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Given recipe of qualitative, kinetic modelling by geometric methods of three-dimensional dendritic crystals. Characteristic features of the perturbations appearing on the surface of a spherical body, leading to different scenarios of the…
Given a projective intersection of two quadrics X in at least 9 variables, the quantitative behaviour of the rational points on X is investigated under the assumption that X contains a pair of conjugate singular points defined over the…
Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are…
A 2-dimensional framework is a straight line realisation of a graph in the Euclidean plane. It is radically solvable if the set of vertex coordinates is contained in a radical extension of the field of rationals extended by the squared edge…
Many questions about triangles and quadrilaterals with rational sides, diagonals and areas can be reduced to solving certain Diophantine equations. We look at a number of such questions including the question of approximating arbitrary…
A famous problem posed by Diophantus was to find sets of distinct positive rational numbers such that the product of any two is one less than a rational square. Such Diophantine sets have been used to construct high rank elliptic curves.…
Mathematical crystal chemistry views crystal structures as the optimal solutions of mathematical optimization problem formalizing inorganic structural chemistry. This paper introduces the minimum and maximum atomic radii depending on the…
This paper, motivated by problems in Diophantine analysis which can be formulated as problems of finding rational points on the intersection of two quadrics, presents an explicit construction of a rationally defined isomorphism (biregular…
In crystallography, a structure is typically represented by the arrangement of atoms in the direct space. Furthermore, space group symmetry and Wyckoff site notations are applied to characterize crystal structures with only a few variables.…
When flat or on a firm mechanical substrate, the atomic composition and atomistic structure of two-dimensional crystals dictate their chemical, electronic, optical, and mechanical properties. These properties change when the two-dimensional…
By using pairs of nontrivial rational solutions of congruent number equation $$ C_N:\;\;y^2=x^3-N^2x, $$ constructed are pairs of rational right (Pythagorean) triangles with one common side and the other sides equal to the sum and…
This paper develops a complete foundational treatment of simplicial complexes from Euclidean spaces through geometric realizations, emphasizing concrete computations, examples, and practical verification methods. Beginning with finite point…
A regular $A_n$-crystal is an edge-colored directed graph, with $n$ colors, related to an irreducible highest weight integrable module over $U_q(sl_{n+1})$. Based on Stembridge's local axioms for regular simply-laced crystals and a…
An algorithm for determining crystal structures from diffraction data is described which does not rely on the usual Fourier-space formulations of atomicity. The new algorithm implements atomicity constraints in real-space, as well as…
The discrete Gaussian model for the surface of a crystal deposited on a disordered substrate is studied by Monte Carlo simulations. A continuous transition is found from a phase with a thermally-induced roughness to a glassy one in which…
In this paper we present some linear algebra behind quadratic parts of quadratically flat complex points of codimension two real submanifold in a complex manifold. Assuming some extra nondegenericity and using the result of Hong, complete…
We have presented computer simulations of conoscopic patterns occurring in optically uniaxial gyrotropic crystals in the vicinity of isotropic point for a number of sets of crystal parameters and orientations. The appearance of special…
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…
For every non-exceptional affine Lie algebra, we explicitly construct a positive geometric crystal associated with a fundamental representation. We also show that its ultra-discretization is isomorphic to the limit of certain perfect…
Fix a finite group $G$. We seek to classify varieties with $G$-action equivariantly birational to a representation of $G$ on affine or projective space. Our focus is odd-dimensional smooth complete intersections of two quadrics, relating…