Die Dupinschen Zykliden im pseudoeuklidischen Raum

In the pseudoeuclidean space¹ E ³ a surface of degree 3 or 4 is calledDupin-cyclide if there exists a (pseudoeuclidean) torus such that a (pseudoeuclidean) spherical inversion maps upon . If the axis of is respectively space-like, isotropic or time-like is calledDupin-r-,-l- or-z-cyclide. ADupin-cyclide is the envelope of two families of spheres, the caustic surfaces of degenerate in caustic curves and the lines of curvature of constitute two (orthogonal) families of circles.     

Inequalities in Products of Minors of Totally Nonnegative Matrices

Let _I,I be the minor of a matrix which corresponds to row set I and column set I. We give a characterization of the inequalities of the form _{I,I K,K J,J L,L} which hold for all totally nonnegative matrices. This generalizes a recent result of Fallat, Gekhtman, and Johnson.     

The Spectral Function and Principal Eigenvalues for Schrödinger Operators

Let m , 0 m₊ in Kato's class. We investigate the spectral function s( + m) where s( + m) denotes the upper bound of the spectrum of the Schrödinger operator + m. In particular, we determine its derivative at 0. If m_- is sufficiently large, we show that there exists a unique ₁ > 0 such that s( + ₁m) = 0. Under suitable conditions on m⁺ it follows that 0 is an eigenvalue of + ₁m with positive eigenfunction.     

Existence for the Cahn-Hilliard phase separation model with a nondifferentiable energy

Summary The Cahn-Hilliard model for phase separation in a binary alloy leads to the equations (I) u_t=w, (II) w= (u)– u with an associated energy functional F(u)=f [(u)+ +¦u¦²/2] dx. In this paper we discuss the existence theory for initial bounday value problems arising from modifications to the Cahn-Hilliard model due to the addition of the non-differentiable term ¦u¦dx to the energy F(u).     

Infinite highly arc transitive digraphs and universal covering digraphs

A digraph (that is a directed graph) is said to be highly arc transitive if its automorphism group is transitive on the set ofs-arcs for eachs0. Several new constructions are given of infinite highly arc transitive digraphs. In particular, for a connected, 1-arc transitive, bipartite digraph, a highly arc transitive digraphDL() is constructed and is shown to be a covering digraph for every digraph in a certain classD() of connected digraphs. Moreover, if is locally finite, thenDL() is a universal covering digraph forD(). Further constructions of infinite highly arc transitive digraphs are given.The second author wishes to acknowledge the hospitality of the Mathematical Institute of the University of Oxford, and the University of Auckland, during the period when the research for this paper was doneResearch supported by the Australian Research Council     

The self-similar solutions to a fast diffusion equation

A quasilinear equation u -x·u/2+f(u)=0 is studied, wheref(u)=–u+u , > 0, 0<. <1, >1 andx R ⁿ. The equation arises from the study of blow-up self-similar solutions of the heat equation _t =+. We prove the existence and non-existence of ground state for various combination of , and . In particular, we prove that when / < forn=1,2 or / < (n + 2) /(n – 2) forn 3 there exists no non-constant positive radial self-similar solution of the parabolic equation, but for many cases where / > (n + 2)/(n – 2) there exists an infinite number of non-constant positive radial self-similar solutions.     

On the distribution of square-full and cube-full integers

LetN _r (x) be the number ofr-full integers x and let _r (x) be the error term in the asymptotic formula forN _r (x). Under Riemann's hypothesis, we prove the estimates ₂(x)x^1/7+, ₃(x)x^97/804+(>0), which improve those of Cao and Nowak. We also investigate the distribution ofr-full andl-free numbers in short intervals (r=2,3). Our results sharpen Krätzel's estimates.     

On the existence of multiple solutions of nonhomogeneous elliptic equations involving critical Sobolev exponents

Letp=2N/(N –2),N 3 be the limiting Sobolev exponent and ^N a bounded smooth domain. We show that for H ^–1(),f satisfies some conditions then–u=c ₁ u ^p–1 +f(x,u) + admits at least two positive solutions.     

Potential transformation methods for large-scale constrained global optimization

Several techniques for global optimization treat the objective functionf as a force-field potential. In the simplest case, trajectories of the differential equationmx=–f sample regions of low potential while retaining the energy to surmount passes which might block the way to regions of even lower local minima. Apotential transformation is an increasing functionV:. It determines a new potentialg=V(f), with the same minimizers asf and new trajectories satisfying . We discuss a class of potential transformations that greatly increase the attractiveness of low local minima.These methods can be applied to constrained problems through the use of Lagrange multipliers. We discuss several methods for efficiently computing approximate Lagrange multipliers, making this approach practical.     

