Mixed Boundary Value Problems for the Helmholtz Equation in a Quadrant

The main objective is the study of a class of boundary value problems in weak formulation where two boundary conditions are given on the half-lines bordering the first quadrant that contain impedance data and oblique derivatives. The associated operators are reduced by matricial coupling relations to certain boundary pseudodifferential operators which can be analyzed in detail. Results are: Fredholm criteria, explicit construction of generalized inverses in Bessel potential spaces, eventually after normalization, and regularity results. Particular interest is devoted to the impedance problem and to the oblique derivative problem, as well.     

Stratifiable spaces defined by pair collections

We define a pair (F,U) to be a closed set F and an open set U such that F ? U. A sequence of pair collections is used to characterize stratifiable spaces instead of a sequence of neighbornets. We introduce a new class of spaces, called regularly stratifiable spaces, which is defined in terms of pair collections. Every stratifiable μ -space is regularly stratifiable, and every regularly stratifiable space has a σ -almost locally finite base, thus is hereditary M₁. J. Nagata's problem for the dimension of M₁ -spaces is answered positively in the class of regularly stratifiable spaces.     

Sensitivity in mixed generalized vector quasiequilibrium problems with moving cones

In this paper, we consider a parametric generalized vector quasiequilibrium problem which is mixed in the sense that several different relations can simultaneously appear in this problem. The moving cones and other data of the problem are assumed to be set-valued maps defined in topological spaces and taking values in topological spaces or topological vector spaces. The main result of this paper gives general verifiable conditions for the solution mapping of this problem to be semicontinuous with respect to a parameter varying in a topological space. The result is proven with the help of notions of cone-semicontinuity of set-valued maps, weaker than the usual concepts of semicontinuity, and an assumption imposed on the set-valued map whose values are the dual cones of the corresponding values of the moving cones.     

Convergence theorems for HL integrable vector-valued functions with applications

In this paper we prove dominated and monotone convergence theorems for HL integrable Banach-valued functions. These results and a fixed point theorem in ordered spaces are then applied to prove existence and comparison results for integral equations of Fredholm type in ordered Banach spaces involving Kurzweil integrals or improper integrals. Results are used also to solve concrete second-order functional boundary value problems involving discontinuities and singularities.     

On wave diffraction by a half‐plane with different face impedances

The impedance wave diffraction problem by a half‐plane screen is revisited in view of its well‐posedness upon different impedance and wave parameters. The problem is analysed with the help of potential and pseudo‐differential operators. Seven conditions between the impedance and wave numbers are found under which the problem will be well‐posed in Bessel potential spaces. In addition, an improvement of the regularity of the solutions is shown for the previous seven conditions. Copyright © 2006 John Wiley & Sons, Ltd.     

Certain classes of weakly infinite-dimensional spaces and topological games

In [V.V. Fedorchuk, Questions on weakly infinite-dimensional spaces, in: E.M. Pearl (Ed.), Open Problems in Topology II, Elsevier, Amsterdam, 2007, pp. 637-645; V.V. Fedorchuk, Weakly infinite-dimensional spaces, Russian Math. Surveys 42 (2) (2007) 1-52] classes w-m-C of weakly infinite-dimensional spaces, 2?m?∞, were introduced. We prove that all of them coincide with the class wid of all weakly infinite-dimensional spaces in the Alexandroff sense. We show also that transfinite dimensions dimwm, introduced in [V.V. Fedorchuk, Questions on weakly infinite-dimensional spaces, in: E.M. Pearl (Ed.), Open Problems in Topology II, Elsevier, Amsterdam, 2007, pp. 637-645; V.V. Fedorchuk, Weakly infinite-dimensional spaces, Russian Math. Surveys 42 (2) (2007) 1-52], coincide with dimension dimw2=dim, where dim is the transfinite dimension invented by Borst [P. Borst, Classification of weakly infinite-dimensional spaces. I. A transfinite extension of the covering dimension, Fund. Math. 130 (1) (1988) 1-25]. Some topological games which are related to countable-dimensional spaces, to C-spaces, and some other subclasses of weakly infinite-dimensional spaces are discussed.     

Admissible control and observation operators for Volterra integral equations

The admissibility of observation operators and control operators for linear Volterra systems is studied by means of the theory of composition operators on Hardy spaces.Under certain assumptions it is shown to be equivalent to admissibility for the classical Cauchy problem. A duality between control and observation operators is also established,extending known results for the Cauchy problem,which is a special case.     

Initial value problems for nonlinear second order impulsive integro-differential equations of mixed type in Banach spaces

In this paper, through solving equations step by step, without any assumption of compactness-type conditions, we obtain unique solution of initial value problems of nonlinear second order impulsive integral-differential in Banach spaces. The results obtained generalize and improve the corresponding results of Guo and Chai in papers [D.J. Guo, Initial value problems for nonlinear second-order impulsive integro-differential equations in Banach spaces, J. Math. Anal. Appl. 200 (1996) 1–13; G.Q. Chai, Initial value problems for nonlinear second order impulsive integro-differential equations in Banach space, Acta Math. Sinica 20 (3) (2000) 351–359 (in Chinese)].     

On nagata's problem for paracompact σ-metric spaces

Let (A) be the characterization of dimension as follows: Ind X?n if and only if X has a σ-closure-preserving base $\text{W}$ such that Ind B(W)?n?1 for every W?$\text{W}$. The validity of (A) is proved for spaces X such that(i) X is a paracompact σ-metric space with a scale {X_i} such that each X_i has a uniformly approaching anti-cover, or(ii) X is a subspace of the product ΠX_i of countably many L-spaces X_i, the notion of which is due to K. Nagami.(i) and (ii) are the partial answers to Nagata's problem wheter (A) holds or not for every M₁-space X.     

