Displaying similar documents to “PDE's for the Dyson, Airy and Sine processes”

The ratio and generating function of cogrowth coefficients of finitely generated groups

Ryszard Szwarc (1998)

Studia Mathematica

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Let G be a group generated by r elements g 1 , … , g r . Among the reduced words in g 1 , … , g r of length n some, say γ n , represent the identity element of the group G. It has been shown in a combinatorial way that the 2nth root of γ 2 n has a limit, called the cogrowth exponent with respect to the generators g 1 , … , g r . We show by analytic methods that the numbers γ n vary regularly, i.e. the ratio γ 2 n + 2 / γ 2 n is also convergent. Moreover, we derive new precise information on the domain of holomorphy of γ(z), the generating function...

Non-vanishing of class group L -functions at the central point

Valentin Blomer (2004)

Annales de l’institut Fourier

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Let K = ℚ ( - D ) be an imaginary quadratic field, and denote by h its class number. It is shown that there is an absolute constant c > 0 such that for sufficiently large D at least c · h ∏ p ∣ D ( 1 - p - 1 ) of the h distinct L -functions L K ( s , χ ) do not vanish at the central point s = 1 / 2 .

On the exponential Orlicz norms of stopped Brownian motion

Goran Peškir (1996)

Studia Mathematica

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Necessary and sufficient conditions are found for the exponential Orlicz norm (generated by ψ p ( x ) = e x p ( | x | p ) - 1 with 0 < p ≤ 2) of m a x 0 ≤ t ≤ τ | B t | or | B τ | to be finite, where B = ( B t ) t ≥ 0 is a standard Brownian motion and τ is a stopping time for B. The conditions are in terms of the moments of the stopping time τ. For instance, we find that ∥ m a x 0 ≤ t ≤ τ | B t | ∥ ψ 1 < ∞ as soon as E ( τ k ) = O ( C k k k ) for some constant C > 0 as k → ∞ (or equivalently ∥ τ ∥ ψ 1 < ∞ ). In particular, if τ ∼ Exp(λ) or | N ( 0 , σ 2 ) | then the last condition is satisfied, and we obtain ∥ m a x 0 ≤ t ≤ τ | B t | ∥ ψ 1 ≤ K √ E ( τ ) with some universal constant...

On linear normal lattices configurations

Mordechay B. Levin, Meir Smorodinsky (2005)

Journal de Théorie des Nombres de Bordeaux

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In this paper we extend Champernowne’s construction of normal numbers in base b to the ℤ d case and obtain an explicit construction of the generic point of the ℤ d shift transformation of the set { 0 , 1 , . . . , b - 1 } ℤ d . We prove that the intersection of the considered lattice configuration with an arbitrary line is a normal sequence in base b .

On the Maximum of a Branching Process Conditioned on the Total Progeny

Kerbashev, Tzvetozar (1999)

Serdica Mathematical Journal

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The maximum M of a critical Bienaymé-Galton-Watson process conditioned on the total progeny N is studied. Imbedding of the process in a random walk is used. A limit theorem for the distribution of M as N → ∞ is proved. The result is trasferred to the non-critical processes. A corollary for the maximal strata of a random rooted labeled tree is obtained.