Investigations on slow versus fast growing: How to majorize slow growing functions nontrivially by fast growing ones

Let T() be the ordinal notation system from Buchholz-Schütte (1988). [The order type of the countable segmentT()₀ is — by Rathjen (1988) — the proof-theoretic ordinal the proof-theoretic ordinal ofACA ₀ + ( ₁ ^l –TR).] In particular let _a denote the enumeration function of the infinite cardinals and leta ₀ a denote the partial collapsing operation on T() which maps ordinals of T() into the countable segment T ₀ of T(). Assume that the (fast growing) extended Grzegorczyk hierarchy and the slow growing hierarchy are defined with respect to the natural system of distinguished fundamental sequences of Buchholz and Schütte (1988) in the following way:

A law of the iterated logarithm for processes with independent increments

ＡＬＡＷＯＦＴＨＥＩＴＥＲＡＴＥＤＬＯＧＡＲＩＴＨＭＦＯＲＰＲＯＣＥＳＳＥＳＷＩＴＨＩＮＤＥＰＥＮＤＥＮＴＩＮＣＲＥＭＥＮＴＳＷＡＮＧＪＩＡＧＡＮＧ（汪嘉冈）（ＥａｓｔＣｈｉｎａＵｎｉｖｅｒｓｉｔｙｏｆＳｃｉｅｎｃｅ＆Ｔｅｃｈｎｏｌｏｇｙ,Ｓｈａｎｇ...     

Graph von Neumann Algebras

In this paper, we will define a graph von Neumann algebra over a fixed von Neumann algebra M, where G is a countable directed graph, by a crossed product algebra = M × _α , where is the graph groupoid of G and α is the graph-representation. After defining a certain conditional expectation from onto its M-diagonal subalgebra we can see that this crossed product algebra is *-isomorphic to an amalgamated free product where = vN(M × _α where is the subset of consisting of all reduced words in {e, e ^–1} and M × _α is a W ^*-subalgebra of as a new graph von Neumann algebra induced by a graph G _e . Also, we will show that, as a Banach space, a graph von Neumann algebra is isomorphic to a Banach space ⊕ where is a certain subset of the set E(G)* of all words in the edge set E(G) of G. The author really appreciates to Prof F. Radulescu and Prof P. Jorgensen for the valuable discussion and kind advice. Also, he appreciates all supports from St. Ambrose Univ.. In particular, he thanks to Prof T. Anderson and Prof V. Vega for the useful conversations and suggestions.     

On vector-valued random Fourier series

Consider a (complex) Banach spaceX, such thatX C_O, and vectors(X _i ) _i ofX. Consider an independent standard normal sequence(g _i ) _i . Then if anX-valued random Fourier series _{|k| n} e ^ikt g _k x _k satisfies

On the computation of modules of long quadrilaterals

We study quadrilateralsQ which are given by two intervals on {:Im = 0} and {:Im = 1}, and two Jordan arcs ₁, ₂, in {:0 Im 1} connecting these two intervals. Many practical problems require the determination of the modulem(Q) ofQ, but ifQ is long, i.e., if

Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents

We are concerned with the semilinear polyharmonic model problem (–)K v = v +v|v| ^s–1 inB,D v|B = 0 for ¦|<-K – 1. HereK ,B is the unit ball in ⁿ,n >2K, is the critical Sobolev exponent. Let ₁ denote the first Dirichlet eigenvalue of (-)^K inB. The existence of a positive radial solutionv is shown for

On realizations related to Weyl operators

The paper deals with the minimal and the maximal realizations (L ^w )^~ and (L ^w ):L ₂L ₂ of linear operators of Weyl type

On <Emphasis Type="Italic">N</Emphasis>-Summations,II

Given any R-semimodule M equipped with a semitopology we construct an N-protosummation for M. If satisfies certain properties, then a similar construction leads to an unconditional N-summation for M, that is an N-summation for M equipped with the trivial prenorm MD over the N-summation (D^N,_D) for D. Conversely any N-protosummation on M gives rise to a topology . If both and satisfy a certain separation property, then and form a Galois connection. Dedicated to my friend and collegue Nico Pumplün on the occasion of his 70th birthdayMathematics Subject Classifications (2000) 16Y60, 54A05.     

О росте вдоль прямой ц елых функций экспоне нциального типа с заданными нулями

The following result is proved. Let=_n} be a sequence of complex numbers with ¦Re _n¦¦ _n ¦, >0, and letg be an entire function of exponential type with a sequence of zeros which satisfies the same condition. There exists an entire function of exponential typef0 such thatf()=0 and ¦f(iy)¦¦g(iy)¦,yR, if and only if there exists a constantM such that for all numbersr andR, 0rR<>, we have

Lattices of interpretability types of varieties

Let be the set of all primes, the field of all algebraic numbers, and Z the set of square-free natural numbers. We consider partially ordered sets of interpretability types such as , and , where AD is a variety of -divisible Abelian groups with unique taking of the pth root _p(x) for every p , is a variety of -modules over a normal field , contained in , and G_n is a variety of n-groupoids defined by a cyclic permutation (12 ...n). We prove that , and are distributive lattices, with and where ub and ub_f are lattices (w.r.t. inclusion) of all subsets of the set and of finite subsets of , respectively.Deceased.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 198–210, March–April, 2005.     

