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

On algebraic structures of numerical integration on vector spaces and manifolds

Numerical analysis of time-integration algorithms has been applying advanced algebraic techniques for more than fourty years. An explicit description of the group of characters in the Butcher-Connes-Kreimer Hopf algebra first appeared in Butcher's work on composition of integration methods in 1972. In more recent years, the analysis of structure preserving algorithms, geometric integration techniques and integration algorithms on manifolds have motivated the incorporation of other algebraic structures in numerical analysis. In this paper we will survey structures that have found applications within these areas. This includes pre-Lie structures for the geometry of flat and torsion free connections appearing in the analysis of numerical flows on vector spaces. The much more recent post-Lie and D-algebras appear in the analysis of flows on manifolds with flat connections with constant torsion. Dynkin and Eulerian idempotents appear in the analysis of non-autonomous flows and in backward error analysis. Non-commutative Bell polynomials and a non-commutative Fa\`a di Bruno Hopf algebra are other examples of structures appearing naturally in the numerical analysis of integration on manifolds.Comment: 42 pages, final versio

Sistem Informasi Geografis Pariwisata Kabupaten Karo Menggunakan Google Maps Berbasis Web

Karo Regency is one of the areas in North Sumatra Province that has the potential to become a tourism area, with the Vision "Realizing Karo Tourism that is advanced, Modern environmentally friendly and highly competitive by maintaining karo cultural values through the broadest participation of the community and business world to increase Regional Original Revenue "The Karo District Government has promoted through mass media such as newspapers, pamphlets, distributing brochures and promoting through blogs. However, this method is not enough to inform tourism extensively towards domestic tourists and foreign tourists. The design and manufacture of Geographic Information Systems (GIS) Tourism is expected to be able to display tourist maps of Karo Regency in the form of a degital map, which can be accessed by tourists. In the Design of Web-Based Karo Tourism Geographic Information System carried out by collecting data, observing and interviewing the Secretary of the Office at the Karo District Tourism and Culture Office and designing tourism maps displayed in this system using Google Maps, the software used in building this application is PHP (Personal Home Page) as a programming language, MySQL as a database server, Macromedia Dreamweaver 8 as the Design Editor. Karo Regency's geographic tourism information system is a system that provides information to tourists about the location of tourist attractions and tourism support facilities in Karo District

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

Analisis Perilaku Keseimbangan Nilai Tukar Riil Di Indonesia

Tujuan artikel ini adalah untuk membuktikan keterkaitan antara variabel ekonomi fundamental dalam kedua sektor moneter dan nyata, dan Behaviour Equilibrium Exchange Rate (BEER) di Indonesia. Sumber data yang digunakan adalah data sekunder (time series) yang bersifat makro dihimpun dari lembaga terkait dengan metode estimasi bulanan dari Januari 2009 sampai Januari 2013. Jumlah sampel yang diestimasi sebanyak 60 data dan model analisis digunakan adalah model BEER. Hasil studi ini menunjukkan bahwa Behaviour Equilibrium Exchange Rate (BEER) dipengaruhi oleh variabel sektor riil (term of trade, produktivitas, dan risiko) dengan nilai sangat tidak signifikan di level 1,17%, variabel sektor moneter (NFA), dan perbedaan antara suku bunga dalam dan luar negeri yang sangat berpengaruh secara signifikan terhadap BEER di level 90,19%.Kata Kunci: Ekonomi, Fundamental, Riil, Moneter ANALYSIS OF REAL EXCHANGE RATE EQUILIBRIUM IN INDONESIA The purpose of this research is to verify the linkages between the fundamental economic variables in both real and monetary sectors, and the Behaviour Equilibrium Exchange Rate (BEER) in Indonesia. The data analysis used is the secondary data (time series), and the estimated data USAge is the monthly data about the real exchange starting from January 2009 until January 2013. Samples 60 units, and the model of analysis used is the BEER model. The result of this study indicated that the Behaviour Equilibrium Exchange Rate (BEER) is influenced by the real sector variables (the terms of trade, productivity, and risk) with the very unsignificant value at the level of 1,17%, and the monetary sector variables (net foreign assets (NFA), and the difference between domestic and foreign interest rates) which very significantly influenced the BEER at the level of 90,19%

One-site density matrix renormalization group and alternating minimum energy algorithm

Given in the title are two algorithms to compute the extreme eigenstate of a high-dimensional Hermitian matrix using the tensor train (TT) / matrix product states (MPS) representation. Both methods empower the traditional alternating direction scheme with the auxiliary (e.g. gradient) information, which substantially improves the convergence in many difficult cases. Being conceptually close, these methods have different derivation, implementation, theoretical and practical properties. We emphasize the differences, and reproduce the numerical example to compare the performance of two algorithms.Comment: Submitted to the proceedings of ENUMATH 201