Dynamics of a population in two patches with dispersal

A two-dimensional discrete system of a species in two patches proposed by Newman et al. is studied. It is shown that the unique interior steady state is globally asymptotically stable if the active population has a Beverton–Holt type growth rate. If the population is also subject to Allee effects, then the system has two interior steady states whenever the density-independent growth rate is large. In addition, the model has period-two solutions if the symmetric dispersal exceeds a critical threshold. For small dispersal, populations may either go extinct or eventually stabilize. However, populations are oscillating over time if dispersal is beyond the critical value and the initial populations are large.     

New conditions for non-stagnation of minimal residual methods

In the solution of large linear systems, a condition guaranteeing that a minimal residual Krylov subspace method makes some progress, i.e., that it does not stagnate, is that the symmetric part of the coefficient matrix be positive definite. This condition results in a well-established worst-case bound for the convergence rate of the iterative method, due to Elman. This bound has been extensively used, e.g., when the linear system comes from discretized partial differential equations, to show that the convergence of GMRES is independent of the underlying mesh size. In this paper we introduce more general non-stagnation conditions, which do not require the symmetric part of the coefficient matrix to be positive definite, and that guarantee, for example, the non-stagnation of restarted GMRES for certain values of the restarting parameter. Work on this paper was supported in part by the U.S. Department of Energy under grant DE-FG02-05ER25672.     

An Atomistic-to-Continuum Analysis of Crystal Cleavage in a Two-Dimensional Model Problem

A two-dimensional atomic mass spring system is investigated for critical fracture loads and its crack path geometry. We rigorously prove that, in the discrete-to-continuum limit, the minimal energy of a crystal under uniaxial tension leads to a universal cleavage law and energy minimizers are either homogeneous elastic deformations or configurations that are completely cracked and do not store elastic energy. Beyond critical loading, the specimen generically cleaves along a unique optimal crystallographic hyperplane. For specific symmetric crystal orientations, however, cleavage might fail. In this case a complete characterization of possible limiting crack geometries is obtained.     

New Korn-type inequalities and regularity of solutions to linear elliptic systems and anisotropic variational problems involving the trace-free part of the symmetric gradient

The aim of? this note is to investigate a regularity theory for minimizers of energies whose density depends on the trace-free part of the symmetric gradient, where integrands of anisotropic growth are considered. An adequate coercive inequality guarantees the existence of minimizers of such energies in suitable Sobolev classes. Moreover, various other Korn-type inequalities are shown, which can be used to prove the smoothness of weak solutions to linear elliptic systems involving the trace-free part of the symmetric gradient. In particular, Campanato-type estimates for solutions to such systems are established so that all tools are available to prove the interior regularity of minimizers of energies depending on the trace-free part of the symmetric gradient.     

Boundary value problems for the one-dimensional Willmore equation

The one-dimensional Willmore equation is studied under Navier as well as under Dirichlet boundary conditions. We are interested in smooth graph solutions, since for suitable boundary data, we expect the stable solutions to be among these. In the first part, classical symmetric solutions for symmetric boundary data are studied and closed expressions are deduced. In the Navier case, one has existence of precisely two solutions for boundary data below a suitable threshold, precisely one solution on the threshold and no solution beyond the threshold. This effect reflects that we have a bending point in the corresponding bifurcation diagram and is not due to that we restrict ourselves to graphs. Under Dirichlet boundary conditions we always have existence of precisely one symmetric solution. In the second part, we consider boundary value problems with nonsymmetric data. Solutions are constructed by rotating and rescaling suitable parts of the graph of an explicit symmetric solution. One basic observation for the symmetric case can already be found in Euler’s work. It is one goal of the present paper to make Euler’s observation more accessible and to develop it under the point of view of boundary value problems. Moreover, general existence results are proved.     

Symmetric patterns in the Dirichlet problem for a two-cell cubic autocatalytor reaction model

By using an approach developed by one of the authors, approximate solutions of the soft periodic boundary conditions for a two-cell reaction diffusion model have been obtained. The system is considered with reactant A and autocatalyst B. The reaction is taken cubic in the autocatalyst in the two-cell with linear exchange through A. The formal exact solution is obtained which is symmetric with respect to the mid-point of the container. Approximate solutions are found through the Picard iterative sequence of solutions constructed after the exact one. It is found that the solution obtained is not unique. When the initial conditions are periodic, the most dominant modes initiate to traveling waves in systems with moderate size. Symmetric configurations forming a parabolic one for large time are observed. In systems of large size, spatially symmetric chaos are produced which are stationary in time. Furthermore, it is found the symmetric pattern formation hold irrespective of the condition of linear instability against small spatial disturbance.     

