Active shielding model for hyperbolic equations

^** Email: s.utyuzhnikov{at}manchester.ac.uk The problem of active shielding (AS) in application to hyperbolicequations is analysed. According to the problem, two domainseffecting each other via distributed source terms are considered.It is required to implement additional sources nearby the commonboundary of the domains in order to "isolate" one domain fromthe action of the other domain. It is important to note thatthe total field of the original sources is only known. In thepaper, the theory of difference potentials is applied to thesystem of hyperbolic equations for the first time. It allowsone to obtain a one-layer AS not requiring any additional computations.Local one-layer and two-layer AS sources are obtained for anarbitrary hyperbolic system. The solution does not require eitherthe knowledge of the Green's function or the specific characteristicsof the sources and medium. The optimal one-layer AS solutionis derived in the case of free space. In particular, the resultsare applicable to the system of acoustics equations. The questionsrelated to a practical realization including the mutual situationof the primary and secondary sources, as well as the measurementpoint, are discussed. The active noise shielding can be realizedvia a one-layer source term requiring the measurements onlyat one layer nearby the domain shielded.     

Why Restricted Additive Schwarz Converges Faster than Additive Schwarz

Recently a variant of the additive Schwarz (AS) preconditioner, the restricted additive Schwarz (RAS) preconditioner has been introduced, and numerical experiments showed that RAS converges faster and requires less communication than AS. We show in this paper how RAS, which is defined at the matrix level, can be interpreted as an iteration at the continuous level of the underlying problem. This interpretation reveals why RAS converges faster than classical AS.This revised version was published online in October 2005 with corrections to the Cover Date.     

Certain qualitative investigation methods for dynamic systems connected with field rotation : PMM vol. 37, n6, 1973, pp. 984–989

We show that the inevitability of realizing bifurcations connected with a double (and triple) limit cycle or with a separatrix loop can, in some cases, be detected from the global evaluation of trajectory behavior under parameter variations by reckoning the sign of the saddle term, and we turn our attention to new possibilities of tracing the bifurcations arising from the use of a monotonic field rotation. The methods of bifurcation theory [1] are widely used in problems of mechanics. However, in the general case, the problem of investigating all possible bifurcations is difficult and regular methods for solving it do not exist. There are no criteria locally connected with the points of the phase space or of the parameter space, which stipulate the actual realizability in a concrete dynamic system of bifurcations connected with a separatrix loop. A similar situation exists for bifurcations connected with the arising of limit cycles from the condensation of trajectories, because we usually know neither the equations of the limit cycle nor the parameter values under which it arises. In a number of the methods used in bifurcation theory an important role is played by the local rotation of the field in a neighborhood of the singular trajectories of the system [1, 2]. In a number of cases the carrying over of this idea to the whole phase space and to the parameter space in the large (realizable in the presence of specific singularities of the system being investigated) permits us not only to trace all bifurcations possible in the system but also to predetermine the disposition of the bifurcation curves or surfaces.     

Sweeping graphs with large clique number

Searching a network for intruders is an interesting and often difficult problem. Sweeping (or edge searching) is one such search model, in which intruders may exist anywhere along an edge. It was conjectured that graphs exist for which the connected sweep number is strictly less than the monotonic connected sweep number. We prove that this is true, and the difference can be arbitrarily large. We also show that the clique number is a lower bound on the sweep number.     

On the block upper-triangularity of undiscounted multi-chain markov decision problems

We show that the LP formulation for an undiscounted multi-chain Markov decision problem can be put in a block upper-triangular form by a polynomial time procedure. Each minimal block (after an appropriate dynamic revision) gives rise to a single-chain Markov decision problem which can be treated independently. An optimal solution to each single-chain problem can be connected by auxiliary dual programs to obtain an optimal solution to a multi-chain problem.     

Tiling simply connected regions with rectangles

In 1995, Beauquier, Nivat, Rémila, and Robson showed that tiling of general regions with two bars is NP-complete, except for a few trivial special cases. In a different direction, in 2005, Rémila showed that for simply connected regions by two rectangles, the tileability can be solved in quadratic time (in the area). We prove that there is a finite set of at most 10⁶ rectangles for which the tileability problem of simply connected regions is NP-complete, closing the gap between positive and negative results in the field. We also prove that counting such rectangular tilings is #P-complete, a first result of this kind.     

