bluuebunny/crossref_metadata_2025_split huggingface dataset:
Fractional powers of operators and Riesz fractional integrals
Authors: Schiavone S. E. 
Published: November 2011

SynopsisIn this paper, a theory of fractional powers of operators due to Balakrishnan, which is valid for certain operators on Banach spaces, is extended to Fréchet spaces. The resultingtheory is shown to be more general than that developed in an earlier approach by Lamb, and is applied to obtain mapping properties of certain Riesz fractional integral operators on spaces of test functions.
M(λ)-computation for singular differential systems
Authors: Kaper Hans G. , Krall Allan M. 
Published: November 2011

SynopsisDepending upon the initial data associated with the fundamental matrix, the function M(λ), used to generate L2-solutions of homogeneous linear differential systems, may vary. We show that there is a matrix bilinear transformation between such functions M(λ) with different initial data and illustrate how the result can be used to simplify the calculation of a specific M(λ)-function for a scalar second-order problem.
Global blow-up of a nonlinear heat equation
Authors: Lacey A. A. 
Published: November 2011

SynopsisSolutions to semilinear parabolic equations of the form ut = Δu + f(u), x in Ω, which blow up at some finite time t* are investigated for “slowly growing” functions f. For nonlinearities such as f(s) = (2 +s)(ln(2 +s))1+b with 0 < b < l,u becomes infinite throughout Ω as t→t* −. It is alsofound that for marginally more quickly growing functions, e.g. f(s) = (2 + s)(ln(2 +s))2, u is unbounded on some subset of Ω which has positive measure, and is unbounded throughout Ω if Ω is a small enough region.
A Liapunov functional for a scalar differential difference equation
Authors: Infante E. F. , Walker J. A. 
Published: November 2011

SynopsisGiven the scalar, retarded differential difference equation x'(t)=ax(t) +bx(t−τ), a quadratic functional in explicit form is obtained that yields necessary and sufficient conditions for the asymptotic stability of this equation. This functional a Liapunov functional, is obtained through the study of the Liapunov functions associated with a difference equation approximation of the difference differential equation. The functional then obtained not only yields necessary and sufficient conditions for asymptotic stability, but provides estimates for rates of decay of the solutions as well as conditions, for asymptotic stability independent of the magnitude of the delay τ.
Bounds for solutions of a class of quasilinear elliptic boundary value problems in terms of the torsion function
Authors: Payne L. E. 
Published: November 2011

SynopsisIn this paper maximum principles are employed to relate solutions of certain classes of nonlinear elliptic problems to solutions of the associated torsion problem. By this method a number of new isoperimetric inequalities are derived. In special cases solutions of the nonlinear problems are also related to solutions of the clamped membrane problem.
V-regular semigroups
Authors: Nambooripad K. S. S. , Pastijn F. 
Published: November 2011

SynopsisA regular semigroup S is called V-regular if for any elements a, b ∈ S and any inverse (ab)′ of ab, there exists an inverse a′ of a and an inverse b′ of b such that (ab)′ = b′a′. A characterization of a V-regular semigroup is given in terms of its partial band of idempotents. The strongly V-regular semigroups form a subclass of the class of V-regular semigroups which may be characterized in terms of their biordered set of idempotents. It is shown that the class of strongly V-regular semigroups comprises the elementary rectangular bands of inverse semigroups (including the completely simple semigroups), a special class of orthodox semigroups (including the inverse semigroups), the strongly regular Baer semigroups (including the semigroups that are the multiplicative semigroup of a von Neumann regular ring), the full transformation semigroup on a set, and the semigroup of all partial transformations on a set.
Diffusion processes with non-smooth diffusion coefficients and their density functions
Authors: Lyons T. J. , Zheng W. A. 
Published: November 2011

SynopsisDenote by Xt an n-dimensional symmetric Markov process associated with an elliptic operatorwhere (aij) is a bounded measurable uniformly positive definite matrix-valued function of x. Let f(x, t) be a measurable function defined on Rn × [0, 1]. In this paper, we prove that f(Xt, t) is a regular Dirichlet process if and only if the following two conditions are satisfied:(i) For almost every and (ii) Let be a sequence of subdivisions of [0,1] so thatThenAs an application of the above result, we prove the following fact: Let p(y, t) be the probability density of the diffusion process Yt, associated with the elliptic operatorwhere (bi) are bounded measurable functions of x and we suppose that . Then, p(Yt, t) is a regular Dirichlet process and therefore p(.,.) satisfies (i) and (ii).
Transformation and canonical forms of functional-differential equations
Authors: Neuman František 
Published: November 2011