Partial Elimination

The partial elimination method of Tuff & Jennings is consideredfor the solution of large sparse sets of simultaneous equationsin which the coefficient matrix is symmetric and positive definite.Proposals are made Jo modify the diagonal elements involvedin the elimination part of the algorithm to ensure stability.Also the iterative part of the algorithm is converted from anaccelerated stationary process to a conjugate gradient technique.Some numerical tests indicate that the method is more efficientthan the standard conjugate gradient method, although more storagespace is required for computer implementation.     

Iterative solvers and inflow boundary conditions for plane sudden expansion flows

Incompressible laminar flow in a symmetric plane sudden expansion is studied numerically. The flow is known to exhibit a stable symmetric solution up to a critical Reynolds number above which symmetry-breaking bifurcation occurs. The aim of the present study is to investigate the effect of using different iterative solvers on the calculation of the bifurcation point. For this purpose, the governing equations for steady two-dimensional incompressible flow are written in terms of a stream function-vorticity formulation. A second order finite volume discretization is applied. Explicit and implicit solvers are used to solve the resulting system of algebraic equations. It is shown that the explicit solver recovers the stable asymmetric solution, while the implicit solver can recover both the unstable symmetric solution or the stable asymmetric solution, depending on whether the initial guess is symmetric or not. It is also found that the type of inflow velocity profile, whether uniform or parabolic, has a significant effect on the onset of bifurcation as uniform inflows tend to stabilize the symmetric solution by delaying the onset of bifurcation to a higher Reynolds number as compared to parabolic inflows.     

Radially symmetric growth of nonnecrotic tumors

The growth of tumors is an important subject in recent research. We present here a mathematical model for the growth of nonnecrotic tumors in all the three regimes of vascularisation. This leads to a free-boundary problem which we treat by means ODE techniques. We prove the existence of a unique radially symmetric stationary solution. It is also shown that, if the initial tumor is radially symmetric, there exists a unique radially symmetric solution of the evolution equation, which exists for all times. The asymptotic behaviour of this solution will be discussed in relation to the parameters characterizing cell proliferation and cell death.     

Bound state solutions of the elliptic sine-Gordon equation

We present a detailed investigation of finite-energy solutions with point-like singularities of the elliptic sine-Gordon equation in a plane. Such solutions are of the bound-state type in the sense of scalar field theory. If the solution has a unique singularity, then it behaves as a soliton-like annular wave packet at a large distance from the singularity. The effective radius of this wave packet is evaluated both analytically and numerically for axially symmetric solutions. The analytical investigation is based on the method of isomonodromy deformations for the third Painlevé equation, which singles out these solutions as separatrices of the manifold of general solutions (with infinite energy). Exact analytical estimates provide a tool for investigating bound-state solutions of the nonintegrable sine-Gordon equation with a nonzero right-hand side. More precisely, for large-intensity fields at the singularity, we derive the critical forcing that allows the existence and stability of a bound state. As an illustration, we consider two applications: a large-area Josephson junction and a nematic liquid crystal in a rotating magnetic field. For each of the examples, we evaluate the critical values of the field that allow finite-energy regimes. These are in good agreement with numerical and experimental data. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 1, pp. 15–31, April, 1997.     

A normalized sparse linear equations solver

Normalized factorization procedures for the solution of large sparse linear finite element systems have been recently introduced in [3]. In these procedures the large sparse symmetric coefficient matrix of irregular structure is factorized exactly to yield a normalized direct solution method. Additionally, approximate factorization procedures yield implicit iterative methods for the finite difference or finite element solution. The numerical implementation of these algorithms is presented here and FORTRAN subroutines for the efficient solution of the resulting large sparse symmetric linear systems of algebraic equations are given.     

Structure of Positive Solutions of Quasilinear Elliptic Equations with Critical and Supercritical Growth

The structure of positive radial solutions to a class of quasilinear elliptic equations with critical and supercritical growth is precisely studied. A large solution and a small solution are obtained for the equations. It is shown that the large solution is unique, its asymptotic behaviour and flat core are also discussed.     

Inequalities for single crystal rod growth by edge-defined film-fed growth (E.F.G.) technique

A boundary value problem in the case of the second order axi-symmetric Young-Laplace differential equation (some of whose solutions describe the static meniscus free surface, i.e. the static liquid bridge free surface between the shaper and the crystal, occurring in single crystal rod growth) is analyzed. The analysis concerns the dependence of the solution of an initial value problem of the equation on a parameter p (the controllable part of the pressure difference Δp across the free surface). Inequalities are established for p which are necessary or sufficient conditions for the existence of a solution which represents a stable and convex free surface of a static meniscus. The analysis is numerically illustrated for the static menisci occurring in the NdYAG laser single crystal rod growth from the melt by edge-defined film-fed growth (E.F.G.) technique. This kind of inequalities can be useful in the experiment planning and technology design.     

