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It is widely recognised that high-stakes assessment can significantly influence what is taught in the classroom. Many argue that high-stakes assessment results in a narrowed curriculum where students learn by rote rather than developing higher cognitive skills. This paper describes a study investigating the various cognitive objectives present from Bloom’s Taxonomy Educational Objectives on the Leaving Certificate biology examination. The study analysed examination papers from the past and current biology syllabuses. Analysis was also carried out to determine the marks being awarded to the different cognitive objectives. The findings show that the examination predominately includes questions that do not promote higher levels of thinking. The majority of the marks on the paper were allocated to the lower objectives of the taxonomy, suggesting students can rely on rote learning to succeed when undertaking the biology examination. This study strongly highlights how high-stake examinations have a narrow scope in terms of student achievement and shows how current biology examination procedures promote low-level learning. This low level of thinking promotes rote learning and regurgitation of facts, requiring little to no understanding of the topics. To prepare students for the working world, there needs to be a shift from only terminal exams to a mixed approach.

The purpose of this paper is twofold. Firstly, certain common fixed point theorems are established via $ \phi $-contractive multivalued mappings involving point-dependent control functions as coefficients in the framework of complex valued metric spaces. Our results improve and extend several results in the existing literature. Moreover, this section is equipped by some illustrative examples in support of our results. Secondly, we point out some slip-ups in the examples of some recent papers Ahmad, Klin-Eam, and Azam (2013) and Kutbi, Ahmad, Azam, and Al-Rawashdeh (2014) based on multivalued contractive mappings in complex valued metric spaces. Our observations are also authenticated with the aid of some appropriate examples. Some rectifications to correct the erratic examples are also suggested.

This short paper characterizes strictly convex sets by the uniqueness of support points (such points are called unique support points or exposed points) under appropriate assumptions. A class of so...

Viewing from polynomials solutions of q-difference equations makes certain generating functions calculated easily. In this short paper, we build the relations between Cigler’s polynomials and q-difference equations. In addition, we deduce generating functions and bilinear generating functions for Cigler’s polynomials. More over, we gain multilinear and mixed generating functions for Cigler’s polynomials. At last, we gain U(n+1) type generating functions for Cigler’s polynomials.

This paper concerns noncooperative difference games with infinite horizon discounted payoffs. More precisely, it is a sequel to a previous paper [A. Fonseca-Morales and O. Hern'andez-Lerma, Potenti...

This paper is concerned with the intrinsic geometric structures of conductive transmission eigenfunctions. The geometric properties of interior transmission eigenfunctions were first studied in [9]. It is shown in two scenarios that the interior transmission eigenfunction must be locally vanishing near a corner of the domain with an interior angle less than $\pi$. We significantly extend and generalize those results in several aspects. First, we consider the conductive transmission eigenfunctions which include the interior transmission eigenfunctions as a special case. The geometric structures established for the conductive transmission eigenfunctions in this paper include the results in [9] as a special case. Second, the vanishing property of the conductive transmission eigenfunctions is established for any corner as long as its interior angle is not $\pi$ when the conductive transmission eigenfunctions satisfy certain Herglotz functions approximation properties. That means, as long as the corner singularity is not degenerate, the vanishing property holds if the underlying conductive transmission eigenfunctions can be approximated by a sequence of Herglotz functions under mild approximation rates. Third, the regularity requirements on the interior transmission eigenfunctions in [9] are significantly relaxed in the present study for the conductive transmission eigenfunctions. Finally, as an interesting and practical application of the obtained geometric results, we establish a unique recovery result for the inverse problem associated with the transverse electromagnetic scattering by a single far-field measurement in simultaneously determining a polygonal conductive obstacle and its surface conductive parameter.

In this paper, the local and global dynamics of a two-stage Bertrand dynamic duopoly model with R&D spillover are studied. This paper aims to study the complex dynamics of the built model by changi...

The paper is devoted to the duality of the Mayer problem for κ-th order viable differential inclusions with endpoint constraints, where κ is an arbitrary natural number. Thus, this paper for constr...

This paper solves a question raised in the paper of Huong, Yao, and Yen [Optimal processes in a parametric optimal economic growth model. Taiwanese J Math. 2020. Available from: https://doi.org/10....

