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Abstract: Abstract Leukaemia accounts for around 3% of all cancer types diagnosed in adults, and is the most common type of cancer in children of paediatric age (typically ranging from 0 to 14 years). There is increasing interest in the use of mathematical models in oncology to draw inferences and make predictions, providing a complementary picture to experimental biomedical models. In this paper we recapitulate the state of the art of mathematical modelling of leukaemia growth dynamics, in time and response to treatment. We intend to describe the mathematical methodologies, the biological aspects taken into account in the modelling, and the conclusions of each study. This review is intended to provide researchers in the field with solid background material, in order to achieve further breakthroughs in the promising field of mathematical biology. PubDate: 2022-05-03

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Abstract: Abstract This paper proposed an effective conjugate gradient parameter via the hybridization of quasi-Newton and conjugate gradient search directions. The method gives a sufficient descent direction using the inexact line search technique. Global convergence analysis of the scheme is obtained using some mild conditions. Numerical experiments with some conjugate gradient methods in the literature show that the proposed approach is very efficient. PubDate: 2022-04-12

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Abstract: Abstract In this paper, we establish the sufficient Karush–Kuhn–Tucker (for short, KKT) conditions of a set-valued optimization problem (P) via contingent epiderivative and \(\rho \) -cone arcwise connectedness assumptions. We also formulate the Mond–Weir type (MWD), Wolfe type (W D), and mixed type (MD) duals of (P) and prove the corresponding weak, strong, and converse duality theorems. PubDate: 2022-04-09

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Abstract: Abstract The purpose of this paper is to introduce a new class of bilevel problem in the frame work of a real Hilbert space. In addition, we introduce an inertial iterative method with a regularization term and we establish the strong convergence of the resulting methods under certain conditions imposed on regularization parameters. Finally, we present some numerical experiments to show the efficiency and applicability of the proposed method. The results obtained in this paper extend, generalize and improve several results in this direction. PubDate: 2022-04-04

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Abstract: Abstract In this paper, we mainly construct and analyze a model of competition between two populations of cells for the glucose in a tissue. We first study the existence and global asymptotic stability of a positive equilibrium point of the model with one population of cells. After that, the sufficient conditions are established for uniform persistence of the competition model from what proceeds and by the mean of the Thieme-Zhao Theorem. Finally, we also give some numerical simulations to illustrate these results and conclusions. PubDate: 2022-03-26

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Abstract: Abstract The nonlinear fractional Boussinesq equations are known as the fractional differential equation class that has an important place in mathematical physics. In this study, a method called \(\Big (\frac{G'}{G^2}\Big )\) -extension method which works well and reveals exact solutions is used to examine nonlinear Boussinesq equations with conformable time-fractional derivative. This method is a very useful approach and extremely utility compared to other analytical methods. With the proposed method, there are three unique types of solutions such as hyperbolic, trigonometric and rational solutions. This approach can similarly be applied to other nonlinear fractional models. PubDate: 2022-03-24

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Abstract: Abstract In this paper we find viscosity solutions to a coupled system composed by two equations, the first one is an obstacle type equation, \(\min \{-\Delta ^1_{\infty }u(x),(u-v)(x)\}=0\) , and the second one is \(- \displaystyle \Delta v(x) + v(x) - u(x)=h(x)\) in \(\Omega \) a smooth bounded domain with Dirichlet boundary conditions \(u(x) = f(x)\) , \(v(x) = g(x)\) for \(x \in \partial \Omega \) . Here \(-\Delta ^1_{\infty }u\) is the \(\infty -\) Laplacian and \(-\Delta v\) is the standard Laplacian. This system is not variational and involves two different elliptic operators. Notice that in the first equation the obstacle is given by the second component of the system that also depends on the first component via the second equation (this system is fully coupled). We prove that there is a two-player zero-sum game played in two different boards with different rules in each board. In the first one one of the players decides to play a round of a Tug-of-War game or to change boards and in the second board we play a random walk with the possibility of changing boards with a positive (but small) probability and a running payoff. We show that this game has two value functions (one for each board) that converge uniformly to the components of a viscosity solution to the PDE system. PubDate: 2022-03-21

