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Yayın On a class of functional equations of the Wiener-Hopf type and their applications in n-part scattering problems(Oxford Univ Press, 2003-12) İdemen, Mehmet Mithat; Alkumru, AliAn asymptotic theory for the functional equation K-phi=f, where K : X-->Y stands for a matrix-valued linear operator of the form K=K1P1+K2P2+...+KnPn, is developed. Here X and Y refer to certain Hilbert spaces, {P-alpha} denotes a partition of the unit operator in X while K-alpha are certain operators from X to Y. One assumes that the partition {P-alpha} as well as the operators K-alpha depend on a complex parameter nu such that all K-alpha are multi-valued around certain branch points at nu=k(+) and nu=k(-) while the inverse operators K-alpha(-1) exist and are bounded in the appropriately cut nu-plane except for certain poles. Then, for a class of {P-alpha} having certain analytical properties, an asymptotic solution valid for \k(+/-)\-->infinity is given. The basic idea is the decomposition of phi into a sum of projections on n mutually orthogonal subspaces of X. The results can be extended in a straightforward manner to the cases of no or more branch points. If there is no branch point or n=2, then the results are all exact. The theory may have effective applications in solving some direct and inverse multi-part boundary-value problems connected with high-frequency waves. An illustrative example shows the applicability as well as the effectiveness of the method.Yayın Influence of the velocity on the energy patterns of moving scatterers(Taylor & Francis, 2004) İdemen, Mehmet Mithat; Alkumru, AliParallel to the developments in the communication through space vehicles achieved during the last two decades, the scattering problems connected with moving objects became more and more important from both theoretical and practical points of view. Same problems are also arisen in point of space science, radio astronomy, radar techniques and particle physics. The earlier investigations available in the open literature concern the analysis of the scattered field pattern and, hence, treat the polarization, frequency shift (Doppler effect), aberration, etc, which are all important from both pure scientific and technological points of view. But, another issue which is also important in regard to the communication, antennas and particle physics is the influence of the motion on the scattered energy patterns which involves the radar cross-section and scattering coefficient. This paper is devoted to this purpose and aims to study the influence of the velocity on the received and scattered energies. Notice that the scattered wave is not time-harmonic even though the incident wave is so because the Lorentz transformation formulas interrelate the space coordinates and time, which makes impossible to extend the notion of radar cross-section to moving bodies. For the sake of simplicity of the mathematical manipulations, only two-dimensional case is taken into account but the method can be adapted by straightforward extensions to other types of scatterer.Yayın Derivation of the Lorentz transformation from the Maxwell equations(VSP BV, Brill Academic Publishers, 2005) İdemen, Mehmet MithatThe Special Theory of Relativity had been established nearly one century ago to conciliate some seemingly contradictory concepts and experimental results such as the Ether, universal time, contraction of dimensions of moving bodies, absolute motion of the Earth, speed of the light, etc. Hence the fundamental revolutionary formulas of the Theory, i.e., the Lorentz Formulas, had been derived first by Einstein by dwelling on a postulate which stipulated the constancy of the speed of the light. To this end he had first postulated that every reference system has a time proper to itself and then redefined the notions of simultaneity, synchronous clocks, time interval, the length of a rod in a system at rest, the length in a moving system, etc. A second postulate of Einstein, which stated that every physical theory is invariant under the Lorentz transformation, enabled him to claim that the Theory of Electromagnetism is correct but the Newtonian Mechanics has to be re-established. Since then the Theory was almost always presented in this way by both Einstein and others except only a few. The aim of this paper is to show that the Lorentz formulas can be derived from the Maxwell equations if one pstulates that the total electric charge of an isolated body does not change if it is in motion. To this end one dwells only on the permanence principle of functional equations, which is not a physical but purely mathematical concept. Thus, from one side the Special Relativity becomes a natural issue (or a part) of the Maxwell Theory and, from the other side, the derivation of the transformation rules pertinent to the electromagnetic field becomes straightforward and easy.Yayın A generalization of the Wiener-Hopf approach to direct and inverse scattering problems connected with non-homogeneous half-spaces bounded by n-part boundaries(Oxford Univ Press, 2000-08) İdemen, Mehmet Mithat; Alkumru, AliThe classical Wiener-Hopf method connected with mixed two-part boundary-value problems is generalized to cover n-part boundaries. To this end one starts from an ad-hoc representation for the Green function, which involves n unknown functions having certain analytical properties. Thus the problem is reduced to a functional equation involving n unknowns, which constitutes a generalization of the classical Wiener-Hopf equation in two unknowns. To solve this latter which cannot be solved exactly when n greater than or equal to 3, one establishes a new method permitting one to obtain the asymptotic expressions valid when the wavelength is sufficiently small as compared with the widths of the inner strips of the boundary. The essentials of the method are elucidated through a concrete inverse scattering problem whose aim is to determine the constitutive electromagnetic parameters of a slab and a half-space bounded by an n-part impedance plane. Some illustrative numerical examples show the applicability as well as the accuracy of the method.
