To David Kazhdan for his 60th birthday " Loxadь sostoit iz trh neravnyh polovin ". 4 The quantum dilogarithm and its properties 26 4.1 The quantum logarithm function and its properties. . 1 " A horse consists of three unequal halves ". cf. A. de Barr, Horse doctor. Moscow 1868. Cluster varieties [FG2] are relatives of cluster algebras [FZI]. Cluster modular groups act by automor-phisms of clust...

We prove that the zeros of a family of extremal modular forms interlace, settling a question of Nozaki. Additionally, we show that the zeros of almost all forms in a basis for the space of weakly holomorphic modular forms of weight k for SL2(Z) interlace on most of the lower boundary of the fundamental domain.

and Applied Analysis 3 the fixed point theory and that a Banach space X has property L if and only if it is reflexive and has the uniform Opial property. For a real vector space X, a function ρ : X → 0,∞ is called amodular if it satisfies the following conditions: i ρ x 0 if and only if x 0; ii ρ αx ρ x for all scalar α with |α| 1; iii ρ αx βy ≤ ρ x ρ y , for all x, y ∈ X and all α, β ≥ 0 with ...

Abstract In order to simulate the complex dynamic behavior of automata system and analyze muzzle disturbance, a modular model was established based on bond space theory. The modules cradle, gun box, barrel projectile were defined, state equations each module deduced in terms momentum vector position change their first derivative. whole is by connecting generalized potential source flow module. ...

We join Hurwitz space constructions and the universal Frattini cover of a nite group The goal is to form and apply generalizations of the towers X p X p X p of modular curves This generalization relies on the appearance of the dihedral group Dp and its companion group Zp f g in the theory of modular curves We replace Dp by any nite group G and p by any prime dividing the order of G The replacem...