A Comparative Result for Multiplicative Functions

In this paper, we compare the asymptotic behavior of _nx f(n) and _nx g(n) for multiplicative functions f and g, respectively, where |f| g. Our results extend relevant theorems by E. Wirsing and G. Hal@aacute;sz. The methods we use are elementary.     

The Logarithmic Frequency of Values of Additive Functions

We consider the weak convergence of distribution functions (_mx 1/ m)^-1 _{m x},f_x(m)x is a set (x 2) of strongly additive functions such that f_x(p){0,1} for each prime number p.     

On a Test Statistic for Linear Trend

Let {W(s)} _s 0 be a standard Wiener process. The supremum of the squared Euclidian norm Y (t)², of the R²-valued process Y(t)=(1/t W(t), {12/t ³ int_{0 t} s dW (s)– {3/t} W(t)), t [, 1], is the asymptotic, large sample distribution, of a test statistic for a change point detection problem, of appearance of linear trend. We determine the asymptotic behavior P {sup _t [, 1] Y(t)² > u as u , of this statistic, for a fixed (0,1), and for a moving = (u) 0 at a suitable rate as u . The statistical interest of our results lie in their use as approximate test levels.     

Divergence of orthogonal series to +∞

An overall bounded orthonormal set of functions _n (x) is constructed for which there exists a series _m =1 a_nn(x) with coefficients a_n=o(ln'n/n), which diverges to + almost everywhere. The bibliography contains 4 references.Translated from Matematicheskie Zametki, Vol. 2, No. 5, pp. 483–494, November, 1967.     

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.     

Stochastic processes with Gaussian interaction of components

Summary LetU(x), x ^d-|0}, be a nonnegative even function such that _x 0U(x)1. In this paper, we consider an infinite system of stochastic process _t (x); x ^d with the following mechanism: at each sitex, after mean 1 exponential waiting time, _t(x) is replaced by a Gaussian random variable with mean _{yx t} (y) U(y-x) and variance 1. It is understood here that all the interactions are independent of one another. The behavior of this system will be investigated and some ergodic theorems will be derived. The results strongly depend whether _{x 0} U(x)<1 or =1.     

Critical sets in parametric optimization

We deal with one-parameter families of optimization problems in finite dimensions. The constraints are both of equality and inequality type. The concept of a generalized critical point (g.c. point) is introduced. In particular, every local minimum, Kuhn-Tucker point, and point of Fritz John type is a g.c. point. Under fairly weak (even generic) conditions we study the set consisting of all g.c. points. Due to the parameter, the set is pieced together from one-dimensional manifolds. The points of can be divided into five (characteristic) types. The subset of nondegenerate critical points (first type) is open and dense in (nondegenerate means: strict complementarity, nondegeneracy of the corresponding quadratic form and linear independence of the gradients of binding constraints). A nondegenerate critical point is completely characterized by means of four indices. The change of these indices along is presented. Finally, the Kuhn-Tucker subset of is studied in more detail, in particular in connection with the (failure of the) Mangasarian-Fromowitz constraint qualification.     

A characterization of brownian motion in a lipschitz domain by its killing distributions

Let (Y _t, Q^x) be a strong Markov process in a bounded Lipschitz domainD with continuous paths up to its lifetime , and let (X _t, P^x) be a Brownian motion inD. IfY _– exists in D andQ ^x(Y_– C)=P^x(X_– C) for all Borel subsetsC of D and allx, thenY is a time change ofX.     

