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The purpose of this paper is to introduce the split combination of variational inclusion problem which combines the concept of the modified variational inclusion problem introduced by Khuangsatung and Kangtunyakarn and the split variational inclusion problem introduced by Moudafi. Using a modified Halpern iterative method, we prove the strong convergence theorem for finding a common solution for the hierarchical fixed point problem and the split combination of variational inclusion problem. The result presented in this paper demonstrates the corresponding result for the split zero point problem and the split combination of variation inequality problem. Moreover, we discuss a numerical example for supporting our result and the numerical example shows that our result is not true if some conditions fail.

A mathematical chaos model for the dynamical behaviors of atmospheric acoustic-gravity waves is considered in this paper. Boundedness and globally attractive sets of this chaos model are studied by means of the generalized Lyapunov function method. The innovation of this paper is that it not only proves this system is globally bounded but also provides a series of global attraction sets of this system. The rate of trajectories entering from the exterior of the trapping domain to its interior is also obtained. Finally, the detailed numerical simulations are carried out to justify theoretical results. The results in this study can be used to study chaos control and chaos synchronization of this chaos system.

In this paper, several relations are obtained among the Riemann zeta and Hurwitz zeta functions, as well as their products. A particular case of these relations give rise to a simple re-derivation of the important results of Katsurada and Matsumoto on the mean square of the Hurwitz zeta function. Also, a relation derived here provides the starting point of a novel approach which, in a series of companion papers, yields a formal proof of the Lindel&ouml, f hypothesis. Some of the above relations motivate the need for analysing the large &alpha, behaviour of the modified Hurwitz zeta function &zeta, 1 ( s , &alpha, ) , s &isin, C , &alpha, &isin, ( 0 , &infin, ) , which is also presented here.

The paper shows that, in looking for exact solutions to nonlinear PDEs, the direct method of functional separation of variables can, in certain cases, be more effective than the method of differential constraints based on the compatibility analysis of PDEs with a single constraint (or the nonclassical method of symmetry reductions based on an invariant surface condition). This fact is illustrated by examples of nonlinear reaction-diffusion and convection-diffusion equations with variable coefficients, and nonlinear Klein&ndash, Gordon-type equations. Hydrodynamic boundary layer equations, nonlinear Schr&ouml, dinger type equations, and a few third-order PDEs are also investigated. Several new exact functional separable solutions are given. A possibility of increasing the efficiency of the Clarkson&ndash, Kruskal direct method is discussed. A generalization of the direct method of the functional separation of variables is also described. Note that all nonlinear PDEs considered in the paper include one or several arbitrary functions.

In this paper the notion of ∗ -Weyl curvature tensor on real hypersurfaces in non-flat complex space forms is introduced. It is related to the ∗ -Ricci tensor of a real hypersurface. The aim of this paper is to provide two classification theorems concerning real hypersurfaces in non-flat complex space forms in terms of ∗ -Weyl curvature tensor. More precisely, Hopf hypersurfaces of dimension greater or equal to three in non-flat complex space forms with vanishing ∗ -Weyl curvature tensor are classified. Next, all three dimensional real hypersurfaces in non-flat complex space forms, whose ∗ -Weyl curvature tensor vanishes identically are classified. The used methods are based on tools from differential geometry and solving systems of differential equations.

This paper focuses on the problem of finite-time stabilization of homogeneous, non-Lipschitz systems with dilations. A key contribution of this paper is the design of a virtual recursive Hölder, non-Lipschitz state feedback which renders the non-Lipschitz systems in the special case dominated by a lower-triangular nonlinear system, finite-time stable. The proof is based on a recursive design algorithm developed recently, to construct the virtual Hölder continuous, finite-time stabilizer as well as a C1 positive definite and proper Lyapunov function that guarantees finite-time stability of the non-Lipschitz non-linear systems.

Using some of the extended fixed point results for Geraghty contractions in b-metric spaces given by Faraji, Savić and Radenović and their idea to apply these results to nonlinear integral equations, in this paper we present some existence and uniqueness conditions for the solution of a nonlinear Fredholm&ndash, Volterra integral equation with a modified argument.

The object of this paper is to derive a double integral in terms of the Hurwitz–Lerch zeta function. Almost all Hurwitz–Lerch zeta functions have an asymmetrical zero-distribution. Special cases are evaluated in terms of fundamental constants. All the results in this work are new.

