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The absolute zeta function for a scheme X of finite type over Z satisfying a certain condition is defined as the limit as p → 1 of the zeta function of X ⊗ F p . In 2016, after calculating absolute zeta functions for a few specific schemes, Kurokawa suggested that an absolute zeta function for a general scheme of finite type over Z should have an infinite product structure which he called the absolute Euler product. In this article, formulating his suggestion using a torsion free Noetherian F 1 -scheme defined by Connes and Consani, we give a proof of his suggestion. Moreover, we show that each factor of the absolute Euler product is derived from the counting function of the F 1 -scheme.

In this paper, the Finite Volume numerical scheme has been used to solve one-dimensional unsteady state and two-dimensional steady-state heat flow problems with the initial condition and Dirichlet boundary condition. Mainly two schemes are used to compute the numerical solutions of unsteady-state flow and one scheme for steady-state flow. The applicability of these schemes is illustrated using simple test problems. The numerical solutions obtained by these schemes are compared with the analytic solutions to check the accuracy of the developed scheme.

We carry out error estimates for a linear, second-order, unconditionally energy stable, semi-discrete time stepping scheme, which is based on the Lagrange Multiplier approach and could also be viewed as a special case of the invariant energy quadratization approach, for the modified phase field crystal equation. At each time step, the proposed scheme only requires solving several linear equations. Rigorous proofs are presented to demonstrate the unique solvability, mass conservation, and unconditional energy stability of the scheme. Various numerical experiments in 2D and 3D are performed to validate the accuracy, unconditional energy stability and mass conservation of the proposed numerical strategy.

This paper is concerned with numerical solutions of one-dimensional (1D) and two-dimensional (2D) nonlinear coupled Schrodinger-Boussinesq equations (CSBEs) by a type of linearly energy- and mass- preserving finite difference methods (EMP-FDMs) because the existing EMP-FDMs for CSBEs are nonlinear and time-consuming, and corresponding theoretical analyses are not easy to generalize high-dimensional problems. Firstly, a linearized EMP-FDM is created for solving 1D CSBEs. By using the discrete energy analysis method, it is shown that this scheme is uniquely solvable and convergent with an order of O ( Δ t 2 + h x 2 ) in L ∞ -, H 1 - and L 2 -norms, and corresponding numerical energy and mass are conservative. Then, by generalizing this type of EMP-FDM, a linearized EMP-FDM is developed for solving 2D CSBEs. Theoretical findings including the convergence, the discrete conservative laws, and the solvability of this numerical algorithm are strictly derived in detail by using the discrete energy analysis method as well. Finally, numerical results confirm the efficiency of our algorithms and the exactness of the theoretical results.

This paper develops a high order semi-implicit weighted compact nonlinear scheme (WCNS) for one- and two-dimensional viscous Burgers’ equations. This semi-implicit WCNS combines a fifth-order WCNS with a third-order implicit-explicit Runge–Kutta (IMEX-RK) time-stepping method. The advection terms of viscous Burgers’ equations are treated explicitly, while the viscosity terms are treated implicitly. Stability analysis shows that the CFL condition of the semi-implicit WCNS is controlled only by the advection terms. Compared to the explicit time-stepping method, the semi-implicit method has the advantage in terms of computational efficiency. Numerical results validate the accuracy and efficiency of the semi-implicit WCNS. This method can also solve viscous Burgers’ equations with large Reynolds numbers and has high-resolution shock-capturing ability.

In this paper, we consider the operator splitting scheme based on barycentric Lagrange interpolation collocation method for the two-dimensional Allen-Cahn equation. The original problem is split into linear and nonlinear subproblems: the linear part is solved by barycentric Lagrange interpolation collocation method in space and Crank-Nicolson scheme in time; the nonlinear part is solved analytically due to the availability of a closed-form solution. The error estimates of the proposed scheme are studied. Numerical experiments are carried out to demonstrate the accuracy and efficiency of the two operator splitting schemes.

We analyze a fully discrete pseudo-spectral scheme for the numerical solution of the initial-periodic boundary value problem for a nonlinear family of Boussinesq systems. The equations are discretized in space by the Fourier collocation method and in time by an explicit fourth order Runge–Kutta scheme. For the resulting fully discrete scheme, we prove error bounds of spectral accuracy in space and of fourth-order accuracy in time.

