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In this paper, the joint distribution of the sum and maximum of independent, not necessarily identically distributed, nonnegative random variables is studied for two cases: (i) continuous and (ii) discrete random variables. First, a recursive formula of the joint cumulative distribution function (CDF) is derived in both cases. Then recurrence relations of the joint probability density function (PDF) and the joint probability mass function (PMF) are given in the former and the latter case, respectively. Interestingly, there is a fundamental difference between the joint PDF and PMF. The proofs are simple and mainly based on the following tools from calculus and discrete mathematics: differentiation under the integral sign (also known as Leibniz’s integral rule), the law of total probability, and mathematical induction. In addition, this work generalizes previous results in the literature, and finally presents several extensions of the methodology. [ABSTRACT FROM AUTHOR]

This study used a mixed-methods design to examine teachers' responses to professional learning on mathematics differentiation. Grade 3 classroom teachers (N = 28) attended four 2-h professional learning sessions and an additional 2 h individually on differentiation and completed pre- and post-surveys with quantitative and short response items assessing their experiences. Although teachers did not report changes in overall differentiation rates, implementation of less extensive practices (i.e., talk moves and mathematical practices) increased. Changes did not appear to be driven by teachers' valuation of differentiation, which was high across surveys. Rather, a thematic analysis indicated that teachers simultaneously weighed benefits, challenges, and alignment with existing practices when making decisions about whether to implement differentiation strategies. [ABSTRACT FROM AUTHOR]

We leverage path differentiability and a recent result on nonsmooth implicit differentiation calculus to give sufficient conditions ensuring that the solution to a monotone inclusion problem will be path differentiable, with formulas for computing its generalized gradient. A direct consequence of our result is that these solutions happen to be differentiable almost everywhere. Our approach is fully compatible with automatic differentiation and comes with the following assumptions which are easy to check (roughly speaking): semialgebraicity and strong monotonicity. We illustrate the scope of our results by considering the following three fundamental composite problem settings: strongly convex problems, dual solutions to convex minimization problems, and primal-dual solutions to min-max problems. [ABSTRACT FROM AUTHOR]

This paper pursues obtaining Jacobi spectral collocation methods to solve Caputo fractional differential equations numerically. We used the shifted Jacobi–Gauss–Lobatto or Jacobi–Gauss–Radau quadrature nodes as the collocation points and derived the fractional differentiation matrices for Caputo fractional derivatives. With the fractional differentiation matrices, the fractional differential equations were transformed into linear systems, which are easier to solve. Two types of fractional differential equations were used for the numerical simulations, and the numerical results demonstrated the fast convergence and high accuracy of the proposed methods. [ABSTRACT FROM AUTHOR]

Research identifying the core features of Growth Mindset Pedagogy (GMP) in Chinese mathematics education is limited. This is also the first time we have examined the barriers to implementing GMP. We used semi-structured interviews to collect data. By analyzing the interview data, we identified four core features of GMP in mathematics classrooms: differentiation as the foundation of pedagogical practice, no stereotypes about mathematics learning, encouraging students to think in a process-oriented manner, and promoting a persistent learning attitude. We also found five barriers to GMP implementation: (a) lack of professional support; (b) lack of time; (c) student differences and class size; (d) ignoring cultural contextual elements; and (e) poor partnership between parents and teachers. In our discussion section, we outlined the implications of these findings for research and practice in educational contexts. [ABSTRACT FROM AUTHOR]

From new integral representations of the n-th derivative of Bessel functions with respect to the order, we derive some reflection formulas for the first and second order derivatives of J ν t and Y ν t for integral order, and for the n-th order derivatives of I ν t and K ν t for arbitrary real order. As an application of the reflection formulas obtained for the first order derivative, we extend some formulas given in the literature to negative integral order. Also, as a by-product, we calculate an integral which does not seem to be reported in the literature. [ABSTRACT FROM AUTHOR]

