Distinguishing non-standard natural numbers in a set theory within Łukasiewicz logic

In , a set theory with the comprehension principle within Łukasiewicz infinite-valued predicate logic, we prove that a statement which can be interpreted as “there is an infinite descending sequence of initial segments of ω” is truth value 1 in any model of , and we prove an analogy of Hájek’s theorem with a very simple procedure.      

From Scalar to Vector Optimization

Initially, second-order necessary optimality conditions and sufficient optimality conditions in terms of Hadamard type derivatives for the unconstrained scalar optimization problem ϕ(x) → min, x ∈ ℝ ^m , are given. These conditions work with arbitrary functions ϕ: ℝ ^m → ℝ, but they show inconsistency with the classical derivatives. This is a base to pose the question whether the formulated optimality conditions remain true when the “inconsistent” Hadamard derivatives are replaced with the “consistent” Dini derivatives. It is shown that the answer is affirmative if ϕ is of class (i.e., differentiable with locally Lipschitz derivative). Further, considering functions, the discussion is raised to unconstrained vector optimization problems. Using the so called “oriented distance” from a point to a set, we generalize to an arbitrary ordering cone some second-order necessary conditions and sufficient conditions given by Liu, Neittaanmaki, Krizek for a polyhedral cone. Furthermore, we show that the conditions obtained are sufficient not only for efficiency but also for strict efficiency.     

A Phase-Space Study of the Quantum Loschmidt Echo in the Semiclassical Limit

The notion of Loschmidt echo (also called “quantum fidelity”) has been introduced in order to study the (in)-stability of the quantum dynamics under perturbations of the Hamiltonian. It has been extensively studied in the past few years in the physics literature, in connection with the problems of “quantum chaos”, quantum computation and decoherence. In this paper, we study this quantity semiclassically (as ), taking as reference quantum states the usual coherent states. The latter are known to be well adapted to a semiclassical analysis, in particular with respect to semiclassical estimates of their time evolution. For times not larger than the so-called “Ehrenfest time” , we are able to estimate semiclassically the Loschmidt Echo as a function of t (time), (Planck constant), and δ (the size of the perturbation). The way two classical trajectories merging from the same point in classical phase-space, fly apart or come close together along the evolutions governed by the perturbed and unperturbed Hamiltonians play a major role in this estimate. We also give estimates of the “return probability” (again on reference states being the coherent states) by the same method, as a function of t and . Submitted: April 27, 2006; Accepted: May 11, 2006     

Elementary properties of free extensions

For any partial groupoid , let Fr be the free extension of to a total groupoid. We show that implies and that the theory of Fr is uniformly recursive in the theory of . These results fail if “groupoid” is replaced by “semigroup”, “commutative semigroup”, “group”, “abelian group”, “semilattice”, “K-lattice” for any nontrivial varietyK of lattices, or “Boolean algebra”. Research supported in part by NSF Grant MCS78-01867. We thank the referee for his valuable comments. Presented by B. Jónsson.     

Multipeak solutions for some singularly perturbed nonlinear elliptic problems on Riemannian manifolds

Given (M, g) a smooth compact Riemannian N-manifold, we prove that for any fixed positive integer K the problem

On the prize-collecting generalized minimum spanning tree problem

The prize-collecting generalized minimum spanning tree problem (PC-GMSTP), is a generalization of the generalized minimum spanning tree problem (GMSTP) and belongs to the hard core of -hard problems. We describe an exact exponential time algorithm for the problem, as well we present several mixed integer and integer programming formulations of the PC-GMSTP. Moreover, we establish relationships between the polytopes corresponding to their linear relaxations and present an efficient solution procedure that finds the optimal solution of the PC-GMSTP for graphs with up 240 nodes.     

