Algebras almost commuting with Clifford algebras in a II∞ factor

We are concerned here with certain Banach algebras of operators contained within a fixed II factor N. These algebras may be thought of as noncommutative classifying spaces for the functor Ext _* ^N The basic objects of study are the algebras A _kN (for n=1, 2,...). Here, we are given an essentially unique representation of the complex Clifford algebra C _k N and the elements of A _k are those operators in N which exactly commute with the first k–1 generators of C _k and also commute with the kth generator modulo a symmetric ideal N. Up to isomorphism, these algebras are periodic with period 2.We determine completely the homotopy types of A ₁ ^–1 and A ₂ ^–1 Here, A ₁ ^–1 is homotopy equivalent to the space of (Breuer) Fredholm operators in N, while A ₂ ^–1 is homotopy equivalent to the group K _N ^–1 ={x N^–1¦ x=1+k, k K_N}. We use these results to compute the K-theory of A ₁ and A ₂.For a fixed C ^*-algebra A, we define abelian groups G _k,N(A) of equivalence classes of homomorphisms AA _k. Letting N = M (H) for a II₁ factor M we define similar abelian groups G _k(A, M) where we replace N by L(E) for countably generated right Hilbert M-modules E with (left) actions C _k L(E). Using ideas of Skandalis, we show that G _k,NG_k(A, M) so that the G _k,N are stable half-exact homotopy functors because the G _k(·, M) are such.In general, we show that G _k(A, M)KK ^k(A, M) and so our theory fits neatly into Kasparov KK-theory. We investigate many interesting examples from our point of view.     

A Note on Primes and Goldbach Numbers in Short Intervals

Let J(N, H) be the Selberg integral and E(x, T) the error term in Kaczorowski-Perelli's weighted form of the classical explicit formula. We prove that the estimate J(N, H) = o(H2 N) is connected with an appropriate estimate of _N ^2N| E(x,T)² dx, uniformly for H and T in some ranges. Moreover, assuming a suitable bound for _N ^2N| E(x,T)|² dx, we also obtain, for all sufficiently large N and H (log N)11/12, that every interval [N,N + H] contains H Goldbach numbers.     

On fixed points of conformal pseudogroups

We show in this paper Theorem 2 that if (H, H ₁) is a pseudogroup generated by a finite numberH ₁ of germs of conformal diffeomorphisms of defined on a sufficiently small discD, which is not linearizable and such that the linear group (L,H ₁)={g(0)/g(H,H ₁)}* is dense in *, then the set of fixed points of the pseudogroup (H, H ₁) is dense inD. This implies the abundance of distinct homotopy classes of loops in leaves of foliations defined in ² by generic polynomial vector fields as well as for germs of holomorphic vector fields in ² beginning with generic jets, both of degree at least 2. These homotopy classes may be realized arbitrarily close to the line at infinity or to 0, respectively. This shows the genericity of polynomial vector fields with infinite Petrovsky-Landis genus ([5]).The idea of the proof is very simple. Ifg is a non-linear conformal diffeomorphism with multiplier =g'(0), then the map obtained by the composition ofg and the linear map with multiplier ^–1 will have at 0 a fixed point of multiplicity at least 2. Since we may approximate ^–1 by elementsh in the pseudogroup and the multiplicity of fixed points satisfy a law of conservation of number, we obtain thath o g has fixed points close to 0. These fixed points appear as a by product of the relative nonlinearity of the generators of the pseudogroup, since linearizable pseudogroups have 0 as an isolated fixed point. The fixed points obtained are not conjugate since they have distinct multipliers.The main technical tool is the angular derivative introduced in [8]. It allows one to split the search for fixed points into two parts: One is to obtain a contraction and the other is to return arbitrarily close to the starting point without modifying the property of contraction. This is carried out since the angular derivative is multiplicative for compositions and is identically 1 for linear maps.Supported by CONACYT-CNRS and CONACYT 3398-E9307.     

