and Magnetoencephalography (MEG) are two important methods for the functional imaging of the brai... more and Magnetoencephalography (MEG) are two important methods for the functional imaging of the brain. For the case of the spherical homogeneous model, we elucidate the mathematical relations of these two methods. In particular, we derive and analyse three different representations for the electric and magnetic potentials, as well as the corresponding electric and magnetic induction fields, namely: integral representations involving the Green and the Neumann kernels, representations in terms of eigenfunction expansions, and closed form expressions. We show that the parts of the EEG and MEG fields in the interior of the brain that are due to the induction current are related via Kelvin's inversion transformation. We also derive closed form expressions for the interior and exterior vector potentials of the corresponding magnetic induction fields. Electrochemically generated neuronal currents in the brain give rise to a magnetic field, which in turn excites an induction current within the conductive brain tissue. This electromagnetic activity of the brain is recorded by measuring the electric potential on the scalp (EEG) and the magnetic induction field at distances 4-6 cm from the head (MEG). There exists an extensive literature on how to utilise the EEG and MEG recordings [13], in order to determine the electric potential and the magnetic induction field outside the head. From the mathematical point of view, the two basic problems for EEG and MEG are the forward problem, namely find the interior and exterior fields in terms of the neuronal current and the inverse problem, namely find the neuronal current in terms of the electric potential or in terms of the magnetic field. The fact that neither of these inverse problems has a unique solution was known to Helmholtz in 1853. Nevertheless, complete quantitative results on the non-uniqueness of the inverse MEG problem were obtained only recently, in [9], [10] for the spherical model
Magnetoencephalography (MEG) is a non-invasive technique that measures the magnetic fields produc... more Magnetoencephalography (MEG) is a non-invasive technique that measures the magnetic fields produced by brain's electrical currents. The basic inverse problem of magnetoencephalography (MEG) consists in estimating the neuronal current in the brain from the measurement of the magnetic field outside the head. The relative inversion algorithms, existing in medical devices, for the identification of excitation sources inside the brain using MEG data, are based on the assumption that the geometry of the brain-head system is spherical. Here, we present the error of identification of the dipole (r 0 , Q) using the inversion algorithm of spherical geometry. In particular, taking magnetoencephalographic measurements from realistic ellipsoidal model and using these data in a spherical model lead to a structural error. For the purpose of this paper, we make numerical investigation for different positions of the source, especially, locating sources at different depths and see how the error depends on depth. These results may have useful implications for the interpretation of the reconstructions obtained via the existing approaches.
IEEE transactions on bio-medical engineering, Jan 20, 2017
The inverse problem of computing the neuronal current density from scalp EEG is highly ill-posed.... more The inverse problem of computing the neuronal current density from scalp EEG is highly ill-posed. In part, this is due to the nonuniqueness of the mapping between current sources and scalp potentials. We develop an explicit formula for the scalp EEG for sources constrained to the cortical surface in terms only of the components of the current that affect the EEG signal. Starting from the quasistatic form of Maxwell's equations, we develop a formula that involves only the "visible" part of the current (i.e. the part of the current that affects the EEG measurements), as well as certain auxiliary functions which depend on the topology and conductivity of the three dimensional domains and , that model the spaces occupied by the cerebrum, cerebrospinal fluid, bone and scalp respectively. we derive expressions for the scalp potential for a general nested topology, as well as for the special case of spherical and ellipsoidal surfaces. We verify that the resulting scalp potent...