In this paper, an asymmetric regression model for censored non-negative data based on the centred exponentiated log-skew-normal and Bernoulli distributions mixture is introduced. To connect the discrete part with the continuous distribution, the logit link function is used. The parameters of the model are estimated by using the likelihood maximum method. The score function and the information matrix are shown in detail. Antibody data from a study of the measles vaccine are used to illustrate applicability of the proposed model, and it was found the best fit to the data with respect to an others models used in the literature.

Recently in various theoretical works, path-breaking progress has been made in recovering the well-known Page Curve of an evaporating black hole with Quantum Extremal Islands, proposed to solve the long-standing black hole information loss problem related to the unitarity issue. Motivated by this concept, in this paper, we study cosmological circuit complexity in the presence (or absence) of Quantum Extremal Islands in the negative (or positive) Cosmological Constant with radiation in the background of Friedmann-Lema$\hat{i}$tre-Robertson-Walker (FLRW) space-time i.e the presence and absence of islands in anti-de Sitter and the de Sitter spacetime having SO(2, 3) and SO(1, 4) isometries respectively. Without using any explicit details of any gravity model, we study the behaviour of the circuit complexity function with respect to the dynamical cosmological solution for the scale factors for the above-mentioned two situations in FLRW space-time using squeezed state formalism. By studying the cosmological circuit complexity, Out-of-Time Ordered Correlators, and entanglement entropy of the modes of the squeezed state, in different parameter spaces, we conclude the non-universality of these measures. Their remarkably different features in the different parameter spaces suggest their dependence on the parameters of the model under consideration., Comment: 75 pages, 29 figures, 4 tables, Dr. Sayantan Choudhury would like to dedicate this work to his lovable father and prime inspiration Professor Manoranjan Choudhury who recently have passed away due to COVID 19. Updated and revised version, Accepted for publication in Symmetry (section: Physics and Symmetry/Asymmetry, Special issue: Manifest and Hidden Symmetries in Field and String Theories)

In this paper we consider systems of three autonomous first-order differential equations x˙=f(x),x=(x,y,z),f=(f1,f2,f3) such that x(t)+y(t)+z(t) is constant for all t. We present some Hamilton–Poisson formulations and integrable deformations. We also analyze the case of Kolmogorov systems. We study from some standard and nonstandard Poisson geometry points of view the three-dimensional Lotka–Volterra system with constant population.

The aim of this paper is to find the Noether symmetries of a generalized Benney-Luke equation. Thereafter, we construct the associated conserved vectors. In addition, we search for exact solutions for the generalized Benney-Luke equation through the extended tanh method. A brief observation on equations arising from a Lagrangian density function with high order derivatives of the field variables, is also discussed.

In this paper, we consider the inverse scattering problem and, in particular, the problem of reconstructing the spectral density associated with the Yukawian potentials from the sequence of the partial-waves fℓ of the Fourier–Legendre expansion of the scattering amplitude. We prove that if the partial-waves fℓ satisfy a suitable Hausdorff-type condition, then they can be uniquely interpolated by a function f˜(λ)∈C, analytic in a half-plane. Assuming also the Martin condition to hold, we can prove that the Fourier–Legendre expansion of the scattering amplitude converges uniformly to a function f(θ)∈C (θ being the complexified scattering angle), which is analytic in a strip contained in the θ-plane. This result is obtained mainly through geometrical methods by replacing the analysis on the complex cosθ-plane with the analysis on a suitable complex hyperboloid. The double analytic symmetry of the scattering amplitude is therefore made manifest by its analyticity properties in the λ- and θ-planes. The function f(θ) is shown to have a holomorphic extension to a cut-domain, and from the discontinuity across the cuts we can iteratively reconstruct the spectral density σ(μ) associated with the class of Yukawian potentials. A reconstruction algorithm which makes use of Pollaczeck and Laguerre polynomials is finally given.

First-order methods such as stochastic gradient descent (SGD) have recently become popular optimization methods to train deep neural networks (DNNs) for good generalization, however, they need a long training time. Second-order methods which can lower the training time are scarcely used on account of their overpriced computing cost to obtain the second-order information. Thus, many works have approximated the Hessian matrix to cut the cost of computing while the approximate Hessian matrix has large deviation. In this paper, we explore the convexity of the Hessian matrix of partial parameters and propose the damped Newton stochastic gradient descent (DN-SGD) method and stochastic gradient descent damped Newton (SGD-DN) method to train DNNs for regression problems with mean square error (MSE) and classification problems with cross-entropy loss (CEL). In contrast to other second-order methods for estimating the Hessian matrix of all parameters, our methods only accurately compute a small part of the parameters, which greatly reduces the computational cost and makes the convergence of the learning process much faster and more accurate than SGD and Adagrad. Several numerical experiments on real datasets were performed to verify the effectiveness of our methods for regression and classification problems.

