One of the main areas of activity of the Applied Mathematics Laboratory of the Hellenic Open University is the scientific research in Applied Mathematics. In particular, the work of the Laboratory is geared toward the development of innovative methods and tools for the understanding and treatment of common diseases such as cancer. The main research areas of the Laboratory are the following:

  • Mathematical analysis of biomedical models and modelling problems arising from Physics, Biology, Medicine and Engineering, such as: blood plasma flow around erythrocytes, blood flow in curved vessels (atherosclerotic or not), drug administration, blood cell precipitation, cancer/tumor progress and growth, Stokes flow through porous medium, and the development of non-invasive techniques for medical diagnosis.
  • Mathematical analysis of imaging techniques of the brain, especially electroencephalography and magnetoencephalography. Analytical methods for determining tensor physical quantities (polarization, magnetic tensor, etc)
  • Wave propagation and scattering problems by a single or by many scatterers of different physical characteristics and geometric shapes. Analysis of wave fields related to scattering in isotropic and anisotropic materials.
  • Straight and inverse scattering problems in acoustics, electromagnetism and thermoelasticity.
  • Initial and boundary value problems in ellipsoidal geometry. Development of techniques for solving boundary value problems in non-convex spaces.
  • Applications of ellipsoidal geometry in geodesy.
  • Mathematical models for the learning process, virtual “environments” and ICT tools for distance learning and learning mathematics. Theories, methods and tools for the Open and Distance Education (ΑεξΑΕ) of Mathematics.
Indicative publications of workshop members are the following:

The well-known low-frequency expansion of the total acoustic field in the exterior of a penetrable spherical scatterer is revisited in view of the Atkinson Wilcox theorem. The corresponding low-frequency approximations of any order are calculated following a purely algebraic algorithmic procedure, based on the spectral decomposition of the problem’s far field pattern. As an indication of its accuracy and effectiveness, the proposed algebraic procedure is shown to recover already known low-frequency coefficients and also to deduce higher order of approximations. The proposed track of calculations leads to a closed-form expression of any such coefficient and has been recently applied on impenetrable spherical scatterers as well, offering equally accurate results. The effectiveness of the proposed procedure indicates an underlying general efficient method applicable to a wider class of starshaped scatterers.

  • Hadjinicolaou, M. and Protopapas, E. (2020) Separability of Stokes Equations in Axisymmetric Geometries. Journal of Applied Mathematics and Physics8, 315-348. DOI: 10.4236/jamp.2020.82026

The blood plasma flow through a swarm of red blood cells in capillaries is modeled as an axisymmetric Stokes flow within inverted prolate spheroidal solid-fluid unitary cells. The solid internal spheroid represents a particle of the swarm, while the external spheroid surrounds the spheroidal particle and contains the analogous amount of fluid that corresponds to the fluid volume fraction of the swarm. Analytical expansions for the components of the flow velocity are obtained by introducing a stream function ψ which satisfies the fourth-order partial differential equation E4ψ = 0. We assume nonslip conditions on the internal inverted spheroidal boundary which is also impermeable, while on the external spheroidal surface, we assume continuity of the tangential velocity component and nil vorticity. In order to solve the problem at hand, we employ the method of Kelvin inversion, under which, the initial problem, formulated in the inverted prolate spheroidal coordinates, is transformed to an equivalent one in the prolate spheroidal coordinates, where the solution space of the equation E4ψ = 0 is already known from our previously published work. The solution for the original problem is obtained by using the inverse Kelvin transformation and the effect of this transform to the Stokes operator (Dassios, IMA J Appl Math 74:427-438, 2009). Finally, the analytical solution for the stream function ψ is given through a series expansion of specific combinations of Gegenbauer functions of mixed order, multiplied by the Euclidean distance on the first and on the third power, in a so-called R-separable form.

  • Hadjinicolaou M., Protopapas E. (2020) A Microscale Mathematical Blood Flow Model for Understanding Cardiovascular Diseases. In: Vlamos P. (eds) GeNeDis 2018. Advances in Experimental Medicine and Biology, vol 1194. Springer, Cham. DOI: 10.1007/978-3-030-32622-7_35

