Solving a Class of Brouwer Fixed-point Problems via a Modified Aggregate Constraint Homotopy Method

In this paper, we provide an aggregate function homotopy interior point method to solve a class of Brouwer fixed-point problems. Compared with the homotopy method （proposed by Yu and Lin, Appl. Math. Comput., 74（1996）, 65）, the main adavantages of this method are as foUows： on the one hand, it can solve the Brouwer fixed-point problems in a broader class of nonconvex subsets Ω in R^n （in this paper, we let Ω={x∈ R^n ： gi（x） ≤0, i= 1,... , m}）; on the other hand, it can also deal with the subsets Ω with larger amount of constraints more effectively.

Modified Homotopy Method for a Class of Brouwer Fixed-point Problems

In this paper, we modify the homotopy method （proposed by Yu and Lin, Appl. Math. Comput., 74（1996）, 65） and hence make the modified method be able to solve Brouwer fixed-point problems in a broader class of nonconvex subsets in Rn. In addition, a simple example is given to show the effectiveness of the modified method.

Traversable Wormholes and the Brouwer Fixed-Point Theorem

The Brouwer fixed-point theorem in topology states that for any continuous mapping f on a compact convex set into itself admits a fixed point, i.e., a point x0 such that f(x0) = x0. Under suitable conditions, this fixed point corresponds to the throat of a traversable wormhole, i.e., b(r0) = r0 for the shape function b = b(r). The possible existence of wormholes can therefore be deduced from purely mathematical considerations without going beyond the existing physical requirements.

Sparse-Grid Implementation of Fixed-Point Fast Sweeping WENO Schemes for Eikonal Equations

Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of fast sweeping schemes,fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order.The resulting iterative schemes have a fast convergence rate to steady-state solutions.Moreover,an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve the inverse operation of any nonlinear local system.Hence,they are robust and flexible,and have been combined with high-order accurate weighted essentially non-oscillatory(WENO)schemes to solve various hyperbolic PDEs in the literature.For multidimensional nonlinear problems,high-order fixed-point fast sweeping WENO methods still require quite a large amount of computational costs.In this technical note,we apply sparse-grid techniques,an effective approximation tool for multidimensional problems,to fixed-point fast sweeping WENO methods for reducing their computational costs.Here,we focus on fixed-point fast sweeping WENO schemes with third-order accuracy(Zhang et al.2006[41]),for solving Eikonal equations,an important class of static Hamilton-Jacobi(H-J)equations.Numerical experiments on solving multidimensional Eikonal equations and a more general static H-J equation are performed to show that the sparse-grid computations of the fixed-point fast sweeping WENO schemes achieve large savings of CPU times on refined meshes,and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids.

A Secure and Effective Energy-Aware Fixed-Point Quantization Scheme for Asynchronous Federated Learning

Asynchronous federated learning(AsynFL)can effectivelymitigate the impact of heterogeneity of edge nodes on joint training while satisfying participant user privacy protection and data security.However,the frequent exchange of massive data can lead to excess communication overhead between edge and central nodes regardless of whether the federated learning(FL)algorithm uses synchronous or asynchronous aggregation.Therefore,there is an urgent need for a method that can simultaneously take into account device heterogeneity and edge node energy consumption reduction.This paper proposes a novel Fixed-point Asynchronous Federated Learning(FixedAsynFL)algorithm,which could mitigate the resource consumption caused by frequent data communication while alleviating the effect of device heterogeneity.FixedAsynFL uses fixed-point quantization to compress the local and global models in AsynFL.In order to balance energy consumption and learning accuracy,this paper proposed a quantization scale selection mechanism.This paper examines the mathematical relationship between the quantization scale and energy consumption of the computation/communication process in the FixedAsynFL.Based on considering the upper bound of quantization noise,this paper optimizes the quantization scale by minimizing communication and computation consumption.This paper performs pertinent experiments on the MNIST dataset with several edge nodes of different computing efficiency.The results show that the FixedAsynFL algorithm with an 8-bit quantization can significantly reduce the communication data size by 81.3%and save the computation energy in the training phase by 74.9%without significant loss of accuracy.According to the experimental results,we can see that the proposed AsynFixedFL algorithm can effectively solve the problem of device heterogeneity and energy consumption limitation of edge nodes.

A Fixed-Point Iterative Method for Discrete Tomography Reconstruction Based on Intelligent Optimization

Discrete Tomography(DT)is a technology that uses image projection to reconstruct images.Its reconstruction problem,especially the binary image(0–1matrix)has attracted strong attention.In this study,a fixed point iterative method of integer programming based on intelligent optimization is proposed to optimize the reconstructedmodel.The solution process can be divided into two procedures.First,the DT problem is reformulated into a polyhedron judgment problembased on lattice basis reduction.Second,the fixed-point iterativemethod of Dang and Ye is used to judge whether an integer point exists in the polyhedron of the previous program.All the programs involved in this study are written in MATLAB.The final experimental data show that this method is obviously better than the branch and bound method in terms of computational efficiency,especially in the case of high dimension.The branch and bound method requires more branch operations and takes a long time.It also needs to store a large number of leaf node boundaries and the corresponding consumptionmatrix,which occupies a largememory space.

