Function Spaces of Posets with Projections

This paper investigates function spaces of structures consisting of a partially ordered set together with some directed family of projections.More precisely, given a fixed directed index set (I,), we consider triples (D,,(p _i ) _iI ) with (D,) a poset and (p _i ) _iI a monotone net of projections of D. We call them (I,)-pop's (posets with projections). Our main purpose is to study structure preserving maps between (I,)-pop's. Such homomorphisms respect both order and projections.Any (I,)-pop is known to induce a uniformity and thus a topology. The set of all homomorphisms between two (I,)-pop's turns out to form an (I,)-pop itself. We show that its uniformity is the uniformity of uniform convergence. This enables us to prove that properties such as completeness and compactness transfer to function pop's.Concerning categorical properties of (I,)-pop's, we will see that we are in a lucky situation from a computer scientist's point of view: we obtain Cartesian closed categories. Moreover, by a D -construction we get (I,)-pop's that are isomorphic to their own exponent. This yields new models for the untyped -calculus.     

Dual-lattice theorems in the geometric approach

A one-to-one correspondence is shown to exist between the lattice of all self-bounded (A, )-controlled invariants contained in and the lattice of all self-hidden (A, )-conditioned invariants containing . This correspondence, stated herein as the main dual-lattice theorem, allows a straightforward derivation of the universal bounds of the lattices, particularly when additional constraints are imposed, such as to contain a given subspace for the elements of the former lattice and to be contained in a given subspace for the elements of the latter. Then, two further minor dual-lattice theorems, dual to each other, are presented, and some connections and applications of the new theory to standard control and observation problems are briefly discussed.     

On the Movement of Transitive Permutation p-Groups

Let G be a transitive permutation group on a set and m a positive integer. If | – | m for every subset of and all g G, then || 2mp/(p – 1) where p is the least odd prime dividing |G|. It was shown by Mann and Praeger [13] that, for p = 3, the 3-groups G which attain this bound have exponent p. In this paper we will show a generalization of this result for any odd primes.AMS Subject Classification (2000), 20BXX     

Position and Height of the Global Maximum of a Twice Differentiable Stochastic Process

For a stochastic process with absolutely continuous sample path derivative, a formula for the joint density of (T, Z), the position and height of the global maximum of in a closed interval, is given. The formula is derived using the generalized Rices formula. The presented result can be applied both to stationary and non-stationary processes under mild assumptions on the process. The formula for the density is explicit but involves integrals that have to be computed using numerical integration. The computation of the density is discussed and some numerical examples are given.     

Epsilon efficiency

This paper considers the extension of -optimality for scalar problems to vector maximization problems, or efficiency problems, which havem objective functions defined on a set .It is shown that the natural extension of the scalar -optimality concepts [viz, given >0, given a solution setS, ifxS there exists an efficient solutiony with f(x)–f(y), and given an efficient solutiony, there exists anxS with f(x)–f(y)] do not hold for some methods used. Six concepts of -efficient sets are introduced and examined, to a very limited extent, in the context of five methods used for generating efficient points or near efficient points.In doing so, a distinction is drawn between methods in which the surrogate optimizations are carried out exactly, and those where terminal -optimal solutions are obtained.The author would like to thank the referees whose thoroughness was extremely helpful for the revised paper.     

Mouvement brownien et formule de Tanaka en analyse

In our previous paper [5], we have obtained a decomposition of f, where f is a function defined on R ^d , that is analogous to the one proved by H. Tanaka in the early sixties for Brownian martingales (the so-called Tanaka formula). The original proofs use purely analytic methods (e.g. the Calderón–Zygmund theory, etc.). In this paper, we give a new proof of our `Tanaka formula in analysis, that is based on probabilistic arguments. The main tools here are Brownian motion, stochastic calculus and Burkholder–Gundy inequalities for martingales. These methods allow us to improve somewhat our previous results, by proving that some significant constants do not depend on the dimension d.     

On Some Structural Properties of Banach Function Spaces and Boundedness of Certain Integral Operators

In this paper the notions of uniformly upper and uniformly lower -estimates for Banach function spaces are introduced. Further, the pair (X, Y) of Banach function spaces is characterized, where X and Y satisfy uniformly a lower -estimate and uniformly an upper -estimate, respectively. The integral operator from X into Y of the form

Start‐up class models in multiple‐class queues with N‐policy

In this paper, we consider multipleclass queueing systems with Npolicy in which the idle server starts service as soon as the number of customers in the startup class reaches threshold N. We consider the cases of FCFS and nonpreemptive priority. We obtain the Laplace–Stieltjes transform of the waiting times of each class of customers. We also show some results for the general behavior of such systems.     

Some remarks on the product of two C α-compact subsets

For a cardinal , we say that a subset B of a space X is C -compact in X if for every continuous function is a compact subset of . If B is a C-compact subset of a space X, then (B, X) denotes the degree of C -compactness of B in X. A space X is called -pseudocompact if X is C -compact into itself. For each cardinal , we give an example of an -pseudocompact space X such that X × X is not pseudocompact: this answers a question posed by T. Retta in Some cardinal generalizations of pseudocompactness Czechoslovak Math. J. 43 (1993), 385–390. The boundedness of the product of two bounded subsets is studied in some particular cases. A version of the classical Glicksberg's Theorem on the pseudocompactness of the product of two spaces is given in the context of boundedness. This theorem is applied to several particular cases.     

