Related papers: Coincidences in 4 dimensions
We show that arithmetic lattices in $\mathrm{SL}_{2}(\mathbb{R})$, stemming from the proper units of an Eichler order in an indefinite quaternion algebra over $\mathbb{Q}$, admit a `small' covering set. In particular, we give bounds on the…
In the present paper, as we did previously in [7], we investigate the relations between the geometric properties of tilings and the algebraic properties of associated relational structures. Our study is motivated by the existence of…
It is known that the groups of Euclidean rotations in dimension 3 (isometries of $S^2$), general Lorentz transformations in dimension 4 (Hyperbolic isometries in dimension 3), and screw motions in dimension 3 can be represented by the…
A coincidence site lattice is a sublattice formed by the intersection of a lattice $\Gamma$ in $\mathbb{R}^d$ with the image of $\Gamma$ under a linear isometry. Such a linear isometry is referred to as a linear coincidence isometry of…
We revisit the subject exploring maps from the space of 4-spinors to 3+1 space-time that commute with the Lorentz transformation. All known mappings have a natural embedding in a higher five dimensional spacetime, and can be succinctly…
Rotations on the 3-dimensional Euclidean vector-space can be represented by real quaternions, as was shown by Hamilton. Introducing complex quaternions allows us to extend the result to elliptic and hyperbolic rotations on the Minkowski…
Quaternionic tori are defined as quotients of the skew field $\mathbb{H}$ of quaternions by rank-4 lattices. Using slice regular functions, these tori are endowed with natural structures of quaternionic manifolds (in fact quaternionic…
In this paper we define infinite-dimensional algebra and its representation, whose basis is naturally identified with semi-infinite configurations of the square ladder model. We also extrapolate the ideas for the cyclic 3-leg triangular…
Motivated by the classical studies on transformations of conjugate nets, we develop the general geometric theory of transformations of their discrete analogues: the multidimensional quadrilateral lattices, i.e. lattices x: Z^N -> R^M, whose…
The relationship between the coincidence indices of a lattice $\Gamma_1$ and a sublattice $\Gamma_2$ of $\Gamma_1$ is examined via the colouring of $\Gamma_1$ that is obtained by assigning a unique colour to each coset of $\Gamma_2$. In…
We theoretically study discrete photonic lattices in more than three dimensions and point out that such systems can exist in continuous three-dimensional (3D) space. We study discrete diffraction in the linear regime, and predict the…
Torus networks of moderate degree have been widely used in the supercomputer industry. Tori are superb when used for executing applications that require near-neighbor communications. Nevertheless, they are not so good when dealing with…
We employ new calculational technique and present complete list of classical $r$-matrices for $D=4$ complex homogeneous orthogonal Lie algebra $\mathfrak{o}(4;\mathbb{C})$, the rotational symmetry of four-dimensional complex space-time.…
We present a method for generating hexagonal aperiodic tilings that are topologically equivalent to the triangular and dice lattices. This approach incorporates aperiodic sequences into the spacing between three sets of grids for the…
Congruence between two n-point sets in 4 dimension can be checked in O(n log n) time. On the way to establishing this result, we revisit several parts of 4-dimensional geometry, such as angles and distances between planes, Hopf fibrations,…
Theories in 5 dimensions with minimal supersymmetry are studied for domain-wall solutions and in the context of the AdS/CFT correspondence. The scalar manifold is a product of a very special real manifold and a quaternionic-Kaehler…
We consider constraints on the S-matrix of any gapped, Lorentz invariant quantum field theory in 3+1 dimensions due to crossing symmetry, analyticity and unitarity. We extremize cubic couplings, quartic couplings and scattering lengths…
Lattice color groups are introduced and used to study the partitioning of a periodically- or quasiperiodically-ordered set of points into N symmetry-related subsets. Applications range from magnetic structure to superlattice ordering in…
The classical dynamics and pulse propagation are presented for a series of lattice-like structures whose spatial dimensionality ranges from one to four: four representing a hyper lattice. The lattices are connected one-dimensional wave…
All indecomposable unimodular hermitian lattices in dimensions 14 and 15 over the ring of integers in $\mathbb{Q}(\sqrt{-3})$ are determined. Precisely one lattice in dimension 14 and two lattices in dimension 15 have minimal norm 3.