Robust clustering around regression lines with high density regions

Robust methods are needed to fit regression lines when outliers are present. In a clustering framework, outliers can be extreme observations, high leverage points, but also data points which lie among the groups. Outliers are also of paramount importance in the analysis of international trade data, which motivate our work, because they may provide information about anomalies like fraudulent transactions. In this paper we show that robust techniques can fail when a large proportion of non-contaminated observations fall in a small region, which is a likely occurrence in many international trade data sets. In such instances, the effect of a high-density region is so strong that it can override the benefits of trimming and other robust devices. We propose to solve the problem by sampling a much smaller subset of observations which preserves the cluster structure and retains the main outliers of the original data set. This goal is achieved by defining the retention probability of each point as an inverse function of the estimated density function for the whole data set. We motivate our proposal as a thinning operation on a point pattern generated by different components. We then apply robust clustering methods to the thinned data set for the purposes of classification and outlier detection. We show the advantages of our method both in empirical applications to international trade examples and through a simulation study.     

Colourings of cubic graphs inducing isomorphic monochromatic subgraphs

A -bisection of a bridgeless cubic graph is a -colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes ( monochromatic components in what follows) have order at most . Ban and Linial Conjectured that every bridgeless cubic graph admits a -bisection except for the Petersen graph. A similar problem for the edge set of cubic graphs has been studied: Wormald conjectured that every cubic graph with has a -edge colouring such that the two monochromatic subgraphs are isomorphic linear forests (ie, a forest whose components are paths). Finally, Ando conjectured that every cubic graph admits a bisection such that the two induced monochromatic subgraphs are isomorphic. In this paper, we provide evidence of a strong relation of the conjectures of Ban-Linial and Wormald with Ando's Conjecture. Furthermore, we also give computational and theoretical evidence in their support. As a result, we pose some open problems stronger than the above-mentioned conjectures. Moreover, we prove Ban-Linial's Conjecture for cubic-cycle permutation graphs. As a by-product of studying -edge colourings of cubic graphs having linear forests as monochromatic components, we also give a negative answer to a problem posed by Jackson and Wormald about certain decompositions of cubic graphs into linear forests.     

An Extension of the Birkhoff Integrability for Multifunctions

A comparison between a set-valued Gould type and simple Birkhoff integrals of bf(X)-valued multifunctions with respect to a non-negative set function is given. Relationships among them and Mc Shane multivalued integrability is given under suitable assumptions.     

Mixed needlets

The construction of needlet-type wavelets on sections of the spin line bundles over the sphere has been recently addressed in Geller and Marinucci (2008) [14], and Geller et al. (2008, 2009) [12] and [13]. Here we focus on an alternative proposal for needlets on this spin line bundle, in which needlet coefficients arise from the usual, rather than the spin, spherical harmonics, as in the previous constructions. We label this system mixed needlets and investigate in full their properties, including localization, the exact tight frame characterization, reconstruction formula, decomposition of functional spaces, and asymptotic uncorrelation in the stochastic case. We outline astrophysical applications.     

Mononuclear heterocyclic rearrangements. Part 4 Synthesis and characterization of the E-isomer phenylhydrazone of 3-benzoyl-5-phenyl-1,2,4-oxadiazole

3-Benzoyl-5-phenyl-1,2,4-oxadiazole (I) with phenylhydrazine in acetic acid gives the two geometrical isomers, phenylhydrazones II-Z and II-E, which have been characterized by uv-visible, ir, and nmr spectra. The possible II-E ? II-Z isomerization as well as the rearrangement of II-Z and of II-E into 2,5-diphenyl-4-benzoylamino-1,2,3-triazole (III) has been pointed out.     

Chaos and reproduction in sea level

Prediction of sea-level is an important task for navigation, coastal engineering and geodetic applications, as well as recreational activities. This study presents a comparison of Chaos theory and Auto-Regressive Integrated Moving Average (ARIMA) techniques for sea level modelling for daily, weekly, 10-day and monthly time scale at the Cocos (Keeling) islands from 1992 to 2001. The state space reconstruction of the unknown underlying process is directly employed from time series data, through Takens delay embedding method: optimal embedding dimension and delay time are obtained from false nearest neighbours and average mutual information techniques, respectively. Optimal values are then used for the estimation of the correlation dimension and the largest Lyapunov exponent, for inspecting possible signatures of chaotic dynamics. We find a positive Lyapunov exponent an evident feature of chaos. Indeed, the nonlinear prediction of sea level, in the period ranging from January 2001 to December 2001, is in an excellent agreement with the data for the same period, evidencing the nonlinear nature of the process. ARIMA method is also used for sea level modelling, for the same time scales; the performances of the two models are compared using such statistical indices as the root mean square error (RMSE) and correlation coefficient (CC). The comparative analyses show that the chaos theory model has a slight edge over ARIMA while both models are in principal acceptable.     

Minimal Order Semihypergroups of Type U on the Right

We study existence and possible uniqueness of special semihypergroups of type U on the right. In particular, we prove that there exists a unique proper semihypergroup of this kind having order 6, apart of isomorphisms; the least order for a hypergroup of type U on the right to have a stable part which is not a subhypergroup is 9; and the minimal cardinality of a proper semihypergroup of that kind whose heart and derived semihypergroup are proper and nontrivial is 12. Contextually, we analyze properties of the kernel of homomorphisms g : H ↦ G, where H is a finite semihypergroup of type U on the right and G is a group. In this way, we obtain results that are immediately applicable both to the heart and to the derived of such semihypergroups.      

A Variational Approach to the Maximization of Preferences Without Numerical Representation

Journal of Optimization Theory and Applications - We introduce a variational approach to study a maximization problem of preferences that cannot be represented by a utility function. In such...