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The preeminent target of present study is to reveal the speed characteristic of ongoing outbreak COVID-19 due to novel coronavirus. On January 2020, the novel coronavirus infection (COVID-19) detected in India, and the total statistic of cases continuously increased to 7 128 268 cases including 109 285 deceases to October 2020, where 860 601 cases are active in India. In this study, we use the Hermite wavelets basis in order to solve the COVID-19 model with time- arbitrary Caputo derivative. The discussed framework is based upon Hermite wavelets. The operational matrix incorporated with the collocation scheme is used in order to transform arbitrary-order problem into algebraic equations. The corrector scheme is also used for solving the COVID-19 model for distinct value of arbitrary order. Also, authors have investigated the various behaviors of the arbitrary-order COVID-19 system and procured developments are matched with exiting developments by various techniques. The various illustrations of susceptible, exposed, infected, and recovered individuals are given for its behaviors at the various value of fractional order. In addition, the proposed model has been also supported by some numerical simulations and wavelet-based results.

The study of the dynamics of the size of a population via mathematical modelling is a problem of interest and widely studied. Traditionally, continuous deterministic methods based on differential equations have been used to deal with this problem. However discrete versions of some models are also available and sometimes more adequate. In this paper, we randomize the Pielou logistic equation in order to include the inherent uncertainty in modelling. Taking advantage of the method of transformation of random variables, we provide a full probabilistic description to the randomized Pielou logistic model via the computation of the probability density functions of the solution stochastic process, the steady state and the time until a certain level of population is reached. The theoretical results are illustrated by means of two examples, the first one consists of a numerical experiment and the second one shows an application to study the diffusion of a technology using real data.

COVID-19 pandemic has affected all aspects of people's lives and disrupted the economy. Forecasting the number of cases infected with this virus can help authorities make accurate decisions on the interventions that must be implemented to control the pandemic. Investigation of the studies on COVID-19 forecasting indicates that various techniques such as statistical, mathematical, and machine and deep learning have been utilized. Although deep learning models have shown promising results in this context, their performance can be improved using auxiliary features. Therefore, in this study, we propose two hybrid deep learning methods that utilize the statistical features as auxiliary inputs and associate them with their main input. Specifically, we design a hybrid method of the multihead attention mechanism and the statistical features (ATT_FE) and a combined method of convolutional neural network and the statistical features (CNN_FE) and apply them to COVID-19 data of 10 countries with the highest number of confirmed cases. The results of experiments indicate that the hybrid models outperform their conventional counterparts in terms of performance measures. The experiments also demonstrate the superiority of the hybrid ATT_FE method over the long short-term memory model.

The first symptomatic infected individuals of coronavirus (Covid-19) was confirmed in December 2020 in the city of Wuhan, China. In India, the first reported case of Covid-19 was confirmed on 30 January 2020. Today, coronavirus has been spread out all over the world. In this manuscript, we studied the coronavirus epidemic model with a true data of India by using Predictor-Corrector scheme. For the proposed model of Covid-19, the numerical and graphical simulations are performed in a framework of the new generalised Caputo sense non-integer order derivative. We analysed the existence and uniqueness of solution of the given fractional model by the definition of Chebyshev norm, Banach space, Schauder's second fixed point theorem, Arzel's-Ascoli theorem, uniform boundedness, equicontinuity and Weissinger's fixed point theorem. A new analysis of the given model with the true data is given to analyse the dynamics of the model in fractional sense. Graphical simulations show the structure of the given classes of the non-linear model with respect to the time variable. We investigated that the mentioned method is copiously strong and smooth to implement on the systems of non-linear fractional differential equation systems. The stability results for the projected algorithm is also performed with the applications of some important lemmas. The present study gives the applicability of this new generalised version of Caputo type non-integer operator in mathematical epidemiology. We compared that the fractional order results are more credible to the integer order results.

[EN] In this paper, a novel technique is formed to obtain the solution of a fractional gas dynamics equation. Some reproducing kernel Hilbert spaces are defined. Reproducing kernel functions of these spaces have been found. Some numerical examples are shown to confirm the efficiency of the reproducing kernel Hilbert space method. The accurate pulchritude of the paper is arisen in its strong implementation of Caputo fractional order time derivative on the classical equations with the success of the highly accurate solutions by the series solutions. Reproducing kernel Hilbert space method is actually capable of reducing the size of the numerical work. Numerical results for different particular cases of the equations are given in the numerical section., This research was partially supported by Spanish Ministerio de Ciencia, Innovacion y Universidades PGC2018-095896-B-C22 and Generalitat Valenciana PROMETEO/2016/089.

