On the Darling-Mandelbrot probability density and the zeros of some incomplete gamma functions

Recently, Mandelbrot has encountered and numerically investigated a probability densityp _d (t) on the nonnegative reals, where, 0D<1. this=" density=" has=" fourier=" transform=">f _d (-is), wheref _d (z)=–Dz ^d (–D, z) and (·.·) is an incomplete gamma function. Previously, Darling had met this density, but had not studied its form. We expressf _d (z) as a confluent hypergeometric function, then locate and approximate its zeros, thereby improving some results of Buchholz. Via properties of Laplace transforms, we approximatep _d (t) asymptotically ast0+ and +, then note some implications asD0+ and 1–.Communicated by Mourad Ismail.     

On the relative Wall-Witt groups

LetR* be a simplicial involutive ring. According to certain involutions onK(R*) and _ε L R _∗, there are 1/2-local splittings and . It is known [2] that _ε L _\ga ^α R _∗, the Wall-Witt group. SupposeI→R S is a split extension of discrete involutive rings withI ²=0, andI is a freeS-bimodule. Then we have and . The trace map Tr: Prim _n ∧*M(I ⊗ ℚ)→ ₀ ρ _n ;I ⊗ ℚ) is an isomorphism. We prove in Lemma 1 that the trace map Tr is ℤ/2-equivariant. In Theorem 2 we show that under a certain assumption the rational relative Wall-Witt group vanishes. Theorem 2 can be extended to a more general case (Theorem 3) by employing Goodwillie’s reduction technique [3]. This work was partially supported by KOSEF under Grant 923-0100-010-1.     

Finite quasihypermetric spaces

Let (X, d) be a compact metric space and let (X) denote the space of all finite signed Borel measures on X. Define I: (X) → ℝ by I(μ) = ∫ _X ∫ _X d(x, y)dμ(x)dμ(y), and set M(X) = sup I(μ), where μ ranges over the collection of measures in (X) of total mass 1. The space (X, d) is quasihypermetric if I(μ) ≦ 0 for all measures μ in (X) of total mass 0 and is strictly quasihypermetric if in addition the equality I(μ) = 0 holds amongst measures μ of mass 0 only for the zero measure. This paper explores the constant M(X) and other geometric aspects of X in the case when the space X is finite, focusing first on the significance of the maximal strictly quasihypermetric subspaces of a given finite quasihypermetric space and second on the class of finite metric spaces which are L ¹-embeddable. While most of the results are for finite spaces, several apply also in the general compact case. The analysis builds upon earlier more general work of the authors [11] [13].      

The relation of dimension,entropy and Lyapunov exponent in random case

We consider random systems generated by two-sided compositions of random surface diffeomorphisms,together with an ergodic Borel probability measure μ.Let D(μω)be its dimension of the sample measure,then we prove a formula relating D(μω)to the entropy and Lyapunov exponents of the random system,where D(μω)is dimHμω,-/dinBμω,or-/dimBμω.     

Curvature, Diameter and Bounded Betti Numbers

In this paper, we introduce the notion of bounded Betti numbers, and show that the bounded Betti numbers of a closed Riemannian n-manifold (M, g) with Ric (M) ≥ -(n - 1) and Diam (M) ≤ D are bounded by a number depending on D and n. We also show that there are only finitely many isometric isomorphism types of bounded cohomology groups (H*(M), || · ||∞) among closed Riemannian manifold (M, g) with K(M) ≥-1 and Diam (M) ≤ D.     

On one representation of analytic functions by harmonic functions

Let u(x) be a function analytic in some neighborhood D about the origin, $ \mathcal{D} Let u(x) be a function analytic in some neighborhood D about the origin, ⊂ ℝ ⁿ . We study the representation of this function in the form of a series u(x) = u ₀(x) + |x|² u ₁(x) + |x|⁴ u ₂(x) + …, where u _k (x) are functions harmonic in . This representation is a generalization of the well-known Almansi formula. Original Russian Text ? V. V. Karachik, 2007, published in Matematicheskie Trudy, 2007, Vol. 10, No. 2, pp. 142–162.     

