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This paper deals with pattern formation in various models described by reaction-diffusion equations. Different spatio-temporal patterns can emerge depending on the kinetics in these models. There are turbulent double rotating spirals, cellular, and target patterns. Reactions are represented by composition changes with a particular probability of grid positions, depending on the occupation of neighboring grid positions. In these cases, pattern formation is self-organized in the oscillatory medium of catalytic oscillators. Self-organizing patterns will only occur in specific reaction condition regimes. An application in nano-structures will be presented. [ABSTRACT FROM AUTHOR]

This paper deals with radial and non-radial solutions for local and non-local Liouville type equations. At first non- degenerate and degenerate mean field equations are studied and radially symmetric solutions to the Dirichlet problem for them are written into explicit form. The Cauchy boundary value problem for nonlinear Laplace equation with several exponential nonlinear- ities is considered and C2 smooth monotonically decreasing radial solution u(r) is found. Moreover, u(r) has logarithmic growth at ∞. Our results are applied to the differential geometry, more precisely, minimal non-superconformal degenerate two dimensional surfaces are constructed in R4 and their Gaussian, respectively normal curvature are written into explicit form. At the end of the paper several examples of local Liouville type PDE with radial coefficients which do not have radial solutions are proposed. [ABSTRACT FROM AUTHOR]

This paper deals with three classes of nonlinear Schrödinger equations (NLSE) having general polynomial nonlinearities. We find travelling wave solutions of soliton type (smooth solitons, peakons and cuspons), periodic ones (smooth, peakons and cuspons) and kinks. In the case of some special polynomials the solutions are expressed up to the inverse map by Gauss and Appell's hypergeometric functions, by some elliptic functions of Jacobi and Weierstrass types etc. In each case the above mentioned solutions of NLSE are obtained into integral form. [ABSTRACT FROM AUTHOR]

Partial Differential Equations (PDE) and Partial Integro-Differential Equations (PIDE) play a major role in financial mathematics and provide a powerful and consistent framework for pricing complex financial derivatives. However, since closed-form solutions are not available in general, numerical methods are often used. The article considers financial models based on exponential Lévy processes, the PIDE derived from the models and different computational methods for solving these PIDE. A proposal for using Cellular Neural Networks (CNN) for solving PIDE is presented. [ABSTRACT FROM AUTHOR]

We discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations. We show that a particular case of SEsM can be used in order to reproduce the methodology of the Inverse Scattering Transform Method for the case of the Burgers equation and Korteweg - de Vries equation. This particular case is connected to use of specific case of Step. 2 of SEsM: reprentation of the solution of the solved nonlinear partial differential equation as expansion as power series containing powers of a "small" parameter є, solving the differential equations arrising from this representation by means of Fourier series and transition from the obtained solution for small values of є to solution for arbitrary finite values of є. [ABSTRACT FROM AUTHOR]

In the paper under consideration we study several model examples of degenerate linear hyperbolic operators - mainly of second order, which are not (micro) locally solvable and do not propagate singularities, i.e. possesses solutions with (micro) locally smooth right-hand sides and stationary singularities on characteristic rays of the principal symbols. These two effects contrast with the properties of the strictly hyperbolic operators local solvability (including of the Cauchy problem) and propagation of singularities. As our paper is a survey on some results of the general theory of ψdo with real values principal symbols we have illustrated it by examples, not by long and complicated theorems that can be found in the corresponding monographs and papers. In some cases of degenerate hyperbolic operators propagation of singularities including branching ones is possible and we discuss it at the end. [ABSTRACT FROM AUTHOR]

In this study we extend a basic model, proposed in [1], which describes dynamical interactions between decision–making components in the public sector and the private sector. We add a new ordinary differential equation to the basic system, which describes dynamics of non-governmental organizations (NGOs) sector and we choose investments in the three sectors to be the common economic variable indicator for the extended system. We consider a particular case of reduced number of system parameters and analize dynamical properties of the equilibrium points of the three–dimensional system for selected values of its parameters. On the basis of this analysis we identify two equilibrium points of the system which evolution corresponds to development of chaos of Shilnikov kind [2]. We find that a such chaos in the three–sector interaction system occurs only for the case, in which the private sector investments dominates on the progress of the public and NGOs investments. We illustrate numerically the development of a chaotic system attractor with increasing the value of an appropriate control parameter. [ABSTRACT FROM AUTHOR]

A credit approval system requires deciding approval or rejection the application for supply credit cards based on some personal data of the applicant. In this paper, based on a data sample of 690 applications from the past for credit card requests, we build a predictor model by using a Deep Learning Toolbox 14.0 in Matlab. In the model's description, detailed analysis, and correction of the input data was carried out, the model was implemented taking into account the structure, training, and testing of the deep neural network. Produced binary classifier could be used for many tasks needful to divide data in two classes by some criteria. Neural networks with different numbers of hidden layers are tested to find the best model for the prediction of credit card approval. [ABSTRACT FROM AUTHOR]