An important advantage of fractional derivatives is that we can formulate models describing much better systems with memory effects. Fractional operators with different memory are related to the different type of relaxation process of the nonlocal dynamical systems. Therefore, we investigate the COVID-19 model with the fractional derivatives in this paper. We apply very effective numerical methods to obtain the numerical results. We also use the Sumudu transform to get the solutions of the models. The Sumudu transform is able to keep the unit of the function, the parity of the function, and has many other properties that are more valuable. We present scientific results in the paper and also prove these results by effective numerical techniques which will be helpful to understand the outbreak of COVID-19.

This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial-boundary problem (P) for the nonlinear diffusion equation in an unbounded domain $\Omega\subset\mathbb{R}^N$ ($N\in{\mathbb N}$), written as \[ \frac{\partial u}{\partial t} + (-\Delta+1)\beta(u) = g \quad \mbox{in}\ \Omega\times(0, T), \] which represents the porous media, the fast diffusion equations, etc., where $\beta$ is a single-valued maximal monotone function on $\mathbb{R}$, and $T>0$. Existence and uniqueness for (P) were directly proved under a growth condition for $\beta$ even though the Stefan problem was excluded from examples of (P). This paper completely removes the growth condition for $\beta$ by confirming Cauchy's criterion for solutions of the following approximate problem (P)$_{\varepsilon}$ with approximate parameter $\varepsilon>0$: \[ \frac{\partial u_{\varepsilon}}{\partial t} + (-\Delta+1)(\varepsilon(-\Delta+1)u_{\varepsilon} + \beta(u_{\varepsilon}) + \pi_{\varepsilon}(u_{\varepsilon})) = g \quad \mbox{in}\ \Omega\times(0, T), \] which is called the Cahn--Hilliard system, even if $\Omega \subset \mathbb{R}^N$ ($N \in \mathbb{N}$) is an unbounded domain. Moreover, it can be seen that the Stefan problem is covered in the framework of this paper.

This paper deals with an adaptation of the Poincare-Lindstedt method for the determination of periodic orbits in three-dimensional nonlinear differential systems. We describe here a general symbolic algorithm to implement the method and apply it to compute periodic solutions in a three-dimensional Lotka-Volterra system modeling a chain food interaction. The sufficient conditions to make secular terms disappear from the approximate series solution are given in the paper.

The aim of this paper is to construct a numerical scheme for solving a class of bilateral free boundaries problem. First, using a shape functional and some regularization terms, an optimal control problem is formulated, in addition, we prove its solution existence's. The first optimality conditions and the shape gradient are computed. the proposed numerical scheme is a genetic algorithm guided conjugate gradient combined with the finite element method, at each mesh regeneration, we perform a mesh refinement in order to avoid any domain singularities. Some numerical examples are shown to demonstrate the validity of the theoretical results, and to prove the robustness of the proposed scheme.

In this paper, we further Meirong Zhang, et al.'s work by computing the number of weighted eigenvalues for Sturm-Liouville equations, equipped with general integrable potentials and Dirac weights, under Dirichlet boundary condition. We show that, for a Sturm-Liouville equation with a general integrable potential, if its weight is a positive linear combination of $n$ Dirac Delta functions, then it has at most $n$ (may be less than $n$, or even be $0$) distinct real Dirichlet eigenvalues, or every complex number is a Dirichlet eigenvalue; in particular, under some sharp condition, the number of Dirichlet eigenvalues is exactly $n$. Our main method is to introduce the concepts of characteristics matrix and characteristics polynomial for Sturm-Liouville problem with Dirac weights, and put forward a general and direct algorithm used for computing eigenvalues. As an application, a class of inverse Dirichelt problems for Sturm-Liouville equations involving single Dirac distribution weights is studied., 23 pages

