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Electronic Research Archive

June 2020 , Volume 28 , Issue 2

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Riemann-Liouville derivative over the space of integrable distributions
María Guadalupe Morales, Zuzana Došlá and Francisco J. Mendoza
2020, 28(2): 567-587 doi: 10.3934/era.2020030 +[Abstract](503) +[HTML](250) +[PDF](371.26KB)
Abstract:

In this paper, we generalize the Riemann-Liouville differential and integral operators on the space of Henstock-Kurzweil integrable distributions, \begin{document}$ D_{HK} $\end{document}. We obtain new fundamental properties of the fractional derivatives and integrals, a general version of the fundamental theorem of fractional calculus, semigroup property for the Riemann-Liouville integral operators and relations between the Riemann-Liouville integral and differential operators. Also, we achieve a generalized characterization of the solution for the Abel integral equation. Finally, we show relations for the Fourier transform of fractional derivative and integral. These results are based on the properties of the distributional Henstock-Kurzweil integral and convolution.

Proof of some conjectures involving quadratic residues
Fedor Petrov and Zhi-Wei Sun
2020, 28(2): 589-597 doi: 10.3934/era.2020031 +[Abstract](427) +[HTML](211) +[PDF](329.14KB)
Abstract:

We confirm several conjectures of Sun involving quadratic residues modulo odd primes. For any prime \begin{document}$ p\equiv 1\ ({\rm{mod}}\ 4) $\end{document} and integer \begin{document}$ a\not\equiv0\ ({\rm{mod}}\ p) $\end{document}, we prove that

and that

where \begin{document}$ (\frac{a}p) $\end{document} is the Legendre symbol, \begin{document}$ \varepsilon_p $\end{document} and \begin{document}$ h(p) $\end{document} are the fundamental unit and the class number of the real quadratic field \begin{document}$ \mathbb Q(\sqrt p) $\end{document} respectively, and \begin{document}$ \{x\}_p $\end{document} is the least nonnegative residue of an integer \begin{document}$ x $\end{document} modulo \begin{document}$ p $\end{document}. Also, for any prime \begin{document}$ p\equiv3\ ({\rm{mod}}\ 4) $\end{document} and \begin{document}$ {\delta} = 1, 2 $\end{document}, we determine

where \begin{document}$ T_m $\end{document} denotes the triangular number \begin{document}$ m(m+1)/2 $\end{document}.

Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity
Xu Liu and Jun Zhou
2020, 28(2): 599-625 doi: 10.3934/era.2020032 +[Abstract](440) +[HTML](214) +[PDF](417.81KB)
Abstract:

In this paper, the initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity is invsitgated. First, we establish the local well-posedness of solutions by means of the semigroup theory. Then by using ordinary differential inequalities, potential well theory and energy estimate, we study the conditions on global existence and finite time blow-up. Moreover, the lifespan (i.e., the upper bound of the blow-up time) of the finite time blow-up solution is estimated.

Revisited bilinear Schrödinger estimates with applications to generalized Boussinesq equations Special Issues
Dan-Andrei Geba and Evan Witz
2020, 28(2): 627-649 doi: 10.3934/era.2020033 +[Abstract](334) +[HTML](194) +[PDF](425.86KB)
Abstract:

In this paper, our goal is to improve the local well-posedness theory for certain generalized Boussinesq equations by revisiting bilinear estimates related to the Schrödinger equation. Moreover, we propose a novel, automated procedure to handle the summation argument for these bounds.

Kirchhoff-type differential inclusion problems involving the fractional Laplacian and strong damping Special Issues
Mingqi Xiang, Binlin Zhang and Die Hu
2020, 28(2): 651-669 doi: 10.3934/era.2020034 +[Abstract](451) +[HTML](207) +[PDF](429.17KB)
Abstract:

The aim of this paper is to investigate the existence of weak solutions for a Kirchhoff-type differential inclusion wave problem involving a discontinuous set-valued term, the fractional \begin{document}$ p $\end{document}-Laplacian and linear strong damping term. The existence of weak solutions is obtained by using a regularization method combined with the Galerkin method.