This paper presents a sequence of variable time step deferred correction (DC) methods constructed recursively from the second-order backward differentiation formula (BDF2) applied to the numerical solution of initial value problems for first-order ordinary differential equations (ODE). The sequence of corrections starts with the BDF2 then considered as DC2. We prove that this improvement from a p-order solution (DCp) results in a p + 1 -order accurate solution (DC p + 1 ). This one-order increment in accuracy holds for the least stringent BDF2 0-stability conditions. If we introduce additional requirements for the ratio of consecutive variable time step sizes, then the order increment is 2, allowing a direct transition from DCp to DC p + 2 . These requirements include the constant time step DCp methods. We also prove that all these DCp methods are A-stable. We briefly discuss two other DC variants to illustrate how a proper transition from DCp to DC p + 1 is critical to maintaining A-stability at all orders. Numerical experiments based on two manufactured (closed-form) solutions confirmed the accuracy orders of the DCp – for DCp, p = 2 , 3 , 4 , 5 – both with constant or alternating time step sizes. We showed that the theoretical conditions required to obtain an increment of orders 1 and 2 are satisfied in practice. Finally, a test case shows that we can estimate the error on the DCp solution with the DC p + 1 solution, and a last test case that our new methods maintain their order of accuracy for a stiff system. [ABSTRACT FROM AUTHOR]

In this paper, we consider the numerical solutions of the multi-dimensional tempered fractional integrodifferential equation. First, the second-order backward differentiation formula (BDF2) and the second-order convolution quadrature rule are utilized for the temporal discretization, and the finite difference method (FDM) and alternating direction implicit (ADI) technique are used for the spatial discretization, in which the ADI algorithm is mainly employed to reduce the computational cost of high-dimensional non-local problems. Then, the fully discrete BDF2 ADI difference scheme for multi-dimensional tempered non-local problems can be obtained. After that, the stability and convergence of the BDF2 ADI difference scheme in multi-dimensional cases are proved by means of the energy method. In the numerical experiments, some examples validate the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigate the modified extended second derivative backward differentiation formulae from second derivative general linear methods point of view. This makes it possible to open some maneuver rooms in developing the methods with superior features by perturbing the abscissa vector of the methods. The proposed methods are constructed to have better accuracy and stability properties in comparison with the original ones. These improvements are verified by giving some numerical experiments. [ABSTRACT FROM AUTHOR]

This paper is concerned with the construction and convergence analysis of novel implicit Peer triplets of two-step nature with four stages for nonlinear ODE constrained optimal control problems. We combine the property of superconvergence of some standard Peer method for inner grid points with carefully designed starting and end methods to achieve order four for the state variables and order three for the adjoint variables in a first-discretize-then-optimize approach together with A-stability. The notion triplets emphasize that these three different Peer methods have to satisfy additional matching conditions. Four such Peer triplets of practical interest are constructed. In addition, as a benchmark method, the well-known backward differentiation formula BDF4, which is only A (73.35 ∘) -stable, is extended to a special Peer triplet to supply an adjoint consistent method of higher order and BDF type with equidistant nodes. Within the class of Peer triplets, we found a diagonally implicit A (84 ∘) -stable method with nodes symmetric in [ 0 ,   1 ] to a common center that performs equally well. Numerical tests with four well established optimal control problems confirm the theoretical findings also concerning A-stability. [ABSTRACT FROM AUTHOR]

This article describes the construction of optical solitons and single traveling wave solutions of Biswas-Arshed equation with the beta time derivative. By using the polynomial complete discriminant system method, a series of traveling wave solutions are constructed, including the rational function solutions, Jacobian elliptic function solutions, hyperbolic function solutions, trigonometric function solutions and inverse trigonometric function solutions. The conclusions of this paper comprise some new and different solutions that cannot be found in existing literature. Using the mathematic software Maple, the 3D and 2D graphs of the obtained traveling wave solutions were also developed. It is worth noting that these traveling wave solutions may motivate us to explore new phenomena which may be appear in optical fiber propagation theory. [ABSTRACT FROM AUTHOR]

We study discrete time maximal regularity in Lebesgue spaces of sequences for time-stepping schemes arising from Lubich's convolution quadrature method. We show minimal properties on the quadrature weights that determines a wide class of implicit schemes. For an appropriate choice of the weights, we are able to identify the θ θ -method as well as the backward differentiation formulas and the L 1 L 1 -scheme. Fractional versions of these schemes, some of them completely new, are also shown, as well as their representation by means of the Grünwald–Letnikov fractional order derivative. Our results extend and improve some recent results on the subject and provide new insights on the basic nature of the weights that ensure maximal regularity. [ABSTRACT FROM AUTHOR]