A Fixed-Point Fast Sweeping WENO Method with Inverse Lax-Wendroff Boundary Treatment for Steady State of Hyperbolic Conservation Laws

Fixed-point fast sweeping WENO methods are a class of efficient high-order numerical methods to solve steady-state solutions of hyperbolic partial differential equations(PDEs).The Gauss-Seidel iterations and alternating sweeping strategy are used to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady-state solutions.A nice property of fixed-point fast sweeping WENO methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems.Hence,they are easy to be applied to a general hyperbolic system.To deal with the difficulties associated with numerical boundary treatment when high-order finite difference methods on a Cartesian mesh are used to solve hyperbolic PDEs on complex domains,inverse Lax-Wendroff(ILW)procedures were developed as a very effective approach in the literature.In this paper,we combine a fifthorder fixed-point fast sweeping WENO method with an ILW procedure to solve steadystate solution of hyperbolic conservation laws on complex computing regions.Numerical experiments are performed to test the method in solving various problems including the cases with the physical boundary not aligned with the grids.Numerical results show highorder accuracy and good performance of the method.Furthermore,the method is compared with the popular third-order total variation diminishing Runge-Kutta(TVD-RK3)time-marching method for steady-state computations.Numerical examples show that for most of examples,the fixed-point fast sweeping method saves more than half CPU time costs than TVD-RK3 to converge to steady-state solutions.

A Proof of Brouwer’s Fixed Point Theorem Using Sperner’s Lemma

This article offers a simple but rigorous proof of Brouwer’s fixed point theorem using Sperner’s Lemma.The general method I have used so far in the proof is mainly to convert the n-dimensional shapes to the corresponding case under the Sperner’s Labeling and apply the Sperner’s Lemma to solve the question.

完备Brouwer格上有限sup-inf合成Fuzzy关系方程解集的一些性质

主要对完备Brouwer格上sup-inf合成的Fuzzy关系方程的解作了深入的讨论。首先讨论了在完备Brouwer格中方程解的情况,并在有解时给出了求解所有极小解的方法。进一步,刻画了sup-inf合成Fuzzy关系方程解集的结构。

无穷观问题的研究(Ⅱ)—从Hausdorff的直觉和Poincaré的名言到Brouwer剧场现象

本组系列论文 ( )～ ( )从数学和认识论的角度系统地研究了无穷观问题的历史发展和现状 ,确立了无穷观背景世界的三分法原则 ,指出了两种无穷观相互排斥的局限性 ,形成了统一两种无穷于同一框架中的基本观点 ,并建立了一个统一实无限与潜无限于同一框架中的公理集合论系统 APAS。

布劳威尔关于向量分布的研究

布劳威尔通过将球面不动点定理和向量场联系起来,推动了布劳威尔不动点定理的产生,也促进了代数拓扑学的发展。基于相关原始文献,围绕布劳威尔关于向量分布的研究,分析布劳威尔对庞加莱指数工作的推广,及其对二维布劳威尔不动点定理新证明方法的思想演变过程,以期厘清布劳威尔将庞加莱的指数工作应用于一般连续向量分布的过程,及二维不动点定理新证明的具体思想。

完备Brouwer格上伪t—模与蕴涵算子

本文引入并讨论Brouwer格L上伪t-模概念,详细研究L上无穷V-分配伪t-模与无穷∧-分配蕴涵之间的关系,证明[0,1]上三类基本的Fuzzy蕴涵与Yager蕴涵都是[0,1]上无穷V-分配伪t-模诱导的蕴涵.此外,我们给出一种生成伪t-模与蕴涵的方法.

Brouwer定理与均衡价格

本文详细论述了在一般均衡体系中均衡价格存在性定理与Brouwer不动点定理是等价的。即,均衡价格π＊就是Brouwer定理中的不动点x并给出证明.这一等价性为计算均衡价格提供了坚实的理论依据,从而肯定了利用Brouwer定理寻找不动点的方法计算均衡价格的计算方法是可靠的,因此具有特别重要的实用意义.

一类带有Hardy项的椭圆型方程正解的存在性

研究了一类带有Hardy项的椭圆型方程在诺伊曼边界条件下正解的存在性.在反应项和边界处均呈指数增长时,利用近似方法和Brouwer不动点定理证明了该方程至少存在一个正解.

Brouwer不动点定理的初等证明

利用Banach不动点定理证明了KKM定理,进而证明了FKKM定理,并给出Brouwer不动点定理的一个简洁初等证明。

基于Brouwer不动点定理的经济均衡问题

根据Vasile I.Istratescu的思路,完善了Brouwer不动点定理的泛函分析证明方法,并运用Brouwer不动点定理证明了纯交易市场下一般经济均衡的存在性.

完备Brouwer格上矩阵的T型{1}-广义逆

利用格上t-模T,给出了完备B rouwer格上矩阵的T型{1}-广义逆的概念,研究了格矩阵的T型{1}-广义逆存在的等价条件,并给出它们的计算方法。

二维Brouwer不动点定理的直观证明

微积分中连续函数的性质和拓扑学中正则不动点的基本结果,给出了二维Brouw-er不动点定理的一个非常直观的证明,且可推广到高维空间。其初等性与直观性可供研究有关问题参考。

带有吸引型奇性的离散周期边值问题多重正解的存在性

基于上下解方法和 Brouwer 度理论,获得如下边值问题 多重正解的存在性,其中 f : (0, +∞) → (0, +∞) 连续,ϕ : Z→ R和r : Z → (0, +∞)为T-周期函数,T > 3为给定的整数,m,µ,是两个正常数,且0 < m ≤1,s ∈ R是参数。

Fuzzy幂集格(非汉字符号)(X)中的Brouwer伪补

讨论Fuzzy幂集格■（X）中的Brouwer伪补的21种型式，并建立了这种伪补的若干性质．