APPLICATION OF MATHEMATICS IN TECHNICAL AND NATURAL SCIENCES: Proceedings of the 2nd International Conference

The Need for a First‐order Quasi Lorentz Transformation
View Description Hide DescriptionSolving electromagnetic scattering problems involving non‐uniformly moving objects or media requires an approximate but consistent extension of Einstein’s Special Relativity theory, originally valid for constant velocities only. For moderately varying velocities a quasi Lorentz transformation is presented. The conditions for form‐invariance of the Maxwell equations, the so‐called “principle of relativity”, are shown to hold for a broad class of motional modes and time scales. A simple example of scattering by a harmonically oscillating mirror is analyzed in detail. Application to generally orbiting objects is mentioned.

Methods for the coupled Stokes‐Darcy problem
View Description Hide DescriptionThe motion of particles in a viscous fluid close to a porous membrane is modelled for the case when particles are large compared with the size of pores of the membrane. The hydrodynamic interactions of one particle with the membrane are detailed here. The model involves Stokes equations for the fluid motion around the particle together with Darcy equations for the flow in the porous membrane and Stokes equations for the flow on the other side of the membrane. Boundary conditions at the fluid‐membrane interface are the continuity of pressure and velocity in the normal direction and the Beavers and Joseph slip condition on the fluid side in the tangential directions. The no‐slip condition applies on the particle. This problem is solved here by two different methods.
The first one is an extended boundary integral method (EBIM). A Green function is derived for the flow close to a porous membrane. This function is non‐symmetric, leading to difficulties hindering the application of the classical boundary integral method (BIM). Thus, an extended method is proposed, in which the unknown distribution of singularities on the particle surface is not the stress, like in the classical boundary integral method. Yet, the hydrodynamic force and torque on the particle are obtained by integrals of this distribution on the particle surface.
The second method consists in searching the solution as an asymptotic expansion in term of a small parameter that is the ratio of the typical pore size to the particle size. The various boundary conditions are taken into account at successive orders: order (0) simply represents an impermeable wall without slip and order (1) an impermeable wall with a peculiar slip prescribed by order (0); at least the 3rd order is necessary to enforce all boundary conditions.
The methods are applied numerically to a spherical particle and comparisons are made with earlier works in particular cases.

Biomathematics and Interval Analysis: A Prosperous Marriage
View Description Hide DescriptionIn this survey paper we focus our attention on dynamical bio‐systems involving uncertainties and the use of interval methods for the modelling study of such systems. The kind of envisioned uncertain systems are those described by a dynamical model with parameters bounded in intervals. We point out to a fruitful symbiosis between dynamical modelling in biology and computational methods of interval analysis. Both fields are presently in the stage of rapid development and can benefit from each other. We point out on recent studies in the field of interval arithmetic from a new perspective—the midpoint‐radius arithmetic which explores the properties of error bounds and approximate numbers. The midpoint‐radius approach provides a bridge between interval methods and the “uncertain but bounded” approach used for model estimation and identification. We briefly discuss certain recently obtained algebraic properties of errors and approximate numbers.

Use of the Vector Finite Element Method for the Solution of Electromagnetic Problems
View Description Hide DescriptionThe vector finite element method (VFEM) is formulated for the time‐harmonic Maxwell’s equations to model microwave integrated circuits and electronic packages at high frequencies. A number of computational challenges often appear during the formulation of these problems ranging from proper excitation of the input ports and reflection‐free truncation of the unbounded infinite domain to accurate modeling of material interfaces and anisotropies. To deal with the challenge of proper excitation/termination of the ports, a generalized eigenvalue problem is formulated at each of the ports in order to obtain the dispersive propagation characteristics and governing modes of the 2‐D structure; these modal characteristics are subsequently used to properly excite and terminate the input and output ports of the 3‐D structure under investigation. In the case where scattering is involved, the unbounded infinite domain is properly truncated using first‐, second‐, or even higher‐order absorbing boundary conditions (ABCs), a perfectly matched layer (PML), or an exact radiation condition based on a boundary‐integral (BI) method. Numerical results on a number of practical engineering applications illustrate the power and effectiveness of the VFEM in solving complex electromagnetic problems.

On the Pulsating Strings in Sasaki‐Einstein Spaces
View Description Hide DescriptionWe study the class of pulsating strings in and Using a generalized ansatz for pulsating string configurations, we find new solutions for this class in terms of Heun functions, and derive the particular case of which was analyzed in arXiv:1006.1539 [hep‐th]. Unfortunately, Heun functions are still little studied, and we are not able to quantize the theory quasi‐classically and obtain the first corrections to the energy. The latter, due to AdS/CFT correspondence, is supposed to give the anomalous dimensions of operators of the gauge theory dual superconformal field theory.

