Related papers: Period functions for Maass wave forms. I

False theta functions are functions that are closely related to classical theta functions and mock theta functions. In this paper, we study their modular properties at all ranks by forming modular completions analogous to modular…

It was shown in previous work that the one-variable $\widehat\mu$-function defined by Zwegers (and Zagier) and his indefinite theta series attached to lattices of signature $(r\!+\!1,1)$ are both Heisenberg harmonic Maa\ss-Jacobi forms. We…

In this article we study the structure of $\Gamma$-invariant spaces of $L^2(\bf R)$. Here $\bf R$ is a second countable LCA group. The invariance is with respect to the action of $\Gamma$, a non commutative group in the form of a semidirect…

We prove an asymptotic formula for a special case of the Gauss hypergeometric function which arises in explicit formulas for moments of Maass form symmetric square L-functions. The resulting formula is uniform in several variables, which is…

Siegel defined zeta functions associated with indefinite quadratic forms, and proved their analytic properties such as analytic continuations and functional equations. Coefficients of these zeta functions are called measures of…

Harmonic maps from S^2 to S^2 are all weakly conformal, and so are represented by rational maps. This paper presents a study of the L^2 metric gamma on M_n, the space of degree n harmonic maps S^2 -> S^2, or equivalently, the space of…

Let $M= \Gamma \setminus \mathbb{H}_d$ be a compact quotient of the $d$-dimensional Heisenberg group $\mathbb{H}_d$ by a lattice subgroup $\Gamma$. We show that the eigenvalue counting function $N(\lambda)$ for any fixed element of a family…

We show a non-vanishing result for the averages of L-functions associated with the orthogonal basis of the space of cusp forms of vector-valued modular forms on the full group. We also show the existence of at least one basis element whose…

In this paper, we discover a secondary term in the asymptotic formula for the mean value of Hecke--Maass special $L$-values $ L (1/2+it_f, f) $ with the average over $f (z)$ in an orthonormal basis of (even or odd) Hecke--Maass cusp forms…

We discuss the geometry of Laplacian eigenfunctions $-\Delta \phi = \lambda \phi$ on compact manifolds $(M,g)$ and combinatorial graphs $G=(V,E)$. The 'dual' geometry of Laplacian eigenfunctions is well understood on $\mathbb{T}^d$…

We investigate non-vanishing properties of $L(f,s)$ on the real line, when $f$ is a Hecke eigenform of half-integral weight $k+{1\over 2}$ on $\Gamma_0(4).$

In this paper, we study polar harmonic Maass forms of negative integral weight. Using work of Fay, we construct Poincar\'e series which span the space of such forms and show that their elliptic coefficients exhibit duality properties which…

Waves play an essential role in many aspects of plasma science, such as plasma manipulation and diagnostics. Due to the complexity of the governing equations, approximate models are often necessary to describe wave dynamics. In this…

Consider $ G:= PSL_2(\R)\equiv T^1\H^2$, a modular group $ \Gamma$, and the homogeneous space $ \Gamma\sm G \equiv T^1(\Gamma\sm\H^2)$. Endow $ G $, and then $ \Gamma\sm G $, with a canonical left-invariant metric, thereby equipping it with…

For any finite simple graph G, the hydrogen identity H=L-L^(-1) holds, where H=(d+d^*)^2 is the sign-less Hodge Laplacian defined by sign-less incidence matrix d and where L is the connection Laplacian. Any spectral information about L…

We prove an $L^2$-$\partial\overline\partial$-Lemma involving smooth square integrable forms on complete K\"ahler manifolds, provided that the unique self-adjoint extension of the Hodge Laplacian on the Hilbert space of $L^2$-forms has a…

Let $\mathcal{P}$ be an $n$-dimensional convex polytope and $\mathcal{S}$ be a hypersurface in $\mathbb{R}^n$. This paper investigates potentials to reconstruct $\mathcal{P}$ or at least to compute significant properties of $\mathcal{P}$ if…

Two properties of Calogero wave functions for rational Calogero models are studied: (i) the representation of the wave functions in terms of the exponential of Lassalle operators, (ii) the $sL(2,\rr)$ structure of the Calogero--Moser wave…

A technique is developed which allows for the detailed mapping of the electronic wave function in two-dimensional electron gases with low-temperature mobilities up to 15E6 cm^2/Vs. Thin ("delta") layers of aluminium are placed into the…

Let $f$ be a fixed (holomorphic or Maass) modular cusp form, with $L$-function $L(f,s)$. We describe an algorithm that computes the value $L(f,1/2+ iT)$ to any specified precision in time $O(1+|T|^{7/8})$.