Global fast and slow solutions of a localized problem with free boundary

In this paper,we consider a localized problem with free boundary for the heat equation in higher space dimensions and heterogeneous environment.For simplicity,we assume that the environment and solution are radially symmetric.First,by using the contraction mapping theorem,we prove that the local solution exists and is unique.Then,some sufficient conditions are given under which the solution will blow up in finite time.Our results indicate that the blowup occurs if the initial data are sufficiently large.Finally,the long time behavior of the global solution is discussed.It is shown that the global fast solution does exist if the initial data are sufficiently small,while the global slow solution is possible if the initial data are suitably large.     

Large positive solutions of semilinear elliptic equations with critical and supercritical growth

Existence and uniqueness of large positive solutions are obtained for some semilinear elliptic equations with critical and supercritical growth on general bounded smooth domains. It is shown that the large positive solution develops a boundary layer. The boundary derivative estimate of the large solution is also established.     

On the Lanczos and Golub–Kahan reduction methods applied to discrete ill‐posed problems

The symmetric Lanczos method is commonly applied to reduce large‐scale symmetric linear discrete ill‐posed problems to small ones with a symmetric tridiagonal matrix. We investigate how quickly the nonnegative subdiagonal entries of this matrix decay to zero. Their fast decay to zero suggests that there is little benefit in expressing the solution of the discrete ill‐posed problems in terms of the eigenvectors of the matrix compared with using a basis of Lanczos vectors, which are cheaper to compute. Similarly, we show that the solution subspace determined by the LSQR method when applied to the solution of linear discrete ill‐posed problems with a nonsymmetric matrix often can be used instead of the solution subspace determined by the singular value decomposition without significant, if any, reduction of the quality of the computed solution. Copyright © 2015 John Wiley & Sons, Ltd.     

A hybrid iterative method for symmetric positive definite linear systems

A hybrid iterative scheme that combines the Conjugate Gradient (CG) method with Richardson iteration is presented. This scheme is designed for the solution of linear systems of equations with a large sparse symmetric positive definite matrix. The purpose of the CG iterations is to improve an available approximate solution, as well as to determine an interval that contains all, or at least most, of the eigenvalues of the matrix. This interval is used to compute iteration parameters for Richardson iteration. The attraction of the hybrid scheme is that most of the iterations are carried out by the Richardson method, the simplicity of which makes efficient implementation on modern computers possible. Moreover, the hybrid scheme yields, at no additional computational cost, accurate estimates of the extreme eigenvalues of the matrix. Knowledge of these eigenvalues is essential in some applications.Research supported in part by NSF grant DMS-9409422.Research supported in part by NSF grant DMS-9205531.     

Interactions of fast and slow waves in hyperbolic systems with two time scales

We consider certain symmetric hyperbolic systems of nonlinear partial differential equations whose solutions vary on two time scales. The large part of the spatial operator is assumed to have constant coefficients, but a nonlinear term multiplying the time derivatives is allowed. We show that if the initial data are not prepared correctly for the suppression of the fast scale motion, but contain errors of amplitude O(?), then the perturbation in the solution will also be of amplitude O(?). Further, if the large part of the spatial operator is nonsingular, we show that the error introduced in the slow scale motion will be of amplitude O(?²), even though fast scale waves of amplitude O(?) will be present in the solution.     

Resonant vibrations in harmonically excited weakly coupled mechanical systems with cyclic symmetry

The resonant vibrations in weakly coupled nonlinear cyclic symmetric structures are studied. These structures consist of weakly coupled identical nonlinear oscillators. A careful bifurcation analysis of the amplitude equations is performed in the fundamental resonance case for an illustrative example consisting of a three particle system. In case of a uniformly distributed excitation, a localized response is identified in which one of the particles exhibits large amplitude motions compared to those of the other particles. In case of single-particle excitation, it is found that for very small coupling strength and large external mistuning, a large stable localized periodic response coexists with an extended small response. With an increase in the coupling strength, multiple extended solutions arise near the exact external resonance via saddle-node bifurcations. Further increase in coupling strength and a decrease in damping results in isolated asymmetric solution branches, which bifurcate from the symmetric solutions via symmetry-breaking bifurcations. The role of coupling strength in creating/destroying localized solutions is discussed.     

On an application of the QMR_SYM method to complex symmetric shifted linear systems

We consider the solution of complex symmetric shifted linear systems. Such systems arise in large scale electronic structure theory and there is a strong need for the fast solution of the systems. In this paper, we describe an algorithm for solving the systems, which is based on the QMR_SYM method for solving complex symmetric linear systems. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)