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The minimal exponent of cones over smooth complete intersection projective varieties
No full textWe compute the minimal exponent of the affine cone over a complete
intersection of smooth projective hypersurfaces intersecting transversely. The
upper bound for the minimal exponent is proved, more generally, in the weighted
homogeneous setting, while the lower bound is deduced from a general lower
bound in terms of a strong factorizing resolution in the sense of Bravo and
Villamayor.Comment: 14 pages; v.2: minor typos fixed to agree with the published versio
Multi-Type Point Cloud Autoencoder: A Complete Equivariant Embedding for Molecule Conformation and Pose
Full text linkRepresentations 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
Spread-out percolation on transitive graphs of polynomial growth
Full text linkLet be a vertex-transitive graph of superlinear polynomial growth. Given
, let be the graph on the same vertex set as , with two vertices
joined by an edge if and only if they are at graph distance at most apart
in . We show that the critical probability for Bernoulli bond
percolation on satisfies as
. This extends work of Penrose and Bollob\'as-Janson-Riordan, who
considered the case .
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
New second-order optimality conditions for directional optimality of a general set-constrained optimization problem
No full textIn this paper we derive new second-order optimality conditions for a very
general set-constrained optimization problem where the underlying set may be
nononvex. We consider local optimality in specific directions (i.e., optimal in
a directional neighborhood) in pursuit of developing these new optimality
conditions. First-order necessary conditions for local optimality in given
directions are provided by virtue of the corresponding directional normal
cones. Utilizing the classical and/or the lower generalized support function,
we obtain new second-order necessary and sufficient conditions for local
optimality of general nonconvex constrained optimization problem in given
directions via both the corresponding asymptotic second-order tangent cone and
outer second-order tangent set. Our results do not require convexity and/or
nonemptyness of the outer second-order tangent set. This is an important
improvement to other results in the literature since the outer second-order
tangent set can be nonconvex and empty even when the set is convex
RDFGraphGen: An RDF Graph Generator based on SHACL Shapes
Full text linkDeveloping 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
Conservation Laws For Every Quantum Measurement Outcome
No full textIn the paradigmatic example of quantum measurements, whenever one measures a
system which starts in a superposition of two states of a conserved quantity,
it jumps to one of the two states, implying different final values for the
quantity that should have been conserved. The standard law of conservation for
quantum mechanics handles this jump by stating only that the total distribution
of the conserved quantity over repeated measurements is unchanged, but states
nothing about individual cases. Here however we show that one can go beyond
this and have conservation in each individual instance. We made our arguments
in the case of angular momentum of a particle on a circle, where many
technicalities simplify, and bring arguments to show that this holds in full
generality. Hence we argue that the conservation law in quantum mechanics
should be rewritten, to go beyond its hitherto statistical formulation, to
state that the total of a conserved quantity is unchanged in every individual
measurement outcome. As a further crucial element, we show that conservation
can be localised at the level of the system of interest and its relevant frame
of reference, and is independent on any assumptions on the distribution of the
conserved quantity over the entire universe.Comment: 9 pages, 1 table, some clarifications particularly in the
introduction/conclusion based on referee feedbac
Target Specific De Novo Design of Drug Candidate Molecules with Graph Transformer-based Generative Adversarial Networks
Full text linkDiscovering 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
Schertz style class invariants for higher degree CM fields
Full text linkSpecial values of Siegel modular functions for 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 yield class invariants under some splitting
conditions on , 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{-system}. Such a system consists of
quadratic polynomials with coefficients in the real-quadratic subfield
satisfying certain congruence conditions modulo . 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
Entropic relations for indistinguishable quantum particles
Full text linkThe von Neumann entropy of a -body reduced density matrix quantifies the entanglement between quantum particles and the remaining ones. In this short paper, we rigorously prove general properties of this entanglement entropy as a function of : it is concave for all and non-decreasing until the midpoint . The results hold for indistinguishable quantum particles and are independent of the statistics.8 pages; revised published version with different title, but same result