This paper presents asymptotic formulas in the case of the following two problems for the {\it Pucci's extremal operators} $\mathcal{M}^\pm$. It is considered the solution $u^\varepsilon(x)$ of $-\varepsilon^2 \mathcal{M}^\pm\left(\nabla ^2 u^\varepsilon\right)+u^\varepsilon=0$ in $\Omega$ such that $u^\varepsilon=1$ on $\Gamma$. Here, $\Omega\subset \mathbb{R}^N$ is a domain (not necessarily bounded) and $\Gamma$ is its boundary. It is also considered $v(x,t)$ the solution of $v_t - \mathcal{M}^\pm\left(\nabla^2 v\right)=0$ in $\Omega\times (0,\infty)$, $v=1$ on $\Gamma\times(0,\infty)$ and $v=0$ on $\Omega\times \{0\}$. In the spirit of their previous works, the authors establish the profiles as $\varepsilon$ or $t\to 0^+$ of the values of $u^\varepsilon(x)$ and $v(x,t)$ as well as of those of their $q$-means on balls touching $\Gamma$. The results represent a further step in the extensions of those obtained by Varadhan and by Magnanini-Sakaguchi in the linear regime., Comment: 15 pages. Minor typos fixed. The article, which is dedicated to the 65th birthday of Sergio Vessella, has been accepted by Applicable Analysis: Special Issue on Partial Differential Equations, Inverse and Ill-Posed Problems, Unique Continuation and Applications. The paper is already available in its online version

The outbreak of COVID-19 was first experienced in Wuhan City, China, during December 2019 before it rapidly spread over globally. This paper has proposed a mathematical model for studying its transmission dynamics in the presence of face mask wearing and hospitalization services of human population in Tanzania. Disease-free and endemic equilibria were determined and subsequently their local and global stabilities were carried out. The trace-determinant approach was used in the local stability of disease-free equilibrium point while Lyapunov function technique was used to determine the global stability of both disease-free and endemic equilibrium points. Basic reproduction number, $R_0 $, was determined in which its numerical results revealed that, in the presence of face masks wearing and medication services or hospitalization as preventive measure for its transmission, $R_0=0.698 $ while in their absence $R_0=3.8 $. This supports its analytical solution that the disease-free equilibrium point $E_0 $ is asymptotically stable whenever $R_0 \lt 1 $, while endemic equilibrium point $E_* $ is globally asymptotically stable for $R_0 \gt 1 $. Therefore, this paper proves the necessity of face masks wearing and hospitalization services to COVID-19 patients to contain the disease spread to the population.

In a number of recent papers, the idea of generalized boundaries has found use in fractal and in multiresolution analysis; many of the papers having a focus on specific examples. Parallel w...

This paper introduces a new mapping that is weaker than nonexpansive mapping. The new mapping is different from the Suzuki’s generalized nonexpansive mapping. This paper introduces a new iteration ...

In this paper, based on the previous work by Bai [On the convergence of the multisplitting methods for the linear complementarity problem. SIAM J Matrix Anal Appl. 1999;21:67–78] and by Zhang et al. [Improved convergence theorems of multisplitting methods for the linear complementarity problem. Appl Math Comput. 2014;243:982–987], we further discuss the multisplitting AOR (MAOR) method for the linear complementarity problem. Some new convergence conditions of the MAOR method are obtained, which are different from those of the aforementioned papers.

This paper presents a non-intrusive framework for integrating existing unsteady partial differential equation (PDE) solvers into a parallel-in-time simultaneous optimization algorithm. The time-parallelization is provided by the non-intrusive software library XBraid, which applies an iterative multigrid reduction technique to the time domain of existing time-marching schemes for solving unsteady PDEs. Its general user-interface has been extended for computing adjoint sensitivities such that gradients of output quantities with respect to design changes can be computed parallel-in-time alongside with the primal PDE solution. In this paper, the primal and adjoint XBraid iterations are embedded into a simultaneous optimization framework, namely the One-shot method. In this method, design updates towards optimality are employed after each state and adjoint update such that optimality and feasibility of the design and the PDE solution are reached simultaneously. The time-parallel optimization method is validated on an advection-dominated flow control problem which shows significant speedup over a classical time-serial optimization algorithm.

In this paper, we study properties of a special class of ordered median functions, called block-functions. These are ordered median functions which belong to a generating binary (row)-vector of the form called a block vector. The aim of this paper is to explicitly determine the simplicial complexes and all steepest descent and ascent directions of descent and ascent cones of ordered median block-function.

We consider the numerical computation of a variational problem that arises from materials science. The target functional is a type of elastic energy that is influenced by obstacles and adhesion. Owing to its strong nonlinearity and discontinuity, the Euler-Lagrange equation is very complicated, and numerical computation of its critical points is difficult. In this paper, we discretize and regularize the target energy as a functional defined on a space of polygonal curves. Moreover, we develop convergence analysis for discrete minimizers in the framework of $\Gamma$-convergence. We first show that the discrete energy functional $\Gamma$-converges to the original one. Then, we establish the compactness property for the sequence of discrete minimizers. These two results allow us to extract a convergent subsequence from the discrete minimizers. We also present some numerical examples in the last section of the paper. Existence of singular local minimizers is suggested by numerical experiments., Comment: 22pages, 7 figures, 5 tables

The paper studies the evolution of the thermomechanical and electric state of a thermoviscoelastic thermistor that is in frictional contact with a reactive foundation. The mechanical process is dynamic, while the electric process is quasistatic. Friction is modeled with a nonmonotone relation between the tangential traction and tangential velocity. Frictional heat generation is taken into account and so is the strong dependence of the electric conductivity on the temperature. The mathematical model for the process is in the form of a system that consists of dynamic hyperbolic subdifferential inclusion for the mechanical state coupled with a nonlinear parabolic equation for the temperature and an elliptic equation for the electric potential. The paper establishes the existence of a weak solution to the problem by using time delays, a priori estimates and a convergence method., 25 pages

In this paper, we solve open problem (5) submitted by Sedaghat in his paper, On third order rational difference equations with quadratic terms, J. Differ. Equ. Appl., 14(8) (2008), pp. 889–897. We also confirm conjecture (6) in the mentioned paper.

In this paper, we introduce a new concept of Poisson Stepanov-like almost automorphy (or Poisson S2-almost automorphy). Under some suitable conditions on the coefficients, we establish the existence and uniqueness of Stepanov-like almost automorphic mild solution to a class of semilinear stochastic differential equations with infinite dimensional Levy noise. We further discuss the global asymptotic stability of these solution. Finally, we give an example to illustrate the theoretical results obtained in this paper.

A highly efficient numerical approach: extended cohesive damage model (ECDM) for predicting multicrack propagation is introduced in this paper. The ECDM is developed within the framework of the eXtended Finite Element Method (XFEM). Unlike XFEM the enriched degrees of freedom (DoFs) are eliminated from the final condensed equilibrium equations in the ECDM. To account for the cohesive crack effect, an equivalent damage scalar relating to a strain field is introduced in terms of energy dissipation. The ECDM is capable of characterizing discontinuities with conventional DoFs only, thus it is significantly efficient in modelling multicrack propagation. The basic formulations, numerical implementation and detailed investigation of the performance of the ECDM through modelling the selected benchmark specimens are given in this paper. This investigation shows the ECDM can effectively guarantee the convergent solutions in nonlinear fracture analysis and can efficiently reduce the computer CPU time in modelling selected fracture benchmark specimens by more than 60% compared to the XFEM in ABAQUS. Therefore, the ECDM is a robust computational approach for predicting multicrack failure mechanism in engineering materials and structures.

An analysis was performed in this paper to study the problem of magnetohydrodynamic squeezing flow of nanofluid between parallel disks. This paper also proposed a reproducing kernel method ...

In the first part of the paper we deal with a second-order evolution variational inequality involving a multivalued term generated by a Clarke subdifferential of a locally Lipschitz potential. For this problem we construct a time-semidiscrete approximation, known as the Rothe scheme. We study a sequence of solutions of the semidiscrete approximate problems and provide its weak convergence to a limit element that is a solution of the original problem. Next, we show that the solution is unique and the convergence is strong. In the second part of the paper, we consider a dynamic visco-elastic problem of contact mechanics. We assume that the contact process is governed by a normal damped response condition with a unilateral constraint and the body is non-clamped. The mechanical problem in its weak formulation reduces to a variational–hemivariational inequality that can be solved by finding a solution of a corresponding abstract problem related to one studied in the first part of the paper. Hence, we a...