Iterated Homology of Simplicial Complexes

We develop an iterated homology theory for simplicial complexes. Thistheory is a variation on one due to Kalai. For a simplicial complex of dimension d – 1, and each r = 0, ...,d, we define rth iterated homology groups of . When r = 0, this corresponds to ordinary homology. If is a cone over , then when r = 1, we get the homology of . If a simplicial complex is (nonpure) shellable, then its iterated Betti numbers give the restriction numbers, h _k,j , of the shelling. Iterated Betti numbers are preserved by algebraic shifting, and may be interpreted combinatorially in terms of the algebraically shifted complex in several ways. In addition, the depth of a simplicial complex can be characterized in terms of its iterated Betti numbers.     

On a property of N-functions

We consider three classes of N-functions: ('), the class of functions satisfying the ' condition, (₂), the class of functions satisfying the ₂ condition, and (M), the class of functions M(u) satisfying the condition: ln M (u) / ln u = p < . We establish the connection between the class of powers and the class of N-functions M(u) which belong to the class (') together with their complementary functions and we also establish the connections between the classes ('), (M) and (₂).Translated from Matematicheskie Zametki, Vol. 4, No. 3, pp. 281–290, 1968.     

On eigenfunctions of nonlinear heat generation

Summary We consider the equation u+ expu=0, >0,u(boundary)0 in the formv= exp (K,v), whereK ^–1=–. We give bounds on for the latter equation to be solvable by the contraction mapping principle, and estimate theL ² norm of the solution so obtained. We also give a bound on for the topological index of the solution to be non-zero and apply Krasnoselskii's results to the least squares method of approximating the solution.

---

Sommario Consideriamo l'equazione u+ expu=0, >0,u(frontiera)=0 nella formav= exp (Kv), doveK ^–1=–. In questo lavoro diamo limitazioni per per cui la seconda equazione e risolubile col metodo delle contrazioni, e diamo una stima della norma inL ² della soluzione cosi ottenuta. Diamo anche una limitazione per per cui l'indice topologico della soluzione diventa non zero, e applichiamo i risultati di Krasnoselskii al metodo dei minimi quadrati per approssimare la soluzione.

     

The spectral bound of Schrödinger operators

Let V: R ^N [0, ] be a measurable function, and >0 be a parameter. We consider the behaviour of the spectral bound of the operator 1/2–V as a function of . In particular, we give a formula for the limiting value as , in terms of the integrals of V over subsets of R ^N on which the Laplacian with Dirichlet boundary conditions has prescribed values. We also consider the question whether this limiting value is attained for finite .     

Colouring a graph frugally

We prove that any graph with maximum degree sufficiently large, has a proper vertex colouring using +1 colours such that each colour class appears at most log⁸ times in the neighbourhood of any vertex. We also show that for 1, the minimum number of colours required to colour any such graph so that each vertex appears at most times in the neighbourhood of any vertex is (+^1+1//), showing in particular that when =log/loglog, such a colouring cannot always be achieved with O() colours. We also provide a polynomial time algorithm to find such a colouring. This has applications to the total chromatic number of a graph.The second two authors were supported by NATO Collaborative Research Grant #CRG950235.     

Noise reduction by gradual memorization

Zusammenfassung Der vorliegende Beitrag beschreibt ein einfaches, nichtlineares, abgetastetes Filter, das «Gradueller Speicher» genannt wird. Das Filter besteht aus einem Speicher, welches neue Eingangswerte nur graduell, d. h. schrittweise, speichert: Bei jedem neuen Eingangswert wird er um einen konstanten Schritt ±s in Richtung des letzteren nachgestellt. Es wird gezeigt, dass eine solche «Graduelle Speicherung» eine Verminderung von Gaussschem Rauschen um einen Faktor gestattet. (=/s, d. h. ist gleich der Streuung des Eingangsrauschens geteilt durch den Schritt s.) Die Wahrscheinlichkeitsfunktion des Speicherausganges wird berechnet und durch eine Gauss-Funktion angenähert. eine Formel wird hergeleitet, welche die richtige Wahl von s in praktischen Fällen gestattet. Im letzten Abschnitt wird das dynamische Verhalten eines graduellen Speichers betrachtet, die Reaktionszeitt _r , welche der Speicher braucht, um einer plötzlichen Änderung U_o im wahren Eingangssignal nachzufolgen, wird berechnet.t _r wird umso grösser, je grösser die Streuung des Eingangsrauschens ist.

