k-regular factors and semi-k-regular factors in bipartite graphs

LetG be a graph, andk1 an integer. LetU be a subset ofV(G), and letF be a spanning subgraph ofG such that deg _F (x)=k for allx V(G)–U. If deg _F (x)k for allxU, thenF is called an upper semi-k-regular factor with defect setU, and if deg _F (x)k for allxU, thenF is called a lower semi-k-regular factor with defect setU. Now letG=(X, Y;E(G)) be a bipartite graph with bipartition (X,Y) such that X=Yk+2. We prove the following two results.(1) Suppose that for each subsetU ₁X such that U ₁=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.(2) Suppose that for each subsetU ₁X such that U ₁=X–1/k+1,G has a lower semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=X–1/k+1,G has a lower semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.     

Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory

Let {X _k , 1 k n} be n independent and real-valued random variables with common subexponential distribution function, and let {k, 1 k n} be other n random variables independent of {X _k , 1 k n} and satisfying a _k b for some 0 < a b < for all 1 k n. This paper proves that the asymptotic relations P (max_{1 m n k=1} ^m _k X _k > x) P (sum _k=1 ⁿ _k X _k > x) sum _k=1 ⁿ P ( _k X _k > x) hold as x . In doing so, no any assumption is made on the dependence structure of the sequence { _k , 1 k n}. An application to ruin theory is proposed.     

On generalized gamma functions related to the Laurent coefficients of the Riemann zeta function

Summary We study a class of generalized gamma functions _k (z) which relate to the generalized Euler constants _k (basically the Laurent coefficients of(s)) as (z) does to the Euler constant. A new series expansion for _k is derived, and the constant term in the asymptotic expansion for log _k (z) is studied in detail. These and related constants are numerically computed for 1 k 15.     

Smoothness of solutions of a nonlinear ode

Smoothness of aC -functionf is measured by (Carleman) sequence {M _k} ₀ ; we sayfC _M [0, 1] if|f ^(k) (t)|CR ^k M _k,k=0, 1, ... withC, R>0. A typical statement proven in this paper isTHEOREM: Let u, b be two C -functions on [0, 1]such that (a) u=u ²+b, (b) |b ^(k) (t)|CR ^k (k!) , >1,k_–.Then |u^(k)(t)|C₁R^k((k–1)!),k_–.The first author acknowledges the hospitality of Mathematical Research Institute of the Ohio State University during his one month visit there in the spring of 1999     

∑II∑ threshold formulas

A II formula has the form, where eachL is either a variable or a negated variable. In this paper we study the computation of threshold functions by II formulas. By combining the proof of the Fredman-Komlós bound [5, 10] and a counting argument, we show that fork andn large andkn/2, every II formula computing the threshold functionT _k ⁿ has size at least exp . Fork andn large andkn ^2/3, we show that there exist II formulas for computingT _k ⁿ with size at most exp .     

Triple Positive Solutions for Multipoint Conjugate Boundary Value Problems

For the nth order nonlinear differential equation y ⁽ⁿ⁾(t)=f(y(t)), t [0,1], satisfying the multipoint conjugate boundary conditions, y ^(j)(a_i) = 0,1 i k, 0 j n _i - 1, 0 =a ₁ < a ₂ < < a _k = 1, and _i=1 ^k n _i =n, where f: [0, ) is continuous, growth condtions are imposed on f which yield the existence of at least three solutions that belong to a cone.     

Properties of theL p-solutions of linear impulsive equations in a Banach space

Summary In the paper conditions for the existence ofL _p-conditions (1 p ) of linear impulsive equations in a Banach space are found.     

Subsets with Restricted Movement

Let G be a finite permutation group on a set with no fixed points in and let m and k be integers with 0 < m < k. For a finite subset of the movement of is defined as move() = max_gG| ^g \ |. Suppose further that G is not a 2-group and that p is the least odd prime dividing |G| and move() m for all k-element subsets of . Then either || k + m or k (7m – 5) / 2, || (9m – 3)/2. Moreover when || > k + m, then move() m for every subset of .     

Montgomery's weighted sieve for dimension two

The most well-known application of Montgomery's weighted sieve is to the so-called Brun-Titchmarsh inequality, which was proved byH. L. Montgomery andR. C. Vaughan in the form (x, k, l)2x((k)log(x/k))^–1 for 1k<x, (k, l)=1, (x, k, l) being the number of primespx andpl modk, (k) being Euler's function. In this paper an upper estimate is given for a certain class of two-dimensional sieve problems, among them bounds for the number of twin primes and the number of Goldbach representations.     

