Prime MP-filter Spaces of Fuzzy Implication Algebras

In this paper,the topological space(PF_(MP)(X),T) based on prime MP-filters of a lattice FI-algebra X is constructed firstly and we proved that it is a compact T_0-space if X with condition(P).Secondly,we restricted T to the set of all maximal MP-filters MF_(MP)(X) of X and concluded that(PF_(MP)(X),T |_(PF_(MP)(X)) )is a compact T_2 space if X with conditions(P) and(S).

A Kind of Identities Involving Complete Bell Polynomials

The purpose of this paper is to establish a formula of higher derivative byFaa di Bruno formula, and apply it to some known results to get some identities involvingcomplete Bell polynomials.

A Generalized Lyapunov-Sylvester Computational Method for Numerical Solutions of NLS Equation with Singular Potential

In the present paper a numerical method is developed to approximate the solution of two-dimensional Nonlinear Schrodinger equation in the presence of a sin- gular potential. The method leads to generalized Lyapunov-Sylvester algebraic opera- tors that are shown to be invertible using original topological and differential calculus issued methods. The numerical scheme is proved to be consistent, convergent and sta- ble using the Lyapunov criterion, lax equivalence theorem and the properties of the generalized Lyapunov-Sylvester operators.

Darcy-Forchheimer flows of copper and silver water nanofluids between two rotating stretchable disks

This investigation describes the nanofluid flow in a non-Darcy porous medium between two stretching and rotating disks. A nanofluid comprises of nanoparticles of silver and copper. Water is used as a base fluid. Heat is being transferred with thermal radiation and the Joule heating. A system of ordinary differential equations is obtained by appropriate transformations. Convergent series solutions are obtained. Effects of various parameters are analyzed for the velocity and temperature. Numerical values of the skin friction coefficient and the Nusselt number are tabulated and examined. It can be seen that the radial velocity is affected in the same manner with both porous and local inertial parameters. A skin friction coefficient depicts the same impact on both disks for both nanofluids with larger stretching parameters.

A Well-Balanced Partial Relaxation Scheme for the Two-Dimensional Saint-Venant System

We develop a new moving-water equilibria preserving partial relaxation(PR)scheme for the two-dimensional(2-D)Saint-Venant systemof shallowwater equations.The new scheme is a 2-D generalization of the one-dimensional(1-D)PR scheme recently proposed in[X.Liu,X.Chen,S.Jin,A.Kurganov,andH.Yu,SIAMJ.Sci.Comput.,42(2020),pp.A2206–A2229].Our scheme is based on the PR approximation,which is designed in two steps.First,the geometric source terms are incorporated into the discharge fluxes,which results in a hyperbolic system with global fluxes.Second,the discharge equations are relaxed so that the nonlinearity is moved into the stiff right-hand side of the four added auxiliary equation.The obtained PR system is then numerically integrated using a semi-discrete hybrid upwind/central-upwind finitevolume method combined with an efficient semi-implicit ODE solver.The new 2-D PR scheme inherits the main advantages of the 1-D PR scheme:(i)no special treatment of the geometric source terms is required,(ii)no nonlinear(cubic)equations should be solved to obtain the point values of the water depth out of the reconstructed equilibriumvariables.The performance of the proposed PR scheme is illustrated on a number of numerical examples,in which we demonstrate that the PR scheme not only capable of exactly preserving quasi 1-D moving-water steady states and accurately capturing their small perturbations,but can also handle genuinely 2-D steady states and their small perturbations in a non-oscillatory manner.

General Synthetic Iterative Scheme for Unsteady Rarefied Gas Flows

In rarefied gas flows,the spatial grid size could vary by several orders of magnitude in a single flow configuration(e.g.,inside the Knudsen layer it is at the order of mean free path of gas molecules,while in the bulk region it is at a much larger hydrodynamic scale).Therefore,efficient implicit numerical method is urgently needed for time-dependent problems.However,the integro-differential nature of gas kinetic equations poses a grand challenge,as the gain part of the collision operator is non-invertible.Hence an iterative solver is required in each time step,which usually takes a lot of iterations in the(near)continuum flow regime where the Knudsen number is small;worse still,the solution does not asymptotically preserve the fluid dynamic limit when the spatial cell size is not refined enough.Based on the general synthetic iteration scheme for steady-state solution of the Boltzmann equation,we propose two numerical schemes to push the multiscale simulation of unsteady rarefied gas flows to a new boundary,that is,the numerical solution not only converges within dozens of iterations in each time step,but also asymptotically preserves the Navier-Stokes-Fourier limit in the continuum flow regime,when the spatial grid is coarse,and the time step is large(e.g.,in simulating the extreme slow decay of two-dimensional Taylor vortex,the time step is even at the order of vortex decay time).The properties of fast convergence and asymptotic preserving of the proposed schemes are not only rigorously proven by the Fourier stability analysis for simplified gas kinetic models,but also demonstrated by several numerical examples for the gas kinetic models and the Boltzmann equation.

