Algebraic and combinatorial aspects of sandpile monoids on directed graphs

The sandpile group of a graph is a well-studied object that combines ideas from algebraic graph theory, group theory, dynamical systems, and statistical physics. A graph's sandpile group is part of a larger algebraic structure on the graph, known as its sandpile monoid. Most of the work on sandpiles so far has focused on the sandpile group rather than the sandpile monoid of a graph, and has also assumed the underlying graph to be undirected. A notable exception is the recent work of Babai and Toumpakari, which builds up the theory of sandpile monoids on directed graphs from scratch and provides many connections between the combinatorics of a graph and the algebraic aspects of its sandpile monoid. In this paper we primarily consider sandpile monoids on directed graphs, and we extend the existing theory in four main ways. First, we give a combinatorial classification of the maximal subgroups of a sandpile monoid on a directed graph in terms of the sandpile groups of certain easily-identifiable subgraphs. Second, we point out certain sandpile results for undirected graphs that are really results for sandpile monoids on directed graphs that contain exactly two idempotents. Third, we give a new algebraic constraint that sandpile monoids must satisfy and exhibit two infinite families of monoids that cannot be realized as sandpile monoids on any graph. Finally, we give an explicit combinatorial description of the sandpile group identity for every graph in a family of directed graphs which generalizes the family of (undirected) distance-regular graphs. This family includes many other graphs of interest, including iterated wheels, regular trees, and regular tournaments.Comment: v2: Cleaner presentation, new results in final section. Accepted for publication in J. Combin. Theory Ser. A. 21 pages, 5 figure

Quantization of Weyl invariant unimodular gravity with antisymmetric ghost fields

The enforcement of the unimodularity condition in a gravity theory by means of a Lagrange multiplier leads, in general, to inconsistencies upon quantization. This is so, in particular, when the classic linear splitting of the metric between the background and quantum fields is used. To avoid the need of introducing such a Lagrange multiplier while using the classic linear splitting, we carry out the quantization of unimodular gravity with extra Weyl symmetry by using Becchi-Rouet-Stora-Tyutin (BRST) techniques. Here, two gauge symmetries are to be gauge-fixed: transverse diffeomorphisms and Weyl transformations. We perform the gauge-fixing of the transverse diffeomorphism invariance by using BRST transformations that involve antisymmetric ghost fields. We show that these BRST transformations are compatible with the BRST transformations needed to gauge-fix the Weyl symmetry, so that they can be combined in a set of transformations generated by a single BRST operator. Newton's law of gravitation is derived within the BRST formalism we put forward as well as the Slavnov-Taylor equation.Comment: 24 pages, 1 table, 1 figur

On the constraint equations in Einstein-aether theories and the weak gravitational field limit

We discuss the set of constraints for Einstein-aether theories, comparing the flat background case with what is expected when the gravitational fields are dynamic. We note potential pathologies occurring in the weak gravitational field limit for some of the Einstein-aether theories

Dual refractive index and viscosity sensing using polymeric nanofibers optical structures

Porous materials have demonstrated to be ideal candidates for the creation of optical sensors with very high sensitivities. This is due both to the possibility of infiltrating the target substances into them and to their notable surface-to-volume ratio that provides a larger biosensing area. Among porous structures, polymeric nanofibers (NFs) layers fabricated by electrospinning have emerged as a very promising alternative for the creation of low-cost and easy-to-produce high performance optical sensors, for example, based on Fabry-Perot (FP) interferometers. However, the sensing performance of these polymeric NFs sensors is limited by the low refractive index contrast between the NFs porous structure and the target medium when performing in-liquid sensing experiments, which determines a very low amplitude of the FP interference fringes appearing in the spectrum. This problem has been solved with the deposition of a thin metal layer (∼ 3 nm) over the NFs sensing layer. We have successfully used these metal-coated FP NFs sensors to perform several real-time and in-flow refractive index sensing experiments. From these sensing experiments, we have also determined that the sponge-like structure of the NFs layer suffers an expansion/compression process that is dependent of the viscosity of the analyzed sample, what thus gives the possibility to perform a simultaneous dual sensing of refractive index and viscosity of a fluid

Recommendations on innovative strategies related to nutrition, health and welfare of small ruminants

The SOLID project (Sustainable Organic Low-Input Dairying) carried out research to improve the sustainability of low-input/organic dairy systems in different ways. This document summarises major challenges in practice for both organic and low input production systems of small ruminants. Workshops and trials explored feed supply (including feeding of by products and pasture irrigation), and health and welfare as important areas for improvement. The report presents some first conclusion from the participatory trials

PART: Pre-trained Authorship Representation Transformer

Authors writing documents imprint identifying information within their texts: vocabulary, registry, punctuation, misspellings, or even emoji usage. Finding these details is very relevant to profile authors, relating back to their gender, occupation, age, and so on. But most importantly, repeating writing patterns can help attributing authorship to a text. Previous works use hand-crafted features or classification tasks to train their authorship models, leading to poor performance on out-of-domain authors. A better approach to this task is to learn stylometric representations, but this by itself is an open research challenge. In this paper, we propose PART: a contrastively trained model fit to learn \textbf{authorship embeddings} instead of semantics. By comparing pairs of documents written by the same author, we are able to determine the proprietary of a text by evaluating the cosine similarity of the evaluated documents, a zero-shot generalization to authorship identification. To this end, a pre-trained Transformer with an LSTM head is trained with the contrastive training method. We train our model on a diverse set of authors, from literature, anonymous blog posters and corporate emails; a heterogeneous set with distinct and identifiable writing styles. The model is evaluated on these datasets, achieving zero-shot 72.39\% and 86.73\% accuracy and top-5 accuracy respectively on the joint evaluation dataset when determining authorship from a set of 250 different authors. We qualitatively assess the representations with different data visualizations on the available datasets, profiling features such as book types, gender, age, or occupation of the author

Counting Arithmetical Structures on Paths and Cycles

Let G be a finite, connected graph. An arithmetical structure on G is a pair of positive integer vectors d, r such that (diag (d) - A) r=0 , where A is the adjacency matrix of G. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices (diag (d) - A)). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients ((2n-1)/(n-1)) , and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles