Intuitionistic weak arithmetic

We construct -framed Kripke models of i₁ and i₁ non of whose worlds satisfies xy(x=2yx=2y+1) and x,yzExp(x, y, z) respectively. This will enable us to show that i₁ does not prove ¬¬xy(x=2yx=2y+1) and i₁ does not prove ¬¬x, yzExp(x, y, z). Therefore, i₁¬¬lop and i₁¬¬i₁. We also prove that HAl₁ and present some remarks about i₂. Mathematics Subject Classification (2000):03F30, 03F55, 03H15.     

A Local Approach to 1-Homogeneous Graphs

Let bea distance-regular graph with diameter d. For vertices x and y of at distancei, 1 i d, we define the setsC _i(x,y) = i–1(x) (y), A _i (x,y) = _i (x) (y) and B _i (x,y) = _i+1(x) (y).Then we say has the CAB_j property,if the partition CAB _i (x,y) = {C _i (x,y),A _i (x,y),B _i (x,y)}of the local graph of y is equitable for each pairof vertices x and y of at distance i j. We show that in with the CAB_j property then the parameters ofthe equitable partitions CAB i(x,y) do not dependon the choice of vertices x and y atdistance i for all i j. The graph has the CAB property if it has the CAB _d property. We show the equivalence of the CAB property and the1-homogeneous property in a distance-regular graph with a ₁ 0. Finally, we classify the 1-homogeneous Terwilligergraphs with c ₂ 2.     

Some fractal sets determined by stable processes

Summary LetY _i be independent stable subordinators in (, ,P) with indices 0< _i <1 andR _i are the ranges ofY _i ,i=1, 2. We are able to find the exact Hausdorff measure and packing measure results for the product setsR ₁×R ₂, and whenever ₁ + ₂ 1/2, we deduce results for the vector sumR ₁R ₂={x+y:xR ₁,yR ₂}.     

The Resolution Method for One Reducible Class of Formulas of the First-Order Modal Logic S4

In the paper, we describe a resolution method for the family of the formulas of the form *ⁱ**(L ₁ L _s ), where i = 0 1 and L _j are modal literals. Negations here stand directly before classical atomic formulas, the formulas may contain constants. We also present the absorption tactics for a set of such formulas.__________Translated from Lietuvos Matematikos Rinkinys, Vol. 44, No. 4, pp. 481–492, October–December, 2004.     

A note on Dilworth's theorem in the infinite case

If is a poset and every antichain is finite, and if the length of the well-founded poset of antichains is less than ² ₁, then is the union of countably many chains. We also compute the length of the poset of antichains in the product of two ordinals, x.     

Nonparametric Estimation of a Conditional Quantile for α-Mixing Processes

Let (X_i,Y _i) be a set of observations form a stationary -mixing process and (x) be the conditional -th quantile of Y given X = x. Several authors considered nonparametric estimation of (x) in the i.i.d. setting. Assuming the smoothness of FF(x), we estimate it by local polynomial fitting and prove the asymptotic normality and the uniform convergence.     

Asymptotic behavior of general M-estimates for regression and scale with random carriers

Summary Let (xini, y _i be a sequence of independent identically distributed random variables, where x _i R ^p and y _i R, and let R ^p be an unknown vector such that y _i =x _i +u _i (^*), where u _i is independent of x _i and has distribution function F(u/), where >0 is an unknown parameter. This paper deals with a general class of M-estimates of regression and scale, ( ^*,^*), defined as solutions of the system: , where r= (y _i –x _i ^1*/)^*, with R ^p ×RR and RR. This class contains estimators of (, ) proposed by Huber, Mallows and Krasker and Welsch. The consistency and asymptotic normality of the general M-estimators are proved assuming general regularity conditions on and and assuming the joint distribution of (x _i , y _i ) to fulfill the model (^*) only approximately.     

Quadratically constrained quadratic programming: Some applications and a method for solution

The problem (QPQR) considered here is: minimizeQ ₁ (x) subject toQ _i (x) 0,i M ₁ {2,...,m},x P R ⁿ, whereQ _i (x), i M {1} M ₁ are quadratic forms with positive semi-definite matrices, andP a compact nonempty polyhedron of Rⁿ. Applications of (QPQR) and a new method to solve it are presented.Letu S={u R ^m;u 0, u _i= l}be fixed;then the problem:iM minimize u _iQ_i (x (u)) overP, always has an optimal solutionx (u), which is either feasible, iM i.e. u C₁ {u S;Q _i (x (u)) 0,i M ₁} or unfeasible, i.e. there exists ani M ₁ withu C {u S; Q_i(x(u)) 0}.Let us defineC _i C_i S _i withS _i {u S; u _i=0}, i M. A constructive method is used to prove that C _i is not empty and thatx (û) withiM û C _i characterizes an optimal solution to (QPQR). Quite attractive numerical results have been reached with this method.