The paper presents an analytical review of theoretical methods for modeling functional nanostructures. The main evolutionary changes in the approaches of quantum-mechanical modeling are described. The foundations of the first-principal theory are considered, including the stationery and time-dependent Schrödinger equations, wave functions, the form of writing energy operators, and the principles of solving equations. The idea and specifics of describing the motion and interaction of nuclei and electrons in the framework of the theory of the electron density functional are presented. Common approximations and approaches in the methods of quantum mechanics are presented, including the Born–Oppenheimer approximation, the Hartree–Fock approximation, the Thomas–Fermi theory, the Hohenberg–Kohn theorems, and the Kohn–Sham formalism. Various options for describing the exchange–correlation energy in the theory of the electron density functional are considered, such as the local density approximation, generalized and meta-generalized gradient approximations, and hybridization of the generalized gradient method. The development of methods of quantum mechanics to quantum molecular dynamics or the dynamics of Car–Parrinello is shown. The basic idea of combining classical molecular modeling with calculations of the electronic structure, which is reflected in the potentials of the embedded atom, is described.

In this paper, new forms of Maxwell’s equations in vector and scalar variants are presented. The new forms are based on the use of Gauss’s theorem for magnetic induction and electrical induction. The equations are formulated in both differential and integral forms. In particular, the new forms of the equations relate to the non-stationary expressions and their integral identities. The indicated methodology enables a thorough analysis of non-stationary boundary conditions on the behavior of electromagnetic fields in multiple continuous regions. It can be used both for qualitative analysis and in numerical methods (control volume method) and optimization. The last Section introduces an application to equations of magnetic fluid in both differential and integral forms.

The main aim of this paper is to numerically solve the first kind linear Fredholm and Volterra integral equations by using Modified Bernstein–Kantorovich operators. The unknown function in the first kind integral equation is approximated by using the Modified Bernstein–Kantorovich operators. Hence, by using discretization, the obtained linear equations are transformed into systems of algebraic linear equations. Due to the sensitivity of the solutions on the input data, significant difficulties may be encountered, leading to instabilities in the results during actualization. Consequently, to improve on the stability of the solutions which imply the accuracy of the desired results, regularization features are built into the proposed numerical approach. More stable approximations to the solutions of the Fredholm and Volterra integral equations are obtained especially when high order approximations are used by the Modified Bernstein–Kantorovich operators. Test problems are constructed to show the computational efficiency, applicability and the accuracy of the method. Furthermore, the method is also applied to second kind Volterra integral equations.

In this paper, we investigate a class of Schrödinger–Poisson systems with critical growth. By the principle of concentration compactness and variational methods, we prove that the system has radially symmetric solutions, which improve the related results on this topic.

In this paper, we study a monotone inclusion problem in the framework of Hilbert spaces. (1) We introduce a new modified Tseng’s method that combines inertial and viscosity techniques. Our aim is to obtain an algorithm with better performance that can be applied to a broader class of mappings. (2) We prove a strong convergence theorem to approximate a solution to the monotone inclusion problem under some mild conditions. (3) We present a modified version of the proposed iterative scheme for solving convex minimization problems. (4) We present numerical examples that satisfy the image restoration problem and illustrate our proposed algorithm’s computational performance.

In this paper, a class of systems of linear and non-linear delay differential equations (DDEs) of first order with time-varying delay is considered. We obtain new sufficient conditions for uniform asymptotic stability of zero solution, integrability of solutions of an unperturbed system and boundedness of solutions of a perturbed system. We construct two appropriate Lyapunov–Krasovskiĭ functionals (LKFs) as the main tools in proofs. The technique of the proofs depends upon the Lyapunov–Krasovskiĭ method. For illustration, two examples are provided in particular cases. An advantage of the new LKFs used here is that they allow to eliminate using Gronwall’s inequality. When we compare our results with recent results in the literature, the established conditions are more general, less restrictive and optimal for applications.

