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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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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 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]

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]

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]

We discuss a model of flow of a substance in a channel of a network and the interaction of this substance with other kinds of substances in the nodes of the channel. The model of interaction can be based on differential or difference equations and different kinds of possible dynamics in the corresponding phase spaces are illustrated. [ABSTRACT FROM AUTHOR]

In this study we extend the generalized reaction-diffiusion model presented in [20],[21], which describes spatio-temporal dynamics of interacting populations. In more detail, we generalize the model system of differential equations for the interaction of three populations in which the growth rates and competition coefficients of populations depend on the number of members of all populations. The model describes several novel features of the interacting agents compared to the well-known classic models in population dynamics. Using particular case of the recently developed SEsM (Simple Equations Method) namely the Modified method of Simplest Equation [5]–[8] and one of its extended versions [8, 9], we obtain a new traveling wave solution of the model system. We assume that nonlinearity in growth rates and interaction coefficients in the generalized model exist according to high density of their individuals. An analytical solution of the extended model is derived. Traveling wave solutions of these equations are of special interest as they describe the motion of wave fronts or the motion of boundary between two different states existing in this system. Numerical simulations of this solution demonstrate propagation of nonlinear waves in the considered extended model. The characteristics of the the obtained traveling wave solution are visualized and discussed. [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]

Lipschitz stability for nonlinear differential equations with non-instantaneous impulses and supremum of the unknown function over a past time interval is studied. The impulses start abruptly at some points and their action continue on given finite intervals. Some sufficient conditions for Lipschitz stability are obtained. Modified Razumikhin method with piecewise continuous functions and comparison results with scalar non-instantaneous equations are applied. [ABSTRACT FROM AUTHOR]

In this paper, we propose the solution of partial Volterra fuzzy integro-differential equation (PVFIDE) with convolution type kernel using two-dimensional fuzzy Sumudu transform (2D-FST) method under Hukuhara differentiability. It is shown that 2D-FST is a simple and reliable approach for solving such equations analytically. Finally, the method is illustrated with example to show the ability of the proposed method. [ABSTRACT FROM AUTHOR]

This article aims to model and study the effect of the strength, time and duration of the restrictive measures for the spread of an infectious disease. The inconveniences, economic losses and gaps in education are the price that society pays to prevent the spread of the virus. It is important that restrictive measures cover the shortest possible time interval, at the most appropriate time, in order to have minimal negative consequences for society, and at the same time to be effective against the spread of the virus. We consider as a basis the SIS compartmental model for the spread of a virus and apply numerical experiments assuming that, unlike the classic model, the transmission rate α is a monotonically decreasing function of time. Numerical experiments show that earlier introduction, greater stringency and a shorter period of adaptation to restrictive measures until they enter into force would lead to a smaller proportion of infected people, a shorter period of implementation of measures and small economic losses. [ABSTRACT FROM AUTHOR]

Caputo fractional differential equations of order q ∈ (1, 2) with impulses at random moments are set up and investigated in this paper. The main characteristic of the studied equations is that the impulses occur at random moments which are exponentially distributed random variables. It leads to a total change of the behavior of the solution which is a stochastic process. Also, we assume at the moments of impulses the lower limit of Caputo fractional derivative is changed. The initial value problem is set up in appropriate way. Sample path solutions are defined as a solutions of Caputo fractional differential equations with determined fixed moments of impulses. P-moment exponential stability is defined and some sufficient conditions for this type of stability are obtained. The study is based on the application of Lyapunov functions and the appropriate its Caputo fractional derivative among the studied fractional equation. [ABSTRACT FROM AUTHOR]

In this paper we study nonlocal semilinear fractional evolution inclusions involving Riemann-Liouville derivative with right-hand side depending on casual operator. The existence of solutions is proved under compactness type conditions on the multivalued term. [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]

For a modified Elman neural network with a periodic input, sufficient conditions for the existence of a periodic output are found by using Mawhin's continuation theorem of coincidence degree theory. An example is given. [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]

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]

We discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations. We show that the much used Homogeneous Balance Method, Extended Homogeneous Balance method and Auxiliary Equation Method are particular cases of SEsM. [ABSTRACT FROM AUTHOR]

We study the existence of ground states for the Beris-Edward system. Under reasonable symmetry assumption we prove the existence, regularity and decay property of the ground states. Using a linearization around these ground state solutions we study the corresponding Cauchy problem with small initial data. Some dispersive estimates for the perturbed linearized problem are established also. [ABSTRACT FROM AUTHOR]

In this paper Harmonic balance technique (HBT) is presented for studying Cellular Nonlinear Network (CNN) model of Meinhardt-Gierer equation (CNN-MG). First, short survey on HBT and CNN is given. Then we prove existence of periodic solutions of CNN-MG model. Computer simulations illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we study travelling wave solutions of the viscoelastic Burgers’ equation. RTD-based Cellular Neural Networks (CNN) model of this equation is presented. Using such realization new wave profiles of the travelling wave solutions are obtained. [ABSTRACT FROM AUTHOR]

In this paper the principle of local activity will be presented for studying complex behavior of different reaction-diffusion CNN models. In our models each cell will be arranged on a two-dimensional square grid and will be connected to adjacent cells through coupling devices that mimic 2-D spatial diffusion and transmit the cell’s state to its neighboring cells, as in conventional CNN. We shall find the edge of chaos regime in Brussaltor CNN model. [ABSTRACT FROM AUTHOR]

They represented most of the strongest research groups in the fields of differential equations and applied mathematics. The conference promoted basic research in mathematics leading to new methods and techniques useful for applications of differential equations. Seventh International conference on New Trends in the Applications of Differential Equations in Sciences (NTADES 2020) hold from 1 st to 4 th September, 2020 in St. Constantin and Helena, Bulgaria. [Extracted from the article]