This paper is concerned with a parabolic-elliptic Keller-Segel system where both diffusive and chemotactic coefficients (motility functions) depend on the chemical signal density. This system was originally proposed by Keller and Segel in \cite{KS-1971-JTB2} to describe the aggregation phase of {\it Dictyostelium discoideum} cells in response to the secreted chemical signal cyclic adenosine monophosphate (cAMP), but the available analytical results are very limited by far. Considering system in a bounded smooth domain with Neumann boundary conditions, we establish the global boundedness of solutions in any dimensions with suitable general conditions on the signal-dependent motility functions, which are applicable to a wide class of motility functions. The existence/nonexistence of non-constant steady states is studied and abundant stationary profiles are found. Some open questions are outlined for further pursues. Our results demonstrate that the global boundedness and profile of stationary solutions to the Keller-Segel system with signal-dependent motilities depend on the decay rates of motility functions, space dimensions and the relation between the diffusive and chemotactic motilities, which makes the dynamics immensely wealthy.

In this paper, a generalized fractional $(q,h)$-Gronwall inequality is investigated. Based on this inequality, we derive the uniqueness theorem and the finite-time stability criterion of nonlinear fractional delay $(q,h)$-difference systems. Several examples are given to illustrate our theoretical result.

The main object of the present paper is to consider the binomial transforms for Horadam quaternion sequences. We gave new formulas for recurrence relation, generating function, Binet formula and some basic identities for the binomial sequence of Horadam quaternions. Working with Horadam quaternions, we have found the most general formula that includes all binomial transforms with recurrence relation from the second order. In the last part, we determined the recurrence relation for this new type of quaternion by working with the iterated binomial transform, which is a different type of binomial transform. WOS:000638679800001 2-s2.0-85104113068

This paper considers minimizers of the following inhomogeneous $L^2$-subcritical energy functional \[E(u):=\int_{\R^N}|\nabla u|^{2}dx-\frac{2}{p+1}\int_{\R^N}m(x)|u|^{p+1}dx,%\ u\in H^{1}(\R^N), \] under the mass constraint $\|u\|^{2}_{2}=M$. Here $N\geq1$, $p\in(1,1+\frac{4}{N})$, $M>0$ and the inhomogeneous term $m(x)$ satisfies $0

Coronavirus disease 2019 (COVID‐19) is a viral disease which is declared as a pandemic by WHO This disease is posing a global threat, and almost every country in the world is now affected by this disease Currently, there is no vaccine for this disease, and because of this, containing COVID‐19 is not an easy task It is noticed that elderly people got severely affected by this disease specially in Europe In the present paper, we propose and analyze a mathematical model for COVID‐19 virus transmission by dividing whole population in old and young groups We find disease‐free equilibrium and the basic reproduction number (R0) We estimate the parameter corresponding to rate of transmission and rate of detection of COVID‐19 using real data from Italy and Spain by least square method We also perform sensitivity analysis to identify the key parameters which influence the basic reproduction number and hence regulate the transmission dynamics of COVID‐19 Finally, we extend our proposed model to optimal control problem to explore the best cost‐effective and time‐dependent control strategies that can reduce the number of infectives in a specified interval of time [ABSTRACT FROM AUTHOR] Copyright of Mathematical Methods in the Applied Sciences is the property of John Wiley & Sons, Inc and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission However, users may print, download, or email articles for individual use This abstract may be abridged No warranty is given about the accuracy of the copy Users should refer to the original published version of the material for the full abstract (Copyright applies to all Abstracts )

In this paper, the semi-rational solutions that causes vector rogue waves and breathers can be obtained by using the Darboux dressing transformation. We studied vector rogue waves and the interaction between rogue waves and light-dark solitons, and observed that during the interaction, due to the interference between the light-dark components of the solitons, a respiration-like structures appears. Besides, it can be observed that the rogue waves and soliton merge together. Moreover, the main characteristics of the interactions between the breathers and bright-dark solitons are displayed with some graphics.

The Markov evolution is studied of an infinite age-structured population of migrants arriving in and departing from a continuous habitat $X \subseteq\mathds{R}^d$ -- at random and independently of each other. Each population member is characterized by its age $a\geq 0$ (time of presence in the population) and location $x\in X$. The population states are probability measures on the space of the corresponding marked configurations. The result of the paper is constructing the evolution $\mu_0 \to \mu_t$ of such states by solving a standard Fokker-Planck equation for this models. We also found a stationary state $\mu$ existing if the emigration rate is separated away from zero. It is then shown that $\mu_t$ weakly converges to $\mu$ as $t\to +\infty$.