The Quarterly Journal of Mechanics and Applied Mathematics, 2004
Many practical applications involve particles (inorganic, organic, biological) with non-spherical... more Many practical applications involve particles (inorganic, organic, biological) with non-spherical but still axisymmetric shapes. The present work is concerned with some interesting aspects of the theoretical analysis of Stokes flow in spheroidal domains. Two different complete representations of Stokes flow are considered here. The first one is obtained through the theory of generalized eigenfunctions, according to which the stream function is expanded in terms of separable and semiseparable eigenfunctions. The second one, valid in non-axisymmetric geometries as well, is the Papkovich-Neuber differential representation, where the velocity and pressure fields are expressed in terms of harmonic spheroidal eigenfunctions. Connection formulae are obtained for the case of axisymmetric flows, which relate the spheroidal harmonic eigenfunctions of the Papkovich-Neuber representation with the semiseparable spheroidal stream eigenfunctions. In the case of axisymmetric spheroidal flows the Papkovich-Neuber approach is equivalent to the Stokes stream function approach, but the three-dimensional representation offers certain important advantages. Particle-in-cell models for Stokes flow through a swarm of particles are of substantial practical interest, because they provide a relatively simple platform for the analytical or semianalytical solution of heat and mass transport problems. The early versions of these models were concerned with spherical particles. For this reason particle-in-cell models for spheroidal particles were developed more recently. The flexibility of the Papkovich-Neuber differential representation is demonstrated by solving the problem of the flow in a fluid cell filling the space between two confocal spheroidal surfaces with Kuwabara-type boundary conditions.
Mathematical Methods in the Applied Sciences, 2012
A detailed mathematical analysis of Stokes flow past a rigid biconcave disc shape, motivated by t... more A detailed mathematical analysis of Stokes flow past a rigid biconcave disc shape, motivated by the flow of blood plasma past a fixed red blood cell (RBC) is developed in the present manuscript. The main objective is to provide a new insight in the theoretical investigation of the flow behaviour of RBCs in pathological situations for which the rheological characteristics of the RBCs change or in medical tests like the erythrocyte sedimentation rate. Here, the RBC is described mathematically by an inverted prolate spheroid having its axis of symmetry along the direction of a uniform flow. The method of Kelvin transformation for Stokes flows and the concept of semiseparable spectral decomposition of Stokes flows in spheroidal geometry are employed to obtain an analytical solution. More specifically, by employing Kelvin's inversion, the exterior boundary value problem in the inverse prolate spheroidal system is mapped to a related problem in the interior of the corresponding prolat...
Mathematical Methods in the Applied Sciences, 2012
A detailed mathematical analysis of Stokes flow past a rigid biconcave disc shape, motivated by t... more A detailed mathematical analysis of Stokes flow past a rigid biconcave disc shape, motivated by the flow of blood plasma past a fixed red blood cell (RBC) is developed in the present manuscript. The main objective is to provide a new insight in the theoretical investigation of the flow behaviour of RBCs in pathological situations for which the rheological characteristics of the RBCs change or in medical tests like the erythrocyte sedimentation rate. Here, the RBC is described mathematically by an inverted prolate spheroid having its axis of symmetry along the direction of a uniform flow. The method of Kelvin transformation for Stokes flows and the concept of semiseparable spectral decomposition of Stokes flows in spheroidal geometry are employed to obtain an analytical solution. More specifically, by employing Kelvin's inversion, the exterior boundary value problem in the inverse prolate spheroidal system is mapped to a related problem in the interior of the corresponding prolat...
Kelvin’s transformation is a non-linear map that, in some sense, preserves harmonicity. This prop... more Kelvin’s transformation is a non-linear map that, in some sense, preserves harmonicity. This property, which was the content of a letter sent by Kelvin to Liouville in 1845, provides a powerful machinery for solving particular potential problems in a very effective way. In the present work, we show that the basic theory can be extended to the biharmonic equation as well to the equations for irrotational and rotational Stokes flow. Hence, biharmonicity, stream functions and bistream functions are also preserved, in some sense, under the Kelvin transformation. We also demonstrate how the Kelvin-type theorems are interconnected with the relative Almansi-type decompositions. These results provide a way to solve analytically many problems in potential theory and Stokes flow which it is impossible to solve by the classical spectral method.
