Stability of the Brascamp-Lieb constant and applications

We prove that the best constant in the general Brascamp-Lieb inequality is a locally bounded function of the underlying linear transformations. As applications we deduce certain very general Fourier restriction, Kakeya-type, and nonlinear variants of the Brascamp-Lieb inequality which have arisen recently in harmonic analysis

Graduating the age-specific fertility pattern using Support Vector Machines

A topic of interest in demographic literature is the graduation of the age-specific fertility pattern. A standard graduation technique extensively used by demographers is to fit parametric models that accurately reproduce it. Non-parametric statistical methodology might be alternatively used for this graduation purpose. Support Vector Machines (SVM) is a non-parametric methodology that could be utilized for fertility graduation purposes. This paper evaluates the SVM techniques as tools for graduating fertility rates In that we apply these techniques to empirical age specific fertility rates from a variety of populations, time period, and cohorts. Additionally, for comparison reasons we also fit known parametric models to the same empirical data sets.age patterns of fertility, graduation techniques, parametric models of fertility, support vector machines

Ricci Flow from the Renormalization of Nonlinear Sigma Models in the Framework of Euclidean Algebraic Quantum Field Theory

The perturbative approach to nonlinear Sigma models and the associated renormalization group flow are discussed within the framework of Euclidean algebraic quantum field theory and of the principle of general local covariance. In particular we show in an Euclidean setting how to define Wick ordered powers of the underlying quantum fields and we classify the freedom in such procedure by extending to this setting a recent construction of Khavkine, Melati and Moretti for vector valued free fields. As a by-product of such classification, we prove that, at first order in perturbation theory, the renormalization group flow of the nonlinear Sigma model is the Ricci flow.Comment: 24 page

Representations of molecules and materials for interpolation of quantum-mechanical simulations via machine learning

Computational study of molecules and materials from first principles is a cornerstone of physics, chemistry and materials science, but limited by the cost of accurate and precise simulations. In settings involving many simulations, machine learning can reduce these costs, sometimes by orders of magnitude, by interpolating between reference simulations. This requires representations that describe any molecule or material and support interpolation. We review, discuss and benchmark state-of-the-art representations and relations between them, including smooth overlap of atomic positions, many-body tensor representation, and symmetry functions. For this, we use a unified mathematical framework based on many-body functions, group averaging and tensor products, and compare energy predictions for organic molecules, binary alloys and Al-Ga-In sesquioxides in numerical experiments controlled for data distribution, regression method and hyper-parameter optimization

Nonnegative Feynman-Kac Kernels in Schr\"{o}dinger's Interpolation Problem

The existing formulations of the Schr\"{o}dinger interpolating dynamics, which is constrained by the prescribed input-output statistics data, utilize strictly positive Feynman-Kac kernels. This implies that the related Markov diffusion processes admit vanishing probability densities only at the boundaries of the spatial volume confining the process. We extend the framework to encompass singular potentials and associated nonnegative Feynman-Kac-type kernels. It allows to deal with general nonnegative solutions of the Schr\"{o}dinger boundary data problem. The resulting stochastic processes are capable of both developing and destroying nodes (zeros) of probability densities in the course of their evolution.Comment: Latex file, 25 p

The algebra of Wick polynomials of a scalar field on a Riemannian manifold

On a connected, oriented, smooth Riemannian manifold without boundary we consider a real scalar field whose dynamics is ruled by $E$, a second order elliptic partial differential operator of metric type. Using the functional formalism and working within the framework of algebraic quantum field theory and of the principle of general local covariance, first we construct the algebra of locally covariant observables in terms of equivariant sections of a bundle of smooth, regular polynomial functionals over the affine space of the parametrices associated to $E$. Subsequently, adapting to the case in hand a strategy first introduced by Hollands and Wald in a Lorentzian setting, we prove the existence of Wick powers of the underlying field, extending the procedure to smooth, local and polynomial functionals and discussing in the process the regularization ambiguities of such procedure. Subsequently we endow the space of Wick powers with an algebra structure, dubbed E-product, which plays in a Riemannian setting the same role of the time ordered product for field theories on globally hyperbolic spacetimes. In particular we prove the existence of the E-product and we discuss both its properties and the renormalization ambiguities in the underlying procedure. As last step we extend the whole analysis to observables admitting derivatives of the field configurations and we discuss the quantum M{\o}ller operator which is used to investigate interacting models at a perturbative level.Comment: 35 page