On the generalized Lie structure of associative algebras

We study the structure of Lie algebras in the category ^H MA ofH-comodules for a cotriangular bialgebra (H, 〈|〉) and in particular theH-Lie structure of an algebraA in ^H MA. We show that ifA is a sum of twoH-commutative subrings, then theH-commutator ideal ofA is nilpotent; thus ifA is also semiprime,A isH-commutative. We show an analogous result for arbitraryH-Lie algebras whenH is cocommutative. We next discuss theH-Lie ideal structure ofA. We show that ifA isH-simple andH is cocommutative, then any non-commutativeH-Lie idealU ofA must contain [A, A]. IfU is commutative andH is a group algebra, we show thatU is in the graded center ifA is a graded domain. Dedicated to the memory of S. A. Amitsur Supported by a Fulbright grant. Supported by NSF grant DMS-9203375.     

Automorphisms of finite groups of bounded rank

LetG be a finite group admitting an automorphismα withm fixed points. Suppose every subgroup ofG isr-generated. It is shown that (1)G has a characteristic soluble subgroupH whose index is bounded in terms ofm andr, and (2) if the orders ofα andG are coprime, then the derived length ofH is also bounded in terms ofm andr. To Professor John Thompson, in honor of his outstanding achievements     

On the evaluation of multivariate polynomials and their derivatives

LetH be a polynomial inn>1 variables over the fields of real or complex numbers. An algorithm is presented here for the simultaneous evaluation ofH and its first and second (F-) derivativesH andH, or of any combination ofH,H,H. The evaluations ofH alone or ofH andH together are of the same order inn andd whered is the degree ofH, while the computation ofH,H, andH isd times this order. The process takes account of the sparsity pattern ofH by using a tree structure induced by the nonzero coefficients. It also allows for simultaneous operation with several polynomials with the same sparsity pattern. The data structure for the method is rather simple in nature and can be adapted easily to specific types of polynomials. Several possible implementations and their complexity are discussed.     

On the isomorphism of commutative group algebras

LetG andH be finite abelian groups and letF be an arbitrary field. One fundamental problem is that of determining necessary and sufficient conditions for the isomorphism of the group algebrasFG andFH. No solution has appeared in the literature. Nevertheless by combining the results of Berman, Perlis and Walker, Cohen, and Deskins and providing connecting arguments a complete solution can be obtained. It is the purpose of this note to present such a solution.     

Remarks on the generic rank of a CR mapping

We study germs of smooth CR mappings between embedded real hypersurfaces in complex spaces of the same dimension. In particular, we are interested in the generic rank of such mappings. IfH:M →M′ is a CR map between two hypersurfacesM andM′, we prove that ifM′ does not contain any germ of a holomorphic manifold then eitherH is constant or the generic rank ofH is odd. We also prove that if there is no formal holomorphic vector field tangent toM, then eitherH is constant or genericallyH is a local diffeomorphism. It follows, as a special case, that ifM andM′ are of D-finite type (in the sense of D’Angelo) thenH is either constant or is generically a local diffeomorphism. Supported by NSF Grant DMS 8901268.     

Greatest common divisors and least common multiples of graphs

A graphH divides a graphG, writtenH|G, ifG isH-decomposable. A graphG without isolated vertices is a greatest common divisor of two graphsG ₁ andG ₂ ifG is a graph of maximum size for whichG|G ₁ andG|G ₂, while a graphH without isolated vertices is a least common multiple ofG ₁ andG ₂ ifH is a graph of minimum size for whichG ₁|H andG ₂|H. It is shown that every two nonempty graphs have a greatest common divisor and least common multiple. It is also shown that the ratio of the product of the sizes of a greatest common divisor and least common multiple ofG ₁ andG ₂ to the product of their sizes can be arbitrarily large or arbitrarily small. Sizes of least common multiples of various pairsG ₁,G ₂ of graphs are determined, including when one ofG ₁ andG ₂ is a cycle of even length and the other is a star.G. C's research was supported in part by the Office of Naval Research, under Grant N00014-91-I-1060     

