We consider on a two-dimensional flat torus T the following equation ∆u + ρ ( eu ∫ T e u − 1 |T | ) = 0. When the fundamental domain of the torus is (0, a) × (0, b) (a ≥ b), we establish that the constants are the unique solutions whenever ρ ≤ { 8π if b a ≥ π4 , 32 b a if b a ≤ π4 , and this result is sharp if b a ≥ π4 . A similar conclusion is obtained for general two-dimensional torus by cons...

Fundamental solutions of dynamic poroelasticity and generalized thermoelasticity are derived in the Laplace transform domain. For poroelasticity, these solutions define the solid displacement field and the fluid pressure in fluid-saturated media due to a point force in the solid and an injection of fluid in the pores. In addition, approximate fundamental solutions for short times are derived by...

We present a mathematical theory of the two-dimensional offset curves from the viewpoint of medial axis transform. We explore the local geometry of the offset curve in relation with the medial axis transform, culminating in the classification of points on the offset curve. We then study the domain decomposition from the viewpoint of offsets, and in particular introduce the concept of monotonic ...

Supersymmetry is used to derive conditions on higher derivative terms in the effective action of type IIB supergravity. Using these conditions, we are able to prove earlier conjectures that certain modular invariant interactions of order (α) relative to the Einstein–Hilbert term are proportional to eigenfunctions of the Laplace operator on the fundamental domain of SL(2,ZZ). We also discuss how...

In this paper we study a geometric coding algorithm for indefinite binary quadratic forms Q for the congruence subgroup Γ(N), with respect to the usual fundamental domain FN , where N is assumed prime. The cycles Q1, . . . , Qn that this algorithm produces are such that the the corresponding paths γ1, . . . , γn in the Riemann surface X0(N)(C) have a nice behaviour around the elliptic points of...

Let W be a compact manifold and let ρ be a representation of its fundamental group into PSL(2,C) Isom(H3). The volume of ρ is defined by taking any ρ-equivariant map from the universal cover W̃ to H and then by integrating the pullback of the hyperbolic volume form on a fundamental domain. This volume does not depend on the choice of the equivariant map because two equivariant maps are always eq...

We show how the (globally supersymmetric) model of Mirabelli and Peskin can be formulated in the boundary (“downstairs” or “interval”) picture. The necessary GibbonsHawking-like terms appear naturally when using (codimension one) superfields. This formulation is free of the δ(0) ambiguities of the orbifold (“upstairs”) picture while describing the same physics since the boundary conditions on t...

We describe a process for defining and computing a fundamental domain in the upper half plane H of a Shimura curve X 0 (N) associated to an order in a quaternion algebra A/Q. A fundamental domain for X 0 (N) realizes a finite presentation of the quaternion unit group, modulo units of its center. We give explicit examples of domains for the curves X 0 (1), X 0 (1), and X 35 0 (1). The first exam...

One of the most popular student modeling techniques currently available is Constraint Based Modeling (CBM), which is based on Ohlsson's theory of learning from performance errors. It focuses on the domain principles to correct faulty knowledge and assumes that a student will reach a correct solution without violating these fundamental domain concepts. However, even though this is a powerful and...

Our main goal in this paper is to construct the first explicit fundamental domain of the Picard modular group acting on the complex hyperbolic space CH . The complex hyperbolic space is a Hermitian symmetric space, its bounded realization is the unit ball in C equipped with the Bergman metric. The Picard modular group is a discontinuous holomorphic automorphism subgroup of SU(2, 1) with Gaussia...