The present work is part of a series of studies conducted by the authors on analytical models of avascular tumour growth that exhibit both geometrical anisotropy and physical inhomogeneity. In particular, we consider a tumour structure formed in distinct ellipsoidal regions occupied by cell populations at a certain stage of their biological cycle. The cancer cells receive nutrient by diffusion from an inhomogeneous supply and they are subject to also an inhomogeneous pressure field imposed by the tumour microenvironment. It is proved that the lack of symmetry is strongly connected to a special condition that should hold between the data imposed by the tumour’s surrounding medium, in order for the ellipsoidal growth to be realizable, a feature already present in other non-symmetrical yet more degenerate models. The nutrient and the inhibitor concentration, as well as the pressure field, are provided in analytical fashion via closed-form series solutions in terms of ellipsoidal eigenfunctions, while their behaviour is demonstrated by indicative plots. The evolution equation of all the tumour’s ellipsoidal interfaces is postulated in ellipsoidal terms and a numerical implementation is provided in view of its solution. From the mathematical point of view, the ellipsoidal system is the most general coordinate system that the Laplace operator, which dominates the mathematical models of avascular growth, enjoys spectral decomposition. Therefore, we consider the ellipsoidal model presented in this work, as the most general analytic model describing the avascular growth in inhomogeneous environment. Additionally, due to the intrinsic degrees of freedom inherited to the model by the ellipsoidal geometry, the ellipsoidal model presented can be adapted to a very populous class of avascular tumours, varying in figure and in orientation.

  • Fragoyiannis, G., Kariotou, F., & Vafeas, P (2020). On the avascular ellipsoidal tumour growth model within a nutritive environment. European Journal of Applied Mathematics, 31(1), 111-142.  DOI: 10.1017/S0956792518000499

The forward problem of magnetoencephalography (MEG) in ellipsoidal geometry has been studied by Dassios and Kariotou [“Magnetoencephalography in ellipsoidal geometry,” J. Math. Phys. 44, 220 (2003) ] using the theory of ellipsoidal harmonics. In fact, the analytic solution of the quadrupolic term for the magnetic induction field has been calculated in the case of a dipolar neuronal current. Nevertheless, since the quadrupolic term is only the leading nonvanishing term in the multipole expansion of the magnetic field, it contains not enough information for the construction of an effective algorithm to solve the inverse MEG problem, i.e., to recover the position and the orientation of a dipole from measurements of the magnetic field outside the head. For this task, the next multipole of the magnetic field is also needed. The present work provides exactly this octapolic contribution of the dipolar current to the expansion of the magnetic induction field. The octapolic term is expressed in terms of the ellipsoidal harmonics of the third degree, and therefore it provides the highest order terms that can be expressed in closed form using long but reasonable analytic and algebraic manipulations. In principle, the knowledge of the quadrupolic and the octapolic terms is enough to solve the inverse problem of identifying a dipole inside an ellipsoid. Nevertheless, a simple inversion algorithm for this problem is not yet known.

  • Bitsouni, V., Gialelis, N., & Ioannis G. Stratis, I. G.  (2020). To appear in Springer Proceedings in Mathematics & Statistics. arXiv: 2008.00828

The well-known low-frequency expansion of the total acoustic field in the exterior of a penetrable spherical scatterer is revisited in view of the Atkinson Wilcox theorem. The corresponding low-frequency approximations of any order are calculated following a purely algebraic algorithmic procedure, based on the spectral decomposition of the problem’s far field pattern. As an indication of its accuracy and effectiveness, the proposed algebraic procedure is shown to recover already known low-frequency coefficients and also to deduce higher order of approximations. The proposed track of calculations leads to a closed-form expression of any such coefficient and has been recently applied on impenetrable spherical scatterers as well, offering equally accurate results. The effectiveness of the proposed procedure indicates an underlying general efficient method applicable to a wider class of starshaped scatterers.

  • Kariotou, F., & Sinikis, D. E. (2018). An accelerated derivation of the acoustic low‐frequency expansion: the penetrable sphere. Mathematical Methods in the Applied Sciences41(3), 973-978. DOI: 10.1002/mma.4024

In the present work we propose an analytical method for the algebraic calculation of the low frequency expansion of the solution of a scalar scattering problem upon a smooth starshaped scatterer. The method is based on the far field expansion theorem, introduced by Atkinson and Wilcox in the middle of the last century, applied in the low frequency realm. In this view, the need for solving elliptical boundary value problems to obtain the low frequency approximations is replaced by an analytical procedure, easily encoded in an algorithm including only algebraic operators. As a demonstration example, the method is applied to the acoustic scattering problem with plane wave excitation upon three different non-penetrable spherical scatterers. The first low frequency approximations are deduced, yielding the validity of the proposed method by both recovering already known results and accurately deriving higher orders of approximation.