Isometries of JB-algebras

A bijective linear mapping between two JB-algebrasA andB is an isometry if and only if it commutes with the Jordan triple products ofA andB. Other algebraic characterizations of isometries between JB-algebras are given. Derivations on a JB-algebraA are those bounded linear operators onA with zero numerical range. For JB-algebras of selfadjoint operators we have: IfH andK are left Hilbert spaces of dimension ≥3 over the same fieldF (the real, complex, or quaternion numbers), then every surjective real linear isometryf fromS(H) ontoS(K) is of the formf(x)=UoxoU ⁻¹ forx inS(H), whereτ is a real-linear automorphism ofF andU is a real linear isometry fromH ontoK withU(λh)=τ(λ)U(h) for λ inF andh inH. Supported by Acción Integrada Hispano-Alemana HA 94 066 B     

Banach-stone theorems and separating maps

LetC(X,E) andC(Y,F) denote the spaces of continuous functions on the Tihonov spacesX andY, taking values in the Banach spacesE andF, respectively. A linear mapH:C(X,E)→C(Y,F) isseparating iff(x)g(x)=0 for allx inX impliesHf(y)Hg(y)=0 for ally inY. Some automatic continuity properties and Banach-Stone type theorems (i.e., asserting that isometries must be of a certain form) for separating mapsH between spaces of real- and complex-valued functions have already been developed. The extension of such results to spaces of vector-valued functions is the general subject of this paper. We prove in Theorem 4.1, for example, for compactX andY, that a linear isometryH betweenC(X,E) andC(Y,F) is a “Banach-Stone” map if and only ifH is “biseparating (i.e,H andH ⁻¹ are separating). The Banach-Stone theorems of Jerison and Lau for vector-valued functions are then deduced in Corollaries 4.3 and 4.4 for the cases whenE andF or their topological duals, respectively, are strictly convex. Research supported by the Fundació Caixa Castelló, MI/25.043/92     

On fixing edges in graph minors

An area of much recent research activity has been involved with tying the presence of certain minors in a matroid to specific elements of this matroid. The aim of this paper is to show that there are exactly two 3-connected simple graphsG with at least four edges and the property that ifH is a 3-connected simple graph havingG as a minor ande andf are edges ofH, thenH has a minor isomorphic toG which containse andf in its edge set. Some extensions of this result are also considered.     

Representation theory of Hopf galois extensions

LetH be a Hopf algebra over the fieldk andB ⊂A a right faithfully flat rightH-Galois extension. The aim of this paper is to study some questions of representation theory connected with the ring extensionB ⊂A, such as induction and restriction of simple or indecomposable modules. In particular, generalizations are given of classical results of Clifford, Green and Blattner on representations of groups and Lie algebras. The stabilizer of a leftB-module is introduced as a subcoalgebra ofH. Very often the stabilizer is a Hopf subalgebra. The special case whenA is a finite dimensional cocommutative Hopf algebra over an algebraically closed field,B is a normal Hopf subalgebra andH is the quotient Hopf algebra was studied before by Voigt using the language of finite group schemes.     

Classification and enumeration of matched free response matrices of linear finite automata

IfH andG are the transfer function matrix and the free response matrix of a linear finite automaton, thenG is called a free response matrix matched toH. GivenH, letG(H) be the set of all possible free response matrices matched toH. A classification and an enumeration on the setG(H) are given in this paper. Supported by National Natural Science Foundation     

On harmonic maps between generalized flag manifolds

The theory of harmonic maps has been developed since the 1960's (see [2]). In recent years, some authors discussed the harmonicity of “homogeneous” maps between Riemannian homogeneous spaces using the theory of Lie groups. LetG andG′ be compact Lie groups,H andH′ their closed subgroups respectively. Assume that a homomorphism θ:G→G′ mapsH intoH′; then there exists an induced mapf _θ:G/H→G′/H′. M.A. Guest gave a necessary and sufficient condition for such a map to be harmonic, whenG/H andG′/H′ are generalized flag manifolds,H=T is a maximal torus andG′ is a unitary group; and he gave some interesting examples (see [3]). We generalize his results to the case of general generalized flag manifoldsG/H, i.e.H is a centralizer of a torus, and give some new examples of harmonic maps. Supported in part by the National Natural Science Foundation of China and K.C. Wong Education Foundation (in Hong Kong).     

