Locally constant dyadic derivatives

We show that in order for a Walsh series to be locally constant it is necessary for certain blocks of that series to sum to zero. As a consequence, we show that a functionf with a somewhat sparse Walsh—Fourier series is necessarily a Walsh polynomial if its strong dyadic derivative is constant on an interval. In particular, if a Rademacher seriesR is strongly dyadically differentiable and if that derivative is constant on any open subset of [0, 1], thenR is a Rademacher polynomial.     

Equivalence of the derived chains corresponding to a boundary-value problem on a finite interval,for polynomial operator pencils

Under investigation is the equivalence of derived chains constructed from root vectors of polynomial pencils of operators acting in a Hilbert space. These derived chains correspond to various boundary—value problems on a finite interval for an operator—differential equation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 1, pp. 83–95, January, 1990.     

Whitham Equations, Bergman Kernel and Lax—Levermore Minimizer

We study multiphase solutions of the Whitham equations. The Whitham equations describe the zero dispersion limit of the Cauchy problem for the Korteweg—de Vries (KdV) equation. The zero dispersion solution of the KdV equation is determined by the Lax—Levermore minimization problem. The minimizer is a measurable function on the real line. When the support of the minimizer consists of a finite number of disjoint intervals to be determined, the minimization problem can be reduced to a scalar Riemann Hilbert (RH) problem. For each fixed x and t 0, the end-points of the contour are determined by the solution of the Whitham equations. The Lax—Levermore minimizer and the solution of the Whitham equations are described in terms of a kernel related to the Bergman kernel. At t = 0 the support of the minimizer consists of one interval for any value of x, while for t > 0, the number of intervals is larger than one in some regions of the (x,t) plane where the multiphase solutions of the Whitham equations develop. The increase of the number of intervals happens whenever the solution of the Whitham equations has a point of gradient catastrophe. For a class of smooth monotonically increasing initial data, we show that the support of the Lax—Levermore minimizer increases or decreases the number of its intervals by one near each point of gradient catastrophe. This result justifies the formation and extinction of the multiphase solutions of the Whitham equations. Furthermore we characterize a class of initial data for which all the points of gradient catastrophe occur only a finite number of times and therefore the support of the Lax—Levermore minimizer consists of a finite number of disjoint intervals for any x and t 0. This corresponds to give an upper bound to the genus of the solution of the Whitham equations. Similar results are obtained for the semi-classical limit of the defocusing nonlinear Schrödinger equation.     

The Maximal and Minimal Ranks of Some Expressions of Generalized Inverses of Matrices

In this paper, we first determine the maximal and minimal ranks of A — BXC with respect to X. Using those results, we then find the maximal and minimal ranks of the expressions AA^– — A^– A — BB^– A — AC^– C and B^– BA — ACC^– with respect to the choice of generalized inverses A^–, B^– and C^–. In particular, we consider the commutativity of A and A^–, A^k and A^–.The research of the author was supported in part by the Natural Sciences and Engineering Research Council of Canada.     

The gap between monotone and non-monotone circuit complexity is exponential

A. A. Razborov has shown that there exists a polynomial time computable monotone Boolean function whose monotone circuit complexity is at leastn ^{c losn} . We observe that this lower bound can be improved to exp(cn ^1/6–o(1)). The proof is immediate by combining the Alon—Boppana version of another argument of Razborov with results of Grötschel—Lovász—Schrijver on the Lovász — capacity, of a graph.     

Regular cyclic coverings of regular affine maps

The regular coverings of regular affine algebraic maps are considered, and a large family of totally ramified coverings—the so-called Steinberg and Accola coverings—are fully classified.     

