Boolean Dimension, Components and Blocks

We investigate the behavior of Boolean dimension with respect to components and blocks. To put our results in context, we note that for Dushnik-Miller dimension, we have that if $\dim(C)\le d$ for every component $C$ of a poset $P$, then $\dim(P)\le \max\{2,d\}$; also if $\dim(B)\le d$ for every block $B$ of a poset $P$, then $\dim(P)\le d+2$. By way of constrast, local dimension is well behaved with respect to components, but not for blocks: if $\text{ldim}(C)\le d$ for every component $C$ of a poset $P$, then $\text{ldim}(P)\le d+2$; however, for every $d\ge 4$, there exists a poset $P$ with $\text{ldim}(P)=d$ and $\dim(B)\le 3$ for every block $B$ of $P$. In this paper we show that Boolean dimension behaves like Dushnik-Miller dimension with respect to both components and blocks: if $\text{bdim}(C)\le d$ for every component $C$ of $P$, then $\text{bdim}(P)\le 2+d+4\cdot2^d$; also if $\text{bdim}(B)\le d$ for every block of $P$, then $\text{bdim}(P)\le 19+d+18\cdot 2^d$.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1712.0609

Dimension and cut vertices: an application of Ramsey theory

Motivated by quite recent research involving the relationship between the dimension of a poset and graph-theoretic properties of its cover graph, we show that for every $d\geq 1$, if $P$ is a poset and the dimension of a subposet $B$ of $P$ is at most $d$ whenever the cover graph of $B$ is a block of the cover graph of $P$, then the dimension of $P$ is at most $d+2$. We also construct examples which show that this inequality is best possible. We consider the proof of the upper bound to be fairly elegant and relatively compact. However, we know of no simple proof for the lower bound, and our argument requires a powerful tool known as the Product Ramsey Theorem. As a consequence, our constructions involve posets of enormous size.Comment: Final published version with updated reference

Post heat treatment effects on double layer metal structures for VLSI applications

The realization of high yield double layer metal systems using wet chemistry processes and the ability to extend yields beyond that attainable with wet chemistry by means of post sintering processes at temperatures below 500 C for potential applications in very large scale integration structures were studied. Yields in excess of 98% and average total contact resistance of less than 150 ohms and 200 ohms were realized for a series of 560 vias of 0.5 X 0.5 mils and 0.2 X 0.2 mils in size, respectively

The cultural parameters of lead poisoning: a medical anthropologist's view of intervention in environmental lead exposure.

This article identifies four culturally shaped sources of lead exposure in human societies: modern and historic technological sources: food habits; culturally defined health beliefs; and beauty practices. Examples of these potential sources of lead poisoning are presented from current cultures. They include the use of lead-glazed cooking pottery in Mexican-American households; folk medical use of lead in Hispanic, Arabic, South Asian, Chinese, and Hmong communities; as well as the use of lead as a cosmetic in the Near East, Southeast Asia, and South Asia. Four interacting cultural conditions that create barriers to the reduction of lead exposure and lead poisoning are identified and discussed. These are knowledge deficiencies, communication resistance, cultural reinterpretations, and incongruity of explanatory models

A characterization of robert's inequality for boxicity

AbstractF.S. Roberts defined the boxicity of a graph G as the smallest positive integer n for which there exists a function F assigning to each vertex x ϵG a sequence F(x)(1),F(x)(2),…, F(x)(n) of closed intervals of R so that distinct vertices x and y are adjacent in G if and only if F(x)(i)∩F(y)(i)≠∅ for i = 1, 2, 3, …, n. Roberts then proved that if G is a graph having 2n + 1 vertices, then the boxicity of G is at most n. In this paper, we provide an explicit characterization of this inequality by determining for each n ⩾ 1 the minimum collection Cn of graphs so that a graph G having 2n + 1 vertices has boxicity n if and only if it contains a graph from Cn as an induced subgraph. We also discuss combinatorial connections with analogous characterization problems for rectangle graphs, circular arc graphs, and partially ordered sets

Embedding finite posets in cubes

AbstractIn this paper we define the n-cube Qn as the poset obtained by taking the cartesian product of n chains each consisting of two points. For a finite poset X, we then define dim2 X as the smallest positive integer n such that X can be embedded as a subposet of Qn. For any poset X we then have log2 |X| ⩽ dim2 X ⩽ |X|. For the distributive lattice L = 2 X, dim2 L = |X| and for the crown Skn, dim2 (Skn) = n + k. For each k ⩾ 2, there exist positive constants c1 and c2 so that for the poset X consisting of all one element and k-element subsets of an n-element set, the inequality c1 log2 n < dim2(X) < c2 log2 n holds for all n with k < n. A poset is called Q-critical if dim2 (X − x) < dim2(X) for every x ϵ X. We define a join operation ⊕ on posets under which the collection Q of all Q-critical posets which are not chains forms a semigroup in which unique factorization holds. We then completely determine the subcollection M ⊆ Q consisting of all posets X for which dim2 (X) = |X|