Supplementing Radicals and Decompositions of Near-Rings

If is a radical of near-rings and is its supplementing radical, then (N)(N) N. We address the issue when (N) (N) = N holds. In the variety F of near-rings in which the constants form an ideal, the assignment c: N N_c is a hereditary Kurosh–Amitsur radical, _c is characterized in terms of distributors and criteria are given for the decomposition N = c(N) c(N). In the subvariety A of all abstract affine near-rings, assigning the maximal torsion ideal (N) is a hereditary Kurosh–Amitsur radical. If such near-rings N A satisfy dcc on principal right ideals, then N splits into a direct sum N = (N) (N) where the additive group of (N) is torsionfree and divisible. Dropping dcc on principal right ideals, an ``essential" decomposition result is proved.     

On σ-discrete borel mappings via quasi-metrics

Let X and Y be metrizable spaces. We show that, for a mapping f : X Y, there exists a quasi-metric X inducing the topology of X such that f regarded as a mapping from (X, max{, ^–1}) to Y is continuous if and only if f in the original topology of X is a -discrete map of Borel class 1. Further, we prove that, for every -discrete mapping f: X Y of Borel class + 1, there exists a compatible quasi-metric on X such that f : (X, max{, ^–1}) Y is of Borel class . We also investigate a more general situation when the range of the mapping under consideration is not necessarily metrizable. In passing, we obtain some results related to the behaviour of absolutely Borel sets and absolutely analytic spaces with respect to compatible quasi-metrics.     

Das Zeitverhalten von einfachen offenen exponentiellen Warteschlangensystemen mit unendlich vielen Warteplätzen

Zusammenfassung Die zeitabhängige (instationäre) Lösung für die Zustandswahrscheinlichkeiten und für einige Kenngrößen von Warteschlangensystemen mit einer Bedienungsstation, unendlich vielen Warteplätzen, exponentiellem Zu- und Abgang und beliebigem Anfangszustand wird bestimmt. Die ZustandswahrscheinlichkeitenP _v (), d. h. die Wahrscheinlichkeiten für Einheiten im System zur Zeit, ergeben sich als Integrale, in denen modifizierteSessel-Funktionen 1. Art auftreten. Der ErwartungswertL () und die VarianzV() der Zahl von Einheiten im System lassen sich als Integrale darstellen, in denen nur die ZustandswahrscheinlichkeitP ₀() auftritt.Für<1 und erreichen die Systeme einen stationären Zustand (für den die Lösung bekannt ist); für1 und giltP _v ()0 für alle, L(),V().Ist>1, dann wachsenL() undV() für große linear mit; ihre Asymptoten werden berechnet. Ist=1, dann wachsenL() und die Standardabweichung() für große mit ; einfache Näherungsformeln werden gefunden.

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Summary The time dependent solution is determined for the state probabilities and for some characteristic values of queuing systems with a single server, an infinite number of waiting places, exponentially distributed inter-arrival and service times, and any initial state. The state probabilitiesP _v (), i.e. the probabilities for units in the system at time, are given in the form of integrals in which modifiedBessel functions of the first kind occur. Integrating the state probalityP ₀() over leads to the meanL() and the varianceV() of the number of units in the system.For<1 and the systems tend to a steady state (for which the solution is known); for1 and we haveP _v ()0 for all, L(),V().If>1 asymptotic expansions for large are found givingL() andV() proportional to. If=1 simple approximate formulas for large are obtained givingL() and the standard deviation() proportional to .

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Vorgel. v.:J. Nitsche.     

A Full Characterization of Multipliers for the Strong ϱ-Integral in the Euclidean Space

We study a generalization of the classical Henstock-Kurzweil integral, known as the strong -integral, introduced by Jarník and Kurzweil. Let be the space of all strongly -integrable functions on a multidimensional compact interval E, equipped with the Alexiewicz norm We show that each element in the dual space of can be represented as a strong -integral. Consequently, we prove that fg is strongly -integrable on E for each strongly -integrable function f if and only if g is almost everywhere equal to a function of bounded variation (in the sense of Hardy-Krause) on E.     