SynopsisFunctional-differential equations, especially linear ones, are considered with respect to global pointwise transformations. Two types of canonical forms for certain classes of these equations are introduced. These transformations and the corresponding canonical forms preserve oscillatory or non-oscillatory behaviour of solutions. They are also suitable for studying both-side solutions of equivalent functional-differential equations.
Travelling-wave solutions to the Korteweg-de Vries-Burgers equation
Authors: Bona J. L. , Schonbek M. E. 
Published: November 2011

SynopsisThe existence and certain qualitative properties of travelling-wave solutions to the Korteweg-de Vries-Burgers equation,are established. The limiting behaviour of these waves, when ε tends to zero and when δ tends to zero is examined together with a singular limit wherein both ε and δ tend to zero.
General edge asymptotics of solutions of second-order elliptic boundary value problems I
Authors: Costabel Martin , Dauge Monique 
Published: November 2011

SynopsisThis is the first of two papers in which we study the singularities of solutions of second-order linear elliptic boundary value problems at the edges of piecewise analytic domains in ℝ3. When the opening angle at the edge is variable, there appears the phenomenon of “crossing” of the exponents of singularities. For this case, we introduce the appropriate combinations of the simple tensor product singularities that allow us to give estimates in ordinary and weighted Sobolev spaces for the regular part of the solution and for the coefficients of the singularities. These combinations appear in a natural way as sections of an analytic bundle above the edge. Their behaviour is described with the help of divided differences of powers of the distance to the edge. The class of operators considered includes second-order elliptic operators with analytic complex-valued coefficients with mixed Dirichlet, Neumann or oblique derivative conditions. With our description of the singularities we are able to remove some restrictive hypotheses that were previously made in other works. In this first part, we prove the basic facts in a simplified framework. Nevertheless the tools we use are essentially the same in the general situation.
General edge asymptotics of solutions of second-order elliptic boundary value problems II
Authors: Costabel Martin , Dauge Monique 
Published: November 2011

SynopsisThis is the second of two papers in which we study the singularities of solutions of second-order linear elliptic boundary value problems at the edges of piecewise analytic domains in ℝ3. When the opening angle at the edge is variable, there appears trie phenomenon of “crossing” of the exponents of singularities. In Part I, we introduced for the Dirichlet problem appropriate combinations of the simple tensor product singularities.In this second part, we extend the results of Part I to general non-homogeneous boundary conditions. Moreover, we show how these combinations of singularities appear in a natural way as sections of an analytic vector bundle above the edge. In the case when the interior operator is the Laplacian, we give a simpler expression of the combined singular functions, involving divided differences of powers of a complex variable describing the coordinates in the normal plane to the edge.
Periodic solutions to second order Hamiltonian systems in an unbounded potential well
Authors: Greco C. 
Published: November 2011

SynopsisIn this paper we give some results on the existence of periodic solutions to the second order Hamiltonian system:where and Ω is an open set of ℝn with non-empty bounded complement ℝn\Ω; we suppose V(t, x) is periodic in t, V(t, x)→ + ∞ as x → ∂Ω and V is super (or sub)-quadratic as |x| → + ∞.
Unstable planes for rank 2 reflexive sheaves
Authors: Miró-Roig Rosa M. 
Published: November 2011

SynopsisThe purpose of this paper is to give a sharp bound for the order of instability of an unstable hyperplane of a rank 2 stable reflexive sheaf E on ℙn, in terras of the Chern classes of E; and a sharp boundfor the order of instability of an unstable hyperplane of a rank 2 nonstable reflexive sheaf E on ℙn, in terms of the Chern classes of E and the order of nonstability of E.
H-measures applied to symmetric systems
Authors: Antonić Nenad 
Published: November 2011

H-measures were recently introduced by Luc Tartar as a tool which might provide better understanding of propagating oscillations. Independently, Patrick Gerard introduced the same objects under the name of microlocal defect measures. Partial differential equations of mathematical physics can often be written in the form of a symmetric system:where Ak and B are matrix functions, while u is an unknown vector function, and f a known vector function. In this work we prove a general propagation theorem for H-measures associated to symmetric systems. This result, combined with the localisation property, is then used to obtain more precise results on the behaviour of H-measures associated to the wave equation, Maxwell's and Dirac's systems, and second-order equations in two variables.
Poincaré series for the occurrence of certain modular representations of GL(n,p) in the symmetric algebra
Authors: Carlisle David P. , Walker Grant 
Published: November 2011

