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Starlike functions have gained popularity both in literature and in usage over the past decade. In this paper, our aim is to examine some useful problems dealing with q-starlike functions. These include the convolution problem, sufficiency criteria, coefficient estimates, and Fekete&ndash, Szeg&ouml, type inequalities for a new subfamily of analytic and multivalent functions associated with circular domain. In addition, we also define and study a Bernardi integral operator in its q-extension for multivalent functions. Furthermore, we will show that the class defined in this paper, along with the obtained results, generalizes many known works available in the literature.

The integral representation of the two-parameter Mittag-Leffler function E &rho, &mu, ( z ) is considered in the paper that expresses its value in terms of the contour integral. For this integral representation, the transition is made from integration over a complex variable to integration over real variables. It is shown that as a result of such a transition, the integral representation of the function E &rho, ( z ) has two forms: the representation &ldquo, A&rdquo, and &ldquo, B&rdquo, Each of these representations has its advantages and drawbacks. In the paper, the corresponding theorems are formulated and proved, and the advantages and disadvantages of each of the obtained representations are discussed.

In Mombeni et al. (2019), Birnbaum-Saunders and Weibull kernel estimators were introduced for the estimation of cumulative distribution functions (c.d.f.s) supported on the half-line $[0,\infty)$. They were the first authors to use asymmetric kernels in the context of c.d.f. estimation. Their estimators were shown to perform better numerically than traditional methods such as the basic kernel method and the boundary modified version from Tenreiro (2013). In the present paper, we complement their study by introducing five new asymmetric kernel c.d.f. estimators, namely the Gamma, inverse Gamma, lognormal, inverse Gaussian and reciprocal inverse Gaussian kernel c.d.f. estimators. For these five new estimators, we prove the asymptotic normality and we find asymptotic expressions for the following quantities: bias, variance, mean squared error and mean integrated squared error. A numerical study then compares the performance of the five new c.d.f. estimators against traditional methods and the Birnbaum-Saunders and Weibull kernel c.d.f. estimators from Mombeni et al. (2019). By using the same experimental design, we show that the lognormal and Birnbaum-Saunders kernel c.d.f. estimators perform the best overall, while the other asymmetric kernel estimators are sometimes better but always at least competitive against the boundary kernel method., 38 pages, 2 tables, 9 figures

In this paper, we introduce a new geometric description of the manifolds of matrices of fixed rank. The starting point is a geometric description of the Grassmann manifold Gr(Rk) of linear subspaces of dimension r&lt, k in Rk, which avoids the use of equivalence classes. The set Gr(Rk) is equipped with an atlas, which provides it with the structure of an analytic manifold modeled on R(k−r)×r. Then, we define an atlas for the set Mr(Rk×r) of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rk) and typical fibre GLr, the general linear group of invertible matrices in Rk×k. Finally, we define an atlas for the set Mr(Rn×m) of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base Gr(Rn)×Gr(Rm) and typical fibre GLr. The atlas of Mr(Rn×m) is indexed on the manifold itself, which allows a natural definition of a neighbourhood for a given matrix, this neighbourhood being proved to possess the structure of a Lie group. Moreover, the set Mr(Rn×m) equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space Rn×m equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space Rn×m, seen as the union of manifolds Mr(Rn×m), as an analytic manifold equipped with a topology for which the matrix rank is a continuous map.

Many of the models used nowadays in mathematical epidemiology, in particular in COVID-19 research, belong to a certain subclass of compartmental models whose classes may be divided into three “(x,y,z)” groups, which we will call respectively “susceptible/entrance, diseased, and output” (in the classic SIR case, there is only one class of each type). Roughly, the ODE dynamics of these models contains only linear terms, with the exception of products between x and y terms. It has long been noticed that the reproduction number R has a very simple Formula in terms of the matrices which define the model, and an explicit first integral Formula is also available. These results can be traced back at least to Arino, Brauer, van den Driessche, Watmough, and Wu (2007) and to Feng (2007), respectively, and may be viewed as the “basic laws of SIR-type epidemics”. However, many papers continue to reprove them in particular instances. This motivated us to redraw attention to these basic laws and provide a self-contained reference of related formulas for (x,y,z) models. For the case of one susceptible class, we propose to use the name SIR-PH, due to a simple probabilistic interpretation as SIR models where the exponential infection time has been replaced by a PH-type distribution. Note that to each SIR-PH model, one may associate a scalar quantity Y(t) which satisfies “classic SIR relations”,which may be useful to obtain approximate control policies.

In this paper we consider the application of several gradient methods to the traffic assignment problem: we search equilibria in the stable dynamics model (Nesterov and De Palma, 2003) and the Beckmann model. Unlike the celebrated Frank--Wolfe algorithm widely used for the Beckmann model, these gradients methods solve the dual problem and then reconstruct a solution to the primal one. We deal with the universal gradient method, the universal method of similar triangles, and the method of weighted dual averages, and estimate their complexity for the problem. Due to the primal-dual nature of these methods, we use a duality gap in a stopping criterion. In particular, we present a novel way to reconstruct admissible flows (i.e.,\ meeting the capacity constraints and induced by flows on the paths) in the stable dynamics model, which provides us with a computable duality gap., 17 pages, 3 figures

In this paper, we propose a reliable treatment for studying the two-wave (symbiotic) solitons of interactions in nonlinear quadratic media. We investigate Schauder's fixed point theorem for proving the existence theorem. Additionally, the uniqueness solution for this system is proved. Also, a highly accurate approximate solution is presented via an iteration algorithm., Comment: 9 pages, 3 figures

In this paper, we study regularity of weak solutions to the incompressible Boussinesq equations in $\mathbb{R}^{3}\times (0,T)$. The main goal is to establish the regularity criterion in terms of one velocity component and the gradient of temperature in Lorentz spaces., Comment: arXiv admin note: text overlap with arXiv:2005.03377

This paper shows how to obtain highly accurate solutions of eighth-order boundary-value problems of linear and nonlinear ordinary differential equations. The presented method is based on the Theory of Functional Connections, and is solved in two steps. First, the Theory of Functional Connections analytically embeds the differential equation constraints into a candidate function (called a $constrained \, expression$) that contains a function that the user is free to choose. This expression always satisfies the constraints, no matter what the free function is. Second, the free-function is expanded as a linear combination of orthogonal basis functions with unknown coefficients. The constrained expression (and its derivatives) are then substituted into the eighth-order differential equation, transforming the problem into an unconstrained optimization problem where the coefficients in the linear combination of orthogonal basis functions are the optimization parameters. These parameters are then found by linear/nonlinear least-squares. The solution obtained from this method is a highly accurate analytical approximation of the true solution. Comparisons with alternative methods appearing in literature validate the proposed approach., 14 pages, 1 figure, 8 tables

This paper is largely a review. It considers two main methods used to study stability and to obtain appropriate quantitative estimates of perturbations of (inhomogeneous) Markov chains with continuous time and a finite or countable state space. An approach is described to the construction of perturbation estimates for the main five classes of such chains associated with queuing models. Several specific models are considered for which the limit characteristics and perturbation bounds for admissible “perturbed” processes are calculated.

This article concerns the expressive power of depth in neural nets with ReLU activations and bounded width. We are particularly interested in the following questions: what is the minimal width $w_{\text{min}}(d)$ so that ReLU nets of width $w_{\text{min}}(d)$ (and arbitrary depth) can approximate any continuous function on the unit cube $[0,1]^d$ aribitrarily well? For ReLU nets near this minimal width, what can one say about the depth necessary to approximate a given function? Our approach to this paper is based on the observation that, due to the convexity of the ReLU activation, ReLU nets are particularly well-suited for representing convex functions. In particular, we prove that ReLU nets with width $d+1$ can approximate any continuous convex function of $d$ variables arbitrarily well. These results then give quantitative depth estimates for the rate of approximation of any continuous scalar function on the $d$-dimensional cube $[0,1]^d$ by ReLU nets with width $d+3.$, Comment: v3. Theorem 3 removed. Comments Welcome. 9p

In this paper, we study the distributed optimal control of a system of three evolutionary equations involving fractional powers of three self-adjoint, monotone, unbounded linear operators having compact resolvents. The system is a generalization of a Cahn&ndash, Hilliard type phase field system modeling tumor growth that has been proposed by Hawkins&ndash, Daarud, van der Zee and Oden. The aim of the control process, which could be realized by either administering a drug or monitoring the nutrition, is to keep the tumor cell fraction under control while avoiding possible harm for the patient. In contrast to previous studies, in which the occurring unbounded operators governing the diffusional regimes were all given by the Laplacian with zero Neumann boundary conditions, the operators may in our case be different, more generally, we consider systems with fractional powers of the type that were studied in a recent work by the present authors. In our analysis, we show the Fr&eacute, chet differentiability of the associated control-to-state operator, establish the existence of solutions to the associated adjoint system, and derive the first-order necessary conditions of optimality for a cost functional of tracking type.

