Preface

This focused issue of the Communications on Applied Mathematics and Computation is in Honour of Prof.Rémi Abgrall on the Occasion of His 61th Birthday.Rémi Abgrall has been a student in mathematics(1981–1985)of Ecole Normale Supérieure de Saint Cloud(now part of ENS Lyon).After his studies in pure mathematics,he changed orientation tofluid mechanics.He did his PhD at the Laboratoire de Météorologie Dynamique(LMD)at Ecole Normale Supérieure under the supervision of Claude Basdevant.He graduated in December 1987 with a thesis on a semi-Lagrangian model of 2D turbulence,refereed by Olivier Pironneau and Marcel Lesieur.

New Fusion Approach of Spatial and Channel Attention for Semantic Segmentation of Very High Spatial Resolution Remote Sensing Images

The semantic segmentation of very high spatial resolution remote sensing images is difficult due to the complexity of interpreting the interactions between the objects in the scene. Indeed, effective segmentation requires considering spatial local context and long-term dependencies. To address this problem, the proposed approach is inspired by the MAC-UNet network which is an extension of U-Net, densely connected combined with channel attention. The advantages of this solution are as follows: 4) The new model introduces a new attention called propagate attention to build an attention-based encoder. 2) The fusion of multi-scale information is achieved by a weighted linear combination of the attentions whose coefficients are learned during the training phase. 3) Introducing in the decoder, the Spatial-Channel-Global-Local block which is an attention layer that uniquely combines channel attention and spatial attention locally and globally. The performances of the model are evaluated on 2 datasets WHDLD and DLRSD and show results of mean intersection over union (mIoU) index in progress between 1.54% and 10.47% for DLRSD and between 1.04% and 4.37% for WHDLD compared with the most efficient algorithms with attention mechanisms like MAU-Net and transformers like TMNet.

Mathematical Model of the Spread of the Coronavirus Disease 2019 (COVID-19) in Burkina Faso

In this paper, we develop a mathematical model of the COVID-19 pandemic in Burkina Faso. We use real data from Burkina Faso National Health Commission against COVID-19 to predict the dynamic of the disease and also the cumulative number of reported cases. We use public policies in model in order to reduce the contact rate, this allows to show how the reduction of the daily report of infectious cases goes, so we would like to draw the attention of decision makers for a rapid treatment of reported cases.

Entropy Formulation for Triply Nonlinear Degenerate Elliptic-Parabolic-Hyperbolic Equation with Zero-Flux Boundary Condition

In this note, we investigated existence and uniqueness of entropy solution for triply nonlinear degenerate parabolic problem with zero-flux boundary condition. Accordingly to the case of doubly nonlinear degenerate parabolic hyperbolic equation, we propose a generalization of entropy formulation and prove existence and uniqueness result without any structure condition.

The Relevance of Biological and Hydrodynamic Timescale in the Growth of Plankton

The relationship between hydrodynamic mesoscale structures and plankton formation in the wake of an island, as well as its interaction with a coastal upwelling, is investigated. Our focus is on the process by which vortices create localized plankton blooms. A basic three-component model for marine ecology was utilized, as well as a coupled system of kinematic flow that mimicked the mesoscale features underlying the island. We show that the prevalence of localized plankton blooms is produced mostly by the prolonged residence times of nutrients and plankton in the island’s vicinity, as well as the confinement of plankton within vortices.

Assessing Safety of Ferry Routes by Ship Handling Model through AHP and Fuzzy Approach

Ferry accidents often occur the result of ship handling difficulty which interfacing human, machine and environment. Therefore, a decision tool model as a comprehensive information system, based on the ship handling difficulty, needs to be developed through the combination of Analytic Hierarchy Process (AHP) and Fuzzy Logic System. The Fuzzy Logic System part consists of ship condition, ship handling facility condition, navigation condition and weather condition. The output of decision tool is the ship handling difficulty level in linguistic form. The simulation of model is conducted at several straits in Indonesia water. The decision tool model could be used as management information system by port authority to monitor the ferry/ship movement in real time regarding the ship handling difficulty. Further, it would be used to take some useful safety operation strategies and safety policies to improve ferry transportation safety at port water and strait water.

ANALYTICAL SMOOTHING EFFECT OF SOLUTION FOR THE BOUSSINESQ EQUATIONS

In this article, we study the analytical smoothing effect of Cauchy problem for the incompressible Boussinesq equations. Precisely, we use the Fourier method to prove that the Sobolev H^1 -solution to the incompressible Boussinesq equations in periodic domain is analytic for any positive time. So the incompressible Boussinesq equations admit exactly same smoothing effect properties of incompressible Navier-Stokes equations.

