bluuebunny/crossref_metadata_2025 huggingface dataset:
Asymptotic behaviour of solutions to the coagulation–fragmentation equations. I. The strong fragmentation case
Authors: Carr J.
Published: November 2011

Synopsis

The discrete coagulation-fragmentation equations are a model for the time-evolution of cluster growth. The processes described by the model are the coagulation of clusters via binary interactions and the fragmentation of clusters. The assumptions made on the fragmentation coefficients in this paper have the physical interpretation that surface effects are not important, i.e. it is unlikely that a large cluster will fragment into two large pieces. Since solutions of the initial-value problem are not unique, we have to restrict the class of solutions. With this restriction, we prove that the fragmentation acts as a strong damping mechanism and we obtain results on the asymptotic behaviour of solutions. The main tool used is an estimate on the moments of admissible solutions.
On the stochastic nonlinear neutron transport equation
Authors: Mokhtar-Kharroubi Mustapha
Published: November 2011

Synopsis

The probability that a neutron leads to a divergent chain reaction in a nuclear reactor is governed by a nonlinear integro-partial-differential equation [1]. A model case of this equation was completely analysed by Pazy and Rabinowitz [2,3]. The purpose of this paper is to extend their results to the general case and to tackle some related topics.
The spectral theory of second order two-point differential operators: I. A priori estimates for the eigenvalues and completeness
Authors: Locker John
Published: November 2011

Synopsis

This paper is the first part in a four-part series which develops the spectral theory for a two-point differential operator L in L2[0, 1] determined by a second order formal differential operator l = −D2 + pD + q and by independent boundary values B1, B2. The differential operator L is classified as belonging to one of five cases, Cases 1–5, according to conditions satisfied by the coefficients of B1, B2. For Cases 1–4 it is shown that if λ = ρ2 is any eigenvalue of L with ∣ρ∣ sufficiently large, then ρ lies in the interior of a horizontal strip (Cases 1–3) or the interior of a logarithmic strip (Case 4), and in each of these cases the generalised eigenfunctions of L are complete in L2[0, 1].
Relaxation of multiple integrals in the spaceBV(Ω, RP)
Authors: Fonseca Irene , Rybka Piotr
Published: November 2011

Synopsis

A characterisation of the surface energy density for the relaxation in V(Ω; Rp) of the functional

is obtained. A lemma of De Giorgi is used to modify a sequence near the boundary without increasing its total energy.
Orders in semiprime rings with minimal condition for principal right ideals
Authors: Fountain John , Gould Victoria
Published: November 2011

Synopsis

An alternative proof of a theorem which characterises orders in semiprime rings with minimal condition is given. The approach used is to make use of the corresponding result for prime rings and is inspired by Herstein's proof of Goldie's theorem on orders in semisimple Artinian rings.
A reaction-diffusion equation on a thin L-shaped domain
Authors: Hale Jack K. , Raugel Geneviève
Published: November 2011

We consider a dissipative reaction–diffusion equation on a thin L-shaped domain (with the thinness measured by a parameter ε); we determine the limit equation for ε = 0 and prove the upper semicontinuity of the global attractors at ε = 0. We also state a lower semicontinuity result. When the limit equation is one-dimensional, we prove convergence of any orbit to a singleton.
On commutative idempotent rings
Authors: Andruszkiewicz R. R. , Puczyłowski E. R.
Published: November 2011

We study the problem when a ring which is an extension of a commutative idempotent ring by a commutative idempotent ring is commutative. In particular, we answer Sands' question showing that the class of commutative idempotent rings whose every homomorphic image has zero annihilator is a maximal but not the largest radical class consisting of commutative idempotent rings.
Counting stationary solutions of the Cahn–Hilliard equation by transversality arguments
Authors: Grinfeld M. , Novick-Cohen A.
Published: November 2011

In this paper we use arguments based on Picard-Fuchs equations and transversality of intersections of level curves to obtain an exact count of the number of stationary solutions of the one-dimensional Cahn-Hilliard equation with a cubic nonlinearity.
Nilpotents and congruences on semigroups of transformations with fixed rank
Authors: Marques-Smith M. Paula O. , Sullivan R. P.
Published: November 2011

In 1988, Howie and Marques-Smith studied Pm, a Rees quotient semigroup of transformations associated with a regular cardinal m, and described the elements which can be written as a product of nilpotents in Pm. In 1981, Marques proved that if Δm denotes the Malcev congruence on Pm, then Pm/Δm is congruence-free for any infinite m. In this paper, we describe the products of nilpotents in Pm when m is nonregular, and determine all the congruences on Pm when m is an arbitrary infinite cardinal. We also investigate when a nilpotent is a product of idempotents.
Dynamics of a nonlinear convection-diffusion equation in multidimensional bounded domains
Authors: Hill Adrian T. , Süli Endre
Published: November 2011

The scalar nonlinear convection-diffusion equation

is considered, for given initial data and zero Dirichlet boundary conditions, in a smooth bounded domain Ω⊂ℝn. The homogeneous viscous Burgers' equation in one dimension is well-known to possess a unique, exponentially attracting equilibrium. These properties are shown to be preserved in the generalisation considered. Furthermore, the equilibrium is shown to be bounded in the maximum norm independently of the function a. The main methods used are maximum principles, and a variational method due to Stampacchia.
Regularity of solutions and interfaces of the porous medium equation via local estimates
Authors: Vazquez Juan Luis
Published: November 2011