A Complexity Analysis of a Smoothing Method Using CHKS-functions for Monotone Linear Complementarity Problems

We consider the standard linear complementarity problem (LCP): Find (x, y) R ²ⁿ such that y = M x + q, (x, y) 0 and x _i y _i = 0 (i = 1, 2, ... , n), where M is an n × n matrix and q is an n-dimensional vector. Recently several smoothing methods have been developed for solving monotone and/or P ₀ LCPs. The aim of this paper is to derive a complexity bound of smoothing methods using Chen-Harker-Kanzow-Smale functions in the case where the monotone LCP has a feasible interior point. After a smoothing method is provided, some properties of the CHKS-function are described. As a consequence, we show that the algorithm terminates in Newton iterations where is a number which depends on the problem and the initial point. We also discuss some relationships between the interior point methods and the smoothing methods.     

Sur la propriétéP *

On the PropertyP _*. LetG be a locally compact group. We give a direct proof of the equivalence of the propertiesP ₁ andP _*, avoiding the use of invariant means onL (G).     

Normalized Solutions of Schrödinger Equations with Potentially Bounded Measures

Given a potentially bounded signed measure on a Brelot space (X,) with Green function G, it is well known that -harmonic functions (i.e., in the classical case, finely continuous versions of solutions to u–u=0) may be very discontinuous. In this paper it is shown that under very general assumptions on G (satisfied for large classes of elliptic second-order linear differential operators) normalized perturbation, however, leads to a Brelot space (X, ) admitting a Green function T (G) which is locally (or even globally) comparable with G and has all properties required of G before. In particular, iterated perturbation is possible. Moreover, intrinsic Hölder continuity of quotients of harmonic functions with respect to the local quasimetric :=(G ^–1+^* G ^–1)/2 yields -Hölder continuity for quotients of -harmonic functions as well.     

The dynamics of turbo-flow

Zusammenfassung Ausgehend von der Tatsache, dass eine Schaufelreihe äquivalent ist einem Kraft- und damit auch einem Wirbelfeld, wird der Begriff der Potential-Kraft-Strömung erweitert. Diese schliesst den «von Mises-Fall» ein, wenn das Euler-Integral im ganzen Feld konstant ist.Die Struktur des Wirbelfeldes, sowohl des freien als auch des gebundenen, wird analysiert. Das Problem der Turbinenströmung reduziert sich auf die Lösung von zwei partiellen Differentialgleichungen, für die ein vollständiger Satz von Randbedingungen gegeben ist; die Rechnung wird auf bekannte numerische Methoden zurückgeführt.Während die üblichen Rechnungsmethoden nur Eintritts- und Austrittsdiagramme liefern, erhalten wir hier die vollständige Geschwindigkeitsverteilung im ganzen Kraftfeld. Die Schaufelform wird angegeben, und es ist möglich, das Zusammenwirken verschiedener Entwurfsparameter zu untersuchen.

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Nomenclature

Symbol a a constantL ² T ^–2 - B blade surface 0 - C velocity vectorLT ^–1 - C speedLT ^–1 - C _a average axial velocityLT ^–1 - d diameterL - F force per unit massLT ^–2 - f blade shape function 0 - g angular momentum per unit mass (g=C r)L ² T ^–1 - h blade heightL - H Bernoulli's constant (stator)L ² T ^–2 - H ^* Bernoulli's constant (rotor)L ² T ^–2 - k a constantL ² T ^–2 - K a parameterL ^–1 T ^–2 - p pressureML ^–1 T ^–2 - r radius, distance from axisL - R ₀ outer radiusL - R _i hub or inner radiusL - s arc lengthL - t timeT - u a unit vector depending on use - W relative velocityLT ^–1 - load functionL ² T ^–2 - family of surfaces 0 - a parameter 0 - r-component of vorticityT ^–1 - -component of vorticityT ^–1 - angle or direction 0 - a parameter 0 - z-component of vorticityT ^–1 - the number 0 - densityML ^–3 - a functionL ^–1 T ^–1 - direction or angular coordinate in rotors 0 - stream functionL ³ T ^–1 - angular velocityT ^–1 - vorticity vectorT ^–1 - vorticityT ^–1 This work, finished in August 1953, was done while the author was at the Aircraft Gas Turbine Division of the General Electric Company, Cincinnati, Ohio.     

