Some classes of projected dynamical systems in Banach spaces and variational inequalities

We introduce some Projected Dynamical Systems based on metric and generalized Projection Operator in a strictly convex and smooth Banach Space. Then we prove that critical points of these systems coincide with the solution of a Variational Inequality.     

Global asymptotic stability in a class of generalized Putnam equations

It was conjectured that for every integer m?3 the unique equilibrium c=1 of the generalized Putnam equation

     

Combinatorial aspects of the sensor location problem

In this paper we address the Sensor Location Problem, that is the location of the minimum number of counting sensors, on the nodes of a network, in order to determine the arc flow volume of all the network. Despite the relevance of the problem from a practical point of view, there are very few contributions in the literature and no combinatorial analysis is performed to take into account particular structure of the network. We prove the problem is -complete in different cases. We analyze special classes of graphs that are particularly interesting from an application point of view, for which we give low order polynomial solution algorithms.     

Partnership formation with age-dependent preferences

We analyze a model of partnership formation in which players’ preferences are based on the age of a prospective partner. There are two classes of individuals, called for convenience here male and female. Males and females are fertile for the same length of time, normalized to one unit. A male enters the mating pool at age 0 and meets prospective partners according to a Poisson process. At equilibrium, he accepts a female if the utility from mating exceeds the expected utility from future search, which depends on the acceptance strategies of all males and females and the corresponding steady-state distribution of age in the pool of unmated individuals. Females face an analogous problem. Mating pairs are only formed by mutual consent and individuals leave the pool of unmated individuals on finding a mating partner or reaching the age of 1. A policy iteration algorithm is used to determine the equilibrium acceptance strategies and the corresponding steady-state distribution of the age of individuals in the mating pool. Two examples are presented.     

Simulating vehicular traffic in a network using dynamic routing

We present a new numerical code which solves the Lighthill – Whitham model, the classic macroscopic model for vehicular traffic flow, in a network with multi-destinations. We use a high-resolution shock-capturing scheme with approximate Riemann solver to solve the partial differential equations of the Lighthill – Whitham theory. These schemes are very efficient, robust and moreover well adapted to simulations of traffic flows. We develop a theory of dynamic routing including a procedure for traffic flow assignment at junctions which reproduces the correct propagation of irregularities and ensures at the same time conservation of the number of vehicles.     

Hölder Continuity of the Unique Solution to Quasiequilibrium Problems in Metric Spaces

We extend and sharpen the result in Ait Mansour and Scrimali (J. Glob. Optim. 40:175–184, 2008) on the Hölder continuity of the solution sets of quasivariational inequalities in Hilbert spaces to the case of quasiequilibrium problems in metric spaces. Applications to some important problems are also provided.     

A new iterative method for equilibrium problems and fixed point problems of infinitely nonexpansive mappings and monotone mappings

In this paper, we introduce a new iterative scheme for finding the common element of the set of common fixed points of infinitely many nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for α-inverse-strongly monotone mapping in Hilbert spaces. We prove that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. This main result improve and extend Plubtieng and Punpaeng’s corresponding result [S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings, Applied Mathematics and Computation 197 (2008), 548–558]. Using this theorem, we obtain three corollaries.     

A modified epidemiological model for computer viruses

Since the computer viruses pose a serious problem to individual and corporative computer systems, a lot of effort has been dedicated to study how to avoid their deleterious actions, trying to create anti-virus programs acting as vaccines in personal computers or in strategic network nodes. Another way to combat viruses propagation is to establish preventive policies based on the whole operation of a system that can be modeled with population models, similar to those that are used in epidemiological studies. Here, a modified version of the SIR (Susceptible-Infected-Removed) model is presented and how its parameters are related to network characteristics is explained. Then, disease-free and endemic equilibrium points are calculated, stability and bifurcation conditions are derived and some numerical simulations are shown. The relations among the model parameters in the several bifurcation conditions allow a network design minimizing viruses risks.     

Order statistics and estimating cardinalities of massive data sets

A new class of algorithms to estimate the cardinality of very large multisets using constant memory and doing only one pass on the data is introduced here. It is based on order statistics rather than on bit patterns in binary representations of numbers. Three families of estimators are analyzed. They attain a standard error of using M units of storage, which places them in the same class as the best known algorithms so far. The algorithms have a very simple internal loop, which gives them an advantage in terms of processing speed. For instance, a memory of only 12 kB and only few seconds are sufficient to process a multiset with several million elements and to build an estimate with accuracy of order 2 percent. The algorithms are validated both by mathematical analysis and by experimentations on real internet traffic.     

An efficient algorithm for optimal design of area traffic control with network flows

An equilibrium network design (EQND) is a problem of finding the optimal design parameters while taking into account the route choice of users. This problem can be formulated as an optimization by taking the user equilibrium traffic assignment as a constraint. In this paper, the methods solving the EQND problem with signal settings are investigated via numerical calculations on two example road networks. An efficient algorithm is proposed in which improvement on a locally optimal search by combining the technique of parallel tangents with the gradient projection method is presented. As it shows, the method combines the locally optimal search and globally search heuristic achieved substantially better performance than did those other approaches.