Parametrized Partitions of Products of Finite Sets*

For every infinite sequence of positive integers and every Borel partition c : ×[]{0, 1} there is H[] and a sequence of subsets of , with |H_i|=m_i for every i, such that c is constant on .* Research partially supported by CNRS-FONACIT Project PI 2000001471. This author thanks the University of Paris VII for hospitality.     

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.     

The differences between Kurepa trees and Jech-Kunen trees

Summary By an ₁ we mean a tree of power ₁ and height ₁. An ₁-tree is called a Kurepa tree if all its levels are countable and it has more than ₁ branches. An ₁-tree is called a Jech-Kunen tree if it has branches for some strictly between ₁ and . In Sect. 1, we construct a model ofCH plus , in which there exists a Kurepa tree with not Jech-Kunen subtrees and there exists a Jech-Kunen tree with no Kurepa subtrees. This improves two results in [Ji1] by not only eliminating the large cardinal assumption for [Ji1, Theorem 2] but also handling two consistency proofs of [Ji1, Theorem 2 and Theorem 3] simultaneously. In Sect. 2, we first prove a lemma saying that anAxiom A focing of size ₁ over Silver's model will not produce a Kurepa tree in the extension, and then we apply this lemma to prove that, in the model constructed for Theorem 2 in [Ji1], there exists a Jech-Kunen tree and there are no Kurepa trees.     

On singular perturbation of a semilinear hyperbolic equation

We consider a hyperbolic version of Eells-Sampson's equation: . This equation is semilinear with respect to space derivative and time derivative. Letu (x) be the solution with initial data u(0) and (0), and putv (t,x)=u (t,x). We show that when the resistance ,V (t,x) converges to a solution of the original parabolic Eells-Sampson's equation: . Note thatv _t(0)= (0) diverges when . We show that this phenomena occurs in more general situations.This article was processed by the author using the Springer-Verlag Pjourlg macro package.     

On convergence of types and processes in Euclidean space

Let be an Euclidean space; Y _n , Z, U random vectors in ; h _n , g _n affine transformations and let þ be a subgroup of the group G of all the in vertible affine transformations, closed relative to G. Suppose that g_n and where Z is nonsingular. The behaviour of _n = h _n g _n ^–1 as n is discussed first. The results are used then to prove that if for all t(0, ), where h _n þ and Z ₁ is nonsingular and nonsymmetric with respect to þ then H, for all t(0,) and is a continuous homomorphism of the multiplicative group of (0, ) into þ. The explicit forms of the possible are shown.     

Zur Interpolation und Integration differenzierbarer periodischer Funktionen

Summary Letx ₀<x ₁<...<x _n–1<x ₀+2 be nodes having multiplicitiesv ₀,...,v _n–1, 1v _k r (0k<n). We approximate the evaluation functional ,x fixed, and the integral respectively by linear functionals of the form and determine optimal weights for the Favard classesW _r C ₂. In the even case of optimal interpolation these weights are unique except forr=1,x(x _k +x _k–1)/2 mod 2. Moreover we get periodic polynomial splinesw _{k, j} (0k<n, 0j<v _k ) of orderr such that are the optimal weights. Certain optimal quadrature formulas are shown to be of interpolatory type with respect to these splines. For the odd case of optimal interpolation we merely have obtained a partial solution.

Bojanov hat in [4, 5] ähnliche Resultate wie wir erzielt. Um Wiederholungen zu vermeiden, werden Resultate, deren Beweise man bereits in [4, 5] findet, nur zitiert     

Some topological properties of ω-covering sets

We prove the following theorems:1. There exists an -covering with the property s ₀.2. Under cov there exists X such that is not an -covering orX \ B is not an -covering].3. Also we characterize the property of being an -covering.     

A Class of Operators on L h 2 . II

Let D be the open unit disk in C, and L _h ² the space of quadratic integrable harmonic functions defined on D. Let be a function in L(D) with the property that (b) = lim _x b,xD (x) for all b D. Define the operator C on L _h ² as follows: C(f) = Q( · f), where Q is the orthogonal projection of L²(D) onto L _h ² . In this paper it is shown that if C is Fredholm, then is bounded away from zero on a neighborhood of D. Also, if C is compact, then |D 0, and the commutator ideal of (D) is K(D), where (D) denotes the norm closed subalgebra of the algebra of all bounded operators on L _h ² generated by , and K(D) is the ideal of compact operators on L _h ² . Finally, the spectrum of classes of operators defined on L _h ² is characterized.     

