No four subsets forming an N

AbstractWe survey results concerning the maximum size of a family F of subsets of an n-element set such that a certain configuration is avoided. When F avoids a chain of size two, this is just Sperner's theorem. Here we give bounds on how large F can be such that no four distinct sets A,B,C,D∈F satisfy A⊂B, C⊂B, C⊂D. In this case, the maximum size satisfies (n⌊n2⌋)(1+1n+Ω(1n2))⩽|F|⩽(n⌊n2⌋)(1+2n+O(1n2)), which is very similar to the best-known bounds for the more restrictive problem of F avoiding three sets B,C,D such that C⊂B, C⊂D

Geometric configuration in robot kinematic design

A lattice of geometries is presented and compared for representing some geometrical aspects of the kinematic design of robot systems and subsystems. Three geometries (set theory, topology and projective geometry) are briefly explored in more detail in the context of three geometric configurations in robotics (robot groupings, robot connectivities and robot motion sensor patterns)

Partitioning the power set of [n][n] into CkC_k-free parts

We show that for $n \geq 3, n\ne 5$, in any partition of $\mathcal{P}(n)$, the set of all subsets of $[n]=\{1,2,\dots,n\}$, into $2^{n-2}-1$ parts, some part must contain a triangle --- three different subsets $A,B,C\subseteq [n]$ such that $A\cap B$, $A\cap C$, and $B\cap C$ have distinct representatives. This is sharp, since by placing two complementary pairs of sets into each partition class, we have a partition into $2^{n-2}$ triangle-free parts. We also address a more general Ramsey-type problem: for a given graph $G$, find (estimate) $f(n,G)$, the smallest number of colors needed for a coloring of $\mathcal{P}(n)$, such that no color class contains a Berge-$G$ subhypergraph. We give an upper bound for $f(n,G)$ for any connected graph $G$ which is asymptotically sharp (for fixed $k$) when $G=C_k, P_k, S_k$, a cycle, path, or star with $k$ edges. Additional bounds are given for $G=C_4$ and $G=S_3$.Comment: 12 page

Sperner type theorems with excluded subposets

Let F be a family of subsets of an n-element set. Sperner's theorem says that if there is no inclusion among the members of F then the largest family under this condition is the one containing all ⌊ frac(n, 2) ⌋-element subsets. The present paper surveys certain generalizations of this theorem. The maximum size of F is to be found under the condition that a certain configuration is excluded. The configuration here is always described by inclusions. More formally, let P be a poset. The maximum size of a family F which does not contain P as a (not-necessarily induced) subposet is denoted by La (n, P). The paper is based on a lecture of the author at the Jubilee Conference on Discrete Mathematics [Banasthali University, January 11-13, 2009], but it was somewhat updated in December 2010. © 2011 Elsevier B.V. All rights reserved

About a certain NP complete problem

In this article we introduce the concept of special decomposition of a set and the concept of special covering of a set under such a decomposition. We study the conditions for existence of special coverings of the sets, under the special decomposition of the set. These conditions of formulated problem have important applications in the field of satisfiability of Boolean functions. Our goal is to study the relationship between sat CNF problem and the problem of existance of special covering of the set. We also study the relationship between classes of computational complexity by searching for special coverings of the sets. We prove, that the decidability of sat CNF problem, in polynomial time reduces to the problem of existence of a special covering of a set. We also prove, that the problem of existence of a special covering of a set, in polynomial time reduces to the decidability of the sat CNF problem. Therefore, the mentioned problems are polynomially equivalent. And then, the problem of existence of a special covering of a set is NP-complete problem

Interpreting the Ionization Sequence in Star-Forming Galaxy Emission-Line Spectra

High ionization star forming (SF) galaxies are easily identified with strong emission line techniques such as the BPT diagram, and form an obvious ionization sequence on such diagrams. We use a locally optimally emitting cloud model to fit emission line ratios that constrain the excitation mechanism, spectral energy distribution, abundances and physical conditions along the star-formation ionization sequence. Our analysis takes advantage of the identification of a sample of pure star-forming galaxies, to define the ionization sequence, via mean field independent component analysis. Previous work has suggested that the major parameter controlling the ionization level in SF galaxies is the metallicity. Here we show that the observed SF- sequence could alternatively be interpreted primarily as a sequence in the distribution of the ionizing flux incident on gas spread throughout a galaxy. Metallicity variations remain necessary to model the SF-sequence, however, our best models indicate that galaxies with the highest and lowest observed ionization levels (outside the range -0.37 < log [O III]/H\b{eta} < -0.09) require the variation of an additional physical parameter other than metallicity, which we determine to be the distribution of ionizing flux in the galaxy.Comment: 41 pages, 17 figures, 9 tables, accepted to MNRA

Densest local packing diversity. II. Application to three dimensions

The densest local packings of N three-dimensional identical nonoverlapping spheres within a radius Rmin(N) of a fixed central sphere of the same size are obtained for selected values of N up to N = 1054. In the predecessor to this paper [A.B. Hopkins, F.H. Stillinger and S. Torquato, Phys. Rev. E 81 041305 (2010)], we described our method for finding the putative densest packings of N spheres in d-dimensional Euclidean space Rd and presented those packings in R2 for values of N up to N = 348. We analyze the properties and characteristics of the densest local packings in R3 and employ knowledge of the Rmin(N), using methods applicable in any d, to construct both a realizability condition for pair correlation functions of sphere packings and an upper bound on the maximal density of infinite sphere packings. In R3, we find wide variability in the densest local packings, including a multitude of packing symmetries such as perfect tetrahedral and imperfect icosahedral symmetry. We compare the densest local packings of N spheres near a central sphere to minimal-energy configurations of N+1 points interacting with short-range repulsive and long-range attractive pair potentials, e.g., 12-6 Lennard-Jones, and find that they are in general completely different, a result that has possible implications for nucleation theory. We also compare the densest local packings to finite subsets of stacking variants of the densest infinite packings in R3 (the Barlow packings) and find that the densest local packings are almost always most similar, as measured by a similarity metric, to the subsets of Barlow packings with the smallest number of coordination shells measured about a single central sphere, e.g., a subset of the FCC Barlow packing. We additionally observe that the densest local packings are dominated by the spheres arranged with centers at precisely distance Rmin(N) from the fixed sphere's center.Comment: 45 pages, 18 figures, 2 table