Nonsmoothness in Machine Learning: Specific Structure,Proximal Identification,and Applications

Set-Valued and Variational Analysis - Nonsmoothness is often a curse for optimization; but it is sometimes a blessing, in particular for applications in machine learning. In this paper, we present...     

连续型数据众数估计方法及应用研究

一些科研单位或者调查机构经常公布关于居民工资收入状况的数据报告,大多数居民觉得收入被提高,并对此提出质疑.针对该问题,提出众数作为公开数据报告的补充,并给出了众数的两种估计方法.最后,采用这两种方法对中国城镇女性居民月工资收入的众数进行了估计.结果表明,要想得到全面准确的收入分布状况,众数是不可或缺的重要统计量,而且两种众数估计方法是简单而行之有效的,这为众数的推广和应用奠定了基础.     

Cycles and paths in semicomplete multipartite digraphs,theorems, and algorithms: a survey

A digraph obtained by replacing each edge of a complete m-partite graph by an arc or a pair of mutually opposite arcs with the same end vertices is calied a semicomplete m-partite digraph. We describe results (theorems and algorithms) on directed walks in semicomplete m-partite digraphs, including some recent results concerning tournaments. © 1995 John Wiley & Sons, Inc.     

Inequalities for Random Utility Models,with Applications to Ranking and Subset Choice Data

Inequalities on orderings of independent random variables are derived in the context of random utility models for ranking and subset choice data. The inequalities can be used to assess whether ranking or subset choice data are consistent with an independent random utility model. The main technique used for the inequalities is association, with conditions for the sharpness for the inequalities coming from identifying when the association inequality is an equality. Applications to real data sets are given.     

Cycles of two-dimensional systems: Computer calculations,proofs, and experiments

One of the central problems in studying small cycles in the neighborhood of equilibrium involves computation of Lyapunov’s quantities. While Lyapunov’s first and second quantities were computed in the general form in the 1940s–1950s, Lyapunov’s third quantity was calculated only for certain special cases. In the present work, we present general formulas for calculation of Lyapunov’s third quantity. Together with the classical Lyapunov method for calculation of Lyapunov’s quantities, which is based on passing to the polar coordinates, we suggest a method developed for the Euclidian coordinates and for the time domain. The calculation of Lyapunov’s quantities by two different analytic methods involving modern software tools for symbolic computing enables us to justify the formulas obtained for Lyapunov’s third quantity. For quadratic systems in which Lyapunov’s first and second quantities vanish, while the third one does not, large cycles were calculated. In the calculations, the quadratic system was reduced to the Liénard equation, which was used to evaluate the domain of parameters corresponding to the existence of four cycles (three “small” cycles and a “large” one). This domain extends the region of parameters obtained by S.L. Shi in 1980 for a quadratic system with four limit cycles.     

Existence and Uniqueness of Limit Cycles for a Polynomial System

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Review of Methods Inspired by Algebraic-Multigrid for Data and Image Analysis Applications

Algebraic Multigrid (AMG) methods were developed originally for numerically solving Partial Differential Equations (PDE), not necessarily on structured grids. In the last two decades solvers inspired by the AMG approach, were developed for non PDE problems, including data and image analysis problems, such as clustering, segmentation, quantization and others. These solvers share a common principle that there is a crosstalk between fine and coarse representations of the problems, with flow of information in both directions, fine-to-coarse and coarse-to-fine. This paper surveys some of these problems and the AMG-inspired algorithms for their solution.     

Statistical,Physical, and Computational Aspects of Massive Data Analysis and Assimilation in Atmospheric Applications

Abstract

The goals and procedures of the most data-intensive operations in atmospheric sciences, including data assimilation and fusion, are introduced. We explore specific problems that result from the expansion in observing systems from conventional to satellite borne and the corresponding transition from small, medium, and large datasets to massive datasets. The satellite data, their volumes, heterogeneity, and structure are described in specific examples. We illustrate that the atmospheric data analysis and assimilation procedures and the satellite data pose unique problems that do not exist in other applications and are not easily addressed by existing methods and tools. Existing solutions are presented and their performance with massive datasets is critically evaluated. We conclude that since the problems are interdisciplinary, a comprehensive solution must be interdisciplinary as well. We note that components of such a solution already exist in statistics, atmospheric, and computational sciences, but that in isolation they often fail to scale up to the massive data challenge. The prospects of synthesizing an interdisciplinary solution which will scale up to the massive data challenge are thus promising.     