  • Kariotou, F., & Sinikis, D. E. (2015). An algebraic calculation method for the acoustic low frequency expansion. Journal of Mathematical Analysis and Applications424(2), 1506-1529. DOI: 10.1016/j.jmaa.2014.12.008

An explicit expression for the rapid computation of the low frequency scattering coefficients is presented. Although the demonstration of the methodology is restricted to the simple case of an acoustically soft spherical scatterer, the introduced main concept can be applied to a wide class of scattering problems. The procedure, based on algebraic calculations only, provides an analytic expression of the corresponding scattering coefficients in a straightforward fashion. The proposed algorithm is readily implemented, furnishing an efficient way for obtaining any order approximation of the total field in the exterior of the scatterer.

  • Kariotou, F., Doschoris, M., & Sinikis, D. E. (2016). An algebraic formula for the accelerated computation of the low frequency scattering coefficients: The case of the acoustically soft sphere. Applied Mathematics and Computation275, 13-23. DOI: 10.1016/j.amc.2015.11.044

The human brain is shaped in the form of an ellipsoid with average semiaxes equal to 6, 6.5 and 9 cm. This is a genuine 3-D shape that reflects the anisotropic characteristics of the brain as a conductive body. The direct electroencephalography problem in such anisotropic geometry is studied in the present work. The results, which are obtained through successively solving an interior and an exterior boundary value problem, are expressed in terms of elliptic integrals and ellipsoidal harmonics, both in Jacobian as well as in Cartesian form. Reduction of our results to spheroidal as well as to spherical geometry is included. In contrast to the spherical case where the boundary does not appear in the solution, the boundary of the realistic conductive brain enters explicitly in the relative expressions for the electric field. Moreover, the results in all three geometrical models reveal that to some extend the strength of the electric source is more important than its location.

  • Kariotou, F. (2004). Electroencephalography in ellipsoidal geometry. Journal of Mathematical Analysis and Applications290(1), 324-342. DOI: 10.1016/j.jmaa.2003.09.066

Electroencephalography (EEG) remains the utmost important technique recording brain activity. The present investigation examines the sensitivity of analytic algorithms employed in order to evaluate EEG data with respect to different brain template models. These algorithms are based on mathematical models built upon certain geometrical and/or physical assumptions regarding the head‐brain shape or the neuronal source. While the ellipsoidal head model provides a realistic approach regarding the interpretation of EEG signals in view of diverse geometries, its eccentricities influence strongly the results. The present study quantifies the deviation of the computed electric potentials when nonconfocal ellipsoids are used to model a homogeneous head conductor. To this end, a correspondence is proposed between points of the different ellipsoids, in view of the Gauss map. The investigation demonstrates that the introduction of nonconfocality imports a highly elevated error rate when EEG recordings are misinterpreted by arriving from different head‐brain models, including ellipsoidal vs spherical ones. Moreover, evidence is presented concerning the location of extrema regarding the errors in the upper brain hemisphere, which could lead to a more precise and accurate protocol regarding the placement of sensors. Although the present work refers to the forward EEG problem, its results may be used under other approaches as well. In particular, it provides the error in the solution of an elliptic boundary value problem with transmission conditions under small perturbations of the eccentricities of its ellipsoidal domain.

  • Doschoris, M., & Kariotou, F. (2018). Error analysis for nonconfocal ellipsoidal systems in the forward problem of electroencephalography. Mathematical Methods in the Applied Sciences41(16), 6793-6813. DOI: 10.1002/mma.5192

The stream function psi for axisymmetric Stokes flow satisfies the well-known equation E”SUP 4″ psi=0. In the present work the complete solution for psi in spheroidal coordinates is obtained as follows. First, the generalized 0-eigenspace of the operator E”SUP 2″ is investigated and a complete set of generalized eigenfunctions is given in closed form, in terms of products of Gegenbauer functions with mixed order. The general Stokes stream function is then represented as the sum of two functions: one from the 0-eigenspace and one from the generalized 0-eigenspace of the operator E”SUP 2″ . The proper solution subspace that provides velocity and vorticity fields is given explicitly. Finally, it is shown how these simple and generalized eigenfunctions reduce to the corresponding spherical eigenfunctions as the focal distance of the spheroidal system tends to zero, in which case the separability is regained. The usefulness of the method is demonstrated by solving the problem of the flow in a fluid cell contained between two confocal spheroidal surfaces with Kuwabara-type boundary conditions.

  • Dassios, G., Hadjinicolaou, M., & Payatakes, A. C. (1994). Generalized eigenfunctions and complete semiseparable solutions for Stokes flow in spheroidal coordinates. Quarterly of Applied Mathematics52(1), 157-191. DOI: 10.1090/qam/1262325

We introduce a complete set of vector harmonic functions in an invariant form, that is, in a form that is independent of any coordinate system. In fact, we define three vector differential operators of the first order which, when they act on a scalar harmonic function they generate three independent vector harmonic functions. Then, we prove the relative independence properties and we investigate the characterization of every harmonic as an irrotational or solenoidal field. We also prove that this set of functions forms a complete set of vector harmonics. Finally, we use these invariant expressions to recover the vector spherical harmonics of Hansen and to introduce vector ellipsoidal harmonics in R3. Our method can be applied to any other coordinate system to produce the corresponding vector harmonics.