Mono-Implicit Runge-Kutta-Nyström Methods with Application to Boundary Value Ordinary Differential Equations

The successful use of mono-implicit Runge—Kutta methods has been demonstrated by several researchers who have employed these methods in software packages for the numerical solution of boundary value ordinary differential equations. However, these methods are only applicable to first order systems of equations while many boundary value systems involve higher order equations. While it is straightforward to convert such systems to first order, several advantages, including substantial gains in efficiency, higher continuity of the approximate solution, and lower storage requirements, are realized when the equations can be treated in their original higher order form. In this paper, we consider generalizations of mono-implicit Runge—Kutta methods, called mono-implicit Runge—Kutta—Nyström methods, suitable for systems of second order ordinary differential equations having the general form, y(t) = f(t,y(t),y(t)), and derive optimal symmetric methods of orders two, four, and six. We also introduce continuous mono-implicit Runge—Kutta—Nyström methods which allow us to provide continuous solution and derivative approximations. Numerical results are included to demonstrate the effectiveness of these methods; savings of 65% are attained in some instances.This revised version was published online in October 2005 with corrections to the Cover Date.     

Convergence of Runge-Kutta Methods for Delay Differential Equations

This paper deals with the adaptation of Runge—Kutta methods to the numerical solution of nonstiff initial value problems for delay differential equations. We consider the interpolation procedure that was proposed in In 't Hout [8], and prove the new and positive result that for any given Runge—Kutta method its adaptation to delay differential equations by means of this interpolation procedure has an order of convergence equal to min {p,q}, where p denotes the order of consistency of the Runge—Kutta method and q is the number of support points of the interpolation procedure.This revised version was published online in October 2005 with corrections to the Cover Date.     

A characterization of totally normal structure

In this note it is proved that a subsetA of a locally convex topological vector spaceE has normal structure with respect to every continuous seminorm onE — called totally normal structure — iff every bounded convex subset ofA is precompact. Some consequences are discussed.     

Asymptotic Homomorphisms of C *-Algebras and C *-Extensions

We present a brief introduction to two theories in the category of C ^*-algebras—theory of asymptotic homomorphisms and theory of extensions—and explain how these theories are related to each other.     

K-greedy algorithms for independence systems

Recently inJenkyns [1976] andKorte/Hausmann [1978] a tight worst-case bound for the well-known greedy heuristic for general independence systems has been deduced. Here some modifications of the greedy heuristic are investigated which allow to enlarge the current independent set by more than one element. It is shown that in spite of the additional enumeration incorporated in these heuristics they can behave worse than the usual greedy heuristic for some problem instances. For one of those modifications the same worst-case bound as mentioned above — but no better one — has been proved again. These results underline the predominant role of the usual greedy heuristic for general independence systems.

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Zusammenfassung Kürzlich wurde inJenkyns [1976] undKorte/Hausmann [1978] eine scharfe Gütegarantie für die bekannte Greedy-Heuristik für allgemeine Unabhängigkeitssysteme hergeleitet. Hier werden nun einige Modifikationen der Greedy-Heuristik untersucht, die es gestatten, die laufende unabhängige Menge durch mehr als ein Element zu vergrößern. Es wird gezeigt, daß sich diese Heuristiken — trotz der zusätzlich in sie eingebauten Enumeration — in Einzelfällen schlechter verhalten können als die gewöhnliche Greedy-Heuristik.Für eine dieser Modifikationen konnte wieder die gleiche Gütegarantie wie die oben erwähnte —aber keine bessere — bewiesen werden. Diese Ergebnisse unterstreichen die beherrschende Rolle der gewöhnlichen Greedy-Heuristik für allgemeine Unabhängigkeitssysteme.

     

Representations ofD-forms

A criterion for module boundedness of a certain class of semihereditary rings — tensor algebras ofD-form — is given.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 4, pp. 572–574, April, 1992.     

Temperature dependence of the coefficient of friction of Ftorlon-graphite fiber

The reduced friction of Ftorlon (fluoroplastic) — graphite fiber against steel on the temperature interval 120–190°C is examined.Khmel'nitskii Technological Institute of Maintenance and Repairs. Translated from Mekhanika Polimerov, No. 6, pp. 1120–1121, November–December, 1973.     