Publisher Summary This chapter discusses the scattering theory; in the physical setting of acoust... more Publisher Summary This chapter discusses the scattering theory; in the physical setting of acoustic wave propagation is a scalar theory where the fields represent excess pressure or velocity potentials. On the other hand, the electromagnetic scattering theory deals with vector fields that represent the electric and the magnetic fields. Two characteristics of electromagnetic wave propagation help in simplifying the corresponding scattering problems. The first is connected to the fact that using the spectral Maxwell equation and knowledge of the electric or the magnetic field, can be readily obtained from the other. The second is due to the fact that both the electric and the magnetic fields are propagating with the same phase velocities. Unfortunately, these simplifying characteristics are missing from the scattering problems that appear in the theory of elasticity. In elasticity, again two waves, the longitudinal (irrotational) and the transverse (solenoidal), propagate at different phase velocities and once one of them is known there is no way to recover the other. For that reason, the scattering problems in elasticity are much harder than the corresponding problems in electromagnetism.
The time evolution of the energy for wave propagation in an infinite ideal magnetofluid is studie... more The time evolution of the energy for wave propagation in an infinite ideal magnetofluid is studied analytically. It is found that equipartition between the kinetic and potential energies is achieved. Equipartition occurs either in finite time or asymptotically as t↠∞ depending upon whether the initial data are compactly supported or have finite energy, respectively.
The Quarterly Journal of Mechanics and Applied Mathematics, 1997
A small resistive scatterer disturbs a spherical time-harmonic field emanating from a point sourc... more A small resistive scatterer disturbs a spherical time-harmonic field emanating from a point source. The incident point-source field is normalized in such a way as to be able to recover the corresponding results for plane-wave incidence. The full low-frequency expansion for the corresponding total field is reduced to an exterior boundary-value problem for the Laplace equation, which has to be solved repeatedly. Exact results for the case of a small resistive sphere are obtained. It is shown that the leading low-frequency approximations for the scattering as well as for the absorption cross-section are increasing functions of the impedance parameter and decreasing functions of the distance of the source from the scatterer. It is also shown that a small sphere scatters and absorbs more energy when it is illuminated by a point-rather than by a plane-wave field establishing the fact that the closer the source of illumination to the scatterer, the stronger the interaction. The leading approximation of.the absorption cross-section is independent of the wavenumber, while the leading approximation of the scattering cross-section is proportional to the second power of the wavenumber. Hence, in the low-frequency realm, absorption is by two orders of magnitude stronger than scattering. Finally, a comparison between point-and plane-wave incidence, based on multipole expansions, is included.
A point generated incident "eld impinges upon a small triaxial ellipsoid which is arbitrarily ori... more A point generated incident "eld impinges upon a small triaxial ellipsoid which is arbitrarily oriented with respect to the point source. The point source "eld is so modi"ed as to be able to recover the corresponding results for plane wave incidence when the source recedes to in"nity. The main dif"culty in solving analytically this low-frequency scattering problem concerns the "tting of the spherical geometry, which characterizes the incident "eld, with the ellipsoidal geometry which is naturally adapted to the scatterer. A series of techniques has been used which lead "nally to analytic solutions for the leading two low-frequency terms of the near as well as the far "eld. In contrast to the near-"eld approximations, which are expressed in terms of ellipsoidal eigenexpansions, the far "eld is furnished by a "nite number of terms. This is very interesting because the constants entering the expressions of the Lamé functions of degree higher than three are not obtainable analytically and therefore, in the near "eld, not even the Rayleigh approximation can be completely obtained. On the other hand, since only a few terms survive at the far "eld, the scattering amplitude and the scattering cross-section are derived in closed form. It is shown that, in practice, if the source is located a distance equal to "ve or six times the biggest semiaxis of the ellipsoid the Rayleigh term of the approximation behaves almost as the incident "eld was a plane wave. The special cases of spheroids, needles, discs, spheres as well as plane wave incidence are recovered. Finally, some theorems concerning monopole and dipole surface potentials are included.