In this paper, we give an approach to the zeta values of a (proper regular) arithmetic scheme X at the integers r>=d:=dim(X), using \'etale cohomology of X with Q_p(r) and Z_p(r)-coefficients., Comment: 62 pages. Remark 2.2 has been added. Proposition 7.3 has been modified. Some details of the proof of Lemma 7.9 have been improved, and many typos have been corrected. To appear in Journal of Number Theory

Concentration properties of functionals of general Poisson processes are studied. Using a modified Phi-Sobolev inequality a recursion scheme for moments is established, which is of independent interest. This is applied to derive moment and concentration inequalities for functionals on abstract Poisson spaces. Applications of the general results in stochastic geometry, namely Poisson cylinder models and Poisson random polytopes, are presented as well. (C) 2021 Elsevier B.V. All rights reserved.

In this paper, an asymmetric cryptosystem with unequal modulus decomposition in the Fourier domain is presented. The input-colour image is decomposed into its red, green, and blue components. Each ...

Kung and Traub’s conjecture indicates that a multipoint iterative scheme without memory and based on m evaluations of functions has an optimal convergence order p = 2 m − 1 . Consequently, a fifth-order iterative scheme requires at least four evaluations of functions. Herein, we derive three novel iterative schemes that have fifth-order convergence and involve four evaluations of functions, such that the efficiency index is E.I.=1.49535. On the basis of the analysis of error equations, we obtain our first iterative scheme from the constant weight combinations of three first- and second-class fourth-order iterative schemes. For the second iterative scheme, we devise a new weight function to derive another fifth-order iterative scheme. Finally, we derive our third iterative scheme from a combination of two second-class fourth-order iterative schemes. For testing the practical application of our schemes, we apply them to solve the van der Waals equation of state.

This paper is concerned with the secure consensus problem of multiagent systems under switching topologies. The studied multiagent systems are affected by both denial-of-service (DoS) attacks and external disturbances . To solve the secure H ∞ consensus problems, some modified definitions are presented. Some graph-based Lyapunov functions , which are based on the solutions of some Lyapunov equations and the graph information , are also designed for the H ∞ performance analysis. Moreover, graph-based frequency and durations have also been presented for attaining the expected system performance. The stabilization controllers are also developed based on the solutions to some Lyapunov equations and an algebraic Riccati equation (ARE), which are easy to acquire. Some simulations are provided to validate the feasibility of the proposed scheme.

In this paper, we study a regularized Lie-Trotter splitting spectral method for a regularized space-fractional logarithmic Schrodinger equation by introducing a small regularized parameter 0 e ≪ 1 . The regularized method can be used to avoid numerical blow-up in the space-fractional logarithmic Schrodinger equation. The regularized space-fractional logarithmic Schrodinger equation is proved to approximate the space-fractional logarithmic Schrodinger equation with linear convergence rate O ( e ) . The proposed numerical scheme can preserve the discrete mass and step-energy for regularized space-fractional logarithmic Schrodinger equation. The first order convergence in time for the numerical method is rigorous proved for the regularized space-fractional logarithmic Schrodinger equation. Due to the appearance of fractional Laplace operator with order α, we can adjust the parameters of order to make the equation show much more dynamic characteristics than the classical logarithmic Schrodinger equation. Numerical simulations for 1D case based on the Fourier spectral approximation in space are presented to validate the theoretical analysis.

Kinetic or Boltzmann schemes are interesting alternatives to the macroscopic numerical methods for solving the hyperbolic conservation laws of gas dynamics. They utilize the particle-based description instead of the wave propagation models. While the continuous particle velocity based upwind schemes were developed in the earlier decades, the discrete velocity Boltzmann schemes introduced in the last decade are found to be simpler and are easier to handle. In this work, we introduce a novel way of introducing discrete velocities which correspond to the physical wave speeds and formulate a discrete velocity Boltzmann scheme for solving Euler equations.