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Abstract: Abstract We present a general class of derivative free iterative methods with optimal order of convergence for solving nonlinear equations. The methodology is based on quadratically convergent Traub–Steffensen scheme and inverse Padé approximation. Unlike that of existing higher order techniques the proposed technique is attractive since it leads to a simple implementation. Numerical examples are provided to confirm the theoretical results and to show the feasibility and efficacy of the new methods. The performance is compared with well established methods in literature. Computational results, including the elapsed CPU-time, confirm the accurate and efficient character of proposed techniques. Moreover, the stability of the methods is checked through complex geometry shown by drawing basins of attraction. PubDate: 2022-03-15

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Abstract: Abstract This work is concerned with a problem of a vibrating system of Timoshenko-type in a bounded one-dimensional domain under Dirichlet boundary conditions with two fractional time delays and two internal frictional dampings. Under a smallness condition on the fractional delay and by using a classical semigroup theory we prove existence and uniqueness of solutions. Furthermore, by a frequency domain approach we prove an exponential stability result. PubDate: 2022-03-12

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Abstract: Abstract This study proposed a new scheme called the Hermite wavelet method (HWM) to find the numerical solutions to the multidimensional fractional coupled Navier–Stokes equation (NSE). This approach is based on the Hermite wavelets approximation with collocation points. Here, we reduce the fractional NSE into a set of nonlinear algebraic equations involving Hermite wavelet unknown coefficients. Convergence analysis is explained through the theorems. Three examples are given to validate the proposed technique’s efficiency and discussed the comparison between the present method solutions with the exact solution. The obtained results are represented through graphs and tables for both integer and fractional order. These results disclose that the existing algorithm offers a better result. PubDate: 2022-03-10

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Abstract: Abstract Recently, motivated by problems in image processing, by the analysis of the peridynamic formulation of the continuous mechanic and by the study of Markov jump processes, there has been an increasing interest in the research of nonlocal partial differential equations. In the last years and with these problems in mind, we have studied some gradient flows in the general framework of a metric random walk space, that is, a Polish metric space (X, d) together with a probability measure assigned to each \(x\in X\) , which encode the jumps of a Markov process. In this way, we have unified into a broad framework the study of partial differential equations in weighted discrete graphs and in other nonlocal models of interest. Our aim here is to provide a summary of the results that we have obtained for the heat flow and the total variational flow in metric random walk spaces. Moreover, some of our results on other problems related to the diffusion operators involved in such processes are also included, like the ones for evolution problems of p-Laplacian type with nonhomogeneous Neumann boundary conditions. PubDate: 2022-03-01

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Abstract: Abstract We review recent rigorous results on the phenomenon of vortex reconnection in classical and quantum fluids. In the context of the Navier–Stokes equations in \(\mathbb {T}^3\) we show the existence of global smooth solutions that exhibit creation and destruction of vortex lines of arbitrarily complicated topologies. Concerning quantum fluids, we prove that for any initial and final configurations of quantum vortices, and any way of transforming one into the other, there is an initial condition whose associated solution to the Gross–Pitaevskii equation realizes this specific vortex reconnection scenario. Key to prove these results is an inverse localization principle for Beltrami fields and a global approximation theorem for the linear Schrödinger equation. PubDate: 2022-03-01

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Abstract: Abstract The Estrada index of a graph/network is defined as the trace of the adjacency matrix exponential. It has been extended to other graph-theoretic matrices, such as the Laplacian, distance, Seidel adjacency, Harary, etc. Here, we describe many of these extensions, including new ones, such as Gaussian, Mittag–Leffler and Onsager ones. More importantly, we contextualize all of these indices in physico-mathematical frameworks which allow their interpretations and facilitate their extensions and further studies. We also describe several of the bounds and estimations of these indices reported in the literature and analyze many of them computationally for small graphs as well as large complex networks. This article is intended to formalize many of the Estrada indices proposed and studied in the mathematical literature serving as a guide for their further studies. PubDate: 2022-03-01

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Abstract: Abstract Franciso Javier Sayas, man of grit and determination, left his hometown of Zaragoza in 2007 in pursuit of a dream, to become a scholar in the USA. I hosted him in Minneapolis, where he spent three long years of an arduous transition before obtaining a permanent position at the University of Delaware. There, he enthusiastically worked on the unfolding of his dream until his life was tragically cut short by cancer, at only 50. In this paper, I try to bring to light the part of his academic life he shared with me. As we both worked on hybridizable discontinuous Galerkin methods, and he wrote a book on the subject, I will tell Javier’s life as it developed around this topic. First, I will show how the ideas of static condensation and hybridization, proposed back in the mid 60s, lead to the introduction of those methods. This background material will allow me to tell the story of the evolution of the hybridizable discontinuous Galerkin methods and describe Javier’s participation in it. Javier faced death with open eyes and poised dignity. I will end with a poem he liked. PubDate: 2022-03-01

