Dynamical Systems, Number Theory and Applications

The following sections are included:

• Preface
• Biographical Note on Armin Leutbecher
• Contents

The following sections are included:

• Einleitung
• Vor 1934
• 1934
• Epilog
• References

In this chapter we continue our study of equivariant minimal Lagrangian surfaces in ℂP², characterizing the rotationally equivariant cases and providing explicit formulae for relevant geometric quantities of translationally equivariant minimal Lagrangian surfaces in terms of Weierstrass elliptic functions.

We prove a structure theorem for the multibrot sets, which are the higher degree analogues of the Mandelbrot set, and give a complete picture of the landing behavior of the rational parameter rays and the bifurcation phenomenon. Our proof is inspired by previous works of Schleicher and Milnor on the combinatorics of the Mandelbrot set; in particular, we make essential use of combinatorial tools such as orbit portraits and kneading sequences. However, we avoid the standard global counting arguments in our proof and replace them by local analytic arguments to show that the parabolic and the Misiurewicz parameters are landing points of rational parameter rays.

The Matovich-Pearson equations are a well-known asymptotic regime of the axisymmetric Navier-Stokes equations with moving boundary. They arise as the slender body approximation of the full equations for a highly viscous, Newtonian fluid and are used to describe the dynamics and evolution of thin, viscous fluid filaments, e.g. in the context of fiber spinning. While these equations can be systematically derived, additional boundary conditions have to be imposed to make the resulting initial-boundary value problem well-posed…

If a diffeomorphism of ℝ² has exactly 2 periodic points and one is attracting while the other is a saddle, it is of interest to find out when the boundary of the basin of attraction is the stable manifold of the saddle. Only in a few special cases of Hénon maps (see for example [5]) has this been proved, although in the standard literature on Hénon maps (see for example the books [1, 6]) it is often said to be true based on computer experiments. In this paper we consider a general family of diffeomorphisms

We investigate interplay between generating series and infinite products including partition numbers, functional equations and modular forms. We apply multiplicative Hecke operators on periodic functions and introduce a new type of functional equations characterizing infinite products of Euler type.

I have been familiar with the name of Armin Leutbecher ever since I discovered his book Zahlentheorie during my student years. “In algebra there are many definitions; some of them are needed.” This remark of a fellow student of Professor Leutbecher, quoted in the preface to that book, perfectly echoes my own sentiments when confronted with abstract algebra for the first time. His book proved valuable to me by providing concrete mathematical material to which the abstract theory could be applied and from which it had grown, especially from the field of quadratic number fields. For this reason I feel honored to have been asked to contribute this article, which may be said to have as its subject the Eisenstein integers, to this volume. I dedicate it to Professor Leutbecher with my best wishes. To state the contents more precisely, this is an expository article on the hexagonal lattice, the Epstein zeta function, and an application to physics. The properties of the lattice and the zeta function are reviewed, then the proof of Montgomery of a minimality property of the Epstein zeta function of the hexagonal lattice is outlined. Finally, the connection to a problem in mathematical physics is sketched.

A set ℱ_q of 3-dimensional subspaces of , the 7-dimensional vector space over the finite field 𝔽_q, is said to form a q-analogue of the Fano plane if every 2-dimensional subspace of is contained in precisely one member of ℱ_q. The existence problem for such q-analogues remains unsolved for every single value of q. Here we report on an attempt to construct such q-analogues using ideas from the theory of subspace codes, which were introduced a few years ago by Koetter and Kschischang in their seminal work on error-correction for network coding. Our attempt eventually fails, but it produces the largest subspace codes known so far with the same parameters as a putative q-analogue. In particular we find a ternary subspace code of new record size 6977, and we are able to construct a binary subspace code of the largest currently known size 329 in an entirely computer-free manner.

We describe integral orthogonal groups of signature (2, n) and results on their generators. Special emphasis is devoted to the cases n = 2 and n = 3 as the latter case corresponds to the paramodular group of degree 2.

One of the most important applications and actual drivers for new insights in the structure of mathematical groups is crystallography. It is also one of the areas were mathematics predicted the physical reality long before experiments could really contribute. This changed about a hundred years ago with the emergence of X-ray diffraction which however turned out to still heavily rely on Mathematics that was now needed to solve the phase problem. New challenges in the interpretation came to light when quasi-crystals were discovered and thus opened the scene for further and deeper mathematical insights…

Given a componentwise continuous semiflow S = S(t, x) on a metric space X, it will be shown that S is continuous at every point (t, x) with t > 0. Furthermore, if X is locally compact or the time of the system is ℕ₀, ℤ or ℝ then S is continuous. For this we give a new, completely elementary proof.

This chapter studies the existence of long-time solutions to the Hamiltonian boundary value problem, and their consistent numerical approximation. Such a boundary value problem is, for example, common in Molecular Dynamics, where one aims at finding a dynamic trajectory that joins a given initial state with a final one, with the evolution being governed by classical (Hamiltonian) dynamics. The setting considered here is sufficiently general so that long time transition trajectories connecting two configurations can be included, provided the total energy E is chosen suitably. In particular, the formulation presented here can be used to detect transition paths between two stable basins and thus to prove the existence of long-time trajectories. The starting point is the formulation of the equation of motion of classical mechanics in the framework of Jacobi's principle; a curve shortening procedure inspired by Birkhoff's method is then developed to find geodesic solutions. This approach can be viewed as a string method.

We analyze the relation of the notion of a pluri-Lagrangian system, which recently emerged in the theory of integrable systems, to the classical notion of variational symmetry, due to E. Noether.