Quadratically constrained quadratic programming: Some applications and a method for solution

The problem (QPQR) considered here is: minimizeQ ₁ (x) subject toQ _i (x) 0,i M ₁ {2,...,m},x P R ⁿ, whereQ _i (x), i M {1} M ₁ are quadratic forms with positive semi-definite matrices, andP a compact nonempty polyhedron of Rⁿ. Applications of (QPQR) and a new method to solve it are presented.Letu S={u R ^m;u 0, u _i= l}be fixed;then the problem:iM minimize u _iQ_i (x (u)) overP, always has an optimal solutionx (u), which is either feasible, iM i.e. u C₁ {u S;Q _i (x (u)) 0,i M ₁} or unfeasible, i.e. there exists ani M ₁ withu C {u S; Q_i(x(u)) 0}.Let us defineC _i C_i S _i withS _i {u S; u _i=0}, i M. A constructive method is used to prove that C _i is not empty and thatx (û) withiM û C _i characterizes an optimal solution to (QPQR). Quite attractive numerical results have been reached with this method.

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Zusammenfassung Die vorliegende Arbeit befaßt sich mit Anwendungen und einer neuen Lösungsmethode der folgenden Aufgabe (QPQR): man minimiere eine konvexe quadratische ZielfunktionQ _i (x) unter Berücksichtigung konvexer quadratischer RestriktionenQ _i (x) 0, iM ₁ {2,...,m}, und/oder linearer Restriktionen.·Für ein festesu S {u R ^m;u 0, u _i=1},M {1} M₁ besitzt das Problem:iM minimiere die konvexe quadratische Zielfunktion u _i Q_i (x (u)) über dem durch die lineareniM Restriktionen von (QPQR) erzeugten, kompakten und nicht leeren PolyederP R ⁿ, immer eine Optimallösungx (u), die entweder zulässig ist: u C₁ {u S;Q ₁ (x (u)) 0,i M ₁} oder unzulässig ist, d.h. es existiert eini M ₁ mitu C_i {u S;Q _i (x(u))0}.Es seien folgende MengenC _i C_i S _i definiert, mitS _i {u S;u _i=0}, i M. Es wird konstruktiv bewiesen, daß C _i 0 undx (û) mitû C _i eine Optimallösung voniM iM (QPQR) ist; damit ergibt sich eine Methode zur Lösung von (QPQR), die sich als sehr effizient erwiesen hat. Ein einfaches Beispiel ist angegeben, mit dem alle Schritte des Algorithmus und dessen Arbeitsweise graphisch dargestellt werden können.

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An earlier version of this paper was written during the author's stay at the Institute for Operations Research, Swiss Federal Institute of Technology, Zürich.     

The classification of finite connected hypermetric spaces

A finite distance spaceX, d d: X ² is hypermetric (of negative type) if a _x a _y d(x, y) 0 for all integral sequences{a _x x X} that sum to 1 (sum to 0).X, d is connected if the set {(x, y)d(x, y) = 1, x, y X} is the edge set for a connected graph onX, and graphical ifd is the path length distance for this graph. Then we proveThe first author was partially supported by NSF grant DMS 8600882.     

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.

     

Asymptotic behavior of the dwell time distribution for a random walk on a positive semi-axis

Let ₁, ₂, ... be a sequence of independent identically distributed random variables with zero means. We consider the functional _n = _k=o ⁿ (S _k ) where S₁=0, S_k= _i=1 ^k _i (k1) and(x)=1 for x0,(x) = 0 for x<0. It is readily seen that _n is the time spent by the random walk S_n, n0, on the positive semi-axis after n steps. For the simplest walk the asymptotics of the distribution P (_n = k) for n and k, as well as for k = O(n) and k/n<1, was studied in [1]. In this paper we obtain the asymptotic expansions in powers of n^–1 of the probabilities P(h_n = nx) and P(nx_{1 n} nx₂) for 0<₁, x = k/n ₂<1, 0<₁x₁2₂<1.Translated from Matematicheskie Zametki, Vol. 15, No. 4, pp. 613–620, April, 1974.The author wishes to thank B. A. Rogozin for valuable discussions in the course of his work.     

Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d ^t and, for each x M, we define a smooth real function _x (t) : (1 + _i (t)), where the _i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d^2t , restricted to v at the point _x ^-t M ⁿ.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that _x ^' (t) 0 for all x M and real t; then, if _x ^t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.     