In this paper, we present new second-order methods with convergence rate $$O\left( k^{-4}\right) $$ O k - 4 , where k is the iteration counter. This is faster than the existing lower bound for this type of schemes (Agarwal and Hazan in Proceedings of the 31st conference on learning theory, PMLR, pp. 774–792, 2018; Arjevani and Shiff in Math Program 178(1–2):327–360, 2019), which is $$O\left( k^{-7/2} \right) $$ O k - 7 / 2 . Our progress can be explained by a finer specification of the problem class. The main idea of this approach consists in implementation of the third-order scheme from Nesterov (Math Program 186:157–183, 2021) using the second-order oracle. At each iteration of our method, we solve a nontrivial auxiliary problem by a linearly convergent scheme based on the relative non-degeneracy condition (Bauschke et al. in Math Oper Res 42:330–348, 2016; Lu et al. in SIOPT 28(1):333–354, 2018). During this process, the Hessian of the objective function is computed once, and the gradient is computed $$O\left( \ln {1 \over \epsilon }\right) $$ O ln 1 ϵ times, where $$\epsilon $$ ϵ is the desired accuracy of the solution for our problem.

In this study, we develop the differential transform method in a new scheme to solve systems of first-order differential equations. The differential transform method is a procedure to obtain the coefficients of the Taylor series of the solution of differential and integral equations. So, one can obtain the Taylor series of the solution of an arbitrary order, and hence, the solution of the given equation can be obtained with required accuracy. Here, we first give some basic definitions and properties of the differential transform method, and then, we prove some theorems for solving the linear systems of first order. Then, these theorems of our system are converted to a system of linear algebraic equations whose unknowns are the coefficients of the Taylor series of the solution. Finally, we give some examples to show the accuracy and efficiency of the presented method.

We show existence of solution for a supercritical nonlinear Schrodinger equation on the whole RN by means of an approximation scheme. We prove a Sobolev embedding corresponding to the variable expo...

We propose a tamed-adaptive Euler-Maruyama approximation scheme for stochastic differential equations with locally Lipschitz continuous, polynomial growth drift, and locally Holder continuous, poly...

We study a filtered Lie splitting scheme for the cubic nonlinear Schrödinger equation. We establish error estimates at low regularity by using discrete Bourgain spaces. This allows us to handle data in H s H^s with 0 > s > 1 0>s>1 overcoming the standard stability restriction to smooth Sobolev spaces with index s > 1 / 2 s>1/2 . More precisely, we prove convergence rates of order τ s / 2 \tau ^{s/2} in L 2 L^2 at this level of regularity.

Various models were proposed to calculate the van der Waals (vdW) forces, but they have generally been complicated and time-consuming. In this paper, a spherical volume elements scheme to calculating the vdW force distributions is proposed, on the basis of the infinitesimal dividing modeling scheme. We apply this scheme to calculate the vdW force distributions between sphere and three types of surfaces with different roughness and also to irregular particles (cylinders, circular cone) orientated differently with respect to the surface. The simulation results reveal that the surface with low roughness has the narrowest force distributions. The average force for the circular cone bottom case is 1, 2, 37 times higher than the case of horizontal, vertical cylinder and circular cone top, respectively. The calculated force distributions are good agreement with exist experimental data. By defining the effective zones of both particles and surface, the computational time of the scheme is an order of magnitude faster than the case of considering the whole surface. The proposed model established the effective zone of the particles in contact from a completely new perspective and significantly simplifies the process of force evaluation.

The aim of present study is to develop a numerical scheme by using the notion of Barycentric Lagrange interpolation based Differential quadrature method to solve the coupled 1D Burgers’ equation. T...

In this paper, two efficient two-grid algorithms with L 1 scheme are presented for solving two-dimensional nonlinear time fractional diffusion equations. The classical L 1 scheme is considered in the time direction, and the two-grid FE method is used to approximate spatial direction. To linearize the discrete equations, the Newton iteration approach and correction technique are applied. The two-grid algorithms reduce the solution of the nonlinear fractional problem on a fine grid to one linear equation on the same fine grid and an original nonlinear problem on a much coarser grid. As a result, our algorithms save total computational cost. Theoretical analysis shows that the two-grid algorithms maintain asymptotically optimal accuracy. Moreover, the numerical experiment presented further confirms the theoretical results.

We present a fast and accurate numerical scheme for approximating weakly singular oscillatory Volterra integral equations (VIEs) of the second kind. The main idea is the combination of the black-bo...

In this paper, we solve analytically the Schrodinger equation for s-wave and arbitrary angular momenta with the Hua potential is investigated respectively. The wave function as well as energy equation are obtained in an exact analytical manner via the Nikiforov Uvarov method using two approximations scheme. Some special cases of this potentials are also studied.