Global behavior of the solutions to nonlinear Klein-Gordon equation with critical initial energy Special Issues
Milena Dimova, Natalia Kolkovska and Nikolai Kutev
2020, 28(2): 671-689 doi: 10.3934/era.2020035 +[Abstract](439) +[HTML](193) +[PDF](361.94KB)
Abstract:

Nonlinear Klein-Gordon equation with combined power type nonlinearity and critical initial energy is investigated. The qualitative properties of a new ordinary differential equation are studied and the concavity method of Levine is improved. Necessary and sufficient conditions for finite time blow up and global existence of the solutions are proved. New sufficient conditions on the initial data for finite time blow up, based on the necessary and sufficient ones, are obtained. The asymptotic behavior of the global solutions is also investigated.

The well-posedness and regularity of a rotating blades equation Special Issues
Lin Shen, Shu Wang and Yongxin Wang
2020, 28(2): 691-719 doi: 10.3934/era.2020036 +[Abstract](436) +[HTML](225) +[PDF](457.43KB)
Abstract:

In this paper, a rotating blades equation is considered. The arbitrary pre-twisted angle, arbitrary pre-setting angle and arbitrary rotating speed are taken into account when establishing the rotating blades model. The nonlinear PDEs of motion and two types of boundary conditions are derived by the extended Hamilton principle and the first-order piston theory. The well-posedness of weak solution (global in time) for the rotating blades equation with Clamped-Clamped (C-C) boundary conditions can be proved by compactness method and energy method. Strong energy estimates are derived under additional assumptions on the initial data. In addition, the existence and regularity of weak solutions (global in time) for the rotating blades equation with Clamped-Free (C-F) boundary conditions are proved as well.

Approximation by multivariate Baskakov–Kantorovich operators in Orlicz spaces
Ling-Xiong Han, Wen-Hui Li and Feng Qi
2020, 28(2): 721-738 doi: 10.3934/era.2020037 +[Abstract](503) +[HTML](195) +[PDF](375.01KB)
Abstract:

Utilizing some properties of multivariate Baskakov–Kantorovich operators and using \begin{document}$ K $\end{document}-functional and a decomposition technique, the authors find two equivalent theorems between the \begin{document}$ K $\end{document}-functional and modulus of smoothness, and obtain a direct theorem in the Orlicz spaces.

Certain $*$-homomorphisms acting on unital $C^{*}$-probability spaces and semicircular elements induced by $p$-adic number fields over primes $p$
Ilwoo Cho
2020, 28(2): 739-776 doi: 10.3934/era.2020038 +[Abstract](284) +[HTML](179) +[PDF](468.15KB)
Abstract:

In this paper, we study the Banach \begin{document}$ * $\end{document}-probability space \begin{document}$ (A\otimes_{\Bbb{C}}\Bbb{LS}, $\end{document} \begin{document}$ \tau_{A}^{0}) $\end{document} generated by a fixed unital \begin{document}$ C^{*} $\end{document}-probability space \begin{document}$ (A, $\end{document} \begin{document}$ \varphi_{A}), $\end{document} and the semicircular elements \begin{document}$ \Theta_{p,j} $\end{document} induced by \begin{document}$ p $\end{document}-adic number fields \begin{document}$ \Bbb{Q}_{p}, $\end{document} for all \begin{document}$ p $\end{document} \begin{document}$ \in $\end{document} \begin{document}$ \mathcal{P}, $\end{document} \begin{document}$ j $\end{document} \begin{document}$ \in $\end{document} \begin{document}$ \Bbb{Z}, $\end{document} where \begin{document}$ \mathcal{P} $\end{document} is the set of all primes, and \begin{document}$ \Bbb{Z} $\end{document} is the set of all integers. In particular, from the order-preserving shifts \begin{document}$ g\times h_{\pm } $\end{document} on \begin{document}$ \mathcal{P} $\end{document} \begin{document}$ \times $\end{document} \begin{document}$ \Bbb{Z}, $\end{document} and \begin{document}$ * $\end{document}-homomorphisms \begin{document}$ \theta $\end{document} on \begin{document}$ A, $\end{document} we define the corresponding \begin{document}$ * $\end{document}-homomorphisms \begin{document}$ \sigma_{(\pm ,1)}^{1:\theta } $\end{document} on \begin{document}$ A\otimes_{\Bbb{C}}\Bbb{LS}, $\end{document} and consider free-distributional data affected by them.