On the existence of a minimum integer representation for?weighted voting systems

A basic problem in the theory of simple games and other fields is to study whether a simple game (Boolean function) is weighted (linearly separable). A second related problem consists in studying whether a weighted game has a minimum integer realization. In this paper we simultaneously analyze both problems by using linear programming. For less than 9 voters, we find that there are 154 weighted games without minimum integer realization, but all of them have minimum normalized realization. Isbell in 1958 was the first to find a weighted game without a minimum normalized realization, he needed to consider 12 voters to construct a game with such a property. The main result of this work proves the existence of weighted games with this property with less than 12 voters. This research was partially supported by Grant MTM 2006-06064 of “Ministerio de Ciencia y Tecnología y el Fondo Europeo de Desarrollo Regional” and SGRC 2005-00651 of “Generalitat de Catalunya”, and by the Spanish “Ministerio de Ciencia y Tecnología” programmes ALINEX (TIN2005-05446 and TIN2006-11345).     

The Poisson Kernel for Hardy Algebras

This note contributes to a circle of ideas that we have been developing recently in which we view certain abstract operator algebras H^∞(E), which we call Hardy algebras, and which are noncommutative generalizations of classical H^∞, as spaces of functions defined on their spaces of representations. We define a generalization of the Poisson kernel, which “reproduces” the values, on , of the “functions” coming from H^∞(E). We present results that are natural generalizations of the Poisson integral formula. They also are easily seen to be generalizations of formulas that Popescu developed. We relate our Poisson kernel to the idea of a characteristic operator function and show how the Poisson kernel identifies the “model space” for the canonical model that can be attached to a point in the disc . We also connect our Poisson kernel to various “point evaluations” and to the idea of curvature. The first named author was supported in part by grants from the National Science Foundation and from the U.S.-Israel Binational Science Foundation. The second named author was supported in part by the U.S.-Israel Binational Science Foundation and by the B. and G. Greenberg Research Fund (Ottawa).     

Solving large set covering problems for crew scheduling

This paper presents a computationally effective heuristic method which produces good-quality solutions for large-scale set covering problems with thousands of constraints and about one million variables. The need to solve such large-scale problems arises from a crew scheduling problem of mass transit agencies where the number of work shifts required has to be minimized. This problem may be formulated as a large-scale non-unicost set covering problem whose rows are trips to be performed while columns stand for round trips. The proposed method is mainly based on lagragian relaxation and sub-gradient optimization. After the reduction of the number of rows and columns by the logical tests, “greedy” heuristic algorithms provide upper and lower bounds which are continuously improved to produce goodquality solutions. Computational results, regarding randomly generated problems and real life problems concerning crew scheduling at Italian Railways Company, show that good-quality solutions can be obtained at an acceptable computational cost. This work was supported by the project “Progetto Finalizzato Transporti 2” of National Research Council of Italy (C.N.R.) contract No. 94.01436PF74 and by “Ferrovie dello Stato S.p.A.”     

Dichotomy laws for the behaviour near the unit element of group representations

A well known “zero-two law" shows that if is a strongly continuous one-parameter group of bounded operators on a Banach space X, and if then Here we discuss analogous problems for general unital representations θ of a topological group G on a unital Banach algebra A. Let 1 be the unit of G, and I the unit element of A. We show that either or if, moreover, θ admits “continuous division by any positive integer”, then, either or Our argument also gives automatic continuity results for representations of abelian Baire groups on a separable Banach algebra and representations of compact non abelian groups on a Banach algebra which are locally bounded and satisfy Received: 8 June 2005; revised: 13 October 2005     

An effective heuristic for large-scale capacitated facility location problems

The Capacitated Facility Location Problem (CFLP) consists of locating a set of facilities with capacity constraints to satisfy the demands of a set of clients at the minimum cost. In this paper we propose a simple and effective heuristic for large-scale instances of CFLP. The heuristic is based on a Lagrangean relaxation which is used to select a subset of “promising” variables forming the core problem and on a Branch-and-Cut algorithm that solves the core problem. Computational results on very large scale instances (up to 4 million variables) are reported.     

