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The generalized de Rham-Hodge theory aspects of Delsarte-Darboux type transformations in multidimension

Anatoliy Samoilenko, Yarema Prykarpatsky, Anatoliy Prykarpatsky (2005)

Open Mathematics

The differential-geometric and topological structure of Delsarte transmutation operators and their associated Gelfand-Levitan-Marchenko type eqautions are studied along with classical Dirac type operator and its multidimensional affine extension, related with selfdual Yang-Mills eqautions. The construction of soliton-like solutions to the related set of nonlinear dynamical system is discussed.

The generic isometry and measure preserving homeomorphism are conjugate to their powers

Christian Rosendal (2009)

Fundamenta Mathematicae

It is known that there is a comeagre set of mutually conjugate measure preserving homeomorphisms of Cantor space equipped with the coinflipping probability measure, i.e., Haar measure. We show that the generic measure preserving homeomorphism is moreover conjugate to all of its powers. It follows that the generic measure preserving homeomorphism extends to an action of (ℚ, +) by measure preserving homeomorphisms, and, in fact, to an action of the locally compact ring 𝔄 of finite adèles. ...

The generic transformation has roots of all orders

Jonathan King (2000)

Colloquium Mathematicae

In the sense of the Baire Category Theorem we show that the generic transformation T has roots of all orders (RAO theorem). The argument appears novel in that it proceeds by establishing that the set of such T is not meager - and then appeals to a Zero-One Law (Lemma 2). On the group Ω of (invertible measure-preserving) transformations, §D shows that the squaring map p: S → S^{2} is topologically complex in that both the locally-dense and locally-lacunary points of p are dense (Theorem 23). The...

The geography of simply-connected symplectic manifolds

Mi Sung Cho, Yong Seung Cho (2003)

Czechoslovak Mathematical Journal

By using the Seiberg-Witten invariant we show that the region under the Noether line in the lattice domain ℤ × ℤ is covered by minimal, simply connected, symplectic 4-manifolds.

The Geometric and Dynamic Essence of Phyllotaxis

P. Atela (2011)

Mathematical Modelling of Natural Phenomena

We present a dynamic geometric model of phyllotaxis based on two postulates, primordia formation and meristem expansion. We find that Fibonacci, Lucas, bijugate and multijugate are all variations of the same unifying phenomenon and that the difference lies in the changes in position of initial primordia. We explore the set of all initial positions and color-code its points depending on the phyllotactic pattern that arises.

The geometry of Calogero-Moser systems

Jacques Hurtubise, Thomas Nevins (2005)

Annales de l’institut Fourier

We give a geometric construction of the phase space of the elliptic Calogero-Moser system for arbitrary root systems, as a space of Weyl invariant pairs (bundles, Higgs fields) on the r -th power of the elliptic curve, where r is the rank of the root system. The Poisson structure and the Hamiltonians of the integrable system are given natural constructions. We also exhibit a curious duality between the spectral varieties for the system associated to a root system, and the Lagrangian varieties for...

The geometry of dented pentagram maps

Boris Khesin, Fedor Soloviev (2016)

Journal of the European Mathematical Society

We propose a new family of natural generalizations of the pentagram map from 2D to higher dimensions and prove their integrability on generic twisted and closed polygons. In dimension d there are d – 1 such generalizations called dented pentagram maps, and we describe their geometry, continuous limit, and Lax representations with a spectral parameter. We prove algebraic-geometric integrability of the dented pentagram maps in the 3D case and compare the dimensions of invariant tori for the dented maps...

The geometry of laminations

Robbert Fokkink, Lex Oversteegen (1996)

Fundamenta Mathematicae

A lamination is a continuum which locally is the product of a Cantor set and an arc. We investigate the topological structure and embedding properties of laminations. We prove that a nondegenerate lamination cannot be tree-like and that a planar lamination has at least four complementary domains. Furthermore, a lamination in the plane can be obtained by a lakes of Wada construction.

The Geometry of Model Spaces for Probability-Preserving Actions of Sofic Groups

Tim Austin (2016)

Analysis and Geometry in Metric Spaces

Bowen’s notion of sofic entropy is a powerful invariant for classifying probability-preserving actions of sofic groups. It can be defined in terms of the covering numbers of certain metric spaces associated to such an action, the ‘model spaces’. The metric geometry of these model spaces can exhibit various interesting features, some of which provide other invariants of the action. This paper explores an approximate connectedness property of the model spaces, and uses it give a new proof that certain...

The geometry of nondegeneracy conditions in completely integrable systems (corrected version of fascicule 4, volume XIV, 2005, p. 705-719)

Nicolas Roy (2006)

Annales de la faculté des sciences de Toulouse Mathématiques

Nondegeneracy conditions need to be imposed in K.A.M. theorems to insure that the set of diophantine tori has a large measure. Although they are usually expressed in action coordinates, it is possible to give a geometrical formulation using the notion of regular completely integrable systems defined by a fibration of a symplectic manifold by lagrangian tori together with a Hamiltonian function constant on the fibers. In this paper, we give a geometrical definition of different nondegeneracy conditions,...

The growth rate and dimension theory of beta-expansions

Simon Baker (2012)

Fundamenta Mathematicae

In a recent paper of Feng and Sidorov they show that for β ∈ (1,(1+√5)/2) the set of β-expansions grows exponentially for every x ∈ (0,1/(β-1)). In this paper we study this growth rate further. We also consider the set of β-expansions from a dimension theory perspective.

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