Related papers: Optimal and non-optimal lattices for non-completel…
We study periodic tessellations of the Euclidean space with unequal cells arising from the minimization of perimeter functionals. Existence results and qualitative properties of minimizers are discussed for different classes of problems,…
We provide theory for computing the lower semi-continuous convex envelope of functionals of the type f(x) plus an l2 misfit, and discuss applications to various non-convex optimization problems. The latter term is a data fit term whereas f…
We study the extension problem on the Sierpinski Gasket ($SG$). In the first part we consider minimizing the functional $\mathcal{E}_{\lambda}(f) = \mathcal{E}(f,f) + \lambda \int f^2 d \mu$ with prescribed values at a finite set of points…
We propose a new formalism to analyse the extremum structure of scale-invariant effective potentials. The problem is stated in a compact matrix form, used to derive general expressions for the stationary point equation and the mass matrix…
We consider a particular instance of the truncated realizability problem on the $d-$dimensional lattice. Namely, given two functions $\rho_1({\bf i})$ and $\rho_2({\bf i},{\bf j})$ non-negative and symmetric on $\mathbb{Z}^d$, we ask…
We propose a method for verifying that a given feasible point for a polynomial optimization problem is globally optimal. The approach relies on the Lasserre hierarchy and the result of Lasserre regarding the importance of the convexity of…
Given a family ${\mathcal F}$ of shapes in the plane, we study what is the lowest possible density of a point set $P$ that pierces (``intersects'', ``hits'') all translates of each shape in ${\mathcal F}$. For instance, if ${\mathcal F}$…
We obtain effective potential of $O(N)$-symmetric $\phi^4$ theory for large $N$ starting with a finite lattice system and taking the thermodynamic limit with great care. In the thermodynamic limit, it is globally real-valued and convex in…
Recent experiments on twisted bilayer graphene show the urgent need for establishing a low-energy lattice model for the system. We use the constrained random phase approximation to study the interaction parameters of such models taking into…
We establish, as $\rho\to 0$, an asymptotic expansion for the minimal Dirichlet energy of $\mathbb S^2$-valued maps outside a finite number of three-dimensional particles of size $\rho$ with fixed centers $x_j\in\mathbb{R}^3$, under general…
A two species interacting system motivated by the density functional theory for triblock copolymers contains long range interaction that affects the two species differently. In a two species periodic assembly of discs, the two species…
We consider in this paper elliptic equations which are perturbations of Laplace's equation by a compactly supported potential. We show that in dimension greater than three for a wide class of potentials all the solutions are globally…
In this paper we consider the problem of characterizing the minimum energy configurations of a finite system of particles interacting between them due to attracting or repulsive forces given by a certain inter molecular potential. We limit…
Densities of particles on $\Rn$ which interact pairwise through an attractive-repulsive power-law potential $W_{\al,\bt}(x) = |x|^\al/\al-|x|^\bt/\bt$ have often been used to explain patterns produced by biological and physical systems. In…
We analyze statistical features of the ``optimization landscape'' in a random version of one of the simplest constrained optimization problems of the least-square type: finding the best approximation for the solution of an overcomplete…
We introduce a general method for designing tailored lattices of magnetic microtraps for ultracold atoms, on the basis of patterned permanently magnetized films. A fast numerical algorithm is used to automatically generate patterns which…
Self-interactions (SIs) are a major problem in density functional approximations and the source of serious divergence from experimental results. Here, we propose to optimize density functional total energies in terms of the effective local…
Based on a characterization of the optimality of a feasible solution of a convex entropy minimization problem, one shows that the feasible solutions obtained using formally the Lagrange multipliers method are optimal.
We have previously used inverse statistical-mechanical methods to optimize isotropic pair interactions with multiple extrema to yield low-coordinated crystal classical ground states (e.g., honeycomb and diamond structures) in d-dimensional…
We consider tight-binding models on Bravais lattices with anisotropic onsite potentials that vary along a given direction and are constant along the transverse one. Inspired by our previous work on flat bands in anti-\(\mathcal{PT}\)…