Finite-dimensional attainable sets for stochastic control systems

Letx _t ^u () be a stochastic control system on the probability space (, ,P) intoR ⁿ. We say that the pointxR ⁿ is (, ) attainable at timet if there exists an admissible controlu such thatP _xo{x _t ^u ()S (x)}, wherex ₀()=x ₀, 0, 10, andS (x) is the closed Euclidean -ball inR ⁿ centered atx. We define the attainable setA (t) to be the set of all pointsxR ⁿ which are (, ) attainable at timet. For a large class of stochastic control systems, it is shown thatA (t) is compact for eacht and continuous as a function oft in an appropriate metric. From this, the existence of stochastic time-optional controls is established for a large class of nonlinear stochastic differential equations.This research was supported by the National Research Council of Canada, Grant No. A-9072.     

Properties of solutions of linear functional-differential equations depending on a parameter

On the segment 0 t1 we study the equation A(d/dt, )x(t) + [F()x](t)=f(t), whereA (d/dt, ) x=x( ⁿ )+A ₁ x(^n–1 +...+ ⁿ A _n x, the matrices A₁,...,A_n are of size m × m, x is an unknown and f a given function with values in the m-dimensional space ^m , F() is a linear operator acting from a Hölder space to a Lebesgue space of vectorfunctions with values in ^m and depending on a complex parameter . We find the set of those at which a one-to-one correspondence is established between the solutions of the given equation and the solutions of the equation A(d/dt, )x(t)=0.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1213–1231, September, 1991.     

Analysis of a class of fractional programming problems

We propose a solution strategy for fractional programming problems of the form max _xx g(x)/ (u(x)), where the function satisfies certain convexity conditions. It is shown that subject to these conditions optimal solutions to this problem can be obtained from the solution of the problem max _xx g(x) + u(x), where is an exogenous parameter. The proposed strategy combines fractional programming andc-programming techniques. A maximal mean-standard deviation ratio problem is solved to illustrate the strategy in action.     

On a property of functional series

The question of the convergence of functional series everywhere in the segment [0, 1] is considered. Let F=f be the set of such functions in [0, 1] for each of which there is a transposition of the series _k=1 f_k(x), which converges to it everywhere in [0, 1]. An example of a series is constructed such that the set F consists just of an identical zero, but _k=1 |f _k (x ₀)ü=,(x₀ [0,1]) for any point of the segment [0, 1].Translated from Matematicheskie Zametki, Vol. 11, No. 5, pp. 481–490, May, 1972.     

Decompositions of factor maps involving bi-closing maps

We describe all possible decompositions of a finite-to-one factor map : _A S, from an irreducible shift of finite type onto a sofic shift, into two maps =, such that the range of is a shift of finite type, and is bi-closing. We also give necessary and sufficient conditions for to be almost topologically conjugate overS to a bi-closing map.     

Ergodicity of nonlinear first order autoregressive models

Criteria are derived for ergodicity and geometric ergodicity of Markov processes satisfyingX _n+1 =f(X _n )+(X _n ) _n+1 , wheref, are measurable, { _n } are i.i.d. with a (common) positive density,E| _n |>. In the special casef(x)/x has limits, , asx– andx+, respectively, it is shown that <1, <1, <1 is sufficient for geometric ergodicity, and that <-1, 1, 1 is necessary for recurrence.     

On the distribution of square-full and cube-full integers

LetN _r (x) be the number ofr-full integers x and let _r (x) be the error term in the asymptotic formula forN _r (x). Under Riemann's hypothesis, we prove the estimates ₂(x)x^1/7+, ₃(x)x^97/804+(>0), which improve those of Cao and Nowak. We also investigate the distribution ofr-full andl-free numbers in short intervals (r=2,3). Our results sharpen Krätzel's estimates.