K-Theory of the Commutator Ideal

We consider for unital c ^*-algebras the short exact sequence 0 I A * _C B A _max B 0and give an explicit formula for the boundary map for separable C ^*-algebras for which the Künneth formula for tensor products holds without Tor-terms. In the course of the article, we study the similar case of an ideal concerning one-relator groups.     

The Volodin model for hermitian algebraic K-theory

We define the Volodin hermitian algebraic K-theory for a (discrete) ring with an involution and show that it is isomorphic to Karoubi's hermitian algebraic K-theory. We also construct the Volodin model X(R _*) of hermitian algebraic K-theory for a simplicial ring R _* and show that it is a homotopy fiber of the map B Ô(R _*)B Ô(R _*)⁺. We also prove the general linear version of this result, which has been claimed in the existing literature, but whose proof was overlooked.     

Décomposition spectrale dans des algèbres munies d'un star-produit

Summary The purpose of this paper is to study, in intrinsic way, the Moyal's product, defined in the flat space R ²ⁿ. This product is defined here with the twisted convolution and the Fourier transform. The S(R ²ⁿ) and L²(R ²ⁿ) spaces are_*5-algebras. Because of this definition, the_*V-product of some tempered distributions is defined. Let O _M ^v be the set of multiplication operators in S(R ²ⁿ). By transposition, the S(R ²ⁿ) space is a right-module on O _M ^v . The support of f_*v g is different from the support of f·g; under large enough hypotheses, there is a Taylor's formula for the star-product function of the v variable. The ^v space of the multiplication operators in L²(R ²ⁿ) is defined here as the space of tempered distributions, the image of which is the set of bounded operators in L²(R ²ⁿ) by the Weyl map. After the study of ^v space, it is possible to show the spectral resolution of the real elements of ^v or of O _M ^v , which satisfies a, probably superfluous, hypothesis.     

Simultaneous reduction of a lattice basis and its reciprocal basis

Given a latticeL we are looking for a basisB=[b ₁, ...b _n ] ofL with the property that bothB and the associated basisB ^*=[b ₁ ^* , ...,b _n ^* ] of the reciprocal latticeL ^* consist of short vectors. For any such basisB with reciprocal basisB ^* let . Håstad and Lagarias [7] show that each latticeL of full rank has a basisB withS(B)exp(c ₁·n ^1/3) for a constantc ₁ independent ofn. We improve this upper bound toS(B)exp(c ₂·(lnn)²) withc ₂ independent ofn.We will also introduce some new kinds of lattice basis reduction and an algorithm to compute one of them. The new algorithm proceeds by reducing the quantity . In combination with an exhaustive search procedure, one obtains an algorithm to compute the shortest vector and a Korkine-Zolotarev reduced basis of a lattice that is efficient in practice for dimension up to 30.     

Unique determination of the convolutions of measures in Rm,m⩾2, by their restriction to a set

Restrictions are indicated on a complex-valued measure in the spaceR ^m ,m 2., under which the n-fold convolution ^n*,n 2, is uniquely determined by its values on any semispacex ₁<r, rR.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 50, pp. 86–90, 1988.     

A nonlinear equation for linear programming

We present a characterization of the normal optimal solution of the linear program given in canonical form max{c ^tx: Ax = b, x 0}. (P) We show thatx ^* is the optimal solution of (P), of minimal norm, if and only if there exists anR > 0 such that, for eachr R, we havex ^* = (rc – A^t_r)₊. Thus, we can findx ^* by solving the following equation for _r A(rc – A^t_r)₊ = b. Moreover,(1/r) _r then converges to a solution of the dual program.On leave from The University of Alberta, Edmonton, Canada. Research partially supported by the National Science and Engineering Research Council of Canada.     

A representation for the remainder of the Maclaurin quadrature formula

Summary We show that the remainder of the Maclaurin quadrature formula belonging to oddn (n+1 is the number of nodes) can be represented asR _n (f)=c _n f ⁽ⁿ⁺¹⁾ (), wheneverf ⁽ⁿ⁺¹⁾ exists and is continuous The corresponding problem for evenn has already been settled by A. Walther in 1925.     