Synopsis

We prove C∞ regularity for moving interfaces of local solutions of the porous medium equation as well as C∞ lateral regularity for the pressure function near such an interface.
A uniqueness proof for the Wulff Theorem
Authors: Fonseca Irene , Müller Stefan
Published: November 2011

Synopsis

The Wulff problem is a generalisation of the isoperimetric problem and is relevant for the equilibrium of (small) elastic crystals. It consists in minimising the (generally anisotropic) surface energy among sets of given volume. A solution of this problem is given by a geometric construction due to Wulff. In the class of sets of finite perimeter this was first shown by J. E. Taylor who, using methods of geometric measure theory, also proved uniqueness. Here a more analytic uniqueness proof is presented. The main ingredient is a sharpened version of the Brunn–Minkowski inequality.
On the trace of the difference of Schrödinger heat semigroups
Authors: Berg M. van den
Published: November 2011

Synopsis

We obtain upper and lower bounds for tr (e−th−etΔ), where H = −Δ + V is a Schrödinger operator on L2 (ℝm), and ℝ is the Laplace operator for ℝm. The bounds are obtained for a class of negative valued Borel measurable potentials with compact support and in L∞(ℝm).
Necessary and sufficient conditions for optimal controls in viscous flow problems
Authors: Fattorini H. O. , Sritharan S. S.
Published: November 2011

A class of optimal control problems in viscous flow is studied. Main results are the Pontryagin maximum principle and the verification theorem for the Hamilton–Jacobi–Bellman equation characterising the feedback problem. The maximum principle is established by two quite different methods.
Convergence of the viscosity method for some nonlinear hyperbolic systems
Authors: Lu Yunguang
Published: November 2011

In this paper some special entropy–entropy flux pairs of Lax type are constructed for nonlinear hyperbolic systems of types (1.1) and (1.2), in which the progression terms are functions of a single variable. The necessary estimates for the major terms are obtained by the use of singular perturbation theory. The special entropies provide a convergence theorem in the strong topology for the artificial viscosity method when applied to the Cauchy problems (1.1), (1.3) and (1.2), (1.3) and used together with the theory of compensated compactness.
Integral maximum principle and its applications
Authors: Grigor'yan Alexander
Published: November 2011

The integral maximum principle for the heat equation on a Riemannian manifold is improved and applied to obtain estimates of double integrals of the heat kernel.
On the absolutely continuous subspaces of Floquet operators
Authors: Huang Min-Jei
Published: November 2011

The purpose of this paper is to describe various subspaces that are closely related to the absolutely continuous subspace of a Floquet operator. This paper generalises and extends several known results.
28 real bitangents: Dedicated to Ian Porteus
Authors: Edge W. L.
Published: November 2011

Although all the coefficients in the equation of a plane algebraic curve may be real numbers, it by no means follows that the equations of all its bitangents are real. But Plücker perceived that this could happen for the 28 bitangents of a nonsingular plane quartic. Where can this be observed in a body of 28 explicit linear equations? This modest note affords an example.
The interaction between bulk energy and surface energy in multiple integrals
Authors: Braides Andrea , Coscia Alessandra
Published: November 2011

This paper is devoted to the study of integral functional denned on the spaceSBV(Ω ℝk) of vector-valued special functions with bounded variation on the open set Ω⊂ℝn, of the form

We suppose only thatfis finite at one point, and thatgis positively 1-homogeneous and locally bounded on the sets ℝk⊗vm, where {v1,…,vR} ⊂Sn−1is a basis of ℝn. We prove that the lower semicontinuous envelope ofFin theL1(Ω;ℝk)-topology is finite and with linear growth on the wholeBV(Ω;ℝk), and that it admits the integral representation

A formula forϕis given, which takes into account the interaction between the bulk energy densityfand the surface energy densityg.
Almost sure convergence of measure-valued branching processes: a critical exponent
Authors: Etheridge Alison M.
Published: November 2011

A large class of measure-valued critical branching processes can be classified in terms of a parameter ρ which arises as a measure of the recurrence of the underlying spatial Markov process. By establishing upper and lower bounds for the total weighted occupation time process, it is shown that if a measure-valued process is started from an invariant measure of its underlying spatial process, then a necessary and sufficient condition for (a.s.) local extinction is that ρ > 0.
Self-similar solutions for diffusion in semiconductors
Authors: Peletier L. A. , Troy W. C.
Published: November 2011

We study the development of concentration profiles in a semi-infinite slab of semiconductor material after impurities have been implanted uniformly through the slab, under the assumption that, at the face of the slab, no impurities can pass and the vacancy concentration is kept at its equilibrium value. It is shown that profiles of self-similar form exist, and their qualitative shape, as well as their asymptotic properties far from the face of the slab, are determined.
Positive solutions for a class of nonpositone problems with concave nonlinearities
Authors: Wang Shin-Hwa
Published: November 2011

