Some Asymptotic Results for the M/M/∞ Queue with Ranked Servers

We consider an M/M/ model with m primary servers and infinitely many secondary ones. An arriving customer takes a primary server, if one is available. We derive integral representations for the joint steady state distribution of the number of occupied primary and secondary servers. Letting =/ be the ratio of arrival and service rates (all servers work at rate ), we study the joint distribution asymptotically for . We consider both m=O(1) and m scaled to be of the same order as . We also give results for the marginal distribution of the number of secondary servers that are occupied.     

A Taylor–Wiles System for Quaternionic Hecke Algebras

Let &ell >3 be a prime. Fix a regular character of F&^{2 ×} of order &–1, and an integer M prime to &. Let fS ₂(₀(M&²)) be a newform which is supercuspidal of type at &. For an indefinite quaternion algebra over Q of discriminant dividing the level of f, there is a local quaternionic Hecke algebra T of type associated to f. The algebra T acts on a quaternionic cohomological module M. We construct a Taylor–Wiles system for M, and prove that T is the universal object for a deformation problem (of type at & and semi-stable outside) of the Galois representation ¯ _f over F¯_& associated to f; that T is complete intersection and that the module M is free of rank 2 over T. We deduce a relation between the quaternionic congruence ideal of type for f and the classical one.     

New sufficient convergence conditions for the secant method

We provide new sufficient conditions for the convergence of the secant method to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses Lipschitz-type and center-Lipschitz-type instead of just Lipschitz-type conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than the earlier ones and under our convergence hypotheses we can cover cases where the earlier conditions are violated.     

Estimation of the Size and Mean Value of a Stigmatized Characteristic of a Hidden Gang in a Finite Population: A Unified Approach

We consider the problem of estimating the size and the mean value of a stigmatized quantitative characteristic of a sub-group (or hidden gang) in a finite population using a unified approach. The proposed method may be useful in estimating the proportion and the mean income of terrorists, hijackers or freedom fighters of a particular country, including those who operate across different countries.     

Anisotropy of the low-cycle tensile fatigue strength and deformability of a glass-reinforced plastic

Experimental results on the anisotropy of the strength and deformability of a 1:1 SVAM oriented glass-reinforced plastic in low-cycle tension in the directions =0, 15, 30, and 45° are discussed. The loading frequency was 1–1.5 cycles/minute. A three-dimensional fatigue strength diagram is constructed in _max - - log N coordinates. It is shown that in the direction =0° the fracture strain depends on stress, whereas at 0° the glass-reinforced plastic fails at almost the same maximum strain, the value of which depends on , but is invariant with respect to _max. It is established that the rate of development of the strain extrema on the linear part of the (N) graph and the stress _max are related by a power-law dependence, whose parameters are determined by the anisotropy of the mechanical properties of the composite.Institute of Mechanics, Academy of Sciences of the Armenian SSR, Erevan. Translated from Mekhanika Polimerov, No. 3, pp. 425–429, May–June, 1976.     

OnU ϕ-sets of trigonometric integrals

, (t) >0 E(–, +),E<, , ¦f(t)¦ (t) xE, f(t)=0 (–, +).     

Asymptotic behavior of constrained stochastic approximations via the theory of large deviations

Summary Let G be a bounded convex set, and _G the projection onto G, and a bounded random process. Projected algorithms of the types , where 0<a _n0, a _n =) occur frequently in applications (among other places) in control and communications theory. The asymptotic convergence properties of {X _n } as 0, n, have been well analyzed in the literature. Here, we use large deviations methods to get a more thorough understanding of the global behavior. Let be a stable point of the algorithm in the sense that X _n in distribution as 0, n. For the unconstrained case, rate of convergence results involve showing asymptotic normality of , and use linearizations about . In the constrained case is often on G, and such methods are inapplicable. But the large deviations method yields an alternative which is often more useful in the applications. The action functionals are derived and their properties (lower semicontinuity, etc.) are obtained. The statistics (mean value, etc.) of the escape times from a neighborhood of are obtained, and the global behavior on the infinite interval is described.Research has been supported in part by the US Army Research Office under Contract #DAAG 29-84-K-0082, and in part by the Office of Naval Research under Contract #N00014-83-K0542Research has been supported in part by the National Science Foundation Grant #ECS 82-11476, and the Air Force Office of Scientific Research under Contract #AF-AFOSR 81-0116     

