Beitrag zur Reduktion des Entscheidungsproblems auf Klassen von Hornformeln mit kurzen Alternationen

Ohne Zusammenfassung Zusatz bei der Korrektur: Ein vollständiger und korrekter Beweis für die Entscheidbarkeit der eingangs angeführten Aanderaaschen Klasse ((0, ), (, , ...)) erscheint demnächst im JSL (S.O. Aanderaa/H.R.Lewis: Prefix classes of Krom formulas). Ebendort wird auch die Reduktionstypeneigenschaft für ((0, ), (0, 0, )) und ((0, )), (0, 0, )) nachgewiesen, während ((0, ), (, )) sich als entscheidbar herausgestellt hat (s. E. Börger: Eine entscheidbare Klasse von Kromformeln. ZMLG 19 (1973), 117–120.) Der Kromsche Reduktionstyp konnte mittlerweile einerseits zu ((0, ), (0, 4)) verschärft werden (s. D. Rödding, E. Börger: The undecidability of (0, 4)-formulae with binary disjunctions, vorgetragen auf dem Logic Coll. Bristol 1973, ein abstract erscheint im JSL), andererseits kündigt H.R.Lewis die Reduktionstypeneigenschaft für ((0, ), (0, 1)) an (s. H.R.Lewis: Krom formulas with one dyadic predicate letter. Notices AMS 20, 5 (1973) A-500, abstr. no. 73T-E78.)Dieser Aufsatz geht aus der Dissertation [2] hervor, die dem Fachbereich Mathematik der Mathematisch-Naturwissenschaftlichen Fakultät der Universität Münster im Sommersemester 1971 vorgelegt worden ist. Die Ergebnisse stammen aus dem Wintersemester 1970/71. Eine Ankündigung der hauptsächlichen Resultate ist in den Notices of the American Mathematical Society 19, 2 (1972) A-333 unter der abstract no. * 72T-E24 erschienen.     

Interpolation of generalized analytical functions of complex variable

A general approach is proposed to the interpolation of x -analytical functions of a complex variable with an arbitrary –,+[Basis x -analytical functions whose imaginary pan is a polynomial in x, and y are obtained in explicit form.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 3–9, 1986.     

On a transformation operator for a system of Sturm-Liouville equations

We prove the existence of a transformation operator with a condition at infinity that sends a solution of the matrix equation^–y + My=²y (M is a constant Hermitian matrix) into a solution of the matrix equation^–y+Q(x)y+My=²y (the matrix function Q(x) is continuously differentiable for 0 x< and it is Hermitian for each x belonging to [0, )); we study some properties of the kernel of the transformation operator.Translated from Matematicheskii Zametki, Vol. 11, No. 5, pp. 559–567, May, 1972.The authors express their thanks to B. M. Levitan for a discussion.     

On the asymptotic and invariantσ-algebras of random walks on locally compact groups

Summary Consider a random walk of law on a locally compact second countable groupG. Let the starting measure be equivalent to the Haar measure and denote byQ the corresponding Markov measure on the space of pathsG . We study the relation between the spacesL (G , ^a ,Q) andL (G , ⁱ ,Q) where ^a and ⁱ stand for the asymptotic and invariant -algebras, respectively. We obtain a factorizationL (G , ^a ,Q) L (G , ⁱ ,Q)L (C) whereC is a cyclic group whose order (finite or infinite) coincides with the period of the Markov shift and is determined by the asymptotic behaviour of the convolution powers ⁿ.     

An update on orthogonal polynomials and weighted approximation on the real line

We briefly review the state of orthogonal polynomials on (–, ), concentrating on analytic aspects, such as asymptotics and bounds on orthogonal polynomials, their zeros and their recurrence coefficients. We emphasize results rather than proofs. We also discuss applications to mean convergence of orthogonal expansions, Lagrange interpolation, Jackson-Bernstein theorems and the weighted incomplete polynomial approximation problem.     

Limiting behavior of weighted central paths in linear programming

We study the limiting behavior of the weighted central paths{(x(), s())} > 0 in linear programming at both = 0 and = . We establish the existence of a partition (B ,N ) of the index set { 1, ,n } such thatx _i() ands _j () as fori B , andj N , andx _N (),s _B () converge to weighted analytic centers of certain polytopes. For allk 1, we show that thekth order derivativesx ^(k) () ands ^(k) () converge when 0 and . Consequently, the derivatives of each order are bounded in the interval (0, ). We calculate the limiting derivatives explicitly, and establish the surprising result that all higher order derivatives (k 2) converge to zero when .     