  • Dassios, G., Kariotou, F., & Vafeas, P. (2013). Invariant vector harmonics. The ellipsoidal case. Journal of Mathematical Analysis and Applications405(2), 652-660. DOI: 10.1016/j.jmaa.2013.03.015

An exact analytic solution for the forward problem in the theory of biomagnetics of the human brain is known only for the (1D) case of a sphere and the (2D) case of a spheroid, where the excitation field is due to an electric dipole within the corresponding homogeneous conductor. In the present work the corresponding problem for the more realistic ellipsoidal brain model is solved and the leading quadrupole approximation for the exterior magnetic field is obtained in a form that exhibits the anisotropic character of the ellipsoidal geometry. The results are obtained in a straightforward manner through the evaluation of the interior electric potential and a subsequent calculation of the surface integral over the ellipsoid, using Lamé functions and ellipsoidal harmonics. The basic formulas are expressed in terms of the standard elliptic integrals that enter the expressions for the exterior Lamé functions. The laborious task of reducing the results to the spherical geometry is also included.

  • Dassios, G., & Kariotou, F. (2003). Magnetoencephalography in ellipsoidal geometry. Journal of Mathematical Physics44(1), 220-241. DOI: 10.1063/1.1522135

Breast cancer is the most frequently diagnosed cancer in women. From mammography, Magnetic Resonance Imaging (MRI), and ultrasonography, it is well documented that breast tumours are often ellipsoidal in shape. The World Health Organisation (WHO) has established a criteria based on tumour volume change for classifying response to therapy. Typically the volume of the tumour is measured on the hypothesis that growth is ellipsoidal. This is the Calliper method, and it is widely used throughout the world. This paper initiates an analytical study of ellipsoidal tumour growth based on the pioneering mathematical model of Greenspan. Comparisons are made with the more commonly studied spherical mathematical models.

  • Dassios, G., Kariotou, F., Tsampas, M., & Sleeman, B. (2012). Mathematical modelling of avascular ellipsoidal tumour growth. Quarterly of applied mathematics70(1), 1-24. DOI: 10.1090/S0033-569X-2011-01240-2

The present work focuses on deriving the evolution equation of a cancer tumour, growing anisotropically in an inhomogeneous host tissue. To this due, a continuous mathematical model is developed in ellipsoidal geometry under widely accepted biological principles, such as that the growth depends on the nutrient distribution and on the pressure field of the surrounding medium and is influenced by the presence of inhibitor factors. The mathematical model consists of three boundary value problems interrelated via a highly nonlinear ordinary differential equation that provides the evolution of the tumor’s exterior boundary. Formulated and solved analytically in the frame of ellipsoidal geometry, the system concludes to the numerical solution of the aforementioned ordinary differential equation, plots of which are included in the present work with respect to different initial tumour sizes.

  • Hadjinicolaou, M., & Kariotou, F. (2010). On the effect of 3D anisotropic tumour growth on modelling the nutrient distribution in the interior of the tumour. Bulletin of the Greek Mathematical Society57, 189-197. DOI: 10.1063/1.4992747

Two main results are included in this paper. The first one deals with the leading asymptotic 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 characterizes 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.

  • Dassios, G., & Kariotou, F. (2004, April). On the exterior magnetic field and silent sources in magnetoencephalography. In Abstract and Applied Analysis (Vol. 2004, No. 4, pp. 307-314). DOI: 10.1155/S1085337504306032

A detailed analysis of the Geselowitz formula for the magnetic induction and for the electric potential fields, due to a localized dipole current density is provided. It is shown that the volume integral, which describes the contribution of the conductive tissue to the magnetic field, exhibits a hyper-singular behaviour at the point where the dipole source is located. This singularity is handled both via local regularization of the volume integral as well as through calculation of the total flux it generates. The analysis reveals that the contribution of the primary dipole to the volume integral is equal to the one third of the magnetic field generated by the primary dipole while the rest is due to the distributed conductive tissue surrounding the singularity. Furthermore, multipole expansion is introduced, which expresses the magnetic field in terms of polyadic moments of the electric potential over the surface of the conductor.