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Zusammenfassung Die vorliegende Arbeit befaßt sich mit Anwendungen und einer neuen Lösungsmethode der folgenden Aufgabe (QPQR): man minimiere eine konvexe quadratische ZielfunktionQ _i (x) unter Berücksichtigung konvexer quadratischer RestriktionenQ _i (x) 0, iM ₁ {2,...,m}, und/oder linearer Restriktionen.·Für ein festesu S {u R ^m;u 0, u _i=1},M {1} M₁ besitzt das Problem:iM minimiere die konvexe quadratische Zielfunktion u _i Q_i (x (u)) über dem durch die lineareniM Restriktionen von (QPQR) erzeugten, kompakten und nicht leeren PolyederP R ⁿ, immer eine Optimallösungx (u), die entweder zulässig ist: u C₁ {u S;Q ₁ (x (u)) 0,i M ₁} oder unzulässig ist, d.h. es existiert eini M ₁ mitu C_i {u S;Q _i (x(u))0}.Es seien folgende MengenC _i C_i S _i definiert, mitS _i {u S;u _i=0}, i M. Es wird konstruktiv bewiesen, daß C _i 0 undx (û) mitû C _i eine Optimallösung voniM iM (QPQR) ist; damit ergibt sich eine Methode zur Lösung von (QPQR), die sich als sehr effizient erwiesen hat. Ein einfaches Beispiel ist angegeben, mit dem alle Schritte des Algorithmus und dessen Arbeitsweise graphisch dargestellt werden können.

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An earlier version of this paper was written during the author's stay at the Institute for Operations Research, Swiss Federal Institute of Technology, Zürich.     

On the ω-limit Set Dichotomy of Cooperating Kolmogorov Systems

The authors study the -limit set dichotomy of the Kolmogorov systems _i=x_i f _i(x)x _i0, 1in with the cooperative and irreducible hypotheses and obtain the quasiconvergence almost everywhere when n=3, which gives an affirmative answer to the open problem by Smith [9, p.72] in the case of n=3.     

Decomposing symmetric exchanges in matroid bases

LetB,B be bases of a matroid, withX B, X B. SetsX,X are asymmetric exchange if(B – X) X and(B – X) X are bases. SetsX,X are astrong serial B-exchange if there is a bijectionf: X X, where for any ordering of the elements ofX, sayx _i ,i = 1, , m, bases are formed by the sets B₀ = B, B_i = (B_i–1 – x_i) f(x _i), fori = 1, , m. Any symmetric exchangeX,X can be decomposed by partitioning X = _i=1 ^m Y_i, X = _i=1 ^m Y_i, X, where (1) bases are formed by the setsB ₀ =B, B _i = (B _i–1 –Y _i ) Y _i ; (2) setsY _i ,Y _i are a strong serialB _i–1 -exchange; (3) properties analogous to (1) and (2) hold for baseB and setsY _i ,Y _i .     

On the extreme spectral properties of Toeplitz matrices generated byL 1 functions with several minima/maxima

In this paper we are concerned with the asymptotic behavior of the smallest eigenvalue ₁ ⁽ⁿ⁾ of symmetric (Hermitian)n ×n Toeplitz matricesT _n (f) generated by an integrable functionf defined in [–, ]. In [7, 8, 11] it is shown that ₁ ⁽ⁿ⁾ tends to essinff =m _f in the following way: ₁ ⁽ⁿ⁾ –m _f 1/n ^2k . These authors use three assumptions:A1)f –m _f has a zero inx =x ₀ of order 2k.A2)f is continuous and at leastC ^2k in a neighborhood ofx ₀.A3)x =x ₀ is the unique global minimum off in [–, ]. In [10] we have proved that the hypothesis of smoothnessA2 is not necessary and that the same result holds under the weaker assumption thatf L ¹[–, ]. In this paper we further extend this theory to the case of a functionf L ¹[–, ] having several global minima by suppressing the hypothesisA3 and by showing that the maximal order 2k of the zeros off –m _f is the only parameter which characterizes the rate of convergence of ₁ ⁽ⁿ⁾ tom _f .     