Lower bounds for increasing complexity of derivations after cut elimination

We denote by C _k ^* the formula. In this paper for all k there is constructed a derivation of C _k ^* with cut, the number of sequents in which depends linearly on k. On the other hand, it is impossible to give an upper bound which is a Kalmar elementary function of k for the number of sequents in any derivation of the formula C _k ^* without cuts, or for the number of disjunctions in a refutation by the method of resolutions of systems of disjunctions corresponding to the negation of the formula C _k ^* . In particular, one can construct a derivation with cut of the formula C ₆ ^* , in which there is contained no more than 253 sequents, but in seeking a derivation of C ₆ ^* by the method of resolutions it is necessary to construct more than 10¹⁹²⁰⁰ disjunctions. With the help of Skolemization and taking out of quantifiers with respect to the formula C _k ^* there is constructed a formula v₀B _k ⁺ (v₀), which satisfies the following conditions: 1) one can construct a derivation with cuts of the formula v₀B _k ⁺ (v₀) in the constructive predicate calculus, the number of sequents in which depends linearly on k; 2) it is impossible to give an upper bound which is a Kalmar elementary function of k of the length of a term t such that the formula B _k ⁺ (t) is derivable.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 137–161, 1979.     

Some properties of the delay probability inM/M/s/s +c systems

We look at an extension of the steady state delay probability inM/M/s/s + c systems to nonintegral number of serverss and queue capacityc, which we call GED function. We show that this function is increasing and concave in the queue capacity. We find that if c 1, the reciprocal of the GED function is convex in the traffic intensity and the GED function is increasing in the traffic intensity if is below some ^* _s,c, and decreasing if is greater than ^* _s,c, Moreover, ^* _s,c is increasing in the number of servers and, fors 1, ^* _i,c=1 p^* _s,c < 2.Research supported by Grant BD/645/90-RM from Junta Nacional de Investigação Científica e Tecnológica.On leave from: Departamento de Matemätica, Instituto Superior Técnico, Av. Rovisco Pais, 1096 Lisboa Codex, Portugal.     

Stochastic Differentiation – A Generalized Approach

The space (D ^*) of Wiener distributions allows a natural Pettis-type stochastic calculus. For a certain class of generalized multiparameter processes X: R ^N (D ^*) we prove several differentiation rules (Itô formulas); these processes can be anticipating. We then apply these rules to some examples of square integrable Wiener functionals and look at the integral versions of the resulting formulas.     

Jordan (Super)Coalgebras and Lie (Super)Coalgebras

We discuss the question of local finite dimensionality of Jordan supercoalgebras. We establish a connection between Jordan and Lie supercoalgebras which is analogous to the Kantor–Koecher–Tits construction for ordinary Jordan superalgebras. We exhibit an example of a Jordan supercoalgebra which is not locally finite-dimensional. Show that, for a Jordan supercoalgebra (J,) with a dual algebra J ^*, there exists a Lie supercoalgebra (L ^c (J), _L ) whose dual algebra (L ^c (J))^* is the Lie KKT-superalgebra for the Jordan superalgebra J ^*. It is well known that some Jordan coalgebra J ⁰ can be constructed from an arbitrary Jordan algebra J. We find necessary and sufficient conditions for the coalgebra (L ^c (J ⁰),_L) to be isomorphic to the coalgebra (Loc(L _in (J)⁰), _L ⁰), where L _in (J) is the adjoint Lie KKT-algebra for the Jordan algebra J.     

Fano'sche Moufang-Ebenen mit Gauss-Figur sind pappos'sch

LetV be a quadrilateral in aMoufang-plane , in which theFano-proposition is valid. Take the pointsP,Q,R respectively in the diagonalsp,q,r ofV and construe the pointsP ^*,Q ^*,R ^* inp,q, r harmonic toP,Q,R with respect to pairs of edges ofV. IfP,Q,R are collinear, so areP ^*,Q ^*,R ^*, if and only if is aPappos-plane. Is V classical, the pointsP ₁ p,Qq,Rr and their harmonic conjugatesP ₁ ^* ,Q ^*,R ^* (construed as above mentioned) lay in a curve of 2^nd order.