  • Dassios, G., & Kariotou, F. (2003). On the Geselowitz formula in biomagnetics. Quarterly of Applied Mathematics61(2), 387-400. DOI: 10.1090/qam/1976377

This work provides an analytic approximation of the electric potential and the magnetic field generated by a dipole source which is located within a spheroidal volume conductor, in the case where the time-dependent variations of both fields are considered to be negligible. The first terms of their multipole expansion are provided in Cartesian coordinates via formulae mediating between the spheroidal and the Cartesian coordinate systems. This work is an attempt to break the complete isotropy of the spherical system by inserting 2-dimensional anisotropy through the spheroidal geometry. This will generalize existing results in an attempt to reveal the physical and geometrical structures as well as to facilitate computation. Both the potential field and the magnetic field, in the exterior of a bounded volume conductor, are experimentally measurable and provide useful data towards an understanding and interpreting of electroencephalographic and magnetoencephalographic data.

It has recently been shown by Fokas and coworkers that if the brain is approximated by a homogeneous sphere, magnetoencephalographic measurements determine only the moments of one of the three scalar functions specifying the electrochemically generated current in the brain. In this letter, we show that this is a generic limitation of MEG. Indeed, this indeterminancy persists in the general case that the sphere is replaced by a starlike conductor.

  • Dassios, G., Fokas, A. S., & Kariotou, F. (2005). On the non-uniqueness of the inverse MEG problem. Inverse Problems21(2), L1. DOI: 10.1088/0266-5611/21/2/L01

Electroencephalography (EEG) is an important clinical tool detecting the electrical activity of the brain. The present work aims to analyse the sensitivity of analytical algorithms that are used to interpret the EEG data obtained from an actual subject, with respect to data obtained from an average head–brain system. We consider the case where the EEG data are misinterpreted as if they referred to a brain with different size than the actual brain under consideration. We employ a homogeneous triaxial ellipsoid to model the head brain system, which offers the best fit realistic approach that permits classical analysis. Utilizing the ellipsoidal coordinate system, two confocal ellipsoids, i.e. their foci remain fixed, are implemented in order to analytically calculate the approximation errors and bounds that occur when solving the EEG problem with respect to an ellipsoidal volume conductor of different size. Our results indicate a high error rate when comparing different sized brains.

  • Doschoris, M., & Kariotou, F. (2018). Quantifying errors during the source localization process in electroencephalography for confocal systems. IMA Journal of Applied Mathematics83(2), 243-260. DOI: 10.1093/imamat/hxx043

A triaxial ellipsoid provides an approximation of the average human brain which is much better than the broadly used spher-ical model. The analytical solution of the forward problem of the Electroencephalography (EEG) with an isolated dipolar source has been already derived and reported in the literature. This solution is expressed in terms of an eigenexpansion in el-lipsoidal harmonics. Nevertheless, this expression was not possible to be handled effectively since no ellipsoidal harmonics of degree higher than seven were available in closed forms. In order to analyze further this problem an effective numerical algorithm has been developed, which generates the ellipsoidal harmonics of arbitrary degree and order in a numerical form. The algorithm has been compared with the known analytical eigenfunctions and the results manifested a perfect coincidence. Finally, this algorithm was used to construct a numerically stable solution of the electric potential on the surface of the head, which is generated by a single dipole of arbitrary position and orientation. The degree of ellipsoidal harmonics needed for a numerically convergent solution goes all the way up to 30 and the result provides a slightly improved model for tackling boundary value problems in ellipsoidal geometry.

Particle-in-cell models are useful in the development of simple but reliable analytical expressions for heat and mass transfer in swarms of particles. Most such models consider spherical particles. Here the creeping flow through a swarm of spheroidal particles, that move with constant uniform velocity in the axial direction through an otherwise quiescent Newtonian fluid, is analyzed with a spheroid-in-cell model. The solid internal spheroid represents a particle of the swarm. The external spheroid contains the spheroidal particle and the amount of fluid required to match the fluid volume fraction of the swarm. The boundary conditions on the (conceptual) external spheroidal surface are similar to those of the sphere-in-cell Happel model [1], namely, nil normal velocity component and shear stress. The stream function is obtained in series form using the recently developed method of semiseparation of variables. It turns out that the first term of the series is sufficient for most engineering applications, so long as the aspect ratio of the spheroids remains within moderate bounds, say ∼1/5<a3<∼5. Analytical expressions for the streamfunction, the velocity components, the vorticity, the drag force acting on each particle, and the permeability of the swarm are obtained. Representative results are presented in graph form and they are compared with those obtained using Kuwabara-type boundary conditions. The Happel formulation is slightly superior because it leads to a particle-in-cell that is self sufficient in mechanical energy.