In this paper a system of nonlinear Riemann–Liouville fractional differential equations with non-instantaneous impulses is studied. We consider a Riemann–Liouville fractional derivative with a changeable lower limit at each stop point of the action of the impulses. In this case the solution has a singularity at the initial time and any stop time point of the impulses. This leads to an appropriate definition of both the initial condition and the non-instantaneous impulsive conditions. A generalization of the classical Lipschitz stability is defined and studied for the given system. Two types of derivatives of the applied Lyapunov functions among the Riemann–Liouville fractional differential equations with non-instantaneous impulses are applied. Several sufficient conditions for the defined stability are obtained. Some comparison results are obtained. Several examples illustrate the theoretical results.

A computational framework for the construction of solutions to linear homogenous partial differential equations (PDEs) with variable coefficients is developed in this paper. The considered class of PDEs reads: ∂p∂t−∑j=0m∑r=0njajrtxr∂jp∂xj=0 F-operators are introduced and used to transform the original PDE into the image PDE. Factorization of the solution into rational and exponential parts enables us to construct analytic solutions without direct integrations. A number of computational examples are used to demonstrate the efficiency of the proposed scheme.

In this paper we first introduce multilinear fractional wavelet transform on Rn×R+n using Schwartz functions, i.e., infinitely differentiable complex-valued functions, rapidly decreasing at infinity. We also give multilinear fractional Fourier transform and prove the Hausdorff–Young inequality and Paley-type inequality. We then study boundedness of the multilinear fractional wavelet transform on Lebesgue spaces and Lorentz spaces.

This paper describes the stability and Hopf bifurcation analysis of the Brusselator system with delayed feedback control in the single domain of a reaction–diffusion cell. The Galerkin analytical technique is used to present a system equation composed of ordinary differential equations. The condition able to determine the Hopf bifurcation point is found. Full maps of the Hopf bifurcation regions for the interacting chemical species are shown and discussed, indicating that the time delay, feedback control, and diffusion parameters can play a significant and important role in the stability dynamics of the two concentration reactants in the system. As a result, these parameters can be changed to destabilize the model. The results show that the Hopf bifurcation points for chemical control increase as the feedback parameters increase, whereas the Hopf bifurcation points decrease when the diffusion parameters increase. Bifurcation diagrams with examples of periodic oscillation and phase-plane maps are provided to confirm all the outcomes calculated in the model. The benefits and accuracy of this work show that there is excellent agreement between the analytical results and numerical simulation scheme for all the figures and examples that are illustrated.

In applications, many states given for a system can be expressed by orthonormal elements, called “state elements,” taken in a separable Hilbert space (called “state space”). The exact nature of the Hilbert space depends on the system, for example, the state space for position and momentum states is the space of square-integrable functions. The symmetries of a quantum system can be represented by a class of unitary operators that act in the Hilbert space. The operators called ladder operators have the effect of lowering or raising the energy of the state. In this paper, we study the spectral properties of a self-adjoint, fourth-order differential operator with a bounded operator coefficient and establish a second regularized trace formula for this operator.

In this paper, a symmetric eight-step predictor method (explicit) of 10th order is presented for the numerical integration of IVPs of second-order ordinary differential equations. This scheme has variable coefficients and can be used as a predictor stage for other implicit schemes. First, we showed the singular P-stability property of the new method, both algebraically and by plotting the stability region. Then, having applied it to well-known problems like Mathieu equation, we showed the advantage of the proposed method in terms of efficiency and consistency over other methods with the same order.

In this paper, we mainly investigate the weak convergence analysis about the error terms which are determined by the discretization for solving the stochastic differential equation (SDE, for short) in forward-backward stochastic differential equations (FBSDEs, for short), which is on the basis of Itô Taylor expansion, the numerical SDE theory, and numerical FBSDEs theory. Under the weak convergence analysis of FBSDEs, we further establish better error estimates of recent numerical schemes for solving FBSDEs.

In this paper, we discuss the approximate controllability for a class of retarded semilinear neutral control systems of fractional order by investigating the relations between the reachable set of the semilinear retarded neutral system of fractional order and that of its corresponding linear system. The research direction used here is to find the conditions for nonlinear terms so that controllability is maintained even in perturbations. Finally, we will show a simple example to which the main result can be applied.