In this paper, we are concerned with the existence of solutions for a critical p-Kirchhoff type problem driven by a nonlocal integro-differential operator:where is a continuous function, is a singular kernel function, is a nonlocal fractional operator, with , , f is a Caratheodory function on satisfying the Ambrosetti–Rabinowitz type condition. Under some suitable assumptions, we obtain the existence of nontrivial solutions for above problem by applying the mountain pass theorem. A distinguished feature of this paper is that M(0) may be zero, which means that the problem is degenerate. Consequently, the main theorem extends in several directions the recent results of Autuori, Fiscella and Pucci [Nonlinear Anal. 2015;125:699–714].

This paper deals with dissipativity of the theoretical solutions and numerical solutions of a general class of advection-reaction-diffusion equations with both fixed and distributed delays. Some sufficient conditions for the dissipativity of the mentioned problems are given. It is shown that adaptations of linear θ-methods and one-leg θ-methods discussed in the present paper have the ability to preserve dissipativity of the underlying systems when $1/2\leq \theta \leq 1$. Finally, numerical experiment is presented to confirm the theoretical results.

In the aspect of numerical methods for incompressible flow problems, there are two different algorithms: semi-implicit method for pressure-linked equations (SIMPLE) series algorithms and the pressure Poisson algorithm. This paper introduced a new discretized pressure Poisson algorithm for the steady incompressible flow based on a nonstaggered grid. Compared with the SIMPLE series algorithms, this paper did not introduce three correction variables. So, there is no need to implement the guess-and-correct procedure for the calculation of pressure and velocity. Compared with the pressure Poisson algorithm, there is no need to calculate unsteady Navier–Stokes equations for steady problems in the new discretized pressure Poisson algorithm. Meanwhile, as the finite volume method and cell-centered grid are used, the governing equation for pressure is obtained from the continuity equation and the boundary conditions for pressure are easily obtained. This new discretized pressure Poisson algorithm was teste...

In this paper, the exponential stability of travelling waves solutions for nonlinear cellular neural networks with distribute delays in the lattice is studied. The weighted energy method and comparison principle are employed to derive the sufficient conditions under which the networks proposed are exponentially stable. Following the study [13] on the existence of the travelling wave solutions in nonlinear delayed cellular neural networks, this paper is focused on the exponential stability of these travelling wave solutions.

In this paper, we present a regression hybrid method that calculates shape sensitivity coefficients for multiscale crack propagation problems with performance measures that are non-differentiable in numerical implementation. These measures are crack propagation speed (or crack speed) defined at atomistic level obtained by solving coupled atomistic/continuum structures using the bridging scale method (BSM). The major contributions of this paper are: first, by analyzing the characteristics of the performance measures of crack speed in design space, this paper verifies for the first time that these measures are theoretically continuous and differentiable with respect to design variables, and as a result, the sensitivity coefficients exist in theory; second, to overcome the non-differentiability of the performance measures in numerical computation due to the finite size of integration time step, this paper proposes a regression hybrid method that calculates the shape sensitivity coefficients of crack speed ...

In this paper, proper minimal elements of a given nonconvex set in a real ordered Banach space are defined utilizing the limiting (Mordukhovich) normal cone. The newly defined points are called limiting proper minimal (LPM) points. It is proved that each LPM is a proper minimal in the sense of Borwein under some assumptions. The converse holds in Asplund spaces. The relation of LPM points with Benson, Henig, super and proximal proper minimal points are established. Under appropriate assumptions, it is proved that the set of robust elements is a subset of the set of LPM points, and the set of LPM points is dense in that of minimal points. Another part of the paper is devoted to scalarization-based and distance function-based characterizations of the LPM points. The paper is closed by some results about LPM solutions of a set-valued optimization problem via variational analysis tools. Clarifying examples are given in addition to the theoretical results.

