Disjoint Blocks in a MOL(6)

2020 ◽  
Vol 76 (1) ◽  
Author(s):  
Dieter Betten
Keyword(s):  
Pairwise Disjoint

AbstractWe prove that the maximal number of pairwise disjoint 4-blocks in a MOL(6) is 3. We recall various proofs for the non-existence of a MOL(6) and show: with the theorem the proofs can be simplified considerably.

scholarly journals On the Number of Perfect Matchings in Random Lifts

2010 ◽  
Vol 19 (5-6) ◽  
pp. 791-817 ◽  
Author(s):  
CATHERINE GREENHILL ◽  
SVANTE JANSON ◽  
ANDRZEJ RUCIŃSKI
Keyword(s):  
Asymptotic Formula ◽  
Complete Graph ◽  
Pairwise Disjoint ◽  
Perfect Matching ◽  
Perfect Matchings ◽  
Lattice Points ◽  
Laplace’S Method ◽  
Second Moment ◽  
Multiple Dimensions ◽  
Laplace's Method

Let G be a fixed connected multigraph with no loops. A random n-lift of G is obtained by replacing each vertex of G by a set of n vertices (where these sets are pairwise disjoint) and replacing each edge by a randomly chosen perfect matching between the n-sets corresponding to the endpoints of the edge. Let XG be the number of perfect matchings in a random lift of G. We study the distribution of XG in the limit as n tends to infinity, using the small subgraph conditioning method.We present several results including an asymptotic formula for the expectation of XG when G is d-regular, d ≥ 3. The interaction of perfect matchings with short cycles in random lifts of regular multigraphs is also analysed. Partial calculations are performed for the second moment of XG, with full details given for two example multigraphs, including the complete graph K4.To assist in our calculations we provide a theorem for estimating a summation over multiple dimensions using Laplace's method. This result is phrased as a summation over lattice points, and may prove useful in future applications.

Generalized reduction theorems for model-theoretic analogs of the class of coanalytic sets

1993 ◽  
Vol 58 (1) ◽  
pp. 81-98
Author(s):  
Shaughan Lavine
Keyword(s):  
Pairwise Disjoint ◽  
Admissible Set ◽  
Effective Version ◽  
Reduction Property ◽  
Uniformization Property ◽  
Weakly Stable

AbstractLet be an admissible set. A sentence of the form is a sentence if φ ∈ (φ is ∨ Φ where Φ is an -r.e. set of sentences from ). A sentence of the form is an , sentence if φ is a sentence. A class of structures is, for example, a ∀1 class if it is the class of models of a ∀1() sentence. Thus ∀1() is a class of classes of structures, and so forth.Let i, be the structure 〈i, <〉, for i > 0. Let Γ be a class of classes of structures. We say that a sequence J1, …, Ji,…, i < ω, of classes of structures is a Γ sequence if Ji ∈ Γ, i < ω, and there is I ∈ Γ such that ∈ Ji, if and only if [],i, where [,] is the disjoint sum. A class Γ of classes of structures has the easy uniformization property if for every Γ sequence J1,…, Ji,…, i < ω, there is a Γ sequence J′t, …, J′i, …, i < ω, such that J′i ⊆ Ji, i < ω, ⋃J′i = ⋃Ji, and the J′i are pairwise disjoint. The easy uniformization property is an effective version of Kuratowski's generalized reduction property that is closely related to Moschovakis's (topological) easy uniformization property.We show over countable structures that ∀1() and ∃2() have the easy uniformization property if is a countable admissible set with an infinite member, that and have the easy uniformization property if α is countable, admissible, and not weakly stable, and that and have the easy uniformization properly. The results proved are more general. The result for answers a question of Vaught(1980).

scholarly journals Rooted $K_4$-Minors

10.37236/3476 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Ruy Fabila-Monroy ◽  
David R. Wood
Keyword(s):  
Planar Graph ◽  
Pairwise Disjoint ◽  
Single Face ◽  
Special Cases ◽  
Connected Subgraphs ◽  
Connected Planar Graph

Let $a,b,c,d$ be four vertices in a graph $G$. A $K_4$ minor rooted at $a,b,c,d$ consists of four pairwise-disjoint pairwise-adjacent connected subgraphs of $G$, respectively containing $a,b,c,d$. We characterise precisely when $G$ contains a $K_4$-minor rooted at $a,b,c,d$ by describing six classes of obstructions, which are the edge-maximal graphs containing no $K_4$-minor rooted at $a,b,c,d$. The following two special cases illustrate the full characterisation: (1) A 4-connected non-planar graph contains a $K_4$-minor rooted at $a,b,c,d$ for every choice of $a,b,c,d$. (2) A 3-connected planar graph contains a $K_4$-minor rooted at $a,b,c,d$ if and only if $a,b,c,d$ are not on a single face.

Diamond Principles, Ideals and the Normal Moore Space Problem

1981 ◽  
Vol 33 (2) ◽  
pp. 282-296 ◽  
Author(s):  
Alan D. Taylor
Keyword(s):  
Topological Space ◽  
Pairwise Disjoint ◽  
Space Problem ◽  
Moore Space ◽  
Image Position

If is a topological space then a sequence (Cα:α < λ) of subsets of is said to be normalized if for every H ⊆ λ there exist disjoint open sets and such thatThe sequence (Cα:α < λ) is said to be separated if there exists a sequence of pairwise disjoint open sets such that for each α < λ. As is customary, we adopt the convention that all sequences (Cα:α < λ) considered are assumed to be relatively discrete as defined in [18, p. 21]: if x ∈ Cα then there exists a neighborhood about x that intersects no Cβ for β ≠ α.

Osculatory Packings by Spheres

1970 ◽  
Vol 13 (1) ◽  
pp. 59-64 ◽  
Author(s):  
David W. Boyd
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Simple Proof ◽  
Pairwise Disjoint ◽  
Exponent Of Convergence ◽  
Residual Set ◽  
Open Set ◽  
Finite Lebesgue Measure

If U is an open set in Euclidean N-space EN which has finite Lebesgue measure |U| then a complete packing of U by open spheres is a collection C={Sn} of pairwise disjoint open spheres contained in U and such that Σ∞n=1|Sn| = |U|. Such packings exist by Vitali's theorem. An osculatory packing is one in which the spheres Sn are chosen recursively so that from a certain point on Sn+1 is the largest possible sphere contained in (Here S- will denote the closure of a set S). We give here a simple proof of the "well-known" fact that an osculatory packing is a complete packing. Our method of proof shows also that for osculatory packings, the Hausdorff dimension of the residual set is dominated by the exponent of convergence of the radii of the Sn.

scholarly journals On the differentiability of hairs for Zorich maps

2017 ◽  
Vol 39 (7) ◽  
pp. 1824-1842 ◽  
Author(s):  
PATRICK COMDÜHR
Keyword(s):  
Pairwise Disjoint ◽  
Exponential Family ◽  
Julia Set ◽  
Exponential Map ◽  
Simple Curves

Devaney and Krych showed that, for the exponential family $\unicode[STIX]{x1D706}e^{z}$, where $0\,<\,\unicode[STIX]{x1D706}\,<\,1/e$, the Julia set consists of uncountably many pairwise disjoint simple curves tending to $\infty$. Viana proved that these curves are smooth. In this article, we consider quasiregular counterparts of the exponential map, the so-called Zorich maps, and generalize Viana’s result to these maps.

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