In this paper we consider the Sturm–Liouville operator d2/dx2 − 1/x on the interval [a, b], a < 0 < b, with Dirichlet boundary conditions at a and b, for which x = 0 is a singular point. In the two components L2(a, 0) and L2(0, b) of the space L2(a, b) = L2(a, 0) ⊕ L2(0, b) we define minimal symmetric operators and describe all the maximal dissipative and self-adjoint extensions of their orthogonal sum in L2(a, b) by interface conditions at x = 0. We prove that the maximal dissipative extensions whose domain contains only continuous functions f are characterized by the interface condition limx→0+(f′(x)−f′(−x)) = γf(0) with γ∈C+∪R or by the Dirichlet condition f(0+) = f(0−) = 0. We also show that the corresponding operators can be obtained by norm resolvent approximation from operators where the potential 1/x is replaced by a continuous function, and that their eigen and associated functions can be chosen to form a Bari basis in L2(a, b).

This paper is concerned with the existence of topological multivortex solutions in (2 + 1) self-dual gauge theories such as the classical abelian Higgs model or the Chern–Simons Higgs gauge theories. A general form of topological multivortex equations is presented with the inclusion of antivortices. Two kinds of solutions are considered; topological vortex solutions and topological vortex–antivortex solutions. We construct these solutions by super and subsolution methods, and derive the exponential decay of solutions at infinity and the quantized integral formula. As an application, we prove the existence of a topological multivortex solutions in a generalized Chern–Simons Higgs theory.

An asymptotic reduction of the Gierer–Meinhardt activator-inhibitor system in the limit of large inhibitor diffusivity and small activator diffusivity ε leads to a singularly perturbed nonlocal reaction-diffusion equation for the activator concentration. In the limit ε → 0, this nonlocal problem for the activator concentration has localized spike-type solutions. In this limit, we analyze the motion of a spike that is confined to the smooth boundary of a two or three-dimensional domain. By deriving asymptotic differential equations for the spike motion, it is shown that the spike moves towards a local maximum of the curvature in two dimensions and a local maximum of the mean curvature in three dimensions. The motion of a spike on a flat segment of a two-dimensional domain is also analyzed, and this motion is found to be metastable. The critical feature that allows for the slow boundary spike motion is the presence of the nonlocal term in the underlying reaction-diffusion equation.

The aim of this paper is to describe a technique based on matched asymptotic expansions that allows us to derive the variation of the stress intensity factors in a homogeneous isotropic elastic medium under plane strain deformation. The case of antiplane shearing is also considered.

This paper introduces an algorithm for calculating all discrete point symmetries of a given partial differential equation with a known nontrivial group of Lie point symmetries. The method enables the user to determine the discrete symmetries with little more effort than is used to find the Lie symmetries. It is used to obtain the discrete point symmetries of Burgers' equation, the spherical Burgers' equation, and the Harry–Dym equation. The method can be extended to some types of nonlocal symmetry; we derive the quasi-local discrete symmetries of a system of PDEs from gas dynamics.

We study an inverse boundary problem for the diffusion equation in ℝ2. Our motivation is that this equation is an approximation of the linear transport equation and describes light propagation in highly scattering media. The diffusion equation in the frequency domain is the nonself-adjoint elliptic equation div(D grad u) - (cμa + iω0) u = 0; ω0 ≠ 0, where D and μa are the diffusion and absorption coefficients. The inverse problem is the reconstruction of D and μa inside a bounded domain using only measurements at the boundary. In the two-dimensional case we prove that the Dirichlet-to-Neumann map, corresponding to any one positive frequency ω0, determines uniquely both the diffusion and the absorption coefficients, provided they are sufficiently slowly-varying. In the null-background case we estimate analytically how large these coefficients can be to guarantee uniqueness of the reconstruction.

We present a model of sedimentation in a subsiding fluvio-deltaic basin with steady sediment supply and unsteady base level. We demonstrate that mass transfer in a fluvio-deltaic basin is analogous to heat transfer in a generalized Stefan problem, where the basin's shoreline represents the phase front. We obtain a numerical solution to the governing equations for sediment transport and deposition in this system via an extension of a deforming-grid technique from the phase-change literature. Through modification of the heat-balance integral method, we also develop a semi-analytical solution, which agrees well with the numerical solution. We construct a space of dimensionless groups for the basin and perform a systematic exploration of this space to illustrate the influence of each group on the shoreline trajectory. Our model results suggest that all subsiding fluvio-deltaic basins exhibit a standard autoretreat shoreline trajectory in which a brief period of shoreline advance is followed by an extended period of shoreline retreat. Base-level cycling produces a shoreline response that varies relative to the autoretreat signal. Contrary to previous studies, we fail to observe either a strong phase shift between shoreline and base level or a pronounced attenuation of the amplitude of shoreline response as the frequency of base-level cycling decreases. However, the amplitude of shoreline response to base-level cycling is a function of the basin's age.

In this paper westudy quasilinear second order boundary value problems with multivalued right hand side and Dirichletboundary conditions. We prove three existence theorems. The first two deal with the“convex” and “nonconvex” problems respectively, while the third establishesthe existence of extremal solutions. For the first two the proof is based on the theory of nonlinearoperators of monotone type, while the proof of the third uses a fixed pointargument.

Let(π_{λ}, ℋ_{λ}) be a unitary highest weight representation ofthe connected Lie group G and ℊ its Lie algebra. Thenℊ contains an invariant closed convex cone W_{\rm{max}}such that, for each X∈W_{\rm{max}}^0, the selfadjoint operatori·dπ_{λ}(X) is bounded from above. We show that for each suchX, the space ℋ_{λ}^{∞} of smooth vectors forthe action of G on ℋ_{λ} coincides with the set𝒟^{∞}(dπ_{λ}(X)) of smooth vectors for the generally unboundedoperator dπ_{λ}(X).