In this paper we introduce the theory of dominant solutions at infinity for nonoscillatory discrete symplectic systems without any controllability assumption. Such solutions represent an opposite concept to recessive solutions at infinity, which were recently developed for such systems by the authors. Our main results include: (i) the existence of dominant solutions at infinity for all ranks in a given range depending on the order of abnormality of the system, (ii) construction of dominant solutions at infinity with eventually the same image, (iii) classification of dominant and recessive solutions at infinity with eventually the same image, (iv) limit characterization of recessive solutions at infinity in terms of dominant solutions at infinity and vice versa, and (v) Reid’s construction of the minimal recessive solution at infinity. These results are based on a new theory of genera of conjoined bases for symplectic systems developed for this purpose in this paper.

The present paper deals with the parabolic-parabolic Keller-Segel equation in the plane inthe general framework of weak (or ``free energy") solutions associated to an initial datum with finite mass $M< 8\pi$, finite second log-moment and finite entropy. The aim of the paper is twofold:(1) We prove the uniqueness of the ``free energy" solution. The proof uses a DiPerna-Lions renormalizing argument which makes possible to get the ``optimal regularity" as well as an estimate of the difference of two possible solutions in the critical $L^{4/3}$ Lebesgue norm similarly as for the $2d$ vorticity Navier-Stokes equation. (2) We prove a radially symmetric and polynomial weighted $L^2$ exponential stability of the self-similar profile in the quasi parabolic-elliptic regime. The proof is based on a perturbation argument which takes advantage of the exponential stability of the self-similar profile for the parabolic-elliptic Keller-Segel equation established by Campos-Dolbeault and Egana-Mischler.

This paper aims at providing a first step toward a qualitative theory for a new class of chemotaxis models derived from the celebrated Keller-Segel system, with the main novelty being that diffusion is nonlinear with flux delimiter features. More precisely, as a prototypical representative of this class we study radially symmetric solutions of the parabolic-elliptic system (see the text). Under the initial condition $u|_{t=0}=u_0>0$ and no-flux boundary conditions in balls $\Omega\subset\mathbb{R}^n$, where $\chi>0$ and $\mu:=\frac{1}{|\Omega|} \int_\Omega u_0$.\abs The main results assert the existence of a unique classical solution, extensible in time up to a maximal $T_{max} \in (0,\infty]$ which has the property that $$\mbox{if} \quad T_{max}0 \ \mbox{ and } m1$ and $m_c:=\infty$ if $\chi\le 1$. That these conditions are essentially optimal will be shown in a forthcoming paper in which ($\star$) will be used to derive complementary results on the occurrence of solutions blowing up in finite time with respect to the norm of $u$ in $L^\infty(\Omega)$.

In this paper, our aim is to show some mean value inequalities for the Wright function, such as Turan-type inequalities, Lazarevic-type inequalities, Wilker-type inequalities and Redheffer-type inequalities. Moreover, we prove monotonicity of ratios for sections of series of Wright functions, the result is also closely connected with Turan-type inequalities. In the end of the paper, we present some other inequalities for the Wright function.

The paper is devoted to properties of set-valued stochastic differential equations. The main result of the paper deals with existence and uniqueness of solutions. Furthermore, a connection between ...

An analytical expression for the optimal control of an Ebola problem is obtained. The analytical solution is found as a first-order approximation to the Pontryagin Maximum Principle via the Euler-Lagrange equation. An implementation of the method is given using the computer algebra system Maple. Our analytical solutions confirm the results recently reported in the literature using numerical methods., Comment: This is a preprint of a paper whose final and definite form is in International Journal for Computational Methods in Engineering Science and Mechanics, ISSN 1550-2287 (Print), 1550-2295 (Online). Paper Submitted 14-Jul-2015; Revised 29-Oct-2015; Accepted for publication 09-Dec-2015

This paper is concerned with stochastic fractional nonlinear Schrodinger equation, which plays a very important role in fractional nonrelativistic quantum mechanics. Due to disturbing and interacting of the fractional Laplacian operator on a bounded interval with white noise, the stochastic fractional nonlinear Schrodinger equation is too complicated to be understood. This paper would explore and analyze this stochastic fractional system. Using a suitable weighted space with some fractional operator skills, it overcame the difficulties coming from the fractional Laplacian operator on a bounded interval. Applying the tightness instead of the common compactness, and combining Prokhorov theorem with Skorokhod embedding theorem, it solved the convergence problem in the case of white noise. It finally established the existence of martingale solutions for the stochastic fractional nonlinear Schrodinger equation on a bounded interval.