$ H^2 $ blowup result for a Schrödinger equation with nonlinear source term
Xuan Liu and Ting Zhang
2020, 28(2): 777-794 doi: 10.3934/era.2020039 +[Abstract](418) +[HTML](308) +[PDF](392.34KB)
Abstract:

In this paper, we consider the nonlinear Schrödinger equation on \begin{document}$ \mathbb{R}^N, N\ge1 $\end{document},

with \begin{document}$ H^2 $\end{document}-subcritical nonlinearities: \begin{document}$ \alpha>0, (N-4)\alpha<4 $\end{document} and Re\begin{document}$ \lambda>0 $\end{document}. For any given compact set \begin{document}$ K\subset\mathbb{R}^N $\end{document}, we construct \begin{document}$ H^2 $\end{document} solutions that are defined on \begin{document}$ (-T, 0) $\end{document} for some \begin{document}$ T>0 $\end{document}, and blow up exactly on \begin{document}$ K $\end{document} at \begin{document}$ t = 0 $\end{document}. We generalize the range of the power \begin{document}$ \alpha $\end{document} in the result of Cazenave, Han and Martel [5]. The proof is based on the energy estimates and compactness arguments.

On Seshadri constants and point-curve configurations
Marek Janasz and Piotr Pokora
2020, 28(2): 795-805 doi: 10.3934/era.2020040 +[Abstract](328) +[HTML](218) +[PDF](299.58KB)
Abstract:

In the note we study the multipoint Seshadri constants of \begin{document}$ \mathcal{O}_{\mathbb{P}^{2}_{\mathbb{C}}}(1) $\end{document} centered at singular loci of certain curve arrangements in the complex projective plane. Our first aim is to show that the values of Seshadri constants can be approximated with use of a combinatorial invariant which we call the configurational Seshadri constant. We study specific examples of point-curve configurations for which we provide actual values of the associated Seshadri constants. In particular, we provide an example based on Hesse point-conic configuration for which the associated Seshadri constant is computed by a line. This shows that multipoint Seshadri constants are not purely combinatorial.

A family of potential wells for a wave equation
Yang Liu and Wenke Li
2020, 28(2): 807-820 doi: 10.3934/era.2020041 +[Abstract](354) +[HTML](200) +[PDF](933.36KB)
Abstract:

In this paper, a family of potential wells are introduced by means of the modified depths of the potential wells. These potential wells are employed to study the initial-boundary value problem for a wave equation. The expression of the depths of the potential wells is derived. Global existence and finite time blow-up of weak solutions with the subcritical initial energy and the critical initial energy are obtained, respectively. Moreover, some numerical simulations of the depths of the potential wells are carried out.

A hybridized weak Galerkin finite element scheme for general second-order elliptic problems Special Issues
Guanrong Li, Yanping Chen and Yunqing Huang
2020, 28(2): 821-836 doi: 10.3934/era.2020042 +[Abstract](401) +[HTML](204) +[PDF](377.34KB)
Abstract:

In this paper, a hybridized weak Galerkin (HWG) finite element scheme is presented for solving the general second-order elliptic problems. The HWG finite element scheme is based on the use of a Lagrange multiplier defined on the element boundaries. The Lagrange multiplier provides a numerical approximation for certain derivatives of the exact solution. It is worth pointing out that a skew symmetric form has been used for handling the convection term to get the stability in the HWG formulation. Optimal order error estimates are derived for the corresponding HWG finite element approximations. A Schur complement formulation of the HWG method is introduced for implementation purpose.

Numerical simulation for 3D flow in flow channel of aeroengine turbine fan based on dimension splitting method Special Issues
Guoliang Ju, Can Chen, Rongliang Chen, Jingzhi Li, Kaitai Li and Shaohui Zhang
2020, 28(2): 837-851 doi: 10.3934/era.2020043 +[Abstract](475) +[HTML](180) +[PDF](1291.07KB)
Abstract:

In this paper, we introduce a dimension splitting method for simulating the air flow state of the aeroengine turbine fan. Based on the geometric model of the fan blade, the dimension splitting method establishes a semi-geodesic coordinate system. Under such coordinate system, the Navier-Stokes equations are reformulated into the combination of membrane operator equations on two-dimensional manifolds and bending operator equations along the hub circle. Using Euler central difference scheme to approximate the third variable, the new form of Navier-Stokes equations is splitting into a set of two-dimensional sub-problems. Solving these sub-problems by alternate iteration, it follows an approximate solution to Navier-Stokes equations. Furthermore, we conduct a numerical experiment to show that the dimension splitting method has a good performance by comparing with the traditional methods. Finally, we give the simulation results of the pressure and flow state of the fan blade.

