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We study the ground states of the extended Gross--Pitaevskii equation with the Lee--Huang--Yang correction from both theoretical and numerical perspectives. Starting from the three-dimensional model, we derive reduced one- and…
We introduce ChebLieNet, a group-equivariant method on (anisotropic) manifolds. Surfing on the success of graph- and group-based neural networks, we take advantage of the recent developments in the geometric deep learning field to derive a…
We show that any closed n-dimensional Riemannian manifold can be embedded by a map constructed from heat kernels at a certain time from a finite number of points. Both this time and this number can be bounded in terms of the dimension, a…
We establish an estimate for the fundamental solution of the heat equation on a closed Riemannian manifold $M$ of dimension at least 3, evolving under the Ricci flow. The estimate depends on some constants arising from a Sobolev imbedding…
In this paper, we consider the one-dimensional semirelativistic Schr\"{o}dinger equation for a particle interacting with $N$ Dirac delta potentials. Using the heat kernel techniques, we establish a resolvent formula in terms of an $N \times…
We study the existence of normalized ground states for the 3D dipolar Bose-Einstein condensate equation with attractive three-body interactions: \begin{align}\label{1} -\Delta u+\beta u+\lambda_1|u|^2 u+\lambda_2…
We present a fully analytical solution to the dynamics of the non-spinning 2.5 post-Newtonian binary problem, accounting for both the long-term (secular) and short-term (oscillatory) temporal behavior, with no restriction on eccentricity.…
We present a nucleus-dependent valence-space approach for calculating ground and excited states of nuclei, which generalizes the shell-model in-medium similarity renormalization group to an ensemble reference with fractionally filled…
Many-body non-equilibrium steady states can be described by a Landau-Ginzburg theory if one allows non-analytic terms in the potential. We substantiate this claim by working out the case of the Ising magnet in contact with a thermal bath…
In machine learning, data is usually represented in a (flat) Euclidean space where distances between points are along straight lines. Researchers have recently considered more exotic (non-Euclidean) Riemannian manifolds such as hyperbolic…
In this work we present an alternative method to construct diverse non-relativistic Chern-Simons supergravity theories in three spacetime dimensions. To this end, we apply the Lie algebra expansion method based on semigroups to a…
In order to study in a regularisation free manner the renormalisability of d=2 supersymmetric non-linear $\si$ models, one has to use the algebraic BRS methods ; moreover, in the absence of an off-shell formulation, one has often to deal…
We point out that the noncommutative selfdual phi^3 model can be mapped to the Kontsevich model, for a suitable choice of the eigenvalues in the latter. This allows to apply known results for the Kontsevich model to the quantization of the…
In this article I give a rigorous construction of the classical and quantum photon field on non-compact manifolds with boundary and in possibly inhomogeneous media. Such a construction is complicated by the presence of zero-modes that may…
In this paper we consider a variational problem related to a model for a nucleon interacting with the $\omega$ and $\sigma$ mesons in the atomic nucleus. The model is relativistic, and we study it in a nuclear physics nonrelativistic limit,…
We develop an alternative description to solve the problem of the ground-state energy of the Lieb-Liniger model that describes one-dimensional bosons with contact repulsion. For this integrable model we express the Lieb integral equation in…
Thin growing tissues (such as plant leaves) can be modelled by a bounded domain $S\subset R^2$ endowed with a Riemannian metric $g$, which models the internal strains caused by the differential growth of the tissue. The elastic energy is…
In this paper, we present a method for denoising and reconstruction of low-dimensional manifold in high-dimensional space. We suggest a multidimensional extension of the Locally Optimal Projection algorithm which was introduced by Lipman et…
In this paper we introduce and analyze the learning scenario of \emph{coupled nonlinear dimensionality reduction}, which combines two major steps of machine learning pipeline: projection onto a manifold and subsequent supervised learning.…
We consider the eigenvalues of the magnetic Laplacian on a bounded domain $\Omega$ of $\mathbb R^2$ with uniform magnetic field $\beta>0$ and magnetic Neumann boundary conditions. We find upper and lower bounds for the ground state energy…