Spread-out percolation on transitive graphs of polynomial growth

Let $G$ be a vertex-transitive graph of superlinear polynomial growth. Given $r>0$, let $G_r$ be the graph on the same vertex set as $G$, with two vertices joined by an edge if and only if they are at graph distance at most $r$ apart in $G$. We show that the critical probability $p_c(G_r)$ for Bernoulli bond percolation on $G_r$ satisfies $p_c(G_r) \sim 1/\mathrm{deg}(G_r)$ as $r\to\infty$. This extends work of Penrose and Bollob\'as-Janson-Riordan, who considered the case $G=\mathbb{Z}^d$. Our result provides an important ingredient in parallel work of Georgakopoulos in which he introduces a new notion of dimension in groups. It also verifies a special case of a conjecture of Easo and Hutchcroft.Comment: 35 page

Schertz style class invariants for higher degree CM fields

Special values of Siegel modular functions for $\operatorname{Sp} (\mathbb{Z})$ generate class fields of CM fields. They also yield abelian varieties with a known endomorphism ring. Smaller alternative values of modular functions that lie in the same class fields (class invariants) thus help to speed up the computation of those mathematical objects. We show that modular functions for the subgroup $\Gamma^0 (N)\subseteq \operatorname{Sp}(\mathbb{Z})$ yield class invariants under some splitting conditions on $N$, generalising results due to Schertz from classical modular functions to Siegel modular functions. We show how to obtain all Galois conjugates of a class invariant by evaluating the same modular function in CM period matrices derived from an \emph{$N$-system}. Such a system consists of quadratic polynomials with coefficients in the real-quadratic subfield satisfying certain congruence conditions modulo $N$. We also examine conditions under which the minimal polynomial of a class invariant is real. Examples show that we may obtain class invariants that are much smaller than in previous constructions

Multi-Type Point Cloud Autoencoder: A Complete Equivariant Embedding for Molecule Conformation and Pose

Representations are a foundational component of any modelling protocol, including on molecules and molecular solids. For tasks that depend on knowledge of both molecular conformation and 3D orientation, such as the modelling of molecular dimers, clusters, or condensed phases, we desire a rotatable representation that is provably complete in the types and positions of atomic nuclei and roto-inversion equivariant with respect to the input point cloud. In this paper, we develop, train, and evaluate a new type of autoencoder, molecular O(3) encoding net (Mo3ENet), for multi-type point clouds, for which we propose a new reconstruction loss, capitalizing on a Gaussian mixture representation of the input and output point clouds. Mo3ENet is end-to-end equivariant, meaning the learned representation can be manipulated on O(3), a practical bonus. An appropriately trained Mo3ENet latent space comprises a universal embedding for scalar and vector molecule property prediction tasks, as well as other downstream tasks incorporating the 3D molecular pose, and we demonstrate its fitness on several such tasks

Target Specific De Novo Design of Drug Candidate Molecules with Graph Transformer-based Generative Adversarial Networks

Discovering novel drug candidate molecules is one of the most fundamental and critical steps in drug development. Generative deep learning models, which create synthetic data given a probability distribution, offer a high potential for designing de novo molecules. However, to be utilisable in real life drug development pipelines, these models should be able to design drug like and target centric molecules. In this study, we propose an end to end generative system, DrugGEN, for the de novo design of drug candidate molecules that interact with intended target proteins. The proposed method represents molecules as graphs and processes them via a generative adversarial network comprising graph transformer layers. The system is trained using a large dataset of drug like compounds and target specific bioactive molecules to design effective inhibitory molecules against the AKT1 protein, which is critically important in developing treatments for various types of cancer. We conducted molecular docking and dynamics to assess the target centric generation performance of the model, as well as attention score visualisation to examine model interpretability. In parallel, selected compounds were chemically synthesised and evaluated in the context of in vitro enzymatic assays, which identified two bioactive molecules that inhibited AKT1 at low micromolar concentrations. These results indicate that DrugGEN's de novo molecules have a high potential for interacting with the AKT1 protein at the level of its native ligands. Using the open access DrugGEN codebase, it is possible to easily train models for other druggable proteins, given a dataset of experimentally known bioactive molecules