The theory of an oscillating cylinder viscometer

The theoretical details of periodic state of motion of a liquid (Newtonian) in a narrow annular gap between long vertical coaxial cylinders is given in a new way.If the inner cylinder is suspended by a delicate torsion wire while the outer cylinder undergoes forced harmonic oscillations about its axis, the amplitude of uniform oscillation attained by the inner cylinder can be calculated. Two equations are derived for this purpose; the first is neither simple nor direct with respect to (coefficient of viscosity of the liquid) and the other constants of the apparatus, but covers all liquids with various densities and viscosities, while the second is more simple and direct but covers only liquids of >1. The theory agrees well with experiment for all values of ₀/₀, where ₀ is the amplitude of deflection of the inner cylinder and ₀ is the amplitude of oscillations inexorably imposed on the outer cylinder. ₀/₀ andV _–d/0, whereV _–d is the velocity of the liquid in the neighbourhood of the curved surface of the inner cylinder, has been plotted as function of the frequency for different values of the constants of the liquid of an available apparatus. Also the effect of density of the liquid has been discussed, and the condition of resonance between the liquid and the inner cylinder is given.

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Zusammenfassung Die Arbeit gibt eine theoretische Behandlung der periodischen Bewegung einer Flüssigkeit in der Lücke zwischen zwei koaxialen Zylindern wieder. Ist der innere Zylinder an einem Torsionsdraht aufgehängt, wobei der äussere Zylinder eine erzwungene Schwingung ausführt, wird eine Berechnung der uniformen Schwingung des inneren Zylinders möglich. Die experimentellen Daten sind gut mit der Theorie vereinbar. Die Kurven, welche die Abhängigkeit von ₀/₀ undV _–d/₀ von der Schwingungszahl darstellen, sind für verschiedene Werte der Konstanten der Flüssigkeit und des Apparates gezeichnet, wobei ₀ die Schwingungsamplitude des inneren Zylinders, ₀ diejenige des äusseren Zylinders undV _–d die Geschwindigkeit der Flüssigkeit in der Nähe der krummen Oberfläche des inneren Zylinders bedeutet. Im weiteren wird der Einfluss der Dichte diskutiert; schliesslich werden die Resonanzbedingungen zwischen der Flüssigkeit und dem inneren Zylinder aufgestellt.

     

Integral representation of linear operators

In this paper we obtain necessary and sufficient conditions in order that a linear operator, acting in spaces of measurable functions, should admit an integral representation. We give here the fundamental results. Let (T_i, _i) (i=1,2) be spaces of finite measure, and let (T,) be the product of these spaces. Let E be an ideal in the space S(T₁, ₁) of measurable functions (i.e., from |e₁||e₂|, e₁ S (T₁, ₁), e₂E it follows that e₁E). THEOREM 2. Let U be a linear operator from E into S(T₂, ₂). The following statements are equivalent: 1) there exists a-measurable kernel K(t,S) such that (Ue)(S)=K(t,S) e(t)d(t) (eE); 2) if 0e_nE (n=1,2,...) and e_n0 in measure, then (Ue_n)(S) 0 ₂ a.e. THEOREM 3. Assume that the function (t,S) is such that for any eE and for s a.e., the ₂-measurable function Y(S)=(t,S)e(t)d ₁(t) is defined. Then there exists a-measurable function K(t,S) such that for any eE we have (t,S)e(t)d ₁(t)=K(t,S)e(t)d ₁(t) ₁a.e.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 47, pp. 5–14, 1974.     

Exponential inequalities for dependent random variables

This paper contains the Kolmogorov-Prokhorov exponential inequalities for dependent random variables, i.e., for-mixing,-mixing and-mixing. As an application, the law of iterated logarithm is established for stationary-mixing sequence under a nearly best assumption.Research supported by National Science Foundation Grant.     

Ein Existenzsatz für Elementarinhalte

An elementary content I is a real valued, non-negative, invariant and monotonous homomorphism on a decomposition structure of elementary figures. The semigroup (H,+,) of abstract classes is introduced (§3) by using the relation of equidecomposability and it's natural generalizations. Each elementary content divides into I= where and are canonical homomorphisms with respect to the relations studied before and : H + is a monotonous homomorphism called content (cf. Satz 3, §3). In §4 (Satz 4) the Existence-Theorem on contents is stated and it is proved in §5. The last section §6 gives the application on Archimedean decomposition structures including the case of volume measurement on polyhedrons.     