The caratheodory number for thek-core

Thek-core of the setS ⁿ is the intersection of the convex hull of all setsA S with ¦SA¦<-k. The Caratheodory number of thek-core is the smallest integerf (d,k) with the property thatx core _{kS, S} ⁿ implies the existence of a subsetT S such thatx core_kT and ¦T¦f (d, k). In this paper various properties off(d, k) are established.Research of this author was partially supported by Hungarian National Science Foundation grant no. 1812.     

Approximations in thermal explosion theory and the nature of the degenerate critical point

Summary In studies of thermal explosion the Frank-Kamenetskii approximation sets exp(–E/RT)=exp(–E/RT ₀)exp (/(1+))exp(–E/RT ₀)exp, where=RT ₀/E i.e. it assumes0. When this approximation is not made, it is known that criticality vanishes for greater than a certain value _*, say. This may occur whether the Arrhenius form is used or some suitable approximation to it; many authors have proposed approximations involving the maximum dimensionless temperature in the reactant. The nature of the degeneracy near the value _* is examined for such approximations in general, some approximations are considered and the results compared.

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Zusammenfassung In Studien von thermischen Explosionen setzt man in der Näherung von Frank-Kamenetskii exp(–E/RT)=exp(–E/RT ₀)exp(/(1+))exp(–E/RT ₀) exp), wobei=RT ₀/E ist, d.h. man nimmt0 an. Wenn diese Näherung nicht benützt wird, so weiß man, daß die Kritikalität verschwindet wenn einen gewissen Wert _* überschreitet. Dies findet man mit Benützung der Formel von Arrhenius oder mit einer Näherung dazu; viele Autoren haben Näherungen vorgeschlagen mit Verwendung der maximalen dimensionslosen Temperatur im Reaktionsgemisch. Es wird für solche Näherungen die Natur der Entartung der Lösung in der Umgebung von _* untersucht; die Resultate für verschiedene Näherungen werden verglichen.

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Résumé Dans les études de la théorie de l'explosion thermale l'approximation de Frank-Kamenetskii pose exp(–E/RT)=exp(–E/RT ₀)exp(/(1+)]exp(–E/RT ₀) exp, avec=RT ₀/E, c'est à dire on admet0.On sait que, en dehors de cette approximation, la limite critique disparaît lorsque dépasse une certaine valeur dénommée _*. Ceci peut se produire soit en utilisant la forme d'Arrhenius ou une approximation adéquate.Plusieurs auteurs ont proposé des approximations utilisant la température maximum non-dimensionelle du réactif. Dans la présente étude on examine d'une façon générale le caractère de la dégénérescence aux alentours de la valeur _* pour ce genre d'approximations. Ensuite on considère quelques approximations particulières et les résultats sont comparés.

     

On convergence subsystems of orthonormal systems

It is proved that for any sequence {R _k} _k=1 of real numbers satisfyingR _kk (k1) andR _k=o(k log₂ k),k, there exists an orthonormal system {n _k(x)} _n=1 ,x (0;1), such that none of its subsystems {n _k(x)} _k=1 withn _kR_k (k1) is a convergence subsystem.     

Harmonic analysis on tori

We consider measurable subsets {ofR}ⁿ with 0<m()<, and we assume that has a spectral set . (In the special case when is also assumed open, may be obtained as the joint spectrum of a family of commuting self-adjoint operators {H _k: 1kn} in L ² () such that each H _k is an extension of i(/x _k) on C _c (), k=1, ..., n.)It is known that is a fundamental domain for a lattice if is itself a lattice. In this paper, we consider a class of examples where is not assumed to be a lattice. Instead is assumed to have a certain inhomogeneous form, and we prove a necessary and sufficient condition for to be a fundamental domain for some lattice in {ofR}ⁿ. We are thus able to decide the question, fundamental domain or not, by considering only properties of the spectrum . Our criterion is obtained as a corollary to a theorem concerning partitions of sets which have a spectrum of inhomogeneous form.Work supported in part by the NSF.Work supported in part by the NSRC, Denmark.     

Asymptotic behavior of the dwell time distribution for a random walk on a positive semi-axis

Let ₁, ₂, ... be a sequence of independent identically distributed random variables with zero means. We consider the functional _n = _k=o ⁿ (S _k ) where S₁=0, S_k= _i=1 ^k _i (k1) and(x)=1 for x0,(x) = 0 for x<0. It is readily seen that _n is the time spent by the random walk S_n, n0, on the positive semi-axis after n steps. For the simplest walk the asymptotics of the distribution P (_n = k) for n and k, as well as for k = O(n) and k/n<1, was studied in [1]. In this paper we obtain the asymptotic expansions in powers of n^–1 of the probabilities P(h_n = nx) and P(nx_{1 n} nx₂) for 0<₁, x = k/n ₂<1, 0<₁x₁2₂<1.Translated from Matematicheskie Zametki, Vol. 15, No. 4, pp. 613–620, April, 1974.The author wishes to thank B. A. Rogozin for valuable discussions in the course of his work.     

