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.     

Large Finite Lattice Packings

A finite lattice packing of a centrally symmetric convex body K in ^d is a family C+K for a finite subset C of a packing lattice of K. For >0 the density (C;K,) is defined by (C;K,) = card C·V(K)/V(conv C+K). Assume that C ⁿ is the optimal packing with given n=card C, n large. It was known that conv C ⁿ is a segment if is less than the sausage radius _s (>0), and the inradius r(conv C ⁿ ) tends to infinity with n if is greater than the critical radius _c ( _s ). We prove that if > _c in ^d , then the shape of conv C ⁿ is not too far from being a ball. In addition, if r(conv C ⁿ ) is bounded but the radius of the largest (d–2)-ball in C ⁿ tends to infinity, then eventually C ⁿ is contained in some k–plane and its shape is not too far from being a k-ball where either k=d–1 or k=d–2. This yields in ³ that if _s << _c , then conv C ⁿ is eventually planar and its shape is not too far from being a disc. As an example, we show that _s = _c if K is a 3-ball, verifying the Strong Sausage Conjecture in this case. On the other hand, if K is the octahedron then _s < _c holds even for general (not only lattice) packings.     

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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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.     

On Ricci eigenvalues of locally homogeneous Riemannian 3-manifolds

We find the necessary and sufficient conditions for three constants ₁, ₂, ₃ ³ to be the principal Ricci curvatures of some 3-dimensional locally homogeneous Riemannian space.The first author was supported by the grant GAR 201/93/0469; the second author was supported by the grant SFS, Project #0401.     

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.     

Energy stability of channel flows

Zusammenfassung Die Energiemethode wird zur Untersuchung der Stabilität einer beliebigen Kanalströmung angewandt. Es wird eine auf Eigenwertabschätzungen basierte qualitative Beschreibung der Fläche (, ) gegeben, welche die Schranke für den Instabilitätsbereich in Abhängigkeit von Längs- bzw. Querwellenzahl der Störungen darstellt. Eine allgemeine untere Grenze für die minimalisierende Querwellenzahl und den zugehörigen Eigenwert (0, ) wird angegeben. Das Problem der Energiestabilität wird dann für eine Klasse von Geschwindigkeitsprofilen mit Wendepunkt gelöst. Die durch numerische Integration erhaltenen Ergebnisse werden mit den Ergebnissen der linearen Stabilitätstheorie verglichen.     

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.     

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     

An Improvement of an Inequality of Fiedler Leading to a New Conjecture on Nonnegative Matrices

Suppose that A is an n × n nonnegative matrix whose eigenvalues are = (A), ₂, ..., _n. Fiedler and others have shown that \det( I -A) ⁿ - ⁿ, for all > with equality for any such if and only if A is the simple cycle matrix. Let a _i be the signed sum of the determinants of the principal submatrices of A of order i × i, i=1, ..., n - 1. We use similar techniques to Fiedler to show that Fiedler's inequality can be strengthened to: for all . We use this inequality to derive the inequality that: . In the spirit of a celebrated conjecture due to Boyle-Handelman, this inequality inspires us to conjecture the following inequality on the nonzero eigenvalues of A: If ₁ = (A), ₂,...,_k are (all) the nonzero eigenvalues of A, then . We prove this conjecture for the case when the spectrum of A is real.     

Perfect equilibria of a model of n-person noncooperative bargaining

We study the set of subgame perfect equilibria associated with then-person noncooperative bargaining mechanism proposed by Hart and Mas-Colell (1992). Our results pertain to transferable utility games. The set of perfect equilibria depends on the parameter representing the continuation probability, . For general TU games, we characterize the set of payoffs from perfect equilibria for (a) small values of and (b) large values of . For symmetric games a complete characterization for all values of is provided.We are grateful to Andreu Mas-Colell for help and encouragement and to two referees for every helpful comments.     

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.     

On the Uniform Summability of Multiple Walsh-Fourier Series

In this paper we prove that if f C (0, 1 ^N ) and the function f is of bounded partial variation, then the N-dimensional Walsh-Fourier series of the function f is uniformly (C,–) summable (₁ +...+ _N < 1, _i > 0, i = 1,...,N) in the sense of Pringsheim. If ₁ +...+ _N = 1, _i > 0, i = 1,2,...,N, then there exists a continuous function f ₀ of bounded partial variation on [0, 1] ^N such that the Cesàro (C,–) means _m ^– (f₀,Õ) of the N-dimensional Walsh-Fourier series of f ₀ diverge over cubes.     