On existence of PI-exponents of unital algebras
Dušan D. Repovš and Mikhail V. Zaicev
2020, 28(2): 853-859 doi: 10.3934/era.2020044 +[Abstract](225) +[HTML](123) +[PDF](288.54KB)
Abstract:

We construct a family of unital non-associative algebras \begin{document}$ \{T_\alpha\vert\; 2<\alpha\in\mathbb R\} $\end{document} such that \begin{document}$ \underline{exp}(T_\alpha) = 2 $\end{document}, whereas \begin{document}$ \alpha\le\overline{exp}(T_\alpha)\le\alpha+1 $\end{document}. In particular, it follows that ordinary PI-exponent of codimension growth of algebra \begin{document}$ T_\alpha $\end{document} does not exist for any \begin{document}$ \alpha> 2 $\end{document}. This is the first example of a unital algebra whose PI-exponent does not exist.

Exponential stability and regularity of compressible viscous micropolar fluid with cylinder symmetry
Zhi-Ying Sun, Lan Huang and Xin-Guang Yang
2020, 28(2): 861-878 doi: 10.3934/era.2020045 +[Abstract](254) +[HTML](116) +[PDF](358.74KB)
Abstract:

This paper is concerned with three-dimensional compressible viscous and heat-conducting micropolar fluid in the domain to the subset of \begin{document}$ R^3 $\end{document} bounded with two coaxial cylinders that present the solid thermoinsulated walls, being in a thermodynamical sense perfect and polytropic. We prove that the regularity and the exponential stability in \begin{document}$ H^2 $\end{document}.

Quasineutral limit for the compressible two-fluid Euler–Maxwell equations for well-prepared initial data
Min Li, Xueke Pu and Shu Wang
2020, 28(2): 879-895 doi: 10.3934/era.2020046 +[Abstract](242) +[HTML](118) +[PDF](347.84KB)
Abstract:

In this paper, we study the quasi-neutral limit for the compressible two-fluid Euler–Maxwell equations for well-prepared initial data. Precisely, we proved the solution of the three-dimensional compressible two-fluid Euler–Maxwell equations converges locally in time to that of the compressible Euler equation as \begin{document}$ \varepsilon $\end{document} tends to zero. This proof is based on the formal asymptotic expansions, the iteration techniques, the vector analysis formulas and the Sobolev energy estimates.

Superclose analysis of a two-grid finite element scheme for semilinear parabolic integro-differential equations Special Issues
Changling Xu and Tianliang Hou
2020, 28(2): 897-910 doi: 10.3934/era.2020047 +[Abstract](255) +[HTML](114) +[PDF](310.06KB)
Abstract:

In this paper, a two-grid finite element scheme for semilinear parabolic integro-differential equations is proposed. In the two-grid scheme, continuous linear element is used for spatial discretization, while Crank-Nicolson scheme and Leap-Frog scheme are ultilized for temporal discretization. Based on the combination of the interpolation and Ritz projection technique, some superclose estimates between the interpolation and the numerical solution in the \begin{document}$ H^1 $\end{document}-norm are derived. Notice that we only need to solve nonlinear problem once in the two-grid scheme, namely, the first time step on the coarse-grid space. A numerical example is presented to verify the effectiveness of the proposed two-grid scheme.

A $ C^0P_2 $ time-stepping virtual element method for linear wave equations on polygonal meshes Special Issues
Jianguo Huang and Sen Lin
2020, 28(2): 911-933 doi: 10.3934/era.2020048 +[Abstract](310) +[HTML](145) +[PDF](1531.97KB)
Abstract:

This paper is concerned with a \begin{document}$ C^0P_2 $\end{document} time-stepping virtual element method (VEM) for solving linear wave equations on polygonal meshes. The spatial discretization is carried out by the VEM while the temporal discretization is obtained based on the \begin{document}$ C^0P_2 $\end{document} time-stepping approach, leading to a fully discrete method. The error estimates in the \begin{document}$ H^1 $\end{document} semi-norm and \begin{document}$ L^2 $\end{document} norm are derived after detailed derivation. Finally, the numerical performance and efficiency of the proposed method is illustrated by several numerical experiments.

