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.     

A note on generating generalized two-dimensional plate and shell theories

Summary Ordinary two dimensional linear shell theory may be formulated by using six differential equations for stress resultants and couples and for middle surface force and moment load intensities, in conjunction with the principle of virtual work, for the derivation of strain displacement relations. The present paper deals with a more general formulation, involving additional two-dimensional equilibrium equations, as a consequence of three-dimensional equations for force and moment stresses, in conjunction with a stipulation of surface force and moment load intensities for two face surfaces in place of the one middle surface. The main intent of the analysis is an illumination of the concept of a mechanical Cosserat-surface theory, in comparison with ordinary two-dimensional shell theory.

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Zusammenfassung Die übliche zweidimensionale Schalentheorie kann formuliert werden, indem man sechs Differentialgleichungen für Spannungsresultierende, Momente, Mittel flächenkraft und Momentlastintensität zusammen mit dem Prinzip der virtuellen Verschiebungen dazu benutzt, um ein System von Dehnungs-Verschiebungsgleichungen abzuleiten. Die vorliegende Arbeit beschäftigt sich mit einer verallgemeinerten Formulierung, mit zusätzlichen zweidimensionalen Gleichgewichtsbedingungen, die von einer dreidimensionalen Theorie für Kraft- und Momentenspannungen abgeleitet werden, zusammen mit einer Vorschrift über zwei Systeme von Ober flächenkräften und Momentlastintensitäten anstelle des einen Mittelfllächenlastsystems. Der Hauptzweck der Analyse ist die Beleuchtung des Begriffs der mechanischen Cosserat-Flächentheorie im Vergleich zur üblichen zweidimensionalen linearen Schalentheorie.

     

Weak fluctuation modes of an electrically and thermally conducting viscous compressible flow in a homogeneous magnetic field

Zusammenfassung Eine Perturbationstheorie erster Ordnung zeigt, daß die Gleichungen, die die Bewegung eines elektrisch und thermisch leitenden zähen kompressiblen Strömungsfeldes bestimmen, welches endliche elektrische Leitfähigkeit hat, vier verschiedene Typen von Störungsfeldern enthalten kann; diese werden von vier unabhängigen partiellen Differentialgleichungen bestimmt. Die drei ersten Typen sind: Wirbel-Stromdichte- und magnetisch-akustische Wellen-Typen. Sie sind hyperbolischer Art und pflanzen sich nicht isotrop als Wellen fort. Der vierte Typ ist ein Entropie-Typ, der sich nicht ausbreitet und der von einer parabolischen Gleichung bestimmt wird. Jeder dieser verschiedenen Typen kann nicht weiter zerlegt werden wie im Falle inkompressibler Strömung. Zusätzlich wird gezeigt, daß die magnetisch-akustischen Wellen im Falle endlicher elektrischer Leitfähigkeit neue und interessante Eigenschaften besitzen.     

Eulerian numbers with fractional order parameters

Summary The aim of this paper is to generalize the well-known Eulerian numbers, defined by the recursion relationE(n, k) = (k + 1)E(n – 1, k) + (n – k)E(n – 1, k – 1), to the case thatn is replaced by . It is shown that these Eulerian functionsE(, k), which can also be defined in terms of a generating function, can be represented as a certain sum, as a determinant, or as a fractional Weyl integral. TheE(, k) satisfy recursion formulae, they are monotone ink and, as functions of , are arbitrarily often differentiable. Further, connections with the fractional Stirling numbers of second kind, theS(, k), > 0, introduced by the authors (1989), are discussed. Finally, a certain counterpart of the famous Worpitzky formula is given; it is essentially an approximation ofx in terms of a sum involving theE(, k) and a hypergeometric function.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

Some properties of the quasimomentum of the one-dimensional Hill operator

In this paper, a study is made of the properties of the energy E(k) of the Hill operator, as a function of the global quasimomentum k, running over the Riemann surface of the quasimomentumK. A study is made of , Ë on the straight line Im k=0, and of the location of the zeros of Ë. For example: 1) the function on the straight line Im k=0 between zeros has only one local maximum or minimum, 2) in the case of an N-zone potential, the function Ë has exactly 6(N–1) zeros inK, N–1 zeros both on the positive (negative) semiaxes, and in each quadrant.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 195, pp. 48–57, 1991.     