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R. Artzy zum 70. Geburtstag zugeeignet     

A general descent framework for the monotone variational inequality problem

We present a framework for descent algorithms that solve the monotone variational inequality problem VIP _v which consists in finding a solutionv ^* _v satisfyings(v ^*)^T(v–v ^*)0, for allv _v. This unified framework includes, as special cases, some well known iterative methods and equivalent optimization formulations. A descent method is developed for an equivalent general optimization formulation and a proof of its convergence is given. Based on this unified logarithmic framework, we show that a variant of the descent method where each subproblem is only solved approximately is globally convergent under certain conditions.This research was supported in part by individual operating grants from NSERC.     

Rigidity theorem for harmonic maps from Riemannian manifold to Grassmannian manifold

We first study the Grassmannian manifoldG _n (R^n+p)as a submanifold in Euclidean space ⁿ (R ^n+p). Then we give a local expression for each map from Riemannian manifoldM toG ⁿ (R ^n+p) ⁿ (R ^n+p), and use the local expression to establish a formula which is satisfied by any harmonic map fromM toG _n (R ^n+p). As a consequence of this formula we get a rigidity theorem.     

Common supports as fixed points

A family S of sets in R ^d is sundered if for each way of choosing a point from rd+1 members of S, the chosen points form the vertex-set of an (r–1)-simplex. Bisztriczky proved that for each sundered family S of d convex bodies in R ^d , and for each partition (S , S ), of S, there are exactly two hyperplanes each of which supports all the members of S and separates the members of S from the members of S . This note provides an alternate proof by obtaining each of the desired supports as (in effect) a fixed point of a continuous self-mapping of the cartesian product of the bodies.     

Trace operators,Feynman distributions,and multiparameter white noise

Within the framework of white noise analysis on the probability space = ^* R ^d R ^M , the recent work by Johnson and Kallianpur on the Hu-Meyer formula, traces, and natural extensions is generalized to the multiparameter case:d>1. Besides providing a more general setting for these topics, the paper gives an alternative definition for the traces, a distributional version of the natural extension, and a generalized Kallianpur-Feynman distribution. The development illustrates how traces and natural extensions are intimately related to Wick products and the change of covariance formula from quantum field theory, as well as to the projective tensor product of Hilbert spaces from functional analysis.     

Eine Fehlerabschätzung zum Quadraturverfahren von Clenshaw und Curtis

Summary LetR _n [f] be the error of the quadrature formula of Clenshaw and Curtis. Narrow bounds for the coefficientc _n in |R _n [f]|c _n (n!)^–1 max |f ⁽ⁿ⁾ (x)| are given. As a consequence is proven, that the Clenshaw-Curtis-method will generally give more accurate results than the method of Filippi.     

BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials

We consider 3-parametric polynomialsP ^* (x; q, t, s) which replace theA _n-series interpolation Macdonald polynomialsP ^* (x; q, t) for theBC _n-type root system. For these polynomials we prove an integral representation, a combinatorial formula, Pieri rules, Cauchy identity, and we also show that they do not satisfy any rationalq-difference equation. Ass the polynomialsP ^* (x; q, t, s) becomeP ^* (x; q, t). We also prove a binomial formula for 6-parametric Koornwinder polynomials.     

Anti-Wick Symbols for Infinite Products in K-Homology

We consider infinite products in K-homology. We study these products in relation with operators on filtered Hilbert spaces, and infinite iterations of universal constructions on C*-algebras. In particular, infinite tensor power of extensions of pseudodifferential operators on R are considered. We extend anti-Wick pseudodifferential operators to infinite tensor products of spaces of the type L ²(R), and compare our infinite tensor power construction with an extension of pseudodifferential operators on R . We show that the K-theory connecting maps coincide. We propose a natural definition of ellipticity for anti-Wick operators on R, compute the corresponding index, and draw some consequences concerning these operators.