Uniform high-order convergence of multiscale finite element computation on a graded recursion for singular perturbation Special Issues
Shan Jiang, Li Liang, Meiling Sun and Fang Su
2020, 28(2): 935-949 doi: 10.3934/era.2020049 +[Abstract](329) +[HTML](123) +[PDF](434.86KB)
Abstract:

A multiscale finite element method is proposed for addressing a singularly perturbed convection-diffusion model. In reference to the a priori estimate of the boundary layer location, we provide a graded recursion for the mesh adaption. On this mesh, the multiscale basis functions are able to capture the microscopic boundary layers effectively. Then with a global reduction, the multiscale functional space may efficiently reflect the macroscopic essence. The error estimate for the adaptive multiscale strategy is presented, and a high-order convergence is proved. Numerical results validate the robustness of this novel multiscale simulation for singular perturbation with small parameters, as a consequence, high accuracy and uniform superconvergence are obtained through computational reductions.

On the mod p Steenrod algebra and the Leibniz-Hopf algebra Special Issues
Neşet Deniz Turgay
2020, 28(2): 951-959 doi: 10.3934/era.2020050 +[Abstract](506) +[HTML](170) +[PDF](310.83KB)
Abstract:

Let \begin{document}$ p $\end{document} be a fixed odd prime. The Bockstein free part of the mod \begin{document}$ p $\end{document} Steenrod algebra, \begin{document}$ \mathcal{A}_p $\end{document}, can be defined as the quotient of the mod \begin{document}$ p $\end{document} reduction of the Leibniz Hopf algebra, \begin{document}$ \mathcal{F}_p $\end{document}. We study the Hopf algebra epimorphism \begin{document}$ \pi\colon \mathcal{F}_p\to \mathcal{A}_p $\end{document} to investigate the canonical Hopf algebra conjugation in \begin{document}$ \mathcal{A}_p $\end{document} together with the conjugation operation in \begin{document}$ \mathcal{F}_p $\end{document}. We also give a result about conjugation invariants in the mod 2 dual Leibniz Hopf algebra using its multiplicative algebra structure.

An adaptive edge finite element method for the Maxwell's equations in metamaterials Special Issues
Hao Wang, Wei Yang and Yunqing Huang
2020, 28(2): 961-976 doi: 10.3934/era.2020051 +[Abstract](425) +[HTML](181) +[PDF](19092.29KB)
Abstract:

In this paper, we study an adaptive edge finite element method for time-harmonic Maxwell's equations in metamaterials. A-posteriori error estimators based on the recovery type and residual type are proposed, respectively. Based on our a-posteriori error estimators, the adaptive edge finite element method is designed and applied to simulate the backward wave propagation, electromagnetic splitter, rotator, concentrator and cloak devices. Numerical examples are presented to illustrate the reliability and efficiency of the proposed a-posteriori error estimations for the adaptive method.

A multi-mode expansion method for boundary optimal control problems constrained by random Poisson equations Special Issues
Jingshi Li, Jiachuan Zhang, Guoliang Ju and Juntao You
2020, 28(2): 977-1000 doi: 10.3934/era.2020052 +[Abstract](266) +[HTML](129) +[PDF](443.55KB)
Abstract:

This paper develops efficient numerical algorithms for the optimal control problem constrained by Poisson equations with uncertain diffusion coefficients. We consider both unconstrained condition and box-constrained condition for the control. The algorithms are based upon a multi-mode expansion (MME) for the random state, the finite element approximation for the physical space and the alternating direction method of multipliers (ADMM) or two-block ADMM for the discrete optimization systems. The compelling aspect of our method is that, target random constrained control problem can be approximated to one deterministic constrained control problem under a state variable substitution equality. Thus, the computing resource, especially the memory consumption, can be reduced sharply. The convergence rates of the proposed algorithms are discussed in the paper. We also present some numerical examples to show the performance of our algorithms.