The problem of conformal transformations of a circle into nonoverlapping regions

Let a, a0, a, be a fixed point in the z-plane, (a, 0, ), the class of all systemsf _k()_l ³ of functions z=f ^k(), k=1, 2, 3, of which the first two map conformally and in a s ingle-sheeted manner the circle ¦¦<1, and the third maps in a similar manner the region ¦¦>1, into pair-wise nonintersecting regions B_k, k=1, 2, 3, containing the points a, 0, and , respectively, so thatf ₁(0)=a,f ₂(0)=0 andf ₃()=. The region of values (a, 0, ) of the system M(¦f ₁'(0)¦, ¦f ₂'(0)¦, 1/¦f ₃'()¦) in the class (a, 0, ) is determined.Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 417–424, October, 1969.     

Trace formulas for pair operators

In this paper we shall study the Fredholm determinant and related trace formulas for a class of operators which correspond to the restriction of integral operators with kernels of the form k(x,y) = (x)g^v(x–y)+[1–(x)]f^v(x–y) to the square |x|,|y| T and shall evaluate the limit as T . Here denotes the indicator function of the right half-line [0,) . The results obtained generalize the well known formulas of M. Kac for the classical convolution operator in which g = f .     

Rates of convergence and optimal spectral bandwidth for long range dependence

Summary For a realization of lengthn from a covariance stationary discrete time process with spectral density which behaves like ^1–2H as 0+ for 1/2<H<1 (apart from a slowly varying factor which may be of unknown form), we consider a discrete average of the periodogram across the frequencies 2j/n,j=1,..., m, wherem andm/n0 asn. We study the rate of convergence of an analogue of the mean squared error of smooth spectral density estimates, and deduce an optimal choice ofm.     

Parametric results for certain infinite-dimensional manifolds

The theory of R-(Q-) manifolds is generalized in two directions. Firstly, an axiomatical approach is proposed to describing various classes of manifolds (so-called K-manifolds) including, along with the indicated classes of Rand Q-manifolds, also, e.g., the manifolds modeled on the space, where is a cardinal. Secondly, all the arguments were carried out in the category Top_B, which makes it possible to carry over from spaces to maps practically all basic results of the theory of R-(Q-) manifolds. Specifically, there are obtained characterization theorems for trivial and microtrivial K-fibrations, theorems on open and closed embeddings, stability thoerems, etc.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 6, pp. 853–859, June, 1991.     

Surface integral and Gauss-Ostrogradskij theorem from the viewpoint of applications

Making use of a surface integral defined without use of the partition of unity, trace theorems and the Gauss-Ostrogradskij theorem are proved in the case of three-dimensional domains with a Lipschitz-continuous boundary for functions belonging to the Sobolev spaces H ^1,p () (1 p < ). The paper is a generalization of the previous author's paper which is devoted to the line integral.     

A quadrature formula for the Hankel transform

We present in this paper a quadrature formula for a certain Fourier-Bessel transform and, closely related to this, for the Hankel transform of order >–1. Such formulas originate in the context of a Galerkin-type projection of the weightedL ₂(–, ; ) space ( is the weight function mentioned below) used to get a discrete representation of a certain physical problem in Quantum Mechanics. The generalized Hermitee polynomialsH ₀ (x),H ₁ (x),..., with weight function (x), are used as the basis on which such a projection takes place. It is shown that theN-dimensional vectors representing certain projected functions as well as the entries of theN×N matrix representing the kernel of that Fourier-Bessel transform, approach the exact functional values at the zeros of theNth generalized Hermitee polynomial whenN.These properties lead to propose this matrix as a finite representation of the kernel of the Fourier-Bessel transform involved in this problem and theN zeros of the generalized Hermitee polynomialH _N (x) as abscissas to yield certain quadrature formulae for this integral and for the related Hankel transform. The error function produced by this algorithm is estimated at theN nodes and its is shown to be of a smaller order than 1/N. This error estimate is valid for piecewise continuous functions satisfying certain integral conditions involving their absolute values. The algorithm is presented with some numerical examples.     

Dissipative Schrödinger Operators with Matrix Potentials

Maximal dissipative Schrödinger operators are studied in L ²((–,);E) (dimE=n<) that the extensions of a minimal symmetric operator with defect index (n,n) (in limit-circle case at – and limit point-case at ). We construct a selfadjoint dilation of a dissipative operator, carry out spectral analysis of a dilation, use the Lax–Phillips scattering theory, and find the scattering matrix of a dilation. We construct a functional model of the dissipative operator, determine its characteristic function in terms of the Titchmarsh–Weyl function of selfadjoint operator and investigate its analytic properties. Finally, we prove a theorem on completeness of the eigenvectors and associated vectors of a dissipative Schrödinger operators.     

On pathwise behavior of multiserver queues

In this paper, by exploiting recent results on the pathwise behavior of the workload process in single server, work conserving queues of theG/G/1/ type, we show that the workload of multiserver, work conserving queues ofG/G/m/ (m<) (andG/G/) queues satisfies an o(t) growth condition, provided that the time average of the work brought into the system is less thanm form < (and finite form=).     

