scholarly journals (Non)-escape of mass and equidistribution for horospherical actions on trees

2021 ◽  
Author(s):  
Corina Ciobotaru ◽  
Vladimir Finkelshtein ◽  
Cagri Sert
Keyword(s):  
Error Term ◽  
Finite Lattice ◽  
The Other ◽  
Point Counting ◽  
Counting Problem ◽  
Uniform Distributions ◽  
Orbit Closures ◽  
Geometrically Finite ◽  
Finite Lattices ◽  
Dense Orbits

AbstractLet G be a large group acting on a biregular tree T and $$\Gamma \le G$$ Γ ≤ G a geometrically finite lattice. In an earlier work, the authors classified orbit closures of the action of the horospherical subgroups on $$G/\Gamma $$ G / Γ . In this article we show that there is no escape of mass and use this to prove that, in fact, dense orbits equidistribute to the Haar measure on $$G/\Gamma $$ G / Γ . On the other hand, we show that new dynamical phenomena for horospherical actions appear on quotients by non-geometrically finite lattices: we give examples of non-geometrically finite lattices where an escape of mass phenomenon occurs and where the orbital averages along a Følner sequence do not converge. In the last part, as a by-product of our methods, we show that projections to $$\Gamma \backslash T$$ Γ \ T of the uniform distributions on large spheres in the tree T converge to a natural probability measure on $$\Gamma \backslash T$$ Γ \ T . Finally, we apply this equidistribution result to a lattice point counting problem to obtain counting asymptotics with exponential error term.

scholarly journals Maximal sublattices and Frattini sublattices of bounded lattices

1997 ◽  
Vol 63 (1) ◽  
pp. 110-127 ◽  
Author(s):  
M. E. Adams ◽  
Ralph Freese ◽  
J. B. Nation ◽  
Jürg Schmid
Keyword(s):  
Positive Integer ◽  
Finite Lattice ◽  
The Other ◽  
Bounded Lattice ◽  
Other Hand ◽  
Finite Lattices

AbstractWe investigate the number and size of the maximal sublattices of a finite lattice. For any positive integer k, there is a finite lattice L with more that ]L]k sublattices. On the other hand, there are arbitrary large finite lattices which contain a maximal sublattice with only 14 elements. It is shown that every bounded lattice is isomorphic to the Frattini sublattice (the intersection of all maximal sublattices) of a finite bounded lattice.

Matchings and Radon transforms in lattices II. Concordant sets

1987 ◽  
Vol 101 (2) ◽  
pp. 221-231 ◽  
Author(s):  
Joseph P. S. Kung
Keyword(s):  
Incidence Matrix ◽  
Finite Lattice ◽  
Möbius Function ◽  
Radon Transforms ◽  
Modular Lattices ◽  
Covering Theorem ◽  
Mobius Function ◽  
Finite Lattices ◽  
Semimodular Lattices

AbstractLet and ℳ be subsets of a finite lattice L. is said to be concordant with ℳ if, for every element x in L, either x is in ℳ or there exists an element x+ such that (CS1) the Möbius function μ(x, x+) ≠ 0 and (CS2) for every element j in , x ∨ j ≠ x+. We prove that if is concordant with ℳ, then the incidence matrix I(ℳ | ) has maximum possible rank ||, and hence there exists an injection σ: → ℳ such that σ(j) ≥ j for all j in . Using this, we derive several rank and covering inequalities in finite lattices. Among the results are generalizations of the Dowling-Wilson inequalities and Dilworth's covering theorem to semimodular lattices, and a refinement of Dilworth's covering theorem for modular lattices.

Sublattices and Initial Segments of the Degrees of Unsolvability

1970 ◽  
Vol 22 (3) ◽  
pp. 569-581 ◽  
Author(s):  
S. K. Thomason
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Lattice Theory ◽  
Initial Segment ◽  
Finite Lattice ◽  
Equivalence Relations ◽  
Finite Lattices ◽  
Image Position ◽  
Degrees Of Unsolvability

