Necessary and sufficient conditions for inclusion relations for absolute summability

We obtain a set of necessary and sufficient conditions for to imply for 1 <k ≤ s < ∞. Using this result we establish several inclusion theorems as well as conditions for the equivalence of and .     

Asymptotic behavior of basic contact process with rapid stirring

In this paper we consider the basic contact process with infection rate λ and stirring rateD. We study the asymptotic behavior of the critical value and survival probability asD→∞.     

Martin boundary of unlimited covering surfaces

LetW be an open Riemann surface and ap-sheeted (1<p<∞) unlimited covering surface ofW. Denote by Δ¹ (resp., ) the minimal Martin boundary ofW (resp., ). For ζ ∈ Δ, let ζ be the (cardinal) number of the set of pionts which lie over ζ and the class of open connected subsetsM ofW such thatM∪{ζ} is a minimal fine neighborhood of ζ. Our main result is the following: , where is the number of components of π^-1 M and π is the projection of ontoW. Moreover, some applications of the above results are discussed whenW is the unit disc.     

Large superuniversal metric spaces

For every uncountable cardinal κ define a metric spaceS to be κ-superuniversal iff for every metric spaceU of cardinality κ, every partial isometry intoS from a subset ofU of cardinality less than κ can be extended to all ofU. We prove that any such space must have cardinality at least , and for each regular uncountable cardinal κ, we construct a κ-superuniversal metric space of cardinality , It is proved that there is a unique κ-superuniversal metric space of cardinality κ iff . Several decomposition theorems are also proved, e.g., every κ-superuniversal space contains a family of disjoint κ-superuniversal subspaces. Finally, we consider some applications to more general topological spaces, to graph theory, and to category theory, and we conclude with a list of open problems.     

Dispersion of Lagrangian trajectories in a random large-scale velocity field

We study the distribution of the distance R(t) between two Lagrangian trajectories in a spatially smooth turbulent velocity field with an arbitrary correlation time and a non-Gaussian distribution. There are two dimensionless parameters, the degree of deviation from the Gaussian distribution α and β=τD, where τ is the velocity correlation time and D is a characteristic velocity gradient. Asymptotically, R(t) has a lognormal distribution characterized by the mean runaway velocity and the dispersion Δ. We use the method of higher space dimensions d to estimate and Δ for different values of α and β. It was shown previously that for β≪ 1 and for β≫ 1. The estimate of Δ is then nonuniversal and depends on details of the two-point velocity correlator. Higher-order velocity correlators give an additional contribution to Δ estimated as αD²τ for β≪1 and αβ/τ for β≫1. For α above some critical value σ_cr, the values of and Δ are determined by higher irreducible correlators of the velocity gradient, and our approach loses its applicability. This critical value can be estimated as for β≪1 and for β≫1. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 3, pp. 456–467, March, 2000.     

Some properties of the absolute galois group of a hilbertian field

LetK be a hilbertian field,G(K) its absolute Galois group. IfK is countable, then for a.a. inG(K) ^e , and there is no intermediate field with . Let ∈G(K) ^e . Then for a.a. in .     

Inequalities for a distribution with monotone hazard rate

Summary LetX be a positive random variable with the survival function and the densityf. LetX have the moments μ=E(X) and μ₂=E(X ²) and put ε=|1-μ₂/2μ²|. Put and . It is proved that the following inequalities hold: , for allx>0, ifq(x) is monotone and that , ifq ₁ (x) is monotone. It is also shown that Brown's inequality which holds wheneverq ₁ (x) is increasing is not valid in general whenq ₁ is decreasing. The Institute of Statistical Mathematics     

The normal subgroup structure of the extended Hecke groups

We consider the extended Hecke groups generated by T(z) = −1/z, S(z) = −1/(z + λ) and R(z) = 1/z with λ ≥ 2. In this paper, firstly, we study the fundamental region of the extended Hecke groups . Then, we determine the abstract group structure of the commutator subgroups , the even subgroup , and the power subgroups of the extended Hecke groups . Also, finally, we give some relations between them.     