Shakes (mainly star shakes and heart shakes) appearing at wood trunk cut are investigated and simulated analytically. Homogeneous, anisotropic, and orthotropic characters are assumed with a limited boundary condition. Some environmental data are attached. Wood dimensions are supplied if possible. Mechanical properties are mainly referred from published database (handbooks, journals, books). Experimental and analytical agreement is found to be good. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic behavior of the Hausdorff distance between Heaviside function and the Epsilon and Omega distribution functions, introduced in [24], [28]. Investigations of "saturation to the horizontal asymptote a = 1, in the Hausdorff sense" is important for professionals working in the field of financial mathematics. We prove precise bound d = ln m m + O (1 m) . One of the most significant goals of any insurance risk activity is to achieve a satisfactory model for the probability distribution of the total claim amount. The estimates for the value of the best Hausdorff approximation of Heaviside function by fixed cumulative distribution function give more insight on the parameters in the strategy "Insurance responsibility". The results are relevant for applied insurance mathematics. Numerical examples using CAS Mathematica, illustrating our results are given. [ABSTRACT FROM AUTHOR]

A model of flow of a substance in a part of a network that consists of a channel and a branch of this channel is discussed. In general this model can be based on differential or difference equations. In this text a difference equations model is considered and studied and it describes how amounts of a substance are distributed along the cells that form the studied part of the network. For the stationary case of motion of the substance a class of long-tail probability distributions is obtained. This class contains famous discrete probability distributions as particular cases. The discussed research has many applications. Several notes about the case of application of the model to channels of human migration are presented. [ABSTRACT FROM AUTHOR]

One of the main properties of solutions of nonlinear neural networks is stability and often the direct Lyapunov method is used to study stability properties. Often the present state depends on its maximum value over a past time interval. It requires the study of a new model, so called model with supremum. Some sufficient conditions for stability of equilibrium of nonlinear neural networks with supremum, time varying self-regulating parameters of all units and time varying functions of the connection between two neurons in the network are obtained. The cases of time varying Lipschitz coefficients as well as non-Lipschitz activation functions are studied. [ABSTRACT FROM AUTHOR]

In this paper we shall study a system of integro-differential equations which arises in nanostructures. We shall approximate it with the Cellular Nanoscale Networks (CNN) architecture. Then we shall study the dynamics of the obtained integro- differential CNN model via the local activity method. We shall determine the region called edge of chaos in which complex behavior can occur. We shall design hybrid control regulator for the CNN model under consideration in order to stabilize the chaotic motion to admissible solutions which is connected in some way to the original behavior of the system. Numerical simulations will be presented which will illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

We study the existence of a least energy solution of the Beris - Edward model for liquid crystals, following the approach of Brezis - Lieb and assuming flow velocity equal to zero. A specific choice of the parameters in the De Gennes energy functional enables one to show that least energy solution exists and can be seen as a minmax point of the energy functional. [ABSTRACT FROM AUTHOR]

In this paper, the direct Lyapunov method is extended to fractional case of Volterra integro-Caputo fractional differential equations with delays. Two types of homogeneous scalar Voterra integro-Caputo fractional differential equations with delays are studied. Three types of fractional derivatives of Lyapunov functions known in the literature are given. Uniform stability of trivial solution of that retarded IFrDEs is investigated using the Razumikhin method and different types of fractional derivatives of Lyapunov functions. Examples to show the application of the theorems are provided. [ABSTRACT FROM AUTHOR]

In this paper, we conduct a comparative study between the homotopy perturbation method (HPM) and Adomian's decomposition method (ADM) for analytic treatment of nonlinear two-dimensional Volterra-Fredholm fuzzy integral equations (2D-VFFIE) and we show that the HPM with a specific convex homotopy is equivalent to the ADM for these type of equations. [ABSTRACT FROM AUTHOR]

A system of nonlinear Riemann-Liouville fractional differential equations with a constant delay is studied. The initial condition is set up similarly to the case of ordinary derivative. Sufficient conditions for finite time stability are obtained. [ABSTRACT FROM AUTHOR]

In this paper we focus our attention on the course of Mathematics in the master degree of the Electrical faculty of the Technical University of Sofia. This subject covers two topics — numerical methods and statistics. Both of these items allow different computer illustrations. We discuss the possibility to change the order of the different topics in the laboratory classes only in order to learn successfully a new programming language. This is very important for the distance education as well. [ABSTRACT FROM AUTHOR]