Efficient numerical methods for elliptic optimal control problems with random coefficient Special Issues
Xiaowei Pang, Haiming Song, Xiaoshen Wang and Jiachuan Zhang
2020, 28(2): 1001-1022 doi: 10.3934/era.2020053 +[Abstract](288) +[HTML](104) +[PDF](883.29KB)
Abstract:

Efficient numerical methods for solving Poisson equation constraint optimal control problems with random coefficient are discussed in this paper. By applying the finite element method and the Monte Carlo approximation, the original optimal control problem is discretized and transformed into an optimization problem. Taking advantage of the separable structures, Algorithm 1 is proposed for solving the problem, where an alternating direction method of multiplier is used. Both computational and storage costs of this algorithm are very high. In order to reduce the computational cost, Algorithm 2 is proposed, where the multi-modes expansion is introduced and applied. Further, for reducing the storage cost, we propose Algorithm 3 based on Algorithm 2. The main idea is that the random term is shifted to the objective functional, which could be computed in advance. Therefore, we only need to solve a deterministic optimization problem, which could reduce all the costs significantly. Moreover, the convergence analyses of the proposed algorithms are established, and numerical simulations are carried out to test the performances of them.

Proof of Sun's conjectural supercongruence involving Catalan numbers Special Issues
Ji-Cai Liu
2020, 28(2): 1023-1030 doi: 10.3934/era.2020054 +[Abstract](233) +[HTML](105) +[PDF](324.81KB)
Abstract:

We confirm a conjectural supercongruence involving Catalan numbers, which is one of the 100 selected open conjectures on congruences of Sun. The proof makes use of hypergeometric series identities and symbolic summation method.

A family of $ q $-congruences modulo the square of a cyclotomic polynomial Special Issues
Victor J. W. Guo
2020, 28(2): 1031-1036 doi: 10.3934/era.2020055 +[Abstract](260) +[HTML](101) +[PDF](258.92KB)
Abstract:

Using Watson's terminating \begin{document}$ _8\phi_7 $\end{document} transformation formula, we prove a family of \begin{document}$ q $\end{document}-congruences modulo the square of a cyclotomic polynomial, which were originally conjectured by the author and Zudilin [J. Math. Anal. Appl. 475 (2019), 1636-646]. As an application, we deduce two supercongruences modulo \begin{document}$ p^4 $\end{document} (\begin{document}$ p $\end{document} is an odd prime) and their \begin{document}$ q $\end{document}-analogues. This also partially confirms a special case of Swisher's (H.3) conjecture.

Asymptotic behaviour of a neural field lattice model with delays Special Issues
Xiaoli Wang, Peter Kloeden and Meihua Yang
2020, 28(2): 1037-1048 doi: 10.3934/era.2020056 +[Abstract](249) +[HTML](112) +[PDF](331.11KB)
Abstract:

The asymptotic behaviour of an autonomous neural field lattice system with delays is investigated. It is based on the Amari model, but with the Heaviside function in the interaction term replaced by a sigmoidal function. First, the lattice system is reformulated as an infinite dimensional ordinary delay differential equation on weighted sequence state space \begin{document}$ \ell_\rho^2 $\end{document} under some appropriate assumptions. Then the global existence and uniqueness of its solution and its formulation as a semi-dynamical system on a suitable function space are established. Finally, the asymptotic behaviour of solution of the system is investigated, in particular, the existence of a global attractor is obtained.

Recursive sequences and girard-waring identities with applications in sequence transformation
Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie and Minghao Chen
2020, 28(2): 1049-1062 doi: 10.3934/era.2020057 +[Abstract](279) +[HTML](143) +[PDF](341.92KB)
Abstract:

We present here a generalized Girard-Waring identity constructed from recursive sequences. We also present the construction of Binet Girard-Waring identity and classical Girard-Waring identity by using the generalized Girard-Waring identity and divided differences. The application of the generalized Girard-Waring identity to the transformation of recursive sequences of numbers and polynomials is discussed.

Two congruences concerning Apéry numbers conjectured by Z.-W. Sun
Chen Wang
2020, 28(2): 1063-1075 doi: 10.3934/era.2020058 +[Abstract](338) +[HTML](147) +[PDF](312.55KB)
Abstract:

Let \begin{document}$ n $\end{document} be a nonnegative integer. The \begin{document}$ n $\end{document}-th Apéry number is defined by

Z.-W. Sun investigated the congruence properties of Apéry numbers and posed some conjectures. For example, Sun conjectured that for any prime \begin{document}$ p\geq7 $\end{document}

and for any prime \begin{document}$ p\geq5 $\end{document}

where \begin{document}$ H_n = \sum_{k = 1}^n1/k $\end{document} denotes the \begin{document}$ n $\end{document}-th harmonic number and \begin{document}$ B_0, B_1, \ldots $\end{document} are the well-known Bernoulli numbers. In this paper we shall confirm these two conjectures.

