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A geometric formulation for stabilization of systems with one degree of underactuation which fully solves the energy shaping problem for these system is given. The results show that any linearly controllable simple mechanical system with…
We review and give elementary proofs of Liouville type properties of harmonic and subharmonic functions in the plane endowed with a complete Riemannian metric, and prove a gap theorem for the possible growth of harmonic functions when this…
In this work we show that there is a direct relationship between a graph's topology and the free energy of a spin system on the graph. We develop a method of separating topological and enthalpic contributions to the free energy, and find…
Inspired by numerical studies of the aggregation equation, we study the effect of regularization on nonlocal interaction energies. We consider energies defined via a repulsive-attractive interaction kernel, regularized by convolution with a…
This work is a geometrical approach to the optimization problem motivated by transportation system management. First, an attempt has been made to furnish a comprehensive account of geometric programming based on the elementary Finsler…
The stationary points of the potential energy function of the \phi^4 model on a two-dimensional square lattice with nearest-neighbor interactions are studied by means of two numerical methods: a numerical homotopy continuation method and a…
We study the structure of the non-perturbative free energy of a one-parameter class of little string theories (LSTs) of A-type in the so-called unrefined limit. These theories are engineered by $N$ M5-branes probing a transverse flat space.…
The free energy balance equation for gyrokinetic fluctuations is derived and applied to instabilities. An additional term due to electromagnetic sources is included. This can provide a simpler way to compute the free energy balance in…
We investigate properties of ($\alpha,\beta$)-harmonic functions. First, we discuss the coefficient estimates for ($\alpha,\beta$)-harmonic functions. In particular, we obtain Heinz's inequality for ($\alpha,\beta$)-harmonic functions,…
An introduction to the minimal supersymmetric Standard Model (MSSM) is given. The motivation for ``low-energy'' supersymmetry is reviewed, and the structure of the MSSM is outlined. In its most general form, the MSSM can be viewed as a…
A two-dimensional free boundary transmission problem arising in the modeling of an electrostatically actuated plate is considered and a representation formula for the derivative of the associated electrostatic energy with respect to the…
We study the problem of finding the one-dimensional structure in a given data set. In other words we consider ways to approximate a given measure (data) by curves. We consider an objective functional whose minimizers are a regularization of…
The control of complex networks has generated a lot of interest in a variety of fields from traffic management to neural systems. A commonly used metric to compare two particular control strategies that accomplish the same task is the…
We study the spectrum of the join of several circulant matrices. We apply our results to compute explicitly the spectrum of certain graphs obtained by joining several circulant graphs.
The aim of this study is to analyze the properties of harmonic fields in the vicinity of rough boundaries where either a constant potential or a zero flux is imposed, while a constant field is prescribed at an infinite distance from this…
Exact expressions of the boundary state and the form factors of the Ising model are used to derive differential equations for the one-point functions of the energy and magnetization operators of the model in the presence of a boundary…
Energy-based models for discrete domains, such as graphs, explicitly capture relative likelihoods, naturally enabling composable probabilistic inference tasks like conditional generation or enforcing constraints at test-time. However,…
In this paper we study the local regularity of almost minimizers of the functional \begin{equation*} J(u)=\int_\Omega |\nabla u(x)|^2 +q^2_+(x)\chi_{\{u>0\}}(x) +q^2_-(x)\chi_{\{u<0\}}(x) \end{equation*} where $q_\pm \in L^\infty(\Omega)$.…
A key question in the analysis of discrete models for material defects, such as vortices in spin systems and superconductors or isolated dislocations in metals, is whether information on boundary energy for a domain can be sufficient for…
We perform a stochastic-homogenization analysis for composite materials exhibiting a random microstructure. Under the assumptions of stationarity and ergodicity, we characterize the Gamma-limit of a micromagnetic energy functional defined…