scholarly journals Growth functions for some uniformly amenable groups

2017 ◽  
Vol 15 (1) ◽  
pp. 502-507
Author(s):  
Janusz Dronka ◽  
Bronislaw Wajnryb ◽  
Paweł Witowicz ◽  
Kamil Orzechowski
Keyword(s):  
Discrete Group ◽  
Amenable Group ◽  
Optimal Growth ◽  
Growth Function ◽  
Constructive Proof ◽  
Amenable Groups ◽  
Growth Functions

Abstract We present a simple constructive proof of the fact that every abelian discrete group is uniformly amenable. We improve the growth function obtained earlier and find the optimal growth function in a particular case. We also compute a growth function for some non-abelian uniformly amenable group.

On a construction of unitary cocycles and the representation theory of amenable groups

1983 ◽  
Vol 3 (1) ◽  
pp. 129-133 ◽  
Author(s):  
Colin E. Sutherland
Keyword(s):  
Representation Theory ◽  
Probability Space ◽  
Amenable Group ◽  
Unitary Representations ◽  
Intertwining Operators ◽  
Type I ◽  
Amenable Groups

AbstractIf K is a countable amenable group acting freely and ergodically on a probability space (Γ, μ), and G is an arbitrary countable amenable group, we construct an injection of the space of unitary representations of G into the space of unitary 1-cocyles for K on (Γ, μ); this injection preserves intertwining operators. We apply this to show that for many of the standard non-type-I amenable groups H, the representation theory of H contains that of every countable amenable group.

A description of the growth of sheep

1998 ◽  
Vol 1998 ◽  
pp. 47-47
Author(s):  
R.M. Lewis ◽  
G.C. Emmans ◽  
G. Simm ◽  
W.S. Dingwall ◽  
J. FitzSimons
Keyword(s):  
Growth Curves ◽  
Growth Function ◽  
Single Equation ◽  
Rate Parameter ◽  
Growth Functions ◽  
Gompertz Equation ◽  
Lumped Parameter ◽  
Highly Correlated ◽  
Mature Size

The idea that an animal of a given kind has, and grows to, a final or mature size is a useful one and several equations have been proposed that describe such growth to maturity (Winsor, 1932; Parks, 1982; Taylor, 1982). The Gompertz is one of these growth functions and describes in a comparatively simple, single equation the sigmoidal pattern of growth. It has 3 parameters, only 2 of which are important - mature size A and the rate parameter B. Estimates of A and B, however, are highly correlated. Considering A and B as a lumped parameter (AB) may overcome this problem. A Gompertz, or any other, growth function is not expected to describe all growth curves. When the environment (e.g., feed, housing) is non-limiting, it may provide a useful and succinct description of growth. The objectives of this study were to examine: (i) if the Gompertz equation adequately describes the growth of two genotypes of sheep under conditions designed to be non-limiting; and, (ii) if the lumped parameter AB has more desirable properties for estimation than A and B separately.

scholarly journals One-dimensional game of life and its growth functions

1992 ◽  
Vol 15 (3) ◽  
pp. 499-508
Author(s):  
Mohammad H. Ahmadi
Keyword(s):  
Power Series ◽  
Rational Function ◽  
Formal Power Series ◽  
Infinite Product ◽  
Growth Function ◽  
The Other ◽  
Arbitrary Integer ◽  
One Dimensional ◽  
Growth Functions ◽  
Formal Power

We start with finitely many1's and possibly some0's in between. Then each entry in the other rows is obtained from the Base2sum of the two numbers diagonally above it in the preceding row. We may formulate the game as follows: Defined1,jrecursively for1, a non-negative integer, andjan arbitrary integer by the rules:d0,j={1     for   j=0,k         (I)0   or   1   for   0<j<kd0,j=0   for   j<0   or   j>k              (II)di+1,j=di,j+1(mod2)   for   i≥0.      (III)Now, if we interpret the number of1's in rowias the coefficientaiof a formal power series, then we obtain a growth function,f(x)=∑i=0∞aixi. It is interesting that there are cases for which this growth function factors into an infinite product of polynomials. Furthermore, we shall show that this power series never represents a rational function.