In this paper we shall prove that every finite lattice is isomorphic to a sublattice of the degrees of unsolvability, and that every one of a certain class of finite lattices is isomorphic to an initial segment of degrees.Acknowledgment. I am grateful to Ralph McKenzie for his assistance in matters of lattice theory.1. Representation of lattices. The equivalence lattice of the set S consists of all equivalence relations on S, ordered by setting θ ≦ θ’ if for all a and b in S, a θ b ⇒ a θ’ b. The least upper bound and greatest lower bound in are given by the ⋃ and ⋂ operations:

scholarly journals On strict sub-Gaussianity, optimal proxy variance and symmetry for bounded random variables

2020 ◽  
Vol 24 ◽  
pp. 39-55
Author(s):  
Julyan Arbel ◽  
Olivier Marchal ◽  
Hien D. Nguyen
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Random Variables ◽  
The Other ◽  
Bounded Support ◽  
Uniform Distributions ◽  
Standard Variance ◽  
Per Se ◽  
The One ◽  
The Relationship

We investigate the sub-Gaussian property for almost surely bounded random variables. If sub-Gaussianity per se is de facto ensured by the bounded support of said random variables, then exciting research avenues remain open. Among these questions is how to characterize the optimal sub-Gaussian proxy variance? Another question is how to characterize strict sub-Gaussianity, defined by a proxy variance equal to the (standard) variance? We address the questions in proposing conditions based on the study of functions variations. A particular focus is given to the relationship between strict sub-Gaussianity and symmetry of the distribution. In particular, we demonstrate that symmetry is neither sufficient nor necessary for strict sub-Gaussianity. In contrast, simple necessary conditions on the one hand, and simple sufficient conditions on the other hand, for strict sub-Gaussianity are provided. These results are illustrated via various applications to a number of bounded random variables, including Bernoulli, beta, binomial, Kumaraswamy, triangular, and uniform distributions.

On finite lattices of degrees of constructibility of reals

1976 ◽  
Vol 41 (2) ◽  
pp. 313-322 ◽  
Author(s):  
Zofia Adamowicz
Keyword(s):  
Standard Model ◽  
Finite Lattice ◽  
Partial Ordering ◽  
Turing Degrees ◽  
Upper Semilattice ◽  
Finite Lattices ◽  
Definition Of ◽  
The Universe

Theorem. Assume that there exists a standard model of ZFC + V = L. Then there is a model of ZFC in which the partial ordering of the degrees of constructibility of reals is isomorphic with a given finite lattice.The proof of the theorem uses forcing. The definition of the forcing conditions and the proofs of some of the lemmas are connected with Lerman's paper on initial segments of the upper semilattice of the Turing degrees [2]. As an auxiliary notion we shall introduce the notion of a sequential representation of a lattice, which slightly differs from Lerman's representation.Let K be a given finite lattice. Assume that the universe of K is an integer l. Let ≤K be the ordering in K. A sequential representation of K is a sequence Ui ⊆ Ui+1 of finite subsets of ωi such that the following holds:(1) For any s, s′ Є Ui, i Є ω, k, m Є l, k ≤Km & s(m) = s′(m) → s(k) = s′(k).(2) For any s Є Ui, i Є ω, s(0) = 0 where 0 is the least element of K.(3) For any s, s′ Є i Є ω, k,j Є l, if k y Kj = m and s(k) = s′(k) & s(j) = s′(j) → s(m) = s′(m), where vK denotes the join in K.

Lattice orbit distribution on ℝ2

2009 ◽  
Vol 30 (4) ◽  
pp. 1201-1214 ◽  
Author(s):  
ARNALDO NOGUEIRA
Keyword(s):  
Ergodic Theory ◽  
Diophantine Approximation ◽  
Error Term ◽  
Absolute Error ◽  
Compact Set ◽  
Linear Action ◽  
Counting Problem ◽  
Asymptotical Behaviour ◽  
Diophantine Property ◽  
The Absolute

AbstractWe study the distribution on ℝ2 of the orbit of a vector under the linear action of SL(2,ℤ). Let Ω⊂ℝ2 be a compact set and x∈ℝ2. Let N(k,x) be the number of matrices γ∈SL(2,ℤ) such that γ(x)∈Ω and ‖γ‖≤k, k=1,2,…. If Ω is a square, we prove the existence of an absolute error term for N(k,x), as k→∞, for almost every x, which depends on the Diophantine property of the ratio of the coordinates of x. Our approach translates the question into a Diophantine approximation counting problem which provides the absolute error term. The asymptotical behaviour of N(k,x) is also obtained using ergodic theory.