We consider the bifurcation of positive solutions of the two-point boundary value problem

where λ> 0 is a real bifurcation parameter, and f ∊ C2 satisfies (fl) f(0) < 0, (f2) f′(s) > 0 for s > 0, (f3) f″(s) < 0 for s > 0 and (f4) limS→+∞f(s) = M where 0 < M ≦+∞. This problem has been studied by Casto and Shivaji under two additional hypotheses (f5) lims→+∞sf′(s) = 0, and (f6)f(θ)/θ < f′(θ), where θ is a positive number satisfying Assuming (fl)−(f6), Castro and Shivaji obtain some existence and nonexistence results and hence partial information on the bifurcation diagram, and they conjecture that this problem has at most two positive solutions. We prove this conjecture. Furthermore, we are able to generalise and improve their results under hypotheses (fl)−(f4). As a corollary, we show that there exists μ1 > 0 such that there exist no positive solutions for 0 <λ <μ1 and at most two positive solutions for μ1≦λ< + ∞, which improves a result of Brunovsky and Chow.
Blow-up for quasilinear heat equations with critical Fujita's exponents
Authors: Galaktionov Victor A.
Published: November 2011

We consider the Cauchy problem for the quasilinear heat equation

where σ > 0 is a fixed constant, with the critical exponent in the source term β = βc = σ + 1 + 2/N. It is well-known that if β ∈(1,βc) then any non-negative weak solution u(x, t)≢0 blows up in a finite time. For the semilinear heat equation (HE) with σ = 0, the above result was proved by H. Fujita [4].

In the present paper we prove that u ≢ 0 blows up in the critical case β = σ + 1 + 2/N with σ > 0. A similar result is valid for the equation with gradient-dependent diffusivity

with σ > 0, and the critical exponent β = σ + 1 + (σ + 2)/N.
A Poincaré–Bendixson theorem for scalar balance laws
Authors: Lyberopoulos Athanasios N.
Published: November 2011

We study the dynamical behaviour, as t → ∞, of admissible weak solutions of the scalar balance law

with x ∊ ≡ ℝ/Lℤ, L > 0, 0 < t < ∞, and f(·) ∊ C2, g(·) ∊ C1. We assume that f(·) is strictly convex, while g(·) is of at most linear growth, has finitely many zeros and changes sign across them. We show that, if u(·,t) stays bounded in L∞(S1), as t → ∞, then it either converges to a constant state or approaches asymptotically a rotating wave, i.e. an admissible weak solution of (1.1) of the form ũ(x − ct), c ∈ ℝ. Hence, the asymptotic state of every bounded solution of (1.1) consists precisely of either an equilibrium or one time-periodic solution. Furthermore, each one of these two alternatives is characterised by the Conley indices of the critical points of the ordinary differential equation .
Schoenberg's exponential Euler spline curves
Authors: Jetter K. , Riemenschneider S. D. , Sivakumar N.
Published: November 2011

Synopsis

The exponential Euler spline curves of Schoenberg are used to derive the correctness of cardinal interpolation by shifted univariate B-splines and the “metric condition” on the bi-infinite Toeplitz matrix of interpolation. Additional monotonicity properties of the associated symbol for interpolation in each of its parameters are also given.
Existence of solutions of boundary value problems for differential equations with deviating arguments, via the topological transversality method
Authors: Tsamatos P. Ch. , Ntouyas S. K.
Published: November 2011

Synopsis

This paper is concerned with the existence of solutions of a general boundary value problem for differential equations with deviating arguments. The results are based on a topological transversality method and rely on a priori bounds on solutions.
Rational equivalence of primitive integral binary quadratic forms
Authors: Arenas A.
Published: November 2011

Synopsis

We compute the number of ℚ-classes of primitive integral binary quadratic forms of any discriminant and characterise when it coincides with the number of genera.
A plane sextic and its five cusps
Authors: Edge W. L.
Published: November 2011

Synopsis

A certain plane sextic of genus 5 was encountered by Humbert and publicised by him [3] in 1894. Its striking geometrical properties clamour for elucidation; this was eventually supplied in 1951. For the canonical curve of genus 5 is the base curve C of a net N of quadrics in projective space [4], and C models a Humbert curve when all the quadrics of N have a common self-polar simplex [1]. The projection of C from one of its chords onto a plane is a 5-nodal sextic, the nodes all becoming cusps when the chord of C becomes a tangent. The properties to be elucidated become clear visually in the projection.

The sextic H described here is a specialisation of the cusped curve; it emerges as linearly dependent on a pair of reducible plane sextics concocted ad hoc.
Lacunary bifurcation for operator equations and nonlinear boundary value problems on ℝN
Authors: Heinz Hans-Peter
Published: November 2011

Synopsis

We consider nonlinear eigenvalue problems of the form Lu + F(u) = λu in a real Hilbert space, where L is a positive self-adjoint linear operator and F is a nonlinearity vanishing to higher order at u = 0. We suppose that there are gaps in the essential spectrum of L and use critical point theory for strongly indefinite functionals to derive conditions for the existence of non-zero solutions for λ belonging to such a gap, and for the bifurcation of such solutions from the line of trivial solutions at the boundary points of a gap. The abstract results are applied to the L2-theory of semilinear elliptic partial differential equations on ℝN. We obtain existence results for the general case and bifurcation results for nonlinear perturbations of the periodic Schrödinger equation.
The Skyrme model for nucleons under spherical symmetry
Authors: McLeod J. B. , Troy W. C.
Published: November 2011

Synopsis

The paper discusses properties of solutions of the differential equation

and in particular the existence and uniqueness of solutions to the boundary-value problem associated with the above equation and the boundary conditions

This equation, first introduced by Skyrme, is a spherically symmetric model for a soliton description of nucleons.
The thermistor problem with conductivity vanishing for large temperature
Authors: Xu Xiangsheng
Published: November 2011