Realizations of regular polytopes

Let be a finite regular incidence-polytope. A realization of is given by an imageV of its vertices under a mapping into some euclidean space, which is such that every element of the automorphism group () of induces an isometry ofV. It is shown in this paper that the family of all possible realizations (up to congruence) of forms, in a natural way, a closed convex cone, which is also denoted by The dimensionr of is the number of equivalence classes under () of diagonals of , and is also the number of unions of double cosets ^{** *–1*} ( ^*), where ^* is the subgroup of () which fixes some given vertex of . The fine structure of corresponds to the irreducible orthogonal representations of (). IfG is such a representation, let its degree bed _G , and let the subgroup ofG corresponding to ^* have a fixed space of dimensionw _G . Then the relations

Universal property of chern character forms of the canonical connection

Let _q,n denote the complex Stiefel bundle over the complex Grassmannian and let ₀ be the universal connection on this bundle. Consider the Chern character form of ₀ defined by the formula where ₀ is the curvature form of the connection ₀. Let M be a manifold of dimension m and a closed 2k-form on M. Suppose, there exists a continuous map which pulls back the cohomology class of ch_k(₀) onto the cohomology class of . We prove that if q and n are greater than certain numbers (which we determine in this paper) then there exists a smooth map such that f*ch_k(₀) = .     

New closure conditions and some problems in loop theory

A loopQ(·) is said to be anA _l-loop (A _r-loop) if x, y Q, l _x,y AutQ (r _x,y AutQ) hold, where

Asymptotic behavior of the spectrum of a pseudodifferential operator with periodic bicharacteristics

Let _j be the eigenvalues of a positive elliptic pseudodifferential operator of order m > 0 on a closed compact d-dimensional C-manifold and let N()=#{j:_j^m}. It is shown that for each > 0 we have

Four-dimensional compact projective planes with doubly transitive action on the fixed pencil

We consider a four-dimensional compact projective plane =( , ) whose collineation group is six-dimensional and solvable with a nilradical N isomorphic to Nil × R, where Nil denotes the three-dimensional, simply connected, non-Abelian, nilpotent Lie group. We assume that fixes a flag pW, acts transitively on _p \{W}, and fixes no point in the set W{p}. We study the actions of and N on and on the pencil _p \{W}, in the case that does not contain a three-dimensional elation group. In the special situation that acts doubly transitively on _p {W}, we will determine all possible planes . There are exactly two series of such planes.     

An explicit Lévy-Hinčin formula for convolution semigroups on locally compact groups

LetG be a locally compact group and (_t)_{t 0} a continuous convolution semigroup of probability measures onG. We show that an operatorN is the infinitesimal generator of (_t)_{t 0} iffN is defined at least on the spaceC ₂(G) of twice right differentiable functions and if

Asymptotics of polynomials orthogonal with respect to varying measures

Letd be a finite positive Borel measure on the interval [0, 2] such that >0 almost everywhere; andW _n be a sequence of polynomials, degW _n =n, whose zeros (w _n ,1,,w _n,n lie in [|z|1]. Let d _n <> for each_nN, whered _n =d/|W _n (e ⁱ )|². We consider the table of polynomials _n,m such that for each fixednN the system _n,m,mN, is orthonormal with respect tod _n . If

An Extension of the Admissibility-Type Conditions for the Exponential Dichotomy of C 0-Semigroups

In the present paper we obtain a sufficient condition for the exponential dichotomy of a strongly continuous, one-parameter semigroup , in terms of the admissibility of the pair . It is already known the equivalence between the -admissibility condition and and the hyperbolicity of a C ₀-semigroup , when we assume a priori that the kernel of the dichotomic projector (denoted here by X ₂) is T(t)-invariant and is an invertible operator. We succeed to prove in this paper that the admissibility of the pair still implies the existence of an exponential dichotomy for a C ₀-semigroup even in the general case where the kernel of the dichotomic projector, X ₂, is not assumed to be T(t)-invariant.      

Wiener-Hopf operators on L_{\omega}^{2}(\mathbb{R}^{+})

Let be a weighted space with weight . In this paper we show that for every Wiener-Hopf operator T on and for every a I, there exists a function such that

Some numerical results on best uniform rational approximation ofx α on [0,1]

Let be a positive number, and letE _n,n (x ;[0,1]) denote the error of best uniform rational approximation from _n,n tox on the interval [0,1]. We rigorously determined the numbers {E _n,n (x ;[0,1])} _n ^=1/30 for six values of in the interval (0, 1), where these numbers were calculated with a precision of at least 200 significant digits. For each of these six values of , Richardson's extrapolation was applied to the products to obtain estimates of