Compact Hankel operators on harmonic Bergman spaces

We study Hankel operators on the harmonic Bergman spaceb ²(B), whereB is the open unit ball inR ⁿ,n2. We show that iff is in then the Hankel operator with symbolf is compact. For the proof we have to extend the definition of Hankel operators to the spacesb ^p(B), 1<p<, and use an interpolation theorem. We also use the explicit formula for the orthogonal projection ofL ²(B, dV) ontob ²(B). This result implies that the commutator and semi-commutator of Toeplitz operators with symbols in are compact.     

Aspects of Number Theory in Terms of Potential Theory

We develop the Potential Theory on the set N^* of all natural numbers 2 associated with the kernel V (resp. V ^*) given by

The monotone follower problem in stochastic decision theory

We consider the problem of optimally tracking the random demandx+w _t, w. Brownian motion, by a nondecreasing process. adapted to the Brownian past, so as to minimize the expected lossE ₀ ^T (x+w_t–_t)dt. The decision problem is reduced to a free boundary one, and the latter is studied and solved for a large class of cost functions().This research was supported in part by the Air Force Office of Scientific Research, under AF-AFOSR 77-3063.     

Homomorphisms into N *

We study the properties of continuous homomorphisms from β S into N ^* and from N ^* into N ^* . We show that the image C of N ^* under a continuous homomorphism which does not arise from the continuous extension of a homomorphism mapping N to itself, has the property that C+C is a singleton.     

Trust region algorithms for the nonlinear least distance problem

The nonlinear least distance problem is a special case of equality constrained optimization. Let a curve or surface be given in implicit form via the equationf(x)=0,x R ^d , and letz R ^d be a fixed data point. We discuss two algorithms for solving the following problem:Find a point x ^* such that f(x ^* )=0and z–x ^*₂ is minimal among all such x. The algorithms presented use thetrust region approach in which, at each iteration, an, approximation to the objective function or merit function is minimized in a given neighborhood (the trust region) of the current iterate. Among other things, this allows one to prove global convergence of the algorithm.Both authors were supported in part by the Deutsche Forschungsgemeinschaft, SFB 350.     

Analysis of Henrici's Transformation for Singular Problems

Henrici's transformation is the underlying scheme that generates, by cycling, Steffensen's method for the approximation of the solution of a nonlinear equation in several variables. The aim of this paper is to analyze the asymptotic behavior of the obtained sequence (s _n ^* ) by applying Henrici's transformation when the initial sequence (s _n ) behaves sublinearly. We extend the work done in the regular case by Sadok [17] to vector sequences in the singular case. Under suitable conditions, we show that the slowest convergence rate of (s _n ^* ) is to be expected in a certain subspace N of R ^p . More precisely, if we write s _n ^* =s _n ^* ,N+s _n ^* ,N, the orthogonal decomposition into N and N , then the convergence is linear for (s _n ^* ,N) but ( _n ^* ,N) converges to the same limit faster than the initial one. In certain cases, we can have N=R ^p and the convergence is linear everywhere.     

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.     

A family of Hermite interpolants by bisection algorithms

A two point subdivision scheme with two parameters is proposed to draw curves corresponding to functions that satisfy Hermite conditions on [a, b]. We build two functionsf andf ¹ on dyadic numbers and for some values of the parameters,f is in ¹ withf ¹=f. Examples are provided which show how different the curves can be.     

