It is shown that a series of recent (2012–2016) generalizations of the notion of contraction (F-contraction, weak F-contraction, etc.) in fact reduce to known notions of contraction (due to Browder, Boyd and Wong, Meir and Keeler, etc.).

We give explicit formulas proving restrictedness of the following Lie (super)algebras: known exceptional simple vectorial Lie (super)algebras in characteristic 3, deformed Lie (super)algebras with indecomposable Cartan matrix, and (under certain conditions) their simple subquotients over an algebraically closed field of characteristic 3, as well as one type of the deformed divergence-free Lie superalgebras with any number of indeterminates in any characteristic.

We give a criterion of tameness and wildness for a einite-dimensional Lie algebra over an algebraicallt closed field.

We study a two-parameter family of nonautonomous ordinary differential equations on the 2-torus. This family models the Josephson effect in superconductivity. We study its rotation number as a function of the parameters and the Arnold tongues (also known as the phase locking domains) defined as the level sets of the rotation number that have nonempty interior. The Arnold tongues of this family of equations have a number of nontypical properties: they exist only for integer values of the rotation number, and the boundaries of the tongues are given by analytic curves. (These results were obtained by Buchstaber–Karpov–Tertychnyi and Ilyashenko– Ryzhov–Filimonov.) The tongue width is zero at the points of intersection of the boundary curves, which results in adjacency points. Numerical experiments and theoretical studies carried out by Buchstaber–Karpov–Tertychnyi and Klimenko–Romaskevich show that each Arnold tongue forms an infinite chain of adjacent domains separated by adjacency points and going to infinity in an asymptotically vertical direction. Recent numerical experiments have also shown that for each Arnold tongue all of its adjacency points lie on one and the same vertical line with integer abscissa equal to the corresponding rotation number. In the present paper, we prove this fact for an open set of two-parameter families of equations in question. In the general case, we prove a weaker claim: the abscissa of each adjacency point is an integer, has the same sign as the rotation number, and does not exceed the latter in absolute value. The proof is based on the representation of the differential equations in question as projectivizations of linear differential equations on the Riemann sphere and the classical theory of linear equations with complex time.

We consider domains in Rn with C1 -boundary and study the following question: For what domains does the Fourier transform of the characteristic function delongs to Lp?

V.I.Arnold classified simple (i.e. having no moduli for the classification) singularities (function germs) and also simple boundary singularities: function germs invariant with respect to the action σ(x1;y1,…,yn)=(−x1;y1,…,yn) of the group Z2. In particular, it was shown that a function germ (a germ of a boundary singularity) is simple if and only if the intersection form (respectively, the restriction of the intersection form to the subspace of anti-invariant cycles) of a germ in 3+4s variables stable equivalent to the one under consideration is negative definite and if and only if the (equivariant) monodromy group on the corresponding space is finite. In a previous paper of the authors, there were obtained analogues of the latter statements for function germs invariant with respect to an arbitrary action of the group Z2 and also for corner singularities. In this paper, we give an analogue of the criterion of simplicity in terms of the intersection form for functions invariant with respect to a number of actions (representations) of the group Z3.

The investigation of decompositions of a permutation into a product of permutations

satisfying certain conditions plays a key role in the study of meromorphic functions or, equivalently,

branched coverings of the 2-sphere; it goes back to A. Hurwitz' work in the late nineteenth century.

In 2000 M. Bousquet-Melou and G. Schaeffer obtained an elegant formula for the number of decompositions

of a permutation into a product of a given number of permutations corresponding to

coverings of genus 0. Their formula has not been generalized to coverings of the sphere by surfaces of

higher genera so far. This paper contains a new proof of the Bousquet-Melou-Schaeffer formula for

the case of decompositions of a cyclic permutation, which, hopefully, can be generalized to positive

genera.