RDFGraphGen: An RDF Graph Generator based on SHACL Shapes

Developing and testing modern RDF-based applications often requires access to RDF datasets with certain characteristics. Unfortunately, it is very difficult to publicly find domain-specific knowledge graphs that conform to a particular set of characteristics. Hence, in this paper we propose RDFGraphGen, an open-source RDF graph generator that uses characteristics provided in the form of SHACL (Shapes Constraint Language) shapes to generate synthetic RDF graphs. RDFGraphGen is domain-agnostic, with configurable graph structure, value constraints, and distributions. It also comes with a number of predefined values for popular schema.org classes and properties, for more realistic graphs. Our results show that RDFGraphGen is scalable and can generate small, medium, and large RDF graphs in any domain.Comment: 11 pages, 2 figure

Generalizations and challenges for the spacetime block-diagonalization

Discovery that gravitational field equations may coerce the spacetime metric with isometries to attain a block-diagonal form compatible with these isometries, was one of the gems built into the corpus of black hole uniqueness theorems. We revisit the geometric background of a block-diagonal metric with isometries, foliation defined by Killing vector fields and the corresponding Godbillon-Vey characteristic class. Furthermore, we analyse sufficient conditions for various matter sources, including scalar, nonlinear electromagnetic and Proca fields, that imply the isometry-compatible block-diagonal form of the metric. Finally, we generalize the theorem on the absence of null electromagnetic fields in static spacetimes to an arbitrary number of spacetime dimensions, wide class of gravitational field equations and nonlinear electromagnetic fields.Comment: 22 pages (ver. 3: equation (51) corrected to m-dimensional case

On the molecular nature of the Ωc(3120)\Omega_c(3120) and its analogy with the Ω(2012)\Omega(2012)

We make a study of the $\Omega_c(3120)$, one of the five $\Omega_c$ states observed by the LHCb collaboration, which is well reproduced as a molecular state from the $\Xi^*_c \bar K$ and $\Omega^*_c \eta$ channels mostly. The state with $J^P = 3/2^-$ decays to $\Xi_c \bar K$ in $D$-wave and we include this decay channel in our approach, as well as the effect of the $\Xi^*_c$ width. With all these ingredients, we determine the fraction of the $\Omega_c(3120)$ width that goes into $\Xi_c \pi \bar K$, which could be a measure of the $\Xi^*_c \bar K$ molecular component, but due to a relatively big binding, compared to its analogous $\Omega(2012)$ state, we find only a small fraction of about 3%, which makes this measurement difficult with present statistics. As an alternative, we evaluate the scattering length and effective range of the $\Xi^*_c \bar K$ and $\Omega^*_c \eta$ channels which together with the binding and width of the $\Omega_c(3120)$ state, could give us an answer to the issue of the compositeness of this state when these magnitudes are determined experimentally, something feasible nowadays, for instance, measuring correlation functions.Comment: 6 pages, 2 figures, 2 tables; References adde

A note on the Schottky problem

In this article, we discuss and survey the recent progress towards the Schottky problem, and make some comments on the relations between the Andr{\'e}-Oort conjecture, Okounkov convex bodies, Coleman's conjecture, stable modular forms, Siegel-Jacobi spaces, stable Jacobi forms and the Schottky problem.Comment: 61 pages. arXiv admin note: text overlap with arXiv:1009.0369, arXiv:1508.03922 by other authors. I added some materials and some references. The title has been change

Dissipation dynamics of a scalar field

We investigate the dissipation rate of a scalar field in the vicinity of the phase transition and the ordered phase, specifically within the universality class of model A. This dissipation rate holds significant physical relevance, particularly in the context of interpreting effective potentials as inputs for dynamical transport simulations, such as hydrodynamics. To comprehensively understand the use of effective potentials and other calculation inputs, such as the functional renormalization group, we conduct a detailed analysis of field dependencies. We solve the functional renormalization group equations on the Schwinger-Keldysh contour to determine the effective potential and dissipation rate for both finite and infinite volumes. Furthermore, we conduct a finite-size scaling analysis to calculate the dynamic critical exponent z. Our extracted value closely matches existing values from the literature.Comment: 17 pages, 6 figures. Code available on Github: https://github.com/laurabatini/flow-equations-code. v2: added a citation, v3: corrected typos, published version from PR

Pointwise estimates for rough operators with applications to Sobolev inequalities

We investigate Sobolev inequalities for several rough operators. We prove that several operators satisfy a pointwise bound by the Riesz potential applied to the gradient. From this inequality, we derive several new Sobolev-type inequalities with an operator on the left-hand side.Comment: v2: typos correcte