  • Dassios, G., Hadjinicolaou, M., Coutelieris, F. A., & Payatakes, A. C. (1995). Stokes flow in spheroidal particle-in-cell models with rappel and Kuwabara boundary conditions. International Journal of Engineering Science33(10), 1465-1490. DOI: 10.1016/0020-7225(95)00010-U

This paper investigates the performance of particle swarm optimization (PSO) and unified particle swarm optimization (UPSO) in magnetoencephalography (MEG) problems. For this purpose, two interesting tasks are considered. The first is the source localisation problem, also called the ‘inverse MEG problem’, where an unknown excitation source has to be identified, based on a set of sensor measurements that can be contaminated by noise. We refer to the second task as ‘forward task for inverse use’. It consists of the detection of the proper coefficients for approximating the magnetic potential through a spherical expansion, as accurately as possible. Also, the study of their behaviour under variations of the number of available measurements is considered. The obtained results are statistically analysed, providing useful insight regarding the applicability of the employed algorithms on such problems. Also, significant indications regarding the behaviour of several intrinsic dependencies of the problem are derived.

  • Parsopoulos, K. E., Kariotou, F., Dassios, G., & Vrahatis, M. N. (2009). Tackling magnetoencephalography with particle swarm optimization. International Journal of Bio-Inspired Computation1(1-2), 32-49. DOI: 10.1504/IJBIC.2009.022772

A well-known mathematical model of radially symmetric tumour growth is revisited in the present work. Under this aim, a cancerous spherical mass lying in a finite concentric nutritive surrounding is considered. The host spherical shell provides the tumor with vital nutrients, receives the debris of the necrotic cancer cells, and also transmits to the tumour the pressure imposed on its exterior boundary. We focus on studying the type of inhomogeneity that the nutrient supply and the pressure field imposed on the host exterior boundary, can exhibit in order for the spherical structure to be supported. It turns out that, if the imposed fields depart from being homogeneous, only a special type of interrelated inhomogeneity between nutrient and pressure can secure the spherical growth. The work includes an analytic derivation of the related boundary value problems based on physical conservation laws and their analytical treatment. Implementations in cases of special physical interest are examined, and also existing homogeneous results from the literature are fully recovered.

  • Kariotou, F., & Vafeas, P. (2012). The avascular tumour growth in the presence of inhomogeneous physical parameters imposed from a finite spherical nutritive environment. International Journal of Differential Equations2012. DOI: 10.1155/2012/175434

This work provides the solution of the direct Electroencephalography (EEG) problem for the complete ellipsoidal shell-model of the human head. The model involves four confocal ellipsoids that represent the successive interfaces between the brain tissue, the cerebrospinal fluid, the skull, and the skin characterized by different conductivities. The electric excitation of the brain is due to an equivalent electric dipole, which is located within the inner ellipsoid. The proposed model is considered to be physically complete, since the effect of the substance surrounding the brain is taken into account. The direct EEG problem consists in finding the electric potential inside each conductive space, as well as at the nonconductive exterior space. The solution of this multitransmission problem is given analytically in terms of elliptic integrals and ellipsoidal harmonics, in such way that makes clear the effect that each shell has on the next one and outside of the head. It is remarkable that the dependence on the observation point is not affected by the presence of the conductive shells. Reduction to simpler ellipsoidal models and to the corresponding spherical models is included.

  • Giapalaki, S. N., & Kariotou, F. (2006, January). The complete ellipsoidal shell-model in EEG imaging. In Abstract and Applied Analysis (Vol. 2006). Hindawi. DOI: 10.1155/AAA/2006/57429

The forward problem of Magnetoencephalography for an ellipsoidal inhomogeneous shell-model of the brain is considered. The inhomogeneity enters through a confocal ellipsoidal shell exhibiting different conductivity than the one of the brain tissue. It is shown that, as far as the leading quadrupolic moment of the exterior magnetic field is concerned, the complicated expression associated with the field itself is the same as in the homogeneous case, while the effect of the shell is focused on the form of the generalized dipole moment. In contrast to the spherical case, where no shell inhomogeneities are “readable” outside the skull, the ellipsoidal shells establish their existence on the exterior magnetic induction field in a way that depends not only on the geometry but also on the conductivity of the shell. The degenerated spherical results are fully recovered.