On the orders of finite semisimple groups

The aim of this paper is to investigate the order coincidences among the finite semisimple groups and to give a reasoning of such order coincidences through the transitive actions of compact Lie groups. It is a theorem of Artin and Tits that a finite simple group is determined by its order, with the exception of the groups (A₃(2), A₂(4)) and(B _n (q), C _n (q)) forn ≥ 3,q odd. We investigate the situation for finite semisimple groups of Lie type. It turns out that the order of the finite group H( ) for a split semisimple algebraic groupH defined over , does not determine the groupH up to isomorphism, but it determines the field under some mild conditions. We then put a group structure on the pairs(H ₁,H ₂) of split semisimple groups defined over a fixed field such that the orders of the finite groups H₁( ) and H₂( ) are the same and the groupsH _i have no common simple direct factors. We obtain an explicit set of generators for this abelian, torsion-free group. We finally show that the order coincidences for some of these generators can be understood by the inclusions of transitive actions of compact Lie groups.     

Asymptotic behavior of the irrational factor

We study the irrational factor function I(n) introduced by Atanassov and defined by , where is the prime factorization of n. We show that the sequence {G(n)/n} _n≧1, where G(n) = Π _ν=1 ⁿ I(ν)^1/n , is convergent; this answers a question of Panaitopol. We also establish asymptotic formulas for averages of the function I(n). Research of the third author is supported in part by NSF grant number DMS-0456615.     

The quasivariety generated by a torsion-free Abelian-by-finite group

Let L_q(qG) be the quasivariety lattice contained in a quasivariety generated by a group G. It is proved that if G is a finitely generated torsion-free group in (i.e., G is an extension of an Abelian group by a group of exponent 2ⁿ), which is a split extension of an Abelian group by a cyclic group, then the lattice L_q(qG) is a finite chain. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 407–427, July–August, 2007.     

Strong rightD-domains

An integral domainR is called rightD-domain if its lattice of all right ideals is distributive. In § 2 a sufficient condition for an integral domainR is given such thatR is a rightD-domain if and only ifR is a leftD-domain. For example each integral domain which is algebraic over its center satisfies this criterion. Furthermore, a rightD-domain is called strong if its lattice of all fractional right ideals is distributive. Examples of strong rightD-domains are given in §4. Each overring of a strong rightD-domain is also a strong rightD-domain whereas arbitrary rightD-domains may have overrings which are no rightD-domains. Section 3 is mainly concerned with the set * of all left invertible fractional right ideals and the mapping :**,II _l ^–1 whereI _l ^–1 denotes the left inverse ofI. For example, equivalent conditions are given for * to be a sublattice of and it is shown that is bijective if and only if (IJ)=(I)+(J) holds for allI,J*. Finally, §5 deals with (right)D-domains which are algebraic over their centersC. It is proved thatR is invariant if and only ifC is a commutative Prüfer domain andR the integral closure ofC inQ(R).     

Weighted mean convergence of Hakopian interpolation on the disk

In this paper, we study weighted mean integral convergence of Hakopian interpolation on the unit disk D. We show that the inner product between Hakopian interpolation polynomial Hn(f;x,y) and a smooth function g(x,y) on D converges to that of f(x,y) and g(x,y) on D when n →∞, provided f(x,y) belongs to C(D) and all first partial derivatives of g(x,y) belong to the space LipαM(0 ＜α≤ 1). We further show that provided all second partial derivatives of g(x,y) also belong to the space LipαM and f(x,y) belongs to C1 (D), the inner product between the partial derivative of Hakopian interpolation polynomial (6)/(6)xHn(f;x,y) and g(x,y) on D converges to that between (6)/(6)xf(x,y) and g(x,y) on D when n →∞.     

Mixed projection methods for systems of variational inequalities

Let H be a real Hilbert space. Let be bounded and continuous mappings where D(F) and D(K) are closed convex subsets of H. We introduce and consider the following system of variational inequalities: find [u ^*,v ^*]∈D(F) × D(K) such that This system of variational inequalities is closely related to a pseudomonotone variational inequality. The well-known projection method is extended to develop a mixed projection method for solving this system of variational inequalities. No invertibility assumption is imposed on F and K. The operators K and F also need not be defined on compact subsets of H.      

Finite Blaschke product and the multiplication operators on Sobolev disk algebra

Let R(D) be the algebra generated in Sobolev space W22(D) by the rational functions with poles outside the unit disk D. In this paper the multiplication operators Mg on R(D) is studied and it is proved that Mg ～ Mzn if and only if g is an n-Blaschke product. Furthermore, if g is an n-Blaschke product, then Mg has uncountably many Banach reducing subspaces if and only if n > 1.     