Baer cotorsion Pairs

LetR be a unital associative ring and two classes of leftR-modules. In [St3] the notion of a ( ) pair was introduced. In analogy to classical cotorsion pairs, a pair (V,W) of subclasses is called a ( ) pair if it is maximal with respect to the classes and the condition Ext _R ¹ (V, W)=0 for all . In this paper we study pairs whereR = ℤ and is the class of all torsion-free abelian groups andT is the class of all torsion abelian groups. A complete characterization is obtained assumingV=L. For example, it is shown that every pair is singly cognerated underV=L. The author was supported by a DFG grant.     

Elementary properties of free extensions

For any partial groupoid , let Fr be the free extension of to a total groupoid. We show that implies and that the theory of Fr is uniformly recursive in the theory of . These results fail if “groupoid” is replaced by “semigroup”, “commutative semigroup”, “group”, “abelian group”, “semilattice”, “K-lattice” for any nontrivial varietyK of lattices, or “Boolean algebra”. Research supported in part by NSF Grant MCS78-01867. We thank the referee for his valuable comments. Presented by B. Jónsson.     

Inverse problem for zeros of certain koebe-related functions

If w₁,…,w _N is a finite sequence of nonzero points in the unit disk, then there are distinct points λ₁,…, λ_N on the unit circle and positive numbers Μ₁,…,Μ _N such that is the zero sequence of the function 1 — . The points λ₁,…, λ_N and numbers Μ₁,…,Μ_N are unique (except for reorderings).     

Asymptotic behavior of the solutions of an integrodifferential system

Summary We consider the system(L): , t ⩾ p, y(t)=f(t), t⩽0, where y is an n-vector and each A_i, B(t) are n × n matrices. System(L) generates a semigroup by means of T_tf(s)=y (t+s, f), f(s) ∈ BC_l(− ∞, 0]. Under some hypotheses concerning the roots ofdet where is the Laplace transform of B(t), the asymptotic behavior of y(t) is discussed. Two typical results are: Theorem 3.1: suppose ∥B(t)∥ ɛ L₁[0, ∞), thendet forRe λ>0 iff for every ɛ>0 there is an M_ɛ>0 such that ∥T_tf∥_l ⩽ ⩽ M_ɛ exp [ɛt]∥f∥_l for t ⩾ 0. Corollary 3.1.1: suppose exp [at]B(t) ∈ ∈ L₁[0, ∞) for some a>0 anddet forRe λ>−a. Then the solution of(L) is exponentially asymptotically stable. Entrata in Redazione il 21 marzo 1975. The author is grateful to ProfessorC. Corduneanu for suggesting this problem and for many helpful discussions during the preparation of the paper.     

Eigenvalue estimate on a compact Riemann manifold

LetM be a compact Riemann manifold with the Ricci curvature ≽ - R(R = const. > 0) . Denote by d the diameter ofM. Then the first eigenvalue λ₁ ofM satisfies . Moreover if , then      

Equivariant D-modules attached to nilpotent orbits in a semisimple Lie algebra

Let _u be a compact Lie algebra and let _u be its complexification. Let ζ^−1/2 be the inverse on the set of regular elements of _u of a square root of the discriminant of . Generalizing a result of W. Lichtenstein in the case _u = (n, ℂ) or (nℝ), we prove that ∂(q).ζ^1/2 is non zero for all harmonic polynomialsq ∈S( ) \ {0}. This fact is deduced from results about equivariantD-modules supported on the nilpotent cone of .     

Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices

Let and be a perturbed eigenpair of a diagonalisable matrixA. The problem is to bound the error in and . We present one absolute perturbation bound and two relative perturbation bounds. The absolute perturbation bound is an extension of Davis and Kahan's sin θ Theorem from Hermitian to diagonalisable matrices. The two relative perturbation bounds assume that and are an exact eigenpair of a perturbed matrixD ₁ AD ₂ , whereD ₁ andD ₂ are non-singular, butD ₁ AD ₂ is not necessarily diagonalisable. We derive a bound on the relative error in and a sin θ theorem based on a relative eigenvalue separation. The perturbation bounds contain both the deviation ofD ₁ andD ₂ from similarity and the deviation ofD ₂ from identity. This work was partially supported by NSF grant CCR-9400921.     

