Percolation and limit theory for the poisson lilypond model

The lilypond model on a point process in d ‐space is a growth‐maximal system of non‐overlapping balls centred at the points. We establish central limit theorems for the total volume and the number of components of the lilypond model on a sequence of Poisson or binomial point processes on expanding windows. For the lilypond model over a homogeneous Poisson process, we give subexponentially decaying tail bounds for the size of the cluster at the origin. Finally, we consider the enhanced Poisson lilypond model where all the balls are enlarged by a fixed amount (the enhancement parameter), and show that for d > 1 the critical value of this parameter, above which the enhanced model percolates, is strictly positive. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012     

Convergence to equilibrium for a fully hyperbolic phase‐field model with Cattaneo heat flux law

We consider a fully hyperbolic phase‐field model in this paper. Our model consists of a damped hyperbolic equation of second order with respect to the phase function χ(t) , which is coupled with a hyperbolic system of first order with respect to the relative temperature θ(t) and the heat flux vector q (t). We prove the well‐posedness of this system subject to homogeneous Neumann boundary condition and no‐heat flux boundary condition. Then, we show that this dynamical system is a dissipative one. Finally, using the celebrated ?ojasiewicz–Simon inequality and by constructing an auxiliary functional, we prove that the solution of this problem converges to an equilibrium as time goes to infinity. We also obtain an estimate of the decay rate to equilibrium. Copyright © 2008 John Wiley & Sons, Ltd.     

Confined steady states of a Vlasov‐Poisson plasma in an infinitely long cylinder

We consider the two‐dimensional Vlasov‐Poisson system to model a two‐component plasma whose distribution function is constant with respect to the third space dimension. First, we show how this two‐dimensional Vlasov‐Poisson system can be derived from the full three‐dimensional model. The existence of compactly supported steady states with vanishing electric potential in a three‐dimensional setting has already been investigated in the literature. We show that these results can easily be adapted to the two‐dimensional system. However, our main result is to prove the existence of compactly supported steady states even with a nontrivial self‐consistent electric potential.     

Simulation of a compositional model for multiphase flow in porous media

In this article, we consider the simulation of a compositional model for three‐dimensional, three‐phase, multicomponent flow in a porous medium. This model consists of Darcy's law for volumetric flow velocities, mass conservation for hydrocarbon components, thermodynamic equilibrium for mass interchange between phases, and an equation of state for saturations. A discretization scheme based on the block‐centered finite difference method for pressures and compositions is developed. Numerical results are reported for the benchmark problem of the third comparative solution project (CSP) organized by the society of petroleum engineers (SPE). © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

Dynamical study of fractional order differential equations of predator‐pest models

To explore the impact of pest‐control strategy through a fractional derivative, we consider three predator‐prey systems by simple modification of Rosenzweig‐MacArthur model. First, we consider fractional‐order Rosenzweig‐MacArthur model. Allee threshold phenomena into pest population is considered for the second case. Finally, we consider additional food to the predator and harvesting in prey population. The main objective of the present investigation is to observe which model is most suitable for the pest control. To achieve this goal, we perform the local stability analysis of the equilibrium points and observe the basic dynamical properties of all the systems. We observe fractional‐order system has the ability to stabilize Rosenzweig‐MacArthur model with low pest density from oscillatory state. In the numerical simulations, we focus on the bistable regions of the second and third model, and we also observe the effect of the fractional order α throughout the stability region of the system. For the third model, we observe a saddle‐node bifurcation due to the additional food and Allee effect to the pest densities. Also, we numerically plot two parameter bifurcation diagram with respect to the harvesting parameter and fractional order of the system. We finally conclude that fractional‐order Rosenzweig‐MacArthur model and the modified Rosenzweig‐MacArthur model with additional food for the predator and harvested pest population are more suitable models for the pest management.     

On a class of renewal risk model with random income

In this paper, we consider a renewal risk process with random premium income based on a Poisson process. Generating function for the discounted penalty function is obtained. We show that the discounted penalty function satisfies a defective renewal equation and the corresponding explicit expression can be obtained via a compound geometric tail. Finally, we consider the Laplace transform of the time to ruin, and derive the closed‐form expression for it when the claims have a discrete K_m distribution (i.e. the generating function of the distribution function is a ratio of two polynomials of order m∈?⁺). Copyright © 2008 John Wiley & Sons, Ltd.     