Subexponential parameterized algorithms for degree-constrained subgraph problems on planar graphs

We present subexponential parameterized algorithms on planar graphs for a family of problems of the following shape: given a graph, find a connected (induced) subgraph with bounded maximum degree and with maximum number of edges (or vertices). These problems are natural generalisations of the Longest Path problem. Our approach uses bidimensionality theory combined with novel dynamic programming techniques over branch decompositions of the input graph. These techniques can be applied to a more general family of problems that deal with finding connected subgraphs under certain degree constraints.     

Two-dimensional problem of electromagnetoelasticity for multiply connected media

We propose a method for the solution of connected two-dimensional and plane problems of electromagnetoelasticity for multiply connected domains. Basic relations of two-dimensional and plane problems are obtained. Generalized complex potentials of electromagnetoelasticity are introduced and investigated. Boundary conditions for their determination and, using them, expressions of main characteristics of the electromagnetoelastic state (stresses, displacements, electromagnetic field intensity vectors, induction vectors, potentials of the electric and magnetic fields) are obtained. We present a solution of the problem for a plate with an elliptic hole or a crack.     

Algebraic substructuring (AS) for eigenvalue and frequency response calculations

Eigenvalue and frequency response calculations are ubiquitous in scientific modeling and engineering analysis. Algebraic substructuring (AS) method is a powerful numerical technique for solving extremely large scale problems. We developed a unified framework and AS code that can solve both problems efficiently. Furthermore, we addressed some of the open problems in this field, including resolving arbitrary eigenmodes, performing high frequency response analysis, accuracy and performance. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Budget and long-range planning in public transportation — An applicative linear programming model

Economic and financial planning is an actual problem for Italian Transport Authorities, since in Italy there are scarce financial resources, to cover either the difference between costs and fares proceeds or investment needs. The proposed model is ‘just tailored’ on public transport, subjected to the Italian laws; it takes into account the particular activities (‘functions’) connected with public transport operating (e.g.: fare collection, service production, maintenance, purchases, inventories, administration). Each function is considered as a ‘module’ with its own constraints, and it is connected with the other ‘modules’, so that we can get a ‘representative’ model. Also the most significant parameters of the ‘public transport operating problem’ (e.g.: vehicle miles, passenger journeys, number of vehicles, number of employees, and so on) are connected one another. Since all relations and constraints, connecting the ‘problem variables’ can be represented as linear, the model structure is based on linear programming; this fact allows to pursue an optimum for one or more objective functions, each of them identifing an operational policy. So it is possible to connect simulation with optimization. A multi-period model can be used for long-range planning, pursuing a multi-year optimum; this is the most significant use. The Administrative Module contains Balance-Sheet, Cash-Flow, Profit and Loss Account, which are ‘constraints’ in the model and are expressed as Italian financial laws require; this structure is one of the main characteristics. Modules' dimensions and additional constraints (e.g. economic or financial ratios, fare level, turnover of inventories, and so on) can be ‘tailored’ on each particular case, driving the model towards more realistic solutions.     

On a Paper by Barden

It is shown that an approach earlier used by the author for classification of closed simply connected 6-manifolds (reduction to the problem of calculating certain bordism groups) can also be applied for easily obtaining the results by Barden (1965) on classification of closed simply connected 5-manifolds. Bibliography: 11 titles.     

Randomized heuristics for handover minimization in mobility networks

A mobile device connects to the cell tower (base station) from which it receives the strongest signal. As the device moves it may connect to a series of towers. The process in which the device changes the base station it is connected to is called handover. A cell tower is connected to a radio network controller (RNC) which controls many of its operations, including handover. Each cell tower handles an amount of traffic and each radio network controller has capacity to handle a maximum amount of traffic from all base stations connected to it. Handovers between base stations connected to different RNCs tend to fail more often than handovers between base stations connected to the same RNC. Handover failures result in dropped connections and therefore should be minimized. The Handover Minimization Problem is to assign towers to RNCs such that RNC capacity is not violated and the number of handovers between base stations connected to different RNCs is minimized. We describe an integer programming formulation for the handover minimization problem and show that state-of-the-art integer programming solvers can solve only very small instances of the problem. We propose several randomized heuristics for finding approximate solutions of this problem, including a GRASP with path-relinking for the generalized quadratic assignment problem, a GRASP with evolutionary path-relinking, and a biased random-key genetic algorithm. Computational results are presented.     