Network Access Control Technology—Proposition to Contain New Security Challenges

Traditional products working independently are no longer sufficient, since threats are continually gaining in complexity, diversity and performance;In order to proactively block such threats we need more integrated information security solution. To achieve this objective, we will analyze a real-world security platform, and focus on some key components Like, NAC, Firewall, and IPS/IDS then study their interaction in the perspective to propose a new security posture that coordinate and share security information between different network security components, using a central policy server that will be the NAC server or the PDP (the Policy Decision Point), playing an orchestration role as a central point of control. Finally we will conclude with potential research paths that will impact NAC technology evolution.

Numerical Study of Natural Convection in an Inclined Porous Cavity

Two-dimensional transient laminar natural convection in a square cavity containing a porous medium and inclined at an angle of 30°is investigated numerically.The vertical walls are differentially heated,and the horizontal walls are adiabatic.The effect of Rayleigh number on heat transfer and on the road to chaos is analyzed.The natural heat transfer and the Darcy Brinkman equations are solved by using a finite volume method and a Tri Diagonal Matrix Algorithm(TDMA).The results are obtained for a porosity equal to 0.45,a Darcy number and a Prandtl respectively equal to 10^(-3)and 0.71;they are analyzed in terms of streamlines,isotherms,phase portrait,attractors,and spectra amplitude as a function of the Rayleigh number.It is found that,as Rayleigh number increases,the natural convection changes from a steady state to a time-periodic state and finally to a chaotic condition.

PENALIZED LEAST SQUARE IN SPARSE SETTING WITH CONVEX PENALTY AND NON GAUSSIAN ERRORS

This paper consider the penalized least squares estimators with convex penalties or regularization norms.We provide sparsity oracle inequalities for the prediction error for a general convex penalty and for the particular cases of Lasso and Group Lasso estimators in a regression setting.The main contribution is that our oracle inequalities are established for the more general case where the observations noise is issued from probability measures that satisfy a weak spectral gap(or Poincaré)inequality instead of Gaussian distributions.We illustrate our results on a heavy tailed example and a sub Gaussian one;we especially give the explicit bounds of the oracle inequalities for these two special examples.

Crystallography in the Spaces E², E³, E⁴, E⁵...N⁰II Isomorphism Classes and Study of Five Crystal Families of Space E⁵

In the paper N0II, we describe some isomorphism classes and we apply their properties to the study of five crystal families of space E5. The names of these families are the following ones (monoclinic di iso squares)-al, decadic-al, (monoclinic di iso hexagons)-al, (rhombotopic cosa=-1/4)-al and rhombotopic cosa=-1/5. The meaning of these names will be given in Paragraphs 5 and 6 with some geometric properties of their cell.

Asymptotic Analysis of Linear and Interval Linear Fractional-Order Neutral Delay Differential Systems Described by the Caputo-Fabrizio Derivative

Asymptotic stability of linear and interval linear fractional-order neutral delay differential systems described by the Caputo-Fabrizio (CF) fractional derivatives is investigated. Using Laplace transform, a novel characteristic equation is derived. Stability criteria are established based on an algebraic approach and norm-based criteria are also presented. It is shown that asymptotic stability is ensured for linear fractional-order neutral delay differential systems provided that the underlying stability criterion holds for any delay parameter. In addition, sufficient conditions are derived to ensure the asymptotic stability of interval linear fractional order neutral delay differential systems. Examples are provided to illustrate the effectiveness and applicability of the theoretical results.

An Immersed Method Based on Cut-Cells for the Simulation of 2D Incompressible Fluid Flows Past Solid Structures

We present a cut-cell method for the simulation of 2D incompressible flows past obstacles.It consists in using the MAC scheme on cartesian grids and imposing Dirchlet boundary conditions for the velocity field on the boundary of solid structures following the Shortley-Weller formulation.In order to ensure local conservation properties,viscous and convecting terms are discretized in a finite volume way.The scheme is second order implicit in time for the linear part,the linear systems are solved by the use of the capacitance matrix method for non-moving obstacles.Numerical results of flows around an impulsively started circular cylinder are presented which confirm the efficiency of the method,for Reynolds numbers 1000 and 3000.An example of flows around a moving rigid body at Reynolds number 800 is also shown,a solver using the PETSc-Library has been prefered in this context to solve the linear systems.

ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR THE POISSON EQUATION WITH A NONLOCAL BOUNDARY OPERATOR

In this work, we investigate the solvability of the boundary value problem for the Poisson equation, involving a generalized Riemann-Liouville and the Caputo derivative of fractional order in the class of smooth functions. The considered problems are generalization of the known Dirichlet and Neumann oroblems with operators of a fractional order.

Evaporation and Combustion of a Drop of Liquid Fuel—A Review

The accelerated depletion of oil reserves and the often exorbitant cost of fossil fuels contribute to the development of fuels from renewable sources. The objective of this work is to analyze the influence of the properties of renewable fuels on their evaporation in natural convection, their combustion and their use in internal combustion engines. A summary of the various numerical and experimental works from the literature has been presented in this work. This work focuses on the numerical modelling of the natural convection evaporation of an isolated drop of a liquid fuel in natural convection. The transfers in the liquid and vapour phases are described by the conservation equations of mass and species, momentum and energy. The main feature of this work is the consideration of advection, azimuthal angle and thickness of the vapour phase of the drop during evaporation of the drop.

WELL-POSEDNESS OF A NONLINEAR MODEL OF PROLIFERATING CELL POPULATIONS WITH INHERITED CYCLE LENGTH

This paper deals with a nonlinear initial boundary values problem derived from a modified version of the so called Lebowitz and Rubinow’s model [16] discussed in [8, 9]modeling a proliferating age structured cell population with inherited properties. We give existence and uniqueness results on appropriate weighted L~p-spaces with 1 ≤ p < ∞ in the case where the rate of cells mortality σ and the transition rate k are depending on the total density of population. General local and nonlocal reproduction rules are considered.

ASYMPTOTIC BEHAVIOR OF GLOBAL SMOOTH SOLUTIONS FOR BIPOLAR COMPRESSIBLE NAVIER-STOKES-MAXWELL SYSTEM FROM PLASMAS

This paper is concerned with the bipolar compressible Navier-Stokes-Maxwell system for plasmas. We investigated, by means of the techniques of symmetrizer and elaborate energy method, the Cauchy problem in R^3. Under the assumption that the initial values are close to a equilibrium solutions, we prove that the smooth solutions of this problem converge to a steady state as the time goes to the infinity. It is shown that the difference of densities of two carriers converge to the equilibrium states with the norm ｜｜·｜｜H^s-1, while the velocities and the electromagnetic fields converge to the equilibrium states with weaker norms than ｜｜·｜｜H^s-1. This phenomenon on the charge transport shows the essential difference between the unipolar Navier-Stokes-Maxwell and the bipolar Navier-Stokes-Maxwell system.

On the Rayleigh-Plateau instability

In this paper,we study the surface instability of a cylindrical pore in the absence of stress.This instability is called the Rayleigh-Plateau instabilty.We consider the model developed by Spencer et al.[18],Kirill et al.[10]and Boutat et al.[2]in the case without stress.We obtain a nonlinear parabolic PDE of order four.We show the local existence and uniqueness of the solution of this problem by using Faedo-Galerkin method.The main results are the global existence of the solution and the convergence to the mean value of the initial data for long time.Numerical tests are also presented in this study.

RENORMALIZED SOLUTIONS OF ELLIPTIC EQUATIONS WITH ROBIN BOUNDARY CONDITIONS

In the present paper, we consider elliptic equations with nonlinear and nonlao mogeneous Robin boundary conditions of the type {-div（B（x,u）△u） = f in Ω,u=0 on Гo, B（x,u）Vu·n^-＋γ（x）h（u） = 9 on Г1,where f and g are the element of L^1（Ω） and L^1（Г1）, respectively. We define a notion of renormalized solution and we prove the existence of a solution. Under additionM assumptions on the matrix field B we show that the renormalized solution is unique.

ENTROPICAL OPTIMAL TRANSPORT,SCHRODINGER'S SYSTEM AND ALGORITHMS

In this exposition paper we present the optimal transport problem of Monge-Ampère-Kantorovitch(MAK in short)and its approximative entropical regularization.Contrary to the MAK optimal transport problem,the solution of the entropical optimal transport problem is always unique,and is characterized by the Schrödinger system.The relationship between the Schrödinger system,the associated Bernstein process and the optimal transport was developed by Léonard[32,33](and by Mikami[39]earlier via an h-process).We present Sinkhorn’s algorithm for solving the Schrödinger system and the recent results on its convergence rate.We study the gradient descent algorithm based on the dual optimal question and prove its exponential convergence,whose rate might be independent of the regularization constant.This exposition is motivated by recent applications of optimal transport to different domains such as machine learning,image processing,econometrics,astrophysics etc..