On the convergence of the multigroup,discrete-ordinates solutions for subcritical transport media

Summary In the multigroup, discrete-ordinates approximations to the linear transport equation, the integration over the directional variable is replaced by a numerical quadrature rule, involving a weighted sum over functional values at selected directions, with the energy dependence discretized by replacing the cross section data by weighted averages over each energy interval. The stability, consistency, and convergence rely fundamentally on the conditions that the maximum fluctuations in the total cross section — and in the expected number of secondary particles arising from each energy level — tend to zero as the energy mesh becomes finer, and as the number of angular nodes becomes infinite. Our analysis is based on using a natural Nyström method of extending the discrete-ordinates, multigroup approximates to all values of the angular and energy variables. Such an extension enables us to employ generalizations of the collectively compact operator approximation theory of P. M. Anselone to deduce stability and convergence of the approximates.This research was started while the first author (H.D.V., Jr.) was an Alexander von Humboldt Research Fellow in the Department of Mathematics, University of Kaiserslautern, Kaiserslautern, Federal Republic of Germany. Generous support was also provided by a Faculty Development Leave from Texas Tech University for the 1982–83 Academic Year.     

Galerkin methods for nonlinear Sobolev equations

Summary We study Galerkin approximations to the solution of nonlinear Sobolev equations with homogeneous Dirichlet boundary condition in two spatial dimensions and derive optimalL ² error estimates for the continuous Crank — Nicolson and Extrapolated Crank — Nicolson approximations.     

Confinement of the infrared divergence for the Mumford process

The Mumford process X is a stochastic distribution modulo constant and cannot be defined as a stochastic distribution invariant in law by dilations. We present two expansions of X—using wavelet bases—in X=X₀+X₁ which allow us to confine the divergence on the “small term” X₁ and which respect the invariance in law by dyadic dilations of the process.     

Some remarks on Maltsev and Goursat categories

Our aim is to analyze and to publicize two interesting properties — well known in universal algebra for varieties — that a regular category, and in particular an exact category, may possess: theMaltsev property, asserting the permutabilitySR=RS of equivalence relations on any object, and the weakerGoursat property, asserting only thatSRS=RSR. We investigate these properties, give various equivalent forms of them, and develop some of their useful consequences.     

Linear homeomorphisms of non-classical Hilbert spaces

Let K be a complete infinite rank valued field. In [4] we studied Norm Hilbert Spaces (NHS) over K i.e. K-Banach spaces for which closed subspaces admit projections of norm ≤ 1. In this paper we prove the following striking properties of continuous linear operators on NHS. Surjective endomorphisms are bijective, no NHS is linearly homeomorphic to a proper subspace (Theorem 3.7), each operator can be approximated, uniformly on bounded sets, by finite rank operators (Theorem 3.8). These properties together — in real or complex theory shared only by finite-dimensional spaces — show that NHS are more ‘rigid’ than classical Hilbert spaces.     

Identities for piecewise polynomial spaces determined by connection matrices

Summary TwoB-spline results — Marsden's identity and the de Boor-Fix dual functionals — are extended to geometrically continuous curves determined by connection matrices.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

Geometric separation and exact solutions for the parameterized independent set problem on disk graphs

We consider the parameterized problem, whether for a given set  of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k non-intersecting disks. For this problem, we expose an algorithm running in time that is—to our knowledge—the first algorithm with running time bounded by an exponential with a sublinear exponent. For λ-precision disk graphs of bounded radius ratio, we show that the problem is fixed parameter tractable with a running time  . The results are based on problem kernelization and a new “geometric ( -separator) theorem” which holds for all disk graphs of bounded radius ratio. The presented algorithm then performs, in a first step, a “geometric problem kernelization” and, in a second step, uses divide-and-conquer based on our new “geometric separator theorem.”