scholarly journals Planar Lattice Subsets with Minimal Vertex Boundary

2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Radhika Gupta ◽  
Ivan Levcovitz ◽  
Alexander Margolis ◽  
Emily Stark
Keyword(s):  
Integer Lattice ◽  
Geometric Characterization ◽  
Planar Lattice ◽  
Minimal Set ◽  
Efficient Sets ◽  
Minimal Sets ◽  
Large Size ◽  
Maximal Size ◽  
Infinite Component

A subset of vertices of a graph is minimal if, within all subsets of the same size, its vertex boundary is minimal. We give a complete, geometric characterization of minimal sets for the planar integer lattice $X$. Our characterization elucidates the structure of all minimal sets, and we are able to use it to obtain several applications. We show that the neighborhood of a minimal set is minimal. We characterize uniquely minimal sets of $X$: those which are congruent to any other minimal set of the same size. We also classify all efficient sets of $X$: those that have maximal size amongst all such sets with a fixed vertex boundary. We define and investigate the graph $G$ of minimal sets whose vertices are congruence classes of minimal sets of $X$ and whose edges connect vertices which can be represented by minimal sets that differ by exactly one vertex. We prove that G has exactly one infinite component, has infinitely many isolated vertices and has bounded components of arbitrarily large size. Finally, we show that all minimal sets, except one, are connected.

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.

scholarly journals Characterization of Cocycle Attractors for Nonautonomous Reaction–Diffusion Equations

2016 ◽  
Vol 26 (08) ◽  
pp. 1650135 ◽  
Author(s):  
C. A. Cardoso ◽  
J. A. Langa ◽  
R. Obaya
Keyword(s):  
Chaotic Dynamics ◽  
Linear Part ◽  
Reaction Diffusion ◽  
Positive Cone ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Minimal Set ◽  
Full Characterization ◽  
The Stability

In this paper, we describe in detail the global and cocycle attractors related to nonautonomous scalar differential equations with diffusion. In particular, we investigate reaction–diffusion equations with almost-periodic coefficients. The associated semiflows are strongly monotone which allow us to give a full characterization of the cocycle attractor. We prove that, when the upper Lyapunov exponent associated to the linear part of the equations is positive, the flow is persistent in the positive cone, and we study the stability and the set of continuity points of the section of each minimal set in the global attractor for the skew product semiflow. We illustrate our result with some nontrivial examples showing the richness of the dynamics on this attractor, which in some situations shows internal chaotic dynamics in the Li–Yorke sense. We also include the sublinear and concave cases in order to go further in the characterization of the attractors, including, for instance, a nonautonomous version of the Chafee–Infante equation. In this last case we can show exponentially forward attraction to the cocycle (pullback) attractors in the positive cone of solutions.

Characterization of a lytic EBP bacteriophage with large size genome against Enterobacter cloacae

Apmis ◽  
2021 ◽  
Author(s):  
Muhammad Asif ◽  
Hafsa Naseem ◽  
Iqbal Ahmed Alvi ◽  
Abdul Basit ◽  
Shafiq‐ur‐Rehman
Keyword(s):  
Enterobacter Cloacae ◽  
Large Size

Growth and characterization of large-size bismuth germanate single crystals by low thermal gradient Czochralski method

1999 ◽  
Vol 197 (4) ◽  
pp. 865-873 ◽  
Author(s):  
R.V Anantha Murthy ◽  
M Ravikumar ◽  
A Choubey ◽  
Krishan Lal ◽  
Lyudmila Kharachenko ◽  
...  
Keyword(s):  
Single Crystals ◽  
Thermal Gradient ◽  
Czochralski Method ◽  
Bismuth Germanate ◽  
Large Size

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.

Geometric characterization of additively manufactured polymer derived ceramics

2017 ◽  
Vol 18 ◽  
pp. 95-102 ◽  
Author(s):  
Jacob M. Hundley ◽  
Zak C. Eckel ◽  
Emily Schueller ◽  
Kenneth Cante ◽  
Scott M. Biesboer ◽  
...  
Keyword(s):  
Geometric Characterization ◽  
Polymer Derived Ceramics

scholarly journals Growth and Characterization of Epsomite Single Crystals Doped with KCl from low Temperature Aqueous Solutions

1970 ◽  
Vol 33 (1) ◽  
pp. 47-54 ◽  
Author(s):  
S Ferdous ◽  
J Podder
Keyword(s):  
Aqueous Solutions ◽  
Single Crystals ◽  
Optical Quality ◽  
Dielectric Studies ◽  
Slow Cooling ◽  
Large Size ◽  
Isothermal Evaporation ◽  
Academy Of Sciences

Highly transparent and well faceted large size epsomite single crystals have been grown in pure form and doped with KCl from aqueous solutions by slow cooling and isothermal evaporation method. The optical quality of the epsomite improves on doping by KCl. Mass growth rates were found to increase with doping of lower concentrations of KCl and then decreases with the higher concentration of KCl. KCl doped epsomite crystal reveals that structures are slightly distorted due to adsorption of Cl- ion into the crystal lattice. DC conductivity along the growth axis for all of the grown crystals increases with temperature in the range of 25 to 70ºC and also increases with the KCl concentration. Dielectric constant is found to be almost independent of frequency up to range of 106Hz. The dielectric studies show the suitability of these grown crystals for optoelectronic applications. DOI: 10.3329/jbas.v33i1.2949 Journal of Bangladesh Academy of Sciences, Vol. 33, No. 1, 47-54, 2009

scholarly journals A characterization of the Morse minimal set

1964 ◽  
Vol 15 (1) ◽  
pp. 70-70 ◽  
Author(s):  
W. H. Gottschalk ◽  
G. A. Hedlund
Keyword(s):  
Minimal Set
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