₁ matrix and extreme U ₁ matrix were successfully used to study quadratic doubly stochastic operators by R. Ganikhodzhaev and F. Shahidi [Linear Algebra Appl., 2010, 432: 24–35], where a necessary condition for a U ₁ matrix to be extreme was given. S. Yang and C. Xu [Linear Algebra Appl., 2013, 438: 3905–3912] gave a necessary and sufficient condition for a symmetric nonnegative matrix to be an extreme U ₁ matrix and investigated the structure of extreme U ₁ matrices. In this paper, we count the number of the permutation equivalence classes of the n × n extreme U ₁ matrices and characterize the structure of the quadratic stochastic operators and the quadratic doubly stochastic operators.