scholarly journals Unified Hanani–Tutte Theorem

10.37236/6663 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Radoslav Fulek ◽  
Jan Kynčl ◽  
Dömötör Pálvölgyi
Keyword(s):  
Common Generalization ◽  
Planar Drawing

We introduce a common generalization of the strong Hanani–Tutte theorem and the weak Hanani–Tutte theorem: if a graph $G$ has a drawing $D$ in the plane where every pair of independent edges crosses an even number of times, then $G$ has a planar drawing preserving the rotation of each vertex whose incident edges cross each other evenly in $D$. The theorem is implicit in the proof of the strong Hanani–Tutte theorem by Pelsmajer, Schaefer and Štefankovič. We give a new, somewhat simpler proof.

On extension of stochastic k-automata

1977 ◽  
Vol 1 (1) ◽  
pp. 231-241
Author(s):  
Sławomir Janicki ◽  
Dominik Szynal
Keyword(s):  
Probability Measure ◽  
Common Generalization ◽  
Stochastic Automata ◽  
Stochastic Automaton ◽  
Chain Type ◽  
Markov’S Chain

There are a great many research works concerning the well-known stochastic automata of Moore, Mealy, Rabin, Turing and others. Recently an automaton of Markov’s chain type has been introduced by Bartoszyński. This automaton is obtained by a generalization of Pawlak’s deterministic machine. The aim of this note is to give a concept of a stochastic automaton of Markov’s generalized chain type. The introduced automaton called a stochastic k-automaton (s.k-a.) is a common generalization of Bartoszyński’s automaton and Grodzki’s deterministic k-machine. By a stochastic k-automaton we mean an ordered triple M k = ⟨ U , a , π ⟩, k ⩾ 1, where U denotes a finite non-empty set, a is a function from Uk to [0, 1] with ∑ v ∈ U k a ( v ) = 1, and π is a function from Uk+1 to [0,1] with ∑ u ∈ U π ( v , u ) = 1 for every v ∈ U k . For all N ⩾ k we can define a probability measure PN on U N = U × U × … × U as follows: P N ( u 1 , u 2 , … , u N ) = a ( u 1 , u 2 , … , u k ) π ( u 1 , u 2 , … , u k + 1 ) π ( u 2 , u 3 , … , u k + 2 ) … π ( u N − k , u N − k + 1 , … , u N ). We deal with the problems of the shrinkage and the extension of a system of s.k-a.’s M k ( i ) = ⟨ U , a ( i ) , π ( i ) ⟩, i = 1 , 2 , … , m , m ⩾ 2. In this note there are given conditions under which an s.k-a. M k = ⟨ U , a , π ⟩ exists and the language of this automaton defined as L M = { ( u 1 , u 2 , u 3 , … ) : ∧ N ⩾ 1 P N ( u l , u 2 , … u N ) > 0 } either contains the languages of all the automata M k ( i ) , i = 1 , 2 , … , m, or this language equals the intersection of all those languages.

scholarly journals Weakly one-based geometric theories

2012 ◽  
Vol 77 (2) ◽  
pp. 392-422 ◽  
Author(s):  
Alexander Berenstein ◽  
Evgueni Vassiliev
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Geometric Theory ◽  
Common Generalization

AbstractWe study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: “weak one-basedness”, absence of type definable “almost quasidesigns”, and “generic linearity”. Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-based ω-categorical geometric theory interprets an infinite vector space over a finite field.

scholarly journals ON REGULAR GROUPS AND FIELDS

2014 ◽  
Vol 79 (3) ◽  
pp. 826-844 ◽  
Author(s):  
TOMASZ GOGACZ ◽  
KRZYSZTOF KRUPIŃSKI
Keyword(s):  
Conjugacy Class ◽  
Classical Group ◽  
Multiplicative Group ◽  
Finite Extension ◽  
Generic Type ◽  
Common Generalization ◽  
Counter Example ◽  
Minimal Groups ◽  
Unbounded Orbit

AbstractRegular groups and fields are common generalizations of minimal and quasi-minimal groups and fields, so the conjectures that minimal or quasi-minimal fields are algebraically closed have their common generalization to the conjecture that each regular field is algebraically closed. Standard arguments show that a generically stable regular field is algebraically closed. LetKbe a regular field which is not generically stable and letpbe its global generic type. We observe that ifKhas a finite extensionLof degreen, thenP(n)has unbounded orbit under the action of the multiplicative group ofL.Known to be true in the minimal context, it remains wide open whether regular, or even quasi-minimal, groups are abelian. We show that if it is not the case, then there is a counter-example with a unique nontrivial conjugacy class, and we notice that a classical group with one nontrivial conjugacy class is not quasi-minimal, because the centralizers of all elements are uncountable. Then, we construct a group of cardinality ω1with only one nontrivial conjugacy class and such that the centralizers of all nontrivial elements are countable.

scholarly journals Ideal versions of the Bolzano-Weierstrass property

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 2963-2973
Author(s):  
Jiakui Yu ◽  
Shuguo Zhang
Keyword(s):  
Common Generalization

Let I, J be ideals on ?, we say that a space X has (I,J)-BW property if every sequence in X contains a J-converging subsequence indexed by an I-positive set. This is a common generalization ofBWlike properties types. By modifying some classic notions, we obtain some characterizations of (I,J)-BW property.

