Non polynomial splines and weakly singular two-point boundary value problems

In the present paper we shall consider an application of simple non-polynomial splines to a numerical solution of a weakly singular two-point boundary value problem:x ^– (x y)=f(x,y), (0<x1) subject toy(0)=0,y(1)=c ₁(1) ory(0)=c ₂,y(1)=c ₃(0<<1). Our collocation method gives a continuously differentiable approximation and isO(h ²)-convergent.     

Analytic Approximations for a Singularly Perturbed Convection—Diffusion Problem with Discontinuous Data in a Half-Infinite Strip

We consider a singularly perturbed convection—diffusion equation, –u+v u=0, defined on a half-infinite strip, (x,y)(0,)×(0,1) with a discontinuous Dirichlet boundary condition: u(x,0)=1, u(x,1)=u(0,y)=0. Asymptotic expansions of the solution are obtained from an integral representation in two limits: (a) as the singular parameter 0⁺ (with fixed distance r to the discontinuity point of the boundary condition) and (b) as that distance r0⁺ (with fixed ). It is shown that the first term of the expansion at =0 contains an error function or a combination of error functions. This term characterizes the effect of discontinuities on the -behavior of the solution and its derivatives in the boundary or internal layers. On the other hand, near the point of discontinuity of the boundary condition, the solution u(x,y) is approximated by a linear function of the polar angle at the point of discontinuity (0,0).     

Der Beweis eines Satzes von G. Choodnovsky

1. If is a weakly compact cardinal then ( ⁺)( ). 2. If is measurable andU a normal ultrafilter then ( ⁺)(U ).

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Diese Arbeit ist ein Teil der Dissertation des Autors. Teilweise gefördert von der DFG.     

A decomposition for the exponential dispersion model generated by the invariant measure on the hyperboloid

For = ₀, ₁, ₂) andx=(x₀, x₁, x₂) in R³, define [,x] = ₀ x ₀ – ₁ x ₁ – ₂ x ₂,C = {x³:x ₀ > 0 and [x, x]>0},R(x)=([x, x]) ^1/2 forx inC andH ₁={xC: x₀>0,R(x)=1}. Define the measure onH ₁ such that if is inC and =R(), then exp (–[,x])(dx = ( exp )^–1. Therefore, is invariant under the action ofSO (1, 2), the connected component ofO(1, 2) containing the identity. We first prove that there exists a positive measure in ³ such that its Laplace transform is ( exp )^– if and only if >1. Finally, for 1 and inC, denotingP(,)(dx) = ( exp )^– exp (–[,x])(dx, we show that ifY ₀,...,Y _n aren+1 independent variables with densityP(,),j=0,...,n and ifS _k =X ₀ + ... +X _k andQ _k =R(S _k) –R(S _k–1) –R(Y _k),k=1,...,n, then then+1 statisticsD _n = [/,S _k ] –R _{k – 1} ),Q ₁,...,Q _n are independent random variables with the exponential () or gamma (1,1/) distribution.This research has been partially funded by NSERC Grant A8947.     

On covering designs with block size 5 and index 5

LetV be a finite set of order . A (, , ) covering design of index and block size is a collection of -element subsets, called blocks, such that every 2-subset ofV occurs in at least blocks. The covering problem is to determine the minimum number of blocks, (, , ), in a covering design. It is well known that , where [x] is the smallest integer satisfyingx[X]. It is shown here that (, 5, 5)=(, 5, 5) for all positive integers 5 with the possible exception of =24, 28, 56, 104, 124, 144, 164, 184.     

Das 0-1-Gesetz der terminalen σ-Algebra bei Harrisirrfahrten

Für homogene M.K., bestehend aus einer aperiodischen rekurrenten Klasse, ist bekannt, da\ die terminale -Algebra das 0–1-Gesetz erfüllt. Der erste Teil dieser Arbeit behandelt die Gültigkeit des 0–1-Gesetzes für transiente Ketten, für die eine Folge ₀, ₁, ₂, ₃,... von ZustÄnden existiert mit P(nX_n= _l+1¦X₀= _l) = 1, N. Das herzuleitende Kriterium wird nur von dieser Folge und den Stoppzeiten, von _l nach _l+1 zu gelangen, abhÄngen. Eine Anwendung auf Harrisirrfahrten wird uns dort konkrete Aussagen liefern.     

On graphs that can be oriented as diagrams of ordered sets

We study some equivalent and necessary conditions for a finite graph to be the covering graph of a (partially) ordered set. For each 1, M. Aigner and G. Prins have introduced a notion of a vertex colouring, here called -good colouring, such that a 1-good colouring is the usual concept and graphs that have a 2-good colouring are precisely covering graphs. We present some inequalities for the corresponding chromatic numbers , especially for x ₂. There exist graphs that satisfy these inequalities for =2 but are not covering graphs. We show also that x ₂ cannot be bounded by a function of x=x ₁. A construction of Neetil and Rödl is used to show that x ₂ is not bounded by a function of the girth.     

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.     