A metric characterisation of compact plane retracts and applications

Summary For compact sets M R², investigating the problem of representing functions V: R² R⁺ in the form V(x)=(x, M), where is a suitable metric on R², a characterisation of retracts is obtained. As for applications, two results in topological dynamics are given.     

Spitzer's condition and ladder variables in random walks

Summary Spitzer's condition holds for a random walk if the probabilities _n =P{ _n > 0} converge in Cèsaro mean to , where 0<<1. We answer a question which was posed both by Spitzer [12] and by Emery [5] by showing that whenever this happens, it is actually true that n converges to . This also enables us to give an improved version of a result in Doney and Greenwood [4], and show that the random walk is in a domain of attraction, without centering, if and only if the first ladder epoch and height are in a bivariate domain of attraction.     

Strong laws of large numbers for random walks associated with a class of one-dimensional convolution structures

We derive strong laws of large numbers for birth and death random walks and random walks on polynomial hypergroups for which the coefficients of the three-term-recurrence formula of the associated orthogonal polynomials satisfy lim _n n ^a (a _n-c_n)= wherea]0, 1[ and >0. We also present these laws for random walks on Sturm-Liouville hypergroups on ₊ for which a corresponding asymptotic condition holds. Our paper supplements articles ofVoit [9] andZeuner [14] in which the casesa=0 anda=1 are considered.This paper was written at Murdoch University in Western Australia while the author held a Feodor Lynen fellowship of the Alexander von Humboldt foundation.     

On sums of two coprimek-th powers

Under the assumption of the Riemann Hypothesis, an asymptotic formula with a sharp error term is established for _{nx k} (n), where _k (n) denotes the number of ways to writen as a sum of twok-th powers of coprime positive integers (k3).     

On summability of weighted Lagrange interpolation. I

The paper is devoted to the study of summability of weighted Lagrange interpolation on the roots of orthogonal polynomials with respect to a weight function w. Starting from the Lagrange interpolation polynomials we shall construct a wide class of discrete processes (using summations) which are uniformly convergent in a suitable Banach space (C,·) of continuous functions (w denotes a weight). We shall give such conditions with respect to w, , (C,·) and to summation methods for which the uniform convergence holds. Error estimates for the approximation will also be considered.     

On Some Proximinal Subspaces of Modular Spaces

Let (, , ) be a complete measure space, L⁰ the vector lattice of -measurable real functions on , : L⁰ [0, )] a lattice semimodular, the corresponding modular space, S₀ the ideal generated by and 0,{\text{ }}\exists {\text{ }}s \in {\text{ }}S_{\text{0}} {\text{ such that }}\rho \left( {\frac{{x - s}}{\user1{\lambda }}} \right) < \infty } \right\}$$ " align="middle" border="0"> . In X consider the distance 0:\rho \left( {\frac{{x - y}}{\user1{\lambda }}} \right) \leqq \user1{\lambda }} \right\}$$ " align="middle" border="0"> and, if is convex, the distances d_L, d_o subordinated to the Luxemburg and Amemiya-Orlicz norms, respectively. We give necessary and sufficient conditions for H(S_o) in order to be proximinal in X with the distances d, d_L and d_o.     

Hecke Algebras of Classical Groups over p-adic Fields II

In the previous part of this paper, we constructed a large family of Hecke algebras on some classical groups G defined over p-adic fields in order to understand their admissible representations. Each Hecke algebra is associated to a pair (J , ) of an open compact subgroup J and its irreducible representation which is constructed from given data = (, P₀, ). Here, is a semisimple element in the Lie algebra of G, P₀ is a parahoric subgroup in the centralizer of in G, and is a cuspidal representation on the finite reductive quotient of P₀. In this paper, we explicitly describe those Hecke algebras when P₀ is a minimal parahoric subgroup, is trivial and is a character.     

