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On the density of the supremum of a stable process
, 2012
"... We study the density of the supremum of a strictly stable Lévy process. Our first goal is to investigate convergence properties of the series representation for this density, which was established recently in [24]. Our second goal is to investigate in more detail the important case when α is ration ..."
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We study the density of the supremum of a strictly stable Lévy process. Our first goal is to investigate convergence properties of the series representation for this density, which was established recently in [24]. Our second goal is to investigate in more detail the important case when α is rational: We derive an explicit formula for the Mellin transform of the supremum. We perform several numerical experiments and discuss their implications. Finally, we state some interesting connections that this problem has to other areas of Mathematics and Mathematical Physics and we also suggest several open problems.
Solvable Discrete Quantum Mechanics: qOrthogonal Polynomials with q
 1 and Quantum Dilogarithm,” arXiv:1406.2768[mathph
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Note on Dilogarithm Identities from Nilpotent Double Affine Hecke Algebras
"... Abstract. Recently Cherednik and Feigin [arXiv:1209.1978] obtained several Rogers–Ramanujan type identities via the nilpotent double affine Hecke algebras (NilDAHA). These identities further led to a series of dilogarithm identities, some of which are known, while some are left conjectural. We conf ..."
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Abstract. Recently Cherednik and Feigin [arXiv:1209.1978] obtained several Rogers–Ramanujan type identities via the nilpotent double affine Hecke algebras (NilDAHA). These identities further led to a series of dilogarithm identities, some of which are known, while some are left conjectural. We confirm and explain all of them by showing the connection with Ysystems associated with (untwisted and twisted) quantum affine Kac–Moody algebras.
Simplex and Polygon Equations
"... It is shown that higher Bruhat orders admit a decomposition into a higher Tamari order, the corresponding dual Tamari order, and a “mixed order”. We describe simplex equations (including the YangBaxter equation) as realizations of higher Bruhat orders. Correspondingly, a family of “polygon equation ..."
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It is shown that higher Bruhat orders admit a decomposition into a higher Tamari order, the corresponding dual Tamari order, and a “mixed order”. We describe simplex equations (including the YangBaxter equation) as realizations of higher Bruhat orders. Correspondingly, a family of “polygon equations ” realizes higher Tamari orders. They generalize the wellknown pentagon equation. The structure of simplex and polygon equations is visualized in terms of deformations of maximal chains in posets forming 1skeletons of polyhedra. The decomposition of higher Bruhat orders induces a reduction of the Nsimplex equation to the (N + 1)gon equation, its dual, and a compatibility equation.