We consider the system (∂/∂t)u = ∆u + σ(u)|∇φ|2, div (σ(u)∇φ) = 0 in a bounded region of ℝN coupled with initial and boundary conditions, where σ(s) ∈ C(ℝ) is nonnegative and σ(u) = 0 if and only if u ≧ a for some a > 0. Owing to the degeneracy involved, solutions of the problem display new phenomena that cannot be incorporated into the classical weak formulation. The notion of a capacity solution introduced in [14,15] is employed to study the problem. It turns out that this notion of a solution is just general enough to encompass the new phenomena involved.
Residual finiteness of free products of combinatorial strict inverse semigroups
Authors: Auinger Karl
Published: November 2011

It is shown that the free product of two residually finite combinatorial strict inverse semigroups in general is not residually finite. In contrast, the free product of a residually finite combinatorial strict inverse semigroup and a semilattice is residually finite.
Projective modules of finite groups with elementary abelian Sylow 3-subgroups of order 9 in characteristic 3
Authors: Koshitani Shigeo
Published: November 2011

Let G be any finite group with elementary abelian Sylow 3-subgroups of order 9, and let F be any field of characteristic 3. Then, the Loewy length of the projective cover of the trivial FG-module is at least 5. This lower bound is the best possible.
A nonlocal Sturm–Liouville eigenvalue problem
Authors: Freitas Pedro
Published: November 2011

A nonlocal eigenvalue problem of the form u″ + a(x)u + Bu = λu with homogeneous Dirichlet boundary conditions is considered, where B is a rank-one bounded linear operator and x belongs to some bounded interval on the real line. The behaviour of the eigenvalues is studied using methods of linear perturbation theory. In particular, some results are given which ensure that the spectrum remains real. A Sturm-type comparison result is obtained. Finally, these results are applied to the study of some nonlocal reaction–diffusion equations.
Uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large
Authors: Guo Zongming , Webb J. R. L.
Published: November 2011

Existence and uniqueness results are proved for positive solutions of a class of quasilinear elliptic equations in a domain Ω⊂ℝN via a generalisation of Serrin's sweeping principle. In the case when Ω is an annulus, it is shown that the solution is radially symmetric.
Convexity of certain integrals of the calculus of variations
Authors: Dacorogna B.
Published: November 2011

Synopsis

In this paper we study the convexity of the integral over the space . We isolate a necessary condition on f and we find necessary and sufficient conditions in the case where f(x, u, u′) = a(u)u′2n or g(u) + h(u′).
A variational approach for a class of singular perturbation problems and applications
Authors: Alikakos Nicholas D. , Simpson Henry C.
Published: November 2011

Synopsis

We study the limit as ε → 0 of global minimisers of functionals of the type

where Ω is an annul us or a ball in ℝn.
Théorème d'existence pour des problèmes variationnels non convexes
Authors: Raymond J. P.
Published: November 2011

Synopsis

On donne dans cet article un théorème d'existence de solutions lipschitziennes pour des problèmes du type:

où Ω est un ouvert convexe borné de ℝn, n ≧ 2, p ≧ 2, aucune hypothèse de convexité n'est faite sur g ou f. On étend de la sorte des ŕesultats d'existence obtenus en dimension 1.

where Ω is a bounded open convex subset of ℝn, n ≧ 2, p ≧ 2; we suppose no assumption of convexity on g or f. In this way we extend existence results proved in dimension 1.)
On the number of conjugacy classes of π-elements in a finite group
Authors: López Antonio Vera , Larrea Ma Concepción
Published: November 2011

Synopsis

In this paper, the number of conjugacy classes of π-elements (respectively non π-elements) of G is analysed in terras of the corresponding numbers of G/N and N, for each N normal subgroup of G. In particular, we generalise well-known results of P. X. Gallagher and C. H. Sah.
Convergence of the strongly damped nonlinear Klein-Gordon equation in ℝn with radial symmetry
Authors: Avrin Joel D.
Published: November 2011

Synopsis

We consider the strongly-damped Klein–Gordon equation in ℝ3 in the case where the initial data possess radial symmetry. With the latter assumption we are able to extend the result of [2] which assumed a bounded spatial domain. Specifically, we construct a global weak solution v of theundamped equation for high powers p which can be approximated arbitrarily closely (for small α) by the global strong solutions of the damped equation found by Aviles and Sandefur [1].
Corrigendum: Enumeration of smooth labelled graphs
Authors: Wright E.M.
Published: November 2011

[Proc. Roy. Soc. Edinburgh Sect. A91 (1982), 205–212]
Erratum: Comparison principles for strongly coupled reaction-diffusion equations
Authors: Sleeman B. D. , Tuma E.
Published: November 2011

[Proc. Roy. Soc. Edinburgh Sect. A106 (1987), 209–219]
Subharmonics near an equilibrium for some second-order Hamiltonian systems
Authors: Felmer Patricio L. , Silva Elves A. de B. e
Published: November 2011

Synopsis

This work is devoted to the study of subharmonic solutions of a second-order Hamiltonian system

near an equilibrium point, say q = 0. The problem of existence of periodic solutions from the global point of view is also considered.

This problem has been studied for the case where the potential is positive and superquadratic. In this work a potential V that has change in sign is considered. The potential is decomposed as

where P is homogeneous, superquadratic and nondegenerate, and is of higher order near 0. In this paper the existence is shown of a sequence of subharmonic solutions of the equation above that converges to the equilibrium, such that their minimal periods converge to infinity.

