Related papers: Reciprocal Space and Crystal Planes in Geometric A…
We explain how following the representation of 3D crystallographic space groups in geometric algebra it is further possible to similarly represent the 162 socalled subperiodic groups of crystallography in geometric algebra. We construct a…
By recasting metrical geometry in a purely algebraic setting, both Euclidean and non-Euclidean geometries can be studied over a general field with an arbitrary quadratic form. Both an affine and a projective version of this new theory are…
This work is a continuation of [1]. As in the previous article, here we will describe some interesting ideas and a lot of new theorems in plane geometry related to them.
This note is an attempt to extend "Geometric Langlands Conjecture" from algebraic curves to algebraic surfaces. We introduce certain Hecke-type operators on vector bundles on an algebraic surface. The crucial observation is that the algebra…
In crystal optics the special status of the rest frame of the crystal means that space-time symmetry is less restrictive of electrodynamic phenomena than it is of static electromagnetic effects. A relativistic justification for this claim…
We consider here square tilings of the plane. By extending the formalism introduced in [3] we build a correspondence between plane maps endowed with an harmonic vector and square tilings satisfying a condition of regularity. In the case of…
The purpose of this paper is to propose the implementation of some methods from algebraic geometry in the theory of gravitation, and more especially in the variational formalism. It has been assumed that the metric tensor depends on two…
Crystal structures can be viewed as assemblies of space-filling polyhedra, which play a critical role in determining material properties such as ionic conductivity and dielectric constant. However, most conventional crystal structure…
Purely real space versions of the differential equations describing the kinematics of a dislocated crystalline medium are considered. The differential geometric structures associated with them are revealed.
The aim of these notes is to present an accessible overview of some topics in classical algebraic geometry which have applications to aspects of discrete integrable systems. Precisely, we focus on surface theory on the algebraic geometry…
We study the reciprocal complement $\mathcal{R}(D)$ of a two-dimensional finitely generated $K$-algebra $D$ by linking it with the properties of a surface with coordinate ring $D$. We give several sufficient criteria to have…
Crystal structures and the Bloch theorem play a fundamental role in condensed matter physics. We extend the static crystal to the dynamic "space-time" crystal characterized by the general intertwined space-time periodicities in $D+1$…
We introduce and study properties of certain new harmonic function spaces on products of upper half-spaces.Norm estimates for the so-called expanded Bergman projections are obtained.Sharp theorems on multipliers acting on certain Sobolev…
In this article we describe the projective representation of Plesken Lie algebras and equivalent central extensions of these algebras. Further it is also shown that there exists a bijective correspondence between second cohomology group,…
In this second part of the paper, dedicated to theories with extra dimensions, a new physical notion about the "tensor length scale" is introduced, based on the gravitational theories with covariant and contravariant metric tensor…
To find a suitable framework for nonlinear crystal optics(NCO), we have revisited linear crystal optics(LCO). At the methodological level, three widely used plane wave bases are compared in terms of eigenanalysis in reciprocal space and…
We study geometric aspects of horizontal 2-plane distributions on the complement of the zero section in the 5-dimensional total space of a rank-3 vector bundle equipped with connection over a surface. We show that any surface in…
Quantum states naturally represent symmetry groups, though often in a projective sense. Intriguingly, the projective nature of crystalline symmetries has remained underexplored until very recently. A series of groundbreaking theoretical and…
We give a brief introduction to the study of the algebraic structures -- and their geometrical interpretations -- which arise in the BRST construction of a conformal string background. Starting from the chiral algebra $\cA$ of a string…
The relation between the properties of a specific crystallographic site and the properties of the full crystal is discussed by using spherical tensors. The concept of spherical tensors is introduced and the way it transforms under the…