In previous work, we considered a four-quadrant Riemann problem for a 2×2 hyperbolic system in which delta shock appears at the initial discontinuity without assuming that each jump of the initial data projects exactly one plane elementary wave. In this paper, we consider the case that does not involve a delta shock at the initial discontinuity. We classified 18 topologically distinct solutions and constructed analytic and numerical solutions for each case. The constructed analytic solutions show the rich structure of wave interactions in the Riemann problem, which coincide with the computed numerical solutions.

Discrete Gaussian sampling is one of the fundamental mathematical tools for lattice-based cryptography. In this paper, we revisit the Bernoulli(-type) sampling for centered discrete Gaussian distributions over the integers, which was proposed by Ducas et al. in 2013. Combining the idea of Karney’s algorithm for sampling from the Bernoulli distribution Be−1/2, we present an improved Bernoulli sampling algorithm. It does not require the use of floating-point arithmetic to generate a precomputed table, as the original Bernoulli sampling algorithm did. It only needs a fixed look-up table of very small size (e.g., 128 bits) that stores the binary expansion of ln2. We also propose a noncentered version of Bernoulli sampling algorithm for discrete Gaussian distributions with varying centers over the integers. It requires no floating-point arithmetic and can support centers of precision up to 52 bits. The experimental results show that our proposed algorithms have a significant improvement in the sampling efficiency as compared to other rejection algorithms.

In this paper, we are concerned with the singular elliptic problems driven by the double phase operator and and Dirichlet boundary conditions. In view of the variational approach, we establish the existence of at least one nontrivial solution and two distinct nontrivial solutions under some general assumptions on the nonlinearity f. Here we use Ricceri’s variational principle and Bonanno’s three critical points theorem in order to overcome the lack of compactness.

Fourier series is a well-established subject and widely applied in various fields. However, there is much less work on double Fourier coefficients in relation to spaces of general double sequences. We understand the space of double Fourier coefficients as an abstract space of sequences and examine relationships to spaces of general double sequences: p-power summable sequences for p = 1, 2, and the Hilbert space of double sequences. Using uniform convergence in the sense of a Cesa`ro mean, we verify the inclusion relationships between the four spaces of double sequences, they are nested as proper subsets. The completions of two spaces of them are found to be identical and equal to the largest one. We prove that the two-parameter Wiener space is isomorphic to the space of Cesa`ro means associated with double Fourier coefficients. Furthermore, we establish that the Hilbert space of double sequence is an abstract Wiener space. We think that the relationships of sequence spaces verified at an intermediate stage in this paper will provide a basis for the structures of those spaces and expect to be developed further as in the spaces of single-indexed sequences.

In this paper, we introduce two novel extragradient-like methods to solve variational inequalities in a real Hilbert space. The variational inequality problem is a general mathematical problem in the sense that it unifies several mathematical models, such as optimization problems, Nash equilibrium models, fixed point problems, and saddle point problems. The designed methods are analogous to the two-step extragradient method that is used to solve variational inequality problems in real Hilbert spaces that have been previously established. The proposed iterative methods use a specific type of step size rule based on local operator information rather than its Lipschitz constant or any other line search procedure. Under mild conditions, such as the Lipschitz continuity and monotonicity of a bi-function (including pseudo-monotonicity), strong convergence results of the described methods are established. Finally, we provide many numerical experiments to demonstrate the performance and superiority of the designed methods.

[EN] In this paper, we introduce two approaches to compute the matrix hyperbolic tangent. While one of them is based on its own definition and uses the matrix exponential, the other one is focused on the expansion of its Taylor series. For this second approximation, we analyse two different alternatives to evaluate the corresponding matrix polynomials. This resulted in three stable and accurate codes, which we implemented in MATLAB and numerically and computationally compared by means of a battery of tests composed of distinct state-of-the-art matrices. Our results show that the Taylor series-based methods were more accurate, although somewhat more computationally expensive, compared with the approach based on the exponential matrix. To avoid this drawback, we propose the use of a set of formulas that allows us to evaluate polynomials in a more efficient way compared with that of the traditional Paterson¿Stockmeyer method, thus, substantially reducing the number of matrix products (practically equal in number to the approach based on the matrix exponential), without penalising the accuracy of the result, This research was funded by the Spanish Ministerio de Ciencia e Innovacion under grant number TIN2017-89314-P.

