### Article

## Geometrical selection in growing needles

We investigate the growth of needles from a flat substrate. We focus on the situation when needles suddenly begin to grow from the seeds randomly distributed on the line. The width of needles is ignored and we additionally assume that (i) the growth rate is the same for all needles; (ii) the direction of the growth of each needle is randomly chosen from the same distribution; (iii) whenever the tip of a needle hits the body of another needle, the former needle freezes, while the latter continues to grow. We elucidate the large time behavior by employing an exact analysis and the Boltzmann equation approach. We also analyze the evolution when seeds are located on a half-line, on a finite interval. Needles growing from the two-dimensional substrate are also examined.

Huntite-family nominally-pure and activated/co-activated LnM3(BO3)4 (Ln = La–Lu, Y; M = Al, Fe, Cr, Ga, Sc) compounds and their-based solid solutions are promising materials for lasers, nonlinear optics, spintronics, and photonics, which are characterized by multifunctional properties depending on a composition and crystal structure. The purpose of the work is to establish stability regions for the rare-earth orthoborates in crystallochemical coordinates (sizes of Ln and M ions) based on their real compositions and space symmetry depending on thermodynamic, kinetic, and crystallochemical factors. The use of diffraction structural techniques to study single crystals with a detailed analysis of diffraction patterns, refinement of crystallographic site occupancies (real composition), and determination of structure–composition correlations is the most efficient and effective option to achieve the purpose. This approach is applied and shown primarily for the rare-earth scandium borates having interesting structural features compared with the other orthoborates. Visualization of structures allowed to establish features of formation of phases with different compositions, to classify and systematize huntite-family compounds using crystallochemical concepts (structure and superstructure, ordering and disordering, isostructural and isotype compounds) and phenomena (isomorphism, morphotropism, polymorphism, polytypism). Particular attention is paid to methods and conditions for crystal growth, affecting a crystal real composition and symmetry. A critical analysis of literature data made it possible to formulate unsolved problems in materials science of rare-earth orthoborates, mainly scandium borates, which are distinguished by an ability to form internal and substitutional (Ln and Sc atoms), unlimited and limited solid solutions depending on the geometric factor.

We have studied and optimized conditions for spontaneous flux growth of TbCr3(BO3)4 crystals. Phase relations in the pseudoternary system TbCr3(BO3)4–K2Mo3O10–B2O3 have been studied in the temperature range 900–1130°C and the single-phase terbium chromium borate crystallization field has been mapped out. It has been shown that increasing the TbCr3(BO3)4 content of the starting high-temperature solution leads to a rhombohedral-to-monoclinic phase transition. Using K2Mo3O10-based high-temperature solutions, we have grown single-phase TbCr3(BO3)4 single crystals or crystals in which the rhombohedral phase (sp. gr. R32) significantly prevails over the monoclinic phase (sp. gr. C2/c). The grown crystals have been characterized by X-ray diffraction techniques, IR spectroscopy, and magnetic measurements.

We consider a variational problem in a planar convex domain, motivated by statistical mechanics of crystal growth in a saturated solution. The minimizers are constructed explicitly and are completely characterized.

The book contains the necessary information from the algorithm theory, graph theory, combinatorics. It is considered partially recursive functions, Turing machines, some versions of the algorithms (associative calculus, the system of substitutions, grammars, Post's productions, Marcov's normal algorithms, operator algorithms). The main types of graphs are described (multigraphs, pseudographs, Eulerian graphs, Hamiltonian graphs, trees, bipartite graphs, matchings, Petri nets, planar graphs, transport nets). Some algorithms often used in practice on graphs are given. It is considered classical combinatorial configurations and their generating functions, recurrent sequences. It is put in a basis of the book long-term experience of teaching by authors the discipline «Discrete mathematics» at the business informatics faculty, at the computer science faculty* *of National Research University Higher School of Economics, and at the automatics and computer technique faculty of National research university Moscow power engineering institute. The book is intended for the students of a bachelor degree, trained at the computer science faculties in the directions 09.03.01 Informatics and computational technique, 09.03.02 Informational systems and technologies, 09.03.03 Applied informatics, 09.03.04 Software Engineering, and also for IT experts and developers of software products.

Huntite-family nominally-pure and activated/co-activated LnM3(BO3)4 (Ln = La–Lu, Y; M = Al, Fe, Cr, Ga, Sc) compounds and their-based solid solutions are promising materials for lasers, nonlinear optics, spintronics, and photonics, which are characterized by multifunctional properties depending on a composition and crystal structure. The purpose of the work is to establish stability regions for the rare-earth orthoborates in crystallochemical coordinates (sizes of Ln and M ions) based on their real compositions and space symmetry depending on thermodynamic, kinetic, and crystallochemical factors. The use of diffraction structural techniques to study single crystals with a detailed analysis of diffraction patterns, refinement of crystallographic site occupancies (real composition), and determination of structure–composition correlations is the most efficient and effective option to achieve the purpose. This approach is applied and shown primarily for the rare-earth scandium borates having interesting structural features compared with the other orthoborates. Visualization of structures allowed to establish features of formation of phases with different compositions, to classify and systematize huntite-family compounds using crystallochemical concepts (structure and superstructure, ordering and disordering, isostructural and isotype compounds) and phenomena (isomorphism, morphotropism, polymorphism, polytypism). Particular attention is paid to methods and conditions for crystal growth, affecting a crystal real composition and symmetry. A critical analysis of literature data made it possible to formulate unsolved problems in materials science of rare-earth orthoborates, mainly scandium borates, which are distinguished by an ability to form internal and substitutional (Ln and Sc atoms), unlimited and limited solid solutions depending on the geometric factor

A method based on the spectral analysis of thermowave oscillations formed under the effect of radiation of lasers operated in a periodic pulsed mode is developed for investigating the state of the interface of multilayered systems. The method is based on high sensitivity of the shape of the oscillating component of the pyrometric signal to adhesion characteristics of the phase interface. The shape of the signal is quantitatively estimated using the correlation coefficient (for a film–interface system) and the transfer function (for multilayered specimens).

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.