Rudiments of an average case complexity theory for piecewise-linear path following algorithms

We examine the efficiency of PL path following algorithms in followingF _T ^-1 (0), whereF _T is the PL approximation, induced by the simplicial triangulationT, to a mapf: ^{n n-1}. In particular, we consider the problem of determining an upper bound on the expected number of pivots made per unit length off ^–1(0) that is approximated. We show that if the sizes of the simplices ofT are sufficiently small, where sufficiently small is an explicitly given quantity dependent on measurements of how nicef is, then the average directional density ofT, as introduced by Todd, really does give a good approximation to the expected number of pivots made, confirming what researchers have believed on intuitive grounds for a decade. Because what constitutes sufficiently small is a precisely given quantity, i.e., non-asymptotic, we are able to provide some rigorous justification for the claim that the expected number of pivots grows only polynomially inn, the number of variables.Several other issues are also examined.Research supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship. This research was performed while the author was a member of the Mathematical Sciences Research Institute, Berkeley, California.     

More on quadratic convergence in interval arithmetic

Necessary and sufficient conditions are given for quadratic convergence in interval arithmetic.Partially supported by NSF grant GJ-797.     

Global bifurcation for a special Hill equation with nonlinear boundary conditions

Summary We introduce a new bifurcation model which consists of a linear second order ordinary differential equation together with nonlocal nonlinear boundary conditions. These are periodic boundary conditions in the case of the trivial solutiony=0 and semi-periodic boundary conditions for y tending to infinity. We use an elementary method to prove the existence of global solution branches connecting the periodic eigenvalues and the trivial solutiony=0 with the semi-periodic eigenvalues and the solutiony=.

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Zusammenfassung Wir stellen ein neues Verzweigungsmodell, bestehend aus einer linearen gewöhnlichen Differentialgleichung zweiter Ordnung und nichtlokalen nichtlinearen Randbedingungen, vor. Periodische Randbedingungen ergeben sich für den Fall der trivialen Lösungy=0 und semi-periodische Randbedingungen falls y gegen unendlich strebt. Mit Hilfe elementarer Methoden weisen wir die Existenz globaler Lösungszweige nach, die die periodischen Eigenwerte und die triviale Lösungy=0 mit den semi-periodischen Eigenwerten und der Lösungy= verbinden.

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Dedicated to the memory of Maria Adelaide Sneider and Johann Kreyenberg

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Supported in part by CONICYT (Grant 89-576) in Concepción/Chile.     

General procedures leading to correlated equilibria

Hart and Mas-Colell [2000] show that if all players play regret-matching strategies, i.e., they play with probabilities proportional to the regrets, then the empirical distribution of play converges to the set of correlated equilibria, and the regrets of every player converge to zero. Here we show that if only one player, say player i, plays with these probabilities, while the other players are not too sophisticated, then the result that player is regrets converge to zero continues to hold. The condition of not too sophisticated essentially says that the effect of one change of action of player i on the future actions of the other players decreases to zero as the horizon goes to infinity. Furthermore, we generalize all these results to a whole class of regret-based strategies introduced in Hart and Mas-Colell [2001]. In particular, these simplify the conditional smooth fictitious play of Fudenberg and Levine [1999].Received: May 2004This is a revision of the authors M.Sc. thesis, May 2000.The author thanks Professor Sergiu Hart for his help and guidance, and the Associate Editor and an anonymous referee for their comments. I am grateful to my parents and wife for everything.     

About generalized ideals in a distributive lattice

Let L be a distributive lattice characterized by a ternary operation (, ,), where (a,b,c)=(ab)(bc)(ac)=(ab)(ac)(bc), a,b,cL. The note considers convex sublattices of L, called generalized ideals of L generated by the operation (, ,). Some remarks have been stated about the graph of a distributive lattice.     