Double-Regularization Proximal Methods, with Complementarity Applications

We consider the variational inequality problem formed by a general set-valued maximal monotone operator and a possibly unbounded “box” in , and study its solution by proximal methods whose distance regularizations are coercive over the box. We prove convergence for a class of double regularizations generalizing a previously-proposed class of Auslender et al. Using these results, we derive a broadened class of augmented Lagrangian methods. We point out some connections between these methods and earlier work on “pure penalty” smoothing methods for complementarity; this connection leads to a new form of augmented Lagrangian based on the “neural” smoothing function. Finally, we computationally compare this new kind of augmented Lagrangian to three previously-known varieties on the MCPLIB problem library, and show that the neural approach offers some advantages. In these tests, we also consider primal-dual approaches that include a primal proximal term. Such a stabilizing term tends to slow down the algorithms, but makes them more robust. This author was partially supported by CNPq, Grant PQ 304133/2004-3 and PRONEX-Optimization.     

Discrete time dynamic traffic assignment models and solution algorithm for managed lanes

“Managed” lanes of highways usually refer to lanes that are not open to all types of vehicles, such as “High Occupancy Vehicles” (HOV) lanes and “High Occupancy Toll” (HOT) lanes, etc. The HOV lanes of highways are reserved only for vehicles with a driver and one or more passengers. Whereas, HOT lanes allow all vehicles but require tolls from the vehicles with no passenger except the driver. In this paper, we present a discrete-time traffic assignment system optimum model to predict the optimal traffic flows on managed lanes at various times in the entire planning horizon. This model minimizes the overall delay (travel time) and belongs to the class of dynamic traffic assignment (DTA) problems. When applied to general networks, DTA problems can be large and difficult to solve, but the problem is manageable when it is applied to a network with managed lanes. In particular, the DTA model in this paper for managed lanes is reduced to a mixed integer program for which several efficient heuristic algorithms exist. This paper also discusses the special properties of the discrete-time DTA model, based upon which a heuristic algorithm is proposed. Numerical results show that this algorithm is efficient for many cases of the managed lane problems.     

Multi-peak solutions for the Hénon equation with slightly subcritical growth

We study the Dirichlet problem for the Hénon equation

Two-edge connected subgraphs with bounded rings: Polyhedral results and Branch-and-Cut

We consider the network design problem which consists in determining at minimum cost a 2-edge connected network such that the shortest cycle (a “ring”) to which each edge belongs, does not exceed a given length K. We identify a class of inequalities, called cycle inequalities, valid for the problem and show that these inequalities together with the so-called cut inequalities yield an integer programming formulation of the problem in the space of the natural design variables. We then study the polytope associated with that problem and describe further classes of valid inequalities. We give necessary and sufficient conditions for these inequalities to be facet defining. We study the separation problem associated with these inequalities. In particular, we show that the cycle inequalities can be separated in polynomial time when K≤4. We develop a Branch-and-Cut algorithm based on these results and present extensive computational results.     

Powers of rationals modulo 1 and rational base number systems

A new method for representing positive integers and real numbers in a rational base is considered. It amounts to computing the digits from right to left, least significant first. Every integer has a unique expansion. The set of expansions of the integers is not a regular language but nevertheless addition can be performed by a letter-to-letter finite right transducer. Every real number has at least one such expansion and a countable infinite number of them have more than one. We explain how these expansions can be approximated and characterize the expansions of reals that have two expansions. The results that we derive are pertinent on their own and also as they relate to other problems in combinatorics and number theory. A first example is a new interpretation and expansion of the constant K(p) from the so-called “Josephus problem.” More important, these expansions in the base allow us to make some progress in the problem of the distribution of the fractional part of the powers of rational numbers. Work partially supported by the CNRS/JSPS contract 13 569, and by the “ACI Nouvelles Interfaces des Mathématiques”, contract 04 312.     

On the radical of Brauer algebras

The radical of the Brauer algebra is known to be non-trivial when the parameter x is an integer subject to certain conditions (with respect to f). In these cases, we display a wide family of elements in the radical, which are explicitly described by means of the diagrams of the usual basis of . The proof is by direct approach for x  =  0, and via classical Invariant Theory in the other cases, exploiting then the well-known representation of Brauer algebras as centralizer algebras of orthogonal or symplectic groups acting on tensor powers of their standard representation. This also gives a great part of the radical of the generic indecomposable -modules. We conjecture that this part is indeed the whole radical in the case of modules, and it is the whole part in a suitable step of the standard filtration in the case of the algebra. As an application, we find some more precise results for the module of pointed chord diagrams, and for the Temperley–Lieb algebra—realised inside —acting on it.

