We present a construction of an entropy-preserving equivariant surjective map from the d-dimensional critical sandpile model to a certain closed, shift-invariant subgroup of T d (the ‘harmonic model’). A similar map is constructed for the dissipative abelian sandpile model and is used to prove uniqueness and the Bernoulli property of the measure of maximal entropy for that model.

Let G be a complex Lie group and ΛG denote the group of maps from the unit circle S into G, of a suitable class. A differentiable map F from a manifold M into ΛG, is said to be of connection order (ba) if the Fourier expansion in the loop parameter λ of the S-family of Maurer-Cartan forms for F , namely F λ dFλ, is of the form P b i=a αiλ . Most integrable systems in geometry are associated to ...

For second harmonic generation in two-dimensional wave-guiding structures composed of segments that are invariant in the longitudinal direction, we develop an efficient numerical method based on the Dirichlet-to-Neumann (DtN) maps of the segments and a marching scheme using two operators and two functions. A Chebyshev collocation method is used to discretize the longitudinal variable for comput...

(1) In Section 2, we define the notion of harmonic maps and quadratic differentials. Then we give a harmonic map proof of Teichmuller’s theorem by Wolf. (See [2], [9]) (2) In Section 3, we explain a compactification of Teichmuller spaces by Wolf using harmonic maps. (See [2], [4], [10], [11]) (3) In Section 4, we review the Morgan-Shalen compactification and the Korevaar-Schoen limit. Then we g...

For integers k ≥ 2, we study two differential operators on harmonic weak Maass forms of weight 2 − k. The operator ξ2−k (resp. D) defines a map to the space of weight k cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms are expected to have transcendental coefficients, we...

The first author proved that the harmonic convolution of a normalized right half-plane mapping with either another normalized right halfplane mapping or a normalized vertical strip mapping is convex in the direction of the real axis. provided that it is locally univalent. In this paper, we prove that in general the assumption of local univalency cannot be omitted. However, we are able to show t...

For integers k ≥ 2, we study two differential operators on harmonic weak Maass forms of weight 2 − k. The operator ξ2−k (resp. D) defines a map to the space of weight k cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms have transcendental coefficients, we show that those...

The paper describes an embedded controller as the basis for high performance navigation tasks. The approach solves Laplace's equation on a discrete grid representing the roadway map. The resulting harmonic potential is free of local minima and a greedy descent minimizes the probability of collision with posted obstacles. In this paper, we will (1) introduce the important properties of harmonic ...

An electric-field-induced second-harmonic-generation signal in a nematic liquid crystal is used to map the electric field in an integrated-circuit-like sample. Since the electric-field-induced second-harmonic-generation signal intensity exhibits a strong dependence on the polarization of the incident laser beam, both the amplitude and the orientation of the electric field vectors can be measure...