Solution of two sequencing problems

The solution of the following problems is offered. Suppose a multiset J (¦J¦=p) is given. For each pair of elements and J, a number 1 P is given. Moreover, if 1 < x<p then x is undefined. If x=1, then x=p. Problem 1. Find the permutation ₁..._F of elements of the multiset J satisfying the following conditions. Let _i, _i=. If i,j < x, thenj <i. If i,j > x, then i<j. Such a permutation is called a PC-schedule. Problem 2. Find a PC-schedule in which the following property holds: if i < x < j, _i=, _j=, then. Such a PC-schedule is called an SC-schedule. The conditions under which these problems have solutions are studied. For their solution an algorithm of shifts is used with the complexity O(¦B(J)¦²¦J¦).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 124, pp. 44–72, 1983.     

Efficiency and the uniform linear minorization of convex functions

LetX be a Banach space, and let {f _i:iI} be a family of proper lower semicontinuous convex functions defined onX, each of whose epigraphs meets a fixed bound subset ofX×. We say that {f _i:iI} is uniformly linearly minorized if there exists a positive scalar such that for alliI andxX, we havef _i(x)–(1+x). We present two very different characterizations of uniform linear minorization for such a family. Using one of these, we show that either strong or weak epi-convergence of a sequence of convex functions at some point in the effective domain of the limit implies, uniform linear minorization for the entire sequence.With 1 Figure     

On quadratic spline interpolation

Using the quadratic spline interpolates(x) fitting the data (x _i,y _i), 0in and satisfying the end conditions_o=y_o, we give formulae approximatingy andy at selected knots by orders up toO(h ⁴).     

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 averaging multisets

We give sufficient conditions to ensure that, given a set , everyxint convM can be represented as a convex combination,x = _{i = 1} ⁿ _i x _i , wherex _i M, _i rational, andn=2s orn=2s–1, respectively.     

Existence of solutions for the Navier-Stokes equations,having an infinite dissipation of energy,in a class of domains with noncompact boundaries

In the domain ³ with two exits at infinity, ₁ = {x:x<g ₁(x ₃),x ₃ > 2} and ₂ = {x:0<x ₃<g ₃(x), x>2}, one investigates the stationary system of Navier-Stokes equations under the boundary condition. One proves existence and uniqueness theorems for the solution of this problem with an infinite Dirichlet integral, under a given flow of the velocity vector across the cross sections of the exits at infinity. As an example one considers the case wheng _i(t) 1.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 110, pp. 180–202, 1981.     

Quadratic spline interpolation on uniform meshes

The interpolation problem at uniform mesh points of a quadratic splines(x _i)=f _i,i=0, 1,...,N ands(x ₀)=f₀ is considered. It is known that s–f=O(h ³) and s–f=O(h ²), whereh is the step size, and that these orders cannot be improved. Contrary to recently published results we prove that superconvergence cannot occur for any particular point independent off other than mesh points wheres=f by assumption. Best error bounds for some compound formulae approximatingf _i andf _i ⁽³⁾ are also derived.     

On a classical Nicoletti boundary value problem

We shall derive existence, uniqueness and comparison results for the functional differential equationx(t)=f(t, x), a. e.tI, with classical Nicoletti boundary conditionsx _i(t_i)=y _iX, iA, whereI is a real interval,A is a nonempty set andX is a Banach space.     

Some analytical properties of γ-convex functions on the real line

This paper deals with the analytical properties of -convex functions, which are defined as those functions satisfying the inequalityf(x ₁ )+f(x ₂ )f(x ₁)+f(x ₂), forx _i [x ₁,x ₂], |x _i –x _i |=, i=1,2, whenever |x ₁–x ₂|>, for some given positive . This class contains all convex functions and all periodic functions with period . In general, -convex functions do not have ideal properties as convex functions. For instance, there exist -convex functions which are totally discontinuous or not locally bounded. But -convex functions possess so-called conservation properties, meaning good properties which remain true on every bounded interval or even on the entire domain, if only they hold true on an arbitrary closed interval with length . It is shown that boundedness, bounded variation, integrability, continuity, and differentiability almost everywhere are conservation properties of -convex functions on the real line. However, -convex functions have also infection properties, meaning bad properties which propagate to other points, once they appear somewhere (for example, discontinuity). Some equivalent properties of -convexity are given. Ways for generating and representing -convex functions are described.This research was supported by the Deutsche Forschungsgemeinschaft. The first author thanks Prof. Dr. E. Zeidler and Prof. Dr. H. G. Bock for their hospitality and valuable support.