This problem is approached from a variational point of view. Existence of subharmonic and periodic solutions is obtained via minimax techniques.
Maximum principles and comparison theorems for semilinear parabolic systems and their applications
Authors: Lu G. , Sleeman B. D.
Published: November 2011

Synopsis

A fundamental comparison theorem is established for general semilinear parabolic systems via the notions of sectorial operators, analytic semigroups and the application of the Tychonoff Fixed Point Theorem. Based on this result, we establish a maximum principle for systems of general parabolic operators and general comparison theorems for parabolic systems with quasimonotone or mixed quasimonotone nonlinearities. These results cover and extend most currently used forms of maximum principles and comparison theorems. A global existence theorem for parabolic systems is derived as an application which, in particular, gives rise to some global existence results for Fujita type systems and certain generalisations.
On homogeneous radicals of semigroup rings of commutative semigroups
Authors: Kelarev A. V.
Published: November 2011

Synopsis

All Archimedean commutative semigroups S are described such that every S-homogeneous hereditary radical is S-normal. It is shown that this result is in a sense unimprovable.
Two representations of a conditioned superprocess
Authors: Evans Steven N.
Published: November 2011

Synopsis

We consider a class of measure-valued Markov processes constructed by taking a superprocess over some underlying Markov process and conditioning it to stay alive forever. We obtain two representations of such a process. The first representation is in terms of an “immortal particle” that moves around according to the underlying Markov process and throws off pieces of mass, which then proceed to evolve in the same way that mass evolves for the unconditioned superprocess. As a consequence of this representation, we show that the tail σ-field of the conditioned superprocess is trivial if the tail σ-field of the underlying process is trivial. The second representation is analogous to one obtained by LeGall in the unconditioned case. It represents the conditioned superprocess in terms of a certain process taking values in the path space of the underlying process. This representation is useful for studying the “transience” and “recurrence” properties of the closed support process.
Regularity of an elliptic problem with a singular nonlinearity
Authors: Gui Changfeng , Lin Fang Hua
Published: November 2011

Synopsis

We study a semilinear boundary value problem with the feature that the vanishing boundary value makes the equation singular. We prove that the positive solution is in general Hölder-continuous up to the boundary and has even better regularity in some special cases.
Generic hyperbolicity for scalar parabolic equations
Authors: Pereira Antonio L.
Published: November 2011

Synopsis

For the reaction diffusion equation

with homogeneous Neumann boundary conditions, we give results on the generic hyperbolicity of equilibria with respect to a for fixed f and with respect to f for fixed a.
Remarks on the equilibrium shape of a Tokamak plasma
Authors: Jianfu Yang
Published: November 2011

Synopsis

The equilibrium of a Tokamak plasma not confined inside a conducting shell is governed by a free boundary value problem. The existence of solutions of the free boundary value problem is discussed.
On the existence and nonexistence of global solutions of a diffusion-advection equation
Authors: Deng Keng
Published: November 2011

Synopsis

We consider the diffusion-advection equation ut = uxx + (ε/(1 − u)β)x(ε >0, β >0), 0 < x < 1, t >0, under the boundary conditions ux + ε/(1 − u)β = 0. We prove that there is a critical number ε(β) such that when ε < ε(β) for certain initial data a global solution exists and converges to the corresponding stationary solution; any solution must quench (u reaches one in finite or infinite time) if ε ≧ε(β). We also show that quenching can only occur at x = 0, and that for each ε > 0 there exist initial data for which the solution quenches in finite time.
Two-weight weak and extra-weak type inequalities for the one-sided maximal operator
Authors: Salvador Pedro Ortega , Pick Luboš
Published: November 2011

Synopsis

Letbe the one-sided maximal operator and let Ф be a convex non-decreasing function on (0, ∞), Ф(0) = 0. We present necessary and sufficient conditions on a couple of weight functions (σ, ϱ) such that the integral inqualities of weak type

and of extra-weak type

hold. Our proofs do not refer to the theory of Orlicz spaces.
Asymptotic behaviour of some reaction-diffusion systems modelling complex combustion on bounded domains
Authors: Avrin Joel D.
Published: November 2011

Synopsis

We consider three models of multiple-step combustion processes on bounded spatial domains. Previously, steady-state convergence results have been established for these models with zero Neumann boundary conditions imposed on the temperature as well as the mass fractions. We retain here throughout the same boundary conditions on the mass fractions, but in our first set of results we establish steady-state convergence results with fixed Dirichlet boundary conditions on the temperature. Next, under certain physically reasonable assumptions, we develop, for two of the models, estimates on the decay rates of both mass fractions to zero, while for the remaining model we develop estimates on the decay rate of one concentration to zero and establish a positive lower bound on the other mass fraction. These results hold under either set of boundary conditions, but when the Dirichlet conditions are imposed on the temperature, we are able to obtain estimates on the rate of convergence of the temperature to its (generally nonconstant) steady-state. Finally, we improve the results of a previous paper by adding a temperature convergence result.
Travelling wave solutions for a balance law
Authors: Mascia C.
Published: November 2011

We study entropy travelling wave solutions for first-order hyperbolic balance laws. Results concerning existence, regularity and asymptotic stability of such solutions are proved for convex fluxes and source terms with simple isolated zeros.
Convergence of regular approximations to the spectra of singular fourth-order Sturm–Liouville problems
Authors: Brown Malcolm , Greenberg Leon , Marletta Marco
Published: November 2011