Weak Convergence Is Not Strong Convergence For Amenable Groups

2001 ◽  
Vol 44 (2) ◽  
pp. 231-241 ◽  
Author(s):  
Joseph M. Rosenblatt ◽  
George A. Willis
Keyword(s):  
Weak Convergence ◽  
Strong Convergence ◽  
Cayley Graphs ◽  
Linear Equations ◽  
Amenable Group ◽  
Amenable Groups

AbstractLet G be an infinite discrete amenable group or a non-discrete amenable group. It is shown how to construct a net (fα) of positive, normalized functions in L1(G) such that the net converges weak* to invariance but does not converge strongly to invariance. The solution of certain linear equations determined by colorings of the Cayley graphs of the group are central to this construction.

MONOID GROWTH FUNCTIONS FOR BRAID GROUPS

1991 ◽  
Vol 01 (02) ◽  
pp. 201-205 ◽  
Author(s):  
MARCUS BRAZIL
Keyword(s):  
Braid Group ◽  
Growth Function ◽  
Braid Groups ◽  
Growth Functions

It is shown that for all n, the braid group on n strings, Bn, has rational growth with respect to a certain set of elements of the group which generate it as a monoid. In particular, the precise growth function for B4 is calculated.

Non-Bernoulli systems with completely positive entropy

2008 ◽  
Vol 28 (1) ◽  
pp. 87-124 ◽  
Author(s):  
A. H. DOOLEY ◽  
V. YA. GOLODETS ◽  
D. J. RUDOLPH ◽  
S. D. SINEL’SHCHIKOV
Keyword(s):  
Infinite Order ◽  
Amenable Group ◽  
Positive Entropy ◽  
Completely Positive ◽  
New Approach ◽  
Amenable Groups ◽  
Uncountable Family ◽  
Cutting And Stacking ◽  
Bernoulli Systems

AbstractA new approach to actions of countable amenable groups with completely positive entropy (cpe), allowing one to answer some basic questions in this field, was recently developed. The question of the existence of cpe actions which are not Bernoulli was raised. In this paper, we prove that every countable amenable groupG, which contains an element of infinite order, has non-Bernoulli cpe actions. In fact we can produce, for any$h \in (0, \infty ]$, an uncountable family of cpe actions of entropyh, which are pairwise automorphically non-isomorphic. These actions are given by a construction which we call co-induction. This construction is related to, but different from the standard induced action. We study the entropic properties of co-induction, proving that ifαGis co-induced from an actionαΓof a subgroup Γ, thenh(αG)=h(αΓ). We also prove that ifαΓis a non-Bernoulli cpe action of Γ, thenαGis also non-Bernoulli and cpe. Hence the problem of finding an uncountable family of pairwise non-isomorphic cpe actions of the same entropy is reduced to one of finding an uncountable family of non-Bernoulli cpe actions of$\mathbb Z$, which pairwise satisfy a property we call ‘uniform somewhat disjointness’. We construct such a family using refinements of the classical cutting and stacking methods.

scholarly journals Homological dimension of elementary amenable groups

2020 ◽  
Vol 2020 (766) ◽  
pp. 45-60
Author(s):  
Peter H. Kropholler ◽  
Conchita Martínez-Pérez
Keyword(s):  
Simple Group ◽  
Characteristic Zero ◽  
Amenable Group ◽  
Solvable Groups ◽  
Complete Information ◽  
Homological Dimension ◽  
Important Paper ◽  
Amenable Groups ◽  
Coefficient Ring ◽  
Special Case

AbstractIn this paper we prove that the homological dimension of an elementary amenable group over an arbitrary commutative coefficient ring is either infinite or equal to the Hirsch length of the group. Established theory gives simple group theoretical criteria for finiteness of homological dimension and so we can infer complete information about this invariant for elementary amenable groups. Stammbach proved the special case of solvable groups over coefficient fields of characteristic zero in an important paper dating from 1970.