Ultrafilter translations

We develop a method for extending results about ultrafilters into a more general setting. In this paper we shall be mainly concerned with applications to cardinality logics. For example, assumingV=L, Gödel's Axiom of Constructibility, we prove that if > then the logic with the quantifier there exist many is (,)-compact if and only if either is weakly compact or is singular of cofinality<. As a corollary, for every infinite cardinals and , there exists a (,)-compact non-(,)-compact logic if and only if either < orcf<cf or < is weakly compact.Counterexamples are given showing that the above statements may fail, ifV=L is not assumed.However, without special assumptions, analogous results are obtained for the stronger notion of [,]-compactness.     

The Reynolds number and large time behaviour for weak solutions of the Navier-Stokes equations

We consider time-dependent perturbationsu of R. Finn's stationary PR-solution of the Navier-Stokes equations, which converges to a constant vectorv as ¦x¦ .We investigate the large time behaviour ofu in the case, that the (energetic) Reynolds numberR _e of has the critical valueR _e =1. We show that perturbationsu with small initial kinetic energy and small driving force become smooth after finite time.     

On local automorphisms of certain quadrics of codimension 2

The article considers nondegenerate quadrics in Cⁿ⁺¹ with codimension 2 that are of the form M={zCⁿ, C²: Im_j=z, z_j; j=1, 2}, where are Hermitian forms, and thje stability groups Aut_x M that preserve the point x. It is proved that if the matrix ¹ is stable and the matrix (¹)^–12 has more than two different eigenvalues, all automorphisms of Aut_x M are linear transformations.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 9–14, July, 1992.     

Interpolation theorems for a family of spanning subgraphs

Let G be a graph with order p, size q and component number . For each i between p – and q, let be the family of spanning i-edge subgraphs of G with exactly components. For an integer-valued graphical invariant if H H is an adjacent edge transformation (AET) implies |(H)-(H^')|1 then is said to be continuous with respect to AET. Similarly define the continuity of with respect to simple edge transformation (SET). Let M _j() and m _j() be the invariants defined by . It is proved that both M _p–() and m _p–(;) interpolate over , if is continuous with respect to AET, and that M _j() and m _j() interpolate over , if is continuous with respect to SET. In this way a lot of known interpolation results, including a theorem due to Schuster etc., are generalized.     

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.     

On the spectrum determined growth assumption and the perturbation ofC 0 semigroups

The spectrum determined growth property ofC ₀ semigroups in a Banach space is studied. It is shown that ifA generates aC ₀ semigroup in a Banach spaceX, which satisfies the following conditions: 1) for any >s(A), sup{R(;A) | Re}<; 2) there is a ₀>(A) such that , xX, and , fX ^*, then (A=s(A). Moreover, it is also shown that ifA=A ₀+B is the infinitesimal generator of aC ₀ semigroup in Hilbert space, whereA ₀ is a discrete operator andB is bounded, then (A)=s(A). Finally the results obtained are applied to wave equation and thermoelastic system.     

On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence

Let M _f(r) and _f(r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let be a continuously differentiable function convex on (–, +) and such that x = o((x)) as x +. We establish that, in order that the equality be true for any entire function f, it is necessary and sufficient that ln (x) = o((x)) as x +.     

Integral averages and oscillation of second order sublinear differential equations

New oscillation criteria are given for the second order sublinear differential equation

Random reals and the relation 171-1171-1171-1

It is consistent that ₁(₁,(:n))² holds in any random extension for n finite and countable.     

On the Harris Recurrence of Iterated Random Lipschitz Functions and Related Convergence Rate Results

A result by Elton⁽⁶⁾ states that an iterated function system

A majorizing semantics for hyperarithmetic sentences

A hyperarithmetic language is considered, obtained by adding to the arithmetic language a special ternary predicateH which acts as the universal predicate for (for some scale of constructive ordinals ). The language expresses a hierarchy {}_< of classes of formulas which is the constructive analog of the initial -section of the classical hyperarithmetic hierarchy. Some properties of this hierarchy are introduced which give a convenient constructive theoryT . It is shown that the majorizing semantics introduced in [1] (for an equivalent variant see [2]) can be extended to the sentences of the language for sentences of the arithmetic language. The basis for the construction of the majorant is the idea (stated in [2]) of relating the majorant to deducibility in systems with an -rule.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 68, pp. 30–37, 1977.