The impedance boundary-value problem of diffraction by a strip

We consider the impedance boundary-value problem for the Helmholtz equation originated by the problem of wave diffraction by an infinite strip with imperfect conductivity. The two possible different situations of real and complex wave numbers are considered. Bessel potential spaces are used to deal with the problem, and the identification of corresponding operators of single and double layer potentials allow a reformulation of the problem into a system of integral equations. The well-posedness of the problem is obtained for a set of impedance parameters (and wave numbers), after the incorporation of some compatibility conditions on the data. At the end, an improvement of the regularity of the solution is derived for the same set of parameters previously considered.     

Weakly singular integral equations with operator-valued kernels and an application to radiation transfer problems

The standard problem of radiation transfer in a bounded regionG ⁿ can be reformulated as a weakly singular integral equation with an unknown functionu: GC(S ^n–1) and a kernelK: ((G × G }x=y}, which is continuously differentiable with respect to the operator strong convergence topology. We take these observations into the basis of an abstract treatment of weakly singular integral equations with (E)-valued kernels, whereE is a Banach space. Our purpose is to characterize the smoothness of the solution by proving that it belongs to special weighted spaces of smooth functions. On the way, realizing the proof techniques, we establish the compactness of the integral operator or its square inL _p (G,E),BC(G,E), and other spaces of interest in numerical analysis as well as in weighted spaces of smooth functions. The smoothness results are specified for the standard problem of radiation transfer as well as for the corresponding eigenvalue problem.     

Dirichlet problem at infinity on Gromov hyperbolic metric measure spaces

We study the asymptotic Dirichlet problem for p-harmonic functions in a very general setting of Gromov hyperbolic metric measure spaces.     

Harbater-Mumford subvarieties of moduli spaces of covers

We define Harbater-Mumford subvarieties, which are special kinds of closed subvarieties of Hurwitz moduli spaces obtained by fixing some of the branch points. We show that, for many finite groups, finding geometrically irreducible HM-subvarieties defined over is always possible. This provides information on the arithmetic of Hurwitz spaces and applies in particular to the regular inverse Galois problem with (almost all) fixed branch points. Profinite versions of our results can also be stated, providing new tools to study the geometry of modular towers and the regular inverse Galois problem for profinite groups.     

An integral equation method for the electromagnetic scattering from a scatterer on an absorbing plane

Consider a time-harmonic electromagnetic plane wave incident on a scatterer on a grounded absorbing plane modelized as an infinite impedance plane. In this paper, a new integral representation formula is rigorously derived. Existence and uniqueness of weak solutions for the model problem are also established. The proof of existence is based on an extension of the Hodge decomposition technique to open boundaries. The results reported in this paper form a basis for numerical solutions of the electromagnetic scattering problem from a scatterer on an absorbing plane.     

The asymptotic behaviour of weak solutions to the forward problem of electrical impedance tomography on unbounded three‐dimensional domains

The forward problem of electrical impedance tomography on unbounded domains can be studied by introducing appropriate function spaces for this setting. In this paper we derive the point‐wise asymptotic behaviour of weak solutions to this problem in the three‐dimensional case. Copyright © 2008 John Wiley & Sons, Ltd.     

Periodic solutions of integro-differential equations in vector-valued function spaces

Operator-valued Fourier multipliers are used to study well-posedness of integro-differential equations in Banach spaces. Both strong and mild periodic solutions are considered. Strong well-posedness corresponds to maximal regularity which has proved very efficient in the handling of nonlinear problems. We are concerned with a large array of vector-valued function spaces: Lebesgue-Bochner spaces Lp, the Besov spaces (and related spaces such as the Hölder-Zygmund spaces Cs) and the Triebel-Lizorkin spaces . We note that the multiplier results in these last two scales of spaces involve only boundedness conditions on the resolvents and are therefore applicable to arbitrary Banach spaces. The results are applied to various classes of nonlinear integral and integro-differential equations.     

Some characterizations of spaces that have transfinite dimension

We characterize separable metrizable spaces that have small transfinite dimension and metrizable spaces that have large transfinite dimension modifying two classical characterizations of countable-dimensional spaces and applying the notion of a strongly point-finite family.     

The automorphism group of certain higher degree forms

We consider symmetric indecomposable d-linear (d>2) spaces of dimension n over an algebraically closed field k of characteristic 0, whose center (the analog of the space of symmetric matrices of a bilinear form) is cyclic, as introduced by Reichstein [B. Reichstein, On Waring’s problem for cubic forms, Linear Algebra Appl. 160 (1992) 1-61]. The automorphism group of these spaces is determined through the action on the center and through the determination of the Lie algebra. Furthermore, we relate the Lie algebra to the Witt algebra.     

On the eigenvalues of principal submatrices of J-normal matrices

The problem of the existence of a J-normal matrix A when its spectrum and the spectrum of some of its (n-1)×(n-1) principal submatrices are prescribed is analyzed. The case of 3×3 matrices is particularly investigated. The results here obtained in the framework of indefinite inner product spaces are in the spirit of those due to Nylen, Tam and Uhlig.     

Characterizations of certain classes of infinite-dimensional metric spaces

We characterize two classes of metric spaces as images under a closed, finite-to-one mapping of a zero-dimensional metric space. In the case of locally finite-dimensional spaces the mapping must be of strong local order, and for strongly countable-dimensional spaces the mapping must have weak local order. The results are analogues to characterizations by K. Morita (of finite-dimensional spaces) and J. Nagata (of countable-dimensional spaces).