Characterizing and Generating Bivariate Empirical Rank Distributions Satisfying Certain Positive Dependence Concepts

Abstract

This article introduces an approach for characterizing the classes of empirical distributions that satisfy certain positive dependence notions. Mathematically, this can be expressed as studying certain subsets of the class S_N of permutations of 1, …, N, where each subset corresponds to some positive dependence notions. Explicit techniques for it-eratively characterizing subsets of S_N that satisfy certain positive dependence concepts are obtained and various counting formulas are given. Based on these techniques, graph-theoretic methods are used to introduce new and more efficient algorithms for constructively generating and enumerating the elements of various of these subsets of S_N. For example, the class of positively quadrant dependent permutations in S_N is characterized in this fashion.     

Novel matrix invariants for characterization of changes of proteomics maps

Previous studies on mathematical characterization of proteomics maps by sets of map invariants were based on the construction of a set of distance-related matrices obtained by matrix multiplication of a single matrix by itself. Here we consider an alternative characterization of proteomics maps based on a set of matrices characterizing local features of an embedded zigzag curve over the map. It is shown that novel invariants can well characterize proteomics maps. Advantages of the novel approach are discussed.     

Physicochemical and Graph Theoretical Descriptors in Developmental Toxicity SAR: A Comparative Study

Abstract

Chemical insults to the developing fetus can lead to growth retardation, malformation, death, and functional deficits. The present study seeks to determine if physicochemical and/or graph theoretical parameters can be used to determine a structure-activity relationship (SAR) for developmental toxicity, and if consistency is observed among the selected features. The biological data utilized consists of a diverse series of compounds evaluated within the Chernoff-Kavlock in vivo mouse assay. Physicochemical parameters calculated correspond to electronic, steric, and transport properties. Graph theoretical parameters calculated include the simple, valence, and kappa indices. Both sets of parameters were independently applied to derive SARs in order to compare the quality of the respective models. Multiple random sampling, without replacement, was utilized to obtain ten training/test partitions. Models were built by linear discriminant analysis, decision trees, and neural networks respectively. Comparisons on identical sets of data were carried out to determine if any of the model building procedures had a significant advantage in terms of predictive performance. Furthermore, comparison of the features selected within and across the model building processes led to the determination of model consistency. Our results indicate that consistent features related to developmental toxicity are observed and that both physicochemical and graph theoretical parameters have equal utility.     

Universal point sets for planar three-trees

For every $n\in \mathbb{N}$, we present a set $S_n$ of $O(n^{3/2}\log n)$ points in the plane such that every planar 3-tree with n vertices has a straight-line embedding in the plane in which the vertices are mapped to a subset of $S_n$. This is the first subquadratic upper bound on the cardinality of universal point sets for planar 3-trees, as well as for the class of 2-trees and serial parallel graphs.     

Large Cross‐Free Sets in Steiner Triple Systems

A cross‐free set of size m in a Steiner triple system is three pairwise disjoint m‐element subsets such that no intersects all the three ‐s. We conjecture that for every admissible n there is an STS(n) with a cross‐free set of size which if true, is best possible. We prove this conjecture for the case , constructing an STS containing a cross‐free set of size 6k. We note that some of the 3‐bichromatic STSs, constructed by Colbourn, Dinitz, and Rosa, have cross‐free sets of size close to 6k (but cannot have size exactly 6k). The constructed STS shows that equality is possible for in the following result: in every 3‐coloring of the blocks of any Steiner triple system STS(n) there is a monochromatic connected component of size at least (we conjecture that equality holds for every admissible n). The analog problem can be asked for r‐colorings as well, if and is a prime power, we show that the answer is the same as in case of complete graphs: in every r‐coloring of the blocks of any STS(n), there is a monochromatic connected component with at least points, and this is sharp for infinitely many n.     

酒类中微量甲醛的快速测定新方法

在FeCl3存在条件下,甲醛与自制蛋白液显色剂能生成紫色化合物,藉此可以定量测定微量甲醛。方法线性范围为1~10μg/10mL,线性相关系数r2=0.996,检出限为0.05μg/mL。用该法测定经活性炭脱色的啤酒和白酒中的甲醛含量,回收率为96.46%~103.32%,相对标准偏差为1.36%~2.09%(n=6)。     

Pseudo-orthogonality for graph 1-Laplacian eigenvectors and applications to higher Cheeger constants and data clustering

The data clustering problem consists in dividing a data set into prescribed groups of homogeneous data. This is an NP-hard problem that can be relaxed in the spectral graph theory, where the optimal cuts of a graph are related to the eigenvalues of graph 1-Laplacian. In this paper, we first give new notations to describe the paths, among critical eigenvectors of the graph 1-Laplacian, realizing sets with prescribed genus. We introduce the pseudo-orthogonality to characterize m₃(G), a special eigenvalue for the graph 1-Laplacian. Furthermore, we use it to give an upper bound for the third graph Cheeger constant h₃(G), that is, h₃(G) 6 m₃(G). This is a first step for proving that the k-th Cheeger constant is the minimum of the 1-Laplacian Raylegh quotient among vectors that are pseudo-orthogonal to the vectors realizing the previous k - 1 Cheeger constants. Eventually, we apply these results to give a method and a numerical algorithm to compute m₃(G), based on a generalized inverse power method.     

On the Differential Polynomial of a Graph

We introduce the differential polynomial of a graph. The differential polynomial of a graph G of order n is the polynomial B(G; x) :=∑?(G)k=-nB_k(G) x~(n+k), where B_k(G) denotes the number of vertex subsets of G with differential equal to k. We state some properties of B(G;x) and its coefficients.In particular, we compute the differential polynomial for complete, empty, path, cycle, wheel and double star graphs. We also establish some relationships between B(G; x) and the differential polynomials of graphs which result by removing, adding, and subdividing an edge from G.     

Halin-图的邻强边染色

图G(V,E)的正常κ-边染色f叫做图G(V,E)的κ-邻强边染色当且仅当任意uv∈E(G)满足f[u]≠f[v],其中,f[u]={f(uw)|uw∈E(G)},称f是G的κ-临强边染色,简记为κ-ASEC.并且x′_as(G)=min{k|κ-ASEC of G}叫做G(V,E)的邻强边色数.本文研究了△(G)≥5的Halin-图的邻强边色数.