In the present paper, we study the normality of the toric rings of stable set polytopes, generators of toric ideals of stable set polytopes, and their Gr&ouml, bner bases via the notion of edge polytopes of finite nonsimple graphs and the results on their toric ideals. In particular, we give a criterion for the normality of the toric ring of the stable set polytope and a graph-theoretical characterization of the set of generators of the toric ideal of the stable set polytope for a graph of stability number two. As an application, we provide an infinite family of stable set polytopes whose toric ideal is generated by quadratic binomials and has no quadratic Gr&ouml, bner bases.

In this paper, a class of neutral type competitive neural networks with mixed time-varying delays and leakage delays on time scales is proposed. Based on the exponential dichotomy of linear dynamic equations on time scales, Banach's fixed point theorem and the theory of calculus on time scales, some sufficient conditions that are independent of the backwards graininess function of the time scale are obtained for the existence and global exponential stability of almost periodic solutions for this class of neural networks. The obtained results are completely new and indicate that both the continuous time and the discrete time cases of the networks share the same dynamical behavior. Finally, an examples is given to show the effectiveness of the obtained results., 30 pages, 4 figures. arXiv admin note: text overlap with arXiv:1508.00818

The sedenions form a 16-dimensional Cayley-Dickson algebra. In this paper, we introduce the Tribonacci and Tribonacci-Lucas sedenions. Furthermore, we present some properties of these sedenions and derive relationships between them., Comment: 17 pages, 1 figure

We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety $X$. The case we concentrate on is when $X$ is a Veronese variety, a Grassmannian or a Segre variety. Not only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skew-symmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most of the known results and the open problems on this fascinating subject., This paper belongs to the special issue "Decomposability of Tensors", ed. Prof. Luca Chiantini

In this paper, we address the general fractional integrals and derivatives with the Sonine kernels on the spaces of functions with an integrable singularity at the point zero. First, the Sonine kernels and their important special classes and particular cases are discussed. In particular, we introduce a class of the Sonine kernels that possess an integrable singularity of power function type at the point zero. For the general fractional integrals and derivatives with the Sonine kernels from this class, two fundamental theorems of fractional calculus are proved. Then, we construct the $n$-fold general fractional integrals and derivatives and study their properties., 19 pages

In this paper, we investigate the existence of solutions for a class of p-Laplacian fractional order Kirchhoff-type system with Riemann&ndash, Liouville fractional derivatives and a parameter &lambda, By mountain pass theorem, we obtain that system has at least one non-trivial weak solution u &lambda, under some local conditions for each given large parameter &lambda, We get a concrete lower bound of the parameter &lambda, and then obtain two estimates of weak solutions u &lambda, We also obtain that u &lambda, &rarr, 0 if &lambda, tends to &infin, Finally, we present an example as an application of our results.

We consider the Cauchy problem ( D ( k ) u ) ( t ) = &lambda, u ( t ) , u ( 0 ) = 1 , where D ( k ) is the general convolutional derivative introduced in the paper (A. N. Kochubei, Integral Equations Oper. Theory 71 (2011), 583&ndash, 600), &lambda, &gt, 0 . The solution is a generalization of the function t ↦ E &alpha, ( &lambda, t &alpha, ) , where 0 &lt, &alpha, &lt, 1 , E &alpha, is the Mittag&ndash, Leffler function. The asymptotics of this solution, as t &rarr, &infin, are studied.

The main aim of this paper is to give a new generalization of Hurwitz-Lerch Zeta function of two variables.Also, we investigate several interesting properties such as integral representations, summation formula and a connection with generalized hypergeometric function. To strengthen the main results we also consider many important special cases., 7 pages

Toric posets are cyclic analogues of finite posets. They can be viewed combinatorially as equivalence classes of acyclic orientations generated by converting sources into sinks, or geometrically as chambers of toric graphic hyperplane arrangements. In this paper we study toric intervals, morphisms, and order ideals, and we provide a connection to cyclic reducibility and conjugacy in Coxeter groups., 28 pages, 8 figures. A 12-page "extended abstract" version appears as [v2]

Every finite simple group P can be generated by two of its elements. Pairs of generators for P are available in the Atlas of finite group representations as (not neccessarily minimal) permutation representations P. It is unusual but significant to recognize that a P is a Grothendieck's dessin d'enfant D and that most standard graphs and finite geometries G-such as near polygons and their generalizations-are stabilized by a D. In our paper, tripods P -- D -- G of rank larger than two, corresponding to simple groups, are organized into classes, e.g. symplectic, unitary, sporadic, etc (as in the Atlas). An exhaustive search and characterization of non-trivial point-line configurations defined from small index representations of simple groups is performed, with the goal to recognize their quantum physical significance. All the defined geometries G' s have a contextuality parameter close to its maximal value 1., 19 pages