Counting disjoint 2-partitions for points in the plane

Disjoint partitions, and its counting, have been widely studied in the literature of optimal partitions and clustering. We give an exact counting on the number of disjoint ordered 2-partitions for n points in general position in R². We also give an exact counting on the maximum number of disjoint 2-partitions, where one part consists of two points, over all sets of n points in R².     

Pseudospectra, critical points and multiple eigenvalues of matrix polynomials

We develop a general framework for perturbation analysis of matrix polynomials. More specifically, we show that the normed linear space L_m(C^n×n) of n-by-n matrix polynomials of degree at most m provides a natural framework for perturbation analysis of matrix polynomials in L_m(C^n×n). We present a family of natural norms on the space L_m(C^n×n) and show that the norms on the spaces C^m+1 and C^n×n play a crucial role in the perturbation analysis of matrix polynomials. We define pseudospectra of matrix polynomials in the general framework of the normed space L_m(C^n×n) and show that the pseudospectra of matrix polynomials well known in the literature follow as special cases. We analyze various properties of pseudospectra in the unified framework of the normed space L_m(C^n×n). We analyze critical points of backward errors of approximate eigenvalues of matrix polynomials and show that each critical point is a multiple eigenvalue of an appropriately perturbed polynomial. We show that common boundary points of components of pseudospectra of matrix polynomials are critical points. As a consequence, we show that a solution of Wilkinson’s problem for matrix polynomials can be read off from the pseudospectra of matrix polynomials.     

A novel fuzzy C-means algorithm to generate diverse and desirable cluster solutions used by genetic-based clustering ensemble algorithms

One of the most significant discussions in the field of machine learning today is on the clustering ensemble. The clustering ensemble combines multiple partitions generated by different clustering algorithms into a single clustering solution. Genetic algorithms are known for their high ability to solve optimization problems, especially the problem of the clustering ensemble. To date, despite the major contributions to find consensus cluster partitions with application of genetic algorithms, there has been little discussion on population initialization through generative mechanisms in genetic-based clustering ensemble algorithms as well as the production of cluster partitions with favorable fitness values in first phase clustering ensembles. In this paper, a threshold fuzzy C-means algorithm, named TFCM, is proposed to solve the problem of diversity of clustering, one of the most common problems in clustering ensembles. Moreover, TFCM is able to increase the fitness of cluster partitions, such that it improves performance of genetic-based clustering ensemble algorithms. The fitness average of cluster partitions generated by TFCM are evaluated by three different objective functions and compared against other clustering algorithms. In this paper, a simple genetic-based clustering ensemble algorithm, named SGCE, is proposed, in which cluster partitions generated by the TFCM and other clustering algorithms are used as the initial population used by the SGCE. The performance of the SGCE is evaluated and compared based on the different initial populations used. The experimental results based on eleven real world datasets demonstrate that TFCM improves the fitness of cluster partitions and that the performance of the SGCE is enhanced using initial populations generated by the TFCM.     

On multiple pattern avoiding set partitions

We study classes of set partitions determined by the avoidance of multiple patterns, applying a natural notion of partition containment that has been introduced by Sagan. We say that two sets S and T of patterns are equivalent if for each n the number of partitions of size n avoiding all the members of S is the same as the number of those that avoid all the members of T.     

On the binary codes with parameters of triply-shortened 1-perfect codes

We study properties of binary codes with parameters close to the parameters of 1-perfect codes. An arbitrary binary (n?=?2 ^m ? 3, 2 ^n-m-1, 4) code C, i.e., a code with parameters of a triply-shortened extended Hamming code, is a cell of an equitable partition of the n-cube into six cells. An arbitrary binary (n?=?2 ^m ? 4, 2 ^n-m , 3) code D, i.e., a code with parameters of a triply-shortened Hamming code, is a cell of an equitable family (but not a partition) with six cells. As a corollary, the codes C and D are completely semiregular; i.e., the weight distribution of such codes depends only on the minimal and maximal codeword weights and the code parameters. Moreover, if D is self-complementary, then it is completely regular. As an intermediate result, we prove, in terms of distance distributions, a general criterion for a partition of the vertices of a graph (from rather general class of graphs, including the distance-regular graphs) to be equitable.     

Smoothness in weakly compactly generated Banach spaces

If $\text{X}{}^1$ is a weakly compactly generated (WCG) Banach space, then X admits an equivalent C¹-smooth norm. If a WCG Banach space X admits a C^k-smooth function with bounded support, then X admits C^k-smooth partitions of unity.     

New interpretations for noncrossing partitions of classical types

We interpret noncrossing partitions of type B and type D in terms of noncrossing partitions of type A. As an application, we get type-preserving bijections between noncrossing and nonnesting partitions of type B, type C and type D which are different from those in the recent work of Fink and Giraldo. We also define Catalan tableaux of type B and type D, and find bijections between them and noncrossing partitions of type B and type D respectively.     

Geometric structure of dimension functions of certain continuous fields

For a continuous field of C^?-algebras A, we give a criterion to ensure that the stable rank of A is one. In the particular case of a trivial field this leads to a characterization of stable rank one, completing accomplishments by Nagisa, Osaka and Phillips. Further, for certain continuous fields of C^?-algebras, we study when the Cuntz semigroup satisfies the Riesz interpolation property, and we also analyze the structure of its functionals. As an application, we obtain a positive answer to a conjecture posed by Blackadar and Handelman in a variety of situations.     