In this paper, we will consider the problem of constructing stable maps between two closed orientable surfaces M and N with a given branch set of curves immersed on N. We will study, from a global point of view, the behavior of its families in different isotopies classes on the space of smooth maps. The main goal is to obtain different relationships between invariants. We will provide a new proof of Quine&rsquo, s Theorem.

Logit, probit and complementary log-log models are the most widely used models when binary dependent variables are available. Conventionally, these models have been frequentists. This paper aims to demonstrate how such models can be implemented relatively quickly and easily from a Bayesian framework using Gibbs sampling Markov chain Monte Carlo simulation methods in WinBUGS. We focus on the modeling and prediction of Down syndrome (DS) and Mental retardation (MR) data from an observational study at Kuwait Medical Genetic Center over a 30-year time period between 1979 and 2009. Modeling algorithms were used in two distinct ways; firstly, using three different methods at the disease level, including logistic, probit and cloglog models, and, secondly, using bivariate logistic regression to study the association between the two diseases in question. The models are compared in terms of their predictive ability via R2, adjusted R2, root mean square error (RMSE) and Bayesian Deviance Information Criterion (DIC). In the univariate analysis, the logistic model performed best, with R2 (0.1145), adjusted R2 (0.114), RMSE (0.3074) and DIC (7435.98) for DS, and R2 (0.0626), adjusted R2 (0.0621), RMSE (0.4676) and DIC (23120) for MR. In the bivariate case, results revealed that 7 and 8 out of the 10 selected covariates were significantly associated with DS and MR respectively, whilst none were associated with the interaction between the two outcomes. Bayesian methods are more flexible in handling complex non-standard models as well as they allow model fit and complexity to be assessed straightforwardly for non-nested hierarchical models.

In this paper, we derive a number of interesting results concerning subordination and superordination relations for certain analytic functions associated with an extension of the Mittag–Leffler function.

The aim of this paper is to present a new method and the tool to validate the numerical results of the Volterra integral equation with discontinuous kernels in linear and non-linear forms obtained from the Adomian decomposition method. Because of disadvantages of the traditional absolute error to show the accuracy of the mathematical methods which is based on the floating point arithmetic, we apply the stochastic arithmetic and new condition to study the efficiency of the method which is based on two successive approximations. Thus the CESTAC method (Controle et Estimation Stochastique des Arrondis de Calculs) and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library are employed. Finding the optimal iteration of the method, optimal approximation and the optimal error are some of advantages of the stochastic arithmetic, the CESTAC method and the CADNA library in comparison with the floating point arithmetic and usual packages. The theorems are proved to show the convergence analysis of the Adomian decomposition method for solving the mentioned problem. Also, the main theorem of the CESTAC method is presented which shows the equality between the number of common significant digits between exact and approximate solutions and two successive approximations.This makes in possible to apply the new termination criterion instead of absolute error. Several examples in both linear and nonlinear cases are solved and the numerical results for the stochastic arithmetic and the floating-point arithmetic are compared to demonstrate the accuracy of the novel method.

In this paper, we further study the generating function involving a variety of special numbers and ploynomials constructed by the second author. Applying the Mellin transformation to this generating function, we define a new class of zeta type functions, which is related to the interpolation functions of the Apostol&ndash, Bernoulli polynomials, the Bernoulli polynomials, and the Euler polynomials. This new class of zeta type functions is related to the Hurwitz zeta function, the alternating Hurwitz zeta function, and the Lerch zeta function. Furthermore, by using these functions, we derive some identities and combinatorial sums involving the Bernoulli numbers and polynomials and the Euler numbers and polynomials.

In this field of research, in order to solve fuzzy fractional differential equations, they are normally transformed to their corresponding crisp problems. This transformation is called the embedding method. The aim of this paper is to present a new direct method to solve the fuzzy fractional differential equations using fuzzy calculations and without this transformation. In this work, the fuzzy generalized Taylor expansion by using the sense of fuzzy Caputo fractional derivative for fuzzy-valued functions is presented. For solving fuzzy fractional differential equations, the fuzzy generalized Euler’s method is introduced and applied. In order to show the accuracy and efficiency of the presented method, the local and global truncation errors are determined. Moreover, the consistency, convergence, and stability of the generalized Euler’s method are proved in detail. Eventually, the numerical examples, especially in the switching point case, show the flexibility and the capability of the presented method The work of J.J. Nieto has been partially supported by Agencia Estatal de Investigación (AEI) of Spain under grant MTM2016-75140-P, co-financed by the European Community fund FEDER, and XUNTA de Galicia under grants GRC2015-004 and ED431C 2019/02 SI

In this paper, we introduce a general procedure to construct the Taylor series development of the inverse of an analytical function, in other words, given y=f(x), we provide the power series that defines its inverse x=hf(y). We apply the obtained results to solve nonlinear equations in an analytic way, and generalize Catalan and Fuss&ndash, Catalan numbers.

