Vol 12 - Issue 3 (Special AUS-ICMS 2020)

Advances in Pure and Applied Mathematics

List of Articles

This special volume of Advances in Pure and Applied Mathematics “APAM” is dedicated to the third International Conference on Mathematics and Statistics (AUS-ICMS2020) held at the American University of Sharjah in the United Arab Emirates, 6th -9th February 2020. The volume consists of eight invited peer reviewed articles covering different areas of mathematics by accomplished mathematicians.

This is an exposition of my keynote talk given at The Third International Conference on Mathematics and Statistics: AUS-ICMS February 2020, Sharjah, UAE. It concerns permutations and permutation matrices, and some of their generalizations. It is not intended to be comprehensive in the topics discussed, but to highlight some aspects. The final section discusses a number of open problems and conjectures. A number of references are provided to get one started on a more comprehensive study.

We consider the problem of the optimal selection of the smoothing parameter $$${h}$$$ in kernel estimation of a trend in nonparametric regression models with strictly stationary errors. We suppose that the errors are stochastic volatility sequences. Three types of volatility sequences are studied: the log-normal volatility, the Gamma volatility and the log-linear volatility with Bernoulli innovations. We take the weighted average squared error (ASE) as the global measure of performance of the trend estimation using $$${h}$$$ and we study two classical criteria for selecting $$${h}$$$ from the data, namely the adjusted generalized cross validation and Mallows-type criteria. We establish the asymptotic distribution of the gap between the ASE evaluated at one of these selectors and the smallest possible ASE. A Monte-Carlo simulation for a log-normal stochastic volatility model illustrates that this asymptotic approximation can be accurate even for small sample sizes.

Main objective of the present paper is to prove solvability of the Dirichlet, Neumann and Mixed boundary value problems for an anisotropic Laplace-Beltrami equation on a hypersurface $$${C}$$$ with the Lipschitz boundary $$${\Gamma=∂C}$$$ in the classical $$${𝕎^1(C)}$$$ space setting.

We study in this article the blow-up of the solution of the generalized Tricomi equation in the presence of two mixed nonlinearities, namely we consider
$$${(Tr) \hspace{1cm} u_{tt}-t^{2m}\Delta u=}$$$|$$${u_t}$$$|$$${^p+}$$$|$$${u}$$$|$$${^q}$$$, $$${\quad \mbox{in}\ \mathbb{R}^N\times[0,\infty)}$$$,
with small initial data, where $$${m ≥ 0}$$$.
For the problem $$${(Tr)}$$$ with $$${m = 0}$$$, which corresponds to the uniform wave speed of propagation, it is known that the presence of mixed nonlinearities generates a new blow-up region in comparison with the case of a one nonlinearity (|$$${u_t}$$$|$$${^p}$$$ or |$$${u}$$$|$$${^q}$$$). We show in the present work that the competition between the two nonlinearities still yields a new blow region for the Tricomi equation $$${(Tr)}$$$ with $$${m ≥ 0}$$$, and we derive an estimate of the lifespan in terms of the Tricomi parameter $$${m}$$$. As an application of the method developed for the study of the equation $$${(Tr)}$$$ we obtain with a different approach the same blow-up result as in [18] when we consider only one time-derivative nonlinearity, namely we keep only |$$${u_t}$$$|$$${^p}$$$ in the right-hand side of $$${(Tr)}$$$.

A general nonlinear model describing the evolution of size-structured populations influenced by coagulation and vertical effects is presented. The nonlinear population dynamics occur in a multi-region setting in which the transfer, coagulation, and vital rates of an individual depend on a vector describing conditions in the environment. The model for these dynamics consists of a system of two-dimensional nonlinear-nonlocal hyperbolic partial differential equations coupled with a system of one-dimensional nonlocal differential equations parametrized by a vertical location coordinate $$${z}$$$ describing the environmental time-dynamics. A finite difference approximation approach is employed to study the wellposedness of the model and convergence of the scheme to the unique weak solution is established. Several examples are presented to illustrate the generality of the model and to motivate applications.

We present a Galerkin-characteristic finite element method for the numerical solution of time-dependent convection-diffusion problems in porous media. The proposed method allows the use of equal-order finite element approximations for all solutions in the problem. In addition, the standard Courant-Friedrichs-Lewy condition is relaxed with the Lagrangian treatment of convection terms, and the time truncation errors are reduced in the diffusion-reaction part. Analysis of convergence and stability of the proposed method is also investigated in this study and error estimates in the $$${L}$$$²-norm are established for the numerical solutions. Numerical performance of the method is examined using two examples to verify the theoretical estimates and to demonstrate the high accuracy and efficiency of the proposed Galerkin-characteristic finite element method.

In this article we investigate some Hessian type equations. Our main aim is to derive new maximum principles for some suitable P-functions, in the sense of L.E. Payne, that is for some appropriate functional combinations of $$${u(x)}$$$ and its derivatives, where $$${u(x)}$$$ is a solution of the given Hessian type equations. To find the most suitable P-functions, we first investigate the special case of a ball, where the solution of our Hessian equations is radial, since this case gives good hints on the best functional to be considered later, for general domains. Next, we construct some elliptic inequalities for the well-chosen P-functions and make use of the classical maximum principles to get our new maximum principles. Finally, we consider some overdetermined problems and show that they have solutions when the underlying domain has a certain shape (spherical or ellipsoidal).

An evolutionary game is usually identified by a smooth (possibly nonlinear) payoff function. In this paper, we propose a model of evolutionary game in which the nonlinear payoff functions are defined by the Ricker models. Namely, in the proposed model, the biological fitness of the pure strategy will increase according to the Ricker model. Motivated by some models of economics of transportation and communication networks, in order to observe the evolutionary bifurcation diagram, we also control the nonlinear payoff functions in two different regimes: positive and negative. One of the interesting feature of the model is that if we switch the controlling parameter from positive to negative regime then the set of local evolutionarily stable strategies (ESSs) changes from one set to another one. We also study the dynamics and stability analysis of the discrete-time replicator equation governed by the proposed nonlinear payoff function. In the long-run time, the following scenario can be observed: (i) in the positive regime, the active dominating pure strategies will outcompete other strategies and only they will survive forever; (ii) in the negative regime, all active pure strategies will coexist together and they will survive forever.