• Volume 99, Issue 2

August 1989,   pages  103-191

• Canonical measures on the moduli spaces of compact Riemann surfaces

We study some explicit relations between the canonical line bundle and the Hodge bundle over moduli spaces for low genus. This leads to a natural measure on the moduli space of every genus which is related to the Siegel symplectic metric on Siegel upper half-space as well as to the Hodge metric on the Hodge bundle.

• Dynamics on Ahlfors quasi-circles

The celebrated theory of Denjoy introduced a topological invariant distinguishingC1 andC2 diffeomorphisms of the circle. AC2 diffeomorphism of the circle cannot have an infinite minimal set other than the circle itself. However, this is possible forC1 diffeomorphisms. In dimension two there is a related invariant distinguishingC2 andC3 diffeomorphisms.

• On the gap between two classes of analytic functions

Two large classes of analytic functions are defined, so that one contains the other. Sharp coefficient bounds for quadratic polynomials falling in the gap between these two classes are given.

• Discrete subgroups of algebraic groups over local fields of positive characteristics

It is shown in this paper that ifG is the group ofk-points of a semisimple algebraic groupG over a local fieldk of positive characteristic such that all itsk-simple factors are ofk-rank 1 and Γ ⊂G is a non-cocompact irreducible lattice then Γ admits a fundamental domain which is a union of translates of Siegel domains. As a consequence we deduce that ifG has more than one simple factor, then Γ is finitely generated and by a theorem due to Venkataramana, it is arithmetic.

• Small fractional parts of additive forms

Letf(x)=θ1x1k+...+θsxsk be an additive form with real coefficients, and ∥α∥ = min {¦α-u¦:uεℤ} denote the distance fromα to the nearest integer. We show that ifθ1,…,θs, are algebraic ands = 4k then there are integersx1,…,xs, satisfying l ≤x1,≤ N and ∥f(x)∥ ≤ NE, withE = − 1 + 2/e.

Whens = λk, 1 ≤λ ≤ 2k, the exponentE may be replaced byλE/4, and if we drop the condition thatθ1,…,θs, be algebraic then the result holds for almost all values of θεℝs. Whenk ≥ 6 is small a better exponent is obtained using Heath-Brown’s version of Weyl’s estimate.

• A generalization of the riemann zeta-function

A generalization of the Riemann zeta-function which has the form$$\zeta _\alpha (s) = \prod\limits_p {\frac{1}{{1 - p^{ - s} + (p + a)^{ - 3} }}}$$ is considered. Analytical properties with respect to s and asymptotic behaviour whena → ∞ are investigated. The correspondingL-function is also discussed. This consideration has an application in the theory ofp-adic strings.

• Generalized absolute Cesaro summability of a Fourier series

In an attempt to study the scope of a theorem due to Pati, the authors have established that φ(t) logK¦tBuV in (0,π)⟹ΣAn(x) is ¦C, 0,β¦ forβ&gt;1, at the pointt = x.

• Fractional integral formulae involving a general class of polynomials and the multivariableH-function

We obtain two fractional integral formulae involving a general class of polynomials and the multivariableH-function. On account of the most general nature of the polynomials and the multivaribleH-function involved herein, our findings provide interesting unifications and extensions of a number of (known and new) results. We have mentioned here only two such results.

• Asymptotic expansions of the Mehler-Fock transform

Asymptotic expansions in the two limitsx → + ∞ andx → 0+ are obtained for the Mehler-Fock transform$$I(x) = \int_0^\infty {P_{ - \tfrac{1}{1} + ir}^m } (cosh x)h(\tau )d\tau ,$$ whereP−1/2+irm (coshx) is the associated Legendre function.

• Asymptotic analysis of some nonlinear problems using Hopf-Cole transform and spectral theory

We consider initial boundary value problems for certain nonlinear scalar parabolic equations. A formula for the unique classical solution by Hopf-Cole transformations is obtained and the asymptotic behaviour of the solution as time goes to ∞ is studied.

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