In this paper, we obtain the existence, uniqueness, and positivity of the solution to delayed stochastic differential equations with jumps. This equation is then applied to model the price movement of the risky asset in a financial market and the Black&ndash, Scholes formula for the price of European option is obtained together with the hedging portfolios. The option price is evaluated analytically at the last delayed period by using the Fourier transformation technique. However, in general, there is no analytical expression for the option price. To evaluate the price numerically, we then use the Monte-Carlo method. To this end, we need to simulate the delayed stochastic differential equations with jumps. We propose a logarithmic Euler&ndash, Maruyama scheme to approximate the equation and prove that all the approximations remain positive and the rate of convergence of the scheme is proved to be 0.5.

Let Fn be the nth Fibonacci number. The order of appearance z(n) of a natural number n is defined as the smallest positive integer k such that Fk&equiv, 0(modn). In this paper, we shall find all positive solutions of the Diophantine equation z(&phi, (n))=n, where &phi, is the Euler totient function.

Kim and Kim (Russ. J. Math. Phys. 26, 2019, 40-49) introduced polyexponential function as an inverse to the polylogarithm function and by this, constructed a new type poly-Bernoulli polynomials. Recently, by using the polyexponential function, a number of generalizations of some polynomials and numbers have been presented and investigated. Motivated by these researches, in this paper, multi-poly-Euler polynomials are considered utilizing the degenerate multiple polyexponential functions and then, their properties and relations are investigated and studied. That the type 2 degenerate multi-poly-Euler polynomials equal a linear combination of the degenerate Euler polynomials of higher order and the degenerate Stirling numbers of the first kind is proved. Moreover, an addition formula and a derivative formula are derived. Furthermore, in a special case, a correlation between the type 2 degenerate multi-poly-Euler polynomials and degenerate Whitney numbers is shown.

This paper proposes a new approach based on the regression framework employing a pivotal quantity to estimate unknown parameters of a Weibull distribution under the progressive Type-II censoring scheme, which provides a closed form solution for the shape parameter, unlike its maximum likelihood estimator counterpart. To resolve serious rounding errors for the exact mean and variance of the pivotal quantity, two different types of Taylor series expansion are applied, and the resulting performance is enhanced in terms of the mean square error and bias obtained through the Monte Carlo simulation. Finally, an actual application example, including a simple goodness-of-fit analysis of the actual test data based on the pivotal quantity, proves the feasibility and applicability of the proposed approach.

The purpose of this paper is to propose an iterative algorithm for solving the split equality common null point problem (SECNP), which is to find an element of the set of common zero points for a finite family of maximal monotone operators in Hilbert spaces. We introduce the concept of bounded linear regularity for the SECNP and construct several sufficient conditions to ensure the linear convergence of the algorithm. Moreover, some numerical experiments are given to test the validity of our results.

In this paper, a discrete-time model for Babesiosis disease, given by means of nonstandard finite difference (NSFD) schemes, is first provided and analyzed. Mathematical analyses show that the provided NSFD schemes preserve the essential (qualitative) dynamical properties of the continuous-time model, namely, positivity and boundedness of the solutions, equilibria, and their stability properties. In particular, the global stability of the disease free equilibrium point is proved by using an appropriate Lyapunov function. As a relevant consequence, we get the dynamic consistency of NSFD schemes in relation to the continuous-time model. Numerical simulations are presented to support the validity of the established theoretical results.

In actuarial mathematics, the claims of an insurance portfolio are often modeled using the collective risk model, which consists of a random number of claims of independent, identically distributed (i.i.d.) random variables (r.v.s) that represent cost per claim. To facilitate computations, there is a classical assumption of independence between the random number of such random variables (i.e., the claims frequency) and the random variables themselves (i.e., the claim severities). However, recent studies showed that, in practice, this assumption does not always hold, hence, introducing dependence in the collective model becomes a necessity. In this sense, one trend consists of assuming dependence between the number of claims and their average severity. Alternatively, we can consider heterogeneity between the individual cost of claims associated with a given number of claims. Using the Sarmanov distribution, in this paper we aim at introducing dependence between the number of claims and the individual claim severities. As marginal models, we use the Poisson and Negative Binomial (NB) distributions for the number of claims, and the Gamma and Lognormal distributions for the cost of claims. The maximum likelihood estimation of the proposed Sarmanov distribution is discussed. We present a numerical study using a real data set from a Spanish insurance portfolio.