Remarks on the descriptive metric characterization of singular sets of analytic functions

This work presents two remarks on the structure of singular boundary sets of functions analytic in the unit diskD: |z|<1. The first remark concerns the conversion of the Plessner theorem. We prove that three pairwise disjoint subsetsE ₁,E ₂, andE ₃ of the unit circle Γ: |z|=1, = Γ, are the setsI(ƒ) of all Plessner points,F(ƒ) of all Fatou points, andE(ƒ) of all exceptional boundary points, respectively, for a function ƒ holomorphic inD if and only ifE ₁ is aG _δ-set andE ₃ is a -set of linear measure zero. In the second part of the paper it is shown that for any -subsetE of the unit circle Γ with a zero logarithmic capacity there exists a one-sheeted function onD whose angular limits do not exist at the points ofE and do exist at all the other points of Γ. Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 56–61, January, 1998.     

Radicals in pseudocomplemented lattices

For a pseudocomplemented latticeL, we prove that the filter D_n(L), 1n<, generated by then-strongly dense elements is contained in everyn-normal filter. Hence, D_n(L)=G_n(L)=Rad_n (L), where G_n(L) is the intersection of all n-normal filters, and Rad_n (L) is the intersection of alln-normal prime filters. Moreover, we prove that a prime filterP is n-normal iff D_n(L)=P. Consequently, for , we have D_n(L)=G_n(L)=Rad_n (L) and therefore iff Rad_n(L)={1} (or iff G_n(L)={1}).Considering the skeleton S(L) ofL, a complete clarification of the relationship between filters ofL and S(L) is given by studying th correspondence FFS(L).We state that D(L) (and that D₁(L), if is an irredundant intersection of maximal filters (resp. of *-maximal filters) iff S(L) is finite.Finally, for we state that the least *-congruence for which is that one generated by D_n(L).Presented by B. Jónsson.Research supported by the I.N.I:C, (Centro de Algebra da Universidade de Lisboa).     

On the dispersion spectrum

A measure for the denseness of sequences (an) mod 1, irrational, is the dispersion constantD() introduced byH. Niederreiter. In this paper the smallest accumulation point ₁ of the set of theD() is determined and all those are explicitely given for whichD () < ₁ holds.     

Lattices of dominions in quasivarieties of Abelian groups

Let M be any quasivariety of Abelian groups, (H) be the dominion of a subgroup H of a group G in M, and L_q(M) be the lattice of subquasivarieties of M. It is proved that (H ) coincides with a least normal subgroup of the group G containing H, the factor group with respect to which is in M. Conditions are specified subject to which the set L(G,H,M) = { (H) | N L_q(M)} forms a lattice under set-theoretic inclusion and the map : L_q(M) L(G,H,M) such that (N) = (H) for any quasivariety N L_q(M)is an antihomomorphism of the lattice L _q (M) onto the lattice L(G, H, M).__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 238–251, March–April, 2005.     

Proof of conjecture involving the second largest signless Laplacian eigenvalue and the index of graphs

Let G = (V, E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the signless Laplacian matrix of G is Q(G) = D(G) + A(G). In [5], Cvetkovi? et al. have given the following conjecture involving the second largest signless Laplacian eigenvalue (q₂) and the index (λ₁) of graph G (see also Aouchiche and Hansen [1]):

     

Global-in-time Uniform Convergence for Linear Hyperbolic-Parabolic Singular Perturbations

We consider the Cauchy problem εu^″ε ＋ δu′ε ＋ Auε = 0, uε（0） = uo, u′ε（0） = ul, where ε 〉 0, δ 〉 0, H is a Hilbert space, and A is a self-adjoint linear non-negative operator on H with dense domain D（A）. We study the convergence of （uε） to the solution of the limit problem ,δu＇ ＋ Au = 0, u（0） = u0. For initial data （u0, u1） ∈ D（A1/2）× H, we prove global-in-time convergence with respect to strong topologies. Moreover, we estimate the convergence rate in the case where （u0, u1）∈ D（A3/2） ∈ D（A1/2）, and we show that this regularity requirement is sharp for our estimates. We give also an upper bound for ｜u′ε（t）｜ which does not depend on ε.     

Closures of finitely generated ideals in Hardy spaces

LetH ^∞ be the algebra of bounded analytic functions in the unit diskD. LetI=I(f ₁,...,f _N) be the ideal generated byf ₁,...,f _N∈H ^∞ andJ=J(f ₁,...,f _N) the ideal of the functionsf∈H ^∞ for which there exists a constantC=C(f) such that |f(z)|≤C(|f ₁ (z)|+...;+|f _N (z)|),z∈D. It is clear that , but an example due to J. Bourgain shows thatJ is not, in general, in the norm closure ofI. Our first result asserts thatJ is included in the norm closure ofI ifI contains a Carleson-Newman Blaschke product, or equivalently, if there existss>0 such that