This paper presents a formally second‐order backward differentiation formula (BDF2) Sinc‐collocation method for solving the Volterra integro‐differential equation with a weakly singular kernel. In the time direction, the time derivative is discretized via the BDF2 and the second‐order convolution quadrature rule is used to approximate the Riemann–Liouville fractional integral term. Then a fully discrete scheme is established via the Sinc approximation based on the double exponential transformation in space. The convergence and stability analysis are derived by the energy method. Numerical examples are provided to illustrate the effectiveness of proposed method and it can be found that our scheme is super‐exponentially convergent in space and order 1 + α convergent in time with 0 < α < 1, respectively. Meanwhile, the numerical results based on the single exponential transformation are compared with the proposed method to illustrate the high accuracy of our method. [ABSTRACT FROM AUTHOR]

In this paper, we consider a general class of positive linear operators satisfying a differential equation and we prove that the Voronovskaya-type formula can be differentiated. [ABSTRACT FROM AUTHOR]

We prove that the two-step backward differentiation formula (BDF) method is stable on arbitrary time grids; while the variable-step three-step backward differentiation formula scheme is stable if almost all adjacent step ratios are less than 2.553. These results relax the severe mesh restrictions in the literature and provide a new understanding of variable-step BDF methods. Our main tools include the discrete orthogonal convolution kernels and an elliptic type matrix norm. [ABSTRACT FROM AUTHOR]

There is a usual multiplicative calculus apart from classical Newton-Leibnitz additive calculus. This article provides a new multiplicative calculus, which includes singular multiplicative differentiation, singular multiplicative Riemann integration, and singular multiplicative Lebesgue integration based on newly introduced multiplicative measure. Properties of multiplicative modulus function have been used. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and discuss multi-objective interval-valued E-convex programming using gH-symmetrical differentiability. We prove nonlinear optimality conditions of Fritz John type for this context and construct an example to verify our results. Furthermore, we define LU-sE-pseudo convexity and LU-sE-quasi convexity for interval-valued functions and study some of their properties. [ABSTRACT FROM AUTHOR]

In this present article, the q-derivative operator and the subordination principle are use to define a class of functions that are analytic and bi-univalent in the open unit disk. Our aim for this class is to obtain the upper bound for the second Hankel determinant for functions in this new subclass of analytic and bi-univalent functions. [ABSTRACT FROM AUTHOR]

We study the complex symmetry of weighted composition-differentiation operators of order n on the weighted Bergman spaces Aα. Several concrete examples are provided. [ABSTRACT FROM AUTHOR]

This paper focuses on the stability analysis of a general class of linear multistep methods for decoupled forward–backward stochastic differential equations (FBSDEs). The general linear multistep methods we consider contain many well-known linear multistep methods from the ordinary differential equation framework, such as Adams, Nyström, Milne--Simpson and backward differentiation formula methods. Under the classical root condition , we prove that general linear multistep methods are mean-square (zero) stable for decoupled FBSDEs with generator function related to both y and z. Based on the stability result, we further establish a fundamental convergence theorem. [ABSTRACT FROM AUTHOR]

This article extends the semidiscrete maximal |$L^p$| -regularity results in Li (2019, Analyticity, maximal regularity and maximum-norm stability of semi-discrete finite element solutions of parabolic equations in nonconvex polyhedra. Math. Comp. , 88 , 1--44) to multistep fully discrete finite element methods for parabolic equations with more general diffusion coefficients in |$W^{1,d+\beta }$|⁠ , where |$d$| is the dimension of space and |$\beta>0$|⁠. The maximal angles of |$R$| -boundedness are characterized for the analytic semigroup |$e^{zA_h}$| and the resolvent operator |$z(z-A_h)^{-1}$|⁠ , respectively, associated to an elliptic finite element operator |$A_h$|⁠. Maximal |$L^p$| -regularity, an optimal |$\ell ^p(L^q)$| error estimate and an |$\ell ^p(W^{1,q})$| estimate are established for fully discrete finite element methods with multistep backward differentiation formulae. [ABSTRACT FROM AUTHOR]

A mixed approximate method is proposed to simulate the generalized Hirota–Satsuma coupled Korteweg–de Vries equations. The equations are represented as linear dispersive equations for the material time derivative along the trajectory of physical particle movements described by the nonlinear Cauchy problem in the Lagrangian viewpoint. The proposed method employs the finite difference method and backward differentiation formula to solve the linear dispersive equations. To solve the nonlinear Cauchy problem, an error correction technique is applied. Moreover, the fifth-order weighted essentially non-oscillatory scheme interpolates solutions of non-grid points to control unnecessary oscillation due to the dispersion of the solution. From numerical experiments, we demonstrate the accuracy and efficiency of the proposed method. The proposed method numerically has the qth-order temporal and 4th-order spatial accuracies in the sense of the error norms L ∞ and L 2 . Consequently, the errors obtained using the proposed method are more accurate even with a larger time step size than those of the compared methods. [ABSTRACT FROM AUTHOR]