In this paper, a comparative study has been made between different algorithms to find the numerical solutions of the fractional-order clock chemical model (FOCCM). The spectral collocation method (SCM) with the shifted Legendre polynomials, the two-stage fractional Runge–Kutta method (TSFRK) and the four-stage fractional Runge–Kutta method (FSFRK) are used to approximate the numerical solutions of FOCCM. Our results are compared with the results obtained for the numerical solutions that are based upon the fundamental theorem of fractional calculus as well as the Lagrange polynomial interpolation (LPI). Firstly, the accuracy of the results is checked by computing the absolute error between the numerical solutions by using SCM, TSFRK, FSFRK, and LPI and the exact solution in the case of the fractional-order logistic equation (FOLE). The numerical results demonstrate the accuracy of the proposed method. It is observed that the FSFRK is better than those by SCM, TSFRK and LPI in the case of an integer order. However, the non-integer orders in the cases of the SCM and LPI are better than those obtained by using the TSFRK and FSFRK. Secondly, the absolute error between the numerical solutions of FOCCM based upon SCM, TSFFRK, FSFRK, and LPI for integer order and non-integer order has been computed. The absolute error in the case of the integer order by using the three methods of the third order is considered. For the non-integer order, the order of the absolute error in the case of SCM is found to be the best. Finally, these results are graphically illustrated by means of different figures.

When a straight cylindrical conductor of finite length is electrostatically charged, its electrostatic potential ϕ depends on the electrostatic charge qe, as expressed by the equation L(qe)=ϕ, where L is an integral operator. Method of moments (MoM) is an excellent candidate for solving L(qe)=ϕ numerically. In fact, considering qe as a piece-wise constant over the length of the conductor, it can be expressed as a finite series of weighted basis functions, qe=&sum, n=1N&alpha, nfn (with weights &alpha, n and N, number of the subsections of the conductor) defined in the L domain so that ϕ becomes a finite sum of integrals from which, considering testing functions suitably combined with the basis functions, one obtains an algebraic system Lmn&alpha, n=gm with dense matrix, equivalent to L(qe)=ϕ. Once solved, the linear algebraic system gets &alpha, n and therefore qe is obtainable so that the electrostatic capacitance C=qe/V, where V is the external electrical tension applied, can give the corresponding electrostatic capacitance. In this paper, a comparison was made among some Krylov subspace method-based procedures to solve Lmn&alpha, n=gm. These methods have, as a basic idea, the projection of a problem related to a matrix A&isin, Rn&times, n, having a number of non-null elements of the order of n, in a subspace of lower order. This reduces the computational complexity of the algorithms for solving linear algebraic systems in which the matrix is dense. Five cases were identified to determine Lmn according to the type of basis-testing functions pair used. In particular: (1) pulse function as the basis function and delta function as the testing function, (2) pulse function as the basis function as well as testing function, (3) triangular function as the basis function and delta function as the testing function, (4) triangular function as the basis function and pulse function as the testing function, (5) triangular function as the basis function with the Galerkin Procedure. Therefore, five Lmn and five pair qe and C were computed. For each case, for the resolution of Lmn&alpha, n=gm obtained, GMRES, CGS, and BicGStab algorithms (based on Krylov subspaces approach) were implemented in the MatLab&reg, Toolbox to evaluate qe and C as N increases, highlighting asymptotical behaviors of the procedures. Then, a particular value for N is obtained, exploiting both the conditioning number of Lmn and considerations on C, to avoid instability phenomena. The performances of the exploited procedures have been evaluated in terms of convergence speed and CPU-times as the length/diameter and N increase. The results show the superiority of BcGStab, compared to the other procedures used, since even if the number of iterations increases significantly, the CPU-time decreases (more than 50%) when the asymptotic behavior of all the procedures is in place. This superiority is much more evident when the CPU-time of BicGStab is compared with that achieved by exploiting Gauss elimination and Gauss&ndash, Seidel approaches.