  • Dassios, G., & Kariotou, F. (2005). The direct MEG problem in the presence of an ellipsoidal shell inhomogeneity. Quarterly of applied mathematics63(4), 601-618. DOI: 10.1090/S0033-569X-05-00971-2

The magnetic induction field in the exterior of an ellipsoidally inhomogeneous, four-conducting-layer model of the human head is obtained analytically up to its quadrupole approximation. The interior ellipsoidal core represents the homogeneous brain while each one of the shells represents the cerebrospinal fluid, the skull and the scalp, all characterized by different conductivities. The inhomogeneities of these four domains, together with the anisotropy imposed by the use of the ellipsoidal geometry, provide the most realistic physical and geometrical model of the brain for which an analytic solution of the biomagnetic forward problem is possible. It is shown that in contrast to the spherical model, where shells of different conductivity are magnetically invisible, the magnetic induction field in ellipsoidal geometry is strongly dependent on the conductivity supports. The fact that spherical shells of different conductivity are invisible has enhanced the common belief that the biomagnetic forward solution does not depend on the conductivity profiles. As we demonstrate in the present work, this is not true. Hence, the proposed multilayered ellipsoidal model provides a qualitative improvement of the realistic interpretation of magnetoencephalography (MEG) measurements. We show that the presence of the shells of different conductivity can be incorporated in the form of the dipole vector for the homogeneous model. Numerical investigations show that the effects of shell inhomogeneities are almost as sound as the level of MEG measurements themselves. The degenerate cases, where either the differences of the conductivities within the shells disappear, or the ellipsoidal geometry is reduced to the spherical one, are also considered.

  • Dassios, G., Giapalaki, S. N., Kandili, A. N., & Kariotou, F. (2007). The exterior magnetic field for the multilayer ellipsoidal model of the brain. Quarterly journal of mechanics and applied mathematics60(1), 1-25. DOI: 10.1093/qjmam/hbl022

The direct problem of magnetoencephalography for an ellipsoidal inhomogeneous one-shell-model of the brain with a dipole source in the shell is studied. The inhomogeneity is due to the attendance of a confocal ellipsoidal shell exhibiting different conductivity than the one of the brain tissue. The magnetic field in the exterior of the conductor is derived. It is shown that it depends strongly on the anisotropy imposed by the use of the ellipsoidal geometry, on the inhomogeneity dictated by the shell and also on the position and on the moment of the dipole source.

The mass transfer rate of pure ammonium nitrate between the aerosol and gas phases was quantified experimentally by the use of the tandem differential mobility analyzer/scanning mobility particle sizer (TDMA/SMPS) technique. Ammonium nitrate particles 80–220 nm in diameter evaporated in purified air in a laminar flow reactor under temperatures of 20–27°C and relative humidities in the vicinity of 10%. The evaporation rates were calculated by comparing the initial and final size distributions. A theoretical expression of the evaporation rate incorporating the Kelvin effect and the effect of relative humidity on the equilibrium constant is developed. The measurements were consistent with the theoretical predictions but there was evidence of a small kinetic resistance to the mass transfer rate. The discrepancy can be explained by a mass accommodation coefficient ranging from 0.8 to 0.5 as temperature increases from 20–27°C. The corresponding timescale of evaporation for submicron NH4NO3 particles in the atmosphere is of the order of a few seconds to 20 min. 

  • Dassios, K. G., & Pandis, S. N. (1999). The mass accommodation coefficient of ammonium nitrate aerosol. Atmospheric Environment33(18), 2993-3003. DOI: 10.1016/S1352-2310(99)00079-5

The forward problem of magnetoencephalography (MEG) in ellipsoidal geometry has been studied by Dassios and Kariotou [“Magnetoencephalography in ellipsoidal geometry,” J. Math. Phys. 44, 220 (2003) ] using the theory of ellipsoidal harmonics. In fact, the analytic solution of the quadrupolic term for the magnetic induction field has been calculated in the case of a dipolar neuronal current. Nevertheless, since the quadrupolic term is only the leading nonvanishing term in the multipole expansion of the magnetic field, it contains not enough information for the construction of an effective algorithm to solve the inverse MEG problem, i.e., to recover the position and the orientation of a dipole from measurements of the magnetic field outside the head. For this task, the next multipole of the magnetic field is also needed. The present work provides exactly this octapolic contribution of the dipolar current to the expansion of the magnetic induction field. The octapolic term is expressed in terms of the ellipsoidal harmonics of the third degree, and therefore it provides the highest order terms that can be expressed in closed form using long but reasonable analytic and algebraic manipulations. In principle, the knowledge of the quadrupolic and the octapolic terms is enough to solve the inverse problem of identifying a dipole inside an ellipsoid. Nevertheless, a simple inversion algorithm for this problem is not yet known.