High-order energy stable schemes of incommensurate phase-field crystal model Special Issues
Kai Jiang and Wei Si
2020, 28(2): 1077-1093 doi: 10.3934/era.2020059 +[Abstract](263) +[HTML](113) +[PDF](3164.56KB)
Abstract:

This article focuses on the development of high-order energy stable schemes for the multi-length-scale incommensurate phase-field crystal model which is able to study the phase behavior of aperiodic structures. These high-order schemes based on the scalar auxiliary variable (SAV) and spectral deferred correction (SDC) approaches are suitable for the \begin{document}$ L^2 $\end{document} gradient flow equation, i.e., the Allen-Cahn dynamic equation. Concretely, we propose a second-order Crank-Nicolson (CN) scheme of the SAV system, prove the energy dissipation law, and give the error estimate in the almost periodic function sense. Moreover, we use the SDC method to improve the computational accuracy of the SAV/CN scheme. Numerical results demonstrate the advantages of high-order numerical methods in numerical computations and show the influence of length-scales on the formation of ordered structures.

Rough semi-uniform spaces and its image proximities Special Issues
Surabhi Tiwari and Pankaj Kumar Singh
2020, 28(2): 1095-1106 doi: 10.3934/era.2020060 +[Abstract](291) +[HTML](167) +[PDF](3941.13KB)
Abstract:

In this paper, we introduce the concept of rough semi-uniform spaces as a supercategory of rough pseudometric spaces and approximation spaces. A completion of approximation spaces has been constructed using rough semi-uniform spaces. Applications of rough semi-uniform spaces in the construction of proximities of digital images is also discussed.

Generating geometric body shapes with electromagnetic source scattering techniques Special Issues
Youzi He, Bin Li, Tingting Sheng and Xianchao Wang
2020, 28(2): 1107-1121 doi: 10.3934/era.2020061 +[Abstract](261) +[HTML](136) +[PDF](1258.3KB)
Abstract:

In this paper, we are concerned with the three-dimensional (3D) geometric body shape generation with several well-selected characteristic values. Since 3D human shapes can be viewed as the support of the electromagnetic sources, we formulate a scheme to regenerate 3D human shapes by inverse scattering theory. With the help of vector spherical harmonics expansion of the magnetic far field pattern, we build on a smart one-to-one correspondence between the geometric body space and the multi-dimensional vector space that consists of all coefficients of the spherical vector wave function expansion of the magnetic far field pattern. Therefore, these coefficients can serve as the shape generator. For a collection of geometric body shapes, we obtain the inputs (characteristic values of the body shapes) and the outputs (the coefficients of the spherical vector wave function expansion of the corresponding magnetic far field patterns). Then, for any unknown body shape with the given characteristic set, we use the multivariate Lagrange interpolation to get the shape generator of this new shape. Finally, we get the reconstruction of this unknown shape by using the multiple-frequency Fourier method. Numerical examples of both whole body shapes and human head shapes verify the effectiveness of the proposed method.

A neural network method for the inverse scattering problem of impenetrable cavities Special Issues
Weishi Yin, Jiawei Ge, Pinchao Meng and Fuheng Qu
2020, 28(2): 1123-1142 doi: 10.3934/era.2020062 +[Abstract](309) +[HTML](108) +[PDF](841.28KB)
Abstract:

This paper proposes a near-field shape neural network (NSNN) to determine the shape of a sound-soft cavity based on a single source and several measurements placed on a curve inside the cavity. The NSNN employs the near-field measurements as input, and the output is the shape parameters of the cavity. The self-attention mechanism is employed to obtain the feature information of the near-field data, as well as the correlations among them. The weights and biases of the NSNN are updated through the gradient descent algorithm, which minimizes the error of the reconstructed shape of the cavity. We prove that the loss function sequence related to the weights is a monotonically bounded non-negative sequence, which indicates the convergence of the NSNN. Numerical experiments show that the shape of the cavity can be effectively reconstructed with the NSNN.

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