Convex Separable Minimization Subject to Bounded Variables

A minimization problem with convex and separable objective function subject to a separable convex inequality constraint and bounded variables is considered. A necessary and sufficient condition is proved for a feasible solution to be an optimal solution to this problem. Convex minimization problems subject to linear equality/linear inequality constraint, and bounds on the variables are also considered. A necessary and sufficient condition and a sufficient condition, respectively, are proved for a feasible solution to be an optimal solution to these two problems. Algorithms of polynomial complexity for solving the three problems are suggested and their convergence is proved. Some important forms of convex functions and computational results are given in the Appendix.     

Invarianz gewisser Operatorenklassen unter nichtkommutativen ergodischen Mitteln

Let be a group of *-automorphisms on the algebra of bounded linear operators on a complex Hilbert space H. Then the strongly closed convex hull of the orbit of any compact operator under consists of compact operators. The same is true if one replaces compact by nuclear, Hilbert-Schmidt or positive Fredholm. We further discuss these results in the framework of the noncommutative mean ergodic theorem of KOVACS and SZ#x00FC;CS and formulate an analogous theorem for the algebra of compact operators on a complex Hilbert space.

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Gefördert von der Deutschen Forschungsgemeinschaft im Rahmen des Forschungsvorhabens Ko 506/1.     

On the uniform (C, 1)-summability of Fourier series and integrability of series with respect to multiplicative orthonormal systems

H (G), f(g)H (G) , (, 1)- OHMC G. , OHMC, A. H. . , . , OHMC, lim supp _n=, , ,n .. . , 117 234 . . -      

Descendants constructed from matter field in Landau-Ginzburg theories coupled to topological gravity

It is argued that gravitational descendants in the theory of topological gravity coupled to topological Landau-Ginzburg theory (not necessarily conformal) can be constructed from matter fields alone (without metric fields and ghosts). In this sense topological gravity is induced. We discuss the mechanism of this effect (that turns out to be connected with K. Saito's higher residue pairing: Kⁱ(_i(₁),₂)=K⁰(₁,₂)), and demonstrate how it works in a simplest nontrivial example: correlator on a sphere with four marked points. We also discuss some results on k-point correlators on a sphere. From the idea of induced topological gravity it follows that the theory of pure topological gravity (without topological matter) is equivalent to the trivial Landau-Ginzburg theory (with quadratic superpotential).Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 2, pp. 307–316, May, 1993.     

Improved Krasnosel'skii theorems for the dimension of the kernel of a starshaped set

We will establish the following improved Krasnosel'skii theorems for the dimension of the kernel of a starshaped set: For each k and d, 0 k d, define f(d,k) = d+1 if k = 0 and f(d,k) = max{d+1,2d–2k+2} if 1 k d.Theorem 1. Let S be a compact, connected, locally starshaped set in R^d, S not convex. Then for a k with 0 k d, dim ker S k if and only if every f(d, k) lnc points of S are clearly visible from a common k-dimensional subset of S.Theorem 2. Let S be a nonempty compact set in R^d. Then for a k with 0 k d, dim ker S k if and only if every f (d, k) boundary points of S are clearly visible from a common k-dimensional subset of S. In each case, the number f(d, k) is best possible for every d and k.     

Axioms forS-pregroups

AnS-pregroup is a subset of a group such that no nontrivial reduced word (a string of length 2 of elements of the set such that the product of no adjacent pair is in the set) represents the identity when placed in the group. To complete the definition we also require anS-pregroup to contain the identity, be closed under inverses and generate the parent group. In [1] John Stallings, coining the term S-pregroups, asked for an internal axiomatic characterization ofS-pregroups. In this paper we describe such a system by modifying the axioms for a pregroup.Presented by J. Mycielski.     

