The notion of travel groupoids was introduced by L. Nebeský in 2006 in connection with a study on geodetic graphs. A travel groupoid is a pair of a set V and a binary operation ∗ on V satisfying two axioms. For a travel groupoid, we can associate a graph. We say that a graph G has a travel groupoid if the graph associated with the travel groupoid is equal to G. Nebeský gave a characterization f...

We apply the partition algorithm to the problem of time-series classification. We assume that the source that generates the time series belongs to a finite set of candidate sources. Classification is based on the computation of posterior probabilities. Prediction error is used to adaptively update the posterior probability of each source. The algorithm is implemented by a hierarchical, modular,...

Let T be a finite entropy, aperiodic automorphism of a nonatomic probability space. We give an elementary proof of the existence of a finite entropy, countable generating partition for T . In this short article we give a simple proof of Rokhlin’s countable generator theorem [Ro], originating from considerations in [Se] which use standard techniques in ergodic theory. We hope that these consider...

The maximum graph bisection problem is a well known graph partition problem. The problem has been proven to be NP-hard. In the maximum graph bisection problem it is required that the set of vertices is divided into two partition with equal number of vertices, and the weight of the edge cut is maximal. This work introduces a connected multidimensional generalization of the maximum bisection prob...

1. The Darboux definition of the Riemann Integral Let f : [a, b]→ R be a it bounded function. Definition 1.1. A partition P of [a, b] is collection of points {x0, · · · , xn} which satisfiy a = x0 < x1 < · · · < xn = b. Denote by ∆xk the length of the interval [xk−1, xk], i.e., ∆xk = xk−xk−1. Then the mesh |P| is by definition equal to max1≤k≤n∆xk. Given a bounded function f and a partition P o...

Given two time series, possibly of different lengths, time warping is a method to construct an optimal alignment obtained by stretching or contracting time intervals. Unlike pairwise alignment of amino acid sequences, classical time warping, originally introduced for speech recognition, is not symmetric in the sense that the time warping distance between two time series is not necessarily equal...

• Permutation tableaux are new objects that come from the enumeration of the totally positive Grassmannian cells [11, 15]. A permutation tableau [12] is a Ferrers diagram together with a filling of the cells with 0’s and 1’s such that the following properties hold: Each column contains at least one 1 and there is no 0 which has a 1 above it in the same column and a 1 to its left in the same row...

We study semirandom k-colorable graphs made up as follows. Partition the vertex set V = {1, . . . , n} randomly into k classes V1, . . . , Vk of equal size and include each Vi-Vj -edge with probability p independently (1 ≤ i < j ≤ k) to obtain a graph G0. Then, an adversary may add further Vi-Vj-edges (i 6= j) to G0, thereby completing the semirandom graph G = G∗n,p,k . We show that if np ≥ max...

This paper considers the problem of clustering a partially observed unweighted graph—i.e., one where for some node pairs we know there is an edge between them, for some others we know there is no edge, and for the remaining we do not know whether or not there is an edge. We want to organize the nodes into disjoint clusters so that there is relatively dense (observed) connectivity within cluster...

An arrangement of k-semilines in the Euclidean (projective) plane or on the 2-sphere is called a k-fan if all semilines start from the same point. A k-fan is an α-partition for a probability measure μ if μ(σi) = αi for each i = 1, . . . , k where {σi}ki=1 are conical sectors associated with the k-fan and α = (α1, . . . , αk). The problem whether for a given collection of measures μ1, . . . , μm...