Embedding a non-embeddable stable plane

In [4], K. Strambach describes a 2-dimensional stable plane admitting =SL₂ as a group of automorphisms such that there exists no -equivarient embedding into a 2-dimensional projective plane. R. Löwen [3] has given a 4-dimensional analogue , admitting =SL₂. He posed the question whether there are embeddings of Strambach's plane into . We show that such embeddings exist, in fact we determine all -equivariant embeddings of 2-dimensional stable planes admitting as atransitive group of automorphisms.     

Convergence of Appell Polynomials of Long Range Dependent Moving Averages in Martingale Differences

We study limit distribution of partial sums S_N,k(t) = _{s = 1} ^{[N t]} A_k(X_s) of Appell polynomials of the long-range dependent moving average process X_t> = _{i t} b_{t - i i}, where {_i} is a strictly stationary and weakly dependent martingale difference sequence, and b_i i^{d - 1} (0 < d < 1/2). We show that if k(1-2 d)<1, then suitably normalized partial sums S_N,k(t) converge in distribution to the kth order Hermite process. This result generalizes the corresponding results of Surgailis, and Avram and Taqqu obtained in the case of the i.i.d. sequence { _i}.     

Finite Representability of ℓp in Quotients of Banach Spaces

A typical result of the paper states that if X is a Banach space with a basis and for some 1pq, the spaces _p and _q are finitely block representable in every block subspace of X, then every block subspace of X admits a block quotient Z such that for every r[p,q], the space _r is finitely block representable in Z. Results of a similar nature are also established for ^N _p-block-sequences and asymptotic spaces.     

An Inequality Involving the Local Eigenvalues of a Distance-Regular Graph

Let denote a distance-regular graph with diameter D 3, valency k, and intersection numbers a _i, b _i, c _i. Let X denote the vertex set of and fix x X. Let denote the vertex-subgraph of induced on the set of vertices in X adjacent X. Observe has k vertices and is regular with valency a ₁. Let _{1 2} ··· _k denote the eigenvalues of and observe ₁ = a ₁. Let denote the set of distinct scalars among ₂, ₃, ..., _k . For let mult denote the number of times appears among ₂, ₃,..., _k . Let denote an indeterminate, and let p ₀, p₁, ...,p _D denote the polynomials in [] satisfying p ₀ = 1 andp _i = c _i+1 p _i+1 + (a _i – c _i+1 + c _i)p _i + b _i p _i–1 (0 i D – 1),where p _–1 = 0. We show where we abbreviate = –1 – b ₁(1+)^–1. Concerning the case of equality we obtain the following result. Let T = T(x) denote the subalgebra of Mat _X ( ) generated by A, E*₀, E*₁, ..., E* _D , where A denotes the adjacency matrix of and E* _i denotes the projection onto the ith subconstituent of with respect to X. T is called the subconstituent algebra or the Terwilliger algebra. An irreducible T-module W is said to be thin whenever dimE* _i W 1 for 0 i D. By the endpoint of W we mean min{i|E* _i W 0}. We show the following are equivalent: (i) Equality holds in the above inequality for 1 i D – 1; (ii) Equality holds in the above inequality for i = D – 1; (iii) Every irreducible T-module with endpoint 1 is thin.     

k-Weakly almost convex groups and π 1 ∞ \tilde M^3

We extend Cannon's notion ofk-almost convex groups which requires that for two pointsx, y on then-sphere in the Cayley graph which can be joined by a pathl ₁ of length k, there is a second pathl ₂ in then-ball, joiningx andy, of bounded length N(k). Ourk-weakly almost convexity relaxes this condition by requiring only thatl ₁ l ₂ bounds a disk of area C ₁(k)n ^{1 - (k)} +C ₂(k). IfM ³ is a closed 3-manifold with 3-weakly almost convex fundamental group, then ₁ .     

Krasnosel'skii numbers for bounded finitely starlike sets inR d

For eachk andd, 1kd, definef(d, d)=d+1 andf(d, k)=2d if 1kd–1. The following results are established:Let be a uniformly bounded collection of compact, convex sets inR ^d . For a fixedk, 1kd, dim {MM in }k if and only if for some > 0, everyf(d, k) members of contain a commonk-dimensional set of measure (volume) at least.LetS be a bounded subset ofR ^d . Assume that for some fixedk, 1kd, there exists a countable family of (k–l)-flats {H _i :i1} inR ^d such that clS S {H_i i 1 } and for eachi1, (clS S) H _i has (k–1) dimensional measure zero. Every finite subset ofS sees viaS a set of positivek-dimensional measure if and only if for some>0, everyf(d,k) points ofS see viaS a set ofk-dimensional measure at least .The numbers off(d,d) andf(d, 1) above are best possible.Supported in part by NSF grant DMS-8705336.