Sugli insiemi di permutazioni strettamente k-transitivi (k ≥ 3)

Summary Let G be a sharply 3-transitive permutation set on a finite set E of even cardinality and let 1 be in G. The following theorems are proved. G is one of the known examples if and only if there exists a non-identity normal subgroup N of G and an element of E such that NG G.G is a group if and only if G for every G and for every G and for every G .By using the classification of finite single groups a result concerning sharply k-transitive permutation sets k>3 is also proved.

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Dedicato a Guido Zappa in occasione del suo 70° compleanno

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Lavoro eseguito nell'ambito dei progetti finanziati dal Ministero della Pubblica Istruzione.     

A perturbation result for the asymptotic behavior of matrix powers

We prove a perturbation result for the asymptotic behavior of the sequence (A ⁿ c) _nN , whereAG|(d), the space of invertibled×d matrices, andc ^d .     

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.     

On a Property of Subexponential Distributions

Let F be a distribution function (d.f.) on [0, ) with finite first moment m >0. We define the integrated tail distribution function F ₁ of F by F ₁(t)=m^-1 ₀ ^t (1- F(u))du, t0. In this paper, we obtain sufficient conditions under which implications FSF ₁S and F ₁S FS hold, where S is the class of subexponential distributions.     

Second-Order Subexponential Sequences and the Asymptotic Behavior of Their De Pril Transform

Suppose that { f(n), n N ₀ } is a sequence of positive real numbers and suppose that the sequence { a(n), n N ₀ } is given by a(0) = 0, and, for n 1, by the convolution equation nf(n) = a* f(n). The resulting sequence is denoted by a(n) = _f (n) and is called the De Pril transform of { f(n), n N ₀ } . In this paper, we consider first- and second-order asymptotic behavior of { _{f (n), n N 0} } for a large class of subexponential sequences { f(n), n N ₀ } . We also discuss some applications.     

Symmetrische Permutationsmengen

A permutation set (M, I) consisting of a setM and a set of permutations ofM, is calledsymmetric, if for any two permutations, the existence of anx M with (x) (x) and ^–1 (x) = ^–1 (x) implies ^–1 = ^–1 , andsharply 3-transitive, if for any two triples (x ₁,x ₂,x ₃), (y ₁,y ₂,y ₃) M ³ with|{x ₁,x ₂,x ₃ }| = |{y ₁,y ₂,y ₃ }| = 3 there is exactly one permutation with(x ₁) =y ₁,(x ₂) =y ₂,(x ₃) =y ₃. The following theorem will be proved.THEOREM.Let (M, ) be a sharply 3-transitive symmetric permutation set with |M|3, such that contains the identity. Then is a group and there is a commutative field K such that and the projective linear group PGL(2, K) are isomorphic.     

Efficiency of Proximal Bundle Methods

We give efficiency estimates for proximal bundle methods for finding f_*min_Xf, where f and X are convex. We show that, for any accuracy <0, these methods find a point x^kX such that f(x^k)–f^* after at most k=O(1/³) objective and subgradient evaluations.     

On improved regularity of weak solutions of some degenerate,Anisotropic elliptic systems

Summary We consider a (possibly) vector-valued function u: R^N, Rⁿ, minimizing the integral , 2-2/(n*1)<p<2, whereD _i u=u/x _i or some more general functional retaining the same behaviour, we prove higher integrability for Du: D₁ u,..., D_n–1 u L^p/(p-1) and D_nu L²; this result allows us to get existence of second weak derivatives: D(D₁ u),...,D(D_n–1u)L² and D(D_n u) L _p.This work has been supported by MURST and GNAFA-CNR.     

Some minimax problems of vector-valued functions

The concepts of cone extreme points, cone saddle points, and cone saddle values are introduced. The relation of inclusion among the sets mini _xX max _yY f(x, y), maxi _yY min _xX f(x, y), and the set of all weak cone saddle values is investigated in the case where the image space ⁿ off is ordered by an acute convex cone.The author is grateful for the useful suggestions and comments given by Prof. K. Tanaka, Niigata University, Niigata, Japan.The author would like to thank the referees for their valuable suggestions on the original draft.     

Stable Subsets and the Existence of a Unit in (Semi)-Prime Rings

Criteria for the existence of a unit in a semiprime, prime, or simple ring and criteria for an idempotent of an arbitrary ring or of a semiprime ring to be central are obtained. In particular, it is shown that a strictly prime ring R in which r Rr for any r R is a ring with unit. In this connection, examples of prime (and even simple) rings are presented such that r Rr rR for any r R but there is no unit. The problem of whether a given ring R has a left unit was reduced earlier by the author to the semiprime case, namely, R has a left unit if and only if r Rr for any element r of the prime radical P(R) and the ring R P(R) has a left unit.     

On a class of functions with equal infima over a domain and its boundary

In this paper, we present a class of functions:f:X such that inf _xX f(x)= , whereX is a nonempty, finitely compact and convex set in a vector space andB _x ={xX: y aff(X){x:[x, y]X={x}. Our main tool is a recent minimax theorem by Ricceri (Ref. 1).     

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.     