On the Computational Complexity of a Rigidity Problem

Consider a structure of flexible joints connected by rigid bars.These bars will constrain the possible motions of the jointsof this structure. By "pinning down" some of the joints so thatthey cannot move further constraints will be added. In thisway the entire structure can be made rigid. A problem consideredby Bolker & Crapo (1977) and others, is that of findingthe minimum number of joints that must be pinned in order tomake a given two- or three-dimensional structure rigid. We considerthe computational complexity of this problem. Lovasz (1980)gives a somewhat complicated but polynomial time procedure forthis problem in the two-dimensional case. In this paper we showthat in three or more dimensions the problem is NP-complete,and so is unlikely to have a polynomial time algorithm.     

The Generic Stability and Existence of Essentially Connected Components of Solutions for Nonlinear Complementarity Problems

The aim of this paper is to develop the general generic stability theory for nonlinear complementarity problems in the setting of infinite dimensional Banach spaces. We first show that each nonlinear complementarity problem can be approximated arbitrarily by a nonlinear complementarity problem which is stable in the sense that the small change of the objective function results in the small change of its solution set; and thus we say that almost all complementarity problems are stable from viewpoint of Baire category. Secondly, we show that each nonlinear complementarity problem has, at least, one connected component of its solutions which is stable, though in general its solution set may not have good behaviour (i.e., not stable). Our results show that if a complementarity problem has only one connected solution set, it is then always stable without the assumption that the functions are either Lipschitz or differentiable.     

Impact of the pheromone trail on the performance of ACO algorithms for solving the <Emphasis Type="Italic">car-sequencing</Emphasis> problem

This paper compares different ant colony optimization algorithms for solving the NP-hard car-sequencing problem, which is of great practical interest. The five algorithms that are compared are the Ant System (AS), the Elitist AS, the Rank-Based AS, the Max–Min AS and the Ant Colony System. These algorithms, which are well known in the literature, differ in the way in which the pheromone trail is managed. The comparative analysis seeks to identify which algorithm best manages the learning process in solving the car-sequencing problem. Moreover, we propose a new structure for the pheromone trail specifically designed to take advantage of the type of constraints found in the car-sequencing problem. The quality of the results obtained with this new form of learning for three problem sets drawn from the literature is superior to that of the best results published and demonstrates the efficiency of this new trail structure.     

Bigraphs versus digraphs via matrices

It was observed by Dulmage and Mendelsohn in their work on matrix reducibility that there is a one-to-one correspondence between bigraphs and digraphs determined by the utilization of the adjacency matrix. In this semiexpository paper we explore the interaction between this correspondence and a theory of matrix decomposability that is developed in several different articles. These results include: (a) a characterization of those bipartite graphs that can be labeled so that the resulting digraph is symmetric; (b) a criterion for the bigraph of a symmetric digraph to be connected; (c) a necessary and sufficient condition for a square binary matrix to be fully indecomposable in terms of its associated bigraph, and (d) matrix criteria for a digraph to be strongly, unilaterally, or weakly connected. We close with an unsolved extermal problem on the number of components of the bigraph of various orientations of a given graph. This leads to new amusing characterizations of trees and bigraphs. Dedicated to the graph-theoretic partnership of Lloyd Dulmage and Nathan Mendelsohn.     

Note on the flux condition and pressure drop in the resolvent problem of the Stokes system

In [4] H. Sohr and the author considered theL ^q-theory of the resolvent problem of the generalized Stokes system in an aperture domain. This type of unbounded domain consists of two disjoint half spaces which are separated from each other by a wall but connected by a hole (aperture) in this wall. Due to this geometry the flux of the velocity field through the hole and the pressure drop at infinity are important physical and mathematical quantities. In this note we show that in order to single out a unique solution of the resolvent problem we must prescribe the flux for largeq, but that for smallq neither the flux nor the pressure drop can be prescribed. Only if the dimension is greater than two there is a certain range of values ofq where we must prescribe either the flux or the pressure drop. As a limit case we also investigate strong solutions of the Stokes system.     

Upper bounds for the forwarding indices of communication networks

The decision version of the forwarding index problem is, given a connected graph G and an integer ξ, to find a way of connecting each ordered pair of vertices by a path so that every vertex is an internal point of at most such paths. The optimization version of the problem is to find the smallest ξ for which a routing of this kind exists. Such a problem arises in the design of communication networks and distributed architectures. A model of parallel computation is represented by a network of processors, or machines processing and forwarding (synchronous) messages to each other, subject to physical constraints bearing on either the number of messages that can be processed by a single machine or the number of messages that can be sent through a connection. It was in this context that the problem was first introduced by Chung et al. (IEEE Trans. Inform. Theory 33 (1987) 224). The aim of this paper is to establish upper bounds for the optimal ξ as a function of the connectivity of the graph.