A polyadic field of rank n is the tensor product of n vector fields. Helmholtz showed that a vect... more A polyadic field of rank n is the tensor product of n vector fields. Helmholtz showed that a vector field, which is a polyadic field of rank 1, is nonuniquely decomposable into the gradient of a scalar function plus the rotation of a vector function. We show here that a polyadic field of rank n is, again nonuniquely, decomposable into a term consisting of n successive applications of the gradient to a scalar function, plus a term that consists of (n-1) successive applications of the gradient to the rotation of a vector function, plus a term that consists of (n-2) successive applications of the gradient to the rotation of the rotation of a dyadic function and so on, until the last (n + l)th term, which consists of n successive applications of the rotation operator to a polyadic function of rank n. Obviously, the n = 1 case recovers the Helmholtz decomposition theorem. For dyadic fields a more symmetric representation is provided and formulae that provide the potential representation functions are given. The special cases of symmetric and antisymmetric dyadics are discussed in detail. Finally, the multidivergence type relations, which reduce the number of independent scalar representation functions to n2, are presented.
We consider the problem of scattering of a longitudinal or a transverse plane elastic wave by a g... more We consider the problem of scattering of a longitudinal or a transverse plane elastic wave by a general ellipsoidal cavity in the low-frequency region. Explicit closed-form solutions for the zeroth- and first-order approximations are provided in terms of the physical and geometric characteristics of the scatterer, as well as the direction cosines of the incidence and observation points. This was made possible with the introduction of an analytical technique based on the Papkovich representations and their interdependence. The leading low-frequency term for the normalized spherical scattering amplitudes and the scattering cross section are also given explicitly. Degenerate ellipsoids corresponding to the prolate and oblate spheroids, the sphere, the needle, and the disc are considered as special cases.
In this paper we prove that, unlike the two-dimensional case, the electric field in the presence ... more In this paper we prove that, unlike the two-dimensional case, the electric field in the presence of closely adjacent spherical perfect conductors does not blow up even though the separation distance between the conducting inclusions approaches zero.
The Quarterly Journal of Mechanics and Applied Mathematics, 2001
The general theory of low-frequency dyadic scattering is developed for the near fields, the far f... more The general theory of low-frequency dyadic scattering is developed for the near fields, the far fields and all the energy functionals associated with scattering problems. The incident field could be any complete dyadic field generated either in the exterior medium of propagation (point source) or at infinity (plane waves). The case of a small rigid sphere, which is illuminated by a plane dyadic field, is solved and the corresponding results for acoustic and elastic scattering are recovered as special cases. In order to solve analytically the sphere problem a special technique had to be developed, which generates Papkovich-type differential representations of dyadic elastostatic displacements. Comparison of numerical results, obtained via the boundary element method, show an amazing accuracy with our analytical results.
Journal of Computational and Applied Mathematics, 1992
Apostolopoulos, T. and G. Dassios, A parallel algorithm for solving the inverse scattering moment... more Apostolopoulos, T. and G. Dassios, A parallel algorithm for solving the inverse scattering moment problem,
The Quarterly Journal of Mechanics and Applied Mathematics, 1998
The inverse problem of determining the geometrical characteristics of a perfectly conducting elli... more The inverse problem of determining the geometrical characteristics of a perfectly conducting ellipsoid or ellipsoidal boss from low-frequency scattering data is considered. It is shown that the orientation and semidiameters of the scatterer can be determined from measurements of the leading-order term in the low-frequency expansion of the electric scattering amplitude for plane wave excitation. For the ellipsoid, measurements corresponding to six directions of incidence and associated polarization provide the necessary data, while three measurements suffice for the ellipsoidal boss on a perfectly conducting base plane.
Two main results are included in this paper. The first one deals with the leading asymp-totic ter... more Two main results are included in this paper. The first one deals with the leading asymp-totic term of the magnetic field outside any conductive medium. In accord with physical reality, it is proved mathematically that the leading approximation is a quadrupole term which means that the conductive brain tissue weakens the intensity of the magnetic field outside the head. The second one concerns the orientation of the silent sources when the geometry of the brain model is not a sphere but an ellipsoid which provides the best possible mathematical approximation of the human brain. It is shown that what char-acterizes a dipole source as “silent ” is not the collinearity of the dipole moment with its position vector, but the fact that the dipole moment lives in the Gaussian image space at the point where the position vector meets the surface of the ellipsoid. The appropriate representation for the spheroidal case is also included. 1. The magnetic field The mathematical theory of magnetoen...