In this paper a finite difference scheme for the one-dimensional double dispersion equation is constructed and studied. The scheme is based on the representation of this equation as a generalized Hamiltonian system. After applying the partitioned Runge-Kutta method with 3-stage Lobatto IIIA and IIIB coefficients, we get a discrete finite difference scheme with approximation error O(h2 + τ4). This scheme is symplectic, i.e. it preserves the symplectic structure of the solution on the discrete level. Numerical experiments are provided for quadratic nonlinearity. Two problems are considered: propagation of a single solitary wave and interaction of two waves traveling toward each other. The numerical results show the convergence of the discrete solution to the exact one with O(h2 + τ4) error. [ABSTRACT FROM AUTHOR]

We discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations. We show that the Hirota method is particular case of SEsM for specific form of the function from the Step. 2 of SEsM and for simple equations from the kind of differential equation for exponential function. We illustrate the methodology by obtaining the three-soliton solution of the Korteweg - de Vries equation, two soliton solution of the nonlinear Schrödinger equation, and the soliton solution of the Ishimori equation for the spin dynamics of ferromagnetic materials. [ABSTRACT FROM AUTHOR]

Vision recognition can be achieved by detecting objects in individual frames and then linking detections across frames. Such an approach can be made very robust to the occasional detection failure: If an object is not detected in a frame but is in previous and following ones, a correct trajectory will nevertheless be produced. By contrast, a false-positive detection in a few frames will be ignored. However, when dealing with a multiple target problem, the linking step results in a difficult optimization problem in the space of all possible families of trajectories. This is usually dealt with by sampling or greedy search based on variants of Dynamic Programming, which can easily miss the global optimum. Here is shown that reformulating that step as a constrained flow optimization results in a convex problem. This particular structure allows to solve it using the k-shortest paths algorithm, which is very fast. This new approach is far simpler formally and algorithmically than existing techniques and demonstrates excellent performance in two very different contexts. [ABSTRACT FROM AUTHOR]

A concurrent-read exclusive-write PRAM algorithm is developed to find the k shortest paths between pairs of vertices in an edge-weighted directed graph. Repetitions of vertices along the paths are allowed. The algorithm computes an implicit representation of the k shortest paths to a given destination vertex from every vertex of a graph with n vertices and m edges in O(m+nklog2k) work and O(log3klog*k+logn(loglogk+log*n)) time, assuming that a shortest path tree rooted at the destination is pre-computed. The paths themselves can be extracted from the implicit representation. Applications to dynamic programming problems including the knapsack problem, sequence alignment, maximum inscribed polygons are described. [ABSTRACT FROM AUTHOR]

We present the detailed dynamical model of a 3D printed walking robot with a minimal number of degrees of freedom (DOF). The robot has only two DOF, but despite that, it is capable of moving forward (or backwards) by walking, rotating to an arbitrary angle, going around obstacles, and even climbing stairs, in accordance with its size. Following a tendency of an increased popularity of robots in the education process, the prototype finds application in specialized educational methods for work with children with autism or development problems. [ABSTRACT FROM AUTHOR]

Dynamic fracture behavior of two nano-cracks in a functionally graded elastic isotropic half-plane under excitation of time-harmonic P-wave is studied. The applied approach is based on the non-hypersingular traction based boundary integral equation method for the graded bulk elastic isotropic solid extended with the non-classical boundary conditions and the localized constitutive law for the matrix-nano-crack interface within the framework of the Gurtin-Murdoch theory. The formulation allows for a quadratic variation of the material properties in depth. The boundary integral equation is treated by using the frequency dependent half-plane Green's function derived analytically by application of an appropriate functional transform, accompanied by a follow-up Fourier transform. The numerical solution provides stress-strain state at any point in the semi-infinite graded solid as well as the displacements from which the stress concentrations are determined. Simulations for two nano-cracks parallel to the free surface of the half-plane are presented and discussed. They reveal the sensitivity of the scattered wave field and dynamic stress concentration factors to the complex influence of nano-cracks interaction phenomena, existence of material gradient and surface elasticity effects. The verification and parametric study illustrate convincingly the applicability of the proposed mechanical model in material science of functional graded composites and in computational dynamic fracture mechanics. [ABSTRACT FROM AUTHOR]

The Chain Ladder Method (CLM) is a widely used method from the insurance companies in order to predict different types of future payments. These important calculations are done a several times in the year because they approximately predict how much will have to be paid for some claims in the different lines of business that are yet to be reported. This paper, in their first part, presents the use of the method to predict the sum of Incurred but not reported type of claims, also known as IBNR and Reported but not settled claims, also known as RBNS. In the second part CLM has been applied to predict reserves only for IBNR. This approach allows to occur in the future and how much money should be booked by the company into a reserve for this kind of future claims. Some numerical experiments have been presented. [ABSTRACT FROM AUTHOR]