We conjecture that for a plasma in a spatial domain with a boundary, the specular reflection effect of the boundary can be approximated by a large magnetic confinement field in the near-boundary region. In this paper, we verify this conjecture for the 1.5D relativistic Vlasov-Maxwell system (RVM) on a bounded domain $\Omega = (0, 1)$ with an external confining magnetic field.

In this paper, we establish existence and finite time blow up of weak solutions of a parabolic equation of p(x)-Laplacian type with the Dirichlet boundary condition. Moreover, we obtain upper and lower bounds for the blow up time of solutions, by employing concavity method and differential inequality technique respectively.

Motivated by gradient methods in optimization theory, we give methods based on $\psi$-fractional derivatives of order $\alpha$ in order to solve unconstrained optimization problems. The convergence of these methods is analyzed in detail. This paper also presents an Adams-Bashforth-Moulton (ABM) method for the estimation of solutions to equations involving $\psi$-fractional derivatives. Numerical examples using the ABM method show that the fractional order $\alpha$ and weight $\psi$ are tunable parameters, which can be helpful for improving the performance of gradient descent methods., Comment: 27 pages

Main aim of the present paper is explore certain physical phenomena by means of $p$-adic probability theory. To overcome this study, we deal with a more general setting to define $p$-adic Gibbs measures. For the sake of simplicity of explanations, we restrict ourselves to the Ising model on the Cayley tree, since such a model has broad theoretical and practical applications. To study $p$-adic quasi Gibbs measures, we reduce the problem to the description of the fixed points of the Ising-Potts mapping. Finding fixed points is not an easy job as in the real setting. Furthermore, the phase transition for the model is established. In the real case, the phase transition yields the the singularity of the limiting Gibbs measures. However, we show that the $p$-adic quasi Gibbs measures do not exhibit the mentioned type of singularity, such kind of phenomena is called strong phase transition. Finally, we deal with the solvability and the number of solutions of ceratin $p$-adic equation depending on several parameters. Such a description allows us to find all possible translation-invariant $p$a-adic quasi Gibbs measures.

In this paper, we consider the generalized approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls. We analyse the relationship between the generalized approximate boundary synchronization and the generalized exact boundary synchronization, give a sufficient condition to realize the generalized approximate boundary synchronization and a necessary condition in terms of Kalman’s matrix, and show the meaning of the number of total controls. Besides, by the generalized synchronization decomposition, we define the generalized approximately synchronizable state, and obtain its properties and a sufficient condition for it to be independent of applied boundary controls.

We consider singularly perturbed convection-diffusion equations on one-dimensional networks (metric graphs) as well as the transport problems arising in the vanishing diffusion limit. Suitable coupling condition at inner vertices are derived that guarantee conservation of mass as well as dissipation of a mathematical energy which allows us to prove stability and well-posedness. For single intervals and appropriately specified initial conditions, it is well-known that the solutions of the convection-diffusion problem converge to that of the transport problem with order $O(\sqrt{\epsilon})$ in the $L^\infty(L^2)$-norm with diffusion $\epsilon \to 0$. In this paper, we prove a corresponding result for problems on one-dimensional networks. The main difficulty in the analysis is that the number and type of coupling conditions changes in the singular limit which gives rise to additional boundary layers at the interior vertices of the network. Since the values of the solution at these network junctions are not known a-priori, the asymptotic analysis requires a delicate choice of boundary layer functions that allows to handle these interior layers.

In this paper, we consider the 3D Navier-Stokes equations in the whole space. We investigate some new inequalities and \textit{a priori} estimates to provide the critical regularity criteria in terms of one directional derivative of the velocity field, namely $\partial_3 \mathbf{u} \in L^p((0,T); L^q(\mathbb{R}^3)), ~\frac{2}{p} + \frac{3}{q} = 2, ~\frac{3}{2}

In this paper, we study Hegselmann-Krause models with a time-variable time delay. Under appropriate assumptions, we show the exponential asymptotic consensus when the time delay satisfies a suitable smallness assumption. Our main strategies for this are based on Lyapunov functional approach and careful estimates on the trajectories. We then study the mean-field limit from the many-individual Hegselmann-Krause equation to the continuity-type partial differential equation as the number N of individuals goes to infinity. For the limiting equation, we prove global-in-time existence and uniqueness of measure-valued solutions. We also use the fact that constants appearing in the consensus estimates for the particle system are independent of N to extend the exponential consensus result to the continuum model. Finally, some numerical tests are illustrated.

