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Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes

Patie Pierre (2009)

Annales de l'I.H.P. Probabilités et statistiques

We first characterize the increasing eigenfunctions associated to the following family of integro-differential operators, for any α, x>0, γ≥0 and fa smooth function on + , 𝐋 ( γ ) f ( x ) = x - α ( σ 2 x 2 f ' ' ( x ) + ( σ γ + b ) x f ' ( x ) + 0 f e - r x - f ( x ) e - r γ + x f ' ( x ) r 𝕀 { r 1 } ν ( d r ) ) , ( 0 . 1 ) where the coefficients b ,σ≥0 and the measure ν, which satisfies the integrability condition ∫0∞(1∧r2)ν(dr)<+∞, are uniquely determined by the distribution of a spectrally negative, infinitely divisible random variable, with characteristic exponent ψ. L(γ) is known to be the infinitesimal generator of a positive...

Intertwining numbers; the n -rowed shapes

Hyoung J. Ko, Kyoung J. Lee (2007)

Czechoslovak Mathematical Journal

A fairly old problem in modular representation theory is to determine the vanishing behavior of the H o m groups and higher E x t groups of Weyl modules and to compute the dimension of the / ( p ) -vector space H o m A ¯ r ( K ¯ λ , K ¯ μ ) for any partitions λ , μ of r , which is the intertwining number. K. Akin, D. A. Buchsbaum, and D. Flores solved this problem in the cases of partitions of length two and three. In this paper, we describe the vanishing behavior of the groups H o m A ¯ r ( K ¯ λ , K ¯ μ ) and provide a new formula for the intertwining number for any...

Invariants for the modular cyclic group of prime order via classical invariant theory

David L. Wehlau (2013)

Journal of the European Mathematical Society

Let 𝔽 be any field of characteristic p . It is well-known that there are exactly p inequivalent indecomposable representations V 1 , V 2 , ... , V p of C p defined over 𝔽 . Thus if V is any finite dimensional C p -representation there are non-negative integers 0 n 1 , n 2 , ... , n k p - 1 such that V i = 1 k V n i + 1 . It is also well-known there is a unique (up to equivalence) d + 1 dimensional irreducible complex representation of S L 2 ( ) given by its action on the space R d of d forms. Here we prove a conjecture, made by R. J. Shank, which reduces the computation of the ring...

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