scholarly journals Characterizing the Absolute Continuity of the Convolution of Orbital Measures in a Classical Lie Algebra

2016 ◽  
Vol 68 (4) ◽  
pp. 841-875 ◽  
Author(s):  
Sanjiv Kumar Gupta ◽  
Kathryn Hare
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Absolute Continuity ◽  
Classical Result ◽  
Continuous Measure ◽  
Empty Interior ◽  
Absolutely Continuous ◽  
Absolutely Continuous Measure ◽  
Orbital Measure ◽  
Classical Lie Algebra

AbstractLet 𝓰 be a compact simple Lie algebra of dimension d. It is a classical result that the convolution of any d non-trivial, G-invariant, orbitalmeasures is absolutely continuous with respect to Lebesgue measure on 𝓰, and the sum of any d non-trivial orbits has non-empty interior. The number d was later reduced to the rank of the Lie algebra (or rank +1 in the case of type An). More recently, the minimal integer k = k(X) such that the k-fold convolution of the orbital measure supported on the orbit generated by X is an absolutely continuous measure was calculated for each X ∈ 𝓰.In this paper 𝓰 is any of the classical, compact, simple Lie algebras. We characterize the tuples (X1 , . . . , XL), with Xi ∊ 𝓰, which have the property that the convolution of the L-orbital measures supported on the orbits generated by the Xi is absolutely continuous, and, equivalently, the sum of their orbits has non-empty interior. The characterization depends on the Lie type of 𝓰 and the structure of the annihilating roots of the Xi. Such a characterization was previously known only for type An.

scholarly journals Certain semigroups embeddabable in topological groups

1982 ◽  
Vol 33 (1) ◽  
pp. 30-39 ◽  
Author(s):  
Heneri A. M. Dzinotyiweyi
Keyword(s):  
Invariant Measures ◽  
Locally Compact Group ◽  
Continuous Measure ◽  
Topological Groups ◽  
Empty Interior ◽  
Locally Compact ◽  
Absolutely Continuous ◽  
Topological Semigroups ◽  
Absolutely Continuous Measure ◽  
Locally Compact Topological Group

AbstractIn this paper we study commutative topological semigroups S admitting an absolutely continuous measure. When S is cancellative we show that S admits a weaker topology J with respect to which (S, J) is embeddable as a subsemigroup with non-empty interior in some locally compact topological group. As a consequence, we deduce certain results related to the existence of invariant measures on S and for a large class of locally compact topological semigroups S, we associate S with some useful topological subsemigroup of a locally compact group.

scholarly journals THE SMOOTHNESS OF ORBITAL MEASURES ON NONCOMPACT SYMMETRIC SPACES

2021 ◽  
pp. 1-20
Author(s):  
SANJIV KUMAR GUPTA ◽  
KATHRYN E. HARE
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Continuous Measure ◽  
Convolution Product ◽  
Absolutely Continuous ◽  
Regular Elements ◽  
Connected Subgroup ◽  
Decay Properties ◽  
Absolutely Continuous Measure ◽  
Special Case

Abstract Let $G/K$ be an irreducible symmetric space, where G is a noncompact, connected Lie group and K is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$ continuous orbital measures has its density function in $L^{2}(G)$ and hence is an absolutely continuous measure with respect to the Haar measure. The number r is approximately the rank of $G/K$ . For the special case of the orbital measures, $\nu _{a_{i}}$ , supported on the double cosets $Ka_{i}K$ , where $a_{i}$ belongs to the dense set of regular elements, we prove the sharp result that $\nu _{a_{1}}\ast \nu _{a_{2}}\in L^{2},$ except for the symmetric space of Cartan class $AI$ when the convolution of three orbital measures is needed (even though $\nu _{a_{1}}\ast \nu _{a_{2}}$ is absolutely continuous).