The paper concerns a new forecast model that includes the class of undiagnosed infected people, and has a multiregion extension, to cope with the in‐time and in‐space heterogeneity of an epidemic. The model is applied to the SARS‐CoV2 (COVID‐19) pandemic that, starting from the end of February 2020, began spreading along the Italian peninsula, by first attacking small communities in north regions, and then extending to the center and south of Italy, including the two main islands. It has proved to be a robust and reliable tool for the forecast of the total and active cases, which can be also used to simulate different scenarios. In particular, the model is able to address a number of issues, such as assessing the adoption of the lockdown in Italy, started from March 11, 2020; the estimate of the actual attack rate; and how to employ a rapid screening test campaign for containing the epidemic.

In this paper, we study the stability of the zero equilibrium and the occurrence of flip bifurcation on the following system of difference equations: \[x_{n+1} =a_1\frac{y_n}{b_1+y_n} +c_1\frac{x_ne^{k_1-d_1x_n}}{1+e^{k_1-d_1x_n}},\]\\ \[y_{n+1} =a_2\frac{z_n}{b_2+z_n} +c_2\frac{y_ne^{k_2-d_2y_n}}{1+e^{k_2-d_2y_n}},\]\\ \[z_{n+1} =a_3\frac{x_n}{b_3+x_n} +c_3\frac{z_ne^{k_3-d_3z_n}}{1+e^{k_3-d_3z_n}}\] where $a_i$, $b_i$, $c_i$, $d_i$, $k_i$, for $i=1,2,3$, are real constants and the initial values $x_0$, $y_0$ and $z_0$ are real numbers. We study the stability of this system in the special case when one of the eigenvalues is equal to -1 and the remaining eigenvalues have absolute value less than 1, using center manifold theory.

In this paper, we consider a region occupied by viscous or inviscid compressible magnetohydrodynamic fluids, and surrounded by vacuum. It is shown that the fluid region will expand at least linearly in time as soon as there are no singularities. The expanding rate is proportional to initial total energy and is inversely proportional to initial mass. The result indicates an interesting fact that the expansion of the viscous monatomic fluids seems similar to the inviscid fluids.

In this paper we consider a class of chemotaxis models with two arbitrary constitutive functions g(u) and f(v). After having performed a complete symmetry group classification with respect to them the reduced systems are derived. By considering g(u) of the logistic form wide classes of exact solutions are found.

Every now and then, there has been natural or man-made calamities Such adversities instigate various institutions to find solutions for them The current study attempts to explore the disaster caused by the micro enemy called coronavirus for the past few months and aims at finding the solution for the system of nonlinear ordinary differential equations to which q?homotopy analysis transform method (q?HATM) has been applied to arrive at effective results In this paper, there are eight nonlinear ordinary differential equations considered and to solve them the advanced fractional operator Atangana-Baleanu (AB) fractional derivative has been applied to produce better understanding The outcomes have been presented in terms of plots Ultimately, the present study assists in examining the real-world models and aids in predicting their behavior corresponding to the parameters considered in the models

The paper deals with the analysis of stable thermo-solutal dendritic growth in the presence of intense convection. The n-fold symmetry of crystalline anisotropy as well as the two- and three-dimensional growth geometries are considered. The steady-state analytical solutions are found with allowance for the convective-type heat and mass exchange boundary conditions at the dendritic surface. A linear morphological stability analysis determining the marginal wavenumber is carried out. The new stability criterion is derived from the solvability theory and stability analysis. This selection criterion takes place in the regions of small undercooling. To describe a broader undercooling diapason, the obtained selection criterion, which describes the case of intense convection, is stitched together with the previously known selection criterion for the conductive-type boundary conditions. It is demonstrated that the stitched selection criterion well describes a broad diapason of experimental undercoolings.