SynopsisThe number of occurrences of the Steinberg representation St of GL(n, p) as a composition factor in the symmetric algebra Fp[x1, … xn] has been determined by several authors. We extend this result to the representations of GL(n, p) which are the closest neighbours of St in the SL(n, p) weight diagram. The method is to play off duality for GL(n, p)-modules against connectivity for M(n, p)-modules. The result is equivalent to determining the cohomology groups of the corresponding indecomposable stable summands of the localisation of an n-fold product of complex projective spaces at the prime p.
Asymptotic behaviour of solutions of hyperbolic conservation laws ut + (um)x = 0 as m → ∞ with inconsistent initial values
Authors: Xu Xiangsheng 
Published: November 2011

SynopsisWe study the behaviour of solutions u = um of ut, + (um)x = 0 for t > 0, x ∊ R, u(x, 0) = u0(x), u0 ≧0, u0 ∊ L1(R) as m → ∞. This is a singular perturbation problem about m = ∞ if u0 > 1 on a set of positive measure. It is shown that the limit exists and satisfies the stationary equation
Symmetry breaking for Δu + 2δe−u = 0 on a disk with general boundary conditions
Authors: Lin Song-Sun 
Published: November 2011

SynopsisWe discuss the radially, symmetric solutions and the symmetry breaking of the equation Δu + 2δe −u = 0 in D and u + b(∂u/∂n) = 0 on ∂D, where D is the unit disk in ℝ2, δ >0 and b is a constant. We prove that for any b < 0, there exists > 0 such that there are exactly two radially symmetric solutions for δ ∊ (0, ), one for δ = and none for δ > δ*b. For , where m is a positive integer, there are (b), k = 1, …, m, such that the equation has symmetry breaking at δ*k (b) on the lower branch of radially symmetric solutions.
Spectral analysis of perturbed multiplication operators occurring in polymerisation chemistry
Authors: Kokholm Niels Jørgen 
Published: November 2011

SynopsisWe consider a mathematical model for the motion of a marked monomer in a system of reacting polymers at equilibrium. A well-posed integro-differential initial value problem for the probability of finding the marked monomer in a molecule of a given length is formulated. We prove exponential convergence of the probability to a unique equilibrium distribution. A quite complete spectral analysis is carried out for a self adjoint operator, which is a perturbation of a multiplication operator by an integral operator and is related to the generator of the time evolution.
The asymptotic behaviour of the solutions of the Kassoy problem with a modified source term
Authors: Budd C. , Qi Y. 
Published: November 2011

SynopsisWe study the asymptotic behaviour as x →∞ of the solutions of the ordinary differential equation problemThis equation generalises the ordinary differential equation obtained by studying the blow-up of the similarity solutions of the semilinear parabolic partial differential equation vt=vxx = ev. We show that if λ≦1, all solutions of (*) tend to —∞ as rapidly as the function —exp (x2/4) (E- solutions). However, if λ>1, then there also exists a solution which tends to –∞, like 2λlog(x) (L-solutions). Thus, the case λ = 1, for which (*) reduces tothe Kassoy equation, is the borderline between two quite different forms of asymptotic behaviour of the function u(x).
Normal and osculating maps for submanifolds of RN
Authors: Gasparini I. Cattaneo , Romani G. 
Published: November 2011

SynopsisLet Mn be a manifold supposed “nicely curved” isometrically immersed in ℝn+p. Starting from a generalised Gauss map associated to the splitting of the normal bundle defined from the values of the fundamental forms of M of order k (k ≧ 0), we give necessary and sufficient conditions for the map to be totally geodesic and harmonic . For k = 0 is the classical Gauss map and our formula reduces to Ruh–Vilm's formula with a more precise formulation due to the consideration of the splitting of the normal bundle.We also give necessary conditions for M, supposed complete, to admit an isometric immersion with . This theorem generalises a theorem of Vilms on the manifolds with second fundamental forms parallel (case k = 0). The result is interesting as the class of manifolds satisfying the condition is larger than the class of manifolds satisfying .
On boundary feedback stabilisability of a viscoelastic beam
Authors: Leugering Guenter 
Published: November 2011

