Convexity properties and algebraic closure operators

1984 ◽  
pp. 113-146 ◽  
Author(s):  
G. L. Cherlin ◽  
H. Volger
Keyword(s):  
Algebraic Closure ◽  
Closure Operators ◽  
Convexity Properties

scholarly journals Neighbourhood lattices – a poset approach to topological spaces

1989 ◽  
Vol 39 (1) ◽  
pp. 31-48 ◽  
Author(s):  
Frank P. Prokop
Keyword(s):  
Topological Space ◽  
Lattice Structure ◽  
Algebraic Closure ◽  
Topological Spaces ◽  
Closure Operators ◽  
Continuity Properties ◽  
Image Function ◽  
Topological Closure ◽  
Arbitrary Lattice ◽  
The Relationship

In this paper neighbourhood lattices are developed as a generalisation of topological spaces in order to examine to what extent the concepts of “openness”, “closedness”, and “continuity” defined in topological spaces depend on the lattice structure of P(X), the power set of X.A general pre-neighbourhood system, which satisfies the poset analogues of the neighbourhood system of points in a topological space, is defined on an ∧-semi-lattice, and is used to define open elements. Neighbourhood systems, which satisfy the poset analogues of the neighbourhood system of sets in a topological space, are introduced and it is shown that it is the conditionally complete atomistic structure of P(X) which determines the extension of pre-neighbourhoods of points to the neighbourhoods of sets.The duals of pre-neighbourhood systems are used to generate closed elements in an arbitrary lattice, independently of closure operators or complementation. These dual systems then form the backdrop for a brief discussion of the relationship between preneighbourhood systems, topological closure operators, algebraic closure operators, and Čech closure operators.Continuity is defined for functions between neighbourhood lattices, and it is proved that a function f: X → Y between topological spaces is continuous if and only if corresponding direct image function between the neighbourhood lattices P(X) and P(Y) is continuous in the neighbourhood sense. Further, it is shown that the algebraic character of continuity, that is, the non-convergence aspects, depends only on the properites of pre-neighbourhood systems. This observation leads to a discussion of the continuity properties of residuated mappings. Finally, the topological properties of normality and regularity are characterised in terms of the continuity properties of the closure operator on a topological space.

The classification of small types of rank ω, Part I

2001 ◽  
Vol 66 (4) ◽  
pp. 1884-1898
Author(s):  
Steven Buechler ◽  
Colleen Hoover
Keyword(s):  
Stability Theory ◽  
Closure Operator ◽  
Algebraic Closure ◽  
Countable Model ◽  
Closure Operators ◽  
Basic Concepts ◽  
Countable Models

Abstract.Certain basic concepts of geometrical stability theory are generalized to a class of closure operators containing algebraic closure. A specific case of a generalized closure operator is developed which is relevant to Vaught's conjecture. As an application of the methods, we proveTheorem A. Let G be a superstate group of U-rank ω such that the generics of G are locally modular and Th(G) has few countable models. Let G− be the group of nongeneric elements of G. G+ = Go + G−. Let Π = {q ∈ S(∅): U(q) < ω}. For any countable model M of Th(G) there is a finite A ⊂ M such thai M is almost atomic over A ∪ (G+ ∩ M) ∪ ⋃p∈Πp(M).

scholarly journals Generating monotone quantities for the heat equation

2019 ◽  
Vol 2019 (756) ◽  
pp. 37-63 ◽  
Author(s):  
Jonathan Bennett ◽  
Neal Bez
Keyword(s):  
Euclidean Space ◽  
Heat Equation ◽  
Positive Solutions ◽  
Algebraic Closure ◽  
Heat Equations ◽  
Differential Inequalities ◽  
Closure Properties ◽  
Rich Variety ◽  
Linear Heat ◽  
Convexity Properties

AbstractThe purpose of this article is to expose and further develop a simple yet surprisingly far-reaching framework for generating monotone quantities for positive solutions to linear heat equations in euclidean space. This framework is intimately connected to the existence of a rich variety of algebraic closure properties of families of sub/super-solutions, and more generally solutions of systems of differential inequalities capturing log-convexity properties such as the Li–Yau gradient estimate. Various applications are discussed, including connections with the general Brascamp–Lieb inequality and the Ornstein–Uhlenbeck semigroup.

scholarly journals On many-sorted algebraic closure operators

2004 ◽  
Vol 266 (1) ◽  
pp. 81-84 ◽  
Author(s):  
Juan Climent Vidal ◽  
Juan Soliveres Tur
Keyword(s):  
Algebraic Closure ◽  
Closure Operators

scholarly journals On the Convexity Preservation of a Quasi C3 Nonlinear Interpolatory Reconstruction Operator on σ Quasi-Uniform Grids

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 310 ◽  
Author(s):  
Pedro Ortiz ◽  
Juan Carlos Trillo
Keyword(s):  
Initial Data ◽  
Numerical Experiments ◽  
Approximation Order ◽  
Piecewise Polynomial ◽  
Nonuniform Grids ◽  
Uniform Grids ◽  
Reconstruction Operator ◽  
Convexity Properties ◽  
Second Degree

This paper is devoted to introducing a nonlinear reconstruction operator, the piecewise polynomial harmonic (PPH), on nonuniform grids. We define this operator and we study its main properties, such as its reproduction of second-degree polynomials, approximation order, and conditions for convexity preservation. In particular, for σ quasi-uniform grids with σ≤4, we get a quasi C3 reconstruction that maintains the convexity properties of the initial data. We give some numerical experiments regarding the approximation order and the convexity preservation.

Digital Jordan Curves and Surfaces with Respect to a Closure Operator

2021 ◽  
Vol 179 (1) ◽  
pp. 59-74
Author(s):  
Josef Šlapal
Keyword(s):  
Direct Product ◽  
Jordan Curve ◽  
Closure Operator ◽  
Jordan Curve Theorem ◽  
Closure Operators ◽  
Ternary Relation ◽  
Digital Space ◽  
Jordan Curves ◽  
Curves And Surfaces ◽  
Digital Plane

In this paper, we propose new definitions of digital Jordan curves and digital Jordan surfaces. We start with introducing and studying closure operators on a given set that are associated with n-ary relations (n > 1 an integer) on this set. Discussed are in particular the closure operators associated with certain n-ary relations on the digital line ℤ. Of these relations, we focus on a ternary one equipping the digital plane ℤ2 and the digital space ℤ3 with the closure operator associated with the direct product of two and three, respectively, copies of this ternary relation. The connectedness provided by the closure operator is shown to be suitable for defining digital curves satisfying a digital Jordan curve theorem and digital surfaces satisfying a digital Jordan surface theorem.

scholarly journals Location and shape of a rectangular facility in ℝn. Convexity properties. Convexity properties

1998 ◽  
Vol 83 (1-3) ◽  
pp. 277-290 ◽  
Author(s):  
E. Carrizosa ◽  
M. Muñoz-Márquez ◽  
J. Puerto
Keyword(s):  
Convexity Properties

scholarly journals Reflective Full Subcategories of the Category ofL-Posets

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Hongping Liu ◽  
Qingguo Li ◽  
Xiangnan Zhou
Keyword(s):  
Special Class ◽  
Full Subcategory ◽  
Closure Operators ◽  
The Relationship ◽  
Equivalent Description

This paper focuses on the relationship betweenL-posets and completeL-lattices from the categorical view. By considering a special class of fuzzy closure operators, we prove that the category of completeL-lattices is a reflective full subcategory of the category ofL-posets with appropriate morphisms. Moreover, we characterize the Dedekind-MacNeille completions ofL-posets and provide an equivalent description for them.

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