We prove some new results which justify the use of interval truncation as a means of regularising a singular fourth-order Sturm–Liouville problem near a singular endpoint. Of particular interest are the results in the so-called lim-3 case, which has no analogue in second-order singular problems.
On a problem of Hartman and Wintner
Authors: Chernyavskaya N.
Published: November 2011

The Hartman–Wintner problem on asymptotic equivalence of fundamental systems of solutions (FSSs) for two Sturm–Liouville equations is studied. The following results are obtained: a criterion of asymptotic equivalence of FSSs, and sufficient conditions of asymptotic equivalence of FSSs which are expressed in terms of the coefficients of the considered equations only.
Estimates on the Green's function of second-order elliptic operators in ℝN
Authors: Hill Adrian T.
Published: November 2011

Sharp upper and lower pointwise bounds are obtained for the Green's function of the equation

for λ> 0. Initially, in a Cartesian frame, it is assumed that . Pointwise estimates for the heat kernel of ut + Lu = 0, recently obtained under this assumption, are Laplace-transformed to yield corresponding elliptic results. In a second approach, the coordinate-free constraint is imposed. Within this class of operators, the equations defining the maximal and minimal Green's functions are found to be simple ODEs when written in polar coordinates, and these are soluble in terms of the singular Kummer confluent hypergeometric function. For both approaches, bounds on are derived as a consequence.
Multiplicity of positive and nodal solutions for nonlinear elliptic problems in RN
Authors: Xie Ziqing
Published: November 2011

We consider the following problem:

where

for x ∈ RN, f(x, t), ft (x, t) ∈ C(RN × R), and f (x, t) ≧ 0 for all x ∈ RN and t ∈ R+, f(x, t) is an odd function of t. We show that if the maximum of Q(x) is achieved at k different points of RN, then for μ large enough the above problem has at least k positive solutions and k nodal solutions.
A relationship between the periodic and the Dirichlet BVPs of singular differential equations
Authors: Zhang Meirong
Published: November 2011

In this paper, a relationship between the periodic and the Dirichlet boundary value problems for second-order ordinary differential equations with singularities is established. This relationship may be useful in explaining the difference between the nonresonance of singular and nonsingular differential equations. Using this relationship, we give in this paper an existence result of positive periodic solutions to singular differential equations when the singular forces satisfy some strong force condition at the singularity 0 and some linear growth condition at infinity.
Shooting method for vortex solutions of a complex-valued Ginzburg–Landau equation
Authors: Chen Xinfu , Elliott Charles M. , Qi Tang
Published: November 2011

In this paper, we study all the stationary solutions of the form u(r)einθ to the complex-valued Ginzburg–Landau equation on the complex plane: here (r, θ) are the polar coordinates, and n is any real number. In particular, we show that there exists a unique solution which approaches to a nonzero constant as r → ∞.
Homogenisation of two-phase emulsions
Authors: Lipton Robert , Vernescu Bogdan
Published: November 2011

We consider an emulsion of two Stokes fluids, one of which is periodically distributed in the form of small spherical bubbles. The effects of surface tension on the bubble boundaries are modelled mathematically, as in the work of G. I. Taylor, by a jump only in the normal component of the traction. For a given volume fraction of bubbles, we consider the two-scale convergence, and in the fine phase limit we find that the bulk flow is described by an anisotropic Stokes fluid. The effective viscosity tensor is consistent with the bulk stress formula obtained by Batchelor [2].
Periodic orbits of difference equations
Authors: Beardon A. F. , Bullett S. R. , Rippon P. J.
Published: November 2011

The real difference equationan+2− (λ|an+1| + μan+1) +an= 0 may be interpreted as a dynamical system Φ:(an,an+1) ↦ (an+1,an+2) acting in the plane. The set ΛPof points (λ, μ) for which the mapping Φ is periodic has a rich structure. In this paper, we derive some geometric properties of ΛP(for example, we show that it is unbounded and uncountable), and we derive criteria for Φ to be periodic. We also investigate when Φ is conjugate to a rotation of the plane, and we describe how the rotation numbers of the corresponding circle maps Φ/|Φ| are related to the structure of ΛP.
The Riemann problem for a system of conservation laws of mixed type with a cubic nonlinearity
Authors: Shearer Michael , Yang Yadong
Published: November 2011

Using the viscosity-capillarity admissibility criterion for shock waves, we solve the Riemann problem for the system of conservation laws

where σ(u) = u3 − u. This system is hyperbolic at (u, v) unless . We find that the Riemann problem has a unique solution for all data in the hyperbolic regions, except for a range of data in the same phase (i.e. on the same side of the nonhyperbolic strip). In the nonunique cases, there are exactly two admissible solutions. The analysis is based upon a formula describing all saddle-to-saddle heteroclinic orbits for a family of cubic vector fields in the plane.
Positive root vectors
Authors: Omladič Matjaž , Omladič Vesna
Published: November 2011

As a generalisation of the well-known result of Perron and Frobenius, it was shown by Rothblum [13] and independently by Richman and Schneider [12] that every nonzero matrix with non-negative entries has a basis of the root space corresponding to the maximal eigenvalue, represented by root vectors with non-negative entries. Krein and Rutman [9] showed that a positive compact nonquasinilpotent operator on a Banach lattice has a positive eigenvector corresponding to its spectral radius. As an extension of both results, we give sufficient conditions on such an operator in order that its spectral subspace corresponding to its spectral radius has a basis made exclusively of positive root vectors.
Uniform asymptotic expansions for oblate spheroidal functions II: negative separation parameter λ
Authors: Dunster T. M.
Published: November 2011