A note on quasi-periodic solutions of some elliptic systems

We extend a recent method of proof of a theorem by Kolmogorov on the conservation of quasi-periodic motion in Hamiltonian systems so as to prove existence of (uncountably many) real-analytic quasi-periodic solutions for elliptic systems u=f _x (u, y), whereu y ^M u(y) ^N ,f=f(x, y) is a real-analytic periodic function and is a small parameter. Kolmogorov's theorem is obtained (in a special case) whenM=1 while the caseN=1 is (a special case of) a theorem by J. Moser on minimal foliations of codimension 1 on a torusT ^M +1. In the autonomous case,f=f(x), the above result holds for any .     

Quadrature formulas obtained by variable transformation

Quadrature formulas suitable for evaluation of improper integrals such as are obtained by means of variable transformations =tanhu and =erfu, and subsequent use of trapezoidal quadrature rule. Error analysis is carried out by the method of contour integral, and the results are confirmed on several concrete examples. Similar formulas are also obtained to accelerate the convergence of infinite integrals by means of variable transformations =sinhu and =tanu.     

Mad families, forcing and the Suslin Hypothesis

Let be a regular cardinal and P a partial ordering preserving the regularity of . If P is (-Baire and) of density , then there is a mad family on killed in all generic extensions (if and) only if below each pP there exists a -sized antichain. In this case a mad family on is killed (if and) only if there exists an injection from onto a dense subset of Ult(P) mapping the elements of onto nowhere dense sets. If 2^<=, then in each generic extension of V, in which is the minimal cardinal obtaining new subsets, some mad family on is killed or an independent subset of appears. Also, the -Suslin Hypothesis holds iff there exists a mad family on which is killed in each generic extension containing new subsets of and preserving P() for <.Mathematics Subject Classification (2000): 03E35, 03E40, 03E65     

The Phase Space of One Generalized Model by Oskolkov

We demonstrate that the phase space of the Cauchy–Dirichlet problem for the equation u _t -u _t =u-K(u)+f is a simple Banach C -manifold.     

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.     

Über das Modell einer Nicht-Newtonschen Flüssigkeit mit Relaxationseigenschaften

This paper deals with the equations ·u=o,u – u – u+u · u+p=f, describing a Non-Newtonian liquid with velocityu, pressure p under the external forcef. The difference to the Navier-Stokes equations consists in the fact that it is assumed here that the external forcef only has an effect on the volume-element of the liquid after the relaxation-time /.     

On the free subset property at singular cardinals

We give a proof ofTheorem 1. Let be the smallest cardinal such that the free subset property Fr (, ₁)holds. Assume is singular. Then there is an inner model with ₁ measurable cardinals.     

On the Densities of Certain Lattice Packings by Parallelepipeds

Given any (non-degenerate) n-dimensional lattice L, let (L) denote the supremum of the numbers such that there exists a lattice packing Q + L of density where Q is some o-symmetric parallelepiped with faces parallel to the coordinate axes. Many efforts have been made to determine or estimate the minimal such density _n taken over all n-dimensional lattices. It is known that 0$$ " align="middle" border="0"> . Here we investigate a sequence of lattices L _n which are known to minimize the function (L) in dimensions n 3 and are likely to provide the minima _n = (L _n ) in certain higher dimensions. We establish the inequality (L _n ) n ^–n/2 which supports the conjecture that lim sup _n ( _n )^{1/(n log n)} is positive.     

Symmetrische Permutationsmengen

A permutation set (M, I) consisting of a setM and a set of permutations ofM, is calledsymmetric, if for any two permutations, the existence of anx M with (x) (x) and ^–1 (x) = ^–1 (x) implies ^–1 = ^–1 , andsharply 3-transitive, if for any two triples (x ₁,x ₂,x ₃), (y ₁,y ₂,y ₃) M ³ with|{x ₁,x ₂,x ₃ }| = |{y ₁,y ₂,y ₃ }| = 3 there is exactly one permutation with(x ₁) =y ₁,(x ₂) =y ₂,(x ₃) =y ₃. The following theorem will be proved.THEOREM.Let (M, ) be a sharply 3-transitive symmetric permutation set with |M|3, such that contains the identity. Then is a group and there is a commutative field K such that and the projective linear group PGL(2, K) are isomorphic.     

A note on extensions of infinitary logic

We show that a strong form of the so called Lindströms Theorem [4] fails to generalize to extensions of L and L : For weakly compact there is no strongest extension of L with the (,)-compactness property and the Löwenheim-Skolem theorem down to . With an additional set-theoretic assumption, there is no strongest extension of L with the (,)-compactness property and the Löwenheim-Skolem theorem down to <.We are indebted to Lauri Hella, Tapani Hyttinen and Kerkko Luosto for useful suggestions.Research partially supported by the United States-Israel Binational Science Foundation. Publication number [ShVa:726]Research partially supported by grant 40734 of the Academy of Finland.     

On the existence of strongly normal ideals overP κ λ

For every uncountable regular cardinal and any cardinal,P denotes the set . Furthermore, < denotes=" the=" binary=" operation=" defined=">P byx<> iffxy¦x<>.By anideal over P we mean a proper, non-principal,-complete ideal overP extending the ideal dual to the filter generated by . For any idealI overP ,I ⁺ denotes the setP –I, andI ^* the filter dual toI.