In this paper, we propose and analyze a variant of the proximal point method for obtaining weakly efficient solutions of convex vector optimization problems in real Hilbert spaces, with respect to a partial order induced by a closed, convex and pointed cone with nonempty interior. The proposed method is a hybrid scheme that combines proximal point type iterations and projections onto some special halfspaces in order to achieve the strong convergence to a weakly efficient solution. To the best of our knowledge, this is the first time that proximal point type method with strong convergence has been considered in the literature for solving vector/multiobjective optimization problems in infinite dimensional Hilbert spaces.

In this paper, we investigate the approximate bound states solutions of Klein–Gordon equation with position-dependent mass for scalar and vector central Hulthen-type potentials by applying an approximation scheme to deal with the centrifugal potential for any l-states. The supersymmetric quantum mechanics (SUSYQM) and shape invariance approaches are used in the calculations. We obtain straightforwardly the relativistic energies and their corresponding normalized wavefunctions, expressed in terms of Jacobi polynomials. Special cases of interest of this problem are also deduced and it is found that our results are in good agreement with those obtained in the literature.

This paper proposes and analyzes a globalized version of the Newton method for finding a singularity of the nonsmooth vector fields. Basically, the new method combines a version of nonsmooth Newton method with a nonmonotone line search strategy. The global convergence analysis of the proposed method as well as results on its rate are established under mild assumptions. Finally, numerical experiments illustrating the practical advantages of the proposed scheme are reported.

The rototranslation group ℛ T is the group comprising rotations and translations of the Euclidean plane which is a 3-dimensional Lie group. In this paper, we use the Riemannian approximation scheme to compute sub-Riemannian limits of the Gaussian curvature for a Euclidean C 2 -smooth surface in the rototranslation group away from characteristic points and signed geodesic curvature for Euclidean C 2 -smooth curves on surfaces. Based on these results, we obtain a Gauss–Bonnet theorem in the rototranslation group.

In this paper, we propose a new linesearch iterative scheme for finding a common solution of split equilibrium and fixed point problems without pseudomonotonicity of the bifunction f in a real Hilbert space. When setting the solution of dual equilibrium problem is nonempty, we obtain a strong convergence theorem which is generated by the iterative scheme. Moreover, we also receive a new linesearch iterative scheme for finding a solution of the split equilibrium problem in suitable assumptions, and report some numerical results to illustrate the convergence of the proposed scheme.

We develop a set of numerical schemes for the Poisson--Nernst--Planck equations. We prove that our schemes are mass conservative, uniquely solvable and keep positivity unconditionally. Furthermore, the first-order scheme is proven to be unconditionally energy dissipative. These properties hold for various spatial discretizations. Numerical results are presented to validate these properties. Moreover, numerical results indicate that the second-order scheme is also energy dissipative, and both the first- and second-order schemes preserve the maximum principle for cases where the equation satisfies the maximum principle., Comment: 24 pages, 10 figures

In this paper, an efficient finite difference scheme is proposed for one dimension and two dimension coupled space fractional nonlinear Schrodinger equations. First, the high-order difference schem...

In the present paper, we study frames for finite-dimensional vector spaces over an arbitrary field. We develop a theory of dual frames in order to obtain and study the different representations of the elements of the vector space provided by a frame. We relate the introduced theory with the classical one of dual frames for Hilbert spaces and apply it to study dual frames for three types of vector spaces: for vector spaces over conjugate closed subfields of the complex numbers (in particular, for cyclotomic fields), for metric vector spaces, and for ultrametric normed vector spaces over complete non-archimedean valued fields. Finally, we consider the matrix representation of operators using dual frames and its application to the solution of operators equations in a Petrov-Galerkin scheme.

A Legendre-tau space-time spectral method is established for the 1-D Maxwell equation. The polynomials of different degrees are used to approximate the electric and magnetic fields, respectively, so that they can be decoupled in computation. Also, the time multi-interval Legendre-tau space-time spectral method is considered to keep the long-time computation stable. Error estimates for the method of single and multi-internal are given, respectively. Moreover, the space-time spectral method is applied to the numerical solutions of the 1-D nonlinear Maxwell equation and describes its implicit-explicit iteration scheme. Numerical examples are compared with some other methods, which verifies the effectiveness of the methods for the 1-D Maxwell equation.

