Approximate transitivity of the ergodic action of the group of finite permutations of on

2018 ◽  
Vol 39 (11) ◽  
pp. 2881-2895
Author(s):  
B. MITCHELL BAKER ◽  
THIERRY GIORDANO ◽  
RADU B. MUNTEANU
Keyword(s):  
Symmetric Group ◽  
Product Space ◽  
Ergodic Action ◽  
Probability Measures ◽  
Natural Action ◽  
Bernoulli Measure ◽  
Approximate Transitivity

In this paper we show that the natural action of the symmetric group acting on the product space $\{0,1\}^{\mathbb{N}}$ endowed with a Bernoulli measure is approximately transitive. We also extend the result to a larger class of probability measures.

Mackey Continuity of Characteristic Functionals

2002 ◽  
Vol 9 (1) ◽  
pp. 83-112
Author(s):  
S. Kwapień ◽  
V. Tarieladze
Keyword(s):  
Product Space ◽  
Locally Convex Space ◽  
Inner Product ◽  
Nuclear Space ◽  
Probability Measures ◽  
Linear Kernel ◽  
Characteristic Functional ◽  
Initial Space ◽  
Radon Probability Measure ◽  
Locally Convex

Abstract Problems of the Mackey-continuity of characteristic functionals and the localization of linear kernels of Radon probability measures in locally convex spaces are investigated. First the class of spaces is described, for which the continuity takes place. Then it is shown that in a non-complete sigmacompact inner product space, as well as in a non-complete sigma-compact metizable nuclear space, there may exist a Radon probability measure having a non-continuous characteristic functional in the Mackey topology and a linear kernel not contained in the initial space. Similar problems for moment forms and higher order kernels are also touched upon. Finally, a new proof of the result due to Chr. Borell is given, which asserts that any Gaussian Radon measure on an arbitrary Hausdorff locally convex space has the Mackey-continuous characteristic functional.

scholarly journals A Class of Nonassociative Algebras

2007 ◽  
Vol 14 (02) ◽  
pp. 313-326 ◽  
Author(s):  
Michel Goze ◽  
Elisabeth Remm
Keyword(s):  
Symmetric Group ◽  
Natural Action ◽  
Nonassociative Algebras

In this paper, we classify the nonassociative algebras whose associator satisfies relations defined by a natural action of the symmetric group of degree 3.

A Comparison of Methods for Constructing Probability Measures on Infinite Product Spaces

1987 ◽  
Vol 30 (3) ◽  
pp. 282-285 ◽  
Author(s):  
Charles W. Lamb
Keyword(s):  
Probability Measure ◽  
Conditional Probability ◽  
Product Space ◽  
Infinite Product ◽  
Probability Measures ◽  
Classical Theorem ◽  
Comparison Of Methods ◽  
Product Spaces ◽  
Probability Functions ◽  
Finite Dimensional

AbstractThe construction, from a consistent family of finite dimensional probability measures, of a probability measure on a product space when the marginal measures are perfect is shown to follow from a classical theorem due to Ionescu Tulcea and known results on the existence of regular conditional probability functions.

Automorphism–invariant measures on ℵ0-categorical structures without the independence property

1996 ◽  
Vol 61 (2) ◽  
pp. 640-652
Author(s):  
Douglas E. Ensley
Keyword(s):  
Boolean Algebra ◽  
Invariant Measures ◽  
Measure Zero ◽  
Probability Measures ◽  
Independence Property ◽  
Natural Action ◽  
Zero Sets ◽  
Finitely Additive Probability ◽  
Additive Probability

AbstractWe address the classification of the possible finitely-additive probability measures on the Boolean algebra of definable subsets of M which are invariant under the natural action of Aut(M). This pursuit requires a generalization of Shelah's forking formulas [8] to “essentially measure zero” sets and an application of Myer's “rank diagram” [5] of the Boolean algebra under consideration. The classification is completed for a large class of ℵ0-categorical structures without the independence property including those which are stable.

scholarly journals THE FREQUENCY SPACE OF A FREE GROUP

2005 ◽  
Vol 15 (05n06) ◽  
pp. 939-969 ◽  
Author(s):  
ILYA KAPOVICH
Keyword(s):  
Conjugacy Class ◽  
Free Group ◽  
Outer Automorphism ◽  
Probability Measures ◽  
Natural Action ◽  
Frequency Space ◽  
Borel Probability

