The classification of homotopy classes of bounded curvature paths

A bounded curvature path is a continuously differentiable piecewise C² path with bounded absolute curvature that connects two points in the tangent bundle of a surface. In this note we give necessary and sufficient conditions for two bounded curvature paths, defined in the Euclidean plane, to be in the same connected component while keeping the curvature bounded at every stage of the deformation. Following our work in [3], [2] and [4] this work finishes a program started by Lester Dubins in [6] in 1961.     

Supervised learning algorithms for multi-class classification problems with partial class memberships

In several application domains such as biology, computer vision, social network analysis and information retrieval, multi-class classification problems arise in which data instances not simply belong to one particular class, but exhibit a partial membership to several classes. Existing machine learning or fuzzy set approaches for representing this type of fuzzy information mainly focus on unsupervised methods. In contrast, we present in this article supervised learning algorithms for classification problems with partial class memberships, where class memberships instead of crisp class labels serve as input for fitting a model to the data. Using kernel logistic regression (KLR) as a baseline method, first a basic one-versus-all approach is proposed, by replacing the binary-coded label vectors with [0,1]-valued class memberships in the likelihood. Subsequently, we use this KLR extension as base classifier to construct one-versus-one decompositions, in which partial class memberships are transformed and estimated in a pairwise manner. Empirical results on synthetic data and a real-world application in bioinformatics confirm that our approach delivers promising results. The one-versus-all method yields the best computational efficiency, while the one-versus-one methods are preferred in terms of predictive performance, especially when the observed class memberships are heavily unbalanced.     

Group classification of systems of diffusion equations

Group classification of a class of systems of diffusion equations is carried out. Arbitrary elements that appear in the system depend on two variables. All forms of the arbitrary elements that provide additional Lie symmetries are determined. Equivalence transformations are used to simplify the analysis. Examples of similarity reductions are presented. Copyright © 2016 John Wiley & Sons, Ltd.     

Large deviations and support theorem for diffusion processes via rough paths

We use the continuity theorem of Lyons for rough paths in the p-variation topology to produce an elementary approach to the large deviation principle and the support theorem for diffusion processes. The proofs reduce to establish the corresponding results for Brownian motion itself as a rough path in the p-variation topology, 2<p<3, and the technical step is to handle the Lévy area in this respect. Some extensions and applications are discussed.     

Supervised box clustering

In this work we address a technique for effectively clustering points in specific convex sets, called homogeneous boxes, having sides aligned with the coordinate axes (isothetic condition). The proposed clustering approach is based on homogeneity conditions, not according to some distance measure, and, even if it was originally developed in the context of the logical analysis of data, it is now placed inside the framework of Supervised clustering. First, we introduce the basic concepts in box geometry; then, we consider a generalized clustering algorithm based on a class of graphs, called incompatibility graphs. For supervised classification problems, we consider classifiers based on box sets, and compare the overall performances to the accuracy levels of competing methods for a wide range of real data sets. The results show that the proposed method performs comparably with other supervised learning methods in terms of accuracy.     

Supervised clustering of variables

In predictive modelling, highly correlated predictors lead to unstable models that are often difficult to interpret. The selection of features, or the use of latent components that reduce the complexity among correlated observed variables, are common strategies. Our objective with the new procedure that we advocate here is to achieve both purposes: to highlight the group structure among the variables and to identify the most relevant groups of variables for prediction. The proposed procedure is an iterative adaptation of a method developed for the clustering of variables around latent variables (CLV). Modification of the standard CLV algorithm leads to a supervised procedure, in the sense that the variable to be predicted plays an active role in the clustering. The latent variables associated with the groups of variables, selected for their “proximity” to the variable to be predicted and their “internal homogeneity”, are progressively added in a predictive model. The features of the methodology are illustrated based on a simulation study and a real-world application.     

Group classification of systems of nonlinear reaction-diffusion equations with triangular diffusion matrix

We complete the group classification of systems of two coupled nonlinear reaction-diffusion equations with general diffusion matrix begun in author’s previous works. Namely, all nonequivalent equations with triangular diffusion matrix are classified. In addition, we describe symmetries of diffusion systems with nilpotent diffusion matrix and additional terms with first-order derivatives. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 3, pp. 395–411, March, 2007.     

Supervised fuzzy pattern recognition

This paper is devoted to the problem of supervised fuzzy pattern recognition. The cases with non-fuzzy and fuzzy labels are considered. Based on the properties of linearly separable fuzzy classes, some algorithms are proposed for building matching functions of these classes. All algorithms are computer oriented and can be implemented for the automatic recognition of fuzzy patterns.     