We address the optimization problem of antenna placement on satellite-mounted interferometric synthetic-aperture instruments. In classic designs, the antennas on satellites are aligned regularly on the satellite’s frame. Inspired by methods proposed for the placement of antennas in astronomical interferometers, such as ALMA or SKA, we explore irregular layouts and show that they are a valid alternative in terms of spatial resolution and reconstruction error. We formalize mathematically the optimization problem of irregularly placed antennas and we show that this kind of arrays can still be calibrated with the same methods used for regular arrays. Finally, this strategy is evaluated in the context of soil moisture and ocean salinity (SMOS) follow-up concepts, such as SMOS-HR (high resolution), for which the new optimized irregular configurations are compared to the regular ones.

In this article, finite element a posteriori error estimates for the linear parabolic integro-differential equation using the two-step backward time descretization formula are explored. For space discretization, we use piecewise linear finite element spaces. The Ritz–Volterra reconstruction operator is used as a raw ingredient to obtain the optimal convergence in space. Further, a novel quadratic space–time reconstruction operator, namely BDF2 operator, is introduced to achieve second-order accuracy in time. Numerical results demonstrate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this work, we propose and analyze a new method for the solution of the three-dimensional evolutionary equation with a nonlocal term. Then the method combines Galerkin finite element methods (FEMs) for the spatial discretization with an alternating direction implicit (ADI) algorithm based on the second-order backward differentiation formula (BDF2), where the Riemann-Liouville (R-L) integral term is approximated via second-order convolution quadrature (CQ) rule. The L 2 -norm stability and convergence are proved. Numerical results confirm the predicted space-time convergence rates. [ABSTRACT FROM AUTHOR]

This article considers the representation of differentiation operators using Daubechies wavelets to solve PDEs numerically. Derivative approximation using this compactly supported wavelets convert the action into a matrix multiplication. The vanishing moments, dilation equation and quadrature formulas play a significant role in the scheme. The computed solution of the PDEs seems to behave better than the results reported in the literature, and we could progress the solution up to a large time-bound. We experimented with Daubechies wavelet filters with six vanishing moments on test problems and summarised the results. [ABSTRACT FROM AUTHOR]

We investigate the permissible growth rates of functions that are distributionally chaotic with respect to differentiation operators. We improve on the known growth estimates for Ddistributionally chaotic entire functions, where growth is in terms of average Lp-norms on spheres of radius r > 0 as r → ∞, for 1 ≤ p ≤ ∞. We compute growth estimates of ∂/∂xk - distributionally chaotic harmonic functions in terms of the average L²-norm on spheres of radius r > 0 as r → ∞. We also calculate sup-norm growth estimates of distributionally chaotic harmonic functions in the case of the partial differentiation operators D α. [ABSTRACT FROM AUTHOR]

In the present work, first of all, a new numerical fractional differentiation formula (called the CF2 formula) to approximate the Caputo-Fabrizio fractional derivative of order a; (0 <α < 1) is developed. It is established by means of the quadratic interpolation approximation using three points (t j-2;y(t j-2)); (t j-1;y(t j-1)), and (t j; y(t j)) on each interval [t j-1; t j] for (j ≥ 2), while the linear interpolation approximation are applied on the first interval [t0; t1]. As a result, the new formula can be formally viewed as a modification of the classical CF1 formula, which is obtained by the piecewise linear approximation for y(t). Both the computational efficiency and numerical accuracy of the new formula is superior to that of the CF1 formula. The coefficients and truncation errors of this formula are discussed in detail. Two test examples show the numerical accuracy of the CF2 formula. The CF1 formula demonstrates that the new CF2 is much more effective and more accurate than the CF1 when solving fractional differential equations. Detailed stability analysis and region stability of the CF2 are also carefully investigated. [ABSTRACT FROM AUTHOR]