We study the global stability of a multistrain SIS model with superinfection and patch structure. We establish an iterative procedure to obtain a sequence of threshold parameters. By a repeated application of a result by Takeuchi et al. [Nonlinear Anal Real World Appl. 2006 7:235-247], we show that these parameters completely determine the global dynamics of the system: for any number of patches and strains with different infectivities, any subset of the strains can stably coexist depending on the particular choice of the parameters. Finally, we return to the special case of one patch examined in [Math Biosci Eng. 2017 14:421-35] and give a correction to the proof of Theorem 2.2 of that paper.

The paper extends some well-known results for analytic functions onto solutions of the Vekua equation $\partial _{\overline{z}}W=aW+b\overline{W}$ regarding the existence and construction of the Bergman kernel and of the corresponding Bergman projection operator.

International audience; This paper introduces the use of the Proper Generalized Decomposition (PGD) method for the optical flow (OF) problem in a classical framework of Sobolev spaces, i.e. optical flow methods including a robust energy for the data fidelity term together with a quadratic penalizer for the regularisation term. A mathematical study of PGD methods is first presented for general regularization problems in the framework of (Hilbert) Sobolev spaces, and their convergence is then illustrated on OF computation. The convergence study is divided in two parts: (i) the weak convergence based on the Brézis-Lieb decomposition, (ii) the strong convergence based on a growth result on the sequence of descent directions. A practical PGD-based OF implementation is then proposed and evaluated on freely available OF data sets. The proposed PGD-based OF approach outperforms the corresponding non-PGD implementation in terms of both accuracy and computation time for images containing a weak level of information, namely low image resolution and/or low Signal-To-Noise Ratio (SNR).

In this paper, we study the existence and multiplicity results of nontrivial positive solutions to a quasilinear elliptic equation in $\RN$, when $N\geq2$, as \begin{equation} \Lp u+u^{p-1}=\lambda\hspace{0.2mm}k(x)u^{r-1}-h(x)u^{q-1}.\nonumber \end{equation} Here, $h(x),k(x)>0$ are Lebesgue measurable functions, $10$ is a parameter.

International audience; In the paper, the authors introduce a notion "multivariate exponential polynomials" which generalize exponential numbers and polynomials, establish explicit formulas, inversion formulas, and recurrence relations for multivariate exponential polynomials in terms of the Stirling numbers of the first and second kinds with the help of the Fàa di Bruno formula, two identities for the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds, construct some determinantal inequalities and product inequalities for multivariate exponential polynomials with the aid of some properties of completely monotonic functions and other known results, derive the logarithmic convexity and logarithmic concavity for multivariate exponential polynomials, and finally find an application of multi-variate exponential polynomials to white noise distribution theory by confirming that multivariate exponential polynomials satisfy conditions for sequences required in white noise distribution theory.

This paper focuses on using the first curvature $\kappa(t)$ of trajectory to describe the stability of linear time-invariant system. We extend the results for two and three-dimensional systems [Y. Wang, H. Sun, Y. Song et al., arXiv:1808.00290] to $n$-dimensional systems. We prove that for a system $\dot{r}(t)=Ar(t)$, (i) if there exists a measurable set whose Lebesgue measure is greater than zero, such that for all initial values in this set, $\lim\limits_{t\to+\infty}\kappa(t)\neq0$ or $\lim\limits_{t\to+\infty}\kappa(t)$ does not exist, then the zero solution of the system is stable; (ii) if the matrix $A$ is invertible, and there exists a measurable set whose Lebesgue measure is greater than zero, such that for all initial values in this set, $\lim\limits_{t\to+\infty}\kappa(t)=+\infty$, then the zero solution of the system is asymptotically stable.

In this paper, we prove the global existence of the weak solution to the mean field kinetic equation derived from the $N$-particle Newtonian system. For $L^1\cap L^\infty$ initial data, the solvability of the mean field kinetic equation can be obtained by using uniform estimates and compactness arguments while the difficulties arising from the non-local non-linear interaction are tackled appropriately using the Aubin-Lions compact embedding theorem.

