Russian Journal of Mathematical Physics最新文献

We consider a Riccati difference equation (Phi(x) + rho(x)/Phi(x-omega) = v(x)) under the assumption that coefficients (rho), (v) are (1)-periodic continuous functions of a real variable and (omega) is an irrational parameter. By using a connection between continued fraction theory and theory of (SL(2,mathbb{R}))-cocycles over irrational rotation, we investigate the problem of existence of continuous solutions to this equation. It is shown that the convergence of a continued fraction representing a solution to the Riccati equation can be expressed in terms of hyperbolicity of the cocycle naturally associated to this continued fraction. We establish sufficient conditions for the uniform hyperbolicity of a (SL(2,mathbb{R}))-cocycle, which imply the convergence of the corresponding continued fraction. The results thus obtained, along with the critical set method, have been applied to a special class of Riccati equations (rho(x)equiv 1, v(x) = g b(x), ggg 1,) to obtain sufficient conditions for the existence of continuous solutions in this case.

DOI 10.1134/S1061920824601538

Spivey showed a recurrence relation for the Bell numbers which are sums of the Stirling numbers of the second kind. Recently, the degenerate Bell polynomials and the degenerate Dowling polynomials were studied, whose coefficients are, respectively, the degenerate Stirling numbers of the second kind and the degenerate Whitney numbers of the second kind. The aim of this paper is to prove Spivey-type recurrence relations for those polynomials. In addition, a recurrence relation of the same type is shown for the degenerate (r)-Bell polynomials.

DOI 10.1134/S1061920825020074

In this paper, the asymptotics of the fundamental solution of a degenerate parabolic equation with a small parameter at the highest derivative is constructed. It is shown that the leading term of the asymptotics contains two phase functions, which is not typical for linear problems. Estimates are provided that relate the leading term of the asymptotics in the general case to the exact solution in the trivial case. The asymptotics is constructed in the form of a formal series in powers of the small parameter. The asymptotics is justified by proving the convergence of the obtained series.

DOI 10.1134/S1061920825600722

Consider two inverse problems for Sturm–Liouville problems on the unit interval. This means that there are two corresponding mappings (F, f) from a Hilbert space of potentials (H) into their spectral data. They are called isomorphic if (F) is a composition of (f) and some isomorphism (U) of (H) onto itself. An isomorphic class is a collection of inverse problems isomorphic to each other. We consider basic Sturm–Liouville problems on the unit interval and on the circle and describe their isomorphic classes of inverse problems. For example, we prove that the inverse problems for the case of Dirichlet and Neumann boundary conditions are isomorphic. The proof is based on nonlinear analysis.

DOI 10.1134/S1061920824601745

In the present paper we combine our previous results in the studies of Lagrangian geometry of the Grassmannian ({rm Gr} (k, n)) with the example of Lagrangian embedding of the full flag variety in the direct product of projective spaces, found by D. Bykov. As the result, we construct a Langrangian immersion of the group ({rm U}(n)), as a submanifold, into the complex Grassmanian ({rm Gr} (n-1, 2 n-1)) equipped with the symplectic form, by the Plücker embedding.

DOI 10.1134/S1061920824601770

The paper studies a wave equation whose velocity has a localized perturbation at some point (x_0). The initial condition has the form of a rapidly oscillating wave packet whose wavelength is not comparable with the scale of the inhomogeneity. In this case, the length of the initial wave is of the order of (varepsilon,) and the width of the localized inhomogeneity is of the order of (sqrt{varepsilon},) where (varepsilon) is a small parameter that tends to 0.

DOI 10.1134/S1061920825600400

For a bundle of compact Lie groups (pcolon {cal G} to B) over a compactum (B) (with the structure group of automorphisms of the corresponding group), we introduce the gauge-equivariant (K)-theory group (K_{{cal G}}^{0}(X; {mathcal A} )) of a bundle (pi_{X}colon X to B) endowed with a continuous action of ({cal G}) constructed using bundles (Eto X) with the typical fiber being a projective finitely generated module over a unital (C^*)-algebra ( {mathcal A} ). The index of a family of gauge-invariant (= ({cal G})-equivariant) Fredholm operators naturally takes values in these groups. We introduce and study products and use them to define the Thom homomorphism in gauge-equivariant (K)-theory and prove that this homomorphism is an isomorphism.

DOI 10.1134/S106192082460168X

Given an essential ideal (Jsubset A) of a (C^*)-algebra (A) and a Hilbert (C^*)-module (M) over (A), we place (M) between two other Hilbert (C^*)-modules over (A), (M_Jsubset Msubset M^J), in such a way that every submodule here is thick, i.e., its orthogonal complement in the greater module is trivial. We introduce the class (mathbb B_J(M)) of (J)-adjointable operators on a Hilbert (C^*)-module (M) over (A) and prove that this class isometrically embeds in the (C^*)-algebras of all adjointable operators both of (M_J) and of (M^J).

DOI 10.1134/S1061920824601782

The so-called ‘hit problem’ initiated by Peterson in [1] as an attempt at better understanding the (E_2)-page of the Adams spectral sequence (operatorname{mod} 2) (that is the cohomology of the Steenrod algebra ( {mathcal{A}_2} )) turned out to be very difficult. The hit problem is to determine a minimal generating set for the cohomology of products of infinite projective spaces ({mathbb R} P^infty) as a module over the Steenrod algebra ( {mathcal{A}_2} ) at the prime 2. The dual problem is to determine the set of ( {mathcal{A}_2} )-annihilated elements in the homology of the same spaces. Anick showed that the set of ( {mathcal{A}_2} )-annihilated elements in the products of infinite projective spaces ({mathbb R} P^infty) forms a free associative algebra [6]. Ault and Singer proved that, for every (k ge 0), the set of (k)-partially ( {mathcal{A}_2} )-annihilated elements in homology of products of ({mathbb R} P^infty) (that is a set of elements that are annihilated by (Sq^{2^i}) for all (i le k)) also forms a free associative algebra.

In this note, we investigate the dual problem at a prime (p>2). In this case, ({mathbb R} P^infty) should be replaced by ({mathbb C} P^infty) if one wants to ignore the action of the Bockstein operation (beta) or by the infinite (p)-lens space (L^infty) to take (beta) into consideration. We prove that, for any (kge 0), a collection of elements in ({mathbb Z}/p)-homology of products of ({mathbb C} P^infty) (or (L^infty)) annihilated by all (P^{p^i}), (ile k), forms a free algebra. The same holds for the collection of elements annihilated by (beta) and all (P^{p^i}), (ile k). We also construct an explicit basis in the subspace (barDelta(0)_{m,*}subset H_*(({mathbb C} P^infty)^{wedge m},{mathbb Z}/p)), (m=1, 2), annihilated by (P^1).

DOI 10.1134/S1061920825600230

A simple nonlocal de Sitter gravity model ((sqrt{dS})) shows good properties on cosmological and galactic scales. Its cosmological solution agrees very well with the experimental data. The rotation curves of spiral galaxies (Milky Way and M33) are also well described by the (sqrt{dS}) model. This article contains a brief overview of earlier results, including a new result on finding an appropriate local description with a scalar field for cosmological solutions.

DOI 10.1134/S1061920824601824