Dynamics of Coupled Cell Networks: Synchrony, Heteroclinic Cycles and Inflation

We consider the dynamics of small networks of coupled cells. We usually assume asymmetric inputs and no global or local symmetries in the network and consider equivalence of networks in this setting; that is, when two networks with different architectures give rise to the same set of possible dynamics. Focussing on transitive (strongly connected) networks that have only one type of cell (identical cell networks) we address three questions relating the network structure to dynamics. The first question is how the structure of the network may force the existence of invariant subspaces (synchrony subspaces). The second question is how these invariant subspaces can support robust heteroclinic attractors. Finally, we investigate how the dynamics of coupled cell networks with different structures and numbers of cells can be related; in particular we consider the sets of possible “inflations” of a coupled cell network that are obtained by replacing one cell by many of the same type, in such a way that the original network dynamics is still present within a synchrony subspace. We illustrate the results with a number of examples of networks of up to six cells.     

Finding Solution Subspaces of the Laplace and Heat Equations Isometric to Spaces of Real Functions,and Some of Their Applications

Mathematical Notes - We single out subspaces of harmonic functions in the upper half-plane coinciding with spaces of convolutions with the Abel–Poisson kernel and subspaces of solutions of...     

The Accumulated Persistence Function,a New Useful Functional Summary Statistic for Topological Data Analysis,With a View to Brain Artery Trees and Spatial Point Process Applications

We start with a simple introduction to topological data analysis where the most popular tool is called a persistence diagram. Briefly, a persistence diagram is a multiset of points in the plane describing the persistence of topological features of a compact set when a scale parameter varies. Since statistical methods are difficult to apply directly on persistence diagrams, various alternative functional summary statistics have been suggested, but either they do not contain the full information of the persistence diagram or they are two-dimensional functions. We suggest a new functional summary statistic that is one-dimensional and hence easier to handle, and which under mild conditions contains the full information of the persistence diagram. Its usefulness is illustrated in statistical settings concerned with point clouds and brain artery trees. The supplementary materials include additional methods and examples, technical details, and the R code used for all examples.     

一类平面三次微分系统极限环的存在性与唯一性

蛤出三次系统{dx/dt=-y(ax^2 bx 1) δx-lx^2 dy/dt=x(ax^2 bx 1)极限环的不存在性，存在性及唯一性的一些充分条件．     

A Data-Reduction Strategy for Splines with Applications to the Approximation of Functions and Data

We present a strategy for reducing the number of knots of agiven B-spline function without perturbing the spline by morethan a given tolerance. The number and location of the remainingknots is determined automatically. Knot removal can be usedsuccessfully to fit a spline to functions and data. We illustratethis with several examples.     

A Storage-Efficient Algorithm for Finding the Regularized Solution of a Large, Inconsistent System of Equations

We propose an algorithm which for any real number r, any k ?l matrix M and any k-vector y, finds the l-vector x which minimizes||x||² – r²||Mx – y||², i.e. it finds the "regularized"solution to the equation Mx = y. (|| || denotes the 2-norm.)The algorithm is iterative with the following properties: (i)in a single step it needs access to only one row of the matrixM, (ii) it needs to store only the present estimate of the solution(size l) and a "residual vector" of size k, (iii) in a singlestep it updates only one component of the residual vector. Becauseof these properties the algorithm has been found useful in "solving"very large inconsistent systems of equations. Convergence ofthe algorithm to the desired solution is proved and the rateof convergence of the algorithm is illustrated. The algorithmis considered here, not because it is believed to be generallysuperior to more commonly used iterative methods, but becausefor certain very large problems it may be the only feasiblemethod from the point of view of storage requirements.     

一类多项式系统极限环存在的充要条件

In this paper, the following polynomial system is discussed: dx/dt=y[1 B(x)],dy/dt=(-x^α δy α1x^2α α2x^αy a5x^2αy)A(1 C(x)].And the necessary and sufficient conditions of existence and uniqueness of limit cycles are obtained.     

Necessary and Sufficient Conditions of Existence and Uniqueness of Limit Cycles for a Polynomial System

In this paper, the following polynomial system is discussed:And the necessary and sufficient conditions of existence and uniqueness of limit cyclesare obtained.