Lattices on simplicial partitions which are not simply connected

In this paper, (d+1)-pencil lattices on simplicial partitions in R^d, which are not simply connected, are studied. It is shown, how the fact that a partition is not simply connected can be used to increase the flexibility of a lattice. A local modification algorithm is developed also to deal with slight partition topology changes that may appear afterwards a lattice has already been constructed.     

Lattices on simplicial partitions

In this paper, (d+1)-pencil lattices on simplicial partitions in R^d are studied. The barycentric approach naturally extends the lattice from a simplex to a simplicial partition, providing a continuous piecewise polynomial interpolant over the extended lattice. The number of degrees of freedom is equal to the number of vertices of the simplicial partition. The constructive proof of this fact leads to an efficient computer algorithm for the design of a lattice.     

The Delay-Capacity product for store-and-forward communication networks: Tree networks

A store-and-forward communication network under a maximal message delay criterion is considered. It is shown that the overall channel capacityC and the associated minimal maximal delayγ, as well as the maximal delayγ and the associated minimal overall capacityC, are characterized by a unique Delay-Capacity (γC) product number. The latter is related to a Delay-Capacity product (γC)⁺ number, uniquely determined solely by the topological structure of the communication network. Basic characteristics of the optimal delay and capacity assignment, a useful algoritm for the calculation of (γC)⁺ and simple upper and lower bounds on (γC)⁺, are derived for store-and-forward tree networks. Synthesis considerations and applications to hierarchical communication networks are noted.     

On Supersymmetric Fermion Lattice Systems

This note provides a C ^*-algebraic framework for supersymmetry. Particularly, we consider fermion lattice models satisfying the simplest supersymmetry relation. Namely, we discuss a restricted sense of supersymmetry without a boson field involved. We construct general supersymmetric C ^*-dynamics in terms of a superderivation and a one-parameter group of automorphisms on the CAR algebra. (We do not introduce Grassmann numbers into our formalism.) We show several basic properties of superderivations on the fermion lattice system. Among others, we establish that superderivations defined on the strictly local algebra are norm-closable. We show a criterion of superderivations on the fermion lattice system for being nilpotent. This criterion can be easily checked and hence yields new supersymmetric fermion lattice models.     

Closure operators and their middle-interchange law

The paper presents a new definition of closure operator which encompasses the standard Dikranjan-Giuli notion, as well as the Bourn-Gran notion of normal closure operator. As is well known, any two closure operators C, D in a category may be composed in, within order, two different ways. For a subobject M→X one may consider DX(CXM) or DCX(M)(M) as the value at M of a new closure operator D⋅C or D?C, respectively. The two binary operations are linked by a lax middle-interchange law. This paper explores situations in which the law holds strictly.     

Building initial partitions through sampling techniques

A variety of iterative clustering algorithms require an initial partition of a dataset as an input parameter. As a rule a good choice of the initial partition is essential for building a high quality final partition. In this note, we generate initial partitions by using small samples of the data. Numerical experiments with k-means like clustering algorithms are reported.     

Smooth partitions of Anosov diffeomorphisms are weak Bernoulli

It is shown that smooth partitions are weak Bernoulli forC ² measure preserving Anosov diffeomorphisms. A related type of coding is defined and an invariant discussed. Supported by the Sloan Foundation and NSF GP-14519.     

Statistical theory of shape under elliptical models and singular value decompositions

The size-and-shape and shape distributions based on non-central and non-isotropic elliptical distributions are derived in this paper by using the singular value decomposition (SVD). The general densities require the computation of new integrals involving zonal polynomials. The invariance of the central shape distribution is also proved. Finally, some particular densities are applied in a classical data of Biology, and the inference based on exact distributions is performed after choosing the best model by using a modified BIC^∗ criterion.     

Entropy transmission in C1-dynamical systems

Three entropies of a state in C¹-dynamical systems are introduced and their relations and dynamical properties are studied. The entropy (information) transmission under a channel between two dynamical systems is considered. We find a condition under which our entropy becomes a dynamical invariant between two systems.     

Embedding smooth and formal diffeomorphisms through the Jordan-Chevalley decomposition

In [Xiang Zhang, The embedding flows of C^∞ hyperbolic diffeomorphisms, J. Differential Equations 250 (5) (2011) 2283-2298] Zhang proved that any local smooth hyperbolic diffeomorphism whose eigenvalues are weakly nonresonant is embedded in the flow of a smooth vector field. We present a new and more conceptual proof of such result using the Jordan-Chevalley decomposition in algebraic groups and the properties of the exponential operator.We characterize the hyperbolic smooth (resp. formal) diffeomorphisms that are embedded in a smooth (resp. formal) flow. We introduce a criterion showing that the presence of weak resonances for a diffeomorphism plus two natural conditions imply that it is not embeddable. This solves a conjecture of Zhang. The criterion is optimal, we provide a method to construct embeddable diffeomorphisms with weak resonances if we remove any of the conditions.