On the composition of finite rotations in $\mathbb{E}^4$

Journal of Modern Dynamics, 2015

We investigate the notion of complex rotation number which was introduced by V. I. Arnold in 1978. Let f : R/Z → R/Z be an orientation preserving circle diffeomorphism and let ω ∈ C/Z be a parameter with positive imaginary part. Construct a complex torus by glueing the two boundary components of the annulus { z ∈ C/Z | 0 < Im(z) < Im(ω) } via the map f + ω. This complex torus is isomorphic to C/(Z + τ Z) for some appropriate τ ∈ C/Z. According to Moldavskis [6], if the ordinary rotation number rot(f + ω 0) is Diophantine and if ω tends to ω 0 non tangentially to the real axis, then τ tends to rot(f + ω 0). We show that the Diophantine and non tangential assumptions are unnecessary: if rot(f + ω 0) is irrational then τ tends to rot(f + ω 0) as ω tends to ω 0. This, together with results of N.Goncharuk [4], motivates us to introduce a new fractal set ("bubbles"), given by the limit values of τ as ω tends to the real axis. For the rational values of rot(f + ω 0), these limits do not necessarily coincide with rot(f + ω 0) and form a countable number of analytic loops in the upper half-plane. Notation: • H = H + is the set of complex numbers with positive imaginary part. • H − is the set of complex numbers with negative imaginary part. • If p/q is a rational number, then p and q are assumed to be coprime. • If x and y are distinct points in R/Z, then (x, y) denotes the set of points z ∈ R/Z − { x, y } such that the three points x, z, y are in increasing order and [x, y]:=(x, y) ∪ { x, y }. • rot(f) ∈ R/Z is a rotation number of an orientation-preserving circle diffeomorphism f .

Transactions of A. Razmadze Mathematical Institute, 2016

It is studied the following problem: for a given function f what kind of may be a set of all rotations γ for which f is not differentiable with respect to γ-rotation of a given basis B? In particular, for translation invariant bases on the plane it is found the topological structure of possible sets of singular rotations.

Annales de l’institut Fourier, 1984

Computer Methods in Applied Mechanics and Engineering, 1997

In this work we discuss some aspects of the three-dimensional finite rotations pertinent to the formulation and computational treatment of the geometrically exact structural theories. Among various possibilities to parameterize the finite rotations, special attention is dedicated to a choice featuring an incremental rotation vector. Some computational aspects pertinent to the implementation of the Newton iterative scheme and the Newmark time-stepping algorithm applied to solving these problems are examined. Representative numerical simulations are presented in order to illustrate the performance of the proposed formulation. 0. Dedication Current strong interest in nonlinear analysis of physics phenomena, nourished by ever-growing computational resources, has long been anticipated by J. Tinsley Oden. In the early 1970s he published a book (see [l]), which served as a road-map for many developments which followed. In this and later works, Dr. Oden recognized that the proper setting for these developments is placed at the crossroads between nonlinear mechanics theories and numerical analysis, helping to establish an independent identity of the scientific discipline of Computational Mechanics. In particular, I have always appreciated a distinct style and mathematical rigor of Tinsley's works, which I believe is the only way to make headway in nonlinear problems. I personally benefited from many works of Tinsley Oden, starting from my PhD thesis at UC Berkeley in the late 1980s where the contact model proposed by Oden and Martins [2] proved very useful for structure-foundation interaction problems I was studying at the time. I am also pleased that my contribution selected for this occasion, regarding some computational aspects in structural theories with finite rotations, appears to be related to a very recent work of Tinsley (see [3]) with an ambitious goal of placing these structural theories in a proper harmony with classical continuum theories and delegating some of the traditional engineer's responsibilities to an adaptive modeling procedure.

Topology and its Applications, 2006

This paper is a study of invariant sets that have "geometric" rotation numbers, which we call rotational sets, for the angle-tripling map σ 3 : T → T, and more generally, the angle-d-tupling map σ d : T → T for d 2. The precise number and location of rotational sets for σ d is determined by d − 1, 1 d -length open intervals, called holes, that govern, with some specifiable flexibility, the number and location of root gaps (complementary intervals of the rotational set of length 1 d ). In contrast to σ 2 , the proliferation of rotational sets with the same rotation number for σ d , d > 2, is elucidated by the existence of canonical operations allowing one to reduce σ d to σ d−1 and construct σ d+1 from σ d by, respectively, removing or inserting "wraps" of the covering map that, respectively, destroy or create/enlarge root gaps.