Effective bounds on differences of singular moduli that are <i>S</i>-units

Bounds on the Effective Moduli of Composite Materials

Bounds on the Effective Moduli of Cracked Materials

We find bounds on the effective elastic moduli of cracked materials in terms of the effective conductivity of such media. These represent the first non-trivial bounds on the effective properties of cracked media which are independent of the shapes and spatial distribution of the cracks. Different approximations for the elastic moduli of cracked media are tested against our bounds. The microgeom...

Geometrical-parameter Bounds on the Effective Moduli of Composites

We study bounds on the effective conductivity and elastic moduli of two-phase isotropic composites that depend on geometrical parameters that take into account up to three-point statistical information concerning the composite microstructure. We summarize existing bounds, apply a special fractional linear transformation to simplify their functional forms, and describe two approaches to improve ...

Traces of Singular Moduli

Introduction. “Singular moduli” is the classical name for the values assumed by the modular invariant j(τ) (or by other modular functions) when the argument is a quadratic irrationality. These values are algebraic numbers and have been studied intensively since the time of Kronecker and Weber. In [5], formulas for their norms, and for the norms of their differences, were obtained. Here we obtai...

Singular Moduli Refined

In this paper, we give a refinement of the work of Gross and Zagier on singular moduli. Let K1 and K2 be two imaginary quadratic fields with relatively prime discriminants d1 and d2, and let F = Q( √ d1d2). Hecke constructed a Hilbert modular Eisenstein series over F of weight 1 whose functional equation forces it to vanish at s = 0. For CM elliptic curves E1 and E2 with complex multiplication ...