φ-Factors for Function Seminorms

Let (, A, ) be a measure space, a function seminorm on M, the space of measurable functions on , and M the space {f M : (f) < }. Every Borel measurable function : [0, ) [0, ) induces a function : M M by (f)(x) = (|f(x)|). We introduce the concepts of -factor and -invariant space. If is a -subadditive seminorm function, we give, under suitable conditions over , necessary and sufficient conditions in order that M be invariant and prove the existence of -factors for . We also give a characterization of the best -factor for a -subadditive function seminorm when is -finite. All these results generalize those about multiplicativity factors for function seminorms proved earlier.     

Discrete Spectrum of Differential Operators in Spectral Gaps under Nonnegative Perturbations of High Order

Let A be a self-adjoint elliptic second-order differential operator, let (, ) be an inner gap in the spectrum of A, and let B(t) = A + tW ^* W, where W is a differential operator of higher order. Conditions are obtained under which the spectrum of the operator B(t) in the gap (, ) is either discrete, or does not accumulate to the right-hand boundary of the spectral gap, or is finite. The quantity N(, A, W, ), (, ), > 0 (the number of eigenvalues of the operator B(t) passing the point (, ) as t increases from 0 to ) is considered. Estimates of N(, A, W, ) are obtained. For the perturbation W ^* W of a special form, the asymptotics of N(, A, W, ) as + is given. Bibliography: 5 titles.     

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.     

Left linear theories — A generalization of module theory

In this paper we introduce left linear theories of exponentN (a set) on the setL as mapsL ×L ^N (l, ) l · L such that for alll L and , L ^N the relation (l · ) =l( · ) holds, where · L ^N is given by ( · )(i) = (i),i N. We assume thatL has a unit, that is an element L ^N withl · =l, for alll L, and · = , for all L ^N . Next, left (resp. right)L-modules andL-M-bimodules and their homomorphisms are defined and lead to categoriesL-Mod, Mod-L, andL-M-Mod. These categories are algebraic categories and their free objects are described explicitly. Finally, Hom(X, Y) andX Y are introduced and their properties are investigated.Herrn Professor Dr. D. Pumplün zum 60. Geburtstag gewidmet     

On the motion of visco-elastic electrically conducting liquid in the presence of a magnetic field

Zusammenfassung Es wird die Strömung einer viskoelastischen, elektrisch leitenden Flüssigkeit untersucht auf Grund der elektromagnetischen Grundgleichung und der für solche Flüssigkeiten geltenden Bewegungsgleichungen. Im besonderen wird die Strömung zwischen parallelen Platten erörtert. Sie ist charakterisiert durch zwei dimensionslose Kennzahlen, die Hartmannsche Zahl und die Elastizitätszahl. Das elastische Verhalten bewirkt eine Verflachung des Geschwindigkeitsprofils und verändert auch den Druck und die Schubspannungen.

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Notations x ₁,x ₂,x ₃ space coordinates - B magnetic induction vector - E electric intensity - H magnetic field - D displacement vector - J current density vector - _e E electrostatic force - V×B current density due to the motion of the fluid - _e V convection current - _e excess charge density - electrical conductivity - magnetic permeability - dielectric constant - =1/ magnetic diffusivity - V=v ¹ i+v ² j+v ³ k velocity vector - v ₁= ₁/ kinematic viscosity - ₁ coefficient of viscosity - _c coefficient of cross-viscosity - v _c = _c / kinematic cross-viscosity - density of the fluid - relaxation time - S _j ⁱ stress-tensor - d _j ⁱ strain-tensor - x, y, z dimensionless coordinates - L spacing between the planes, taken as the characteristic length - U ₀ characteristic velocity - f non-dimensional velocity - H non-dimensionalx-component of the magnetic field.     

Compact elements in the lattice of radicals

Summary A radical R, in the sense of Kurosh--Amitsur, is said to be compact if, given any collection of radicals X such that R &le;;VX, we have R &le;;VX&apos; for some finite subcollection X&apos; of X. A ring A is said to be radical compact if the lower radical on the singleton {A} is compact. This paper explores the relationship between radical compact rings and rings satisfying certain finiteness conditions. Closure properties of the class of all radical compact rings are also investigated.