Positive Geometries and Differential Forms with Non-Logarithmic Singularities I

Positive geometries encode the physics of scattering amplitudes in flat space-time and the wavefunction of the universe in cosmology for a large class of models. Their unique canonical forms, providing such quantum mechanical observables, are characterised by having only logarithmic singularities along all the boundaries of the positive geometry. However, physical observables have logarithmic singularities just for a subset of theories. Thus, it becomes crucial to understand whether a similar paradigm can underlie their structure in more general cases. In this paper we start a systematic investigation of a geometric-combinatorial characterisation of differential forms with non-logarithmic singularities, focusing on projective polytopes and related meromorphic forms with multiple poles. We introduce the notions of covariant forms and covariant pairings. Covariant forms have poles only along the boundaries of the given polytope; moreover, their leading Laurent coefficients along any of the boundaries are still covariant forms on the specific boundary. Whereas meromorphic forms in covariant pairing with a polytope are associated to a specific (signed) triangulation, in which poles on spurious boundaries do not cancel completely, but their order is lowered. These meromorphic forms can be fully characterised if the polytope they are associated to is viewed as the restriction of a higher dimensional one onto a hyperplane. The canonical form of the latter can be mapped into a covariant form or a form in covariant pairing via a covariant restriction. We show how the geometry of the higher dimensional polytope determines the structure of these differential forms. Finally, we discuss how these notions are related to Jeffrey-Kirwan residues and cosmological polytopes.Comment: 47 pages, figures in Tik

Cabetian socialism: More's utopian heritage in nineteenth-century Spain

En 1840 se publica Voyage en Icarie una obra que revela uno de los proyectos de socialismo utópico más importantes del siglo XIX. Su autor, Étienne Cabet, probablemente se había inspirado en la lectura de Utopía de Tomás Moro. Esta lectura produjo en el político francés tal impresión que decidió cambiar totalmente su estrategia política. A los pocos años de la primera edición del Viaje a Icaria, un grupo de republicanos catalanes emprende la publicación por entregas de esta obra en el periódico La Fraternidad, y empieza a difundir las ideas de Cabet a través de la publicación de los artículos de Le Populaire. Es el inicio del grupo cabetiano más importante fuera de Francia. El objetivo de este texto es analizar el legado que Moro tuvo en el proyecto utópico de Cabet para ver, efectivamente, que propuestas del político inglés seguían siendo actuales a los ojos de los pensadores utópicos del siglo XIX; sobre todo a la luz del interés que despertó la propuesta utópica cabetiana en el contexto político españolVoyage en Icarie was published in 1840, this is a book that reveals one of the most important utopian’s projects of the 19th century. His author, Étienne Cabet, was influenced by More’s Utopia in such a manner that he decided to change completely his political strategy. A few years after the publication of The Voyage to Icaria’s first edition, a group of Catalans Republicans decided to publish this book in Spanish in the weekly newspaper La Fraternidad. They also started to propagate Cabet’s ideas through the publication of articles of the newspaper Le Populaire. This is the beginning of the most important icarian group out of France. The purpose of this text is to investigate More’s legacy for this utopian project, and find out which proposals of the English chancellor could be valid for a utopian thinker of the 19th century, mainly due to the interest aroused by the icarian proposal in SpainEste trabajo se ha realizado en el marco del proyecto HAR2015-65957-P del Plan Nacional de I+D+i MINECO-FEDER (Historia del futuro: la utopía y sus alternativas en los horizontes de expectativa del mundo contemporáneo, siglos XIX-XXI

Uniform sampling of steady states in metabolic networks: heterogeneous scales and rounding

The uniform sampling of convex polytopes is an interesting computational problem with many applications in inference from linear constraints, but the performances of sampling algorithms can be affected by ill-conditioning. This is the case of inferring the feasible steady states in models of metabolic networks, since they can show heterogeneous time scales . In this work we focus on rounding procedures based on building an ellipsoid that closely matches the sampling space, that can be used to define an efficient hit-and-run (HR) Markov Chain Monte Carlo. In this way the uniformity of the sampling of the convex space of interest is rigorously guaranteed, at odds with non markovian methods. We analyze and compare three rounding methods in order to sample the feasible steady states of metabolic networks of three models of growing size up to genomic scale. The first is based on principal component analysis (PCA), the second on linear programming (LP) and finally we employ the lovasz ellipsoid method (LEM). Our results show that a rounding procedure is mandatory for the application of the HR in these inference problem and suggest that a combination of LEM or LP with a subsequent PCA perform the best. We finally compare the distributions of the HR with that of two heuristics based on the Artificially Centered hit-and-run (ACHR), gpSampler and optGpSampler. They show a good agreement with the results of the HR for the small network, while on genome scale models present inconsistencies.Comment: Replacement with major revision

Cluster Adjacency for m=2 Yangian Invariants

11 pages, 3 figuresWe classify the rational Yangian invariants of the $m=2$ toy model of $\mathcal{N}=4$ Yang-Mills theory in terms of generalised triangles inside the amplituhedron $\mathcal{A}_{n,k}^{(2)}$. We enumerate and provide an explicit formula for all invariants for any number of particles $n$ and any helicity degree $k$. Each invariant manifestly satisfies cluster adjacency with respect to the $Gr(2,n)$ cluster algebra.Peer reviewe

Multi-objective reinforcement learning with continuous pareto frontier approximation

This paper is about learning a continuous approximation of the Pareto frontier in Multi-Objective Markov Decision Problems (MOMDPs). We propose a policy-based approach that exploits gradient information to generate solutions close to the Pareto ones. Differently from previous policy-gradient multi-objective algorithms, where n optimization routines are used to have n solutions, our approach performs a single gradient-ascent run that at each step generates an improved continuous approximation of the Pareto frontier. The idea is to exploit a gradient-based approach to optimize the parameters of a function that defines a manifold in the policy parameter space so that the corresponding image in the objective space gets as close as possible to the Pareto frontier. Besides deriving how to compute and estimate such gradient, we will also discuss the non-trivial issue of defining a metric to assess the quality of the candidate Pareto frontiers. Finally, the properties of the proposed approach are empirically evaluated on two interesting MOMDPs