Regular minimal sets over the circle and the ellis minimal set

1972 ◽  
Vol 6 (1-2) ◽  
pp. 145-163
Author(s):  
Richard Freiman
Keyword(s):  
Minimal Set ◽  
Minimal Sets

Minimal Sets on Graphs and Dendrites

2003 ◽  
Vol 13 (07) ◽  
pp. 1721-1725 ◽  
Author(s):  
Francisco Balibrea ◽  
Roman Hric ◽  
L'ubomír Snoha
Keyword(s):  
Topological Structure ◽  
Partial Characterization ◽  
Minimal Set ◽  
Full Characterization ◽  
Minimal Sets ◽  
Continuous Maps

The topological structure of minimal sets of continuous maps on graphs, dendrites and dendroids is studied. A full characterization of minimal sets on graphs and a partial characterization of minimal sets on dendrites are given. An example of a minimal set containing an interval on a dendroid is given.

Pseudoprojective strongly minimal sets are locally projective

1991 ◽  
Vol 56 (4) ◽  
pp. 1184-1194 ◽  
Author(s):  
Steven Buechler
Keyword(s):  
Predicate Symbol ◽  
Infinite Dimension ◽  
Elementary Extension ◽  
Morley Rank ◽  
Minimal Set ◽  
Unary Predicate ◽  
Minimal Sets

AbstractLet D be a strongly minimal set in the language L, and D′ ⊃ D an elementary extension with infinite dimension over D. Add to L a unary predicate symbol D and let T′ be the theory of the structure (D′, D), where D interprets the predicate D. It is known that T′ is ω-stable. We proveTheorem A. If D is not locally modular, then T′ has Morley rank ω.We say that a strongly minimal set D is pseudoprojective if it is nontrivial and there is a k < ω such that, for all a, b ∈ D and closed X ⊂ D, a ∈ cl(Xb) ⇒ there is a Y ⊂ X with a ∈ cl(Yb) and ∣Y∣ ≤ k. Using Theorem A, we proveTheorem B. If a strongly minimal set D is pseudoprojective, then D is locally projective.The following result of Hrushovski's (proved in §4) plays a part in the proof of Theorem B.Theorem C. Suppose that D is strongly minimal, and there is some proper elementary extension D1 of D such that the theory of the pair (D1, D) is ω1-categorical. Then D is locally modular.

One theorem of Zil′ber's on strongly minimal sets

1985 ◽  
Vol 50 (4) ◽  
pp. 1054-1061 ◽  
Author(s):  
Steven Buechler
Keyword(s):  
Main Part ◽  
Minimal Set ◽  
Minimal Sets ◽  
Image Position

AbstractSuppose D ⊂ M is a strongly minimal set definable in M with parameters from C. We say D is locally modular if for all X, Y ⊂ D, with X = acl(X ∪ C)∩D, Y = acl(Y ∪ C) ∩ D and X ∩ Y ≠ ∅,We prove the following theorems.Theorem 1. Suppose M is stable and D ⊂ M is strongly minimal. If D is not locally modular then inMeqthere is a definable pseudoplane.(For a discussion of Meq see [M, §A].) This is the main part of Theorem 1 of [Z2] and the trichotomy theorem of [Z3].Theorem 2. Suppose M is stable and D, D′ ⊂ M are strongly minimal and nonorthogonal. Then D is locally modular if and only if D′ is locally modular.

scholarly journals Quality of Minimal Sets of Prime Implicants of Boolean Functions

2017 ◽  
Vol 63 (2) ◽  
pp. 165-169
Author(s):  
V. C. Prasad
Keyword(s):  
Power Consumption ◽  
Boolean Function ◽  
Boolean Functions ◽  
New Technique ◽  
Minimal Set ◽  
Reduce Power Consumption ◽  
Minimal Sets ◽  
Prime Implicants

Abstract Two new problems are posed and solved concerning minimal sets of prime implicants of Boolean functions. It is well known that the prime implicant set of a Boolean function should be minimal and have as few literals as possible. But it is not well known that min term repetitions should also be as few as possible to reduce power consumption. Determination of minimal sets of prime implicants is a well known problem. But nothing is known on the least number of (i) prime implicants (ii) literals and (iii) min term repetitions , any minimal set of prime implicants will have. These measures are useful to assess the quality of a minimal set. They are then extended to determine least number of prime implicants / implicates required to design a static hazard free circuit. The new technique tends to give smallest set of prime implicants for various objectives.

scholarly journals DMP in strongly minimal sets

2007 ◽  
Vol 72 (3) ◽  
pp. 1019-1030 ◽  
Author(s):  
Assaf Hasson ◽  
Ehud Hrushovski
Keyword(s):  
Finite Cover ◽  
Minimal Set ◽  
Minimal Sets ◽  
Minimal Theory

AbstractWe construct a strongly minimal set which is not a finite cover of one with DMP. We also show that for a strongly minimal theory T, generic automorphisms exist iff T has DMP, thus proving a conjecture of Kikyo and Pillay.