We analyze the structure of the frequency spaceQ(F) of a nonabelian free group F = F(a1,…,ak) consisting of all shift-invariant Borel probability measures on ∂F and construct a natural action of Out(F) on Q(F). In particular we prove that for any outer automorphism ϕ of F the conjugacy distortion spectrum of ϕ, consisting of all numbers ‖ϕ(w)‖/‖w‖, where w is a nontrivial conjugacy class, is the intersection of ℚ and a closed subinterval of ℝ with rational endpoints.

scholarly journals Ewens Sampling and Invariable Generation

2018 ◽  
Vol 27 (6) ◽  
pp. 853-891 ◽  
Author(s):  
GERANDY BRITO ◽  
CHRISTOPHER FOWLER ◽  
MATTHEW JUNGE ◽  
AVI LEVY
Keyword(s):  
Symmetric Group ◽  
Fixed Number ◽  
Probability Measures ◽  
Random Permutations ◽  
Positive Probability ◽  
Ewens Sampling Formula ◽  
Sampling Formula ◽  
Special Case ◽  
The Way

We study the number of random permutations needed to invariably generate the symmetric group Sn when the distribution of cycle counts has the strong α-logarithmic property. The canonical example is the Ewens sampling formula, for which the special case α = 1 corresponds to uniformly random permutations.For strong α-logarithmic measures and almost every α, we show that precisely ⌈(1−αlog2)−1⌉ permutations are needed to invariably generate Sn with asymptotically positive probability. A corollary is that for many other probability measures on Sn no fixed number of permutations will invariably generate Sn with positive probability. Along the way we generalize classic theorems of Erdős, Tehran, Pyber, Łuczak and Bovey to permutations obtained from the Ewens sampling formula.

scholarly journals Ergodic measures on spaces of infinite matrices over non-Archimedean locally compact fields

2017 ◽  
Vol 153 (12) ◽  
pp. 2482-2533
Author(s):  
Alexander I. Bufetov ◽  
Yanqi Qiu
Keyword(s):  
Probability Measures ◽  
Symmetric Matrices ◽  
Infinite Matrices ◽  
Locally Compact ◽  
Natural Action ◽  
Ring Of Integers ◽  
Compact Field ◽  
Ergodic Measures

Let$F$be a non-discrete non-Archimedean locally compact field and${\mathcal{O}}_{F}$the ring of integers in$F$. The main results of this paper are the classification of ergodic probability measures on the space$\text{Mat}(\mathbb{N},F)$of infinite matrices with entries in$F$with respect to the natural action of the group$\text{GL}(\infty ,{\mathcal{O}}_{F})\times \text{GL}(\infty ,{\mathcal{O}}_{F})$and the classification, for non-dyadic$F$, of ergodic probability measures on the space$\text{Sym}(\mathbb{N},F)$of infinite symmetric matrices with respect to the natural action of the group$\text{GL}(\infty ,{\mathcal{O}}_{F})$.

scholarly journals Generating Hard Tautologies Using Predicate Logic and the Symmetric Group

1998 ◽  
Vol 5 (19) ◽  
Author(s):  
Søren Riis ◽  
Meera Sitharam
Keyword(s):  
Symmetric Group ◽  
Second Order ◽  
Proof Complexity ◽  
Finite Collection ◽  
Propositional Formula ◽  
Algebraic Proof ◽  
Natural Action ◽  
Existential Sentence ◽  
One To One ◽  
Propositional Proof