A horn sentence for involution lattices of quasiorders

The quasiorders of a setA form a lattice Quord(A) with an involution ^–1={x, y: y, x}. Some results in [1] and Chajda and Pinus [2] lead to the problem whether every lattice with involution can be embedded in Quord(A) for some setA. Using the author's approach to the word problem of lattices (cf. [3]), which also applies for involution lattices, it is shown that the answer is negative.Research supported by the Hungarian National Foundation for Scientific Research (OTKA), under grant no. T 7442.     

On monodromy of complex projective structures

Summary We prove that for any nonelementary representation : ₁(S SL (2, )) of the fundamental group of a closed orientable hyperbolic surfaceS there exists a complex projective structure onS with the monodromy .Oblatum IV-1993 & 24-IV-1994     

Density functions for prime and relatively prime numbers

Letr ^*(x) denote the maximum number of pairwiserelatively prime integers which can exist in an interval (y,y+x] of lengthx, and let ^*(x) denote the maximum number ofprime integers in any interval (y,y+x] whereyx. Throughout this paper we assume the primek-tuples hypothesis. (This hypothesis could be avoided by using an alternative sievetheoretic definition of ^*(x); cf. the beginning of Section 1.) We investigate the differencer ^*(x)—^*(x): that is we ask how many more relatively prime integers can exist on an interval of lengthx than the maximum possible number of prime integers. As a lower bound we obtainr ^*(x)—^*(x)<x ^c for somec>0 (whenx). This improves the previous lower bound of logx. As an upper bound we getr ^*(x)—^*(x)=o[x/(logx)²]. It is known that ^*(x)—(x)>const.[x/(logx)²];.; thus the difference betweenr ^*(x) and ^*(x) is negligible compared to ^*(x)—(x). The results mentioned so far involve the upper bound or maximizing sieve. In Section 2, similar comparisons are made between two types of minimum sieves. One of these is the erasing sieve, which completely eliminates an interval of lengthx; and the other, introduced by Erdös and Selfridge [1], involves a kind of minimax for sets of pairwise relatively prime numbers. Again these two sieving methods produce functions which are found to be closely related.     

Cocycles d'Euler et de Maslov

Conclusion Nous espérons avoir convaincu le lecteur qu'il peut être utile de considérer la classe de Maslov comme une classe bornée. Dans [Gh], nous avons montré que la classe d'Euler bornée pour un groupe d'homéomorphismes directs du cercle rend compte de la dynamique topologique de ce groupe. Existe-t-il un résultat analogue pour Sp(2n,)? En d'autres termes, soit un groupe discret et ₁, ₂ deux représentations de dans Sp(2n,). On suppose que les cocycles ₁ ^* et ₂ ^* définissent la même classe bornée. Que peut-on en conclure sur ₁ et ₂?Par ailleurs, l'article [At l] traite aussi d'invariants sur SL(2,) différents de ceux que nous avons considérés, comme par exemple les fonctionsL de Shimizu. Est-il possible de les faire rentrer naturellement dans notre cadre?

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Singularities of the Green function of a random walk on a discrete group

LetX be a countable discrete group and let be an irreducible probability onX. The radius of convergence of the Green function is finite, and independent ofx. Let 0} \right\}$$ " align="middle" border="0"> be the period of . We show that for eachxX the singularities of the analytic functionzG(x; z) on the circle {z:|z|=} are precisely the points e ^2ik/d k=0, ...,d–1. In particular, is the only singularity on the circle in the aperiodic cased=1 (which occurs, for example, when (e)>0). This affirms a conjecture ofLalley [5]. When is symmetric, i.e., (x ^–1)=(x) for allxX, d is either 1 or 2. As another particular case of our result, we see that- is then a singularity ofzG (x; z) if and only ifd=2, in which caseX is bicolored. This answers a question ofde la Harpe, Robertson andValette [2].     

Schallwellen in Gasen mit festen Teilchen

Summary Small solid particles contained in a gas are unable to follow rapid changes of velocity and temperature of the gas immediately. These relaxation effects give rise to dispersion and attenuation of sound waves passing through the mixture. These phenomena are treated in the case in which the mean state of the flow is a constant equilibrium state as well as in two special cases in which the mean velocity difference of both media does not vanish. Using these results, a condition is derived which has to be satisfied for shock waves to be possible in the mixture.

