Related papers: A shape evolution model under affine transformatio…
In the present paper we investigate the mechanics of systems of affinely-rigid bodies, i.e., bodies rigid in the sense of affine geometry. Certain physical applications are possible in modelling of molecular crystals, granular media, and…
Having in view some applications in nanophysics, in particular in nanophysics of materials, we develop new dynamical models of structured bodies with affine internal degrees of freedom. In particular, we construct some models where not only…
We introduce a class of examples which provide an affine generalization of the nonholonomic problem of a convex body rolling without slipping on the plane. We investigate dynamical aspects of the system such as existence of first integrals,…
We classify affine varieties with an action of a connected, reductive algebraic group such that the group is isomorphic to an open orbit in the variety. This is accomplished by associating a set of one-parameter subgroups of the group to…
Discussed is a model of collective and internal degrees of freedom with kinematics based on affine group and its subgroups. The main novelty in comparison with the previous attempts of this kind is that it is not only kinematics but also…
Discrete models have a long tradition in engineering, including finite state machines, Boolean networks, Petri nets, and agent-based models. Of particular importance is the question of how the model structure constrains its dynamics. This…
The purpose of this publication is to derive and discuss equations of motion of affinely rigid (homogeneously deformable) body moving in Euclidean space of general dimension $n$. Our aim is to present some analytical methods and to discuss…
In this paper we study the general affine differential geometry of surfaces in affine space $A^3$. For a regular elliptical surface we define a moving frame of minimal order and get the complete system of differential invariants. As an…
We study the property of a normal scheme, that the complement of every hypersurface is an affine scheme. To this end we introduce the affine class group. It is a factor group of the divisor class group and measures the deviation from this…
Presented is description of kinematics and dynamics of material points with internal degrees of freedom moving in a Riemannian manifold. The models of internal degrees of freedom we concentrate on are based on the orthogonal and affine…
Convex geometry has recently attracted great attention as a framework to formulate general probabilistic theories. In this framework, convex sets and affine maps represent the state spaces of physical systems and the possible dynamics,…
We report on the erosion of flat linoleum "pebbles" under steady rotation in a slurry of abrasive grit. To quantify shape as a function of time, we develop a general method in which the pebble is photographed from multiple angles with…
The simple realistic model of the tippe top is considered. An averaged system of equations of motion is obtained in special evolutionary variables. Through the qualitative analysis of this system the general features of the motion of the…
We propose a geometrically motivated mathematical model which reveals the key features of coastal and fluvial fragment shape evolution from the earliest stages of the abrasion. Our \textit{collisional polygon model} governs the evolution…
Models for fluid deformable surfaces provide valid theories to describe the dynamics of thin fluidic sheets of soft materials. To use such models in morphogenesis and development requires to incorporate active forces. We consider active…
This paper studies a longitudinal shape transformation model in which shapes are deformed in response to an internal growth potential that evolves according to an advection reaction diffusion process. This model extends prior works that…
We illustrate shape mode analysis as a simple, yet powerful technique to concisely describe complex biological shapes and their dynamics. We characterize undulatory bending waves of beating flagella and reconstruct a limit cycle of…
We give a survey of the theory of affine spheres, emphasizing the convex cases and relationsships to Monge-Ampere equations and geometric structures on manifolds.
The goal of this survey article is to explain and elucidate the affine structure of recent models appearing in the rough volatility literature, and show how it leads to exponential-affine transform formulas.
We introduce a class of objects which we call 'affine surfaces'. These provide families of foliations on surfaces whose dynamics we are interested in. We present and analyze a couple of examples, and we define concepts related to these in…