Optimal quadratures for analytic functions

For integrals _–1 ¹ w(x)f(x)dx with and with analytic integrands, we consider the determination of optimal abscissasx _i ^o and weightsA _i ^o , for a fixedn, which minimize the errorE _n (f)= _–1 ¹ w(x)f(x)dx – _i ⁼¹ⁿ A _i f(x _i ) over an appropriate Hilbert spaceH ²(E ; w(z)) of analytic functions. Simultaneously, we consider the simpler problem of determining intermediate-optimal weightsA _i *, corresponding to (preassigned) Gaussian abscissasx _i ^G , which minimize the quadrature error. For eachw(x), the intermediate-optimal weightsA _i * are obtained explicitly, and these come out proportional to the corresponding Gaussian weightsA _i ^G . In each case,A _i ^G =A _i *+O( ^–4n ), . For , a complete explicit solution for optimal abscissas and weights is given; in fact, the set {x _i ^G ,A _i *;i=1,...,n} to provides the optimal abscissas and weights. For otherw(x), we study the closeness of the set {x _i ^G ,A _i *;i=1,...,n} to the optimal solution {x _i ^o ,A _i ^o ;i=1,...,n} in terms of _n (), the maximum absolute remainder in the second set ofn normal equations. In each case, _n () is, at least, of the order of ^–4n for large.     

On the convexity of the sum of the first eigenvalues of operators depending on a real parameter

Let ₁ (k) ₂ (k) be the eigenvalues of an operator of a certain type depending on a real parameterk. The paper shows that under certain requirements on the operator and on the nature of its dependence onk, the sum ₁ (k)++ _N (k) is a concave function ofk, for any positive integerN.

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Zusammenfassung Seien ₁ (k) ₂ (k) die Eigenwerte eines von einem reellen Parameterk abhängigen Operators. Man zeigt, daß unter gewissen Voraussetzungen über den Operator und seine Abhängigkeit vonk die Summe ₁ (k)++ _N (k) für jedesN eine konkave Funktion vonk ist.

     

Preconditioners for Ill-Conditioned Toeplitz Matrices

This paper is concerned with the solution of systems of linear equations A _N x = b, where denotes a sequence of positive definite Hermitian ill-conditioned Toeplitz matrices arising from a (real-valued) nonnegative generating function f C ₂ with zeros. We construct positive definite Hermitian preconditioners M _N such that the eigenvalues of M _N ^–1 A _N are clustered at 1 and the corresponding PCG-method requires only O(N log N) arithmetical operations to achieve a prescribed precision. We sketch how our preconditioning technique can be extended to symmetric Toeplitz systems, doubly symmetric block Toeplitz systems with Toeplitz blocks and non-Hermitian Toeplitz systems. Numerical tests confirm the theoretical expectations.     

Dually Degenerate Varieties and the Generalization of a Theorem of Griffiths–Harris

The dual variety X^* for a smooth n-dimensional variety X of the projective space P^N is the set of tangent hyperplanes to X. In the general case, the variety X^* is a hypersurface in the dual space (P^N)^*. If dimX^*<N–1, then the variety X is called dually degenerate. The authors refine these definitions for a variety XP^N with a degenerate Gauss map of rankr. For such a variety, in the general case, the dimension of its dual variety X^* is N–l–1, where l=n–r, and X is dually degenerate if dimX^*<N–l–1. In 1979 Griffiths and Harris proved that a smooth variety XP^N is dually degenerate if and only if all its second fundamental forms are singular. The authors generalize this theorem for a variety XP^N with a degenerate Gauss map of rankr. Mathematics Subject Classification (2000) 53A20.     

Estimates of the independence number of a hypergraph and the Ryser conjecture

For the Legendre-Sobolev orthonormal polynomials depending on the parametersM≥0,N≥0, weighted and uniform estimates on the orthogonality interval are obtained. It is shown that, unlike the Legendre orthonormal polynomials, the polynomials forM>0,N>0 decay at the rate ofn ^−3/2 at the points 1 and −1. The values of are calculated. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 871–880, December, 1997. Translated by N. K. Kulman