Sequential selection of random vectors under a sum constraint

2004 ◽  
Vol 41 (1) ◽  
pp. 131-146
Author(s):  
Mario Stanke
Keyword(s):  
Continuous Distribution ◽  
Center Of Gravity ◽  
Continuous Measure ◽  
Random Vectors ◽  
Expected Number ◽  
Absolutely Continuous ◽  
Selection Policy ◽  
Sequential Selection ◽  
Absolutely Continuous Measure ◽  
Selection Of

We observe a sequence X1, X2,…, Xn of independent and identically distributed coordinatewise nonnegative d-dimensional random vectors. When a vector is observed it can either be selected or rejected but once made this decision is final. In each coordinate the sum of the selected vectors must not exceed a given constant. The problem is to find a selection policy that maximizes the expected number of selected vectors. For a general absolutely continuous distribution of the Xi we determine the maximal expected number of selected vectors asymptotically and give a selection policy which asymptotically achieves optimality. This problem raises a question closely related to the following problem. Given an absolutely continuous measure μ on Q = [0,1]d and a τ ∈ Q, find a set A of maximal measure μ(A) among all A ⊂ Q whose center of gravity lies below τ in all coordinates. We will show that a simplicial section {x ∈ Q | 〈x, θ〉 ≤ 1}, where θ ∈ ℝd, θ ≥ 0, satisfies a certain additional property, is a solution to this problem.

Absolutely continuous measure for a jump-type Fleming–Viot process

2012 ◽  
Vol 82 (3) ◽  
pp. 557-564 ◽  
Author(s):  
Telles Timóteo da Silva ◽  
Marcelo Dutra Fragoso
Keyword(s):  
Continuous Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Measure

Sequential selection of random vectors under a sum constraint

2004 ◽  
Vol 41 (01) ◽  
pp. 131-146
Author(s):  
Mario Stanke
Keyword(s):  
Continuous Distribution ◽  
Center Of Gravity ◽  
Continuous Measure ◽  
Random Vectors ◽  
Expected Number ◽  
Absolutely Continuous ◽  
Selection Policy ◽  
Sequential Selection ◽  
Absolutely Continuous Measure ◽  
Selection Of

We observe a sequence X 1, X 2,…, X n of independent and identically distributed coordinatewise nonnegative d-dimensional random vectors. When a vector is observed it can either be selected or rejected but once made this decision is final. In each coordinate the sum of the selected vectors must not exceed a given constant. The problem is to find a selection policy that maximizes the expected number of selected vectors. For a general absolutely continuous distribution of the X i we determine the maximal expected number of selected vectors asymptotically and give a selection policy which asymptotically achieves optimality. This problem raises a question closely related to the following problem. Given an absolutely continuous measure μ on Q = [0,1] d and a τ ∈ Q, find a set A of maximal measure μ(A) among all A ⊂ Q whose center of gravity lies below τ in all coordinates. We will show that a simplicial section { x ∈ Q | 〈 x , θ 〉 ≤ 1}, where θ ∈ ℝ d , θ ≥ 0, satisfies a certain additional property, is a solution to this problem.

scholarly journals The size of characters of exceptional lie groups

2004 ◽  
Vol 77 (2) ◽  
pp. 233-248 ◽  
Author(s):  
Kathryn E. Hare ◽  
Karen Yeats
Keyword(s):  
Lie Groups ◽  
Classical Result ◽  
Continuous Measure ◽  
Compact Lie Groups ◽  
Absolutely Continuous ◽  
Exceptional Lie Groups

AbstractPointwise bounds for characters of representations of the compact, connected, simple, exceptional Life groups are obtained. It is a classical result that if μ is a central, continuous measure on such a group, then μdimG is absolutely continuous. Our estimates on the size of characters allow us to prove that the exponent, dimension of G, can be replaced by approximately the rank of G. Similar results were obtained earlier for the classical, compact Lie groups.

scholarly journals Un exemple de transformation dilatante et C1 par morceaux de l'intervalle, sans probabilité absolument continue invariante

1989 ◽  
Vol 9 (1) ◽  
pp. 101-113 ◽  
Author(s):  
P. Gora ◽  
B. Schmitt
Keyword(s):  
Probability Measure ◽  
Modulus Of Continuity ◽  
Invariant Probability Measure ◽  
Negative Answer ◽  
Continuous Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Measure ◽  
Absolutely Continuous Invariant

AbstractWe construct a transformation on the interval [0, 1] into itself, piecewiseC1 and expansive, which doesn't admit any absolutely continuous invariant probability measure (a.c.i.p.).So in this case we give a negative answer to a question by Anosov: is C1 character sufficient for the existence of absolutely continuous measure?Moreover, in our example,ƒ' has a modulus of type K/(|1+|log|x‖); it is known that a modulus of continuity of type K/(1+|log|x‖)1+γ, γ>0 implies the existence of a.c.i.p..