Nonparametric estimation of an affinity measure between two absolutely continuous distributions with hypotheses testing applications

LetF andG denote two distribution functions defined on the same probability space and are absolutely continuous with respect to the Lebesgue measure with probability density functionsf andg, respectively. A measure of the closeness betweenF andG is defined by: . Based on two independent samples it is proposed to estimate λ by , whereF _n (x) andG _n (x) are the empirical distribution functions ofF(x) andG(x) respectively and and are taken to be the so-called kernel estimates off(x) andg(x) respectively, as defined by Parzen [16]. Large sample theory of is presented and a two sample goodness-of-fit test is presented based on . Also discussed are estimates of certain modifications of λ which allow us to propose some test statistics for the one sample case, i.e., wheng(x)=f ₀ (x), withf ₀ (x) completely known and for testing symmetry, i.e., testingH ₀:f(x)=f(−x).     

Remarks on dispersionless KP,KdV, and 2D gravity

Summary We use the SDiff(2) framework of Takasaki and Takebe and the (L, M) program (L is the Lax operator andMω=ω_λ) to show that =semiclassical limit ofM is , where ( ) are action angle variables in the Gibbons-Kodama theory of Hamilton-Jacobi type for dispersionless KP. We also show is the semiclassical limit ofWxW ⁻¹ (W is the gauge operator), whereG=WxW ⁻¹ is a quantity studied by the author in an earlier paper in connection with symmetries. We give then a semiclassical version of the Jevicki-Yoneya action principle for 2D gravity, where again arises in calculations, and this yields directly the Landau-Ginsburg equation that corresponds to the semiclassical limit of an integrated string equation. For KdV we also show how inverse scattering data are connected to Hamiltonians for dispersionless KdV. We also discuss Hirota bilinear formulas relative to the dispersionless hierarchies and establish various limiting formulas.     

Some proximity and sensitivity results in quadratic integer programming

We show that for any optimal solution for a given separable quadratic integer programming problem there exist an optimal solution for its continuous relaxation such that wheren is the number of variables and(A) is the largest absolute subdeterminant of the integer constraint matrixA. Also for any feasible solutionz, which is not optimal for the separable quadratic integer programming problem, there exists a feasible solution having greater objective function value and with . We further prove, under some additional assumptions, that the distance between a pair of optimal solutions to an integer quadratic programming problem with right hand side vectorsb andb, respectively, depends linearly on b–b₁. Finally the validity of all the results for nonseparable mixed-integer quadratic programs is established. The proximity results obtained in this paper are extensions of some of the results described in Cook et al. (1986) for linear integer programming.This research was partially supported by Natural Sciences and Engineering Research Council of Canada Grant 5-83998.     

The conformal radius as a function and its gradient image

Let Ω be a domain in with three or more boundary points in andR(w, Ω) the conformal, resp. hyperbolic radius of Ω at the pointw ε Ω/{∞}. We give a unified proof and some generalizations of a number of known theorems that are concerned with the geometry of the surface in the case that the Jacobian of ∇R(w, Ω), the gradient ofR, is nonegative on Ω. We discuss the function ∇R(w, Ω) in some detail, since it plays a central role in our considerations. In particular, we prove that ∇R(w, Ω) is a diffeomorphism of Ω for four different types of domains. This work was supported by a grant of the Deutsche Forschungsgemeinschaft for F. G. Avkhadiev.     

Forbidden paths and cycles in ordered graphs and matrices

Assume % MathType!End!2!1! and let Ω⊂R ^N(N≥4) be a smooth bounded domain, 0∈Ω. We study the semilinear elliptic problem: % MathType!End!2!1!. By investigating the effect of the coefficientQ, we establish the existence of nontrivial solutions for any λ>0 and multiple positive solutions with λ,μ>0 small.