Solvability Criterion for Convolution Equations on a Half-Strip

We obtain the criterion of solvability of homogeneous convolution equation in a half-strip. Proof is based on a new decomposition property of the weighted Hardy space. This result has relations to the spectral analysis-synthesis problem, cyclicity problem, information theory. All data generated or analysed during this study are included in this published article.

Solvability criterion of the control problem of an asynchronous spectrum of linear almost periodic systems with the trivial averaging of the coefficient matrix

A linear control system with an almost periodic matrix of coefficients and the control in the form of feedback linear in phase variables is considered. It is assumed that the feedback coefficient is almost periodic and its frequency module, i. e. the smallest additive group of real numbers, including all the Fourier exponents of this coefficient, is contained in the frequency module of the coefficient matrix. The system under consideration is studied in the case of a zero average value of the matrix of coefficients. For the described class of systems, the control problem of the spectrum of irregular oscillations (asynchronous spectrum) with a target set of frequencies is solved. This task is as follows: to construct such a control from an admissible set so that the system closed by this control has almost periodic solutions, the set of Fourier exponents (frequency spectrum) that are contained in a predetermined subset; the intersection of the solution frequency modules and the coefficient matrix is trivial. The necessary and sufficient conditions for solvability of the control problem of the asynchronous spectrum are obtained.

On the n × n × n Rubik's Cube

We state and prove the “First law of Cubology”, i.e. the solvability criterion, for the n × n × n Rubik's Cube.

Thermo-solutal and kinetic modes of stable dendritic growth with different symmetries of crystalline anisotropy in the presence of convection

Motivated by important applications in materials science and geophysics, we consider the steady-state growth of anisotropic needle-like dendrites in undercooled binary mixtures with a forced convective flow. We analyse the stable mode of dendritic evolution in the case of small anisotropies of growth kinetics and surface energy for arbitrary Péclet numbers and n -fold symmetry of dendritic crystals. On the basis of solvability and stability theories, we formulate a selection criterion giving a stable combination between dendrite tip diameter and tip velocity. A set of nonlinear equations consisting of the solvability criterion and undercooling balance is solved analytically for the tip velocity V and tip diameter ρ of dendrites with n -fold symmetry in the absence of convective flow. The case of convective heat and mass transfer mechanisms in a binary mixture occurring as a result of intensive flows in the liquid phase is detailed. A selection criterion that describes such solidification conditions is derived. The theory under consideration comprises previously considered theoretical approaches and results as limiting cases. This article is part of the theme issue ‘From atomistic interfaces to dendritic patterns’. This article is part of the theme issue ‘From atomistic interfaces to dendritic patterns’.