and Magnetoencephalography (MEG) are two important methods for the functional imaging of the brai... more and Magnetoencephalography (MEG) are two important methods for the functional imaging of the brain. For the case of the spherical homogeneous model, we elucidate the mathematical relations of these two methods. In particular, we derive and analyse three different representations for the electric and magnetic potentials, as well as the corresponding electric and magnetic induction fields, namely: integral representations involving the Green and the Neumann kernels, representations in terms of eigenfunction expansions, and closed form expressions. We show that the parts of the EEG and MEG fields in the interior of the brain that are due to the induction current are related via Kelvin's inversion transformation. We also derive closed form expressions for the interior and exterior vector potentials of the corresponding magnetic induction fields. Electrochemically generated neuronal currents in the brain give rise to a magnetic field, which in turn excites an induction current within the conductive brain tissue. This electromagnetic activity of the brain is recorded by measuring the electric potential on the scalp (EEG) and the magnetic induction field at distances 4-6 cm from the head (MEG). There exists an extensive literature on how to utilise the EEG and MEG recordings [13], in order to determine the electric potential and the magnetic induction field outside the head. From the mathematical point of view, the two basic problems for EEG and MEG are the forward problem, namely find the interior and exterior fields in terms of the neuronal current and the inverse problem, namely find the neuronal current in terms of the electric potential or in terms of the magnetic field. The fact that neither of these inverse problems has a unique solution was known to Helmholtz in 1853. Nevertheless, complete quantitative results on the non-uniqueness of the inverse MEG problem were obtained only recently, in [9], [10] for the spherical model
Magnetoencephalography (MEG) is a non-invasive technique that measures the magnetic fields produc... more Magnetoencephalography (MEG) is a non-invasive technique that measures the magnetic fields produced by brain's electrical currents. The basic inverse problem of magnetoencephalography (MEG) consists in estimating the neuronal current in the brain from the measurement of the magnetic field outside the head. The relative inversion algorithms, existing in medical devices, for the identification of excitation sources inside the brain using MEG data, are based on the assumption that the geometry of the brain-head system is spherical. Here, we present the error of identification of the dipole (r 0 , Q) using the inversion algorithm of spherical geometry. In particular, taking magnetoencephalographic measurements from realistic ellipsoidal model and using these data in a spherical model lead to a structural error. For the purpose of this paper, we make numerical investigation for different positions of the source, especially, locating sources at different depths and see how the error depends on depth. These results may have useful implications for the interpretation of the reconstructions obtained via the existing approaches.
IEEE transactions on bio-medical engineering, Jan 20, 2017
The inverse problem of computing the neuronal current density from scalp EEG is highly ill-posed.... more The inverse problem of computing the neuronal current density from scalp EEG is highly ill-posed. In part, this is due to the nonuniqueness of the mapping between current sources and scalp potentials. We develop an explicit formula for the scalp EEG for sources constrained to the cortical surface in terms only of the components of the current that affect the EEG signal. Starting from the quasistatic form of Maxwell's equations, we develop a formula that involves only the "visible" part of the current (i.e. the part of the current that affects the EEG measurements), as well as certain auxiliary functions which depend on the topology and conductivity of the three dimensional domains and , that model the spaces occupied by the cerebrum, cerebrospinal fluid, bone and scalp respectively. we derive expressions for the scalp potential for a general nested topology, as well as for the special case of spherical and ellipsoidal surfaces. We verify that the resulting scalp potent...