The fraction of large random trees representing a given Boolean function in implicational logic

We consider the logical system of Boolean expressions built on the single connector of implication and on positive literals. Assuming all expressions of a given size to be equally likely, we prove that we can define a probability distribution on the set of Boolean functions expressible in this system. Then we show how to approximate the probability of a function f when the number of variables grows to infinity, and that this asymptotic probability has a simple expression in terms of the complexity of f. We also prove that most expressions computing any given function in this system are “simple”, in a sense that we make precise. The probability of all read‐once functions of a given complexity is also evaluated in this model. At last, using the same techniques, the relation between the probability of a function and its complexity is also obtained when random expressions are drawn according to a critical branching process. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Consecutive k-out-of-n systems with maintenance

A consective k-out-of-n system consists of n linearly or cycliccally ordered components such that the system fails if and only if at least k consecutive components fail. In this paper we consider a maintained system where each component is repaired independently of the others according to an exponential distribution. Assuming general lifetime distributions for system's components we prove a limit theorem for the time to first failure of both linear and circular systems.     

Effort Games and the Price of Myopia

We consider Effort Games, a game‐theoretic model of cooperation in open environments, which is a variant of the principal‐agent problem from economic theory. In our multiagent domain, a common project depends on various tasks; carrying out certain subsets of the tasks completes the project successfully, while carrying out other subsets does not. The probability of carrying out a task is higher when the agent in charge of it exerts effort, at a certain cost for that agent. A central authority, called the principal, attempts to incentivize agents to exert effort, but can only reward agents based on the success of the entire project. We model this domain as a normal form game, where the payoffs for each strategy profile are defined based on the different probabilities of carrying out each task and on the boolean function that defines which task subsets complete the project, and which do not. We view this boolean function as a simple coalitional game, and call this game the underlying coalitional game. We suggest the Price of Myopia (PoM) as a measure of the influence the model of rationality has on the minimal payments the principal has to make in order to motivate the agents in such a domain to exert effort. We consider the computational complexity of testing whether exerting effort is a dominant strategy for an agent, and of finding a reward strategy for this domain, using either a dominant strategy equilibrium or using iterated elimination of dominated strategies. We show these problems are generally #P‐hard, and that they are at least as computationally hard as calculating the Banzhaf power index in the underlying coalitional game. We also show that in a certain restricted domain, where the underlying coalitional game is a weighted voting game with certain properties, it is possible to solve all of the above problems in polynomial time. We give bounds on PoM in weighted voting effort games, and provide simulation results regarding PoM in another restricted class of effort games, namely effort games played over Series‐Parallel Graphs (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A primal‐dual method for the Meyer model of cartoon and texture decomposition

In this paper, we study the original Meyer model of cartoon and texture decomposition in image processing. The model, which is a minimization problem, contains an l₁‐based TV‐norm and an l_∞‐based G‐norm. The main idea of this paper is to use the dual formulation to represent both TV‐norm and G‐norm. The resulting minimization problem of the Meyer model can be given as a minimax problem. A first‐order primal‐dual algorithm can be developed to compute the saddle point of the minimax problem. The convergence of the proposed algorithm is theoretically shown. Numerical results are presented to show that the original Meyer model can decompose better cartoon and texture components than the other testing methods.     

Cycle Double Covers in Cubic Graphs having Special Structures

In the first part of this article, we employ Thomason's Lollipop Lemma 25 to prove that bridgeless cubic graphs containing a spanning lollipop admit a cycle double cover (CDC) containing the circuit in the lollipop; this implies, in particular, that bridgeless cubic graphs with a 2‐factor F having two components admit CDCs containing any of the components in the 2‐factor, although it need not have a CDC containing all of F. As another example consider a cubic bridgeless graph containing a 2‐factor with three components, all induced circuits. In this case, two of the components may separately be used to start a CDC although it is uncertain whether the third component may be part of some CDC. Numerous other corollaries shall be given as well. In the second part of the article, we consider special types of bridgeless cubic graphs for which a prominent circuit can be shown to be included in a CDC. The interest here is the proof technique and therefore we only give the simplest case of the theorem. Notably, we show that a cubic graph that consists of an induced 2k‐circuit C together with an induced 4k‐circuit T and an independent set of 2k vertices, each joined by one edge to C and two edges to T, has a CDC starting with T.     

