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Том 50, № 1 (2016)
- Год: 2016
- Статей: 11
- URL: https://journals.rcsi.science/0016-2663/issue/view/14558
Article
Differential Forms on Quasihomogeneous Noncomplete Intersections
Аннотация
In this article, we discuss a few simple methods for computing the Poincaré series of modules of differential forms given on quasihomogeneous noncomplete intersections of various types. Among them are curves associated with a semigroup, bouquets of such curves, affine cones over rational or elliptic curves, and normal determinantal and toric varieties, including some types of quotient singularities, as well as cones over the Veronese embedding of projective spaces or over the Segre embedding of products of projective spaces, rigid singularities, fans, etc. In many cases, correct formulas can be derived without resorting to analysis of complicated resolvents or using computer systems of algebraic calculations. The obtained results allow us to compute the basic invariants of singularities in an explicit form by means of elementary operations on rational functions.
1-16
Birational Darboux Coordinates on (Co)Adjoint Orbits of GL(N, ℂ)
Аннотация
The set of all linear transformations with a fixed Jordan structure J is a symplectic manifold isomorphic to the coadjoint orbit O(J) of the general linear group GL(N, C). Any linear transformation can be projected along its eigenspace onto a coordinate subspace of complementary dimension. The Jordan structure \(\tilde J\) of the image under the projection is determined by the Jordan structure J of the preimage; consequently, the projection \(O\left( J \right) \to O\left( {\tilde J} \right)\) is a mapping of symplectic manifolds.
It is proved that the fiber ℰ of the projection is a linear symplectic space and the map \(O\left( J \right)\tilde \to E \times O\left( {\tilde J} \right)\) is a birational symplectomorphism. Successively projecting the resulting transformations along eigensubspaces yields an isomorphism between O(J) and the linear symplectic space being the direct product of all fibers of the projections. The Darboux coordinates on O(J) are pullbacks of the canonical coordinates on this linear symplectic space.
Canonical coordinates on orbits corresponding to various Jordan structures are constructed as examples.
17-30
How to Approach Nonstandard Boundary Value Problems
Аннотация
A new approach to nonstandard boundary value problems is suggested. For such problems, we construct equivalent inclusions with surjective operators and study the solvability of these inclusions. The paper consists of two parts. The first part deals with problems in which the right-hand side of the equation is a Lipschitz mapping (Section 3); in the second part (Section 4), this mapping is completely continuous with respect to a surjective operator A. The paper also gives examples of how our theorems can be applied when studying nonstandard boundary value problems.
31-38
On Unital Full Amalgamated Free Products of Quasidiagonal C*-Algebras
Аннотация
In the paper, we consider the question as to whether a unital full amalgamated free product of quasidiagonal C*-algebras is itself quasidiagonal. We give a sufficient condition for a unital full amalgamated free product of quasidiagonal C*-algebras with amalgamation over a finite-dimensional C*-algebra to be quasidiagonal. By applying this result, we conclude that the unital full free product of two AF algebras with amalgamation over a finite-dimensional C*-algebra is AF if there exists a faithful tracial state on each of the two AF algebras such that the restrictions of these states to the common subalgebra coincide.
39-47
Omega-Limit Sets of Generic Points of Partially Hyperbolic Diffeomorphisms
Аннотация
We prove that, for any Eu ⊕ Ecs partially hyperbolic C2 diffeomorphism, the ω-limit set of a generic (with respect to the Lebesgue measure) point is a union of unstable leaves. As a corollary, we prove a conjecture made by Ilyashenko in his 2011 paper that the Milnor attractor is a union of unstable leaves. In the paper mentioned above, Ilyashenko reduced the local generecity of the existence of a “thick” Milnor attractor in the class of boundary-preserving diffeomorphisms of the product of the interval and the 2-torus to this conjecture.
48-53
Commuting Differential Operators of Rank 2 with Polynomial Coefficients
Аннотация
Self-adjoint commuting differential operators with polynomial coefficients are considered. These operators form a commutative subalgebra of the first Weyl algebra. New examples of commuting differential operators of rank 2 are found.
54-61
Brief Communications
Holomorphic Minorants of Plurisubharmonic Functions
Аннотация
Let φ be a plurisubharmonic function on a pseudoconvex domain D in an n-dimensional complex space. We show that there exists a nonzero holomorphic function f on D such that some local mean value of φ with logarithmic additional terms majorizes log|f|. A similar problem is discussed for a locally integrable function on D in terms of balayage of positive measures.
62-65
The Strong Suslin Reciprocity Law and Its Applications to Scissor Congruence Theory in Hyperbolic Space
Аннотация
We prove the strong Suslin reciprocity law conjectured by A. B. Goncharov and describe its corollaries for the theory of scissor congruence of polyhedra in hyperbolic space. The proof is based on the study of Goncharov’s conjectural description of certain rational motivic cohomology groups of a field. Our main result is a homotopy invariance theorem for these groups.
66-70
On Homogenization for Non-Self-Adjoint Periodic Elliptic Operators on an Infinite Cylinder
Аннотация
We consider an operator Aε on L2(\({\mathbb{R}^{{d_1}}} \times {T^{{d_2}}}\)) (d1 is positive, while d2 can be zero) given by Aε = −div A(ε−1x1,x2)∇, where A is periodic in the first variable and smooth in a sense in the second. We present approximations for (Aε − μ)−1 and ∇(Aε − μ)−1 (with appropriate μ) in the operator norm when ε is small. We also provide estimates for the rates of approximation that are sharp with respect to the order.
71-75
Adiabatic Evolution Generated by a Schrödinger Operator with Discrete and Continuous Spectra
Аннотация
In the paper, we consider the one-dimensional nonstationary Schrödinger equation with a potential slowly depending on time. It is assumed that the corresponding stationary operator depending on time as a parameter has a finite number of negative eigenvalues and absolutely continuous spectrum filling the positive semiaxis. A solution close at some moment to an eigenfunction of the stationary operator is studied. We describe its asymptotic behavior in the case where the eigenvalues of the stationary operator move to the edge of the continuous spectrum and, having reached it, disappear one after another.
76-79
On Nearly Subadditive Maps
Аннотация
Several results on stability in spirit of Ulam and Hyers are obtained for classes of maps from groups to lattices satisfying the subadditivity condition
80-82