Compact pseudodifferential operators whose symbol fails to be smooth with respect to x on a given set are considered. Conditions under which Weyl’s law of spectral asymptotics remains valid for such operators are obtained. The results are applied to operators with symbols such that their order of decay as |ξ| → ∞ is a nonsmooth function of x.

We prove the attainability of the best constant in the fractional Hardy-Sobolev inequality with a boundary singularity for the spectral Dirichlet Laplacian. The main assumption is the average concavity of the boundary at the origin. A similar result has been proved earlier for the conventional Hardy-Sobolev inequality.

We study the center of the universal enveloping algebra of the strange Lie superalgebra q(N). We obtain an analogue of the well-known Perelomov-Popov formula [6] for the central elements of this algebra—an expression of the central characters through the highest weight parameters.

It is proved that any Bethe subalgebra corresponding to a regular semisimple element in a Yangian is a maximal commutative subalgebra and, moreover, the centralizer of its quadratic part. As a consequence, a description of such subalgebras as the traces of the R-matrix over all finite-dimensional representations of the Yangian is obtained.

Rules for calculating the densities of Borel measures which are absolutely continuous with respect to a positive nonatomic Radon measure are considered. The Borel measures are generated by composite functions which depend on continuous functions of bounded variation defined on an interval. Questions related to the absolute continuity of Borel measures generated by composite functions with respect to

We compute the Coulomb branch of a multiloop quiver gauge theory for the quiver with a single vertex, r loops, one-dimensional framing, and dim V = 2. We identify it with a Slodowy slice in the nilpotent cone of the symplectic Lie algebra of rank r. Hence it possesses a symplectic resolution with 2r fixed points with respect to a Hamiltonian torus action. We also identify its flavor deformation with

This paper describes a class of spectral curves and gives explicit formulas for the Darboux coordinates of the Hitchin systems of types Al, Bl, and Cl on hyperelliptic curves. The current state of the problem in the case of the systems of type Dl is described.

S. V. Chmutov, M. E. Kazarian, and S. K. Lando have recently introduced a class of graph invariants, which they called shadow invariants (these invariants are graded homomorphisms from the Hopf algebra of graphs to the Hopf algebra of polynomials in infinitely many variables). They proved that, after an appropriate rescaling of the variables, the result of the averaging of almost every such invariant

The variational problem on the equilibrium of a two-phase elastic medium is studied for conditions of the Signorini type. The strong convergence of its solutions to single-phase states as the temperature unboundedly increases is proved. A sufficient condition for the existence of phase transition temperatures for one-sided problems is given. A one-dimensional example illustrating the results is presented

We consider generalized Hausdorff operators and introduce the notion of the symbol of such an operator. Using this notion, we describe, under some natural conditions, the structure and investigate important properties (such as invertibility, spectrum, and norm) of normal generalized Hausdorff operators on Lebesgue spaces over ℝn. As an example we consider Cesàro operators.

Relative projection constants and strong unicity constants for a certain class of projection operators on the space l 2 n∞ are found. The maximum values of strong unicity constants are calculated for the projection operators with unit norm on certain codimension-2 subspaces formed by using hyperplanes in l 2 n∞ .

In this paper we strengthen the result of Fomin and Zelevinsky (2002) on the Laurent phenomenon for Somos-4 and Somos-5 sequences.

A new Caristi-type inequality is considered and Caristi’s fixed point theorem for mappings of complete metric spaces is developed (in both the single- and set-valued cases). On the basis of this development mappings of complete metric spaces which are contractions with respect to a function of two vector arguments are studied. This function is not required to be a metric or even a continuous function

Let T be a self-adjoint operator on a Hilbert space H with domain \(\mathscr{D}(T)\). Assume that the spectrum of T is contained in the union of disjoint intervals Δk = [α2k−1,α2k], k ∈ ℤ, the lengths of the gaps between which satisfy the inequalities $${\alpha _{2k + 1}} - {\alpha _{2k}}\geqslant b{\rm{|}}{\alpha _{2k + 1}} + {\alpha _{2k}}{{\rm{|}}^p}\;\;\;\;{\rm{for}}\;{\rm{some}}\;\;{\rm{b}} >

We give a complete solution of the inverse problem for finite Jacobi operators with matrix-valued coefficients.

We consider arbitrary noncompact hyperbolic Riemann surfaces of finite area. For such surfaces, we obtain identities relating the discrete spectrum of the Laplace operator to the resonance spectrum (formed by the poles of the scattering matrix). These identities depend on the choice of a test function. We indicate a class of admissible test functions and consider two examples corresponding to specific

This paper is devoted to the classical problem of the inversion of ultraelliptic integrals given by basic holomorphic differentials on a curve of genus 2. Basic solutions F and G of this problem are obtained from a single-valued 4-periodic meromorphic function on the Abelian covering W of the universal hyperelliptic curve of genus 2. Here W is the nonsingular analytic curve W = {u =(u1, u3) ∈ ℂ2: σ(u)

The asymptotic behavior at large times of the solution of the Cauchy problem for the Airy equation—a third-order evolutionary equation—is established. We assume that the initial function is locally Lebesgue integrable and has a power-law asymptotics at infinity. For the solution in the form of a convolution integral with the Airy function, we use the auxiliary parameter method and the regularization

We consider foliations of compact complex manifolds by analytic curves. We suppose that the line bundle tangent to the foliation is negative. We show that, in the generic case, there exists a finitely smooth homomorphism holomorphic on the fibers and mapping fiberwise the manifold of universal coverings over the leaves passing through a given transversal B onto some domain with continuous boundary

We obtain a qualitative characterization of the convergence rate of the averages (with respect to the Margulis measure) of C2 functions over the iterations of domains in unstable manifolds of a topologically mixing C3 Anosov diffeomorphism with oriented invariant foliations. For this purpose, we extend the constructions of Margulis and Bufetov and introduce holonomy invariant families of finitely additive