Prior bounds on generalized derivatives of solutions of second-order quasilİnear elliptic equations

For the equations u–dºa(x,u u) = f, u–f = a(x, u, u)ºd²u f = Re f L¹ (Re^l, d^lx) L (R^lx) C (Rⁱ, d^lx) with smooth coefficients in R^l we establish prior bounds on the first and second generalized derivatives of their solutions in the spaces L^2p (R^l, d^lx), L^p(Rl, d^lx), 1 < p < , respectively.Kiev Polytechnical Institute. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 68, pp. 52–60, 1989.     

Subsets of a Hilbert space,having finite hausdorff dimension

Let X₂, X₂ be Hilbert spaces, X₂ X₁, X₂ is dense in X₁, the imbedding is compact,m X₂, dim_H ⁱ m and h⁽ⁱ⁾(m) are the Hausdorff dimension and the limit capacity (information dimension) of the setm with respect to the metrics of the spaces X_i (i=1, 2). Two examples are constructed. 1) An example of a setm bounded in X₂, such that: a) h⁽¹⁾(m) < (and, consequently, dim_H ¹ m); b)m cannot be covered by a countable collection of sets, compact in X₂ (and, consequently, dim_H ² m=). 2) an Example of a setm, compact in X₂, such that h⁽¹⁾(m) < and h⁽²⁾(m)=.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 163, pp. 154–165, 1987.     

Remarks on ergodicity of stationary irreducible transient Markov chains

Summary Consider a stationary process {X _n(), – < n < . If the measure of the process is finite (the measure of the whole sample space finite), it is well known that ergodicity of the process {X _n(), - < n < and of each of the subprocesses {X _n(), 0 n < , {X _n(), – < n 0 are equivalent (see [3]). We shall show that this is generally not true for stationary processes with a sigma-finite measure, specifically for stationary irreducible transient Markov chains. An example of a stationary irreducible transient Markov chain {X _n(), - < n <} with {itX_n(), 0 n < < ergodic but {X _n(), < n 0 nonergodic is given. That this can be the case has already been implicitly indicated in the literature [4]. Another example of a stationary irreducible transient Markov chain with both {X _n(), 0 n < and {itX _n(),-< < n 0} ergodic but {X _n(), - < n < nonergodic is presented. In fact, it is shown that all stationary irreducible transient Markov chains {X _n(), - < n < < are nonergodic.This research was supported in part by the Office of Naval Research.John Simon Guggenheim Memorial Fellow.     

OnH 2 minimization of theH ∞ suboptimal solutions

In this paper we show that theH ² minimization of theH suboptimal solutions for a class of suboptimalH distance problems can be reduced to a finite dimensional nonlinear optimization problem. This extends a result of [7] where the same problem is considered in the Caratheodory-Schur interpolation case.     

Several Constructions of Exact Sequences

The article presents a construction of a 5-term relative exact sequence in purely categorical terms and the Mayer–Vietoris sequence for weak -groupoids of R. Street. In the first half of the paper, it is shown that several well-known exact sequences can be obtained, using the Z-diagram described at the beginning of the article. There are number of categories that are close to the category of topological spaces in the sense of homotopy theory. The objects of such a category can be viewed as weak -groupoids in a very intuitive way. Therefore, the language of weak -groupoids seems to be very convenient to make explicit constructions in categories of this type. The article presents a combinatorial construction of the path space of a weak -groupoid and applies the machinery of Sec. 1 to obtain the long exact sequence of a homotopy fiber. The construction is quite natural and gives information on the structure of the relative terms. In the last section, the Mayer–Vietoris sequence of a fiber square of weak -groupoids is obtained under some conditions. Of course, this construction makes sense for topological spaces or for any category mentioned above. But the statement of the conditions is more natural in terms of weak -groupoids. This sequence generalizes the sequence of fibration, and in this case the conditions given in the article are equivalent to the homotopy lifting property. Bibliography: 6 titles.     

K-theory of reduced C*-algebras and rapidly decreasing functions on groups

We associate to any length function L on a group a space of rapidly decreasing functions on (in the l ² sense), denoted by H _L (). When H _L () is contained in the reduced C*-algebra C _r ^* () of (), then it is a dense *-subalgebra of C _r ^* () and we prove a theorem of A. Connes which asserts that under this hypothesis H _L () has the same K-theory as C _r ^* (). We introduce another space of rapidly decreasing functions on (in the l ¹ sense), denoted by H _L ^1, (), which is always a dense *-subalgebra of the Banach algebra l ¹(), and we show that H _L ^1, () has the same K-theory as l ¹().