In [3], uniform asymptotic expansions were derived for solutions of the oblate spheroidal wave equation (z2 − 1)d2p/dz2 + 2zdp/dz − (λ + μ2/(z2 − 1)) p = 0, for the case where the parameter μ is real and non-negative, the separation parameter λ is real and positive, and γ is purely imaginary (γ = iu). As u → ∞, uniform asymptotic expansions were derived involving elementary, Airy and Bessel functions, these being valid in certain subdomains of the complex z plane. In this paper the complementary case, where λ is real and negative, is considered. Asymptotic expansions are derived which are valid in certain subdomains of the half-plane |arg (z)| ≦ π/2, uniformly valid for u → ∞ with λ /u2 fixed and negative, and 0 ≦ μ/u ≦ − ½λ /u2 − δ, where δ is an arbitrary positive constant. Explicit error bounds are available for all the approximations.
Realisation of ordinary differential equations by retarded functional differential equations in neighbourhoods of equilibrium points
Authors: Faria Teresa , Magalhães Luis T.
Published: November 2011

This paper addresses the realisation of ordinary differential equations (ODEs) by retarded functional differential equations (FDEs) in finite-dimensional invariant manifolds, locally around equilibrium points. A necessary and sufficient condition for realisability of C1 vector fields is established in terms of their linearisations at the equilibrium.

It is also shown that any arbitrary finite jet of vector fields of ODEs can be realised without any further restrictions than those imposed by the realisability of its linear term, a fact of relevance for discussing the flows defined by FDEs around singularities, and their bifurcations. Besides, it is proved that such a realisation can always be achieved with FDEs whose nonlinearities are defined in terms of a finite number of delayed values of the solutions.
Boundary value problems on noncompact intervals
Authors: O'Regan Donal
Published: November 2011

Existence results are established for second-order boundary value problems for ordinary differential equations on non-compact intervals.
On the eigenvalue asymptotics for a nonselfadjoint elliptic problem involving an indefinite weight
Authors: Faierman M.
Published: November 2011

We derive asymptotic formulae for the distribution functions of the real parts of the eigenvalues of an oblique derivative problem involving an indefinite weight function.
Application of two parameter eigencurves to Sturm–Liouville problems with eigenparameter-dependent boundary conditions
Authors: Binding P. A. , Browne Patrick J.
Published: November 2011

Oscillation, comparison and asymptotic theory for the Sturm-Liouville problem

with 1/p, q, r ε L1 ([0, 1]), p, r > 0, are studied subject to eigenvalue-dependent boundary conditions

This continues previous work on cases with (− 1)j δj ≦ 0 where δj = ajdj − bjcj. We now consider the remaining sign conditions for δj, exploiting the interplay between the graph of cot θ− (λ, 1), for a modified Prüfer angle θ−, and the eigencurves of a related two-parameter problem.
Colour algebras and Cayley–Dickson algebras
Authors: Elduque Alberto , Myung Hyo Chul
Published: November 2011

Forms of the colour algebra introduced by Domokos and Kövesi-Domokos are studied by relating them to the well-known Cayley–Dickson algebras. Automorphisms groups and derivation algebras of these algebras are also determined.
Existence of two solutions of nonlinear elliptic equations with critical Sobolev exponents and mixed boundary conditions
Authors: Huang Feimin
Published: November 2011

Let Ω be a bounded domain in Rn(n ≧ 3) with Lipschitz-continuous boundary, ∂Ω = Γ0∪Γ1. In this paper we consider the following problem:

where φ ∈ L2 (Γ1), φ ≢ 0 on Γ1 and γ is the unit outward normal and p = 2n/(n − 2) = 2* is the critical exponent for the Sobolev embedding . We prove that for φ ∈ L2(Γ1) satisfying suitable conditions, the problem admits two solutions.
Sampling and interpolation for a lacunary spectrum
Authors: Lyubarskii Yu. , Spitkovsky I.
Published: November 2011

We reduce the problem of constructing a sampling and interpolating set for the space of functions with limited multi-band spectra to a problem of invertibility of certain convolution operators on a system of intervals, and obtain an example of such a set located in a horizontal strip along the real axis. We also study the question of sampling of a signal with two-banded spectra via its values at the union of two arithmetic progressions.
Stability and instability of some nonlinear dispersive solitary waves in higher dimension
Authors: de Bouard Anne
Published: November 2011

We study the stability of positive radially symmetric solitary waves for a three dimensional generalisation of the Korteweg de Vries equation, which describes nonlinear ion-acoustic waves in a magnetised plasma, and for a generalisation in dimension two of the Benjamin–Bona–Mahony equation.
Stability and characteristic wavelength of planar interfaces in the large diffusion limit of the inhibitor
Authors: Taniguchi Masaharu , Nishiura Yasumasa
Published: November 2011

A characteristic wavelength and its parametric dependency are studied for planar interfaces of activator-inhibitor systems as well as their stability in two-dimensional space. When an unstable planar interface is slightly perturbed in a random way, it develops with a characteristic wavelength, that is, the fastest-growing one. A natural question is to ask under what conditions this characteristic wavelength remains finite and approaches a positive definite value as the width of interface, say ε, tends to zero. In this paper, we show that the fastest-growing wavelength has a positive limit value as ε tends to zero for the system:

This is a fundamental fact for stuyding the domain size of patterns in higher-space dimensions.
An exotic totally real minimal immersion of S3 in ℂP3 and its characterisation
Authors: Chen B.-Y. , Dillen F. , Verstraelen L. , Vrancken L.
Published: November 2011

In a previous paper, B.-Y. Chen defined a Riemannian invariant δ by subtracting from the scalar curvature at every point of a Riemannian manifold the smallest sectional curvature at that point, and proved, for a submanifold of a real space form, a sharp inequality between δ and the mean curvature function. In this paper, we extend this inequality to totally real submanifolds of a complex space form. As a consequence, we obtain a metric obstruction for a Riemannian manifold Mn to admit a minimal totally real (i.e. Lagrangian) immersion into a complex space form of complex dimension n. Next we investigate three-dimensional submanifolds of the complex projective space ℂP3 which realise the equality in the inequality mentioned above. In particular, we construct and characterise a totally real minimal immersion of S3 in ℂP3.
Heat invariants for odd-dimensional hemispheres
Authors: van den Berg M. , Gilkey Peter B.
Published: November 2011

We compute the heat content asymptotics of odd dimensional hemispheres in closed form.
Least energy solutions for elliptic equations in unbounded domains
Authors: del Pino Manuel A. , Felmer Patricio L.
Published: November 2011

In this paper we study the existence of least energy solutions to subcritical semilinear elliptic equations of the form

where Ω is an unbounded domain inRNandfis aC1function, with appropriate superlinear growth. We state general conditions on the domain Ω so that the associated functional has a nontrivial critical point, thus yielding a solution to the equation. Asymptotic results for domains stretched in one direction are also provided.
Approximation of relaxed Dirichlet problems by boundary value problems in perforated domains
Authors: Maso Gianni Dal , Malusa Annalisa
Published: November 2011

Given an elliptic operator L on a bounded domain Ω ⊆ Rn, and a positive Radon measure μ on Ω, not charging polar sets, we discuss an explicit approximation procedure which leads to a sequence of domains Ωh ⊇ Ω with the following property: for every f ∈ H−1(Ω) the sequence uh of the solutions of the Dirichlet problems Luh = f in Ωh, uh = 0 on ∂Ωh, extended to 0 in Ω\Ωh, converges to the solution of the “relaxed Dirichlet problem” Lu + μu = f in Ω, u = 0 on ∂Ω.
Inequalities for a function involving its integral and derivative
Authors: Li Horng Jaan , Yeh Cheh Chih
Published: November 2011

We give a concise approach to generalising the inequalities of Wirtinger, Hardy, Weyl and Opial by using the well-known inequality: if X and Y are non-negative, then

for p > 1 (0 < p < 1), respectively.
On an elliptic equation related to the blow-up phenomenon in the nonlinear Schrödinger equation
Authors: Johnson Russell , Pan Xingbin
Published: November 2011

Synopsis

This paper is devoted to the study of the asymptotic behaviour of radial solutions to an elliptic equation in ℝn. The equation is derived from the blow-up problem in the non-linear Schrödinger equation.
Large-frequency global regularity for the incompressible Navier–Stokes equation
Authors: Avrin Joel D.
Published: November 2011

We obtain global existence and regularity of strong solutions to the incompressible Navier–Stokes equations for a variety of boundary conditions in such a way that the initial and forcing data can be large in the high-frequency eigenspaces of the Stokes operator. We do not require that the domain be thin as in previous analyses. But in the case of thin domains (and zero Dirichlet boundary conditions) our results represent a further improvement and refinement of previous results obtained.
Some second-order integral inequalities of generalized Hardy type
Authors: Florkiewicz B. , Wojteczek K.
Published: November 2011

A uniform method of obtaining various types of integral inequalities involving a function and its first derivative is extended to integral inequalities involving a function and its second derivative. Specifically, some quadratic integral inequalities of generalized Hardy type involving a function and its second derivative are derived and examined. The functions for which the inequalities hold are characterized by boundary conditions.
Symmetric groupoids, free categories andE*-unitary inverse monoids
Authors: Leech Jonathan
Published: November 2011

Symmetric groupoids which classify the monomorphism contexts of objects in arbitrary categories are studied, along with their connections to symmetric inverse monoids and symmetric inverse algebras. Particular attention is given to symmetric groupoids of objects in free categories and to the inverse algebras induced from them by 0-closure. These graph algebras generalize both the class of polycyclic semigroups and the class of combinatorial ω-semigroups with adjoined zeros. Since all such algebras areE*-unitary, an analogue of McAlister's theory of.E-unitary inverse monoids is developed for a special class ofE*-unitary inverse monoids and then illustrated on the class of graph algebras.
Periodic solutions of planar systems with two delays
Authors: Ruan Shigui , Wei Junjie
Published: November 2011

In this paper, we consider a planar system with two delays:

ẋ1(t) = −a0x1(t) + a1F1 (x1(t − τ1), x2(τ−t2)).

ẋ2(t) = −b0x2(t) + b1F2 (x1(t − τ1), x2(t − τxs2)).

Firstly, linearized stability and local Hopf bifurcations are studied. Then, existence conditions for non-constant periodic solutions are derived using degree theory methods. Finally, a simple neural network model with two delays is analysed as an example.