Let A be a noetherian connectedgraded ring with a balanced dualizing complex R. If A has cohomological dimension and Krull dimension2, then

(1) R is Auslander;

(2) \rm{Cdim} M=\rm{Kdim}M for all noetherian graded A-modules M.

In particular, ifA is AS-Gorenstein of injective and Krull dimension 2, then

(3)A is Auslander-Gorenstein;

(4) A is 2-pure with aself-injective artinian quotient ring;

(5) A has a residuecomplex.

(1,3,4) generalize a result of Levasseur [7, 5.13] and (5) generalizes aresult of Ajitabh [1, 3.12].

An interestingclass of submanifolds of Hermitian manifolds is the class of slant submanifolds which are submanifoldswith constant Wirtinger angle. In [1–4,7,8] slantsubmanifolds of complex projective and complex hyperbolic spaces have been investigated. Inparticular, it was shown that there exist many proper slant surfaces in CP^2 and inCH^2 and many proper slant minimal surfaces in C2. In contrast,in the first part of this paper we prove that there do not exist proper slant minimal surfaces inCP^2 and in CH^2. In the second part, we present a generalconstruction procedure for obtaining the explicit expressions of such slant submanifolds. By applyingthis general construction procedure, we determine the explicit expressions of special slant surfacesof CP^2 and of CH^2. Consequently, we are able to completelydetermine the slant surface which satisfies a basic equality. Finally, we apply the constructionprocedure to prove that special θ-slant isometric immersions of a hyperbolicplane into a complex hyperbolic plane are not unique in general.

Let X be acomplex infinite dimensional Banach space. We use σ_a(T) andσ_{ea}(T), respectively, to denote the approximate point spectrum and theessential approximate point spectrum of a bounded operator T onX. Also, \pi _a(T) denotes the set <$>{\rm{iso}σ_a(T)\backslash σ_{ea}(T)}<$>. An operator T onX obeys the a-Browder's theorem provided that<$>σ_{ea}(T) =σ_a(T\,)\backslash π_a(T)<$>. We investigateconnections between the Browder's theorems, the spectral mapping theorem and spectralcontinuity.

The structure of an inverse monoid can bedetermined by the complete set of Schützenberger graphs of a presentation. Necessary andsufficient conditions for a collection of inverse X-graphs to be the complete set ofSchützenberger graphs of some inverse monoid presentation are established and decidabilityresults are obtained. Conditions for a single inverse X-graph to be a Schu¨tzenbergergraph for some presentation are also obtained, and both problems are restricted to the case ofClifford monoids and E-unitary inverse monoids. Decidability and undecidability results are obtainedfor the case of finite graphs. It is also proved that the problem of embedding a finite inverseX-graph in the Cayley graph of a group is undecidable.

A well known result of B. Osofsky assertsthat if R is a left (or right) perfect, left and right selfinjective ring thenR is quasi-Frobenius. It was subsequently conjectured by Carl Faith that every left(or right) perfect, left selfinjective ring is quasi-Frobenius. While several authors have proved theconjecture in the affirmative under some restricted chain conditions, the conjecture remains open evenif R is a semiprimary, local, left selfinjective ring withJ(R)^3=0. In this paper we construct a local ring R withJ(R)^3=0 and characterize when R is artinian or selfinjective interms of conditions on a bilinear mapping from a D-D-bimodule toD, where D is isomorphic to R/J(R). Our workshows that finding a counterexample to the Faith conjecture depends on the existence of aD-D-bimodule over a division ring Dsatisfying certain topological conditions.

In this paperwe study fundamental properties of spectra of log-hyponormal operators on a Hilbert space. Inparticular we show that log-hyponormal operators are normaloid and quasi-similar log-hyponormaloperators have the same spectra and essential spectra.

We give a discrete analogueof the harmonic morphism between two Riemannian manifolds. Roughly speaking, this is a mapping betweentwo graphs preserving local harmonic functions. We characterize harmonic morphisms in terms ofhorizontal conformality. Many examples including coverings, non-complete extended p-sums andcollapsings are given. Introducing the horizontal and vertical Laplacians, the Green kernel estimatesare obtained for the harmonic morphism. As applications, a general and sharp estimate of the Greenkernel for an infinite tree is obtained.

In this note we investigate the diophantine equation<$$>x^y\pm y^x=\prod n_i!<$$>

where x and y are odd and greater than 1. We prove that this equation has no integer solutions.

We say that the operator T ona Hilbert space H into itself is strongly stable if \left\VertT^nx\right\Vert →0 as n→∞, for allx∈H. If T is a contraction, then T issaid to be cs-stable if T has C_0 completelynon-unitary part. This note considers the strong stability of operators A⊗Band the p-hyponormality of operators A⊗B. It is shown that thecontraction A⊗B is cs-stable if and only if so are the contractionscA and c^{−1}B for some scalar c andA⊗B is p-hyponormal if and only if A andB are. We also characterize p-hyponormal A⊗B forwhich the commutator |A⊗B|_{2p}−|A^*⊗B^*|^{2p} iscompact.

In the presentpaper, for a large family of topological semigroups namely, compactly cancellative and rightcancellative foundation semigroups S, we study the topological centers of the Banach algebrasLUC(S)* and M_a(S)**. We also give a generalizationof a known result of Lau and Lorsert by showing that for such topological semigroups the topologicalcenter of LUC(S)* (M_a(S)**, respectively) is thesame as M(S) (M_a(S),respectively).