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

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

TINGKAT KEPUASAN PELAYANAN ADMINISTRASI KEPESERTAAN PADA PESERTA JKN DI BPJS KESEHATAN KANTOR CABANG MANADO

Memastikan kepuasan peserta adalah aspek penting untuk dipertimbangkan saat terlibat dalam upaya jaminan kualitas layanan. Penelitian ini bertujuan untuk menilai tingkat kepuasan terhadap pelayanan administrasi kepesertaan yang diberikan kepada peserta JKN di Kantor BPJS Kesehatan Cabang Manado. Penelitian yang dilakukan termasuk dalam kategori penelitian kuantitatif, dengan menggunakan pendekatan survei deskriptif. Penelitian berlangsung pada bulan Februari sampai dengan Juli tahun 2023. Populasi sasaran penelitian ini terdiri dari data individu peserta program JKN dan melakukan kunjungan ke Kantor BPJS Kesehatan Cabang Manado pada bulan Januari sampai dengan Desember 2022. Jumlah total kunjungan yang tercatat selama periode ini berjumlah 15.183. Besar sampel penelitian ini adalah 95 orang, dipilih melalui non-probability sampling dengan teknik accidental sampling selama periode penelitian. Analisis yang dilakukan dalam penelitian ini difokuskan pada analisis data univariat, bertujuan untuk memberikan gambaran karakteristik dari masing-masing variabel penelitian, antara lain tangibles (bukti langsung), reliability (kehandalan), responsiveness (ketanggapan), empathy (empati), dan assurance (jaminan). ). Berdasarkan temuan penelitian ini dapat disimpulkan bahwa sebagian besar responden menyatakan puas dengan pelayanan administrasi kepesertaan yang diberikan oleh BPJS Kesehatan Kantor Cabang Manado. Kepuasan ini dikaitkan dengan lima dimensi utama kualitas pelayanan, yaitu bukti fisik, keandalan, daya tanggap, empati, dan jaminan

Applying a Precautionary Approach to Mobile Contact Tracing for COVID-19: The Value of Reversibility

The COVID-19 pandemic presents unprecedented challenges to public health decision-making. Specifically, the lack of evidence and the urgency with which a response is called for, raise the ethical challenge of assessing how much (and what kind of) evidence is required for the justification of interventions in response to the various threats we face. Here we discuss the intervention of introducing technology that aims to trace and alert contacts of infected persons-contact tracing (CT) technology. Determining whether such an intervention is proportional is complicated by complex trade-offs and feedback loops. We suggest that the resulting uncertainties necessitate a precautionary approach. On the one hand, precautionary reasons support CT technology as a means to contribute to the prevention of harms caused by alternative interventions, or COVID-19 itself. On the other hand, however, both the extent to which such technology itself present risks of serious harm, as well as its effectiveness, remain unclear. We therefore argue that a precautionary approach should put reversibility of CT technology at the forefront. We outline several practical implications

Survey on Plum pox virus in Norway

In 1998 Plum pox virus (PPV) was detected for the first time in Norway. Virus-like symptoms were observed on several trees in a collection of plum cultivars at Njøs Research Station in the Sogn og Fjordane County in West Norway. The Norwegian Food Safety Authority and the Norwegian Crop Research Institute immediately started surveying other variety collections around the country, nuclear stock material and orchards in all important plum-growing areas. Since 1998 we have surveyed the main part of the commercial plum orchards in Norway. About 75 000 individual trees have been tested. About 1 % of the trees have been found infected by PPV. Only the PPV-D strain has been found. It is suspected that the main infection source was infected plums or apricots imported to Njøs around 1970 or earlier. In most plum orchards in Norway, the spread of PPV by aphids is relatively slow. Therefore, we expect to be able to eradicate PPV from commercial plum orchards in the near future. The eradication work is continuing.Keywords: Plum pox virus, surve

Continuous and discrete Clebsch variational principles

The Clebsch method provides a unifying approach for deriving variational principles for continuous and discrete dynamical systems where elements of a vector space are used to control dynamics on the cotangent bundle of a Lie group \emph{via} a velocity map. This paper proves a reduction theorem which states that the canonical variables on the Lie group can be eliminated, if and only if the velocity map is a Lie algebra action, thereby producing the Euler-Poincar\'e (EP) equation for the vector space variables. In this case, the map from the canonical variables on the Lie group to the vector space is the standard momentum map defined using the diamond operator. We apply the Clebsch method in examples of the rotating rigid body and the incompressible Euler equations. Along the way, we explain how singular solutions of the EP equation for the diffeomorphism group (EPDiff) arise as momentum maps in the Clebsch approach. In the case of finite dimensional Lie groups, the Clebsch variational principle is discretised to produce a variational integrator for the dynamical system. We obtain a discrete map from which the variables on the cotangent bundle of a Lie group may be eliminated to produce a discrete EP equation for elements of the vector space. We give an integrator for the rotating rigid body as an example. We also briefly discuss how to discretise infinite-dimensional Clebsch systems, so as to produce conservative numerical methods for fluid dynamics