Semilattices of left reversible semigroups

Let S be a cancellative semigroup which is a semilattice of left reversible semigroups S, . This article studies the relationship between the group of quotients G of S and the groups of quotients G of S, . It is shown that G is the maximum group homomorphic image of an inverse semigroup which is a semilattice of groups G (up to isomorphism).The technique used here which involves the use of Ore's quotients also applies to the study of the maximum group homomorphic image of a semigroup which is a semilattice of inverse semigroups.     

Inner functions and related spaces of pseudocontinuable functions

Let be an inner function, let C, ¦¦=1. Then the harmonic function [(+)]/(–)] is the Poisson integral of a singular measure D. N. Clark's known theorem enables us to identify in a natural manner the space H² H² with the space L² ( ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 7–33, 1989.     

Properties of certain integral operators

Two integral operatorsP andQ for analytic functions in the open unit disk are introduced. The object of the present paper is to derive some properties of integral operatorsP andQ .     

Sum of Divisors in a Ring of Gaussian Integers

We construct an asymptotic formula for a sum function for _a (), where _a () is the sum of the ath powers of the norms of divisors of the Gaussian integer on an arithmetic progression ₀ (mod ) and in a narrow sector ₁ arg < ₂. For this purpose, we use a representation of _a (n) in the form of a series in the Ramanujan sums.     

Concretely Locally Presentable and Locally Generated Categories

We give concrete versions of the characterizations of locally -presentable (resp. -generated) categories as categories of models of (resp. weakly) -ary limit-theories and of (resp. weakly) -ary essentially algebraic theories. By concrete we mean that starting with a category of -structures, the theories obtained are extensions of the original ones, and the equivalences of categories are concrete isomorphisms. -presentable and -generated objects are also investigated from this viewpoint.     

Growth of length of sequential derivation transformed into natural one

We consider the (&, )-fragment of the intuitionistic propositional calculus. It is proved that under the standard transformation of a Gentzen derivation into a natural derivation(), the length of (())2^2·length( ). There is constructed a sequence of Gentzen derivations of length _i, for which the length of (( _i))2^{1/3·length(i}), which shows that the upper bound obtained is not too weak.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 192–196, 1979.     

Partially ordered permutation groups

In this note we discuss permutation groups (G, ) in which the set admits aG-invariant order. By aG-invariant partial order (G-partial order) we mean a partial order < of such that < implies g<g, for all and in andg inG. If the set admits aG-partial order which is a total order, then (G, ) is an O-permutation group (orderable permutation group).The main concern of this paper is the development of a foundation for partially ordered permutation groups analogous to the existing one for partially ordered groups, as found in Fuchs [2].     

Lower functions for asymmetric Lévy processes

Summary Let {X _t } be a ¹ process with stationary independent increments and its Lévy measurev be given byv{yy>x}=x ^–L ₁ (x), v{yy<–x}=x ^–L ₂ (x) whereL ₁,L ₂ are slowly varying at 0 and and 0<1. We construct two types of a nondecreasing functionh(t) depending on 0<<1 or =1 such that lim inf a.s. ast 0 andt for some positive finite constantC.This research is partialy supported by a grant from Korea University     

Normalformen für Bewegungen in hyperbolischen Räumen

If K is an Euclidean field and f a symmetric bilinear form of index 1, there is for each element O₄ (K,f) an orthogonal decomposition of the corresponding metric vector space V₄(K,f) into — invariant subspaces U_i such that dim U_i 3. Thus it is possible to determine nomal form for and to decompose the characteristic polynomial X into irreducible factors.     

On Fractional Brownian Processes

We use Liouville spaces in order to prove the existence of some different fractional -Brownian motion ( 0 < 1 ), or fractional ( , )-Brownian sheets. There are also applications to the Wiener stochastic integral with respect to these -Brownian.     

On the weak convergence of stochastic processes without discontinuities of the second kind

Summary Subspaces D , > 0, of D[0, 1] are defined and given complete metrics d which are stronger than the Prokhorov metric. The spaces (D d ) are shown to be separable, and their pre-compact subsets are characterized. A condition which is known to guarantee weak pre-compactness of sets of probability measures over D[0, 1] is shown to also guarantee weak pre-compactness of probability measures over D for appropriate values of . Applications are made to the weak convergence of measures induced by stochastic processes, and some examples are included.     

Parabolic subgroups of Chevalley groups over a semilocal ring

Let G be the Chevalley group over a commutative semilocal ring R which is associated with a root system . The parabolic subgroups of G are described in the work. A system =() of ideals in R ( runs through all roots of the system ) is called a net of ideals in the commutative ring R if ₊ for all those roots and for which + is also a root. A net is called parabolic if =R for >0. The main theorem: under minor additional assumptions all parabolic subgroups of G are in bijective correspondence with all parabolic nets . The paper is related to two works of K. Suzuki in which the parabolic subgroups of G are described under more stringent conditions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 43–58, 1978.     