The hydromagnetic viscous boundary layer at the free surface of a rotating baroclinic fluid

Steady, linear hydromagnetic boundary layer at the free surface of a rotating baroclinic fluid is analysed under Boussinesq approximation. The flow is driven by the buoyancy forces which exist due to the density variations associated with baroclinicity. Meridional circulations of mass flux and electric current flux occur due to the boundary layer suction as well as due to the hydromagnetic effects. The boundary layer suction dominates and the dynamics of the flow reduce to that of nonmagnetic case for ²<O(E), whereE is the Ekman number and ² (=B ₀ ² /) is the magnetic interaction parameter, which represents the ratio of hydromagnetic to Coriolis forces. When ²>O(E), meridional circulation induced in the interior due to hydromagnetic effects is stronger than that can be supported by the boundary layer. The electric current flux through the interior is stronger for any value of than that can be supported by the boundary layer. Further it is argued that hydromagnetic effects suppress the laminated flow (v/z0) in the interior. As a result, the meridional circulations are strong when =O(1).

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Zusammenfassung Die stationäre lineare hydromagnetische Grenzschicht an der freien Oberfläche eines rotierenden, baroklinischen Fluids wird mit Hilfe der Boussinesq-Näherung untersucht. Die Strömung wird durch die Auftriebskräfte angetrieben, die durch die baroklinischen Dichte-Variationen verursacht werden. Die meridionale Zirkulation des Massenstromes und des elektrischen Stromes wird durch die Grenzschicht-Saugwirkung und durch hydromagnetische Effekte erzeugt. Die Grenzschicht-Saugwirkung dominiert und die Strömung reduziert sich auf den nicht-magnetischen Fall für ²<O(E); dabei istE die Ekman-Zahl und ² (=B ₀ ² /) ist der magnetische Kopplungs-Parameter, der das Verhältnis der hydromagnetischen zu den Coriolis-Kräften gibt. Wenn ²>O(E) ist, wird die meridionale Zirkulation, die durch hydromagnetische Effekte erzeugt worden ist, stärker als die, welche durch Grenzschichten erzeugt werden kann. Der elektrische Strom im Innern ist für alle Werte von größer als der Wert den die Grenzschicht verursacht. Weiterhin wird gezeigt, daß hydromagnetische Effekte die Laminal-Strömung (v/z0) im Innern unterdrückt. Es ergibt sich, daß die meridionalen Zirkulationen stark sind für =O(1).

     

Convergence of sequential and asynchronous nonlinear paracontractions

Summary We establish the convergence of sequential and asynchronous iteration schemes for nonlinear paracontracting operators acting in finite dimensional spaces. Applications to the solution of linear systems of equations with convex constraints are outlined. A first generalization of one of our convergence results to an infinite pool of asymptotically paracontracting operators is also presented.Research supported in part by Sonderforschungsbereich 343 Diskrete Strukturen in der MathematikResearch supported in part by NSF Grant DMS-9007030 and by Sonderforschungsbereich 343 Diskrete Strukturen in der Mathematik, Fakultät für Mathematik at the Universität BielefeldResearch supported in part by U.S. Air Force Grant AFOSR-88-0047, by NSF Grants DMS-8901860 and DMS-9007030, and by Sonderforschungsbereich 343 Diskrete Strukturen in der Mathematik, Fakultät für Mathematik at the Universität Bielefeld     

A Nonlinear Problem Associated to the Motion of a Membrane

A nonlinear model associated to the motion of a membrane is considered as limit of a sequence of approximate models, for which a global existence and uniqueness theorem can be proved. The paper investigates the relationship between the solutions of the real and approximate models.     

0-1-sequences of Toeplitz type

Summary 0-1-sequences are constructed by successive insertion of a periodic sequence of symbols 0, 1 and hole into the holes of the sequence already constructed. Assuming that finally all holes are filled with symbols 0, 1, an almost periodic point in shift space results. Under certain conditions, it is even strictly ergodic. It is proved that the attached invariant measure has pure point spectrum, and a rather explicit expression for eigenvectors is obtained.