Radially Symmetric Functions as Fixed Points of some Logarithmic Operators

In this note we show that for f C((0,); R₊) C¹ ((0,)) with support in [0,), if a function u C¹(R²) is such that support (u⁺) is compact and u(x) = R² f(u(y)) log 1/(|x-y|)dy x, then u is radial. This result is important for some free boundary problems in R² or some axisymmetric ones in Rⁿ.     

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.     

Asymptotics of Solutions of the Sturm–Liouville Equation with Respect to a Parameter

On a finite segment [0, l], we consider the differential equation

Generalized sequencing problem “Towers of Hanoi”

ForpN certain integer-valued functionsA _p (x), defined forx N {0}, are studied. These functions occur in a functional equation system corresponding to a generalized version of the transportation game Towers of Hanoi and their values may be interpreted as minimum numbers of moves. An explicit representation ofA _p (x) is given and so-called minimum partitions ofx with respect top are determined for allx N. The minimum partitions ofx are of interest concerning the realisation of the minimum number of moves by optimal policies.

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Zusammenfassung Es werden fürp N gewisse ganzzahlige für allex N {0} erklärte FunktionenA _p (x) untersucht, die bei einer Verallgemeinerung des unter dem Namen Türme zu Hanoi bekannten Transportspiels in einem entsprechenden Funktionalgleichungssystem auftreten und deren Funktionswerte sich als Mindestzugzahlen interpretieren. Es werden fürA _p (x) eine explizite Darstellung und sogenannte Minimalzerlegungen vonx bezüglichp für allex N bestimmt. Die Minimalzerlegungen vonx spielen eine besondere Rolle bei der Angabe von optimalen Strategien zur Realisierung der Mindestzugzahl.

     

On semiregular conditional distributions

Let (,A,P) denote some probability space and some sub--algebra ofA. It is shown that there exists a semiregular versionQ (A),A, , of the conditional distributionP(A|), AA, i.e., Q (A), (AA fixed) is andAQ (A),AA ( fixed), is a probability charge satisfyingQ (N)=0, , for allP-zero setsN, if and only ifL ₁(,P|) has a lifting, which exists for any sub--algebra ofA ifL ₁(,A P) is separable. Separability ofL ₁(,A,P) implies also the existence of a strongly semiregular versionQ (A),A, , ofP(A|), A , i.e., Q (A), (AA fixed), is -measurable andAQ (A),A ( fixed), is a probability charge. Furthermore,P can be written as P ₁+(1–)P ₂, 01, whereP ₁ are probability measures onA such thatP ₁(A|),AA, has a semiregular version vanishing for anyP-zero setN andP ₂ is singular with respect to any probability measure onA of the type ofP ₁. In the case 0<<1 the probability measuresP _j ,j=1, 2, are uniquely determined. The decomposition can be carried over to the case, where the additional condition thatQ (N)=0 for all and anyP-zero setN is valid, is omitted respectively semiregularity is replaced by (i) strong semiregularity, or (ii) classical regularity. In the last mentioned case (ii) the decomposition is multiplicative.     

Random rotations of the Wiener path

Summary Let (W, H, ) be an abstract Wiener space and letR(w) be a strongly measurable random variable with values in the set of isometries onH. Suppose that Rh is smooth in the Sobolev sense and that it is a quasi-nilpotent operator onH for everyhH. It is shown that (R(w)h) is again a Gaussian (0, |h| _H ² )-random variable. Consequently, if (e _i ,i)W ^* is a complete, orthonormal basis ofH, then defines a measure preserving transformation, a rotation, onW. It is also shown that if for some strongly measurable, operator valued (onH) random variableR, (R(w+k)h) is (0, |h| _H ² )-Gaussian for allk, hH, thenR is an isometry and Rh is quasi-nilpotent for allHH. The relation between the stochastic calculi for these Wiener pathsw and , as well as the conditions of the inverbibility of the map are discussed and the problem of the absolute continuity of the image of the Wiener measure under Euclidean motion on the Wiener space (i.e. composed with a shift) is studied.The research of the second author was supported by the Fund for the Promotion of Research at the TechnionDedicated to the memory of Albert Badrikian     

Variations of Constrained Domain Functionals Associated with Boundary-Value Problems

We study variational formulas for maximizers for domain functionalsF(x0, u(x0)), x0, and F(x,u(x))dxover all Lipschitz domains satisfying the constraintg(x) dx=1. Here, u is the solution ofa diffusion equation in . Functional variations arecomputed using domain variations which preserve the constraint exactly. Weshow that any maximizer solves a moving boundary problem for the diffusionequation. Further, we show that, for problems with symmetry, the optimaldomains are balls.     

Orthogonality and additivity modulo a subgroup

Summary LetE be a real inner product space of dimension at least 2,F a topological Abelian group, andK a discrete subgroup ofF. Assume also thatF is continuously divisible by 2 (that is, the functionu 2u is a homeomorphism ofF ontoF). Iff: E F fulfils the conditionf(x + y) – f(x) – f(y) K for all orthogonalx, y E and is continuous at the origin then there exist continuous additive functionsa: R F andA: E F such thatf(x) – a(x ²)– A(x) K for everyx E. Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.