We find identities between theta constants with rational characteristics evaluated at period matrix of R, a cyclic 3 sheeted cover of the sphere with 3k branch points λ1...λ3k. These identities follow from Thomae formula [BR]. This formula expresses powers of theta constants as polynomials in λ1...λ3k. We apply the representation of the symmetric group to find relations between the polynomials ...

We study the Dirichlet boundary value problem for the p-Laplacian of the form −∆pu− λ1|u|p−2u = f in Ω, u = 0 on ∂Ω, where Ω ⊂ N is a bounded domain with smooth boundary ∂Ω, N 1, p > 1, f ∈ C(Ω) and λ1 > 0 is the first eigenvalue of ∆p. We study the geometry of the energy functional Ep(u) = 1 p ∫

In this thesis, we studied eigenvalues of directed and undirected graphs and their applications. In the first part, a detailed study of the largest eigenvalue of the normalized Laplace operator ∆ for undirected graphs was presented. In contrast to the smallest nontrivial eigenvalue λ1, the largest eigenvalue λn−1 has not been studied systematically before. However, it is well-known that λ1 can ...

We consider elliptic equations in bounded domains Ω ⊂ R with nonlinearities which have exponential growth at +∞ (subcritical and critical growth respectively) and linear growth λ at −∞, with λ > λ1, the first eigenvalue of the Laplacian. We prove that such equations have at least two solutions for certain forcing terms; one solution is negative, the other one is sign-changing. Some critical gro...

In this note we discuss a method of order 1 + √ 3 for computing the smallest eigenvalue λ1 of a symmetric and positive definite Toeplitz matrix. It generalizes and improves a method introduced in [7] which is based on rational Hermitean interpolation of the secular equation. Taking advantage of a further rational approximation of the secular equation which is essentially for free and which yiel...

Introduction We begin with the description of a geometric construction in the real line. We consider p positive numbers λ1, . . . , λp < 1, and choose p disjoint closed intervals ∆1, . . . , ∆p with length λ1, . . . , λp. For each k = 1, . . . , p, we choose again p disjoint closed intervals ∆k1, . . . , ∆kp ⊂ ∆k with length λkλ1, . . . , λkλp. Iterating this procedure, for each integer n we ob...

The heavy-quark expansion for inclusive semi-leptonic B decays introduces Λ̄ and λ1, which are matrix elements in heavy-quark effective field theory. We review how they can be obtained from an analysis of the heavy quark mass dependence of heavy-light meson masses in lattice QCD. We present preliminary results for the bottom quark mass, mb, using Λ̄ and λ1 for the Bs meson from a 2+1 sea-flavor u...

2 Autonomous differential equations Lemma 1. If A ∈ B(R), then det(I + A+ o( )) = 1 + trA+ o( ) as → 0. Proof. Let λ1, . . . , λn be the eigenvalues of A, repeated according to algebraic multiplicity. For > 0, the eigenvalues of I + A + o( ) repeated according to algebraic multiplicity are 1 + λ1 + o( ), . . . , 1 + λn + o( ), as → 0. The determinant of a linear map R → R is the product of its ...

where dVg is the volume measure on M with respect to the Kähler metric g. When M is compact and ∆g is uniformly elliptic, λ1(∆g) is the first positive eigenvalue of ∆g with Dirichlet boundary condition. A lot of research has been done on its upper and lower bound estimates and its impact on geometry and physics (see for examples, the lecture notes of P. Li [8] and the paper of S. Udagawa [16] a...