Models, Algorithms, and Technologies for Network Analysis
This volume contains a selection of contributions from the "First International Conference in Network Analysis," held at the University of Florida, Gainesville, on December 14-16, 2011. The remarkable diversity of fields that take advantage of Network Analysis makes the endeavor of gathering up-to-date material in a single compilation a useful, yet very difficult, task. The purpose of this volume is to overcome this difficulty by collecting the major results found by the participants and combining them in one easily accessible compilation.
The notion of a metric modular on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces and Orlicz spaces, were recently introduced and studied by the author [Chistyakov: Dokl. Math. 73(1):32–35, 2006 and Nonlinear Anal. 72(1):1–30, 2010]. In this chapter we present yet one more application of the metric modulars theory to the existence of fixed points of modular contractive maps in modular metric spaces. These are related to contracting generalized average velocities rather than metric distances, and the successive approximations of fixed points converge to the fixed points in the modular sense, which is weaker than the metric convergence. We prove the existence of solutions to a Carathéodory-type differential equation with the right-hand side from the Orlicz space. Metric modular, Modular convergence, Modular contraction, Fixed point, Mapping of finite f-variation, Carathéodory-type differential equation
The distribution of the sum of independent random variables plays an important role in many problems of applied mathematics. In this paper we concentrate on the case when random variables have a continuous distribution with a discontinuity (or a probability mass) at a certain point r. Such a distribution arises naturally in actuarial mathematics when a responsibility or a retention limit is applied to every claim payment. An analytical expression for the distribution of the sum of i.i.d. random variables, which have a uniform distribution with a discontinuity, is reported.
In this paper we introduce a new pattern-based approach within the Linear Assignment Model with the purpose to design heuristics for a combinatorial optimization problem (COP). We assume that the COP has an additive (separable) objective function and the structure of a feasible (optimal) solution to the COP is predefined by a collection of cells (positions) in an input file. We define a pattern as a collection of positions in an instance problem represented by its input file (matrix). We illustrate the notion of pattern by means of some well known problems in COP among them the Linear Ordering Problem, Cell Formation Problem (CFP) just to mention a couple. The CFP is defined on a Boolean input matrix which rows represent machines and columns - parts. The CFP consists in finding three optimal objects: a block-diagonal collection of rectangles, a rows (machines) permutation, and a columns (parts) permutation such that the grouping efficacy is maximized. The suggested heuristic combines two procedures: the pattern-based procedure to build an initial solution and an improvement procedure to obtain a final solution with high grouping efficacy for the CFP. Our computational experiments with the most popular set of 35 benchmark instances show that our heuristic outperforms all well known heuristics and returns either the best known or improved solutions to the CFP.
Dynamics of solitons in the frame of the extended nonlinear Schr¨odinger equation (NSE) taking into account stimulated Raman scattering (SRS) and inhomogeneous second-order dispersion (SOD) is considered. Compensation of soliton Raman self-wave number downshift in media with increasing second-order linear dispersion is shown. Quasi-soliton solution with small wave number spectrum variation, amplitude and extension are found analytically in adiabatic approximation and numerically. The soliton is considered as the equilibrium of SRS and increasing SOD. For dominate SRS soliton wave number spectrum tends to long wave region. For dominate increasing SOD soliton wave number spectrum tends to shortwave region.