Enumeration of inequivalent irreducible Goppa codes

We consider irreducible Goppa codes over F_q of length qⁿ defined by polynomials of degree r, where q is a prime power and n,r are arbitrary positive integers. We obtain an upper bound on the number of such codes.     

Note on the application of the reduced graph model in conjunction with search trees to the enumeration ofKekulé structures

The reduced graph model, when used in conjunction with the search trees method, provides a novel combinatorial procedure for the enumeration and generation ofKekulé structures. The procedure is suited for large benzenoid hydrocarbons consisting of cata- and thin peri-condensed parts.

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Zur Anwendung des Reduced Graph Model im Zusammenhang mit Search Trees zur Ermittlung der Anzahl möglicherKekulé-Strukturen

Zusammenfassung Das Modell erlaubt mit der im Titel genannten Kombination eine neuartige Methode zur Ermittlung und Generierung vonKekulé-Strukturen. Das Verfahren ist für große benzoide Kohlenwasserstoffe geeignet, die aus cata- und (dünnen) peri-kondensierten Teilstrukturen bestehen.

     

取代金刚烷的异构体计数

采用图论方法对单一种取代基取代的金刚烷和烷基取代的金刚烷构型、手性构型和非手性构型数进行了计算,结果分别以母函数形式和表格形式给出。     

Stories about groups and sequences

The main theme of this article is that counting orbits of an infinite permutation group on finite subsets or tuples is very closely related to combinatorial enumeration; this point of view ties together various disparate stories. Among these are reconstruction problems, the relation between connected and arbitrary graphs, the enumeration of N-free posets, and some of the combinatorics of Stirling numbers.Dedicated to Hanfried Lenz on the occasion of his 80th birthday.     

Pre-service elementary teachers’ strategies for tiling and relating area units

It has been established that preservice elementary school teachers (PSTs) often employ procedural methods when solving measurement problems without conceptual understanding or flexibility, but a significant gap exists in the literature identifying why. Through the lens of discrete and continuous interpretations of area, this study extends the research base by describing strategies PSTs use to tile a two-dimensional space with varying size tiles and what these strategies imply about PSTs’ conceptions of area measurement. These strategies and implied conceptions enable further discussion on the multiple purposes of the area model as an illustrative measure for mathematics throughout the elementary school curriculum.     

Undirected simple connected graphs with minimum number of spanning trees

We show that for positive integers n, m with n(n−1)/2≥m≥n−1, the graph L_n,m having n vertices and m edges that consists of an (n−k)-clique and k−1 vertices of degree 1 has the fewest spanning trees among all connected graphs on n vertices and m edges. This proves Boesch’s conjecture [F.T. Boesch, A. Satyanarayana, C.L. Suffel, Least reliable networks and reliability domination, IEEE Trans. Commun. 38 (1990) 2004-2009].     

Random graphs on surfaces

Counting labelled planar graphs, and typical properties of random labelled planar graphs, have received much attention recently. We start the process here of extending these investigations to graphs embeddable on any fixed surface S. In particular we show that the labelled graphs embeddable on S have the same growth constant as for planar graphs, and the same holds for unlabelled graphs. Also, if we pick a graph uniformly at random from the graphs embeddable on S which have vertex set {1,…,n}, then with probability tending to 1 as n→∞, this random graph either is connected or consists of one giant component together with a few nodes in small planar components.     

Analyzing the Sensitivity of Generalized Linear Models to Incomplete Outcomes via the IDE Algorithm

Incomplete data models typically involve strong untestable assumptions about the missing data distribution. As inference may critically depend on them, the importance of sensitivity analysis is well recognized. Molenberghs, Kenward, and Goetghebeur proposed a formal frequentist approach to sensitivity analysis which distinguishes ignorance due to unintended incompleteness from imprecision due to finite sampling by design. They combine both sources of variation into uncertainty. This article develops estimation tools for ignorance and uncertainty concerning regression coefficients in a complete data model when some of the intended outcome values are missing. Exhaustive enumeration of all possible imputations for the missing data requires enormous computational resources. In contrast, when the boundary of the occupied region is of greatest interest, reasonable computational efficiency may be achieved via the imputation towards directional extremes (IDE) algorithm. This is a special imputation method designed to mark the boundary of the region by maximizing the direction of change of the complete data estimator caused by perturbations to the imputed outcomes. For multi-dimensional parameters, a dimension reduction approach is considered. Additional insights are obtained by considering structures within the region, and by introducing external knowledge to narrow the boundary to useful proportions. Special properties hold for the generalized linear model. Examples from a Kenyan HIV study will illustrate the points.