We introduce methods to generate uniform families of hard propositional tautologies. The tautologies are essentially generated from a single propositional formula by a natural action of the symmetric group Sn.<br />The basic idea is that any Second Order Existential sentence Psi can be systematically translated into a conjunction phi of a finite collection of clauses such that the models of size n of an appropriate Skolemization Psi~   are in one-to-one correspondence with the satisfying assignments to phi_n: the Sn-closure of phi, under a natural action of the symmetric group Sn. Each phi_n is a CNF and thus has depth at most 2. The size of the phi_n's is bounded by a polynomial in n. Under the assumption NEXPTIME |= co- NEXPTIME, for any such sequence phi_n for which the spectrum S := {n : phi_n satisfiable} is NEXPTIME-complete, the tautologies not phi_(n not in S) do not have polynomial length proofs in any propositional proof system.<br /> Our translation method shows that most sequences of tautologies being studied in propositional proof complexity can be systematically generated from Second Order Existential sentences and moreover, many natural mathematical statements can be converted into sequences of propositional tautologies in this manner.<br /> We also discuss algebraic proof complexity issues for such sequences of tautologies. To this end, we show that any Second Order Existential sentence  Psi    can be systematically translated into a finite collection of polynomial equations Q = 0 such that the models of size n of an appropriate skolemization Psi~   are in one-to-one correspondence with the solutions to Qn = 0: the Sn-closure of Q = 0, under a natural action of the symmetric group Sn. The degree of Qn is the same as that of Q, and hence is independent of n, and the number of variables is no more than a polynomial in n. Furthermore, using results in [19] and [20], we briefly describe how, for the corresponding sequences of tautologies phi_n, the rich structure of the Sn closed, uniformly generated, algebraic systems Qn has profound consequences on the algebraic proof complexity of phi_n.

scholarly journals Space-distribution PDEs for path independent additive functionals of McKean–Vlasov SDEs

2020 ◽  
Vol 23 (03) ◽  
pp. 2050018
Author(s):  
Panpan Ren ◽  
Feng-Yu Wang
Keyword(s):  
Product Space ◽  
Space Variable ◽  
Space Distribution ◽  
Nonlinear Pdes ◽  
Probability Measures ◽  
Girsanov Transformation ◽  
Additive Functionals ◽  
Drift Term ◽  
Path Independence ◽  
Girsanov Transformations

Let [Formula: see text] be the space of probability measures on [Formula: see text] with finite second moment. The path independence of additive functionals of McKean–Vlasov SDEs is characterized by PDEs on the product space [Formula: see text] equipped with the usual derivative in space variable and Lions’ derivative in distribution. These PDEs are solved by using probabilistic arguments developed from Ref. 2. As a consequence, the path independence of Girsanov transformations is identified with nonlinear PDEs on [Formula: see text] whose solutions are given by probabilistic arguments as well. In particular, the corresponding results on the Girsanov transformation killing the drift term derived earlier for the classical SDEs are recovered as special situations.

scholarly journals GENERALIZED BESSEL AND FRAME MEASURES

2020 ◽  
Vol 35 (1) ◽  
pp. 217
Author(s):  
Fariba Zeinal Zadeh Farhadi ◽  
Mohammad Sadegh Asgari ◽  
Mohammad Reza Mardanbeigi ◽  
Mahdi Azhini
Keyword(s):  
Product Space ◽  
Borel Measure ◽  
Finite Measure ◽  
Inner Product Space ◽  
Inner Product ◽  
Probability Measures ◽  
Bessel Sequence ◽  
Finite Borel Measure ◽  
Conjugate Exponents ◽  
Finite Measures

Considering a finite Borel measure $ \mu $ on $ \mathbb{R}^d $, a pair of conjugate exponents $ p, q $, and a compatible semi-inner product on $ L^p(\mu) $, we have introduced $ (p,q) $-Bessel and $ (p,q) $-frame measures as a generalization of the concepts of Bessel and frame measures. In addition, we have defined the notions of $ q $-Bessel sequence and $ q$-frame in the semi-inner product space $ L^p(\mu) $. Every finite Borel measure $\nu$ is a $(p,q)$-Bessel measure for a finite measure $ \mu $. We have constructed a large number of examples of finite measures $ \mu $ which admit infinite $ (p,q) $-Bessel measures $ \nu $. We have showed that if $ \nu $ is a $ (p,q) $-Bessel/frame measure for $ \mu $, then $ \nu $ is $ \sigma $-finite and it is not unique. In fact, by using the convolutions of probability measures, one can obtain other $ (p,q) $-Bessel/frame measures for $ \mu $. We have presented a general way of constructing a $ (p,q) $-Bessel/frame measure for a given measure.

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