scholarly journals Coboundary equations of eventually expanding transformations

2003 ◽  
Vol 67 (1) ◽  
pp. 39-50
Author(s):  
Young-Ho Ahn
Keyword(s):  
Measurable Function ◽  
Finite Subgroup ◽  
Unit Interval ◽  
Continuous Measure ◽  
Ergodic Transformation ◽  
Skew Product ◽  
Markov Condition ◽  
Absolutely Continuous ◽  
Circle Group ◽  
Absolutely Continuous Measure

Let T be an eventually expansive transformation on the unit interval satisfying the Markov condition. The T is an ergodic transformation on (X, ß, μ) where X = [0, 1), ß is the Borel σ-algebra on the unit interval and μ is the T invariant absolutely continuous measure. Let G be a finite subgroup of the circle group or the whole circle group and φ: X → G be a measurable function with finite discontinuity points. We investigate ergodicity of skew product transformations Tφ on X × G by showing the solvability of the coboundary equation φ(x) g (Tx) = λg (x), |λ| = 1. Its relation with the uniform distribution mod M is also shown.

scholarly journals CONVOLUTION OF ORBITAL MEASURES IN SYMMETRIC SPACES

2011 ◽  
Vol 83 (3) ◽  
pp. 470-485 ◽  
Author(s):  
BOUDJEMÂA ANCHOUCHE ◽  
SANJIV KUMAR GUPTA
Keyword(s):  
Positive Integer ◽  
Symmetric Space ◽  
Lie Algebra ◽  
Haar Measure ◽  
Symmetric Spaces ◽  
Double Coset ◽  
Cartan Decomposition ◽  
Absolutely Continuous ◽  
Orbital Measure ◽  
Noncompact Symmetric Space

AbstractLet G/K be a noncompact symmetric space, Gc/K its compact dual, 𝔤=𝔨⊕𝔭 the Cartan decomposition of the Lie algebra 𝔤 of G, 𝔞 a maximal abelian subspace of 𝔭, H be an element of 𝔞, a=exp (H) , and ac =exp (iH) . In this paper, we prove that if for some positive integer r, νrac is absolutely continuous with respect to the Haar measure on Gc, then νra is absolutely continuous with respect to the left Haar measure on G, where νac (respectively νa) is the K-bi-invariant orbital measure supported on the double coset KacK (respectively KaK). We also generalize a result of Gupta and Hare [‘Singular dichotomy for orbital measures on complex groups’, Boll. Unione Mat. Ital. (9) III (2010), 409–419] to general noncompact symmetric spaces and transfer many of their results from compact symmetric spaces to their dual noncompact symmetric spaces.

A Product Formula and a Non-Negative Poisson Kernel for Racah-Wilson Polynomials

1980 ◽  
Vol 32 (6) ◽  
pp. 1501-1517 ◽  
Author(s):  
Mizan Rahman
Keyword(s):  
Orthogonal Polynomials ◽  
Series Representation ◽  
General System ◽  
Continuous Measure ◽  
Absolutely Continuous ◽  
Angular Momenta ◽  
Special Cases ◽  
Absolutely Continuous Measure ◽  
Wilson Polynomials ◽  
Special Case

Physicists have long been using Racah's [7] 6-j symbols as a representation for the addition coefficients of three angular momenta. Racah himself discovered a series representation of the 6-j symbol which can be expressed as a balanced 4F3 series of argument 1, that is, a generalized hypergeometric function such that the sum of the 3 denominator parameters exceeds that of the 4 numerator parameters by 1. What Racah does not seem to have realized or, perhaps, cared to investigate, is that his 4F3 functions, with variables and parameters suitably identified, form a system of orthogonal polynomials in a discrete variable. The orthogonality of 6-j symbols as an orthogonality of 4F3 polynomials was recognized much later by Biedenharn et al. [3] in some special cases. Recently J. Wilson [13, 14] introduced a very general system of orthogonal polynomials expressible as balanced 4F3 functions of argument 1 orthogonal with respect to an absolutely continuous measure and/or a discrete weight function. Wilson's polynomials contain Racah's 6-j symbols as a special case. These polynomials might rightfully be credited to Wilson alone, but justice might be better served if we call them Racah-Wilson polynomials.

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