Eulerian numbers with fractional order parameters

Summary The aim of this paper is to generalize the well-known Eulerian numbers, defined by the recursion relationE(n, k) = (k + 1)E(n – 1, k) + (n – k)E(n – 1, k – 1), to the case thatn is replaced by . It is shown that these Eulerian functionsE(, k), which can also be defined in terms of a generating function, can be represented as a certain sum, as a determinant, or as a fractional Weyl integral. TheE(, k) satisfy recursion formulae, they are monotone ink and, as functions of , are arbitrarily often differentiable. Further, connections with the fractional Stirling numbers of second kind, theS(, k), > 0, introduced by the authors (1989), are discussed. Finally, a certain counterpart of the famous Worpitzky formula is given; it is essentially an approximation ofx in terms of a sum involving theE(, k) and a hypergeometric function.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

On a generalized Eulerian distribution

The distribution with probability function p _k(n, , ) = A _{n, k}(, )/(+ )^[p], k = 0, 1, 2, ..., n, where the parameters and are positive real numbers, A _{n, k} (, ) is the generalized Eulerian number and ( + )^[n] = ( + )( + +1) ... ( + +n – 1), introduced and discussed by Janardan (1988, Ann. Inst. Statist. Math., 40, 439–450), is further studied. The probability generating function of the generalized Eulerian distribution is expressed by a generalized Eulerian polynomial which, when expanded suitably, provides the factorial moments in closed form in terms of non-central Stirling numbers. Further, it is shown that the generalized Eulerian distribution is unimodal and asymptotically normal.     

A chord-stretching map of a convex loop is an isometry

Let ⁿ be n-dimensional Euclidean space, and let : [0, L] ⁿ and : [0, L] ⁿ be closed rectifiable arcs in ⁿ of the same total length L which are parametrized via their arc length. is said to be a chord-stretched version of if for each 0s tL, |(t)–(s)| |(t)–(s)|. is said to be convex if is simple and if ([0, L]) is the frontier of some plane convex set. Individual work by Professors G. Choquet and G. T. Sallee demonstrated that if were simple then there existed a convex chord-stretched version of . This result led Professor Yang Lu to conjecture that if were convex and were a chord-stretched version of then and would be congruent, i.e. any chord-stretching map of a convex arc is an isometry. Professor Yang Lu has proved this conjecture in the case where and are C ² curves. In this paper we prove the conjecture in general.     

A Periodicity Theorem in K-Theory for Finite Covariant Systems

Let A be a complex Banach algebra. In this paper, we prove a periodicity theorem that K(S ² A, G, )K(A, G,), where (A,G,) is a finite covariant system and K(A,G,) is an Abelian group associated with (A,G,).     

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.     

On cyclic biquadratic fields related to a problem of Hasse

In this note we shall prove that there exist infinitely many cyclic biquadratic fieldsK whose integral bases are neither {1, , ², } nor {1, , , ³) for any numbers , inK. Next, we shall construct infinitely many cyclic biquadratic fieldsK which have the index 1, but still have not the integral basis {1, , ², ³) for every inK. Finally we shall give a class of biquadratic fields for a problem of Hasse concerning an integral basis.     

Percentage Points of the Largest Among Student's T Random Variable

Let us consider k( 2) independent random variables U₁, . . . ,U_k where U_i is distributed as the Student's t random variable with a degree of freedom m_i, i=1, . . . ,k. Here, m₁, . . . ,m_k are arbitrary positive integers. We denote m=(m₁, . . . ,m_k) and U_k:k=max {U₁, . . . ,U_k}, the largest Student's t random variable. Having fixed 0< <1, let a a(k,) and h_m h_m (k,) be two positive numbers for which we can claim that (i) ^k(a)–^k(–a)=1–, and (ii) P{–h_m U_k:k h_m}=1–. Then, we proceed to derive a Cornish–Fisher expansion (Theorem 3.1) of the percentage point h_m. This expansion involves a as well as expressions such as _i=1 ^k m_i ^–1, _i=1 ^km_i ^–2, and _i=1 ^k m_i ^–3. The corresponding approximation of h_m is shown to be remarkably accurate even when k or m₁, . . . ,m_k are not very large.     

Untere und obere Schranken für den tiefsten Eigenwert der elastisch gestützten Membran

Zusammenfassung Es werden untere und obere Schranken für den tiefsten Eigenwert ₁() der elastisch gestützten schwingenden Membran hergeleitet. Die elastische Bindung der Membran am Rande wird durch charakterisiert, und wird als Parameter betrachtet.Die Verwendung des klassischen Rayleigh-Prinzipes liefert obere Schranken, mit Hilfe eines konvexen FunktionalsJ() erhält man obere und untere Schranken. Eine Zerlegungsmethode endlich gibt eine untere Schranke für ₁().

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Summary This article is concerned with the determination of upper and lower bounds for the lowest eigenvalue ₁() of the elastically supported vibrating membrane. The elastic support on the boundary is characterized by which is regarded as a parameter.The classical Rayleigh-Principle gives upper bounds. The use of a convex functionalJ() yields upper and lower bounds for ₁(). A method of decomposition leads to a lower bound for ₁().

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Neu-Technikum, Buchs SG