Subgroups of prescribed finite index in linear groups

LetR be a commutative ring,M a finitely generatedR-module andG a subgroup of Aut _R M. Under either of the following conditions, for every positive integerd there is a normal subgroupH ofG of finite index such thatG/H contains an element of orderd. (a)G is infinite and finitely generated. (b)R is finitely generated as a ring andG is not unipotent-by-finite. This extends recent work of A. Lubotzky.     

On the algebraic independence in the selberg class

We prove that a functionF of the Selberg class ℐ is ab-th power in ℐ, i.e.,F=H ^b for someHσ ℐ, if and only ifb divides the order of every zero ofF and of everyp-componentF _p. This implies that the equationF ^a=G^b with (a, b)=1 has the unique solutionF=H ^b andG=H ^a in ℐ. As a consequence, we prove that ifF andG are distinct primitive elements of ℐ, then the transcendence degree of ℂ[F,G] over ℂ is two.     

ELLIPTIC H^2-VOLTERRA PROJECTION AND THE H^1-GALERKIN METHODS FOR THE INTEGRO-DIFFERENTIAL EQUATIONS OF EVOLUTION

This paper studies H^1-Galerkin methods for the integro-differential equations of evolution. The elliptic H^2-Volterra projection is induced and then used in the derivations of error estimates for semi-discrete and full-discrete H^1-Galerkin methods.The optimal L^2, H^1 and H^2 norm error estimates are obtained.     

Singularities of 3-dimensional varieties admitting an ample effective divisor of Kodaira dimension zero

For a normal threefoldX with an effective Cartier divisorH, which is a minimal model of Kodaira dimension zero, we prove that eitherX is a generalized cone overH, orX has quadruple singularities andH is either a K3 surface, or an Enriques surface. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 617–625, April, 1996. The author is extremely grateful to V. A. Iskovskikh and Yu. G. Prokhorov for valuable advice and fruitful discussions. This research was partially supported by the International Science Foundation under grant No. M90 000.     

Commutants of nest algebras modulo the compact operators

Any operatorx which commutes modulo the compact operators with a nest algebra is of the form λI+C, where λ is a scalar andC is a compact operator. Any derivation from a nest algebra on a Hilbert spaceH into the compact operators onH is implemented by a compact operator. Any derivation on a quasitriangular operator algebra is inner.     

A linearity theorem for group actions on spheres with applications to homotopy representations

LetG be a finite group andX an equivariantZ/|G|-homology sphere. By Smith-theory the fixed point setX ^H for ap-subgroupH is aZ/p-homology sphere of dimensiond(H).     

Minimizing noisy functionals in hilbert space: An extension of the Kiefer-Wolfowitz procedure

LetH be an infinite-dimensional real separable Hilbert space andR the real line. Given an unknown functionalg:HR that can only be observed with random error, we consider a recursive method to locate an extremum ofg. Applications to optimal stochastic control are presented.     

On the variational convergence of fuzzy sets in metric spaces

Kaleva [9] has studied the relationships between the metric convergencesH andD of fuzzy convex sets on Euclidean spaces. The distanceH between two fuzzy set is given by Hausdorff distance of their sendographs, whileD is the supremum of the Hausdorff distances of the level sets corresponding to the fuzzy sets. The aim of this paper is to compareH andD with the variational convergence, called γ-convergence (see De Giorgi and Franzoni [3]). Our analysis which is carried out in the setting of metric spaces (not necessarily locally compact or vector spaces), improves Kaleva's results.

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Sunto Kaleva ha investigato in [9] le relazioni esistenti tra due convergenze metriche, detteH eD, di sottoinsiemi fuzzy di spazi euclidei finito-dimensionali. In questo articolo le convergenzeH eD (la loro definizione dipende dalla distanza di Hausdorff tra insiemi compatti) sono confrontate con la convergenza variazionale, detta γ-convergenza, introdotta da De Giorgi and Franzoni in [3] nel contesto degli spazi topologici. Tale confronto con la γ-convergenza (vedi Teorema 3.7), svolto nell'ambito degli spazi metrici (non necessariamente, localmente compatti o lineari) migliora ed estende i precedenti risultati di Kaleva.