The paper proposes the use of a simulation based on the Monte Carlo method to determine the waiting time of a passenger vehicles in urban passenger transport. The method is based on real data on the operation of urban passenger transport in the conditions of Ruse, Bulgaria. The given test results were obtained with the software product Matlab R2017b. Adjustments to the vehicle schedules may be made on the basis of the results obtained from the test. These corrections lead to the development of more accurate timetables, which has a positive impact on the quality of the transport service. [ABSTRACT FROM AUTHOR]

We present new extended Strichartz estimates for the solutions to the heat equation with inhomogeneous nonlinearity in the mass-energy intercritical framework in space dimension d ≥ 1. As an application we show local and global well-posedness in the Strichartz space Lq((0,T); Lr(ℝd)), both in the focusing and the defocusing case, assuming the initial data are in the Sobolev space Hσ(ℝd)), with σ ∈ (0, 1]. [ABSTRACT FROM AUTHOR]

In this study we discuss existence of solitary wave solutions of a famous higher-order model evolution equation arising from the water wave theory. This evolution equation describes several nonlinear natural phenomena such as propagation of long waves in shallow water over a flat surface or propagation of long gravity-capillary waves. We obtain several exact analytical solutions of this equation by applying a particular case of the Simplest Equations Method (SEsM).Numerical simulations of the obtained solutions include various kinds of solitary waves depending on a key model parameter. [ABSTRACT FROM AUTHOR]

In [1] the authors considered the following new two–parameter power Garima (PG) c.d.f.: M (t) = 1 − θ t α + θ + 2 θ + 2 e − θ t α , where α, θ ∈ R+, t > 0. Also of interest to the specialists is the task of approximating the Heaviside function ht0 (t) where t0 is the median with the new cumulative function in the Hausdorff sense. We define a new family of recurrence generated transmuted power Garima (TPG) c.d.f. M i + 1 (t) = M i (t) (μ i + 1 + 1 − μ i + 1 M i (t)) , i = 0 , 1 , 2 , ... ,. Some properties and applications to the specific data are given. Numerical examples, illustrating our results are presented using programming environment CAS Mathematica. [ABSTRACT FROM AUTHOR]

In [1] H. Bakouch consider a G–family of extended cumulative distribution function (cdf): F a (t) = a G (t) a + 1 − G (t) ; a > 0 , where G(t) is the baseline cdf. In particular case G(t) = 1 − e−kt, k > 0 we find the following Extended–Bakouch Half–Logistic cdf (EBHL-cdf): F a (t) = a (1 − e − k t ) a + e − k t . Similar to our previous studies [2]–[4], in this article we will define and analyze in detail the following new family: F a (t ; a 0 ... , a n) = a (1 − e − F (t) ) a + e − F (t) ; F (t) = ∑ i = 0 n a i t i ; a 0 = 0. We will call this family the "Extended–Bakouch Half–Logistic cdf with polynomial variable transfer" (EBHLPVT– cdf). Illustrating our results the following datasets are fitted [5] using CAS MATHEMATICA: "Data BG Total Cases COVID–19"; "Data BG Total Deaths COVID–19". [ABSTRACT FROM AUTHOR]

New Hardy type inequalities in sectorial area and as a limit in an exterior of a ball are proved. Sharpness of the inequalities is shown as well. [ABSTRACT FROM AUTHOR]

Trading with securities and stock indices gains a lot of popularity these days. Every investor's dream is to know the future prices of financial instruments. This paper attempts to apply a Markov chain model to forecast the trends in the stock prices. Markov chain models are obtained for the prices of 3 different stocks based on probability transition matrix and initial state vector. Quantitative data on the daily closing share prices of the stocks is obtained for the period 01.01.2019-31.12.2019. The analyzed stocks are further mixed up in an optimal risk portfolio. An analysis of risk aversion coefficient and how it influences the choice of complete portfolio is made. Such an approach could be applied in future studies on this matter as well as in the practice of portfolio managers and investors. [ABSTRACT FROM AUTHOR]

In this study, we consider the d-dimensional polyharmonic matrix operator H (l , V) u = (− Δ) l u + V (x) u , where (−Δ)l is a diagonal s×s matrix, whose diagonal elements are the scalar polyharmonic operators, V is the operator of multiplication by a symmetric s×s matrix, V(x) is periodic with respect to an arbitrary lattice and s ≥ 2, x = (x 1 , x 2 , ⋯ , x d) ∈ ℝ d , d ≥ 2 , 1 2 < l < 1. We obtain the high energy asymptotics for the eigenvalues of this operator which lies near the diffraction planes. [ABSTRACT FROM AUTHOR]