Sampling and multilevel coarsening algorithms for fast matrix approximations

This paper addresses matrix approximation problems for matrices that are large, sparse, and/or representations of large graphs. To tackle these problems, we consider algorithms that are based primarily on coarsening techniques, possibly combined with random sampling. A multilevel coarsening technique is proposed, which utilizes a hypergraph associated with the data matrix and a graph coarsening strategy based on column matching. We consider a number of standard applications of this technique as well as a few new ones. Among standard applications, we first consider the problem of computing partial singular value decomposition, for which a combination of sampling and coarsening yields significantly improved singular value decomposition results relative to sampling alone. We also consider the column subset selection problem, a popular low‐rank approximation method used in data‐related applications, and show how multilevel coarsening can be adapted for this problem. Similarly, we consider the problem of graph sparsification and show how coarsening techniques can be employed to solve it. We also establish theoretical results that characterize the approximation error obtained and the quality of the dimension reduction achieved by a coarsening step, when a proper column matching strategy is employed. Numerical experiments illustrate the performances of the methods in a few applications.     

Optimal Control for an SEIR Epidemic Model with Nonlinear Incidence Rate

The objective of this paper is to explore possible impacts of nonlinearity of functional responses and a number of compartments of an infection disease model on principal qualitative properties of the optimal controls. To address this issue, we consider optimal controls for an Susceptible‐Exposed‐Infectious‐Removed (SEIR) model of an endemically persisting infectious disease. We assume that the incidence rate is given by an unspecified nonlinear function constrained by a few biologically motivated conditions. For this model, we consider five controls (which comprise all controls that are possible for this model) with a possibility of acting simultaneously, and establish principal qualitative properties of the controls. A comparison with a similar SIR model is provided.     

Universality and extremal aging for dynamics of spin glasses on subexponential time scales

We consider random hopping time (RHT) dynamics of the Sherrington‐Kirkpatrick (SK) model and p‐spin models of spin glasses. For any of these models and for any inverse temperature β > 0 we prove that, on time scales that are subexponential in the dimension, the properly scaled clock process (time‐change process) of the dynamics converges to an extremal process. Moreover, on these time scales, the system exhibits aging‐like behavior, which we call extremal aging. In other words, the dynamics of these models ages as the random energy model (REM) does. Hence, by extension, this confirms Bouchaud's REM‐like trap model as a universal aging mechanism for a wide range of systems that, for the first time, includes the SK model. © 2011 Wiley Periodicals, Inc.     

Non‐homogeneous Navier–Stokes systems with order‐parameter‐dependent stresses

We consider the Navier–Stokes system with variable density and variable viscosity coupled to a transport equation for an order‐parameter c. Moreover, an extra stress depending on c and ?c, which describes surface tension like effects, is included in the Navier–Stokes system. Such a system arises, e.g. for certain models of granular flows and as a diffuse interface model for a two‐phase flow of viscous incompressible fluids. The so‐called density‐dependent Navier–Stokes system is also a special case of our system. We prove short‐time existence of strong solution in L^q‐Sobolev spaces with q>d. We consider the case of a bounded domain and an asymptotically flat layer with a combination of a Dirichlet boundary condition and a free surface boundary condition. The result is based on a maximal regularity result for the linearized system. Copyright © 2010 John Wiley & Sons, Ltd.     

Commuting projections on graphs

Motivated by the increasing importance of large‐scale networks typically modeled by graphs, this paper is concerned with the development of mathematical tools for solving problems associated with the popular graph Laplacian. We exploit its mixed formulation based on its natural factorization as product of two operators. The goal is to construct a coarse version of the mixed graph Laplacian operator with the purpose to construct two‐level, and by recursion, a multilevel hierarchy of graphs and associated operators. In many situations in practice, having a coarse (i.e., reduced dimension) model that maintains some inherent features of the original large‐scale graph and respective graph Laplacian offers potential to develop efficient algorithms to analyze the underlined network modeled by this large‐scale graph. One possible application of such a hierarchy is to develop multilevel methods that have the potential to be of optimal complexity. In this paper, we consider general (connected) graphs and function spaces defined on its edges and its vertices. These two spaces are related by a discrete gradient operator, ‘Grad’ and its adjoint, ‘ ? Div’, referred to as (negative) discrete divergence. We also consider a coarse graph obtained by aggregation of vertices of the original one. Then, a coarse vertex space is identified with the subspace of piecewise constant functions over the aggregates. We consider the ?₂‐projection Q_H onto the space of these piecewise constants. In the present paper, our main result is the construction of a projection π_H from the original edge‐space onto a properly constructed coarse edge‐space associated with the edges of the coarse graph. The projections π_H and Q_H commute with the discrete divergence operator, that is, we have Div π_H = Q_H div. The respective pair of coarse edge‐space and coarse vertex‐space offer the potential to construct two‐level, and by recursion, multilevel methods for the mixed formulation of the graph Laplacian, which utilizes the discrete divergence operator. The performance of one two‐level method with overlapping Schwarz smoothing and correction based on the constructed coarse spaces for solving such mixed graph Laplacian systems is illustrated on a number of graph examples. Copyright © 2013 John Wiley & Sons, Ltd.     

