Homeomorphisms of S1 and Factorization

For each $n>0$ there is a one complex parameter family of homeomorphisms of the circle consisting of linear fractional transformations “conjugated by $z \to z^n$”. We show that these families are free of relations, which determines the structure of “the group of homeomorphisms of finite type”. We next consider factorization for more robust groups of homeomorphisms. We refer to this as root subgroup factorization (because the factors correspond to root subgroups). We are especially interested in how root subgroup factorization is related to triangular factorization (i.e., conformal welding) and correspondences between smoothness properties of the homeomorphisms and decay properties of the root subgroup parameters. This leads to interesting comparisons with Fourier series and the theory of Verblunsky coefficients.

An Improvement to Exact Reanalysis Algorithm for Local Non-topological Structural Modifications

A new direct reanalysis algorithm for local high-rank structural modifications, based on triangular factorization, has recently been developed. In this work, an improvement is proposed for further reduction of the workload so that the algorithm becomes more efficient and also suitable for low-rank modifications. Compared to the original algorithm, firstly, an alternative formula for updating the triangular factors is derived to avoid repetitive calculations for certain low-rank cases. Secondly, to maximize the efficiency, a combined algorithm is proposed that estimates the workloads of the original and alternative algorithms for each row individually before numerical calculations and selects the one with smaller workload. Numerical experiments show that compared with full factorization, the combined algorithm is significantly more efficient and expected to save up to 75% of execution time in our numerical examples. The new method can be easily implemented and applied to engineering problems, especially to local and step-by-step modification of structures.

A direct topological reanalysis algorithm based on updating matrix triangular factorization

Purpose The purpose of this paper is to establish and implement a direct topological reanalysis algorithm for general successive structural modifications, based on the updating matrix triangular factorization (UMTF) method for non-topological modification proposed by Song et al. [Computers and Structures, 143(2014):60-72]. Design/methodology/approach In this method, topological modifications are viewed as a union of symbolic and numerical change of structural matrices. The numerical part is dealt with UMTF by directly updating the matrix triangular factors. For symbolic change, an integral structure which consists of all potential nodes/elements is introduced to avoid side effects on the efficiency during successive modifications. Necessary pre- and post processing are also developed for memory-economic matrix manipulation. Findings The new reanalysis algorithm is applicable to successive general structural modifications for arbitrary modification amplitudes and locations. It explicitly updates the factor matrices of the modified structure and thus guarantees the accuracy as full direct analysis while greatly enhancing the efficiency. Practical implications Examples including evolutionary structural optimization and sequential construction analysis show the capability and efficiency of the algorithm. Originality/value This innovative paper makes direct topological reanalysis be applicable for successive structural modifications in many different areas.

Loops in SL(2, ℂ) and root subgroup factorization

In previous work, we proved that for a [Formula: see text]-valued loop having the critical degree of smoothness (one half of a derivative in the [Formula: see text] Sobolev sense), the following are equivalent: (1) the Toeplitz and shifted Toeplitz operators associated to the loop are invertible, (2) the loop has a unique triangular factorization, and (3) the loop has a unique root subgroup factorization. For a loop [Formula: see text] satisfying these conditions, the Toeplitz determinant [Formula: see text] and shifted Toeplitz determinant [Formula: see text] factor as products in root subgroup coordinates. In this paper, we observe that, at least in broad outline, there is a relatively simple generalization to loops having values in [Formula: see text]. The main novel features are that (1) root subgroup coordinates are now rational functions, i.e. there is an exceptional set and associated uniqueness issues, and (2) the noncompactness of [Formula: see text] entails that loops are no longer automatically bounded, and this (together with the exceptional set) complicates the analysis at the critical exponent.