A Note on Interpolation Functions of the Frobenious‐Euler Numbers
View Description Hide DescriptionThe main purpose of this paper is to construct partial twisted zeta‐type function, which interpolates q‐generalized Frobenius‐Euler numbers and twisted q‐generalized Frobenius‐Euler numbers. We give some applications related to this function.

Bounds for the Kernel Dimension of Singular Integral Operators with Carleman Shift
View Description Hide DescriptionUpper bounds for the kernel dimension of singular integral operators with orientation‐preserving Carleman shift are obtained. This is implemented by using some estimates which are derived with the help of certain explicit operator relations. In particular, the interplay between classes of operators with and without Carleman shifts have a preponderant importance to achieve the mentioned bounds.

Wiener‐Hopf plus Hankel Operators with Almost Periodic Symbols on Weighted Lebesgue Spaces
View Description Hide DescriptionWe obtain an invertibility characterization for Wiener‐Hopf plus Hankel operators with almost periodic symbols on weighted Lebesgue spaces where and w belongs to a subclass of Muckenhoupt weights. The method is essentially based on a factorization concept for almost periodic functions within the Fourier multipliers on Additionally, formulas for one‐sided and the two‐sided inverses of the Wiener‐Hopf plus Hankel operators under study are also obtained.

On the Solvability of a Problem of Wave Diffraction by a Union of a Strip and a Half‐Plane
View Description Hide DescriptionWe consider a problem of wave diffraction by a union of an infinite strip and an half‐plane characterized by higher order boundary conditions. From the mathematical point of view, the problem is formulated as a boundary value problem for the Helmholtz equation within a Bessel potential space framework. Operator theoretical methods are used to deal with this problem and, as a consequence, several convolution type operators are constructed and associated to the problem. A Fredholm characterization of those operators is obtained for certain smoothness space orders.

Accelerated Iterations for Finding the Soft‐constrained Stochastic Nash Games Equilibrium
View Description Hide DescriptionIn this paper, the stochastic Nash games for weakly coupled large‐scale systems with state‐dependent noise are considered. The considered stochastic algebraic Riccati equations is quite different from the existing results in the sense that the equations have the additional linear term. The numerical algorithm based on Lyapunov iterations for solving the set of cross‐coupled stochastic algebraic Riccati equations is derived by Mukaidani (American Control Conference, June 2008, USA 4232–4237). We modify this method and derive new iterations based on the solution of linear matrix equations with linear rate of convergence. We carry out numerical experiments to illustrate the effectiveness of the considered iterations.

Cubic Convergence for Nonlinear Differential Equations with Initial Time Difference
View Description Hide DescriptionIn this paper we have applied the method of generalized quasilinearization to initial value problem of differential equations with initial time difference. We have obtained lower and upper monotone sequences that converge uniformly to the unique solution of the problem. Furthermore, we have proven that the rate of convergence is cubic.

Mathematical Models and Distribution of the Zeros of the Duffing Equation and More General Equations
View Description Hide DescriptionHere we investigate the oscillation behaviour of the equation generalizing two recent results of Petrova. The Duffing equation is The common assumptions are that and δ∈R are constants and T is a large enough constant, since we compare and Also, in the general case we suppose that all the delays are nonnegative constants as well as
Further, we make several approaches, concerned with the mathematical intuition based on the distribution of the zeros of wide classes of second order ordinary differential equations. The first one treats the bifurcation theory. The second one is connected with proper numerical methods.

Modeling of Line Shapes using Continuous Time Random Walk Theory
View Description Hide DescriptionIn order to provide a general framework where the Stark broadening of atomic lines in plasmas can be calculated, we model the plasma stochastic electric field by using the CTRW approach [1,2]. This allows retaining non Markovian terms in the Schrödinger equation averaged over the electric field fluctuations. As an application we consider a special case of a non separable CTRW process, the so called Kangaroo process [3]. An analytic expression for the line profile is finally obtained for arbitrary waiting time distribution functions. An application to the hydrogen Lyman α line is discussed.

Temperature Model of a New High‐powered Laser
View Description Hide DescriptionThe radial temperature in the cross‐section of a strontium copper bromide vapor laser is studied on the basis of a previously constructed mathematical model. The model has been expanded so as to allow for changes to the structural elements of the tube, in order to develop a new laser with an increased power output. The limits of the maximum allowed operating temperature of the gas have been established. The numerical results of the simulations have been presented.