The numerical approximation for the Landau-Lifshitz equation, which models the dynamics of the magnetization in a ferromagnetic material, is taken into consideration. This highly nonlinear equation, with a non-convex constraint, has several equivalent forms, and involves solving an auxiliary problem in the infinite domain. All these features have posed interesting challenges in developing numerical methods. In this paper, we first present a fully discrete semi-implicit method for solving the Landau-Lifshitz equation based on the second-order backward differentiation formula and the one-sided extrapolation (using previous time-step numerical values). A projection step is further used to preserve the length of the magnetization. Subsequently, we provide a rigorous convergence analysis for the fully discrete numerical solution by the introduction of two sets of approximated solutions where one set of solutions solves the Landau-Lifshitz equation and the other is projected onto the unit sphere. Second-order accuracy in both time and space is obtained provided that the spatial step-size is the same order as the temporal step-size. And also, the unique solvability of the numerical solution without any assumption for the step-size in both time and space is theoretically justified, using a monotonicity analysis. All these theoretical properties are verified by numerical examples in both 1D and 3D spaces. [ABSTRACT FROM AUTHOR]

Both in New Zealand and internationally, there has been a focus on the use of differentiation in mathematics instruction to raise achievement levels and provide equitable outcomes. New Zealand has a long history of the use of ability grouping to provide differentiation. Recently, this practice has been challenged in a large scale professional learning and development (PLD) initiative entitled Developing Mathematical Inquiry Communities (DMIC) which focuses on shifting to strength-based, capability focused heterogeneous grouping practices. This article draws on a case study of five teachers involved in the DMIC PLD initiative to examine the enabling factors and barriers to shifting teacher beliefs in relation to mathematical ability grouping. The findings indicate the persistence of teacher beliefs related to fixed ability levels in mathematics. Assessment practices focusing on narrow domains in regard to success in mathematics appeared to act as a barrier to changing teacher beliefs. In contrast, a focus on strengths that students brought to a mathematical task and collaborative group-work appeared to be a key enabling factor to change in teacher beliefs and practices. A key implication of the article is that changes to assessment practices are required alongside changes to pedagogical practices to support teachers to move from gap gazing to recognising multiple strengths in the mathematics classroom. [ABSTRACT FROM AUTHOR]

In order to transform mathematics classrooms for students with disabilities, teachers must design based on learners at the margins. Universal Design for Learning (UDL) is a framework built on these principles (Meyer et al., 2014), but lacks a robust process through which educators can reimagine mathematics. We theorize that when educators merge the principles of Universal Design for Learning in mathematics with the process of Design Thinking, more accessible designs for curriculum, routines, tools, and spaces would result. Using design research methodology, we analysed a 6-week summer course for mathematics educators on UDL and Design Thinking. Data sources included class video recordings and transcripts, class artefacts, as well as pre- and post-participant surveys. Findings indicate shifts in participants' knowledge of UDL from a passive to an active role, empathy as a central guiding principle, and the potential of using Design Thinking as a process to enact UDL. [ABSTRACT FROM AUTHOR]

Tasks play a central role in the mathematics classroom. Especially when teaching in a heterogeneous mathematics class, teachers should be able to find, select, modify, and assign tasks adequately. The focus in this paper is on adaptivity to cognitively activate all learners at the individual level, and on teachers' abilities to allow for such adaptivity by means of selecting appropriate tasks. More specifically, when planning lesson phases for practice and consolidation, teachers may consider which tasks have differentiation potential and can thus be completed by all students at the same time but at different levels. In order to analyse teachers' strategies in this regard, we investigated the task features that teachers may consider when assessing the differentiation potential of exercise tasks. We first deductively constructed rating categories based on the literature on instructionally relevant features of adaptive tasks. Then, we inductively extended and refined the constructed categories by analysing teachers' reasoning based on a sample of 78 in-service teachers at secondary schools. We validated the resulting 22 categories by determining interrater reliability. Our findings indicate that teachers consider a broad spectrum of task features when analysing the differentiation potential of tasks. However, only some of these features are directly relevant with regard to using adaptive tasks as a differentiation strategy. Our results also show that many teachers arrived at conclusions about the differentiation potential of tasks that were different from task-design experts. Based on our findings regarding teachers' perspectives on the differentiation potential of tasks and on certain task features, we discuss how these findings may have arisen and how important the knowledge about the deep structure of adaptive tasks is for teachers' professional development. [ABSTRACT FROM AUTHOR]

For their several attractive features from the viewpoint of the numerical computations, linear barycentric rational interpolants have been recently used to construct various numerical methods for solving different classes of equations. In this paper, we introduce a family of linear multistep second derivative methods together with a starting procedure based on barycentric rational interpolants. The order of convergence and linear stability properties of the proposed methods have been investigated. To validate the theoretical results and efficiency of the methods, some numerical experiments have been provided. [ABSTRACT FROM AUTHOR]