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Bezeichnungen a Schallgeschwindigkeit im Gas - b gefrorene Schallgeschwindigkeit in der Mischung, in der im ungestörten Zustand Geschwindgikeitsgleichgewicht herrscht - c Gleichgewichtsschallgeschwindigkeit in der Mischung - c ₁ teilgefrorene Schallgeschwindigkeit bei Geschwindigkeitsgleichgewicht - c ₂ teilgefrorene Schallgeschwindigkeit bei Temperaturgleichgewicht in der Mischung, in der im ungestörten Zustand Geschwindigkeitsgleichgewicht herrscht - c _p isobare spezifische Wärme des Gases - c _P spezifische Wärme des Teilchenmaterials - c _w Widerstandskoeffizient - E=/1– Verhältnis der Volumenanteile von Teilchen und Gas - h _G Enthalpie pro Masseneinheit des Gases - k=k _r +i k _i Wellenzahl - Nu Nusselt-Zahl - p Gesamtdruck, bzw. Gasdruck - Pr Prandtl-Zahl des Gases (Pr=c _p /) - r Teilchenradius - Re _P Reynolds-Zahl für die Teilchen (Rep=2 _G |u–|/u) - s _G Entropie pro Masseneinheit des Gases - s _P Entropie pro Masseneinheit des Teilchenmaterials - s _M Entropie pro Masseneinheit der Mischung bei Geschwindigkeitsgleichgewicht - t Zeit - T _G Temperatur des Gases - T _P Temperatur der Teilchen - u Gasgeschwindigkeit - v Teilchengeschwindigkeit - x Ortskoordinate - Verhältnis zwischen isobarer und isochorer spezifischer Wärme des Gases - =c _P /c _p Verhältnis der spezifischen Wärme des Teilchenmaterials zur isobaren spezifischen Wärme des Gases - Volumenanteil der Teilchen - Wärmeleitfähigkeit des Gases - dynamische Scherzähigkeit des Gases - _G=1/_G spezifisches Volumen des Gases - _G Dichte des Gases, bezogen auf das Gasvolumen - _P Dichte des Teilchenmaterials - =_P/_G Verhältnis der beiden Materialdichten - ₁ Relaxationszeit für den Geschwindigkeitsausgleich durch Reibung zwischen Gas und Teilchen - ₂ Relaxationszeit für den Temperaturausgleich durch Wärmeübergang zwischen Gas und Teilchen - Frequenz     

Nonlinear semigroups and age-dependent population models

Summary This paper treats the nonlinear age-dependent population problem (1)(0,a)=(a), a I; (2)(t, 0)=F((t, ·)), t0; (3) ,t0,where I is the age range of the population, (t, ·) is the unknown age density at time t, is the known initial age distribution, and the functionals F and G are nonlinear. The problems of existence, uniqueness, continuous dependence upon initial values, and the positivity of solutions are investigated using the method of nonlinear semigroups.Supported in part by the National Science Foundation Grant NSF 75-06332A01.     

Large-Deviation Local Theorem for Additive Functionals of a Markov Gaussian Field. I

We prove a local limit theorem (LLT) on Cramer-type large deviations for sums S _V = _t V ( _t ), where _t , t Z , 1, is a Markov Gaussian random field, V Z , and is a bounded Borel function. We get an estimate from below for the variance of S _V and construct two classes of functions , for which the LLT of large deviations holds.     

Symmetric balanced ternary designs with ϱ1 = 1 or 2

Summary We prove the following two non-existence theorems for symmetric balanced ternary designs. If ₁ = 1 and 0 (mod 4) then eitherV = + 1 or 4₂ – + 1 is a square and (4₂ – + 1) divides ² – 1. If ₁ = 2 thenV = ((m + 1)/2) ² + 2,K = (m ² + 7)/4 and = ((m – 1)/2)² + 1 wherem 3 (mod 4). An example belonging to the latter series withV = 18 is constructed.