Physical Meaning of Physical‐mesomechanical Formulation of Deformation and Fracture
View Description Hide DescriptionPhysical mesomechanics is a relatively recent theory of deformation and fracture. Based on the physical principle known as gauge invariance, it has derived field equations that describe the displacement behavior in plastically deforming materials. By the nature of the way the gauge invariance is applied, the theory is capable of describing all stages of deformation on the same theoretical basis. Previously, we analyzed this formalism from the mathematical point of view, and discussed the dynamics resulting from the field equations. In this paper, we focus on the physical meaning of the field equations and related dynamics. It is shown that plastic deformation is characterized as the dynamics where materials’ longitudinal resistive force is proportional to the local velocity as opposed to the elastic regime where the longitudinal resistive force is proportional to the displacement. Consequently, in the plastic regime, part of the work done by external force is not stored in the material as potential energy but dissipated as heat. In this dynamics, a quantity called the deformation charge plays an important role. From the gauge theoretical viewpoint, the deformation charge is associated with the charge of symmetry. From the viewpoint of conventional mechanics, it is normal strain whose energy is unrecoverable to the mechanical field. Experimental results that support these arguments are present.

A New Method of Prompt Fission Neutron Energy Spectrum Unfolding
View Description Hide DescriptionThe prompt neutron emission in spontaneous fission of has been investigated applying digital signal electronics along with associated digital signal processing algorithms. The goal was to find out the reasons of a long time existing discrepancy between theoretical calculations and the measurements of prompt fission neutron (PFN) emission dependence on the total kinetic energy (TKE) of fission fragments (FF). On the one hand the (sf) reaction is one of the main references for nuclear data, on the other hand the understanding of PFN emission mechanism is very important for nuclear fission theory. Using a twin Frisch‐grid ionization chamber for fission fragment (FF) detection and a NE213‐equivalent neutron detector in total about fission fragment‐neutron coincidences have been registered. Fission fragment kinetic energy, mass and angular distribution, neutron time‐of‐flight and pulse shape have been investigated using a 12 bit waveform digitizer. The signal waveforms have been analyzed using digital signal processing algorithms. For the first time the dependence of the number of emitted neutrons as a function of total kinetic energy (TKE) of the fragments is in very good agreement with theoretical calculations in the range of TKE from 140–220 MeV.

Tomographic Reconstruction of Nodular Images from Incomplete Data
View Description Hide DescriptionIn many problems of medical imaging very limited amount of tomographic data can be collected. There are well developed mathematical theories proving that unless certain a priori information is known about the image, some of these limited data sets will be insufficient to generate images of acceptable quality. While in various cases such a priori information is readily available, the translation of this knowledge into mathematical constraints and their subsequent incorporation into image reconstruction algorithms has proved to be a formidable task, and there are very few results on this matter. In this paper we introduce a method that uses the knowledge of image patterns consisting of nodules to stabilize the image reconstruction in incomplete data x‐ray tomography. The technique uses a modified expectation‐maximization algorithm to cluster singularities in Radon domain, and recover the centers and radii of a finite number of nodules in the image.

A Case of Multi‐vector and Multi‐host Epidemiological Model: Bartonella Infection
View Description Hide DescriptionWe consider a compartmental model for the Bartonella infection on rodents. More precisely, on the co‐occurring populations of Rattus rattus and Rattus norvegicus where the vectors are two species of ectoparasites, namely ticks and fleas. As usual for such models a key stage is the modelling of the forces of infection. While the vital dynamics and the progression of the infection within each of the four species are sufficiently well known to determine the rest of the transfer rates, there is practically no data on the probability of infection. In order to determine appropriate values for the coefficients of the forces of infection we solve an optimal control problem where the objective function is the norm of the difference between the observed and the predicted by the model equilibrium infection prevalence rates in the four species. Within this setting the conjecture that the higher prevalence of the infection in Rattus norvegicus can be explained solely by their higher ectoparasite load is tested and disproved.

Analysis of Types of Oscillations in Goodwin’s Model of Business Cycle
View Description Hide DescriptionTypes of solutions of the Goodwin business cycle model with the fixed investment time lag have been numerically studied. It is shown that the long‐periodic Goodwin’s oscillations are excited by the independent investment A in case A exceeds a threshold. If A falls below the threshold, then there are only sawtooth oscillations with a period equal to the investment time lag. Near the threshold, the time behavior of the income is irregular.