bluuebunny/crossref_metadata_2025_split huggingface dataset:
Examples of degenerate symmetric differential operators with infinite deficiency indices in L2(ℝm)
Authors: Orochko Yurii B. 
Published: November 2011

For an unbounded self-adjoint operator A in a separable Hilbert space ℌ and scalar real-valued functions a(t), q(t), r(t), t ∊ ℝ, consider the differential expressionacting on ℌ-valued functions f(t), t ∊ ℝ, and degenerating at t = 0. Let Sp denotethe corresponding minimal symmetric operator in the Hilbert space (ℝ) of ℌ-valued functions f(t) with ℌ-norm ∥f(t)∥ square integrable on the line. The infiniteness of the deficiency indices of Sp, 1/2 < p < 3/2, is proved under natural restrictions on a(t), r(t), q(t). The conditions implying their equality to 0 for p ≥ 3/2 are given. In the case of a self-adjoint differential operator A acting in ℌ = L2(ℝn), the first of these results implies examples of symmetric degenerate differential operators with infinite deficiency indices in L2(ℝm), m = n + 1.
Regularity of almost extremal solutions of Monge–Ampère equations
Authors: Urbas John I. E. 
Published: November 2011

SynopsisWe show that for a large class of Monge-Ampère equations, generalised solutions on a uniformly convex domain Ω⊂ℝn are classical solutions on any pre-assigned subdomain Ω′⋐Ω, provided the solution is almost extremal in a suitable sense. Alternatively, classical regularity holds on subdomains of Ω which are sufficiently distant from ∂Ω. We also show that classical regularity may fail to hold near ∂Ω in the nonextremal case. The main example of the class of equations considered is the equation of prescribed Gauss curvature.
The existence of a positive solution of semilinear elliptic equations with limiting Sobolev exponent
Authors: Wang Shixiao 
Published: November 2011

SynopsisOur paper concerns the existence of a positive solution for the equation:A new condition, which guarantees the existence of a solution of the above equation, has been established. It has also given some sharp information in the cases where: (1) a(x) = λ = const. and Ω is a “thin” domain; (2) Ω is a ball and a(x) is a radially symmetrical function.
Homogeneous polynomial splines
Authors: Goodman T. N. T. , Lee S. L. 
Published: November 2011

SynopsisWe construct functions which are piecewise homogeneous polynomials in the positive octant in three dimensions. These give a rich and elegant theory which combines properties of polynomial box splines see [6] and the references therein) with the explicit representation of simple exponential box splines [11], while enjoying complete symmetry in the three variables. By a linear transformation followed by a projection on suitable planes, one obtains piecewise polynomial functions of two variables on a mesh formed by three pencils of lines. The vertices of these pencils may be finite or one or two may be infinite, i.e. the corresponding pencils may comprise parallel lines. As a limiting case, all three vertices become infinite and one recovers polynomial box splines on a three-direction mesh.
Strongly elliptic boundary integral equations for electromagnetic transmission problems
Authors: Costabel Martin , Stephan Ernst P. 
Published: November 2011

SynopsisWe study a boundary integral equation method for transmission problems for strongly elliptic differential operators, which yields a strongly elliptic system of pseudodifferential operators and which therefore can be used for numerical computations with Galerkin's procedure. The method is shown to work for the vector Helmholtz equation in ℝ3 with electromagnetic transmission conditions. We propose a slightly modified system of boundary values in order for the corresponding bilinear form to be coercive over H1. We analyse the boundary integral equations using the calculus of pseudodifferential operators. Here the concept of the principal symbol is used to derive existence and regularity results for the solution.
A combinatorial property of finite full transformation semigroups
Authors: Howie John M. , Robertson Edmund F. , Schein Boris M. 
Published: November 2011

SynopsisLet E be the set of idempotents in the semigroup Singn of singular self-maps of N = {1, …, n}. Let α ∊ Singn. Then α ∊ E2 if and only if for every x in im α the set xα−1 either contains x or contains an element of (im α)′.Write rank α for |im α| and fix α for |{x ∊ N: xa = x}|. Define (x, xα, xα2) to be an admissible α-triple if x ∊ (im α)′, xα3 ≠ xα2. Let comp α (the complexity of α) be the maximum number of disjoint admissible α-triples. Then α ∊ E3 if and only if
On the domains of minimal and maximal operators for regularisable singular differential expressions
Authors: Race D. 
Published: November 2011

SynopsisCertain classical differential expressions which are singular at a finite end-point (or at an interior point) can be represented as regular, scalar quasi-differential expressions, the best-known examples being the Boyd Equation and Laplace Tidal Wave Equation. We show here that in all such cases the domains of the minimal and maximal operators in the appropriate weighted Hilbert space, for the regularised expression, coincide with the corresponding domains for the expression in its original, singular form.This is contrasted with a known property of the corresponding expression domains. Whereas for an expressionM, the operator domains contain only functions y for which bothyandMylie in the appropriate Hilbert space, the expression domain comprises a much larger set of functions with no such restrictions beyond those necessary forMyto exist as a function. In the second-order case, the expression domain of the regularisation of a singular expression is known to be a strict subset of the original expression domain, contrasting with the results proved here for the operator domains.
Asymptotic behaviour of solutions to the coagulation–fragmentation equations. I. The strong fragmentation case
Authors: Carr J. 
Published: November 2011

SynopsisThe 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

SynopsisThe 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

SynopsisThis 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

SynopsisA characterisation of the surface energy density for the relaxation in V(Ω; Rp) of the functionalis 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

SynopsisAn 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.
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 equationis 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.
A uniqueness proof for the Wulff Theorem
Authors: Fonseca Irene , Müller Stefan 
Published: November 2011

SynopsisThe 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

SynopsisWe 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).
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 formWe 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 representationA formula forϕis given, which takes into account the interaction between the bulk energy densityfand the surface energy densityg.
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 problemwhere 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 equationwhere 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 diffusivitywith 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 lawwith 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

SynopsisThe 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

SynopsisThis 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.
A plane sextic and its five cusps
Authors: Edge W. L. 
Published: November 2011

SynopsisA 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

SynopsisWe 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

SynopsisThe paper discusses properties of solutions of the differential equationand in particular the existence and uniqueness of solutions to the boundary-value problem associated with the above equation and the boundary conditionsThis equation, first introduced by Skyrme, is a spherically symmetric model for a soliton description of nucleons.
Convexity of certain integrals of the calculus of variations
Authors: Dacorogna B. 
Published: November 2011

SynopsisIn 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′).
Théorème d'existence pour des problèmes variationnels non convexes
Authors: Raymond J. P. 
Published: November 2011

SynopsisOn 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

SynopsisIn 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

SynopsisWe 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].
Subharmonics near an equilibrium for some second-order Hamiltonian systems
Authors: Felmer Patricio L. , Silva Elves A. de B. e 
Published: November 2011

SynopsisThis work is devoted to the study of subharmonic solutions of a second-order Hamiltonian systemnear 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 aswhere 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

SynopsisA 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.
Two representations of a conditioned superprocess
Authors: Evans Steven N. 
Published: November 2011

SynopsisWe 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.
On the existence and nonexistence of global solutions of a diffusion-advection equation
Authors: Deng Keng 
Published: November 2011

SynopsisWe 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

SynopsisLetbe 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 typeand of extra-weak typehold. Our proofs do not refer to the theory of Orlicz spaces.