Algebraic objects generated by topological structure

The survey examines the properties of the Tikhonov space X that depend on the topological properties of the function space C(X) with pointwise topology and of the free topological group F(X). It covers the part of the theory of topological groups dealing with topological invariants and topological classification of topological groups. Continuous images of topological groups are considered in the context of studying the actions of topological groups on spaces.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 25, pp. 141–198, 1987.     

Hopf-algebraic structure of combinatorial objects and differential operators

A Hopf-algebraic structure on a vector space which has as basis a family of trees is described, and we survey some applications of this structure to combinatorics and to differential operators. Some possible future directions for this work are indicated. Supported in part by NASA Grant NAG 2-513. Supported in part by NSF Grant DMS 870-1085.     

Unconditional bases and the order structure

The paper deals with an analytic characterization of unconditional bases, a characterization of the unconditional character of a Schauder base for the associated cone and a number of counterexamples supporting various conditions appearing in the theorems.     

Periodic jordan rings and order structure

The aim of this paper is to obtain information about a periodic Jordan ring by using only properties of its idempotent elements. Osborn proves that a power-associative periodic ring having only one nonzero idempotent element is a division ring. so associative. He also proves that a periodic Jordan ring is a subdirect product of simple periodic Jordan rings and that a simple periodic Jordan ring is either a periodic field or a Jordan ring of capacity 2. Using these results we obtain some necessary and suficient conditions for a periodic Jordan ring to be associative, and these conditions are only given in terms of the idempotent elements. We also characterize the periodic Jordan ring which are a direct product of periodic fields and simple periodic Jordan rings of capacity two.     

Higher‐order discontinuous Galerkin method for pyramidal elements using orthogonal bases

We study finite elements of arbitrarily high‐order defined on pyramids for discontinuous Galerkin methods. We propose a new family of high‐order pyramidal finite elements using orthogonal basis functions which can be used in hybrid meshes including hexahedra, tetrahedra, wedges, and pyramids. We perform a comparison between these orthogonal functions and nodal functions for affine and non‐affine elements. Different strategies for the inversion of the mass matrix are also considered and discussed. Numerical experiments are conducted for the three dimensional Maxwell's equations. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Random order coalition structure values

Recently, attention has been focused on generalizations of the Shapley value obtained by relaxing the symmetry postulate. Shapley defined the class of weighted values and these have been characterized by Kalai and Samet. Random order values, treated by Weber, provide the most general approach to values without symmetry. This paper extends the random order idea to games with coalition structures. The symmetric CS value was defined by Owen; axiomatic characterizations have been given by Owen and Hart and Kurz. Levy and McLean extended their work and analyzed various classes of weighted CS values. The random order CS values of this paper include all the CS values described above as special cases.     

Locally piecewise affine functions and their order structure

Piecewise affine functions on subsets of \(\mathbb R^m\) were studied in Aliprantis et al. (Macroecon Dyn 10(1):77–99, 2006), Aliprantis et al. (J Econometrics 136(2):431–456, 2007), Aliprantis and Tourky (Cones and duality, 2007), Ovchinnikov (Beitr\(\ddot{\mathrm{a}}\)ge Algebra Geom 43:297–302, 2002). In this paper we study a more general concept of a locally piecewise affine function. We characterize locally piecewise affine functions in terms of components and regions. We prove that a positive function is locally piecewise affine iff it is the supremum of a locally finite sequence of piecewise affine functions. We prove that locally piecewise affine functions are uniformly dense in \(C(\mathbb R^m)\), while piecewise affine functions are sequentially order dense in \(C(\mathbb R^m)\). This paper is partially based on Adeeb (Locally piece-wise affine functions, 2014)     

On the dependence structure of order statistics

Given a random sample from a continuous variable, it is observed that the copula linking any pair of order statistics is independent of the parent distribution. To compare the degree of association between two such pairs of ordered random variables, a notion of relative monotone regression dependence (or stochastic increasingness) is considered. Using this concept, it is proved that for i<j, the dependence of the jth order statistic on the ith order statistic decreases as i and j draw apart. This extends earlier results of Tukey (Ann. Math. Statist. 29 (1958) 588) and Kim and David (J. Statist. Plann. Inference 24 (1990) 363). The effect of the sample size on this type of dependence is also investigated, and an explicit expression is given for the population value of Kendall's coefficient of concordance between two arbitrary order statistics of a random sample.     