The Quarterly Journal of Mechanics and Applied Mathematics, 2004
Many practical applications involve particles (inorganic, organic, biological) with non-spherical... more Many practical applications involve particles (inorganic, organic, biological) with non-spherical but still axisymmetric shapes. The present work is concerned with some interesting aspects of the theoretical analysis of Stokes flow in spheroidal domains. Two different complete representations of Stokes flow are considered here. The first one is obtained through the theory of generalized eigenfunctions, according to which the stream function is expanded in terms of separable and semiseparable eigenfunctions. The second one, valid in non-axisymmetric geometries as well, is the Papkovich-Neuber differential representation, where the velocity and pressure fields are expressed in terms of harmonic spheroidal eigenfunctions. Connection formulae are obtained for the case of axisymmetric flows, which relate the spheroidal harmonic eigenfunctions of the Papkovich-Neuber representation with the semiseparable spheroidal stream eigenfunctions. In the case of axisymmetric spheroidal flows the Papkovich-Neuber approach is equivalent to the Stokes stream function approach, but the three-dimensional representation offers certain important advantages. Particle-in-cell models for Stokes flow through a swarm of particles are of substantial practical interest, because they provide a relatively simple platform for the analytical or semianalytical solution of heat and mass transport problems. The early versions of these models were concerned with spherical particles. For this reason particle-in-cell models for spheroidal particles were developed more recently. The flexibility of the Papkovich-Neuber differential representation is demonstrated by solving the problem of the flow in a fluid cell filling the space between two confocal spheroidal surfaces with Kuwabara-type boundary conditions.
Mathematical Methods in the Applied Sciences, 2012
A detailed mathematical analysis of Stokes flow past a rigid biconcave disc shape, motivated by t... more A detailed mathematical analysis of Stokes flow past a rigid biconcave disc shape, motivated by the flow of blood plasma past a fixed red blood cell (RBC) is developed in the present manuscript. The main objective is to provide a new insight in the theoretical investigation of the flow behaviour of RBCs in pathological situations for which the rheological characteristics of the RBCs change or in medical tests like the erythrocyte sedimentation rate. Here, the RBC is described mathematically by an inverted prolate spheroid having its axis of symmetry along the direction of a uniform flow. The method of Kelvin transformation for Stokes flows and the concept of semiseparable spectral decomposition of Stokes flows in spheroidal geometry are employed to obtain an analytical solution. More specifically, by employing Kelvin's inversion, the exterior boundary value problem in the inverse prolate spheroidal system is mapped to a related problem in the interior of the corresponding prolat...
Mathematical Methods in the Applied Sciences, 2012
A detailed mathematical analysis of Stokes flow past a rigid biconcave disc shape, motivated by t... more A detailed mathematical analysis of Stokes flow past a rigid biconcave disc shape, motivated by the flow of blood plasma past a fixed red blood cell (RBC) is developed in the present manuscript. The main objective is to provide a new insight in the theoretical investigation of the flow behaviour of RBCs in pathological situations for which the rheological characteristics of the RBCs change or in medical tests like the erythrocyte sedimentation rate. Here, the RBC is described mathematically by an inverted prolate spheroid having its axis of symmetry along the direction of a uniform flow. The method of Kelvin transformation for Stokes flows and the concept of semiseparable spectral decomposition of Stokes flows in spheroidal geometry are employed to obtain an analytical solution. More specifically, by employing Kelvin's inversion, the exterior boundary value problem in the inverse prolate spheroidal system is mapped to a related problem in the interior of the corresponding prolat...
Kelvin’s transformation is a non-linear map that, in some sense, preserves harmonicity. This prop... more Kelvin’s transformation is a non-linear map that, in some sense, preserves harmonicity. This property, which was the content of a letter sent by Kelvin to Liouville in 1845, provides a powerful machinery for solving particular potential problems in a very effective way. In the present work, we show that the basic theory can be extended to the biharmonic equation as well to the equations for irrotational and rotational Stokes flow. Hence, biharmonicity, stream functions and bistream functions are also preserved, in some sense, under the Kelvin transformation. We also demonstrate how the Kelvin-type theorems are interconnected with the relative Almansi-type decompositions. These results provide a way to solve analytically many problems in potential theory and Stokes flow which it is impossible to solve by the classical spectral method.