  • Dassios, G., Hadjiloizi, D., & Kariotou, F. (2009). The octapolic ellipsoidal term in magnetoencephalography. Journal of mathematical physics50(1), 013508. DOI: 10.1063/1.3036183

The computational singular perturbation (CSP) method is employed for the solution of stiff PDEs and for the acquisition of the most important physical understanding. The usefulness of the method is demonstrated by analyzing a transient reaction-diffusion problem. It is shown that in the regions where the solution exhibits smooth spatial slopes, a simple nonstiff system of equations can be used instead of the full governing equations. From the simplified system, which is numerically provided by CSP and whose structure varies with space and time, important physical information comes to light. The relation of this method to the class of asymptotic expansion methods is explored. It is shown that the CSP results are identical to the ones obtained by the asymptotic methods. The identifications of the nondimensional parameters and the tedious manipulations needed by the asymptotic methods are performed by programmable numerical or analytic computations specified by CSP. Preliminary numerical results are presented validating the theoretical aspects of the proposed algorithm and providing a measure of its usefulness and its accuracy.

  • Hadjinicolaou, M., & Goussis, D. A. (1998). Asymptotic solution of stiff PDEs with the CSP method: the reaction diffusion equation. SIAM Journal on Scientific Computing20(3), 781-810. DOI: 10.1137/S1064827596303995

Ο σκοπός, η φιλοσοφία, οι αρχές, το περιεχόμενο, τα μέσα και οι μέθοδοι διδασκαλίας των Μαθηματικών, στο πλαίσιο της Ανοικτής και εξ Αποστάσεως Εκπαίδευσης (ΑεξΑΕ) παρουσιάζονται και αναλύονται στην περίπτωση του Προγράμματος Σπουδών «Μεταπτυχιακές σπουδές στα Μαθηματικά» ΜΣΜ. Από τα πρώτα οργανωμένα μεταπτυχιακά προγράμματα σπουδών Ελληνικών Πανεπιστημίων για τα μαθηματικά, προσφέρεται από το 2006 από το ΕΑΠ, με στόχο την κάλυψη των αναγκών της μαθηματικής κοινότητας για αναβάθμιση των γνώσεων και συνέχιση των σπουδών σε μεταπτυχιακό επίπεδο. Το πρόγραμμα ΜΣΜ, υπηρετεί το σκοπό αυτό προσφέροντες δύο κατευθύνσεις. Η μία οδηγεί σε ειδίκευση με έμφαση στην ιστορική εξέλιξη και διδακτική των μαθηματικών, ενώ η δεύτερη σε ειδίκευση με έμφαση στη μαθηματική προτυποποίηση και τις εφαρμογές των Μαθηματικών στις επιστήμες και την τεχνολογία.

Στα μεταπτυχιακά προγράμματα, η διδακτική των μαθηματικών εστιάζει στις θεωρίες μάθησης της γνωστικής ψυχολογίας. Το πρόγραμμα αυτό, συμπληρώνει τη διάσταση αυτή με τη συνιστώσα της ιστορικής εξέλιξης των μαθηματικών εννοιών και της φιλοσοφίας των μαθηματικών, διευρύνοντας έτσι το θεωρητικό υπόβαθρο των μεταπτυχιακών φοιτητών στα αντίστοιχα επιστημονικά πεδία, συνεισφέροντας στη βαθύτερη κατανόηση της σύλληψης, της κατασκευής, και λειτουργίας των μαθηματικών εννοιών και οντοτήτων.

Η δεύτερη κατεύθυνση του προγράμματος, ακολουθεί μια διαφορετική προσέγγιση, που απορρέει από τις σύγχρονες τάσεις και απόψεις, οι οποίες θεωρούν σημαντική την ενσωμάτωση της μαθηματικής μοντελοποίησης στη μαθηματική εκπαίδευση. Η μαθηματική μοντελοποίηση περιγράφεται ως η διαδικασία μέσω της οποίας ένα φυσικό φαινόμενο περιγράφεται με «μαθηματική γλώσσα» και διατυπώνεται ως μαθηματικό πρόβλημα. Στη συνέχεια εφαρμόζονται μαθηματικές θεωρίες, μέθοδοι και τεχνικές για να μελετηθεί και επιλυθεί αναλυτικά ή αριθμητικά. Η ενσωμάτωση αυτής της προσέγγισης στο πρόγραμμα, συνεισφέρει στην καλλιέργεια της μαθηματικής σκέψης, και της ερευνητικής αναζήτησης, αναβαθμίζει τις μαθηματικές γνώσεις των μεταπτυχιακών φοιτητών, και τους εφοδιάζει με μεθοδολογικά εργαλεία και δεξιότητες αξιοποιήσιμες εκπαιδευτικά αλλά και ερευνητικά, μέσω εκπόνησης εργασιών και διδακτορικών διατριβών.

Η μεθοδολογία και οι αρχές της Ανοικτής και εξ Αποστάσεως εκπαίδευσης
εξυπηρετούν τους σκοπούς του προγράμματος, παρέχοντας το κατάλληλο πλαίσιο.