In this paper, we develop and compare four kinds of fully discrete continuous Galerkin finite element methods for a fractional heat transport equation describing the deep underground unsteady flow. We adopt continuous Galerkin finite element methods for the spatial discretization and the backward Euler together with linear interpolation (L1), second‐order backward differentiation formula (BDF2), weighted and shifted Grünwald difference (WSGD), and Crank‐Nicolson (CN) time‐stepping methods for time discretization. We derive the stability estimates and a priori error estimates for the semidiscrete and fully discrete finite element schemes. Finally, numerical comparisons of the four families of fully discrete finite element methods are presented to illustrate the effectiveness of the studied numerical schemes. [ABSTRACT FROM AUTHOR]

Existing negative results on invalidity of analogues of classical Density and Differentiation Theorems in infinite dimensional spaces are considerably strengthened by a construction of a Gaussian measure γ on a separable Hilbert space H for which the Density Theorem fails uniformly, i.e., there is a set M⊂H of positive γ-measure such that. [ABSTRACT FROM AUTHOR]

We revisit the formalism for analytic derivative couplings between excited states in time-dependent density functional theory (TDDFT). We derive and implement these couplings using quadratic response theory, then numerically compare this response-theory formulation to couplings implemented previously based on a pseudo-wavefunction formalism and direct differentiation of the Kohn-Sham determinant. Numerical results, including comparison to full configuration interaction calculations, suggest that the two approaches perform equally well for many molecular systems, provided that the underlying DFT method affords accurate potential energy surfaces. The response contributions are found to be important for certain systems with high symmetry, but can be calculated with only a moderate increase in computational cost beyond what is required for the pseudo-wavefunction approach. In the case of spin-flip TDDFT, we provide a formal proof that the derivative couplings obtained using response theory are identical to those obtained from the pseudo-wavefunction formulation, which validates our previous implementation based on the latter formalism. [ABSTRACT FROM AUTHOR]

This article presents an efficient stabilized finite element scheme for solving viscoelastic flow problems at highWeissenberg numbers. The velocity and pressure variables in the momentum balance equations are uncoupled using an incremental fractional step method based on the second-order backward differentiation formula. The pressure gradient projection stabilization technique and the discrete elastic-viscous-split-stress formulation are introduced into the scheme, in explicit versions, to circumvent the LBB constraints. For the constitutive equation, a square root transformation is first applied to preserve the positive definiteness of the polymer conformation tensor, and then the streamline upwind/Petrov--Galerkin method and the second-order Runge--Kutta scheme are implemented for spatial and time discretizations. Three benchmark problems are tested and numerical results have revealed the accuracy and convergence of the scheme. What is more, since the presented scheme enables the use of equal low-order interpolations for all variables, and requires no iterative process, it is computationally efficient and easy to be implemented. [ABSTRACT FROM AUTHOR]

In this paper, a 2‐step backward differentiation formula (BDF2) Galerkin method is investigated for semilinear parabolic equation. More precisely, the second‐order time‐stepping scheme is used for time discretization and the piecewise linear continuous Galerkin method is employed for spatial discretization, respectively. Optimal error estimates in L2 and H1‐norms are obtained without any restriction on the time‐step size, while previous works always require certain conditions on time step‐size. The key to our analysis is to derive a uniform boundness of the numerical solution in energy norm so as to avoid the inverse inequality used in the usual convergence analysis of the finite element methods. Numerical experiments are carried out to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

Many initial value problems are difficult to be solved using ordinary, explicit step-by-step methods because most of these problems are considered stiff. Certain implicit methods, however, are capable of solving stiff ordinary differential equations (ODEs) usually found in most applied problems. This study aims to develop a new numerical method, namely the high order variable step variable order block backward differentiation formula (VSVO-HOBBDF) for the main purpose of approximating the solutions of third order ODEs. The computational work of the VSVO-HOBBDF method was carried out using the strategy of varying the step size and order in a single code. The order of the proposed method was then discussed in detail. The advancement of this strategy is intended to enhance the efficiency of the proposed method to approximate solutions effectively. In order to confirm the efficiency of the VSVO-HOBBDF method over the two ODE solvers in MATLAB, particularly ode15s and ode23s, a numerical experiment was conducted on a set of stiff problems. The numerical results prove that for this particular set of problem, the use of the proposed method is more efficient than the comparable methods. VSVO-HOBBDF method is thus recommended as a reliable alternative solver for the third order ODEs. [ABSTRACT FROM AUTHOR]

