On the Lie enveloping algebra of a post-Lie algebra

We consider pairs of Lie algebras $g$ and $\bar{g}$, defined over a common vector space, where the Lie brackets of $g$ and $\bar{g}$ are related via a post-Lie algebra structure. The latter can be extended to the Lie enveloping algebra $U(g)$. This permits us to define another associative product on $U(g)$, which gives rise to a Hopf algebra isomorphism between $U(\bar{g})$ and a new Hopf algebra assembled from $U(g)$ with the new product. For the free post-Lie algebra these constructions provide a refined understanding of a fundamental Hopf algebra appearing in the theory of numerical integration methods for differential equations on manifolds. In the pre-Lie setting, the algebraic point of view developed here also provides a concise way to develop Butcher's order theory for Runge--Kutta methods.Comment: 25 page

Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras

Bell polynomials appear in several combinatorial constructions throughout mathematics. Perhaps most naturally in the combinatorics of set partitions, but also when studying compositions of diffeomorphisms on vector spaces and manifolds, and in the study of cumulants and moments in probability theory. We construct commutative and noncommutative Bell polynomials and explain how they give rise to Fa\`a di Bruno Hopf algebras. We use the language of incidence Hopf algebras, and along the way provide a new description of antipodes in noncommutative incidence Hopf algebras, involving quasideterminants. We also discuss M\"obius inversion in certain Hopf algebras built from Bell polynomials.Comment: 37 pages, final version, to appear in IJA

Representative factor generation for the interactive visual analysis of high-dimensional data

Datasets with a large number of dimensions per data item (hundreds or more) are challenging both for computational and visual analysis. Moreover, these dimensions have different characteristics and relations that result in sub-groups and/or hierarchies over the set of dimensions. Such structures lead to heterogeneity within the dimensions. Although the consideration of these structures is crucial for the analysis, most of the available analysis methods discard the heterogeneous relations among the dimensions. In this paper, we introduce the construction and utilization of representative factors for the interactive visual analysis of structures in high-dimensional datasets. First, we present a selection of methods to investigate the sub-groups in the dimension set and associate representative factors with those groups of dimensions. Second, we introduce how these factors are included in the interactive visual analysis cycle together with the original dimensions. We then provide the steps of an analytical procedure that iteratively analyzes the datasets through the use of representative factors. We discuss how our methods improve the reliability and interpretability of the analysis process by enabling more informed selections of computational tools. Finally, we demonstrate our techniques on the analysis of brain imaging study results that are performed over a large group of subjects

Post-Lie Algebras and Isospectral Flows

In this paper we explore the Lie enveloping algebra of a post-Lie algebra derived from a classical $R$-matrix. An explicit exponential solution of the corresponding Lie bracket flow is presented. It is based on the solution of a post-Lie Magnus-type differential equation

A serological survey of ruminant livestock in Kazakhstan during post-Soviet transitions in farming and disease control

The results of a serological survey of livestock in Kazakhstan, carried out in 1997–1998, are reported. Serum samples from 958 animals (cattle, sheep and goats) were tested for antibodies to foot and mouth disease (FMD), bluetongue (BT), epizootic haemorrhagic disease (EHD), rinderpest (RP) and peste des petits ruminants (PPR) viruses, and to Brucella spp. We also investigated the vaccination status of livestock and related this to changes in veterinary provision since independence in 1991. For the 2 diseases under official surveillance (FMD and brucellosis) our results were similar to official data, although we found significantly higher brucellosis levels in 2 districts and widespread ignorance about FMD vaccination status. The seroprevalence for BT virus was 23%, and seropositive animals were widespread suggesting endemicity, despite the disease not having being previously reported. We found a few seropositives for EHDV and PPRV, which may suggest that these diseases are also present in Kazakhstan. An hierarchical model showed that seroprevalence to FMD and BT viruses were clustered at the farm/village level, rather than at a larger spatial scale. This was unexpected for FMD, which is subject to vaccination policies which vary at the raion (county) level