Asymptotic expansion for harmonic functions in the half‐space with a pressurized cavity

In this paper, we address a simplified version of a problem arising from volcanology. Specifically, as a reduced form of the boundary value problem for the Lamé system, we consider a Neumann problem for harmonic functions in the half‐space with a cavity C. Zero normal derivative is assumed at the boundary of the half‐space; differently, at ?C, the normal derivative of the function is required to be given by an external datum g, corresponding to a pressure term exerted on the medium at ?C. Under the assumption that the (pressurized) cavity is small with respect to the distance from the boundary of the half‐space, we establish an asymptotic formula for the solution of the problem. Main ingredients are integral equation formulations of the harmonic solution of the Neumann problem and a spectral analysis of the integral operators involved in the problem. In the special case of a datum g, which describes a constant pressure at ?C, we recover a simplified representation based on a polarization tensor. Copyright © 2016 John Wiley & Sons, Ltd.     

Attractors of reaction‐diffusion systems in unbounded domains and their spatial complexity

The nonlinear reaction‐diffusion system in an unbounded domain is studied. It is proven that, under some natural assumptions on the nonlinear term and on the diffusion matrix, this system possesses a global attractor ?? in the corresponding phase space. Since the dimension of the attractor happens to be infinite, we study its Kolmogorov's ?‐entropy. Upper and lower bounds of this entropy are obtained. Moreover, we give a more detailed study of the attractor for the spatially homogeneous RDE in ?ⁿ. In this case, a group of spatial shifts acts on the attractor. In order to study the spatial complexity of the attractor, we interpret this group as a dynamical system (with multidimensional “time” if n > 1) acting on a phase space ??. It is proven that the dynamical system thus obtained is chaotic and has infinite topological entropy. In order to clarify the nature of this chaos, we suggest a new model dynamical system that generalizes the symbolic dynamics to the case of the infinite entropy and construct the homeomorphic (and even Lipschitz‐continuous) embedding of this system into the spatial shifts on the attractor. Finally, we consider also the temporal evolution of the spatially chaotic structures in the attractor and prove that the spatial chaos is preserved under this evolution. © 2003 Wiley Periodicals, Inc.     

Large deviations for a catalytic Fleming‐Viot branching system

We consider a jump‐diffusion process describing a system of diffusing particles that upon contact with an obstacle (catalyst) die and are replaced by an independent offspring with position chosen according to a weighted average of the remaining particles. The obstacle is a bounded nonnegative function V(x) and the birth/death mechanism is similar to the Fleming‐Viot critical branching. Since the mass is conserved, we prove a hydrodynamic limit for the empirical measure, identified as the solution to a generalized semilinear (reaction‐diffusion) equation, with nonlinearity given by a quadratic operator. A large‐deviation principle from the deterministic hydrodynamic limit is provided. The upper bound is given in any dimension, and the lower bound is proven for d = 1 and V bounded away from 0. An explicit formula for the rate function is provided via an Orlicz‐type space. © 2006 Wiley Periodicals, Inc.     

Maximizing set function formulation of two scheduling problems

In this note we consider two problems: (1) Schedulingn jobs non-preemptively on a single machine to minimize total weighted earliness and tardiness (WET). (2) Schedulingn jobs nonpreemptively on two parallel identical processors to minimize weighted mean flow time (WMFT). A new approach for these problems is presented. The approach is based on a problem of maximizing a submodular set function. Heuristic algorithm for the problems also is presented.