Comparing Polynomial and von Bertalanffy Growth Functions for Fitting Tag–Recapture Data

1994 ◽  
Vol 51 (8) ◽  
pp. 1689-1691 ◽  
Author(s):  
William S. Hearn ◽  
George M. Leigh
Keyword(s):  
Growth Function ◽  
Growth Increment ◽  
Bias Reduction ◽  
Release Time ◽  
Von Bertalanffy ◽  
Growth Functions ◽  
Straight Line ◽  
Von Bertalanffy Growth Function ◽  
Multiple Measurements ◽  
The One

The properties of polynomial and von Bertalanffy growth functions are compared for analysing data from tag–recapture experiments in which fish are recaptured once. For the quadratic and von Bertalanffy growth functions, explicit formulae are obtained for the expected growth increment in terms of length-at-release, time-at-liberty, and the function parameters. If the least-squares fitting technique is used the von Bertalanffy function fits tag–recapture data with no more bias (probably less) than any other growth function, including polynomial growth functions. A bias-reduction technique for fitting the von Bertalanffy growth function to tag–recapture data is not applicable to other growth functions. We conclude that, apart from the straight line, the von Bertalanffy growth function is the one with the most desirable mathematical and statistical properties for fitting to tag–recapture data. The matter of the function that best characterises the way a specific fish species grows can be adequately addressed only by analyses of multiple measurements of individual fish.

Tilings of amenable groups

2019 ◽  
Vol 2019 (747) ◽  
pp. 277-298 ◽  
Author(s):  
Tomasz Downarowicz ◽  
Dawid Huczek ◽  
Guohua Zhang
Keyword(s):  
Fixed Points ◽  
Topological Entropy ◽  
Finite Subset ◽  
Dimensional Space ◽  
Amenable Group ◽  
Free Action ◽  
Amenable Groups ◽  
Entropy Zero

Abstract We prove that for any infinite countable amenable group G, any {\varepsilon>0} and any finite subset {K\subset G} , there exists a tiling (partition of G into finite “tiles” using only finitely many “shapes”), where all the tiles are {(K,\varepsilon)} -invariant. Moreover, our tiling has topological entropy zero (i.e., subexponential complexity of patterns). As an application, we construct a free action of G (in the sense that the mappings, associated to elements of G other than the unit, have no fixed points) on a zero-dimensional space, such that the topological entropy of this action is zero.

scholarly journals HIGH DENSITY PIECEWISE SYNDETICITY OF PRODUCT SETS IN AMENABLE GROUPS

2016 ◽  
Vol 81 (4) ◽  
pp. 1555-1562 ◽  
Author(s):  
MAURO DI NASSO ◽  
ISAAC GOLDBRING ◽  
RENLING JIN ◽  
STEVEN LETH ◽  
MARTINO LUPINI ◽  
...  
Keyword(s):  
Lower Bound ◽  
Finite Subset ◽  
Amenable Group ◽  
High Density ◽  
Amenable Groups ◽  
Quantitative Version ◽  
Banach Density ◽  
Piecewise Syndetic ◽  
Product Sets

AbstractM. Beiglböck, V. Bergelson, and A. Fish proved that if G is a countable amenable group and A and B are subsets of G with positive Banach density, then the product set AB is piecewise syndetic. This means that there is a finite subset E of G such that EAB is thick, that is, EAB contains translates of any finite subset of G. When G = ℤ, this was first proven by R. Jin. We prove a quantitative version of the aforementioned result by providing a lower bound on the density (with respect to a Følner sequence) of the set of witnesses to the thickness of EAB. When G = ℤd, this result was first proven by the current set of authors using completely different techniques.

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