On Fraenkel’s <Emphasis Type="Italic">N</Emphasis>-Heap Wythoff’s Conjectures

The N-heap Wythoffs game is a two-player impartial game with N piles of tokens of sizes Players take turns removing any number of tokens from a single pile, or removing (a₁,..., a_N) from all piles - a_i tokens from the i-th pile, providing that where is the nim addition. The first player that cannot make a move loses. Denote all the P-positions (i.e., losing positions) by Two conjectures were proposed on the game by Fraenkel [7]. When are fixed, i) there exists an integer N₁ such that when . ii) there exist integers N₂ and _2 such that when , the golden section.In this paper, we provide a sufficient condition for the conjectures to hold, and subsequently prove them for the three-heap Wythoffs game with the first piles having up to 10 tokens.AMS Subject Classification: 91A46, 68R05.     

A generalization of addition formulae

Let R denote the set of reals, J a real interval and X a real linear space. We determine the functions g : X J, M : J R and H : J ² R satisfying the equationg(x+M(g(x))y)=H(g(x),g(y)),under the assumptions that g is continuous on rays, M is continuous, and H is symmetric. As a consequence we obtain characterizations of some groups and semigroups.     

Controllability and extremality in nonconvex differential inclusions

LetF:[0, T]×R ⁿ 2 ^{R n} be a set-valued map with compact values; let :R ⁿ R ^m be a locally Lipschitzian map,z(t) a given trajectory, andR the reachable set atT of the differential inclusion . We prove sufficient conditions for (z(T))intR and establish necessary conditions in maximum principle form for (z(T))(R). As a consequence of these results, we show that every boundary trajectory is simultaneously a Pontryagin extremal, Lagrangian extremal, and relaxed Lagrangian extremal.The author is grateful to an anonymous referee for his valuable remarks and comments which have helped to improve the paper.The paper was written while the author was visiting the laboratory of Prof. S. Suzuki, Department of Mechanical Engineering, Sophia University, Tokyo, Japan.     

Strictly and Roughly Convexlike Functions

A function is said to be strictly and roughly convexlike with respect to the roughness degree r > 0 (for short, strictly r-convexlike) provided that, for all x ₀, x ₁ D satisfying ||x ₀ – x ₁|| > r, there exists a ]0, 1[ such that

Sensitivity analysis of the greatest eigenvalue of a symmetric matrix via the ɛ-subdifferential of the associated convex quadratic form

LetA(·) be ann × n symmetric affine matrix-valued function of a parameteruR ^m , and let (u) be the greatest eigenvalue ofA(u). Recently, there has been interest in calculating (u), the subdifferential of atu, which is useful for both the construction of efficient algorithms for the minimization of (u) and the sensitivity analysis of (u), namely, the perturbation theory of (u). In this paper, more generally, we investigate the Legendre-Fenchel conjugate function of (·) and the -subdifferential (u) of atu. Then, we discuss relations between the set (u) and some perturbation bounds for (u).The author is deeply indebted to Professor J. B. Hiriart-Urruty who suggested this study and provided helpful advice and constant encouragement. The author also thanks the referees and the editors for their substantial help in the improvement of this paper.     

Perturbation des résolvantes et représentation des mesures excessives

Résumé Etant donnée une résolvante V=(V )_>0 sous-markovienne sur un espace mesurable (X, B) de noyau initial V propre; on étudie alors le balayage des mesures surmédianes au moyen de la résolvante perturbée V de V par une function mesurable positive bornée sur X.Dans le cas où (X, E _v) est un espace de balayage, on montre que toute mesure excessive vérifiant , s'écrit d'une manière unique sous la forme =V où est une mesure positive sur X.     

The Brownian loop soup

We define a natural conformally invariant measure on unrooted Brownian loops in the plane and study some of its properties. We relate this measure to a measure on loops rooted at a boundary point of a domain and show how this relation gives a way to chronologically add Brownian loops to simple curves in the plane.Cornell University; Research supported in part by the National Science FoundationUniversité Paris-Sud and IUFMathematics Subject Classification (2000):60J65, 81T40     

A Simple Proof of the Borel Extension Theorem and Weak Compactness of Operators

Let T be a locally compact Hausdorff space and let C ₀(T) be the Banach space of all complex valued continuous functions vanishing at infinity in T, provided with the supremum norm. Let X be a quasicomplete locally convex Hausdorff space. A simple proof of the theorem on regular Borel extension of X-valued -additive Baire measures on T is given, which is more natural and direct than the existing ones. Using this result the integral representation and weak compactness of a continuous linear map u: C ₀(T) X when c ₀ X are obtained. The proof of the latter result is independent of the use of powerful results such as Theorem 6 of [6] or Theorem 3 (vii) of [13].     

On Colimits in Categories of Relations

We study (finite) coproducts and colimits of -chains in Rel(C), the 2-category of relations over a given category C. The former exist and are the same as in C provided that C is extensive. The latter do not exist for example in Rel(Set). However, the canonical construction of those colimits in the category of sets can be generalized to Rel(Set). The canonical cocone is shown to satisfy a 2-categorical universal property, namely that of an lax adjoint cooplimit. Sufficient conditions for any base category C to admit the construction are given.A necessary and sufficient condition for the construction to yield colimits of -chains in the category of maps of Rel(C) is also given.