Improved finite-lattice method for estimating the zero-temperature properties of two-dimensional lattice models

1996 ◽  
Vol 74 (1-2) ◽  
pp. 54-64 ◽  
Author(s):  
D. D. Betts ◽  
S. Masui ◽  
N. Vats ◽  
G. E. Stewart
Keyword(s):  
Spontaneous Magnetization ◽  
Finite Lattice ◽  
Quantum Spin ◽  
Square Lattice ◽  
Two Dimensional ◽  
Zero Temperature ◽  
Quantum Spin Systems ◽  
Lattice Method ◽  
Dimensional Lattice ◽  
Finite Lattices

The well-known finite-lattice method for the calculation of the properties of quantum spin systems on a two-dimensional lattice at zero temperature was introduced in 1978. The method has now been greatly improved for the square lattice by including finite lattices based on parallelogram tiles as well as the familiar finite lattices based on square tiles. Dozens of these new finite lattices have been tested and graded using the [Formula: see text] ferromagnet. In the process new and improved estimates have been obtained for the XY model's ground-state energy per spin, ε0 = −0.549 36(30) and spontaneous magnetization per spin, m = 0.4349(10). Other properties such as near-neighbour, zero-temperature spin–spin correlations, which appear not to have been calculated previously, have been estimated to high precision. Applications of the improved finite-lattice method to other models can readily be carried out.

On finite lattices of degrees of constructibility

1977 ◽  
Vol 42 (3) ◽  
pp. 349-371 ◽  
Author(s):  
Zofia Adamowicz
Keyword(s):  
Finite Lattice ◽  
Partial Ordering ◽  
Generic Model ◽  
Crucial Point ◽  
Generic Models ◽  
Ordinal Numbers ◽  
Finite Lattices ◽  
Definition Of ◽  
Two Stages ◽  
The Given

We shall prove the following theorem:Theorem. For any finite lattice there is a model of ZF in which the partial ordering of the degrees of constructibility is isomorphic with the given lattice.Let M be a standard countable model of ZF satisfying V = L. Let K be the given finite lattice. We shall extend M by forcing.The paper is divided into two parts. The first part concerns the definition of the set of forcing conditions and some properties of this set expressible without the use of generic filters.We define first a representation of a lattice and then the set of conditions. In Lemmas 1, 2 we show that there are some canonical isomorphisms between some conditions and that a single condition has some canonical automorphisms.In Lemma 3 and Definition 7 we show some methods of defining conditions. We shall use those methods in the second part to define certain conditions with special properties.Lemma 4 gives a connection between the sets P and Pk (see Definitions 4 and 5). It is next employed in the second part in Lemma 10 in an essential way.Indeed, Lemma 10 is necessary for Lemma 13, which is the crucial point of the whole construction. Lemma 5 is also employed in Lemma 13 (exactly in its Corollary).The second part of the paper is devoted to the examination of the structure of degrees of constructibility in a generic model. First, we show that degrees of some “sections” of a generic real (Definition 9) form a lattice isomorphic with K. Secondly, we show that there are no other degrees in the generic model; this is the most difficult property to obtain by forcing. We prove, in two stages, that it holds in our generic models. We first show, by using special properties of the forcing conditions, that sets of ordinal numbers have no other degrees. Then we show that the degrees of sets of ordinals already determine the degrees of other sets.

scholarly journals Expected Winning Probabilities in Sequential Truels under Uniform Distributions

2019 ◽  
Vol 8 (6) ◽  
pp. 79
Author(s):  
David R Hare ◽  
Faisal Kaleem
Keyword(s):  
The Other ◽  
Computational Approaches ◽  
Uniform Distributions

In this paper we examine the expected probabilities of survival of each of the three participants in sequential truels assuming uniform distributions for their marksmanships. We start by discussing the two most common sequential truels, one in which everyone has to attempt to eliminate someone on each turn, and the other in which the weakest marksman has the option to abstain. We conjecture that the expected winning probabilities in sequential truels cannot be calculated analytically, and so we estimate these probabilities using multiple computational approaches. We also calculate the expected winning probabilities of each of the participants for one fixed marksmanship at a time. At the end we show that as the three marksmanships approach 0, the sequential truel approaches the simultaneous truel and this explains some aspects of the sequential truel for small marksmanships.

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