In this paper, the application of Ledley antecedence solutions in designing state feedback stabilizers of generic logic systems has been proposed. To make the method feasible, two modifications are made to the original Ledley antecedence solution theory: (i) the preassigned logical functions have been extended from being a set of equations to an admissible set; (ii) the domain of arguments has been extended from the whole state space to a restricted subset. In the proposed method, state feedback controls are considered as a set of extended Ledley antecedence solutions for a designed iterative admissible sets over their corresponding restricted subsets. Based on this, an algorithm has been proposed to verify the solvability, and simultaneously to provide all possible state feedback stabilizers when the problem is solvable. All stabilizers are optimal, which stabilize the logic systems from any initial state to the destination state/state set in the shortest time. The method is firstly demonstrated on Boolean control networks to achieve point stabilization. Then, with some minor modifications, the proposed method is also proven to be applicable to set stabilization problems. Finally, it is shown that in $k$-valued and mix-valued logical systems, the proposed method remains effective., Comment: 10 pages, 3 figures

© 2021 The Authors. The aim of this paper is to develop a layer potential theory in L2-based weighted Sobolev spaces on Lipschitz bounded and exterior domains of Rn , n ≥ 3, for the anisotropic Stokes system with L∞ viscosity tensor coefficient satisfying an ellip- ticity condition for symmetric matrices with zero matrix trace. To do this, we explore equivalent mixed variational formulations and prove the well-posedness of some transmission problems for the anisotropic Stokes system in Lipschitz domains of Rn, with the given data in L2-based weighted Sobolev spaces. These results are used to define the volume (Newtonian) and layer potentials and to obtain their properties. Then, we analyze the well-posedness of the exterior Dirichlet and Neumann problems for the anisotropic Stokes system with L∞ symmetrically elliptic tensor coefficient by representing their solutions in terms of the obtained volume and layer potentials. EPSRC grant EP/M013545/1: "Mathematical Analysis of Boundary-Domain Integral Equations for Nonlinear PDEs"; Babeş-Bolyai University research grant AGC35124/31.10.2018; Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy-EXC 2075-390740016.

International audience; This paper aims at understanding the nature of the subwavelength resonant frequencies of dielectric particles with high refractive indices. It is proved that for an arbitrary shaped particle, these subwavelength resonant frequencies can be expressed in terms of the eigenvalues of the Newtonian potential associated with its shape. The enhancement of the scattered field at the resonant frequencies is shown. The hybridization of the subwavelength resonant frequencies of a dimer consisting of high refractive index dielectric nanoparticles is also characterized.

In this paper, we introduce two golden ratio algorithms with new stepsize rules for solving pseudomonotone and Lipschitz variational inequalities in finite dimensional Hilbert spaces. The presented stepsize rules allow the resulting algorithms to work without the prior knowledge of the Lipschitz constant of operator. The first algorithm uses a sequence of stepsizes which is previously chosen, diminishing and non-summable. While the stepsizes in the second one are updated at each iteration and by a simple computation. A special point is that the sequence of stepsizes generated by the second algorithm is separated from zero. The convergence as well as the convergence rate of the proposed algorithms are established under some standard conditions. Also, we give several numerical results to show the behavior of the algorithms in comparisons with other algorithms.

[EN] In this paper, we consider the problem of solving initial value problems and boundary value problems through the point of view of its continuous form. It is well known that in most cases these types of problems are solved numerically by performing a discretization and applying the finite difference technique to approximate the derivatives, transforming the equation into a finite-dimensional nonlinear system of equations. However, we would like to focus on the continuous problem, and therefore, we try to set the domain of existence and uniqueness for its analytic solution. For this purpose, we study the semilocal convergence of a Newton-type method with frozen first derivative in Banach spaces. We impose only the assumption that the Frechet derivative satisfies the Lipschitz continuity condition and that it is bounded in the whole domain in order to obtain appropriate recurrence relations so that we may determine the domains of convergence and uniqueness for the solution. Our final aim is to apply these theoretical results to solve applied problems that come from integral equations, ordinary differential equations, and boundary value problems., Spanish Ministry of Science and Innovation. Grant Number: MTM2014- 52016-C2-2-P Generalitat Valenciana Prometeo. Grant Number: 2016/089