Rate of convergence in the central limit theorem of the UH-statistics

The problem of estimating the rate of convergence in the central limit theorem for the U-statistics with values in the Hilbert space is considered. Bounds of the order o(n^–1/2) if the kernel satisfies the nondegenerate condition and E³ < .Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 8, pp. 1043–1050, August, 1990.     

Automorphisms of prime order of curves

If an order q of an automorphism of an algebraic curve of genus g2 is prime, then q2g+1. In this paper we determine all curves having an automorphism of order 2g+1 when 2g+1 is prime.     

Normal bundles of rational curves inP 3

LetP ^r=P _k ^r be the projective space over an algebraically closed ground field k. Let X be a rational space cur ve of degree n with only ordinary singularities. Since X is rational, the normal bundleN of X inP ³ splits inN = _{1 2} where ₁, and ₂ are line bundles, and we have deg ₁ + deg ₂ = 4n – 2. We consider the non-negative integer defined by 2 = |deg ₁ – deg ₂|. The aim of this paper is to determine all possible values of and to describe the variety parametrizing all twisted rational curves inP ³ with only ordinary singularities for a fixed degree n and fixed .The paper was supported by C.N.R., while both authors were members of GNSAGA     

Regularization of Nonlinear Ill-Posed Variational Inequalities and Convergence Rates

Let H be a Hilbert space and K be a nonempty closed convex subset of H. For f H, we consider the (ill-posed) problem of finding u K for which 0 for all v K, where A : H H is a monotone (not necessarily linear) operator. We study the approximation of the solutions of the variational inequality by using the following perturbed variational inequality: for f H, f – f , find u, K for which 0 for all v K, where , , and are positive parameters, and K, a perturbation of the set K, is a nonempty closed convex set in H. We establish convergence and a rate O(1 / 3) of convergence of the solutions of the regularized variational inequalities to a solution of the original variational inequality using the Mosco approximation of closed convex sets, where A is a weakly differentiable inverse-strongly-monotone operator.     

Structures of topological type

In this paper we continue the study of structures of various types initiated by the author in the earlier paper Structures of extensions (Ref. Zh. Mat., 1974, 4A361). The present paper is devoted to the so-called structure of topological type. By a structure of topological type on the set X is meant a topological structure, defined on some set obtained from X, and possibly additional sets, by a totally ordered sequence of operations of unions of sets, products of sets, and passage to the set of subsets. We study certain structures of topological type: bitopological (Sec. 2) and settopological (Sec. 3). A bitopological structure on the set X is any topological structure on the set X×X, and a bitopological space is a pair (X,). This concept is a natural extension of the concept of a bitopological space as a set X on which there are given two topological structures ₁ and ₂-these structures define a structure =₁×₂ on the set X×X. A settopological structure on the set X is any topological structure on the set={A¦A. There are given representations of piecewise-linear structures (Sec. 4) and smooth structures (Sec. 5) as settopological structures.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 83, pp. 5–62, 1979.     

Order of a Function on the Bruschlinsky Group

For an arbitrary function F defined on the group of homotopy classes of mappings of a finite polyhedron X to the circle and taking values in an Abelian group Q, the notion of order is defined. The order ord F is compared with the algebraic degree of F. It is proved that ord Fdeg F and deg Fdim X ord F. The inequality ord Fdeg F is proved in the case where Q is torsion-free or ord F1. Bibliography: 2 titles.     

Solution of two sequencing problems

The solution of the following problems is offered. Suppose a multiset J (¦J¦=p) is given. For each pair of elements and J, a number 1 P is given. Moreover, if 1 < x<p then x is undefined. If x=1, then x=p. Problem 1. Find the permutation ₁..._F of elements of the multiset J satisfying the following conditions. Let _i, _i=. If i,j < x, thenj <i. If i,j > x, then i<j. Such a permutation is called a PC-schedule. Problem 2. Find a PC-schedule in which the following property holds: if i < x < j, _i=, _j=, then. Such a PC-schedule is called an SC-schedule. The conditions under which these problems have solutions are studied. For their solution an algorithm of shifts is used with the complexity O(¦B(J)¦²¦J¦).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 124, pp. 44–72, 1983.     

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.     

A generalized Laplace transform of generalized functions

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