Boolean algebras, splitting theorems, and sets
Michael Morley, Robert Soare (1975)
Fundamenta Mathematicae
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Michael Morley, Robert Soare (1975)
Fundamenta Mathematicae
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Jan Waszkiewicz (1974)
Colloquium Mathematicae
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Ernest J. Cockayne, Stephen Finbow (2004)
Discussiones Mathematicae Graph Theory
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For each vertex s of the vertex subset S of a simple graph G, we define Boolean variables p = p(s,S), q = q(s,S) and r = r(s,S) which measure existence of three kinds of S-private neighbours (S-pns) of s. A 3-variable Boolean function f = f(p,q,r) may be considered as a compound existence property of S-pns. The subset S is called an f-set of G if f = 1 for all s ∈ S and the class of f-sets of G is denoted by . Only 64 Boolean functions f can produce different classes , special cases...
Rostislav Černý (2006)
Kybernetika
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Consider a stationary Boolean model with convex grains in and let any exposed lower tangent point of be shifted towards the hyperplane by the length of the part of the segment between the point and its projection onto the covered by . The resulting point process in the halfspace (the Laslett’s transform of ) is known to be stationary Poisson and of the same intensity as the original Boolean model. This result was first formulated for the planar Boolean model (see N. Cressie...
Rainer Schimming (1996)
Discussiones Mathematicae Graph Theory
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A special relational structure, called genealogical tree, is introduced; its social interpretation and geometrical realizations are discussed. The numbers of all abstract genealogical trees with exactly n+1 nodes and k leaves is found by means of enumeration of code words. For each n, the form a partition of the n-th Catalan numer Cₙ, that means .
Piotr Nayar (2014)
Colloquium Mathematicae
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We consider Boolean functions defined on the discrete cube equipped with a product probability measure , where and γ = √(α/β). This normalization ensures that the coordinate functions are orthonormal in . We prove that if the spectrum of a Boolean function is concentrated on the first two Fourier levels, then the function is close to a certain function of one variable. Our theorem strengthens the non-symmetric FKN Theorem due to Jendrej, Oleszkiewicz and Wojtaszczyk. Moreover,...
Eric Hall, Kyriakos Keremedis (2013)
Bulletin of the Polish Academy of Sciences. Mathematics
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(i) The statement P(ω) = “every partition of ℝ has size ≤ |ℝ|” is equivalent to the proposition R(ω) = “for every subspace Y of the Tychonoff product the restriction |Y = Y ∩ B: B ∈ of the standard clopen base of to Y has size ≤ |(ω)|”. (ii) In ZF, P(ω) does not imply “every partition of (ω) has a choice set”. (iii) Under P(ω) the following two statements are equivalent: (a) For every Boolean algebra of size ≤ |ℝ| every filter can be extended to an ultrafilter. (b) Every Boolean...
W. B. Vasantha Kandasamy (1992)
Publications du Département de mathématiques (Lyon)
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Jerzy Płonka (2001)
Discussiones Mathematicae - General Algebra and Applications
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Let τ: F → N be a type of algebras, where F is a set of fundamental operation symbols and N is the set of all positive integers. An identity φ ≈ ψ is called biregular if it has the same variables in each of it sides and it has the same fundamental operation symbols in each of it sides. For a variety V of type τ we denote by the biregularization of V, i.e. the variety of type τ defined by all biregular identities from Id(V). Let B be the variety of Boolean algebras of type , where...
Kamal Kabyl, Abdelhafid Berrachedi, Éric Sopena (2015)
Czechoslovak Mathematical Journal
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The cubical dimension of a graph is the smallest dimension of a hypercube into which is embeddable as a subgraph. The conjecture of Havel (1984) claims that the cubical dimension of every balanced binary tree with vertices, , is . The 2-rooted complete binary tree of depth is obtained from two copies of the complete binary tree of depth by adding an edge linking their respective roots. In this paper, we determine the cubical dimension of trees obtained by subdividing twice...
Ivan Chajda, Filip Švrček (2011)
Discussiones Mathematicae - General Algebra and Applications
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We study unitary rings of characteristic 2 satisfying identity for some natural number p. We characterize several infinite families of these rings which are Boolean, i.e., every element is idempotent. For example, it is in the case if or or for a suitable natural number n. Some other (more general) cases are solved for p expressed in the form or where q is a natural number and .
Laura Fontanella, Sy David Friedman (2015)
Fundamenta Mathematicae
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We force from large cardinals a model of ZFC in which and both have the tree property. We also prove that if we strengthen the large cardinal assumptions, then in the final model even satisfies the super tree property.
Dominique Jeulin (2016)
Applications of Mathematics
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Models of random sets and of point processes are introduced to simulate some specific clustering of points, namely on random lines in and and on random planes in . The corresponding point processes are special cases of Cox processes. The generating distribution function of the probability distribution of the number of points in a convex set and the Choquet capacity are given. A possible application is to model point defects in materials with some degree of alignment. Theoretical...
Zhi-Hong Sun, Lin-Lin Wang, Yi-Li Wu (2015)
Colloquium Mathematicae
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Let T¹ₙ = (V,E₁) and T²ₙ = (V,E₂) be the trees on n vertices with , and . For p ≥ n ≥ 5 we obtain explicit formulas for ex(p;T¹ₙ) and ex(p;T²ₙ), where ex(p;L) denotes the maximal number of edges in a graph of order p not containing L as a subgraph. Let r(G₁,G₂) be the Ramsey number of the two graphs G₁ and G₂. We also obtain some explicit formulas for , where i ∈ 1,2 and Tₘ is a tree on m vertices with Δ(Tₘ) ≤ m - 3.