An asymptotics for the head wave in the Cauchy problem for the pseudodifferential equation corresponding to a difference scheme for the wave equation with localized initial data is constructed. Dispersive effects appear in the head wave, depending on the relation between the parameters of the problem. In the case of strong dispersive effects, the uniform asymptotic behavior of the head wave in a sufficiently large neighborhood of the leading front is constructed. Moreover, such an asymptotics can be represented in terms of Jacobi theta-functions in the case where the initial function has the form of a Gaussian exponential.

In this paper, a novel system of Maxwell–Schrodinger equations with nonlocal effect in metamaterials is derived from the Drude model, hydrodynamical model and Schrodinger equation. A leap-frog finite element scheme, which can be solved one by one efficiently, is constructed by presenting a group of initial values. This scheme is proved to be stable conditionally in energy norm. It is confirmed that the error convergent rate is O ( τ 2 + h r ) by splitting the proof into three parts, where τ is the time step, h is the mesh size and r is the maximum total degree of polynomials in finite element spaces. Finally, some numerical results are given to verify the theories.

This brief proposes a new discrete-time realization of the super-twisting algorithm based observer. While keeping the property of chattering suppression, the estimation precision of the proposed realization scheme is consistent with the continuous-time super-twisting algorithm (STA), that is, in the absence of noises, the estimation errors satisfy the standard asymptotical accuracy level, i.e., $|e_{1}| \sim \mu _{1}h^{2}$ and $|e_{2}|\sim \mu _{2}h$ with sampling time $h$ and two positive constants $\mu _{1}$ and $\mu _{2}$ . Comparing to the conventional explicit Euler method, the proposed scheme is more exact in the term of smaller constants $\mu _{1}$ and $\mu _{2}$ . The superiority of the proposed scheme is demonstrated through an example of typical buck converter circuit system.

A new Finite Difference complete in phase method with vanished derivatives of the Phase-lag up to order five for the initial value problems with periodical and/or oscillating solutions is produced in the present paper.

We have recently introduced a new cell-centered Lax-Wendroff HLL hybrid scheme for Lagrangian hydrodynamics [Fridrich et al. J. Comp. Phys. 326 (2016) 878-892] with results presented only on logical rectangular quadrilateral meshes. In this study we present an improved version on unstructured meshes, including uniform triangular and hexagonal meshes and non-uniform triangular and polygonal meshes. The performance of the scheme is verified on Noh and Sedov problems and its second-order convergence is verified on a smooth expansion test.Finally the choice of the scalar parameter controlling the amount of added artificial dissipation is studied.

It considered a non-homogeneous allocation scheme of distinguishable particles by different cells. We proved gaussian and Poisson limit theorems for the number of empty cells from the first $$K$$ cells. These results we apply to study the type I error and the type II error in select analogs of the empty box test.

This paper develops a high order adaptive scheme for solving nonlinear Schrodinger equations. The solutions to such equations often exhibit solitary wave and local structures, which makes adaptivity essential in improving the simulation efficiency. Our scheme uses the ultra-weak discontinuous Galerkin (DG) formulation and belongs to the framework of adaptive multiresolution schemes. Various numerical experiments are presented to demonstrate the excellent capability of capturing the soliton waves and the blow-up phenomenon.

In this paper, we construct and study an iterative process involving a finite family of non-self mappings defined on a nonempty closed convex subset of a real Hilbert space. We approximate a common fixed point of a finite family of $$k_i$$ -strictly pseudocontractive non-self mappings for $$i=1,2,\ldots ,N$$ by strong or weak convergence of the scheme depending on the nature of the iteration parameter. Also, we obtain strong convergence results for approximating a common fixed point of a finite family of $$k_i$$ -strictly pseudocontractive non-self mappings for $$i=1,2,\ldots , N$$ under appropriate conditions. Moreover, we prove strong convergence results to a common fixed point of a finite family of quasi-nonexpansive non-self mappings. The results obtained in this paper improve, generalize and complement many of the results in the literature.

In this paper, we propose a direct Eulerian generalized Riemann problem (GRP) scheme for a blood flow model in arteries. It is an extension of the Eulerian GRP scheme, which is developed by Ben-Art...