wolfram cellular automata are considered and their operation is demonstrated using an example of traffic flow simulation. For the class of one-dimensional elementary cellular automata, the concept of linearity is introduced in the language of Zhegalkin operators. An algorithm for finding linear Zhegalkin operators with multipliers of three variables is presented. The algorithm is implemented in Python.

A photonic topological insulator is a structure that isolates radiation in the bulk rather than at the edge (surface). Paradoxically, applications of such an insulator focus on its conducting edge states, which are robust against structural defects. We suggest a tiling photonic topological insulator constructed from identical prism resonators connected to each other. The light beam circulates inside the tiling bulk without propagation. However, we experimentally demonstrate a topologically-protected propagating state due to the disconnected faces of the edge resonators. The investigated state is robust against removing or attaching prism resonators. Moreover, the protection principle is phase-free and therefore highly scalable both in wavelength and resonator size. The tiling is suggested for active topological photonic devices and laser arrays.

A topological insulator is a material that exhibits the properties of a conductor on the surface and of an insulator in the bulk. The Rudner game is a simplified model of a topological insulator implemented on a two-dimensional photonic lattice of resonators, which is described in the language of tricolor four-cycle two-dimensional Wolfram cellular automata. It is a case of a regular two-dimensional lattice, in which each cell is colored in one of three colors (for a photonic topological insulator, these colors mean the presence of a photon in a resonator, the absence of a photon, and a topological insulator boundary). By setting the transformation rule for each cell, depending on the state of the nearest neighbors and the cell itself, for equal discrete time intervals we obtain a cellular automaton. In this study, the Rudner game is rewritten equivalently in terms of operators in the Zhegalkin polynomial ring with coefficients in a field consisting of three elements.

Total internal reflection occurs at the interface between two media with different refractive indices during propagation of light rays from a medium with a higher refractive index to a medium with a lower refractive index. If the thickness of the second medium is comparable to a specific light wavelength, then total internal reflection is violated partially or completely. Based on the frustrated total internal reflection, herein we discuss a two-dimensional photonic topological insulator in an array consisting of triangular, quadrangular, or hexagonal transparent prism resonators with a narrow gap between them. An array of prism resonators allows topologically stable edge solutions (eigenwaves) similar to those studied in ring resonators. Moreover, total internal reflection occurs at different angles of incidence of light. This makes it possible to obtain a set of fundamentally new edge solutions. The light is presumably concentrated at the surface; however, in the new solutions it penetrates relatively deep into the photonic topological insulator and excites several layers of prisms positioned beyond the surface. Remarkably, the direction of light propagation is precisely biased, and therefore new solutions exhibit lower symmetry than the resonator array symmetry.

In physics, a topological insulator is a material that simultaneously exhibits the properties of a conductor on the surface and an insulator in the bulk. An abstract model of a twodimensional topological insulator is described in terms of tricolour cellular automata and excitations of a topological insulator are classified.

In this paper we consider the reductant of the dihedral group Dn, consisting of a set of axial symmetries, and the sphere S2 as a reductant of the group SU(2,C) ∼= S3 (the group of unit quaternions). By introducing the Sabinin’s multiplication on the reductant of Dn, we get a quasigroup with unit

We prove that if u is a locally Lipschitz continuous function on an open set \(\mathcal {X} \subset \mathbb {R}^{n+1}\) satisfying the nonlinear heat equation \(\partial _t u = \Delta (|u|^{p-1} u)\), \(p > 1\), weakly away from the zero set \(u^{-1} (0)\) in \(\mathcal {X}\), then u is a weak solution to this equation in all of \(\mathcal {X}\).

The aim of this paper is to bring together two areas which are of great importance for the study of overdetermined boundary value problems. The first area is homological algebra which is the main tool in constructing the formal theory of overdetermined problems. And the second area is the global calculus of pseudodifferential operators which allows one to develop explicit analysis.

In order to study the Toeplitz algebras related to a Dirac operators in a neighborhood of a closed bounded domain D with smooth boundary in C-n we introduce a singular Cauchy type integral. We compute its principal symbol, thus initiating the index theory.

We prove a Fedosov index formula for the index of Toeplitz operators connected with the Hardy space of solutions to an elliptic system of first-order partial differential equations in a bounded domain in R-n with smooth boundary.

Let be a bounded domain in , , with a smooth boundary . We indicate appropriate Sobolev spaces of negative smoothness to study the non-homogeneous Cauchy problem for an elliptic differential complex of first order operators. In particular, we describe traces on of the tangential part and the normal part of a (vector)-function from the corresponding Sobolev space and give an adequate formulation of the problem. If the Laplacians of the complex satisfy the uniqueness condition in the small then we obtain necessary and sufficient solvability conditions of the problem and produce formulae for its exact and approximate solutions. For the Cauchy problem in the Lebesgue space , we construct the approximate and exact solutions to the Cauchy problem with the maximal possible regularity. Moreover, using Hilbert space methods, we construct Carleman’s formulae for a (vector-) function from the Sobolev space by its Cauchy data on a subset and the values of in modulo the null-space of the Cauchy problem. Some instructive examples for elliptic complexes of operators with constant coefficients are considered.

In this paper we consider a simple formula for analytic continuation in a domain D of special form.

We study the Cauchy problem for the Laplace equation in a cylindrical domain with data on a part of it's boundary which is a cross-section of the cylinder. On reducing the problem to the Cauchy problem for the wave equation in a complex domain and using hyperbolic theory we obtain explicit formulas for the solution, thus developing the classical approach of Hans Lewy (1927).