This paper presents an approach to solving truss topology optimization problem with small uncertainty in the locations of the structural nodes. The nodal locations in the truss are assumed to be random, and the probabilistic method is used here to deal with the uncertainty. The objective of the optimization problem is to minimize the mean compliance of the truss structure under nodal location uncertainty. It is a well-acknowledged barrier to compute the inverse of the structural stiffness matrix which involves variations in the optimization problem. In this paper, based on Neumann series expansion, this optimization problem can be recast into a simpler deterministic structural optimization problem. In order to avoid the sensitivity calculations for the objective function, the proportional topology optimization method which shows comparable efficiency and accuracy with gradient-based method is used. The numerical examples demonstrate the effectiveness and high efficiency of the proposed approach, and f...

In part I of this paper, a new physical-based computational approach for the solution of convection heat transfer problems on co-located non-orthogonal grids in the context of an element-based finite volume method was discussed. The test problems are presented here, in part II of the paper. These problems include five steady two-dimensional convection heat transfer problems. In all test cases, the convergence history, the required under-relaxations for the iterative solution of the linearized equations, and the order of accuracy of the method are discussed and the streamlines as well as isotherms are presented. The computational results show that the proposed method is second order accurate and might occasionally need mild under-relaxation in relatively complex problems. Excellent match between the computational results and the corresponding reliable published results is observed.

The purpose of this paper is to introduce a new physical-based computational approach for the solution of convection heat transfer problems on co-located non-orthogonal grids in the context of an element-based finite volume method. The approach has already been presented in the context of two-dimensional incompressible flow problems without heat transfer. It has been shown that the pressure–velocity coupling on co-located grids can be correctly modeled via the so-called method of proper closure equations (MPCE). Here, MPCE is extended to the numerical simulation of natural, forced, and mixed convection heat transfer problems. It is shown that the couplings between pressure, velocity, and temperature can be conveniently handled on co-located grids by resorting again to the modified forms of the governing equations, i.e., the proper closure equations. The set of discrete equations is solved in a fully coupled manner in this study. Here, in part I of the paper, only the basic methodology is described...

In this paper, we develop and analyze an adaptive multiscale approach for heterogeneous problems in perforated domains. In many applications, these problems have a multiscale nature arising because of the perforations, their geometries, the sizes of the perforations, and configurations. In this paper, we present a general offline/online procedure, which can adequately and adaptively represent the local degrees of freedom and derive appropriate coarse-grid equations. The main contributions of this paper are (1) the rigorous analysis of the offline approach (2) the development of the online procedures and their analysis (3) the development of adaptive strategies. We present an online procedure, which allows adaptively incorporating global information and is important for a fast convergence when combined with the adaptivity. Our methodology allows adding and guides constructing new online multiscale basis functions adaptively in appropriate regions. We present the convergence analysis of the online adaptive enrichment algorithm for the Stokes system. In particular, we show that the online procedure has a rapid convergence with a rate related to the number of offline basis functions, and one can obtain fast convergence by a sufficient number of offline basis functions, which are computed in the offline stage. To illustrate the performance of our method, we present numerical results with both small and large perforations. We see that only a few (1 or 2) online iterations can significantly improve the offline solution.

The present paper, based on the vorticity–velocity formulation of the Navier–Stokes equations, proposes an immersed boundary method for the simulation of heat transfer problems within a geometrically complex domain. The desired boundary conditions are imposed by the direct modification of the initial conditions of vorticity transport and energy equations using smooth interpolations. The time advancement of both transport equations is performed by the explicit fourth-order Runge–Kutta method. One of the main objectives of this paper is to present global smooth interpolations to evaluate the local Nusselt number. The forced convection of moving and fixed circular cylinders, natural convection problem in complex geometries, and the mixed convection between two concentric cylinders—at various Reynolds numbers—are studied.