Complete Study on a Bi-Center Problem for the Z2-Equivariant Cubic Vector Fields

For the planar Z2-equivariant cubic systems having twoelementary focuses, the characterization of a bi-center problem and shortened expressions of the first six Liapunov constants are completely discussed. The necessary and sufficient conditions for the existence of the bi-center are obtained. All possible first integrals are given. Under small Z2-equivariant cubic perturbations, the conclusion that there exist at most 12 small-amplitude limit cycles with the scheme （6 II 6） is proved.

A Rate of Convergence of Physics Informed Neural Networks for the Linear Second Order Elliptic PDEs

In recent years,physical informed neural networks(PINNs)have been shown to be a powerful tool for solving PDEs empirically.However,numerical analysis of PINNs is still missing.In this paper,we prove the convergence rate to PINNs for the second order elliptic equations with Dirichlet boundary condition,by establishing the upper bounds on the number of training samples,depth and width of the deep neural networks to achieve desired accuracy.The error of PINNs is decomposed into approximation error and statistical error,where the approximation error is given in C2 norm with ReLU^(3)networks(deep network with activation function max{0,x^(3)})and the statistical error is estimated by Rademacher complexity.We derive the bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm with ReLU^(3)network,which is of immense independent interest.

A New Earthquake Location Method Based on the Waveform Inversion

In this paper,a new earthquake location method based on the waveform inversion is proposed.As is known to all,the waveform misfit function under the L2 measure is suffering from the cycle skipping problem.This leads to a very small convergence domain of the conventional waveform based earthquake location methods.In present study,by introducing and solving two simple sub-optimization problems,we greatly expand the convergence domain of the waveform based earthquake location method.According to a large number of numerical experiments,the new method expands the range of convergence by several tens of times.This allows us to locate the earthquake accurately even from some relatively bad initial values.

Optimal Superconvergence of Energy Conserving Local Discontinuous Galerkin Methods for Wave Equations

This paper is concerned with numerical solutions of the LDG method for 1D wave equations.Superconvergence and energy conserving properties have been studied.We first study the superconvergence phenomenon for linear problems when alternating fluxes are used.We prove that,under some proper initial discretization,the numerical trace of the LDG approximation at nodes,as well as the cell average,converge with an order 2k+1.In addition,we establish k+2-th order and k+1-th order superconvergence rates for the function value error and the derivative error at Radau points,respectively.As a byproduct,we prove that the LDG solution is superconvergent with an order k+2 towards the Radau projection of the exact solution.Numerical experiments demonstrate that in most cases,our error estimates are optimal,i.e.,the error bounds are sharp.In the second part,we propose a fully discrete numerical scheme that conserves the discrete energy.Due to the energy conserving property,after long time integration,our method still stays accurate when applied to nonlinear Klein-Gordon and Sine-Gordon equations.

ASYMPTOTIC BEHAVIOR OF HARMONIC MAPS FROM COMPLETE MANIFOLDS

In this paper,the author considers a class of complete noncompact Riemannian manifoldswhich satisfy certain conditions on Ricci curvature and volume comparison. It is shown thatany harmonic map with finite energy from such a manifold M into a normal geodesic ball inanother manifold N must be asymptotically constant at the infinity of each large end of M. Arelated existence theorem for harmonic maps is established.

Three-dimensional oscillation of an acoustic microbubble between two rigid curved plates

Understanding the near boundary acoustic oscillation of microbubbles is critical for the effective design of ultrasonic biomedical devices and surface cleaning technologies.Accordingly,this study investigates the three-dimensional microbubble oscillation between two curved rigid plates experiencing a planar acoustic field using boundary integral method(BIM).The numerical model is validated via comparison with the nonlinear oscillation of the bubble governed by the modified Rayleigh-Plesset equation and with the axisymmetric model for an acoustic microbubble in infinite fluid domain.Then,the influence of the wave direction and horizontal standoff distance(h)on the bubble dynamics(including jet velocity,jet direction,centroid movement,total energy,and Kelvin impulse)were evaluated.It was concluded that the jet velocity,the maximum radius and the total energy of the bubble are not significantly influenced by the wave direction,while the jet direction and the high-pressure region depend strongly on it.More importantly,it was found that the jet velocity and the high-pressure region around the jet in acoustic bubble are drastically larger than their counterparts in the gas bubble.