scholarly journals Fusions of Description Logics and Abstract Description Systems

2002 ◽  
Vol 16 ◽  
pp. 1-58 ◽  
Author(s):  
F. Baader ◽  
C. Lutz ◽  
H. Sturm ◽  
F. Wolter
Keyword(s):  
Large Class ◽  
Description Logics ◽  
General Setting ◽  
Modal Logics ◽  
Common Generalization ◽  
Abstract Description ◽  
Normal Modal Logics ◽  
Combining Logics

Fusions are a simple way of combining logics. For normal modal logics, fusions have been investigated in detail. In particular, it is known that, under certain conditions, decidability transfers from the component logics to their fusion. Though description logics are closely related to modal logics, they are not necessarily normal. In addition, ABox reasoning in description logics is not covered by the results from modal logics. In this paper, we extend the decidability transfer results from normal modal logics to a large class of description logics. To cover different description logics in a uniform way, we introduce abstract description systems, which can be seen as a common generalization of description and modal logics, and show the transfer results in this general setting.

Gorenstein flat modules relative to injectively resolving subcategories

2021 ◽  
pp. 2250099
Author(s):  
Zenghui Gao ◽  
Wan Wu
Keyword(s):  
Homological Dimension ◽  
Common Generalization ◽  
Flat Modules ◽  
Resolving Subcategory ◽  
Homological Dimensions ◽  
Gorenstein Flat

Let [Formula: see text] be an injectively resolving subcategory of left [Formula: see text]-modules. We introduce and study [Formula: see text]-Gorenstein flat modules as a common generalization of some known modules such as Gorenstein flat modules (Enochs, Jenda and Torrecillas, 1993), Gorenstein AC-flat modules (Bravo, Estrada and Iacob, 2018). Then we define a resolution dimension relative to the [Formula: see text]-Gorensteinflat modules, investigate the properties of the homological dimension and unify some important properties possessed by some known homological dimensions. In addition, stability of the category of [Formula: see text]-Gorensteinflat modules is discussed, and some known results are obtained as applications.

scholarly journals How pigeons couple three-dimensional elbow and wrist motion to morph their wings

2017 ◽  
Vol 14 (133) ◽  
pp. 20170224 ◽  
Author(s):  
Amanda K. Stowers ◽  
Laura Y. Matloff ◽  
David Lentink
Keyword(s):  
Three Dimensional ◽  
Columba Livia ◽  
Micro Computed Tomography ◽  
Wrist Motion ◽  
Motion Data ◽  
Morphing Wings ◽  
Computed Tomography Scans ◽  
Motion Paths ◽  
Wing Morphing ◽  
Planar Drawing

Birds change the shape and area of their wings to an exceptional degree, surpassing insects, bats and aircraft in their ability to morph their wings for a variety of tasks. This morphing is governed by a musculoskeletal system, which couples elbow and wrist motion. Since the discovery of this effect in 1839, the planar ‘drawing parallels’ mechanism has been used to explain the coupling. Remarkably, this mechanism has never been corroborated from quantitative motion data. Therefore, we measured how the wing skeleton of a pigeon ( Columba livia ) moves during morphing. Despite earlier planar assumptions, we found that the skeletal motion paths are highly three-dimensional and do not lie in the anatomical plane, ruling out the ‘drawing parallels’ mechanism. Furthermore, micro-computed tomography scans in seven consecutive poses show how the two wrist bones contribute to morphing, particularly the sliding ulnare. From these data, we infer the joint types for all six bones that form the wing morphing mechanism and corroborate the most parsimonious mechanism based on least-squares error minimization. Remarkably, the algorithm shows that all optimal four-bar mechanisms either lock, are unable to track the highly three-dimensional bone motion paths, or require the radius and ulna to cross for accuracy, which is anatomically unrealistic. In contrast, the algorithm finds that a six-bar mechanism recreates the measured motion accurately with a parallel radius and ulna and a sliding ulnare. This revises our mechanistic understanding of how birds morph their wings, and offers quantitative inspiration for engineering morphing wings.

scholarly journals AFFINE ACTIONS ON NON-ARCHIMEDEAN TREES

2013 ◽  
Vol 23 (02) ◽  
pp. 217-253 ◽  
Author(s):  
SHANE O. ROURKE
Keyword(s):  
Abelian Group ◽  
Heisenberg Group ◽  
Isometric Action ◽  
Length Functions ◽  
Common Generalization ◽  
New Class ◽  
Discrete Heisenberg Group ◽  
Affine Action ◽  
Affine Actions

We initiate the study of affine actions of groups on Λ-trees for a general ordered abelian group Λ; these are actions by dilations rather than isometries. This gives a common generalization of isometric action on a Λ-tree, and affine action on an ℝ-tree as studied by Liousse. The duality between based length functions and actions on Λ-trees is generalized to this setting. We are led to consider a new class of groups: those that admit a free affine action on a Λ-tree for some Λ. Examples of such groups are presented, including soluble Baumslag–Solitar groups and the discrete Heisenberg group.

A common generalization of curvature homogeneity theories

2020 ◽  
Vol 111 (1) ◽  
Author(s):  
Corey Dunn ◽  
Alexandro Luna ◽  
Sammy Sbiti
Keyword(s):  
Common Generalization ◽  
Curvature Homogeneity
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