Large time behavior for the incompressible magnetohydrodynamic equations in half‐spaces

The weighted L^r‐asymptotic behavior of the strong solution and its first‐order spacial derivatives to the incompressible magnetohydrodynamic (MHD) equations is established in a half‐space. Further, the L^∞‐decay rates of the second‐order spatial derivatives of the strong solution are derived by using the Stokes solution formula and employing a decomposition for the nonlinear terms in MHD equations. Copyright © 2014 John Wiley & Sons, Ltd.     

On a parabolic‐elliptic chemotaxis system with periodic asymptotic behavior

We study a parabolic‐elliptic chemotactic PDEs system, which describes the evolution of a biological population “u” and a chemical substance “v” in a bounded domain . We consider a growth term of logistic type in the equation of “u” in the form μu(1 ? u + f(t,x)). The function “f,” describing the resources of the systems, presents a periodic asymptotic behavior in the sense where f ^? is independent of x and periodic in time. We study the global existence of solutions and its asymptotic behavior. Under suitable assumptions on the initial data and f ^?, if the constant chemotactic sensitivity χ satisfies we obtain that the solution of the system converges to a homogeneous in space and periodic in time function.     

Finite‐dimensional asymptotic behavior in a thermosyphon including the Soret effect

We analyse the dynamics of a fluid transporting a soluble substance in the interior of a closed loop of arbitrary geometry and subjected to the action of gravity and natural convection. After obtaining the governing equations and analysing the well posedness of the system we prove the existence of a global attractor. Finally, using inertial manifold techniques, we obtain an explicit reduced system of ODE's that describes the asymptotic behaviour of the full system. Copyright © 1999 John Wiley & Sons, Ltd.     

Tetravalent half‐edge‐transitive graphs and non‐normal Cayley graphs

Let X be a vertex‐transitive graph, that is, the automorphism group Aut(X) of X is transitive on the vertex set of X. The graph X is said to be symmetric if Aut(X) is transitive on the arc set of X. suppose that Aut(X) has two orbits of the same length on the arc set of X. Then X is said to be half‐arc‐transitive or half‐edge‐transitive if Aut(X) has one or two orbits on the edge set of X, respectively. Stabilizers of symmetric and half‐arc‐transitive graphs have been investigated by many authors. For example, see Tutte [Canad J Math 11 (1959), 621–624] and Conder and Maru?i? [J Combin Theory Ser B 88 (2003), 67–76]. It is trivial to construct connected tetravalent symmetric graphs with arbitrarily large stabilizers, and by Maru?i? [Discrete Math 299 (2005), 180–193], connected tetravalent half‐arc‐transitive graphs can have arbitrarily large stabilizers. In this article, we show that connected tetravalent half‐edge‐transitive graphs can also have arbitrarily large stabilizers. A Cayley graph Cay(G, S) on a group G is said to be normal if the right regular representation R(G) of G is normal in Aut(Cay(G, S)). There are only a few known examples of connected tetravalent non‐normal Cayley graphs on non‐abelian simple groups. In this article, we give a sufficient condition for non‐normal Cayley graphs and by using the condition, infinitely many connected tetravalent non‐normal Cayley graphs are constructed. As an application, all connected tetravalent non‐normal Cayley graphs on the alternating group A₆ are determined. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Semiclassical asymptotic behavior of ground state for the two‐component Hartree system

We consider the semiclassical asymptotic behaviors of ground state solution for the following two‐component Hartree system: which is originated from the study on cold atoms of boson and fermion system with long‐range interaction. Under the assumption by detailed compactness analysis, we prove that there is a β₀>0 such that if β<β₀, the system has a ground state solution. For this solution, the energy estimates and the decay rates are presented, and the asymptotic profiles as ε→0 are displayed in details for β<0 and β>0, respectively. Furthermore, we show that for β<0, the phase separation phenomenon may occur.     

Geometry of the uniform infinite half‐planar quadrangulation

We give a new construction of the uniform infinite half‐planar quadrangulation with a general boundary (or UIHPQ), analogous to the construction of the UIPQ presented by Chassaing and Durhuus, which allows us to perform a detailed study of its geometry. We show that the process of distances to the root vertex read along the boundary contour of the UIHPQ evolves as a particularly simple Markov chain and converges to a pair of independent Bessel processes of dimension 5 in the scaling limit. We study the “pencil” of infinite geodesics issued from the root vertex as reported by Curien, Ménard, and Miermont and prove that it induces a decomposition of the UIHPQ into 3 independent submaps. We are also able to prove that balls of large radius around the root are on average 7/9 times as large as those in the UIPQ, both in the UIHPQ and in the UIHPQ with a simple boundary; this fact we use in a companion paper to study self‐avoiding walks on large quadrangulations.     

Learning finite binary sequences from half‐space data

The problem of inferring a finite binary sequence w *∈{−1, 1}ⁿ is considered. It is supposed that at epochs t=1, 2,…, the learner is provided with random half‐space data in the form of finite binary sequences u ^(t)∈{−1, 1}ⁿ which have positive inner‐product with w *. The goal of the learner is to determine the underlying sequence w * in an efficient, on‐line fashion from the data { u ^(t), t≥1}. In this context, it is shown that the randomized, on‐line directed drift algorithm produces a sequence of hypotheses {w^(t)∈{−1, 1}ⁿ, t≥1} which converges to w * in finite time with probability 1. It is shown that while the algorithm has a minimal space complexity of 2n bits of scratch memory, it has exponential time complexity with an expected mistake bound of order Ω(e^0.139n). Batch incarnations of the algorithm are introduced which allow for massive improvements in running time with a relatively small cost in space (batch size). In particular, using a batch of 𝒪(n log n) examples at each update epoch reduces the expected mistake bound of the (batch) algorithm to 𝒪(n) (in an asynchronous bit update mode) and 𝒪(1) (in a synchronous bit update mode). The problem considered here is related to binary integer programming and to learning in a mathematical model of a neuron. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 14, 345–381 (1999)     

