scholarly journals An Energy Bound in the Affine Group

2020 ◽  
Author(s):  
Giorgis Petridis ◽  
Oliver Roche-Newton ◽  
Misha Rudnev ◽  
Audie Warren
Keyword(s):  
Affine Group ◽  
Affine Transformations ◽  
Constant Multiple ◽  
General Field ◽  
Finite Set ◽  
Group A ◽  
Point Set ◽  
Energy Bound ◽  
Line Meeting ◽  
Plane Point

Abstract We prove a nontrivial energy bound for a finite set of affine transformations over a general field and discuss a number of implications. These include new bounds on growth in the affine group, a quantitative version of a theorem by Elekes about rich lines in grids. We also give a positive answer to a question of Yufei Zhao that for a plane point set $P$ for which no line contains a positive proportion of points from $P$, there may be at most one line, meeting the set of lines defined by $P$ in at most a constant multiple of $|P|$ points.

A note on minimal coverings of groups by subgroups

2001 ◽  
Vol 71 (2) ◽  
pp. 159-168 ◽  
Author(s):  
R. A. Bryce ◽  
L. Serena
Keyword(s):  
Maximal Subgroups ◽  
Soluble Groups ◽  
Finite Set ◽  
Group A ◽  
Abelian Subgroups

AbstractA cover for a group is a finite set of subgroups whose union is the whole group. A cover is minimal if its cardinality is minimal. Minimal covers of finite soluble groups are categorised; in particular all but at most one of their members are maximal subgroups. A characterisation is given of groups with minimal covers consisting of abelian subgroups.

scholarly journals Rascal finite sets

2004 ◽  
Vol 35 (4) ◽  
pp. 351-358
Author(s):  
Pier Luigi Papini
Keyword(s):  
Hilbert Space ◽  
Dimensional Space ◽  
Infinite Dimensional Space ◽  
Finite Sets ◽  
Infinite Dimensional ◽  
Small Cardinality ◽  
Finite Set ◽  
Point Set

In this paper we consider finite sets in a normed, infinite dimensional space. First, we study the following problem: given a finite set $F$, does there exist a sphere containing $F$ on its surface? We indicate some results and we collect some examples concerning this problem, also for sets of small cardinality. Then we give an example of a three-point set, in a Hilbert space, without incenter.

Point Set Registration With Similarity and Affine Transformations Based on Bidirectional KMPE Loss

2019 ◽  
pp. 1-12 ◽  
Author(s):  
Yang Yang ◽  
Dandan Fan ◽  
Shaoyi Du ◽  
Muyi Wang ◽  
Badong Chen ◽  
...  
Keyword(s):  
Affine Transformations ◽  
Point Set Registration ◽  
Point Set

scholarly journals Axiomatic Characterizations of a Proportional Influence Measure for Sequential Projects with Imperfect Reliability

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 247
Author(s):  
Andries van van Beek ◽  
Peter Borm ◽  
Marieke Quant
Keyword(s):  
Finite Number ◽  
Task Group ◽  
Influence Measure ◽  
Axiomatic Characterizations ◽  
Finite Set ◽  
Group A ◽  
The Relationship

We define and axiomatically characterize a new proportional influence measure for sequential projects with imperfect reliability. We consider a model in which a finite set of players aims to complete a project, consisting of a finite number of tasks, which can only be carried out by certain specific players. Moreover, we assume the players to be imperfectly reliable, i.e., players are not guaranteed to carry out a task successfully. To determine which players are most important for the completion of a project, we use a proportional influence measure. This paper provides two characterizations of this influence measure. The most prominent property in the first characterization is task decomposability. This property describes the relationship between the influence measure of a project and the measures of influence one would obtain if one divides the tasks of the project over multiple independent smaller projects. Invariance under replacement is the most prominent property of the second characterization. If, in a certain task group, a specific player is replaced by a new player who was not in the original player set, this property states that this should have no effect on the allocated measure of influence of any other original player.

scholarly journals Partial associativity and rough approximate groups

2020 ◽  
Vol 30 (6) ◽  
pp. 1583-1647
Author(s):  
W. T. Gowers ◽  
J. Long
Keyword(s):  
Recent Result ◽  
Binary Operation ◽  
Multiplication Table ◽  
Additive Combinatorics ◽  
Positive Proportion ◽  
Finite Set ◽  
Group A ◽  
Metric Approximation ◽  
Metric Group

