Hierarchical Model-Based Clustering for Large Datasets

In recent years, hierarchical model-based clustering has provided promising results in a variety of applications. However, its use with large datasets has been hindered by a time and memory complexity that are at least quadratic in the number of observations. To overcome this difficulty, this article proposes to start the hierarchical agglomeration from an efficient classification of the data in many classes rather than from the usual set of singleton clusters. This initial partition is derived from a subgraph of the minimum spanning tree associated with the data. To this end, we develop graphical tools that assess the presence of clusters in the data and uncover observations difficult to classify. We use this approach to analyze two large, real datasets: a multiband MRI image of the human brain and data on global precipitation climatology. We use the real datasets to discuss ways of integrating the spatial information in the clustering analysis. We focus on two-stage methods, in which a second stage of processing using established methods is applied to the output from the algorithm presented in this article, viewed as a first stage.     

The number of spanning trees in an (r,s)‐semiregular graph and its line graph

For a graph G, a “spanning tree” in G is a tree that has the same vertex set as G. The number of spanning trees in a graph (network) G, denoted by t(G), is an important invariant of the graph (network) with lots of decisive applications in many disciplines. In the article by Sato (Discrete Math. 2007, 307, 237), the number of spanning trees in an (r, s)‐semiregular graph and its line graph are obtained. In this article, we give short proofs for the formulas without using zeta functions. Furthermore, by applying the formula that enumerates the number of spanning trees in the line graph of an (r, s)‐semiregular graph, we give a new proof of Cayley's Theorem. © 2013 Wiley Periodicals, Inc.     

Prospects of anionic nanolipoplexes in nanotherapy: transmission electron microscopy and light scattering studies

Currently nanosystems composed of polynucleotides and lipid vesicles (nanolipoplexes) are considered to be promising tools for gene therapeutics. Successful in vivo application of these vectors depends on their physicochemical, technological and biological characteristics including morphology, size distribution, molecular interactions and stability. Anionic nanoliposomes (DPPC:DCP:CHOL) were prepared by two different techniques, namely the conventional thin-film hydration method followed by extrusion, and the heating method (HM), in which no volatile solvent or detergent is used. A non-viral and non-cationic gene transfer vector was constructed by incorporating plasmid DNA (pcDNA3.1/His B/lacZ) to the HM-nanoliposomes by the electrostatic mediation of Ca²⁺ ions. Transfection efficiency of the nanolipoplexes was evaluated using a human bronchial epithelial cell line (16HBE14o-) in the presence of serum. Particle characterisation, stability of the formulations and lipid–DNA interaction studies were performed using transmission electron microscopy (TEM) and light scattering. TEM pictures of nanolipoplexes showed presence of two to four closely packed vesicles with signs of fusion. Efficient delivery of plasmid DNA and subsequent β-galactosidase expression was achieved using the anionic nanolipoplexes. Transfection efficiency increased with lipid:DNA ratio up to 7:1 (w/w), where transfection efficiency was 12-fold higher than in untreated cells. Further increase in lipid ratio decreased transfection. These nanolipoplexes appear to be safe, stable and efficient in the protection and delivery of DNA to different cells and tissues.     

Addendum to: “Hamilton-laceable bi-powers of locally finite bipartite graphs” [Discrete Math. 345 (2022) 112777]

In this addendum we give a short and easy negative answer to the two questions raised in the original article.     

The effect of a market factor on information flow between stocks using the minimal spanning tree

We empirically investigated the effects of market factors on the information flow created from N(N−1)/2 linkage relationships among stocks. We also examined the possibility of employing the minimal spanning tree (MST) method, which is capable of reducing the number of links to N−1. We determined that market factors carry important information value regarding information flow among stocks. Moreover, the information flow among stocks showed time-varying properties according to the changes in market status. In particular, we noted that the information flow increased dramatically during periods of market crises. Finally, we confirmed, via the MST method, that the information flow among stocks could be assessed effectively with the reduced linkage relationships among all links among stocks from the perspective of the overall market.     

Diffraction reflection of light on natural periodic inhomogeneities of a crystal with open surface of wave vectors

The forms of the surface of wave vectors for an extraordinary wave in the presence and absence of absorption are discussed. The normal transmission of light through a plane-parallel layer of a crystal with an open surface of wave vectors is considered. The diffraction reflection of light on natural periodic inhomogeneities of a crystal is studied. The frequency dependence of the reflection coefficient is calculated.     

Majoration de sommes trigonométriques modulo pn

We give a generic estimation of trigonometric sums defined over closed sub-schemes with semi-stable reduction of the standard affine scheme modulo pn(n 2). We use Greenberg realisation to reduce to trigonometric sums defined over smooths sub-schemes of a finite product of Witt vectors over the finite field of p elements. Using the cohomological interpretation of this sums over a finite field, the sum is directly related to the Fourier–Deligne transformation of the dual pairs of Witt vectors. We deduce the estimation from the properties of the Fourier–Deligne transformation on simple perverse sheaves and pure sheaves.     

Multi-sample cluster analysis using Akaike's Information Criterion

Summary Multi-sample cluster analysis, the problem of grouping samples, is studied from an information-theoretic viewpoint via Akaike's Information Criterion (AIC). This criterion combines the maximum value of the likelihood with the number of parameters used in achieving that value. The multi-sample cluster problem is defined, and AIC is developed for this problem. The form of AIC is derived in both the multivariate analysis of variance (MANOVA) model and in the multivariate model with varying mean vectors and variance-covariance matrices. Numerical examples are presented for AIC and another criterion calledw-square. The results demonstrate the utility of AIC in identifying the best clustering alternatives. This research was supported by Office of Naval Research Contract N00014-80-C-0408, Task NR042-443 and Army Research Office Contract DAAG 29-82-K-0155, at the University of Illinois at Chicago.     

Spanning network games

We study fundamental properties of monotone network enterprises which contain public vertices and have positive and negative costs on edges and vertices. Among the properties studied are the nonemptiness of the core, characterization of nonredundant core constraints, ease of computation of the core and the nucleolus, and cases of decomposition of the core and the nucleolus. Received December 1994/Final version March 1998     

Spectra of Some Interesting Combinatorial Matrices Related to Oriented Spanning Trees on a Directed Graph

The Laplacian of a directed graph G is the matrix L(G) = O(G) –, A(G) where A(G) is the adjaceney matrix of G and O(G) the diagonal matrix of vertex outdegrees. The eigenvalues of G are the eigenvalues of A(G). Given a directed graph G we construct a derived directed graph D(G) whose vertices are the oriented spanning trees of G. Using a counting argument, we describe the eigenvalues of D(G) and their multiplicities in terms of the eigenvalues of the induced subgraphs and the Laplacian matrix of G. Finally we compute the eigenvalues of D(G) for some specific directed graphs G. A recent conjecture of Propp for D(H _n ) follows, where H _n stands for the complete directed graph on n vertices without loops.