---

List of Symbols Real valueX=U ₀+U Input signal (true value + noise) - Real valueU ₀ Normalized valueu ₀=U ₀/s True input signal - Real valueU Normalized valueu=U/s Input noise - Real value s Step of memory - Real valueY=U ₀+R Output signal (true value + noise) - Real valueR Normalized valuer=R/s Output noise - Real value Normalized value Average momentary value of output signal - Real value Normalized value =/s Standard deviation of Gaussian noise at input - Normalized value Standard deviation of Gaussian noise at output - Normalized valueb Parameter of logistical distribution - Real value U ₀ Normalized valuea=U ₀/s Step of true input value - Real valuet Normalized value =t/t Time - Real value t=T Normalized value 1 Sampling interval - Real value Normalized value Average momentary velocity of output signal - Real valueU _tol Prescribed tolerance interval for output signal - Real value Prescribed probability (that the output signal remains within the tolerance interval) - Real valuew(U) Normalized valuew(u) Probability density function (input) - Real value (R) Normalized value (r) Probability (density) function (output) - Normalized valueq _r =1-P _r Cumulative distribution function - Normalized valueK=r/ Momentary value of output noise, expressed in multiples of the standard deviation - Real valueQ Normalized valueQ=/ Noise reduction factor - Real valuet _r Normalized value _r =t _r /t Time of reaction - Normalized valuez Increase in reaction time due to input noise     

Positive Solutions for a Singular Nonlinear Problem on a Bounded Domain in R 2

For a bounded regular Jordan domain in R ², we introduce and study a new class of functions K() related on its Green function G. We exploit the properties of this class to prove the existence and the uniqueness of a positive solution for the singular nonlinear elliptic equation u+(x,u)=0, in D(), with u=0 on and uC(), where is a nonnegative Borel measurable function in ×(0,) that belongs to a convex cone which contains, in particular, all functions (x,t)=q(x)t ^–,>0 with nonnegative functions qK(). Some estimates on the solution are also given.     

G-Uniformities,LR-Proximities and Hypertopologies

We look closely at the relationships between hit-and-miss and proximal hit-and-miss -topologies, in the setting of proximity spaces. We provide equivalent conditions that force comparisons among proximal hit-and-miss -topologies determined by different proximities. We pay attention to these topologies when consists of the family of all closed balls of a proximity space, and we study their interplay with the Wijsman convergence expressed in proximity spaces. Finally we study the supremum of all Wijsman convergences and of all proximal ball topologies when X is at most regular, and the infimum of all Wijsman convergences when X is at least Tychonoff.     

Configurational conditions and digraphs

Configurational conditions (Schließungsaussagen) of a noncommutative space will be developped from pairs (, ) of digraphs where is a partial digraph of . In this way we obtain an extensive generalization of Pfalzgraf 's q-simplex-conditions Sim_q.In Memoriam Hans Zassenhaus     

Sequential gradient-restoration algorithm for the minimization of constrained functions—Ordinary and conjugate gradient versions

The problem of minimizing a functionf(x) subject to the constraint (x)=0 is considered. Here,f is a scalar,x ann-vector, and aq-vector. Asequential algorithm is presented, composed of the alternate succession of gradient phases and restoration phases.In thegradient phase, a nominal pointx satisfying the constraint is assumed; a displacement x leading from pointx to a varied pointy is determined such that the value of the function is reduced. The determination of the displacement x incorporates information at only pointx for theordinary gradient version of the method (Part 1) and information at both pointsx and for theconjugate gradient version of the method (Part 2).In therestoration phase, a nominal pointy not satisfying the constraint is assumed; a displacement y leading from pointy to a varied point is determined such that the constraint is restored to a prescribed degree of accuracy. The restoration is done by requiring the least-square change of the coordinates.If the stepsize of the gradient phase is ofO(), then x=O() and y=O(²). For sufficiently small, the restoration phase preserves the descent property of the gradient phase: the functionf decreases between any two successive restoration phases.This research, supported by the NASA Manned Spacecraft Center, Grant No. NGR-44-006-089, and by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-828-67, is a condensation of the investigations reported in Refs. 1 and 2.