Publisher Summary This chapter discusses the scattering theory; in the physical setting of acoust... more Publisher Summary This chapter discusses the scattering theory; in the physical setting of acoustic wave propagation is a scalar theory where the fields represent excess pressure or velocity potentials. On the other hand, the electromagnetic scattering theory deals with vector fields that represent the electric and the magnetic fields. Two characteristics of electromagnetic wave propagation help in simplifying the corresponding scattering problems. The first is connected to the fact that using the spectral Maxwell equation and knowledge of the electric or the magnetic field, can be readily obtained from the other. The second is due to the fact that both the electric and the magnetic fields are propagating with the same phase velocities. Unfortunately, these simplifying characteristics are missing from the scattering problems that appear in the theory of elasticity. In elasticity, again two waves, the longitudinal (irrotational) and the transverse (solenoidal), propagate at different phase velocities and once one of them is known there is no way to recover the other. For that reason, the scattering problems in elasticity are much harder than the corresponding problems in electromagnetism.
The time evolution of the energy for wave propagation in an infinite ideal magnetofluid is studie... more The time evolution of the energy for wave propagation in an infinite ideal magnetofluid is studied analytically. It is found that equipartition between the kinetic and potential energies is achieved. Equipartition occurs either in finite time or asymptotically as t↠∞ depending upon whether the initial data are compactly supported or have finite energy, respectively.
The Quarterly Journal of Mechanics and Applied Mathematics, 1997
A small resistive scatterer disturbs a spherical time-harmonic field emanating from a point sourc... more A small resistive scatterer disturbs a spherical time-harmonic field emanating from a point source. The incident point-source field is normalized in such a way as to be able to recover the corresponding results for plane-wave incidence. The full low-frequency expansion for the corresponding total field is reduced to an exterior boundary-value problem for the Laplace equation, which has to be solved repeatedly. Exact results for the case of a small resistive sphere are obtained. It is shown that the leading low-frequency approximations for the scattering as well as for the absorption cross-section are increasing functions of the impedance parameter and decreasing functions of the distance of the source from the scatterer. It is also shown that a small sphere scatters and absorbs more energy when it is illuminated by a point-rather than by a plane-wave field establishing the fact that the closer the source of illumination to the scatterer, the stronger the interaction. The leading approximation of.the absorption cross-section is independent of the wavenumber, while the leading approximation of the scattering cross-section is proportional to the second power of the wavenumber. Hence, in the low-frequency realm, absorption is by two orders of magnitude stronger than scattering. Finally, a comparison between point-and plane-wave incidence, based on multipole expansions, is included.
A point generated incident "eld impinges upon a small triaxial ellipsoid which is arbitrarily ori... more A point generated incident "eld impinges upon a small triaxial ellipsoid which is arbitrarily oriented with respect to the point source. The point source "eld is so modi"ed as to be able to recover the corresponding results for plane wave incidence when the source recedes to in"nity. The main dif"culty in solving analytically this low-frequency scattering problem concerns the "tting of the spherical geometry, which characterizes the incident "eld, with the ellipsoidal geometry which is naturally adapted to the scatterer. A series of techniques has been used which lead "nally to analytic solutions for the leading two low-frequency terms of the near as well as the far "eld. In contrast to the near-"eld approximations, which are expressed in terms of ellipsoidal eigenexpansions, the far "eld is furnished by a "nite number of terms. This is very interesting because the constants entering the expressions of the Lamé functions of degree higher than three are not obtainable analytically and therefore, in the near "eld, not even the Rayleigh approximation can be completely obtained. On the other hand, since only a few terms survive at the far "eld, the scattering amplitude and the scattering cross-section are derived in closed form. It is shown that, in practice, if the source is located a distance equal to "ve or six times the biggest semiaxis of the ellipsoid the Rayleigh term of the approximation behaves almost as the incident "eld was a plane wave. The special cases of spheroids, needles, discs, spheres as well as plane wave incidence are recovered. Finally, some theorems concerning monopole and dipole surface potentials are included.