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Abstract: Abstract We develop an extended convergence ball for an efficient eighth order method to obtain numerical solutions of Banach space valued nonlinear models. Convergence of this algorithm has previously been shown using assumptions up to the ninth derivative. However, in our convergence theorem, we use only the first derivative. As a consequence, in contrast to previous ideas, the results on calculable error bounds, convergence radius and uniqueness zone for the solution are provided. Furthermore, this scheme is applied to several complex polynomials and related attraction basins are displayed. The results of numerical tests are presented and compared with the earlier technique. We arrive at the conclusion that the suggested analysis produces much larger convergence radii in all tests. Hence, we expand the convergence domain of this iterative formula. PubDate: 2022-02-23 DOI: 10.1007/s40324-022-00287-0

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Abstract: Abstract The general method of graph coarsening or graph reduction has been a remarkably useful and ubiquitous tool in scientific computing and it is now just starting to have a similar impact in machine learning. The goal of this paper is to take a broad look into coarsening techniques that have been successfully deployed in scientific computing and see how similar principles are finding their way in more recent applications related to machine learning. In scientific computing, coarsening plays a central role in algebraic multigrid methods as well as the related class of multilevel incomplete LU factorizations. In machine learning, graph coarsening goes under various names, e.g., graph downsampling or graph reduction. Its goal in most cases is to replace some original graph by one which has fewer nodes, but whose structure and characteristics are similar to those of the original graph. As will be seen, a common strategy in these methods is to rely on spectral properties to define the coarse graph. PubDate: 2022-01-10 DOI: 10.1007/s40324-021-00282-x

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Abstract: Abstract The general method of graph coarsening or graph reduction has been a remarkably useful and ubiquitous tool in scientific computing and it is now just starting to have a similar impact in machine learning. The goal of this paper is to take a broad look into coarsening techniques that have been successfully deployed in scientific computing and see how similar principles are finding their way in more recent applications related to machine learning. In scientific computing, coarsening plays a central role in algebraic multigrid methods as well as the related class of multilevel incomplete LU factorizations. In machine learning, graph coarsening goes under various names, e.g., graph downsampling or graph reduction. Its goal in most cases is to replace some original graph by one which has fewer nodes, but whose structure and characteristics are similar to those of the original graph. As will be seen, a common strategy in these methods is to rely on spectral properties to define the coarse graph. PubDate: 2022-01-10 DOI: 10.1007/s40324-021-00282-x

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Abstract: Abstract A stochastic Lagrangian model for simulating the dynamics and rheology of a Brownian multi-particle system interacting with a simple liquid medium is presented. The discrete particle model is formulated within the GENERIC framework for Non-Equilibrium Thermodynamics and therefore it satisfies discretely the First/Second Laws of Thermodynamics and the Fluctuation Dissipation Theorem (FDT). Long-range fluctuating hydrodynamics interactions between suspended particles are described by an explicit solvent model. To this purpose, the Smoothed Dissipative Particle Dynamics method is adopted, which is a GENERIC-compliant Lagrangian meshless discretization of the fluctuating Navier–Stokes equations. In dense multi-particle systems, the average inter-particle distance is typically small compared to the particle size and short-range hydrodynamics interactions play a major role. In order to bypass an explicit—computationally costly—solution for these forces, a lubrication correction is introduced based on semi-analytical expressions for spheres under Stokes flow conditions. We generalize here the lubrication formalism to Brownian conditions, where an additional thermal-lubrication contribution needs to be taken into account in a way that discretely satisfies FDT. The coupled lubrication dynamics is integrated in time using a generalized semi-implicit splitting scheme for stochastic differential equations. The model is finally validated for a single particle diffusion as well as for a Brownian multi-particle system under homogeneous shear flow. Results for the diffusional properties as well as the rheological behavior of the whole suspension are presented and discussed. PubDate: 2022-01-08 DOI: 10.1007/s40324-021-00280-z

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Abstract: We present an abstract framework for the eigenvalue approximation of a class of non-coercive operators. We provide sufficient conditions to guarantee the spectral correctness of the Galerkin scheme and to obtain optimal rates of convergence. The theory is applied to the convergence analysis of mixed finite element approximations of the elasticity and Stokes eigensystems. PubDate: 2022-01-04 DOI: 10.1007/s40324-021-00279-6