The structure of maximal tori in spin groups of type DlD_l

We determine the abelian invariants of the maximal tori in the finite spin groups of type $D_l

An initial-boundary value problem in a strip for two-dimensional Zakharov-Kuznetsov-Burgers equation

An initial-boundary value in a strip with homogeneous Dirichlet boundary conditions for two-dimensional Zakharov-Kuznetsov-Burgers equation is considered. Results on global well-posedness and long-time decay of solutions in Sobolev spaces are established.Comment: 16 page

Cosheaves

The categories pCS(X,Pro(k)) of precosheaves and CS(X,Pro(k)) of cosheaves on a small Grothendieck site X, with values in the category Pro(k) of pro-k-modules, are constructed. It is proved that pCS(X,Pro(k)) satisfies the AB4 and AB5* axioms, while CS(X,Pro(k)) satisfies AB3 and AB5*. Homology theories for cosheaves and precosheaves, based on quasi-projective resolutions, are constructed and investigated

Embedding central extensions of simple linear groups into wreath products

We find a criterion for the embedding of a nonsplit central extension of $PSL_n(q)$ with kernel of prime order into the permutation wreath product that corresponds to the action on the projective space

Precosheaves of pro-sets and abelian pro-groups are smooth

Let $\mathbb{D}$ be the category of pro-sets (or abelian pro-groups). It is proved that for any Grothendieck site $X$, there exists a reflector from the category of precosheaves on $X$ with values in $\mathbb{D}$ to the full subcategory of cosheaves. In the case of precosheaves on topological spaces, it is proved that any precosheaf is smooth, i.e. is locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory

Dynamic Transposition of Melodic Sequences on Digital Devices

A method is proposed which enables one to produce musical compositions by using transposition in place of harmonic progression. A transposition scale is introduced to provide a set of intervals commensurate with the musical scale, such as chromatic or just intonation scales. A sequence of intervals selected from the transposition scale is used to shift instrument frequency at predefined times during the composition which serves as a harmonic sequence of a composition. A transposition sequence constructed in such a way can be extended to a hierarchy of sequences. The fundamental sound frequency of an instrument is obtained as a product of the base frequency, instrument key factor, and a cumulative product of respective factors from all the harmonic sequences. The multiplication factors are selected from subsets of rational numbers, which form instrument scales and transposition scales of different levels. Each harmonic sequence can be related to its own transposition scale, or a single scale can be used for all levels. When composing for an orchestra of instruments, harmonic sequences and instrument scales can be assigned independently to each musical instrument. The method solves the problem of using just intonation scale across multiple octaves as well as simplifies writing of instrument scores.Comment: 13 pages, 5 figures, 3 music sample

Cosheafification

It is proved that for any Grothendieck site $X$, there exists a coreflection (called $\mathbf{cosheafification}$) from the category of precosheaves on $X$ with values in a category $\mathbf{K}$, to the full subcategory of cosheaves, provided either $\mathbf{K}$ or $\mathbf{K}^{op}$ is locally presentable. If $\mathbf{K}$ is cocomplete, such a coreflection is built explicitly for the (pre)cosheaves with values in the category \mathbf{Pro}% \left( \mathbf{K}\right) of pro-objects in $\mathbf{K}$. In the case of precosheaves on topological spaces, it is proved that any precosheaf with values in $\mathbf{Pro}\left( \mathbf{K}\right)$ is $\mathbf{smooth}$, i.e. is strongly locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory

An initial-boundary value problem in a strip for a two-dimensional equation of Zakharov-Kuznetsov type

An initial-boundary value problem in a strip with homogeneous Diriclet boundary conditions for two-dimensional generalized Zakharov-Kuznetsov equation is considered. In particular, dissipative and absorbing degenerate terms can be supplemented to the original Zakharov-Kuznetsov equation. Results on global existence, uniqueness and long-time decay of weak silutions are established.Comment: 26 page

Properties of element orders in covers for L(n,q) and U(n,q)

We show that if a finite simple group G isomorphic to PSL(n,q) or PSU(n,q), where either $n\ne 4$, or q is prime or even, acts on a vector space over a field of the defining characteristic of G, then the corresponding semidirect product contains an element whose order is distinct from every element order of G. As a consequence, we prove that the group PSL(n,q), where $n\ne 4$ or q prime or even, is recognizable by spectrum from its covers thus giving a partial positive answer to Problem 14.60 from the Kourovka notebook

Thermal vacancies in random alloys in the single-site mean-field approximation

A formalism for the vacancy formation energies in random alloys within the single-site mean-filed approximation, where vacancy-vacancy interaction is neglected, is outlined. It is shown that the alloy configurational entropy can substantially reduce the concentration of vacancies at high temperatures. The energetics of vacancies in random Cu-0.5Ni-0.5 alloy is considered as a numerical example illustrating the developed formalism. It is shown that the effective formation energy is increases with temperature, however, in this particular system it is still below the mean value of the vacancy formation energy which would correspond to the vacancy formation energy in a homogeneous model of a random alloy, such as given by the coherent potential approximation.Comment: 5 pages, 3 figure