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Vol 53, No 4 (2019)
- Year: 2019
- Articles: 10
- URL: https://journals.rcsi.science/0016-2663/issue/view/14593
Article
Coulomb Branch of a Multiloop Quiver Gauge Theory
Abstract
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 a base change of the full Slodowy slice.
241-249
Invariants of Framed Graphs and the Kadomtsev—Petviashvili Hierarchy
Abstract
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 over all graphs turns into a linear combination of single-part Schur functions and, thereby, becomes a τ-function of an integrable Kadomtsev-Petviashvili hierarchy. We prove a similar assertion for the Hopf algebra of framed graphs. At the same time, we show that there is no such an analogue for a number of other Hopf algebras of a similar nature, in particular, for the Hopf algebras of weighted graphs, chord diagrams, and binary delta-matroids. Thus, it turns out that the Hopf algebras of graphs and framed graphs are distinguished among the graded Hopf algebras of combinatorial nature.
250-260
On the Structure of Normal Hausdorff Operators on Lebesgue Spaces
Abstract
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.
261-269
Behavior of Solutions of One-Sided Variational Problems on Phase Transitions in Continuum Mechanics at High Temperatures
Abstract
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.
270-280
Densities of Measures as an Alternative to Derivatives for Measurable Inclusions
Abstract
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 a positive Radon measure and rules for calculating the densities of Borel measures generated by composite functions with respect to a positive nonatomic Radon measure are studied.
281-290
Spectral Curves of the Hyperelliptic Hitchin Systems
Abstract
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.
291-303
Brief Communication
An Analogue of the Perelomov-Popov Formula for the Lie Superalgebra q(N)
Abstract
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.
304-308
The Maximality of Certain Commutative Subalgebras in Yangians
Abstract
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.
309-312
The Asymptotic Behavior of Singular Numbers of Compact Pseudodifferential Operators with Symbol Nonsmooth in Spatial Variables
Abstract
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.
313-316
On the Attainability of the Best Constant in Fractional Hardy-Sobolev Inequalities Involving the Spectral Dirichlet Laplacian
Abstract
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.
317-321