Numerical treatment of degenerate diffusion equations via Feller's boundary classification,and applications

A numerical method is devised to solve a class of linear boundary‐value problems for one‐dimensional parabolic equations degenerate at the boundaries. Feller theory, which classifies the nature of the boundary points, is used to decide whether boundary conditions are needed to ensure uniqueness, and, if so, which ones they are. The algorithm is based on a suitable preconditioned implicit finite‐difference scheme, grid, and treatment of the boundary data. Second‐order accuracy, unconditional stability, and unconditional convergence of solutions of the finite‐difference scheme to a constant as the time‐step index tends to infinity are further properties of the method. Several examples, pertaining to financial mathematics, physics, and genetics, are presented for the purpose of illustration. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Rainbow paths

A k-rainbow path in a graph with colored edges is a path of length k where each edge has a different color. In this note, we settle the problem of obtaining a constructive k-coloring of the edges of K_n in which one may find, between any pair of vertices, a large number of internally disjoint k-rainbow paths. In fact, our construction obtains the largest possible number of paths. This problem was considered in a less general setting by Chartrand et al. (2007) [6].     

Approximation Methods for Supervised Learning

Let ρ be an unknown Borel measure defined on the space Z := X × Y with X ⊂ ℝ^d and Y = [-M,M]. Given a set z of m samples z_i =(x_i,y_i) drawn according to ρ, the problem of estimating a regression function f_ρ using these samples is considered. The main focus is to understand what is the rate of approximation, measured either in expectation or probability, that can be obtained under a given prior f_ρ ∈ Θ, i.e., under the assumption that f_ρ is in the set Θ, and what are possible algorithms for obtaining optimal or semioptimal (up to logarithms) results. The optimal rate of decay in terms of m is established for many priors given either in terms of smoothness of f_ρ or its rate of approximation measured in one of several ways. This optimal rate is determined by two types of results. Upper bounds are established using various tools in approximation such as entropy, widths, and linear and nonlinear approximation. Lower bounds are proved using Kullback-Leibler information together with Fano inequalities and a certain type of entropy. A distinction is drawn between algorithms which employ knowledge of the prior in the construction of the estimator and those that do not. Algorithms of the second type which are universally optimal for a certain range of priors are given.     

Graceful labellings of paths

It is easily shown that every path has a graceful labelling, however, in this paper we show that given almost any path P with n vertices then for every vertex v∈V(P) and for every integer i∈{0,…,n-1} there is a graceful labelling of P such that v has label i. We show precisely when these labellings can also be α-labellings. We then extend this result to strong edge-magic labellings. In obtaining these results we make heavy use of π-representations of α-labellings and review some relevant results of Kotzig and Rosa.     

Ramification of rough paths

The stack of iterated integrals of a path is embedded in a larger algebraic structure where iterated integrals are indexed by decorated rooted trees and where an extended Chen's multiplicative property involves the Dürr-Connes-Kreimer coproduct on rooted trees. This turns out to be the natural setting for a non-geometric theory of rough paths.     

Reconstructing edge-disjoint paths

For an undirected graph G=(V,E), the edge connectivity values between every pair of nodes of G can be succinctly recorded in a flow-equivalent tree that contains the edge connectivity value for a linear number of pairs of nodes. We generalize this result to show how we can efficiently recover a maximum set of disjoint paths between any pair of nodes of G by storing such sets for a linear number of pairs of nodes. At the heart of our result is an observation that combining two flow solutions of the same value, one between nodes s and r and the second between nodes r and t, into a feasible flow solution of value f between nodes s and t, is equivalent to solving a stable matching problem on a bipartite multigraph.Our observation, combined with an observation of Chazelle, leads to a data structure, which takes O(n^3.5) time to generate, that can construct the maximum number λ(u,v) of edge-disjoint paths between any pair (u,v) of nodes in time O(α(n,n)λ(u,v)n) time.     

Skew Motzkin paths

In this paper, we study the class S of skew Motzkin paths, i.e., of those lattice paths that are in the first quadrat, which begin at the origin, end on the x-axis, consist of up steps U =(1, 1),down steps D =(1,-1), horizontal steps H =(1, 0), and left steps L =(-1,-1), and such that up steps never overlap with left steps. Let S_n be the set of all skew Motzkin paths of length n and let 8_n = |S_n|. Firstly we derive a counting formula, a recurrence and a convolution formula for sequence{8_n}n≥0. Then we present several involutions on S_n and consider the number of their fixed points.Finally we consider the enumeration of some statistics on S_n.