SynopsisIt is shown that a cantilevered beam with weak viscoelastic damping of Boltzmann-type can be uniformly stabilised by velocity feedback applied as a shearing force at the free end of the beam. Estimates for the viscoelastic energy are derived using the energy multiplier method. The energy decay is related to the decay of the relaxation modulus associated with the viscoelastic material.
A nonlinear pseudoparabolic equation
Authors: Ang Dang Dinh , Thanh Tran 
Published: November 2011

SynopsisThe authors prove results on uniqueness and global existence of initial and boundary value problems for the nonlinear pseudoparabolic equationwith nonhomogeneous boundary conditions. A salient feature of the paper is that F and its partial derivatives are allowed to be unbounded. In the special case b(x, t)= α2 (a positive constant), it is proved that the corresponding solution uα, under appropriate conditions on the data (which are satisfied, for example, by the Benjamin–Bona–Mahony equation), uα→ u0 the solution corresponding to β = 0, on sufficiently small time interval. A result on the asymptotic behaviour of the solution is given for t → ∞.
Real roots of polynomials and right inverses for partial differential operators in the space of tempered distributions
Authors: Langenbruch Michael 
Published: November 2011

SynopsisLetP(D)be a partial differential operator with constant coefficients. IfP(D)has a continuous linear right inverse in the space of tempered distributions, thenPis the product of a polynomial without real roots and a real polynomial admitting a right inverse. If the polynomialPis real and irreducible, thenP(D)admits a right inverse in the tempered distributions if and only ifP(×)has the property of zeros of R. Thorn.
On a linear, partly hyperbolic model of viscoelastic flow past a plate
Authors: Fraenkel L. E. 
Published: November 2011

SynopsisThe paper presents solutions, for a class of shear relaxation functions, ofa linear problem formulated by Joseph [8] to elucidate the steady, supercritical flow of a viscoelastic fluid past a semi-infinite flat plate. The velocity (U, 0) at infinity is parallel to the plate, and‘supercritical flow ‘means that U is greater than the propagation speed C of shear waves. As a result, the vorticity satisfies a hyperbolic equation and is confined to the region downstream of a shock wave from the leading edge of the plate. The disturbance velocity fieldextends upstream of the shock and is continuous across it. In contrast to the case of a Newtonian fluid, the solutions are unique under the condition that the functions representing the vorticity on the two sides of the platebelong to a certain Banach space.
A variational approach to a nonlinear Steklov problem
Authors: Ermens B. , Mawhin J. 
Published: November 2011

SynopsisWe study the Steklov problem which consists in finding a complex function w(z) = U(z) + iV(z) holomorphic in the open unit disc G of the complex plane, continuous on its closure, such that V(0) = 0 and verifying on its boundary the condition(d/ds)V(eis)+g(s, U(eis))=h(s),where g and h are given functions. Using the Hilbert transform, the problem is reduced to the search for periodic solutions of an equivalent singular integro-differential equation which is treated by the direct method of the calculus of variations. When g(s,.) is non-decreasing, we obtain a necessary and sufficient condition for the solvability. The case of a non-monotone nonlinearity is also considered.
Complicated dynamics of scalar reaction diffusion equations with a nonlocal term
Authors: Fiedler Bernold , Poláčik Peter 
Published: November 2011

SynopsisWe consider the dynamics of scalar equations ut, = uxx + f(x, u) + c(x)α(u), 0 < x < l, where α denotes some weighted spatial average and Dinchlet boundary conditions are assumed. Prescribing f, c, α appropriately, it is shown that complicated dynamics can occur. Specifically, linearisations at equilibria can have any number of purely imaginary eigenvalues. Moreover, the higher order terms of the reduced vector field in an associated centre manifold can be prescribed arbitrarily, up to any finite order. These results are in marked contrast with the case α = 0, where bounded solutions are known to converge to equilibrium.
Serrin integrals and second order problems of plasticity
Authors: Cesari L. , Yang Wei H. 
Published: November 2011

SynopsisWe use the modern tools of the duality principles and the calculus of variations to formulate, analyse and solve a class of plasticity problems involving second order partial derivatives. The Serrin-type integrals can most appropriately facilitate the existence statements for the extrema from either side of the duality relation in a larger class of BV functions, and interpret the solutions with possible discontinuities on sets of measure zero. The exact solutions of a beam and numerical solutions of a circular plate are presented to demonstrate the theoretical conclusions.
Asymptotic stability criteria for delay-differential equations
Authors: Becker L.C. , Burton T.A. 
Published: November 2011

