Equality in a result of Kleitman

AbstractAn upset is a set U of subset of a finite set. S such that if U ⊆ V and U ϵ U, then V ϵ U. A downset D is defined analogously. In 1966, Kleitman (J. Combin. Theory 1 (1966), 153–155) proved that if U and D are arbitrary up- and downsets, respectively, then |U| |D| ⩾ 2|S| |U ∩ D|. In this note, we show that a necessary and sufficient condition for equality to hold is: for every minimal element U of U and every maximal element D of D, U ⊆ D. This result is extended to some related inequalities

On Essential and Inessential Polygons in Embedded Graphs

AbstractIn this article, we present a number of results of the following type: A given subgraph of an embedded graph either is embedded in a disc or it has a face chain containing a non-contractible closed path. Our main application is to prove that any two faces of a 4-representative embedding are simultaneously contained in a disc bounded by a polygon. This result is used to prove the existence of ⌊(r−1)/8⌋ pairwise disjoint, pairwise homotopic non-contractible separating polygons in an r -representative orientable embedding. Our proof of this latter result is simple and mechanical

The smallest matrix of given period and primitive roots of unity

AbstractA nonsingular matrix A has period n if An = I but Ak ≠ I for 0 < k < n. We investigate the number rK(n), which is the smallest r such that there is an r × r matrix, with entries in the field K, that has period n. We compute this number as a function of the common degree θK(j) of the irreducible factors of the cyclotomic polynomial cj(x). Thus, we are led to an investigation of roots of unity in order to better understand the function θ

Ultrasonic Sizing of Cracks in Web Geometries

Evaluation of the critical nature of interior cracks in turbine rotor component web regions in order to assess the remaining service life of the parts requires accurate determination of the crack sizes in order to perform fracture mechanics analysis. This analysis is important both in order to retire critically defective parts and in order to return components to service if detected flaws are sub-critical. The purpose of this paper is to evaluate several techniques for sizing internal cracks in planar geometries

Primal graphs with small degrees

AbstractIt has previously been shown that there is a unique set Π of primal graphs such that every graph has an edge-decomposition into non-isomorphic elements of Π and that the only decomposition of an element of Π into non-isomorphic elements of Π is the obvious one. Here it is shown that there are infinitely many elements of Π even among graphs having a relatively simple structure. On the other hand, within this same class of graphs, we show that ‘most’ of them are not primal

Estimates of Eddy Current Response to Subsurface Cracks from 2-D Finite Element Code Predictions

Using a two dimensional finite element code, the response of a U-core eddy current probe was computed for a subsurface flaw in a stainless steel medium. Next, using a three dimensional scattering model, the change in coil impedance was calculated for the same situation. From a comparison of these two results, it was concluded that the two dimensional finite element code overestimates the eddy current sensor response for the practical problem at hand by a factor of 10. This agreed well with the result obtained using an approximate technique described in this paper to estimate the true response from two dimensional calculations. Application of such desensitization factor should allow the two dimensional calculations to be effectively used in design studies

Arrangements, circular arrangements and the crossing number of C7×Cn

AbstractMotivated by the problem of determining the crossing number of the Cartesian product Cm×Cn of two cycles, we introduce the notion of an (m,n)-arrangement, which is a generalization of a planar drawing of Pn+1×Cm in which the two “end cycles” are in the same face of the remaining n cycles. The main result is that every (m,n)-arrangement has at least (m−2)n crossings. This is used to show that the crossing number of C7×Cn is 5n, in agreement with the general conjecture that the crossing number of Cm×Cn is (m−2)n, for 3⩽m⩽n

Organizational change and development: the case for evidence-based practice

This chapter first discusses the complexities of change in organizations and why so many OCD programs fail and makes the case for change agents to become evidence-based in their change agency practice. The author then offers a definition of evidence-based organizational change and development (EBOCD) and outlines the types of “best evidence” that can be used to inform and shape the formulation and implementation of OCD strategies and to critically evaluate the associated processes and change agency practices. Various distinctive evidence-based initiatives for OCD are discussed and several case examples from the United Kingdom are presented. The chapter closes with a discussion of the specific merits of “design science,” “professional partnership” research, and “replication” researchChapter

On the non-orientable genus of a 2-connected graph

AbstractIn earlier works, additivity theorems for the genus and Euler genus of unions of graphs at two points have been given. In this work, the analogous result for the non-orientable genus is given. If Σ is obtained from the sphere by the addition of k>0 crosscaps, define γ(Σ) to be k. For a graph G, define γ(G) to be the least element in the set {γ(Σ) | G embeds in Σ}.Theorem. Let H1 and H2 be connected graphs such that H1 ∩ H2 consists of the isolated vertices v and w. Then, for some μ ϵ −1, 0, 1, 2, γ(H1 ∪ H2) = γ(H1) + γ(H2) + μ.A formula for μ is given