scholarly journals Linear equations over multiplicative groups, recurrences, and mixing III

2017 ◽  
Vol 38 (7) ◽  
pp. 2625-2643 ◽  
Author(s):  
H. DERKSEN ◽  
D. MASSER
Keyword(s):  
Prime Ideal ◽  
Equivalence Relation ◽  
Linear Equations ◽  
Several Variables ◽  
Natural Equivalence ◽  
Ring Of Polynomials ◽  
Multiplicative Groups

Given an algebraic $\mathbf{Z}^{d}$-action corresponding to a prime ideal of a Laurent ring of polynomials in several variables, we show how to find the smallest order $n+1$ of non-mixing. It is known that this is determined by the non-mixing sets of size $n+1$, and we show how to find these in an effective way. When the underlying characteristic is positive and $n\geq 2$, we prove that there are at most finitely many classes under a natural equivalence relation. We work out two examples, the first with five classes and the second with 134 classes.

EDGE-TRANSITIVE GRAPH AUTOMORPHISMS AND PERIODIC SURFACE HOMEOMORPHISMS

1999 ◽  
Vol 09 (09) ◽  
pp. 1803-1813 ◽  
Author(s):  
JÉRÔME E. LOS ◽  
ZBIGNIEW H. NITECKI
Keyword(s):  
Equivalence Relation ◽  
Complete List ◽  
Transitive Graph ◽  
Periodic Surface ◽  
Natural Equivalence ◽  
Surface Homeomorphisms ◽  
Edge Transitive ◽  
Automorphism Of A Graph

An automorphism of a graph is edge-transitive if it acts transitively on the set of geometric edges (components of the complement of the vertices), or equivalently, if there is no nontrivial invariant subgraph. Every such automorphism can be embedded as the restriction to an invariant spine of some orientation-preserving periodic homeomorphism of a punctured surface. We find all the edge-transitive graph automorphisms and for each, find a complete list (up to a natural equivalence relation) of the possible ways that it can be embedded in a periodic homeomorphism.

Cauchy Points of Metric Locales

1989 ◽  
Vol 41 (5) ◽  
pp. 830-854 ◽  
Author(s):  
B. Banaschewski ◽  
A. Pultr
Keyword(s):  
Equivalence Relation ◽  
Metric Spaces ◽  
Equivalence Classes ◽  
Classical Description ◽  
Natural Approach ◽  
Precise Meaning ◽  
Natural Equivalence ◽  
Cauchy Sequences

A natural approach to topology which emphasizes its geometric essence independent of the notion of points is given by the concept of frame (for instance [4], [8]). We consider this a good formalization of the intuitive perception of a space as given by the “places” of non-trivial extent with appropriate geometric relations between them. Viewed from this position, points are artefacts determined by collections of places which may in some sense by considered as collapsing or contracting; the precise meaning of the latter as well as possible notions of equivalence being largely arbitrary, one may indeed have different notions of point on the same “space”. Of course, the well-known notion of a point as a homomorphism into 2 evidently fits into this pattern by the familiar correspondence between these and the completely prime filters. For frames equipped with a diameter as considered in this paper, we introduce a natural alternative, the Cauchy points. These are the obvious counterparts, for metric locales, of equivalence classes of Cauchy sequences familiar from the classical description of completion of metric spaces: indeed they are decreasing sequences for which the diameters tend to zero, identified by a natural equivalence relation.

HYPERSTABILITY OF GENERALISED LINEAR FUNCTIONAL EQUATIONS IN SEVERAL VARIABLES

2020 ◽  
Vol 102 (2) ◽  
pp. 293-302
Author(s):  
THEERAYOOT PHOCHAI ◽  
SATIT SAEJUNG
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Functional Equations ◽  
Linear Functional ◽  
Linear Equations ◽  
General Linear ◽  
Several Variables ◽  
Linear Functional Equations

Zhang [‘On hyperstability of generalised linear functional equations in several variables’, Bull. Aust. Math. Soc.92 (2015), 259–267] proved a hyperstability result for generalised linear functional equations in several variables by using Brzdęk’s fixed point theorem. We complete and extend Zhang’s result. We illustrate our results for general linear equations in two variables and Fréchet equations.

scholarly journals FINITARY REDUCIBILITY ON EQUIVALENCE RELATIONS

2016 ◽  
Vol 81 (4) ◽  
pp. 1225-1254 ◽  
Author(s):  
RUSSELL MILLER ◽  
KENG MENG NG
Keyword(s):  
Equivalence Relation ◽  
Equivalence Relations ◽  
Arithmetical Hierarchy ◽  
Natural Equivalence ◽  
Image Position ◽  
The Way

AbstractWe introduce the notion of finitary computable reducibility on equivalence relations on the domainω. This is a weakening of the usual notion of computable reducibility, and we show it to be distinct in several ways. In particular, whereas no equivalence relation can be${\rm{\Pi }}_{n + 2}^0$-complete under computable reducibility, we show that, for everyn, there does exist a natural equivalence relation which is${\rm{\Pi }}_{n + 2}^0$-complete under finitary reducibility. We also show that our hierarchy of finitary reducibilities does not collapse, and illustrate how it sharpens certain known results. Along the way, we present several new results which use computable reducibility to establish the complexity of various naturally defined equivalence relations in the arithmetical hierarchy.

The Effects of Fluid Entry Conditions on the Performance of Vapour-Jet Compressors

1978 ◽  
Vol 5 (1) ◽  
pp. 9-14 ◽  
Author(s):  
N. Galanis ◽  
G. Faucher ◽  
Nguyen Mau Phung
Keyword(s):  
Experimental Data ◽  
Numerical Solution ◽  
Linear Equations ◽  
Gas Analysis ◽  
Phase Changes ◽  
Perfect Gas ◽  
Expansion Process ◽  
One Dimensional ◽  
Several Variables ◽  
Non Linear Equations

An earlier one-dimensional frictionless perfect-gas analysis of jet ejectors is generalized to account for phase changes during the expansion process and is used to evaluate the performance of vapour-jet compressors operating with initially saturated freon-12 or n-butane. The results, obtained by numerical solution of the non-linear equations, show better agreement with experimental data than the predictions based on perfect-gas relations and indicate that the performance depends upon several variables which are not accounted for by the perfect-gas model.

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