SynopsisThis paper is concerned with the problem of showing uniform stability and equiasymptotic stability of thezero solution of functional differential equations with either finite or infinite delay. The investigations are based on Liapunov's direct method and attention is focused on those equations whose right-hand sides are unbounded for bounded state variables.
On the vibrations of a square membrane
Authors: Komornik V. 
Published: November 2011

SynopsisWe establish that if Ω is an open square and if P is an arbitrary point of Ω, then the solutions of the wave equation with Dirichlet boundary condition utt − Δu = 0 in R × Ω, u = 0 on R × Γ can remain strictly positive during an arbitrarily large time. In fact stronger results are proved, considering several points of Ω simultaneously and prescribing different, more general, behaviour of the solution at these points.
On the asymptotic behaviour at infinity of solutions of the traction boundary value problem
Authors: Orazov B.B. 
Published: November 2011

SynopsisKorn's inequalities are proved for star-shaped domains and it is shown how the constants in these inequalities depend on the dimensions of the domain. These inequalities are then used to prove a generalisation of Saint-Venant's Principle for nonlinear elasticity and additionally to establish the asymptotic behaviour of solutions to the traction boundary value problem for a non-prismatic cylinder.
Remarks concerning a free boundary problem arising in the theory of liquid drops and in plasma physics
Authors: Barrett John W. , Elliott Charles M. 
Published: November 2011

SynopsisWe consider a generalisation of the liquid drop problem, introduced in [1, Part II], by allowing the upper and lower surfaces to have different surface tension coefficients γv and γu. We study the existence, uniqueness and regularity of this problem. In addition, we show that as γv/γu →0, the solution of this problem converges to the solution of the “plasma problem”.
On a uniqueness theorem of L. Amerio and G. Prouse
Authors: Haraux Alain 
Published: November 2011

SynopsisWe give two generalizations of a theorem of L. Amerio and G. Prouse concerning uniqueness of the almost-periodic solution of the wave equation with a local multivalued damping term and almost-periodic forcing.Counterexamples are given which show that these results may fail if some hypotheses of the theorems are dropped. They also show that the two generalizations are relatively independent.
Resolution of a semilinear equation in L1
Authors: Gallouët Thierry , Morel Jean-Michel 
Published: November 2011

SynopsisLet Ω = ℝN or Ω be a bounded regular open set of ℝN and let γ(x, S): Ω × ℝ → ℝ be a continuous nondecreasing function in s, measurable in x, such that γ(x, 0) = 0 almost everywhere. We solve, for f ∈ L1(Ω), the problem (P): −Δu + γ(., u) = f in Ω, u = 0 on ∂Ω. (In fact, for this result, instead of assuming that γ is nondecreasing in s we need only that γ(x, s)s≧0.) We deduce an ‘almost’ necessary and sufficient condition on , in order that (P) has a solution. Roughly speaking, this condition is f = −ΔV + g, with g ∈ L1(Ω) and γ(., V) ∈ L1(Ω)
On the semigroups of partial one-to-one order-decreasing finite transformations
Authors: Umar Abdullahi 
Published: November 2011

SynopsisLet In be the symmetric inverse semigroup on Xn = {1,…, n}, let Sln be the subsemigroup of strictly partial one-to-one self-maps of Xn and let = { α ∊ SIn: x} ≦ x = U = ∅= be the semigroup of all partial one-to-one decreasing maps including the empty or zero map of Xn. In this paper it is shown that is an (irregular, for n ≧ 2) type A semigroup with n D*-classes and D* = I*. Further, it is shown that is generated by the n(n + l)/2 quasi-idempotents in
The equation utt − Δu = |u|p for the critical value of p
Authors: Schaeffer Jack 
Published: November 2011

SynopsisThe equation utt − Δu = |u|p is considered in two and three space dimensions. Smooth Cauchy data of compact support are given at t = 0. For the case of three space dimensions, John has shown that solutions with sufficiently small data exist globally in time if but that small data solutions blow up in finite time if Glassey has shown the two dimensional case is similar. This paper shows that small data solutions blow up in finite time when p is the critical value, in three dimensions and in two.