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Том 51, № 4 (2017)
- Год: 2017
- Статей: 11
- URL: https://journals.rcsi.science/0016-2663/issue/view/14573
Article
Logarithmic differential forms on varieties with singularities
Аннотация
In the article we introduce the notion of logarithmic differential forms with poles along a Cartier divisor given on a variety with singularities, discuss some properties of such forms, and describe highly efficient methods for computing the Poincaré series and generators of modules of logarithmic differential forms in various situations. We also examine several concrete examples by applying these methods to the study of divisors on varieties with singularities of many types, including quasi-homogeneous complete intersections, normal, determinantal, and rigid varieties, and so on.
245-254
On Lie’s problem and differential invariants of ODEs y″ = F(x, y)
Аннотация
A point classification of ordinary differential equations of the form y″ = F(x, y) is considered. The algebra of differential invariants of the action of the point symmetry pseudogroup on the right-hand sides of equations of the form y″ = F(x, y) is calculated, and Lie’s problem on the point equivalence of such equations is solved.
255-262
On properties of topological pressure
Аннотация
We consider a parametric family of continuous maps of a compact metric space which continuously depend on a parameter ranging over a metric space. The topological pressure of maps in any such family is studied as a function of the parameter from the viewpoint of the Baire classification of functions.
263-269
A trace formula and application to Stark Hamiltonians with nonconstant magnetic fields
Аннотация
In this paper we generalize a trace formula due to D. Robert involving the spectral shift function. We then give applications to the study of the high-energy and the semiclassical asymptotics of the spectral shift function for a Stark Hamiltonian with nonconstant magnetic field.
270-282
On unconditional bases of reproducing kernels in Fock-type spaces
Аннотация
The existence of unconditional bases of reproducing kernels in the Fock-type spaces Fφ with radial weights φ is studied. It is shown that there exist functions φ(r) of arbitrarily slow growth for which ln r = o(φ(r)) as r → ∞ and there are no unconditional bases of reproducing kernels in the space Fφ. Thus, a criterion for the existence of unconditional bases cannot be given only in terms of the growth of the weight function.
283-292
293-299
Combinatorics of a statistical model constructed from the 2 × n square lattice
Аннотация
Various weight functions for the model of a 2×n square lattice are defined so that the graded Euler characteristic for the complex corresponding to this model remain equal to the usual one. Differentials preserving these gradings and determining one-dimensional cohomology are also specified.
300-305
Brief Communications
On real solutions of systems of equations
Аннотация
Systems of equations f1 = ··· = fn−1 = 0 in ℝn = {x} having the solution x = 0 are considered under the assumption that the quasi-homogeneous truncations of the smooth functions f1,..., fn−1 are independent at x ≠ 0. It is shown that, for n ≠ 2 and n ≠ 4, such a system has a smooth solution which passes through x = 0 and has nonzero Maclaurin series.
306-309
Conditions for invertibility and Hurwitz stability
Аннотация
In a generalization of Fiedler’s theorem, a block condition for the invertibility of an operator and an estimate for the operator matrix of the inverse operator are presented. A block condition for an operator to be Hurwitz is also given, which contains an estimate of the spectral abscissa of the operator.
310-315
On products of nuclear operators
Аннотация
The possibility of factoring a product of nuclear operators through operators in the von Neumann–Schatten class is considered. In particular, generally, the product of two nuclear operators can be factored only through a Hilbert–Schmidt operator.
316-317
On the equivalence of certain statements on fixed points of contractions
Аннотация
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.).
318-321