Interactive Visual Analysis of Heterogeneous Cohort Study Data

Cohort studies in medicine are conducted to enable the study of medical hypotheses in large samples. Often, a large amount of heterogeneous data is acquired from many subjects. The analysis is usually hypothesis-driven, i.e., a specific subset of such data is studied to confirm or reject specific hypotheses. In this paper, we demonstrate how we enable the interactive visual exploration and analysis of such data, helping with the generation of new hypotheses and contributing to the process of validating them. We propose a data-cube based model which handles partially overlapping data subsets during the interactive visualization. This model enables seamless integration of the heterogeneous data, as well as linking spatial and non-spatial views on these data. We implemented this model in an application prototype, and used it to analyze data acquired in the context of a cohort study on cognitive aging. We present case-study analyses of selected aspects of brain connectivity by using the prototype implementation of the presented model, to demonstrate its potential and flexibility

Backward error analysis and the substitution law for Lie group integrators

Butcher series are combinatorial devices used in the study of numerical methods for differential equations evolving on vector spaces. More precisely, they are formal series developments of differential operators indexed over rooted trees, and can be used to represent a large class of numerical methods. The theory of backward error analysis for differential equations has a particularly nice description when applied to methods represented by Butcher series. For the study of differential equations evolving on more general manifolds, a generalization of Butcher series has been introduced, called Lie--Butcher series. This paper presents the theory of backward error analysis for methods based on Lie--Butcher series.Comment: Minor corrections and additions. Final versio

On post-Lie algebras, Lie--Butcher series and moving frames

Pre-Lie (or Vinberg) algebras arise from flat and torsion-free connections on differential manifolds. They have been studied extensively in recent years, both from algebraic operadic points of view and through numerous applications in numerical analysis, control theory, stochastic differential equations and renormalization. Butcher series are formal power series founded on pre-Lie algebras, used in numerical analysis to study geometric properties of flows on euclidean spaces. Motivated by the analysis of flows on manifolds and homogeneous spaces, we investigate algebras arising from flat connections with constant torsion, leading to the definition of post-Lie algebras, a generalization of pre-Lie algebras. Whereas pre-Lie algebras are intimately associated with euclidean geometry, post-Lie algebras occur naturally in the differential geometry of homogeneous spaces, and are also closely related to Cartan's method of moving frames. Lie--Butcher series combine Butcher series with Lie series and are used to analyze flows on manifolds. In this paper we show that Lie--Butcher series are founded on post-Lie algebras. The functorial relations between post-Lie algebras and their enveloping algebras, called D-algebras, are explored. Furthermore, we develop new formulas for computations in free post-Lie algebras and D-algebras, based on recursions in a magma, and we show that Lie--Butcher series are related to invariants of curves described by moving frames.Comment: added discussion of post-Lie algebroid

Pulmonary Nodule Classification in Lung Cancer from 3D Thoracic CT Scans Using fastai and MONAI

We construct a convolutional neural network to classify pulmonary nodules as malignant or benign in the context of lung cancer. To construct and train our model, we use our novel extension of the fastai deep learning framework to 3D medical imaging tasks, combined with the MONAI deep learning library. We train and evaluate the model using a large, openly available data set of annotated thoracic CT scans. Our model achieves a nodule classification accuracy of 92.4% and a ROC AUC of 97% when compared to a “ground truth” based on multiple human raters subjective assessment of malignancy. We further evaluate our approach by predicting patient-level diagnoses of cancer, achieving a test set accuracy of 75%. This is higher than the 70% obtained by aggregating the human raters assessments. Class activation maps are applied to investigate the features used by our classifier, enabling a rudimentary level of explainability for what is otherwise close to “black box” predictions. As the classification of structures in chest CT scans is useful across a variety of diagnostic and prognostic tasks in radiology, our approach has broad applicability. As we aimed to construct a fully reproducible system that can be compared to new proposed methods and easily be adapted and extended, the full source code of our work is available at https://github.com/MMIV-ML/Lung-CT-fastai-2020