This paper is concerned with efficient numerical methods for solving the time-dependent scattering and inverse scattering problems of acoustic waves in a locally perturbed half-plane. By symmetric continuation, the scattering problem is reformulated as an equivalent symmetric problem defined in the whole plane. The retarded potential boundary integral equation method is modified to solve the forward problem. Then we consider the inverse scattering problem of determinating the local perturbation from the measured scattered data. The time domain linear sampling method is employed to deal with the inverse problem. The computation schemes proposed in this paper are relatively simple and easy to implement. Several numerical examples are presented to show the effectiveness of the proposed methods.

In this paper we introduce the definition of partial action on small $k$-categories generalizing the similar well known notion of partial actions on algebras. The point of view of partial action which we use in this paper is the one which was introduced by Exel in his work on $C^*$-algebras, see \cite{E}. Various generalizations were done afterward, see \cite{CJ, DEP, DE, DFP}. Also we define the notion partial skew category. We prove similar results to the ones in \cite{CM}. Finally we show a result given conditions for a partial action to have a globalization., Comment: 17 pages

In this note, we make some comments about the paper of Alheety and Kibria (2014) and correct the wrongly proved Theorems in that paper. © 2017 Taylor & Francis Group, LLC.

In this paper, we consider the asymptotic behaviour of the solution for a damped Rosenau equation on . We apply a variant of Riesz–Rellich criteria, which involves the Littlewood–Paley projection operators, to prove that the damped Rosenau equation possesses a global attractor in for any . Moreover, the global attractor is contained in , if the time-independent source term is in and the initial data are in . Our results establish the regularity of the global attractor for the damped Rosenau equation in fractional order Sobolev space, which is a new ingredient in this paper.

In this paper, we present an isothermal model for describing damage and fatigue by the use of the Ginzburg–Landau (G–L) equation. Fatigue produces progressive damage, which is related with a variation of the internal structure of the material. The G–L equation studies the evolution of the order parameter, which describes the constitutive arrangement of the system and, in this framework, the evolution of damage. The thermodynamic coherence of the model is proved. In the last part of the work, we extend the results of the paper to a nonisothermal system, where fatigue contains thermal effects, which increase the damage of materials.

This paper suggests a new stratified randomized response model based on Kuk's [Biometrika (1990), 77, 2, pp.436–438] model that has Neyman allocation and considerable gain in precision. It has been identified that the stratified randomized response models due to Kim and Warde (2004), Kim and Elam's (2005), and Kim and Elam's (2007) are members of the proposed model. It is shown that the proposed model is more efficient than Kuk's (1990) model both theoretically and empirically. The results of this paper are also extended in the situation when trials are repeated.

This paper proposes a mathematical method to solve the equilibrium equations of a membrane with rigid and cable boundaries for the so-called prestressing phase. More exactly, no constrained thrust network approach is implemented and the problem is studied starting directly from its continuous formulation. In this sense, the membrane is represented by a regular surface with negative Gaussian curvature, and its boundary by a set of regular curves, whose curvature depends on the structural elements, rigid or cable. The equilibrium is expressed by means of boundary differential problems in terms of the membrane shape and its stress tensor. The membrane-cable interface equilibrium requires taking into account a singular condition which makes the problem more complex. Therefore, the corresponding mathematical problem is defined and analysed, and its principal qualitative aspects are discussed in this paper. Moreover, a numerical resolution procedure is proposed and applied.

In this paper, we review many recent developments and further applications of nonstandard finite difference (NSFD) methods encountered in the past decade. In particular, it is a follow up article of the one published in 2005 [K.C. Patidar, On the use of non-standard finite difference methods, J. Differ. Equ. Appl. 11 (2005), pp. 735–758]. It also includes those research contributions in this field that are very significant and published prior to the above article but were not included in the above paper simply because we did not have access to them when we wrote the above article. We also give a detailed account on various definitions/notions of NSFD methods appeared in the literature in past two decades. All contributions are listed chronologically except that in some instances we have grouped certain works to show connectivity in those fields. While categorizing these research contributions, we considered a number of different application areas. Moreover, due to space limitations, firstly, we have not i...