Extremal problems of Bloch function spaces

Letf be analytic in a hyperbolic region . The Bloch constant _f off is defined by , where (z)|dz| is the Poincaré metric in . Suppose is hyperbolic and where . Then for allf withf() , we have _f 1/(). In this paper we study the extremal functions defined by _f =1/() and the existence of those functions.Supported by the National Natural Science Foundation of China.     

On the positive solutions of some nonlinear diffusion problems

We consider the nonlinear diffusion equationu _t –a(x, u _{x x} )+b(x, u)=g(x, u) with initial boundary conditions andu(t, 0)=u(t, 1)=0. Here,a, b, andg denote some real functions which are monotonically increasing with respect to the second variable. Then, the corresponding stationary problem has a positive solution if and only if(0, ^*) or(0, ^*]. The endpoint ^* can be estimated by , where ₁ u denotes the first eigenvalue of the stationary problem linearized at the pointu. The minimal positive steady state solutions are stable with respect to the nonlinear parabolic equation.

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Zusammenfassung Wir betrachten die nichtlineare Diffusionsgleichungu _t –a(x, u _x ) _x +b(x, u)=g(x, u) mit Randbedingungen undu (t, 0)=u (t, 1)=0. Dabei sinda, b, undg monoton wachsende Funktionen bzgl. des zweiten Argumentes. Das zugehörige stationäre Problem hat genau dann eine positive Lösung, falls (0, ^*) oder(0, ^*]. Der Endpunkt ^* kann durch abgeschätzt werden, wobei ₁ u den ersten Eigenwert des an der Stelleu linearisierten stationären Problems bezeichnet. Die minimale positive stationäre Lösung ist stabil bzgl. der obigen nichtlinearen parabolischen Gleichung.

     

Positivity and Negativity of Solutions to a Schrödinger Equation in ℝ N

Weak L ² -solutions u of the Schrödinger equation, –u + q(x) u – u = f(x) in L ² , are represented by a Fourier series using spherical harmonics in order to prove the following strong maximum and anti-maximum principles in (N 2): Let ₁ denote the positive eigenfunction associated with the principal eigenvalue ₁ of the Schrödinger operator . Assume that the potential q(x) is radially symmetric and grows fast enough near infinity, and f is a `sufficiently smooth' perturbation of a radially symmetric function, f 0 and 0 f / C const a.e. in . Then u is ₁-positive for - < < ₁ (i.e., u c ₁ with c const > 0) and ₁-negative for ₁ < < ₁ + (i.e., u –c₁ with c const > 0), where > 0 is a number depending on f. The constant c > 0 depends on both and f.     

Estimation of the Intensity of the Flow of Nonmonotone Refusals in the Queuing System (≤ λ)/G/m

We consider a queuing system ()/G/m, where the symbol () means that, independently of prehistory, the probability of arrival of a call during the time interval dtdoes not exceed dt. The case where the queue length first attains the level r m+ 1 during a busy period is called the refusal of the system. We determine a bound for the intensity ₁(t) of the flow of homogeneous events associated with the monotone refusals of the system, namely, ₁(t) = O( ^{r+ 1}₁ ^{m– 1} _{r– m+ 1}), where _k is the kth moment of the service-time distribution.     

The existence of some resolvable block designs with divisibility into groups

This paper proves the existence of resolvable block designs with divisibility into groups GD(v; k, m; ₁, ₂) without repeated blocks and with arbitrary parameters such that ₁ = k, (v–1)/(k–1) ₂ v^k–2 (and also ₁ k/2, (v–1)/(2(k–1)) ₂ v^k–2 in case k is even) k 4 andp=1 (mod k–1), k < p for each prime divisor p of number v. As a corollary, the existence of a resolvable BIB-design (v, k, ) without repeated blocks is deduced with X = k (and also with = k/2 in case of even k) k , where a is a natural number if k is a prime power and=1 if k is a composite number.Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 623–634, April, 1976.     

On the Mean Value of the Index of Composition of an Integer

For each integer n 2, let be the index of composition of n, where . For convenience, we write (1)=(1)=1. We obtain sharp estimates for and , as well as for and . Finally we study the sum of running over shifted primes.Research supported in part by a grant from NSERC.Research supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA.     

Décroissance rapide de la distribution fλ

Let X be an open subset of ⁿ and (f₁, ...,f_p): X ^p be a holomorphic mapping. We prove that if (x⁰,0, ⁰) T^* × ^p does not belong to the characteristic variety of the _X []-module _X[]f, then there exists a conic neighborhood V × of (x⁰, ⁰) such the function is rapidely decreasing in | Im | for with Re bounded, for any (n,n)-form of class C with compact support in V. The following partial converse of this result is also established: if for all (n,n)-forms of class C with compact support in X, then .     