Oscillation and non‐oscillation results for solutions of perturbed half‐linear equations

The purpose of this paper is to describe the oscillatory properties of second‐order Euler‐type half‐linear differential equations with perturbations in both terms. All but one perturbations in each term are considered to be given by finite sums of periodic continuous functions, while coefficients in the last perturbations are considered to be general continuous functions. Since the periodic behavior of the coefficients enables us to solve the oscillation and non‐oscillation of the considered equations, including the so‐called critical case, we determine the oscillatory properties of the equations with the last general perturbations. As the main result, we prove that the studied equations are conditionally oscillatory in the considered very general setting. The novelty of our results is illustrated by many examples, and we give concrete new corollaries as well. Note that the obtained results are new even in the case of linear equations.     

Non-existence of higher-dimensional pseudoholomorphic submanifolds

 In this paper we prove that only pseudoholomorphic curves appear as J-invariant submanifolds of generic almost complex manifolds (M,J). We also prove there exist no non-trivial automorphisms or submersions of such manifolds. On the other hand we show that abundance of 1-jets of PH-submanifolds, automorphisms or submersions implies integrability of the almost complex structure. Received: 19 April 2002 / Revised version: 10 October 2002 Published online: 14 February 2003     

Fisher–KPP equation with advection on the half‐line

We consider the Fisher–KPP equation with advection: u_t=u_xx?βu_x+f(u) on the half‐line x∈(0,∞), with no‐flux boundary condition u_x?βu = 0 at x = 0. We study the influence of the advection coefficient ?β on the long time behavior of the solutions. We show that for any compactly supported, nonnegative initial data, (i) when β∈(0,c₀), the solution converges locally uniformly to a strictly increasing positive stationary solution, (ii) when β∈[c₀,∞), the solution converges locally uniformly to 0, here c₀ is the minimal speed of the traveling waves of the classical Fisher–KPP equation. Moreover, (i) when β > 0, the asymptotic positions of the level sets on the right side of the solution are (β + c₀)t + o(t), and (ii) when β≥c₀, the asymptotic positions of the level sets on the left side are (β ? c₀)t + o(t). Copyright © 2016 John Wiley & Sons, Ltd.     

The spectrum of a weighted Laplacian in the half‐space

J. Banasiak In this paper, we deal with spectral properties of a weighted Laplacian in the half‐space when a Dirichlet or a Neumann boundary condition is imposed. After proving that the spectrum is discrete under suitable assumptions, we give explicit formulae of eigenvalues and eigenfunctions in a specific case. In particular, the obtained eigenfunctions are rational or pseudo‐rational and have remarkable orthogonality properties. These results suggest the use of the discovered functions for approximating solutions of elliptic problems in the half‐space. Copyright © 2015 John Wiley & Sons, Ltd.     

Boundary value problems on half‐line for second‐order nonlinear impulsive differential equations

We obtain sufficient conditions for existence and uniqueness of solutions of boundary value problems on half‐line for a class of second‐order nonlinear impulsive differential equations. Our technique is different than the traditional ones, as it is based on asymptotic integration method involving principal and nonprincipal solutions. Examples are provided to illustrate the relevance of the results.     

Gromov convergence of pseudoholomorphic disks

The goal of this paper is to give a self-contained exposition of Gromov compactness for pseudoholomorphic disks in compact symplectic manifolds. The proof leads naturally to the concept of stable maps which was first introduced by M. Kontsevich. Our definition of stable maps for disks is based on the one given by D. McDuff and D. Salamon for spheres. We also generalize the notion of Gromov convergence to the case of disks. We show that the homotopy class is preserved under Gromov convergence. Dedicated to Vladimir Arnold     

Harmonic and monogenic potentials in low dimensional Euclidean half‐space

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space was constructed recently, including a higher dimensional analogue of the logarithmic function in the complex plane. Their distributional limits at the boundary were also determined. In this paper, the potentials and their distributional boundary values are calculated in dimensions 3 and 4, dimensions for which the expressions in general dimension break down. Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical solution of nonlinear three‐point boundary value problem on the positive half‐line

In this paper, we present an efficient numerical algorithm to solve the three‐point boundary value problem on the half‐line based on the reproducing kernel theorem. Considering the boundary conditions including a limit form, a new weighted reproducing kernel space is established to overcome the difficulty. By applying reproducing property and existence of the orthogonal basis in the weighted reproducing kernel space, the approximate solution is constructed by the orthogonal projection of the exact solution. Convergence has also been discussed. We demonstrate the accuracy of the method by numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.     

Study of generalized eigenfunctions of a perturbed isotropic elastic half‐space

We consider the self‐adjoint operator governing the propagation of elastic waves in a perturbed isotropic half‐space (perturbation with compact support of a homogeneous isotropic half‐space) with a free boundary condition. We propose a method to obtain, numerical values included, a complete set of generalized eigenfunctions that diagonalize this operator. The first step gives an explicit representation of these functions using a perturbative method. The unbounded boundary is a new difficulty compared with the method used by Wilcox [25], who set the problem in the complement of bounded open set. The second step is based on a boundary integral equations method which allows us to compute these functions. For this, we need to determine explicitly the Green's function of (A₀ – ω²), where A₀ is the self‐adjoint operator describing elastic waves in a homogeneous isotropic half‐space. Copyright © 2000 John Wiley & Sons, Ltd.