危重症患者中谵妄的归因病死率：前瞻性队列研究

目的明确在危重症患者中因谵妄而导致的病死率。设计前瞻性队列研究。背景2011年1月至2013年7月，荷兰一个有32张混合床位的重症医疗病房（ICU）。研究对象连续收住ICU至少24小时的1112例成人患者。暴露由受过培训的观察人员使用经过验证的评估工具每天评估患者的谵妄状态。应用Logistic回归分析和竞争风险生存分析校正基线变量，应用边际结构模型分析校正谵妄发作前疾病严重程度进展造成的混杂因素影响。

非线性分析及其应用：纪念拉克斯米堪沙姆诞辰80周年 第2卷

《非线性分析和应用论文集》是为纪念V．Lakshmikantham诞辰80周年而编著的，本书是其中的第2卷。

Two-Dimensional Parabolic Inverse Source Problem with Final Overdetermination in Reproducing Kernel Space

A new method of the reproducing kernel Hilbert space is applied to a twodimensional parabolic inverse source problem with the final overdetermination. The exact and approximate solutions are both obtained in a reproducing kernel space. The approximate solution and its partial derivatives are proved to converge to the exact solution and its partial derivatives, respectively. A technique is proposed to improve some existing methods. Numerical results show that the method is of high precision, and confirm the robustness of our method for reconstructing source parameter.

非线性分析及其应用：纪念拉克斯米堪沙姆诞辰80周年

《非线性分析和应用论文集》是为V．Lakshmikantham诞辰80周年而编著的。本书是其中第1卷。

A Unified FastMemory-Saving Time-SteppingMethod for Fractional Operators and Its Applications

Time-dependent fractional partial differential equations typically require huge amounts of memory and computational time,especially for long-time integration,which taxes computational resources heavily for high-dimensional problems.Here,we first analyze existing numerical methods of sum-of-exponentials for approximating the kernel function in constant-order fractional operators,and identify the current pitfalls of such methods.In order to overcome the pitfalls,an improved sum-of-exponentials is developed and verified.We also present several sumof-exponentials for the approximation of the kernel function in variable-order fractional operators.Subsequently,based on the sum-of-exponentials,we propose a unified framework for fast time-stepping methods for fractional integral and derivative operators of constant and variable orders.We test the fast method based on several benchmark problems,including fractional initial value problems,the time-fractional Allen-Cahn equation in two and three spatial dimensions,and the Schr¨odinger equation with nonreflecting boundary conditions,demonstrating the efficiency and robustness of the proposed method.The results show that the present fast method significantly reduces the storage and computational cost especially for long-time integration problems.

Effective Lower Bounds of Wrap-Around L_2-Discrepancy on Three-Level Combined Designs

How to obtain an effective design is a major concern of scientific research. This topic always involves high-dimensional inputs with limited resources. The foldover is a quick and useful technique in construction of fractional designs, which typically releases aliased factors or interactions.This paper takes the wrap-around L_2-discrepancy as the optimality measure to assess the optimal three-level combined designs. New and efficient analytical expressions and lower bounds of the wraparound L_2-discrepancy for three-level combined designs are obtained. The new lower bound is useful and sharper than the existing lower bound. Using the new analytical expression and lower bound as the benchmarks, the authors may implement an effective algorithm for constructing optimal three-level combined designs.

Multi-Phase Segmentation Using Modified Complex Cahn-Hilliard Equations

In this paper,we propose a novel PDE-based model for the multi-phase segmentation problem by using a complex version of Cahn-Hilliard equations.Specifically,we modify the original complex system of Cahn-Hilliard equations by adding the mean curvature term and the fitting term to the evolution of its real part,which helps to render a piecewisely constant function at the steady state.By applying the K-means method to this function,one could achieve the desired multiphase segmentation.To solve the proposed system of equations,a semi-implicit finite difference scheme is employed.Numerical experiments are presented to demonstrate the feasibility of the proposed model and compare our model with other related ones.