scholarly journals On Finite Groups With ‘Hidden’ Primes

1971 ◽  
Vol 12 (3) ◽  
pp. 287-300 ◽  
Author(s):  
L. G. Kovács ◽  
Joachim Neubüser ◽  
B. H. Neumann
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Cyclic Group ◽  
Finite Groups ◽  
Infinite Order ◽  
Proper Subset ◽  
Minimal Set ◽  
Minimal Sets ◽  
Starting Point

The starting point of this investigation was a question put to us by Martin B. Powell: If the prime number p divides the order of the finite group G, must there be a minimal set of generators of G that contains an element whose order is divisible by p? A set of generators of G is minimal if no set with fewer elements generates G. A minimal set of generators is clearly irredundant, in the sense that no proper subset of it generates G; an irredundant set of generators, however, need not be minimal, as is easily seen from the example of a cyclic group of composite (or infinite) order. Powell's question can be asked for irredundant instead of minimal sets of generators; it turns out that the answer is not the same in these two cases. A different formulation, together with some notation, may make the situation clearer.

scholarly journals Utilization of the Karnaugh Map in Exploring Cause-effect Relations Modeled by Partially-defined Boolean Functions

2021 ◽  
pp. 117-137
Author(s):  
Ali Muhammad Ali Rushdi ◽  
Raid Salih Badawi
Keyword(s):  
Boolean Functions ◽  
Qualitative Comparative Analysis ◽  
Systematic Investigation ◽  
Exact Matching ◽  
Minimal Set ◽  
Minimal Sets ◽  
Prime Implicants ◽  
Switching Functions ◽  
Programming Techniques ◽  
Karnaugh Map

This paper utilizes a modern regular and modular eight-variable Karnaugh map in a systematic investigation of cause-effect relationships modeled by partially-defined Boolean functions (PDBF) (known also as incompletely specified switching functions). First, we present a Karnaugh-map test that can decide whether a certain variable must be included in a set of supporting variables of the function, and, otherwise, might enforce the exclusion of this variable from such a set. This exclusion is attained via certain don’t-care assignments that ensure the equivalence of the Boolean quotient w.r.t. the variable, and that w.r.t. its complement, i.e., the exact matching of the half map representing the internal region of the variable, and the remaining half map representing the external region of the variable, in which case any of these two half maps replaces the original full map as a representation of the function. Such a variable exclusion might be continued w.r.t. other variables till a minimal set of supporting variables is reached. The paper addresses a dominantly-unspecified PDBF to obtain all its minimal sets of supporting variables without resort to integer programming techniques. For each of the minimal sets obtained, standard map methods for extracting prime implicants allow the construction of all irredundant disjunctive forms (IDFs). According to this scheme of first identifying a minimal set of supporting variables, we avoid the task of drawing prime-implicant loops on the initial eight-variable map, and postpone this task till the map is dramatically reduced in size. The procedure outlined herein has important ramifications for the newly-established discipline of Qualitative Comparative Analysis (QCA). These ramifications are not expected to be welcomed by the QCA community, since they clearly indicate that the too-often strong results claimed by QCA adherents need to be checked and scrutinized.

The flow near non-trivial minimal sets on 2-manifolds

1990 ◽  
Vol 108 (3) ◽  
pp. 569-573 ◽  
Author(s):  
Konstantin Athanassopoulos
Keyword(s):  
Continuous Flow ◽  
Qualitative Behaviour ◽  
Minimal Set ◽  
Minimal Sets ◽  
First Results

In this paper we give a description of the qualitative behaviour of the orbits near a non-trivial compact minimal set of a continuous flow on a 2-manifold. The first results in this direction were obtained in [1] and the present paper can be regarded as a continuation of that work. The main result can be stated as follows:Theorem 1·1. Let (ℝ, M, f) be a continuous flow on a 2-manifold M and A ⊂ M a non-trivial compact minimal set.

scholarly journals Minimal sets on tori

1984 ◽  
Vol 4 (4) ◽  
pp. 499-507 ◽  
Author(s):  
Daniel Berend
Keyword(s):  
Hausdorff Dimension ◽  
Commutative Semigroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Dimensional Torus ◽  
Minimal Set ◽  
Circle Group ◽  
Minimal Sets ◽  
Necessary And Sufficient

AbstractLet Σ be a commutative semigroup of continuous endomorphisms of the r-dimensional torus. Generalizing a result of Furstenberg dealing with the circle group, necessary and sufficient conditions are given here for Σ to possess the following property: Any Σ-minimal set consists of torsion elements. Semigroups not having this property are shown to admit minimal sets of positive Hausdorff dimension.

D-function of a minimal set and an extension of Sharkovskii's theorem to minimal sets

1992 ◽  
Vol 12 (2) ◽  
pp. 365-376 ◽  
Author(s):  
Xiangdong Ye
Keyword(s):  
Finite Type ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Topological Invariant ◽  
Minimal Set ◽  
Subshift Of Finite Type ◽  
Minimal Sets ◽  
Continuous Mappings

AbstractLetXbe a compact Hausdorff space,f∈C0(X, X) andA⊂Xa minimal set off. We first introduce a new topological invariant, the D-function of a minimal set, by the investigation of the decomposition of the minimal set A under the action offn,n∈N. Then important properties about the invariant and the existence of minimal set with a given D-function in some subshift of finite type are discussed. Finally Sharkovskii's theorem is generalized to minimal sets of continuous mappings from the interval into itself.

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