[EN] Bladder cancer is one of the most common malignant diseases in the urinary system and a highly aggressive neoplasm. The prognosis is not favorable usually, and its evolution for particular patients is very difficult to find out. In this paper, we propose a dynamic mathematical model that describes the bladder tumor growth and the immune response evolution. This model is customized for a single patient, determining appropriate model parameter values via model calibration. Due to the uncertainty of the tumor evolution, using the calibrated model parameters, we predict the tumor size and the immune response evolution over the next few months assuming three different scenarios: favorable, neutral, and unfavorable. In the former, it is not expected any trace of the cancer in the middle of September 2018 (after 16 mo). In the neutral scenario, at the same date, a 7- to 8-mm tumor is expected. In the worst case, a 40-mm tumor is expected. The patient was cited on 10 September 2018 to check the tumor size, and according to the doctors, there was no sign of recurrence. It seems that we are in the favorable scenario. The patient will be called again for follow-up in mid-2019., This work has been supported by the Ministerio de Economía, Industria y Competitividad grant MTM2017-89664-P.

We consider Sturm-Liouville operators on geometrical graphs without cycles (trees) with singular potentials from the class $W_2^{-1}$. We suppose that the potentials are known on a part of the graph, and study the so-called partial inverse problem, which consists in recovering the potentials on the remaining part of the graph from some parts of several spectra. The main results of the paper are the uniqueness theorem and a constructive procedure for the solution of the partial inverse problem. Our method is based on the completeness and the Riesz-basis property of special systems of vector functions, and the reduction of the partial inverse problem to the complete one on a part of the graph.

In this paper, by means of Banach fixed point theorem, we investigate the existence and Ulam--Hyers--Rassias stability of the non-instantaneous impulsive integrodifferential equation by means of $\psi$-Hilfer fractional derivative. In this sense, some examples are presented, in order to consolidate the results obtained., Comment: 15 pages

In this paper, we study several aspects of solitary wave solutions of the rotation Benjamin-Ono equation. By solving a minimization problem on the line, we construct a family of even travelling waves $\psi_{c,\gamma}$. We also study the strong convergence of this family and we establish the uniqueness of $\psi_{c,\gamma}$ for $\gamma$ small enough. Note that this improves the results in [5] where the stability of the set of ground states is proven.

In this paper we study the stability of a tree-shaped network of elastic strings and beams with some feedbacks at the ends. The whole system is asymptotically stable. Moreover, the energy of the solution decay exponentially to zero if there is no beam following a string ( from the root to the leaves) and decay polynomially if not. We use a frequency domain method from the semigroups theory.

In this paper, we study the following fractional Schr\"{o}dinger-Poisson system involving competing potential functions \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u=K(x)f(u)+Q(x)|u|^{2_s^{\ast}-2}u, & \hbox{in $\mathbb{R}^3$,} \varepsilon^{2t}(-\Delta)^t\phi=u^2,& \hbox{in $\mathbb{R}^3$,} \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $f$ is a function of $C^1$ class, superlinear and subcritical nonlinearity, $2_s^{\ast}=\frac{6}{3-2s}$, $s>\frac{3}{4}$, $t\in(0,1)$, $V(x)$ $K(x)$ and $Q(x)$ are positive continuous function. Under some suitable assumptions on $V$, $K$ and $Q$, we prove that there is a family of positive ground state solutions with polynomial growth for sufficiently small $\varepsilon>0$, of which it is concentrating on the set of minimal points of $V(x)$ and the sets of maximal points of $K(x)$ and $Q(x)$. The methods are based on the Nehari manifold, arguments of Brezis-Nirenberg and concentration compactness of P. L. Lions.

This paper presents a novel model to detect the COVID-19 infected person from a Markovian feedback persons in a limited department capacity. The persons arrive one by one to the department and the balking and the retention of reneged person approaches are considered. There exists one server presents the service to these persons according to first-come, first-served (FCFS) discipline. An efficient and novel algorithm is presented to get the exact value of the probability of n persons in the department at any time interval. This algorithm depends on the Laplace transform to solve a probabilistic dynamical system of differential equations. By considering the exponential detection function and if the probability of the infected person in the department is equal to the probability of each one, then this algorithm is useful to obtain the detection probability of the infected one. Under steady state, the detection probability of the infected person is described. The usefulness of this model is illustrated for different capacities by using a numerical example to describe the behavior of probabilities of the persons in the department, the detection probabilities of the infected person as functions in time, and the mean time to detection.