Asymptotic properties of local densities of measures generated by semimartingales

For families of probability measures (P , )) generated by semimartingales, we consider the local density)(y, )= _t (y, )) _t0 of a, measureP _y with respect to the measureP whose logarithm is the difference of a local martingale and a positive predictable increasing locally bounded process. Conditions are obtained under which the relations and hold, wherey _t depends in some way ont, while _t ast . Applications of these relations are exhibited and an example is given when the hypotheses of the theorems proved can be verified.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 48–55, 1986.     

On a property of the entire dirichlet series with decreasing coefficients

The classS ^* (A) of the entire Dirichlet series is studied, which is defined for a fixed sequence by the conditions 0 _n + and _n (1n⁺(1/a _n )) imposed on the parameters _n, where is a positive continuous function on (0, +) such that (x) + and x/(x) + asx + . In this class, the necessary and sufficient conditions are given for the relation (InM(,F))(In (,F)) to hold as +, where , and is a positive continuous function increasing to + on (0,+), forwhich ln (x) is a concave function and(lnx) is a slowly increasing function.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 6, pp. 843–853, June. 1993.     

A general minimax theorem

This paper is concerned with minimax theorems for two-person zero-sum games (X, Y, f) with payofff and as main result the minimax equality inf supf (x, y)=sup inff (x, y) is obtained under a new condition onf. This condition is based on the concept of averaging functions, i.e. real-valued functions defined on some subset of the plane with min {x, y}< (x, y)x, y} forx y and (x, x)=x. After establishing some simple facts on averaging functions, we prove a minimax theorem for payoffsf with the following property: Forf there exist averaging functions and such that for any x₁, x₂ X, > 0 there exists x₀ X withf (x₀, y) > f (x₁,y),f (x₂,y))– for ally Y, and for any y₁, y₂ Y, > 0 there exists y₀ Y withf (x, y₀) (f (x, y₁),f (x, y₂))+. This result contains as a special case the Fan-König result for concave-convex-like payoffs in a general version, when we take linear averaging with (x, y)=x+(1–)y, (x, y)=x+(1–)y, 0 <, < 1.Then a class of hide-and-seek games is introduced, and we derive conditions for applying the minimax result of this paper.

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Zusammenfassung In dieser Arbeit werden Minimaxsätze für Zwei-Personen-Nullsummenspiele (X, Y,f) mit Auszahlungsfunktionf behandelt, und als Hauptresultat wird die Gültigkeit der Minimaxgleichung inf supf (x, y)=sup inff (x, y) unter einer neuen Bedingung an f nachgewiesen. Diese Bedingung basiert auf dem Konzept mittelnder Funktionen, d.h. reellwertiger Funktionen, welche auf einer Teilmenge der Ebene definiert sind und dort der Eigenschaft min {x, y} < < (x, y)x, y} fürx y, (x, x)=x, genügen. Nach der Herleitung einiger einfacher Aussagen über mittelnde Funktionen beweisen wir einen Minimaxsatz für Auszahlungsfunktionenf mit folgender Eigenschaft: Zuf existieren mittelnde Funktionen und, so daß zu beliebigen x₁, x₂ X, > 0 mindestens ein x₀ X existiert mitf (x₀,y) (f (x ₁,y),f (x₂,y)) – für alley Y und zu beliebigen y₁, y₂ Y, > 0 mindestens ein y₀ Y existiert mitf (x, y₀) (f (x, y₁),f (x, y ₂))+ für allex X. Dieses Resultat enthält als Spezialfall den Fan-König'schen Minimaxsatz für konkav-konvev-ähnliche Auszahlungsfunktionen in einer allgemeinen Version, wenn wir lineare Mittelung mit (x, y)=x+(1–)y, (x, y)= x+(1–)y, 0 <, < 1, betrachten.Es wird eine Klasse von Suchspielen eingeführt, welche mit dem vorstehenden Resultat behandelt werden können.

     

A complete asymptotic expansion of the jacobi functions with error bounds

In this paper we give a complete asymptotic expansion of the Jacobi functions ^{(, )} (t) as + . The method we employed to get the complete expansion follows that of Olver in treating similar problems. By using a Gronwall-Bellman type inequality for an improper integral in which the integrand is an unbounded function and contains a parameter, we get an error bound of the asymptotic approximation which is different from that of Olver's.     

Subordination des résolvantes

Résumé Soit (V ₎₀ une résolvante définie sur un espace mesurable telle que le noyau initial est borné; on trouve une condition nécéssaire et suffisante pour qu'un noyau borné U possède une résolvante (U ₎₀ telle que U V pour tout 0. On donne plusieurs applications de ce résultat.     