Many hyperbolic and kinetic equations contain a non-stiff convection/transport part and a stiff relaxation/collision part (characterized by the relaxation or mean free time ε). To solve this type of problems, implicit-explicit (IMEX) multistep methods have been widely used and their performance is understood well in the non-stiff regime (ε = O(1)) and limiting regime (ε → 0). However, in the intermediate regime (say, ε =O(Δ t)), uniform accuracy has been reported numerically without a complete theoretical justification (except some asymptotic or stability analysis). In this work, we prove the uniform accuracy - an optimal a priori error bound - of a class of IMEX multistep methods, IMEX backward differentiation formulas (IMEX-BDF), for linear hyperbolic systems with stiff relaxation. The proof is based on the energy estimate with a new multiplier technique. For nonlinear hyperbolic and kinetic equations, we numerically verify the same property using a series of examples. [ABSTRACT FROM AUTHOR]

We present uniform and Lp mixed Caputo-Bochner abstract generalized fractional Landau inequalities over R of fractional orders 2 < α ≤ 3. These estimate the size of first and second derivatives of a composition with a Banach space valued function over R. We give applications when α = 2:5. [ABSTRACT FROM AUTHOR]

Let I ⊆ R be a nonempty open subinterval. We say that a two-variable mean M : I × I → R enjoys the balancing property if, for all x , y ∈ I , the equality 1 M (M (x , M (x , y)) , M (M (x , y) , y)) = M (x , y) holds. The above equation has been investigated by several authors. The first remarkable step was made by Georg Aumann in 1935. Assuming, among other things, that M is analytic, he solved (1) and obtained quasi-arithmetic means as solutions. Then, two years later, he proved that (1) characterizes regular quasi-arithmetic means among Cauchy means, where, the differentiability assumption appears naturally. In 2015, Lucio R. Berrone, investigating a more general equation, having symmetry and strict monotonicity, proved that the general solutions are quasi-arithmetic means, provided that the means in question are continuously differentiable. The aim of this paper is to solve (1), without differentiability assumptions in a class of two-variable means, which contains the class of Matkowski means. [ABSTRACT FROM AUTHOR]

We establish the generalized Evans–Krylov and Schauder type estimates for nonlocal fully nonlinear elliptic equations with rough kernels of variable orders. In contrast to the fractional Laplacian type operators having a fixed order of differentiability σ ∈ (0 , 2) , the operators under consideration have variable orders of differentiability. Since the order is not characterized by a single number, we consider a function φ describing the variable orders of differentiability, which is allowed to oscillate between two functions r σ 1 and r σ 2 for some 0 < σ 1 ≤ σ 2 < 2. By introducing the generalized Hölder spaces, we provide C φ ψ estimates that generalizes the standard Evans–Krylov and Schauder type C σ + α estimates. [ABSTRACT FROM AUTHOR]

In this note, we provide changes in the proof of Theorem 5 in [1] and present the modified version of this Theorem. [ABSTRACT FROM AUTHOR]

The article informs that mathematics, machine learning and artificial intelligence have received tremendous attention in which learn about science, the press and the public potential has a revolution in science. Topics include distinguish between subject area, the distinction is made in public discourse it is often more about marketing than differentiation in terms of content; and Canonical examples are visual systems in autonomous driving.

The aim of this paper is to develop and analyze high-order time stepping schemes for approximately solving semilinear subdiffusion equations. We apply the convolution quadrature generated by k-step backward differentiation formula (BDFk) to discretize the time-fractional derivative with order α ∈ (0, 1) and modify the starting steps in order to achieve optimal convergence rate. This method has already been well-studied for the linear fractional evolution equations in Jin, Li, and Zhou [SIAM J. Sci. Comput., 39 (2017), pp. A3129-A3152], while the numerical analysis for the nonlinear problem is still missing in the literature. By splitting the nonlinear potential term into an irregular linear part and a smoother nonlinear part and using the generating function technique, we prove that the convergence order of the corrected BDFk scheme is O(τmin(k,1+2α-ε)) without imposing further assumption on the regularity of the solution. Numerical examples are provided to support our theoretical results. [ABSTRACT FROM AUTHOR]