Related papers: A Multidimensional Gauss Map
Area preserving maps provide the simplest and most accurate means to visualize and quantify the behavior of nonlinear systems. Convenience of the mapping equations of motion for investigation of transition to chaotic behavior in dynamics of…
"Physical theories of fundamental significance tend to be gauge theories. These are theories in which the physical system being dealt with is described by more variables than there are physically independent degree of freedom. The…
Harmonic maps are nonlinear extensions of harmonic functions. They are critical points of natural energy functionals between Riemannian manifolds. Such type of problems appear in Physics, Geometry of Finance and the study of regularity and…
In one dimension, the theory of the $G$-normal distribution is well-developed, and many results from the classical setting have a nonlinear counterpart. Significant challenges remain in multiple dimensions, and some of what has already been…
We first introduce the class of quasi-algebraically stable meromorphic maps of $\P^k.$ This class is strictly larger than that of algebraically stable meromorphic self-maps of $\P^k.$ Then we prove that all maps in the new class enjoy a…
Causal graph dynamics are transformations over graphs that capture two important symmetries of physics, namely causality and homogeneity. They can be equivalently defined as continuous and translation invariant transformations or functions…
We describe dynamical properties of a map $\mathfrak{F}$ defined on the space of rational functions. The fixed points of $\mathfrak{F}$ are classified and the long time behavior of a subclass is described in terms of Eulerian polynomials.
Continuous limits of discrete systems with long-range interactions are considered. The map of discrete models into continuous medium models is defined. A wide class of long-range interactions that give the fractional equations in the…
Measures generated by Iterated Function Systems composed of uncountably many one--dimensional affine maps are studied. We present numerical techniques as well as rigorous results that establish whether these measures are absolutely or…
A dynamical map is a map which takes one density operator to another. Such a map can be written in an operator-sum representation (OSR) using a spectral decomposition. The method of the construction applies to more general maps which need…
Determining the range of complex maps plays a fundamental role in the study of several complex variables and operator theory. In particular, one is often interested in determining when a given holomorphic function is a self-map of the unit…
Contour maps are widely used to display estimates of spatial fields. Instead of showing the estimated field, a contour map only shows a fixed number of contour lines for different levels. However, despite the ubiquitous use of these maps,…
We study singularities of Gauss maps of fronts and give characterizations of types of singularities of Gauss maps by geometric properties of fronts which are related to behavior of bounded principal curvatures. Moreover, we investigate…
A precise meaning is given to the notion of continuous iteration of a mapping. Usual discrete iterations are extended into a dynamical flow which is a homotopy of them all. The continuous iterate reveals that a dynamical map is formend by…
Any continuous, transitive, piecewise monotonic map is determined up to a binary choice by its dimension module with the associated finite sequence of generators. The dimension module by itself determines the topological entropy of any…
We generalize the two dimensional Lozi map in order to systematically obtain piece-wise continuous maps in three and higher dimensions. Similar to higher-dimensional generalizations of the related Henon map, these higher-dimensional Lozi…
Multidimensional scaling (MDS) is a dimensionality reduction tool used for information analysis, data visualization and manifold learning. Most MDS procedures embed data points in low-dimensional Euclidean (flat) domains, such that…
In regular dynamics, discrete maps are model presentations of discrete dynamical systems, and they may approximate continuous dynamical systems. Maps are used to investigate general properties of dynamical systems and to model various…
Isoradial graphs are a natural generalization of regular graphs which give, for many models of statistical mechanics, the right framework for studying models at criticality. In this survey paper, we first explain how isoradial graphs…
The notion of microscopic state of the system at a given moment of time as a point in the phase space as well as a notion of trajectory is widely used in classical mechanics. However, it does not have an immediate physical meaning, since…