More on limit theorems for iterates of probability measures on semigroups and groups

Summary In this paper, we continue earlier works of one of the authors on vague convergence of the sequence _k,n= _k+1 *...* _n, where _n is a sequence of probability measures on semigroups or groups. Typical results in this paper are: Theorem. Let S be a locally compact noncompact second countable group such that being the support of a probability measure on S. Suppose there exists an open set V with compact closure such that x ^–1 Vx=V for every xS. Then for all compact sets K, sup{ ⁿ (Kx): xS0 as n. Theorem. Let S be an at most countable discrete group. Let _n be a sequence of probability measures on S. Then for all nonnegative integers k, the sequence _k,n converges vaguely to some probability measure if and only if there exists a finite subgroup G such that the series and for any proper subgroup G of G and any choice of elements g_n in S, the series . A sufficient condition for the vague convergence of the sequence _k,n to a probability measure is that (i) there exists a finite subgroup G such that and (ii) _n(e)>s>0 for all n, e being the identity.The author was supported by NSF grant MCS77-03639     

Identities of an algebra of triangular matrices

This paper deals with the ideals of identities of certain associative algebras over a field F of characteristic zero. An algebra W of matrices of the form ,,,M, where and , are F-algebras with unity and M is a (,)-bimodule, is considered. Under certain natural restrictions on M one obtains the equality of ideals of identities T(W)=T()T(), if [[x₁,x₂], x₃[x₄,x₅]]T().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 7–27, 1982.     

On eigenvalues of perturbed quadratic matrix polynomials

Among other results, it is shown that ifC andK are arbitrary complexn×n matrices and if det( ₀ ² I₀ C+K)=0 for some ₀0 (resp. ₀=0), then the Newton diagram of the polynomialt(, ) = det(² I+(1+)C+K expanded in (–₀) and , has at least a point on or below the linex+y=b (resp. has no expanded in (–₀) and , has at least a point on or below the of 0 as an eigenvalue of ₀ ² I+₀ C+K. These are extensions of similar results deu to H. Langer, B. Najman, and K. Veseli proved for diagonable matricesC, and shed light on the eigenvalues of the perturbed quadratic matrix polynomials. Our proofs are independent and seem to be simpler     

Unipotent Brauer Character Values of GL(n, \mathbb{F}_q ) and the Forgotten Basis of the Hall Algebra

We give a formula for the values of irreducible unipotent p-modular Brauer characters of at unipotent elements, where p is a prime not dividing q, in terms of (unknown!) weight multiplicities of quantum GL _n and certain generic polynomials S _,(q). These polynomials arise as entries of the transition matrix between the renormalized Hall-Littlewood symmetric functions and the forgotten symmetric functions. We also provide an alternative combinatorial algorithm working in the Hall algebra for computing S _,(q).     

Numerical implementation of coherence for the example of the “Swiss Cross”

Summary We consider the two-dimensional Helmholtz equation u+u=0 inD with the boundary conditionsu=0 on D. D is the Swiss Cross — a region consisting of five unit squares. A method based on the concept of Coherence is utilized to determine an approximation for the first eigenvalue= ₁ more accurate than calculated by classical difference methods. The numerical result is used to illustrate isoperimetric upper and lower bounds for ₁, and to test some conjectures on its relations with torsional rigidity.Dedicated to the memory of Professor Lathar Collatz     

Absolute values of the coefficients of the polynomials in Weierstrass's approximation theorem

The following problem, bound up with Weierstrass's classical approximation theorem, is solved definitively: to determine the sequence of positive numbersM _k such that, for anyf(z)c[0,1] and > 0 there exists the polynomial that f–P< and _k <M _k ,k=1, ...,n.Translated from Matematicheskii Zametki, Vol. 22, No. 2, pp. 269–276, August, 1977.     

On the width of ordered sets and Boolean algebras

Thewidth (chain number) of a partial order P, < is the smallest cardinal such that ¦A¦< 1 + whenever A is an antichain (chain) in P. We prove that, if a partial order (P, <) has width and cf()=, then P contains antichains A_n (n<) such that ¦A ₀¦<¦A₁¦ <...<={¦A_n¦: n < < } and either A₀1 A₂< ... or A₀>A₁ >A₂> ... A similar structure result is obtained for partial orders with chain number if cf()=. As an application we solve a problem of van Douwen, Monk and Rubin [1] by showing that if a Boolean algebra has width , thencf() .This work has been partially supported by NATO grant No. 339/84.Presented by Bjarni Jonsson.     

Best mean-square approximation of functions of m variables

Let be the best mean-square approximation of a functionf(x) L₂(R_m) (m=1, 2, ...) by integral functions of the exponential spherical type (in the sense of thel _q metric, 0>0, where(f,/; l _p)L₂(R_m) is the spherical (in the sense of the metricl _p, 0f(x) L₂(R_m). For the quantity two-sided estimates are obtained which are uniform in the parameters m, q, and p. Similar results are also obtained in the case of q=p=2 for classes of functions W _f2 (R_m) (=1,2,...).Translated from Matematicheskie Zametki, Vol. 14, No. 6, pp. 913–924, December, 1973.The author would like to express his deep gratitude to N. I. Chernykh under whose guidance this work has been carried out.