Related papers: On the geometry of a dislocated medium
Some aspects of the multidimensional soliton geometry are considered.
We develop a rigorous framework for modelling the geometry equilibration of crystalline defects. We formulate the equilibration of crystal defects as a variational problems on a discrete energy space and establish qualitatively sharp…
The geometry of the $q$-deformed line is studied. A real differential calculus is introduced and the associated algebra of forms represented on a Hilbert space. It is found that there is a natural metric with an associated linear connection…
Some differential equations are considered in the context of Synthetic Differential Geometry. Here, this means that not only nilpotent infinitesimals, but also the formation of function spaces, is exploited. In particular, we utilize…
We analyze geometrical structures necessary to represent bulk and surface interactions of standard and substructural nature in complex bodies. Our attention is mainly focused on the influence of diffuse interfaces on sharp discontinuity…
Mostly aimed at an audience with backgrounds in geometry and homological algebra, these notes offer an introduction to derived geometry based on a lecture course given by the second author. The focus is on derived algebraic geometry, mainly…
We explain how to translate several recent results in derived algebraic geometry to derived differential geometry. These concern shifted Poisson structures on NQ-manifolds, Lie groupoids, smooth stacks and derived generalisations, and…
The structure and properties of possible $q$-Minkowski spaces is discussed, and the corresponding non-commutative differential calculi are developed in detail and compared with already existing proposals. This is done by stressing its…
We introduce an original approach to geometric calculus in which we define derivatives and integrals on functions which depend on extended bodies in space--that is, paths, surfaces, and volumes etc. Though this theory remains to be fully…
The field of Materials Science is concerned with, e.g., properties and performance of materials. An important class of materials are crystalline materials that usually contain ``dislocations'' -- a line-like defect type. Dislocation…
A short introduction to the mathematical methods and technics of differential algebras and modules adapted to the problems of mathematical and theoretical physics is presented.
The behavior of dynamical system interacting with non-equilibrium medium is investigated. Formally exact kinetic equations are derived for the statistical operator of the dynamical system and the macroscopic parameters of the medium. In the…
We review very briefly the main mathematical structures and results in some important areas of Quantum Mechanics involving PDEs and formulate open problems.
A general relativistic description of a disk rotating at constant angular velocity is given. It is argued that conceptually this direct approach poses fewer problems than the special relativistic one. For observers on the disk, the geometry…
We develop the notion of crystal in the context of derived algebraic geometry, and to connect crystals to more classical objects such as D-modules.
Differentiable physics provides a new approach for modeling and understanding the physical systems by pairing the new technology of differentiable programming with classical numerical methods for physical simulation. We survey the rapidly…
We formulate the laws governing the dynamics of a crystalline solid in which a continuous distribution of dislocations is present. Our formulation is based on new differential geometric concepts, which in particular relate to Lie groups. We…
The central structure in various versions of noncommutative geometry is a differential calculus on an associative algebra. This is an analogue of the calculus of differential forms on a manifold. In this short review we collect examples of…
This paper describes the foundations of a differential geometry of a quaternionic curves. The Frenet-Serret equations and the evolutes and evolvents of a particular quaternionic curve are accordingly determined. This new formulation takes…
The relation between differential geometry of surfaces and some Heisenberg ferromagnet models is considered.