Numerical implementation for reconstruction of inhomogeneous conductivities via generalized polarization tensors

This paper deals with numerical methods for reconstruction of inhomogeneous conductivities. We use the concept of Generalized Polarization Tensors to do reconstruction. Basic resolution and stability analysis are presented. Least‐square norm methods with respect to Generalized Polarization Tensors are used for reconstruction of conductivities. Finally, reconstruction of three different types of conductivities in the plane is demonstrated. Copyright © 2014 John Wiley & Sons, Ltd.     

Sharp estimate of electric field from a conductive rod and application

We are concerned with the quantitative study of the electric field perturbation due to the presence of an inhomogeneous conductive rod embedded in a homogenous conductivity. We sharply quantify the dependence of the perturbed electric field on the geometry of the conductive rod. In particular, we accurately characterize the localization of the gradient field (i.e., the electric current) near the boundary of the rod where the curvature is sufficiently large. We develop layer‐potential techniques in deriving the quantitative estimates and the major difficulty comes from the anisotropic geometry of the rod. The result complements and sharpens several existing studies in the literature. It also generates an interesting application in EIT (electrical impedance tomography) in determining the conductive rod by a single measurement, which is also known as the Calderón's inverse inclusion problem in the literature.     

Conformal and generalized concircular mappings of Einstein-Weyl manifolds

In this article, after giving a necessary and sufficient condition for two Einstein-Weyl manifolds to be in conformal correspondence, we prove that any conformal mapping between such manifolds is generalized concircular if and only if the covector field of the conformal mapping is locally a gradient. Using this fact we deduce that any conformal mapping between two isotropic Weyl manifolds is a generalized concircular mapping. Moreover, it is shown that a generalized concircularly flat Weyl manifold is generalized concircular to an Einstein manifold and that its scalar curvature is prolonged covariant constant.     

Construction of inclusions with vanishing generalized polarization tensors by imperfect interfaces

We investigate the problem of planar conductivity inclusion with imperfect interface conditions. We assume that the inclusion is simply connected. The presence of the inclusion causes a perturbation in the incident background field. This perturbation admits a multipole expansion of which coefficients we call as the generalized polarization tensors (GPTs), extending the previous terminology for inclusions with perfect interfaces. We derive explicit matrix expressions for the GPTs in terms of the incident field, material parameters, and geometry of the inclusion. As an application, we construct GPT-vanishing structures of general shape that result in negligible perturbations for all uniform incident fields. The structure consists of a simply connected core with an imperfect interface. We provide numerical examples of GPT-vanishing structures obtained by our proposed scheme.     

Fixed point of contractive mappings in generalized metric spaces

We prove a fixed point theorem for contractive mappings of Boyd and Wong type in generalized metric spaces, a concept recently introduced in [BRANCIARI, A.: A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, Publ. Math. Debrecen 57 (2000), 31–37].      

Bergman spaces of higher‐dimensional hyperbolic polyhedron‐type domains I

In this paper we develop explicit formulas for the Green's function and the monogenic reproducing Bergman kernel function of some hyperbolic polyhedron‐type domains that generalize the fundamental domain of the modular group SL(2,?) to higher dimensions. Copyright © 2005 John Wiley & Sons, Ltd.     

Ishikawa iterative process with errors for nonlinear equations of generalized monotone type in Banach spaces

The concept of the operators of generalized monotone type is introduced and iterative approximation methods for a fixed point of such operators by the Ishikawa and Mann iteration schemes {xn} and {yn} with errors is studied. Let X be a real Banach space and T : D ? X → 2^D be a multi‐valued operator of generalized monotone type with fixed points. A new general lemma on the convergence of real sequences is proved and used to show that {xn} converges strongly to a unique fixed point of T in D. This result is applied to the iterative approximation method for solutions of nonlinear equations with generalized strongly accretive operators. Our results generalize many of know results. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Characterizations of generalized convexities via generalized directional derivatives

The paper gives characterizations of convexity, quasiconvexity, invexity and pseudoconvexity for a (radially) upper-semicontinuous function f in a topological vector space via appropriate properties of a bifunction which is majorized by the upper radial derivative of f and which stands for a generalized derivative of some sort.     

Lipschitz estimates for generalized commutators of fractional integrals with rough kernel

Let (0 < α < n) be the generalized commutator generated by fractional integral with rough kernel and the m–th order remainder of the Taylor formula of a function A. In this paper, the (L^p, L^r) (r > 1) boundedness, the weak (L¹, L^{n/(n–α–β)}) boundedness and the (L^p, ?^{β, ∞}_p) boundedness of are discussed, when D^γA belongs to the Lipschitz function spaces.     

Some regularity results to the generalized Emden–Fowler equation with irregular data

We deal with the generalized Emden–Fowler equation f^″(x) + g(x)f^?θ(x) = 0, where belongs to L^p((a,b)). We obtain a priori estimates for the solutions, as well as information about their asymptotic behavior near boundary points. As a tool, we derive new nonlinear variants of first‐order and second‐order Poincaré inequalities, which are based on strongly nonlinear multiplicative inequalities obtained recently by first author and Peszek. Copyright © 2014 John Wiley & Sons, Ltd.     

Gap functions for a system of generalized vector quasi-equilibrium problems with set-valued mappings

In this paper, some gap functions for three classes of a system of generalized vector quasi-equilibrium problems with set-valued mappings (for short, SGVQEP) are investigated by virtue of the nonlinear scalarization function of Chen, Yang and Yu. Three examples are then provided to demonstrate these gap functions. Also, some gap functions for three classes of generalized finite dimensional vector equilibrium problems (GFVEP) are derived without using the nonlinear scalarization function method. Furthermore, a set-valued function is obtained as a gap function for one of (GFVEP) under certain assumptions.      

Horseshoe in a modified Van der Pol–Duffing circuit

In this paper, we study a modified Van der Pol–Duffing circuit, and present a rigorous verification for existence of chaos in this system. The arguments are given in the manner of computer-assisted proof by using topological horseshoe theory.     

Asymptotic behavior of spectral of Neumann‐Poincaré operator in Helmholtz system

In this paper, we are concerned with the asymptotic behavior of the Neumann‐Poincaré operator in Helmholtz system. By analyzing the asymptotic behavior of spherical Bessel function near the origin and/or approach higher order, we prove the asymptotic behavior of spectral of Neumann‐Poincaré operator when frequency is small enough and/or the order is large enough. The results show that spectral of Neumann‐Poincaré operator is continuous at the origin and converges to 0 from the complex plane in general.     

Computation of periodic orbits in three‐dimensional Lotka‐Volterra systems

This paper deals with an adaptation of the Poincaré‐Lindstedt method for the determination of periodic orbits in three‐dimensional nonlinear differential systems. We describe here a general symbolic algorithm to implement the method and apply it to compute periodic solutions in a three‐dimensional Lotka‐Volterra system modeling a chain food interaction. The sufficient conditions to make secular terms disappear from the approximate series solution are given in the paper.     

Explicit analytical solutions of the generalized Burger and Burger–Fisher equations by homotopy perturbation method

In this article, we apply the homotopy perturbation method (HPM) to obtain approximate analytical solutions of the generalized Burger and Burger‐Fisher (B–F) equations. Several numerical examples are given to illustrate the efficiency of the HPM. Comparison of the result obtained by the present method with exact solution reveals that the accuracy and fast convergence of the new method. It is predicted that the HPM can be found wide application in engineering problems. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009