AbstractSuppose that a binary operation $$\circ $$ ∘ on a finite set X is injective in each variable separately and also associative. It is easy to prove that $$(X,\circ )$$ ( X , ∘ ) must be a group. In this paper we examine what happens if one knows only that a positive proportion of the triples $$(x,y,z)\in X^3$$ ( x , y , z ) ∈ X 3 satisfy the equation $$x\circ (y\circ z)=(x\circ y)\circ z$$ x ∘ ( y ∘ z ) = ( x ∘ y ) ∘ z . Other results in additive combinatorics would lead one to expect that there must be an underlying ‘group-like’ structure that is responsible for the large number of associative triples. We prove that this is indeed the case: there must be a proportional-sized subset of the multiplication table that approximately agrees with part of the multiplication table of a metric group. A recent result of Green shows that this metric approximation is necessary: it is not always possible to obtain a proportional-sized subset that agrees with part of the multiplication table of a group.

On time changes in a digraph

1965 ◽  
Vol 2 (01) ◽  
pp. 101-118
Author(s):  
T. N. Bhargava
Keyword(s):  
Communication Networks ◽  
Incidence Matrix ◽  
Specific Activity ◽  
Evolutionary Process ◽  
The Other ◽  
Finite Set ◽  
Group A ◽  
Time Changes ◽  
Set Of Points ◽  
Ordered Pairs

The object of this paper is to present a probabilistic model for analyzing changes through time in a binary dyadic relation on a finite set of points. The total relation on the set takes the form of an aggregate of directed binary dyadic relations between ordered pairs of points belonging to the set; equivalently, the total relation on the set can be represented by means of a digraph or an incidence matrix isomorphic with the total relation. Such a relation, changing in its structure as time proceeds, is a reasonable mathematical model, for example, for the evolution of inter-relationships of the members of a social or any other group. A group of this kind is organized for a specific activity involving some sort of “communication” from one member to the other, and may be observed at successive discrete points in time generating statistics on the evolutionary process. (For a detailed treatment, see [3].) As a matter of fact, under suitable assumptions, the model presented here has potentialities for application in those situations which can be represented mathematically in terms of a finite set of points and an all-or-none relationship between ordered pairs of these points. Some of the other examples are communication networks, ecology, animal sociology, and management sciences (see [5]).

scholarly journals COMPUTING THE DIAMETER OF A POINT SET

2002 ◽  
Vol 12 (06) ◽  
pp. 489-509 ◽  
Author(s):  
GRÉGOIRE MALANDAIN ◽  
JEAN-DANIEL BOISSONNAT
Keyword(s):  
Experimental Study ◽  
Simple Algorithm ◽  
Hybrid Algorithms ◽  
Maximum Distance ◽  
Worst Case ◽  
Recent Approach ◽  
Finite Set ◽  
Point Set ◽  
Set Of Points

Given a finite set of points [Formula: see text] in ℝd, the diameter of [Formula: see text] is defined as the maximum distance between two points of [Formula: see text]. We propose a very simple algorithm to compute the diameter of a finite set of points. Although the algorithm is not worst-case optimal, an extensive experimental study has shown that it is extremely fast for a large variety of point distributions. In addition, we propose a comparison with the recent approach of Har-Peled5 and derive hybrid algorithms to combine advantages of both approaches.

On time changes in a digraph

1965 ◽  
Vol 2 (1) ◽  
pp. 101-118 ◽  
Author(s):  
T. N. Bhargava
Keyword(s):  
Communication Networks ◽  
Incidence Matrix ◽  
Specific Activity ◽  
Evolutionary Process ◽  
The Other ◽  
Finite Set ◽  
Group A ◽  
Time Changes ◽  
Set Of Points ◽  
Ordered Pairs

The object of this paper is to present a probabilistic model for analyzing changes through time in a binary dyadic relation on a finite set of points. The total relation on the set takes the form of an aggregate of directed binary dyadic relations between ordered pairs of points belonging to the set; equivalently, the total relation on the set can be represented by means of a digraph or an incidence matrix isomorphic with the total relation. Such a relation, changing in its structure as time proceeds, is a reasonable mathematical model, for example, for the evolution of inter-relationships of the members of a social or any other group. A group of this kind is organized for a specific activity involving some sort of “communication” from one member to the other, and may be observed at successive discrete points in time generating statistics on the evolutionary process. (For a detailed treatment, see [3].) As a matter of fact, under suitable assumptions, the model presented here has potentialities for application in those situations which can be represented mathematically in terms of a finite set of points and an all-or-none relationship between ordered pairs of these points. Some of the other examples are communication networks, ecology, animal sociology, and management sciences (see [5]).

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