A clustering algorithm for item assignment in a synchronized zone order picking system

In a synchronized zone order picking system, all the zones process the same order simultaneously. There may be some idle time when the zone pickers wait until all the pickers complete the current order. This paper develops a heuristic algorithm to balance the workload among all pickers so that the utilization of the order picking system is improved and to reduce the time needed for fulfilling each requested order. A similarity measurement, using customer orders, of any two items is first presented for measuring the co-appearance of both items in the same order. With this similarity measurement, a natural cluster model, which is a relaxation of the well-studied NP-hard homogeneous cluster model, is constructed. The heuristic algorithm is then proposed to solve the model for locating all the items into distinct zones. Finally, empirical data and simulation experiments verify that the objectives of the item cluster model are achieved.     

Kernel-based fuzzy clustering and fuzzy clustering: A comparative experimental study

In this study, we present a comprehensive comparative analysis of kernel-based fuzzy clustering and fuzzy clustering. Kernel based clustering has emerged as an interesting and quite visible alternative in fuzzy clustering, however, the effectiveness of this extension vis-à-vis some generic methods of fuzzy clustering has neither been discussed in a complete manner nor the performance of clustering quantified through a convincing comparative analysis. Our focal objective is to understand the performance gains and the importance of parameter selection for kernelized fuzzy clustering. Generic Fuzzy C-Means (FCM) and Gustafson–Kessel (GK) FCM are compared with two typical generalizations of kernel-based fuzzy clustering: one with prototypes located in the feature space (KFCM-F) and the other where the prototypes are distributed in the kernel space (KFCM-K). Both generalizations are studied when dealing with the Gaussian kernel while KFCM-K is also studied with the polynomial kernel. Two criteria are used in evaluating the performance of the clustering method and the resulting clusters, namely classification rate and reconstruction error. Through carefully selected experiments involving synthetic and Machine Learning repository (http://archive.ics.uci.edu/beta/) data sets, we demonstrate that the kernel-based FCM algorithms produce a marginal improvement over standard FCM and GK for most of the analyzed data sets. It has been observed that the kernel-based FCM algorithms are in a number of cases highly sensitive to the selection of specific values of the kernel parameters.     

Algebraic structure of association schemes of prime order

Finite groups of prime order must be cyclic. It is natural to ask what about association schemes of prime order. In this paper, we will give an answer to this question. An association scheme of prime order is commutative, and its valencies of nontrivial relations and multiplicities of nontrivial irreducible characters are constant. Moreover, if we suppose that the minimal splitting field is an abelian extension of the field of rational numbers, then the character table is the same as that of a Schurian scheme.     

Symbolic dynamics and Arnold diffusion

We consider hyperbolic tori of three degrees of freedom initially hyperbolic Hamiltonian systems. We prove that if the stable and unstable manifold of a hyperbolic torus intersect transversaly, then there exists a hyperbolic invariant set near a homoclinic orbit on which the dynamics is conjugated to a Bernoulli shift. The proof is based on a new geometrico-dynamical feature of partially hyperbolic systems, the transversality-torsion phenomenon, which produces complete hyperbolicity from partial hyperbolicity. We deduce the existence of infinitely many hyperbolic periodic orbits near the given torus. The relevance of these results for the instability of near-integrable Hamiltonian systems is then discussed. For a given transition chain, we construct chain of hyperbolic periodic orbits. Then we easily prove the existence of periodic orbits of arbitrarily high period close to such chain using standard results on hyperbolic sets.     

On the correlation structure of exponential order statistics and some extensions

In this paper, we carry out a theoretical study of the correlation coefficients between exponential order statistics and their monotonicity properties. Then, aided by a Monte Carlo simulation study, we make some conjectures about the correlation structure of order statistics from a larger class of distributions.