“Ahi quanto a dir che sia è cosa dura lo radical dell’algebra di Brauer pur se’l pensier già muove a congettura” N. Barbecue, “Scholia”

Partially supported by the European RTN “LieGrits”, contract no. MRTN-CT-2003-505078, and by the Italian PRIN 2005 “Moduli e teorie di Lie”.     

Extremal Problems For Transversals In Graphs With Bounded Degree

We introduce and discuss generalizations of the problem of independent transversals. Given a graph property , we investigate whether any graph of maximum degree at most d with a vertex partition into classes of size at least p admits a transversal having property . In this paper we study this problem for the following properties : “acyclic”, “H-free”, and “having connected components of order at most r”. We strengthen a result of [13]. We prove that if the vertex set of a d-regular graph is partitioned into classes of size d+⌞d/r⌟, then it is possible to select a transversal inducing vertex disjoint trees on at most r vertices. Our approach applies appropriate triangulations of the simplex and Sperner’s Lemma. We also establish some limitations on the power of this topological method. We give constructions of vertex-partitioned graphs admitting no independent transversals that partially settles an old question of Bollobás, Erdős and Szemerédi. An extension of this construction provides vertex-partitioned graphs with small degree such that every transversal contains a fixed graph H as a subgraph. Finally, we pose several open questions. * Research supported by the joint Berlin/Zurichgrad uate program Combinatorics, Geometry, Computation, financed by the German Science Foundation (DFG) and ETH Zürich. † Research partially supported by Hungarian National Research Fund grants T-037846, T-046234 and AT-048826.     

Two-level evolutionary approach to the survivable mesh-based transport network topological design

The complete topology design problem of survivable mesh-based transport networks is to address simultaneously design of network topology, working path routing, and spare capacity allocation based on span-restoration. Each constituent problem in the complete design problem could be formulated as an Integer Programming (IP) and is proved to be NP\mathcal{NP} -hard. Due to a large amount of decision variables and constraints involved in the IP formulation, to solve the problem directly by exact algorithms (e.g. branch-and-bound) would be impractical if not impossible. In this paper, we present a two-level evolutionary approach to address the complete topology design problem. In the low-level, two parameterized greedy heuristics are developed to jointly construct feasible solutions (i.e., closed graph topologies satisfying all the mesh-based network survivable constraints) of the complete problem. Unlike existing “zoom-in”-based heuristics in which subsets of the constraints are considered, the proposed heuristics take all constraints into account. An estimation of distribution algorithm works on the top of the heuristics to tune the control parameters. As a result, optimal solution to the considered problem is more likely to be constructed from the heuristics with the optimal control parameters. The proposed algorithm is evaluated experimentally in comparison with the latest heuristics based on the IP software CPLEX, and the “zoom-in”-based approach on 28 test networks problems. The experimental results demonstrate that the proposed algorithm is more effective in finding high-quality topologies than the IP-based heuristic algorithm in 21 out of 28 test instances with much less computational costs, and performs significantly better than the “zoom-in”-based approach in 19 instances with the same computational costs.     

Spectral conditions for admissibility and observability of wave systems: applications to finite element schemes

In this article, we derive uniform admissibility and observability properties for the finite element space semi-discretizations of , where A ₀ is an unbounded self-adjoint positive definite operator with compact resolvent. To address this problem, we present a new spectral approach based on several spectral criteria for admissibility and observability of such systems. Our approach provides very general admissibility and observability results for finite element approximation schemes of , which stand in any dimension and for any regular mesh (in the sense of finite elements). Our results can be combined with previous works to derive admissibility and observability properties for full discretizations of . We also present applications of our results to controllability and stabilization problems. The author was partially supported by the “Agence Nationale de la Recherche” (ANR), Project C-QUID, number BLAN-3-139579.