A polyadic field of rank n is the tensor product of n vector fields. Helmholtz showed that a vect... more A polyadic field of rank n is the tensor product of n vector fields. Helmholtz showed that a vector field, which is a polyadic field of rank 1, is nonuniquely decomposable into the gradient of a scalar function plus the rotation of a vector function. We show here that a polyadic field of rank n is, again nonuniquely, decomposable into a term consisting of n successive applications of the gradient to a scalar function, plus a term that consists of (n-1) successive applications of the gradient to the rotation of a vector function, plus a term that consists of (n-2) successive applications of the gradient to the rotation of the rotation of a dyadic function and so on, until the last (n + l)th term, which consists of n successive applications of the rotation operator to a polyadic function of rank n. Obviously, the n = 1 case recovers the Helmholtz decomposition theorem. For dyadic fields a more symmetric representation is provided and formulae that provide the potential representation functions are given. The special cases of symmetric and antisymmetric dyadics are discussed in detail. Finally, the multidivergence type relations, which reduce the number of independent scalar representation functions to n2, are presented.
We consider the problem of scattering of a longitudinal or a transverse plane elastic wave by a g... more We consider the problem of scattering of a longitudinal or a transverse plane elastic wave by a general ellipsoidal cavity in the low-frequency region. Explicit closed-form solutions for the zeroth- and first-order approximations are provided in terms of the physical and geometric characteristics of the scatterer, as well as the direction cosines of the incidence and observation points. This was made possible with the introduction of an analytical technique based on the Papkovich representations and their interdependence. The leading low-frequency term for the normalized spherical scattering amplitudes and the scattering cross section are also given explicitly. Degenerate ellipsoids corresponding to the prolate and oblate spheroids, the sphere, the needle, and the disc are considered as special cases.
In this paper we prove that, unlike the two-dimensional case, the electric field in the presence ... more In this paper we prove that, unlike the two-dimensional case, the electric field in the presence of closely adjacent spherical perfect conductors does not blow up even though the separation distance between the conducting inclusions approaches zero.
The Quarterly Journal of Mechanics and Applied Mathematics, 2001
The general theory of low-frequency dyadic scattering is developed for the near fields, the far f... more The general theory of low-frequency dyadic scattering is developed for the near fields, the far fields and all the energy functionals associated with scattering problems. The incident field could be any complete dyadic field generated either in the exterior medium of propagation (point source) or at infinity (plane waves). The case of a small rigid sphere, which is illuminated by a plane dyadic field, is solved and the corresponding results for acoustic and elastic scattering are recovered as special cases. In order to solve analytically the sphere problem a special technique had to be developed, which generates Papkovich-type differential representations of dyadic elastostatic displacements. Comparison of numerical results, obtained via the boundary element method, show an amazing accuracy with our analytical results.
Journal of Computational and Applied Mathematics, 1992
Apostolopoulos, T. and G. Dassios, A parallel algorithm for solving the inverse scattering moment... more Apostolopoulos, T. and G. Dassios, A parallel algorithm for solving the inverse scattering moment problem,
The Quarterly Journal of Mechanics and Applied Mathematics, 1998
The inverse problem of determining the geometrical characteristics of a perfectly conducting elli... more The inverse problem of determining the geometrical characteristics of a perfectly conducting ellipsoid or ellipsoidal boss from low-frequency scattering data is considered. It is shown that the orientation and semidiameters of the scatterer can be determined from measurements of the leading-order term in the low-frequency expansion of the electric scattering amplitude for plane wave excitation. For the ellipsoid, measurements corresponding to six directions of incidence and associated polarization provide the necessary data, while three measurements suffice for the ellipsoidal boss on a perfectly conducting base plane.
Two main results are included in this paper. The first one deals with the leading asymp-totic ter... more Two main results are included in this paper. The first one deals with the leading asymp-totic term of the magnetic field outside any conductive medium. In accord with physical reality, it is proved mathematically that the leading approximation is a quadrupole term which means that the conductive brain tissue weakens the intensity of the magnetic field outside the head. The second one concerns the orientation of the silent sources when the geometry of the brain model is not a sphere but an ellipsoid which provides the best possible mathematical approximation of the human brain. It is shown that what char-acterizes a dipole source as “silent ” is not the collinearity of the dipole moment with its position vector, but the fact that the dipole moment lives in the Gaussian image space at the point where the position vector meets the surface of the ellipsoid. The appropriate representation for the spheroidal case is also included. 1. The magnetic field The mathematical theory of magnetoen...
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Papers by George Dassios