Vol 101, No 5-6 (2017)
- Year: 2017
- Articles: 36
- URL: https://journals.rcsi.science/0001-4346/issue/view/8957
Volume 101, Number 5, May, 2017
Finding the coefficients in the new representation of the solution of the Riemann–Hilbert problem using the Lauricella function
Abstract
The solution of the Riemann–Hilbert problem for an analytic function in a canonical domain for the case in which the data of the problem is piecewise constant can be expressed as a Christoffel–Schwartz integral. In this paper, we present an explicit expression for the parameters of this integral obtained by using a Jacobi-type formula for the Lauricella generalized hypergeometric function FD(N). The results can be applied to a number of problems, including those in plasma physics and the mechanics of deformed solids.
A conditional functional limit theorem for decomposable branching processes with two types of particles
Abstract
Consider a critical decomposable branching process with two types of particles in which particles of the first type give birth, at the end of their life, to descendants of the first type, as well as to descendants of the second type, while particles of the second type produce only descendants of the same type at the time of their death. We prove a functional limit theorem describing the distribution for the total number of particles of the second type appearing in the process in time Nt, 0 ≤ t < ∞, given that the number of particles of the first type appearing in the process during its evolution is N.
Lie algebras with Abelian centralizers
Abstract
In the paper, finite-dimensional real Lie algebras for which the centralizers of all nonzero element are Abelian are studied. These Lie algebras are also characterized by the transitivity condition for the commutation relation for two nonzero elements. A complete description of these Lie algebras up to isomorphism is given. Some results concerning the relationship between the aforementioned Lie algebras and the Lie algebras of vector fields whose orbits are one-dimensional are considered.
Uniform asymptotics of the boundary values of the solution in a linear problem on the run-up of waves on a shallow beach
Abstract
We consider the Cauchy problem with spatially localized initial data for a two-dimensional wave equation with variable velocity in a domain Ω. The velocity is assumed to degenerate on the boundary ∂Ω of the domain as the square root of the distance to ∂Ω. In particular, this problems describes the run-up of tsunami waves on a shallow beach in the linear approximation. Further, the problem contains a natural small parameter (the typical source-to-basin size ratio) and hence admits analysis by asymptotic methods. It was shown in the paper “Characteristics with singularities and the boundary values of the asymptotic solution of the Cauchy problem for a degenerate wave equation” [1] that the boundary values of the asymptotic solution of this problem given by a modified Maslov canonical operator on the Lagrangian manifold formed by the nonstandard characteristics associatedwith the problemcan be expressed via the canonical operator on a Lagrangian submanifold of the cotangent bundle of the boundary. However, the problem as to how this restriction is related to the boundary values of the exact solution of the problem remained open. In the present paper, we show that if the initial perturbation is specified by a function rapidly decaying at infinity, then the restriction of such an asymptotic solution to the boundary gives the asymptotics of the boundary values of the exact solution in the uniform norm. To this end, we in particular prove a trace theorem for nonstandard Sobolev type spaces with degeneration at the boundary.
Two nontrivial solutions of boundary-value problems for semilinear Δγ-differential equations
Abstract
In this paper, we study the existence of multiple solutions for the boundary-value problem
Weighted inequalities for a general commutator associated to a singular integral operator satisfying a variant of Hörmander’s condition
Abstract
In this paper, weighted inequalities for a certain general commutator associated to a singular integral operator satisfying a variant of Ho¨ rmander’s condition on Lebesgue spaces are obtained. To do this, some weighted sharp maximal function inequalities for the commutator are proved.
On the second cohomology of an algebraic group and of its lie algebra in a positive characteristic
Abstract
Necessary and sufficient isomorphism conditions for the second cohomology group of an algebraic group with an irreducible root system over an algebraically closed field of characteristic p ≥ 3h − 3, where h stands for the Coxeter number, and the corresponding second cohomology group of its Lie algebra with coefficients in simple modules are obtained, and also some nontrivial examples of isomorphisms of the second cohomology groups of simple modules are found. In particular, it follows from the results obtained here that, among the simple algebraic groups SL2(k), SL3(k), SL4(k), Sp4(k), and G2, nontrivial isomorphisms of this kind exist for SL4(k) and G2 only. For SL4(k), there are two simple modules with nontrivial second cohomology and, for G2, there is one module of this kind. All nontrivial examples of second cohomology obtained here are one-dimensional.
Adiabatic approximation for a model of cyclotron motion
Abstract
Aspecific problem is used to illustrate the limits of the approach resulting in an adiabatic approximation. The system of differential equations modeling the cyclotron motion of a charged relativistic particle in the field of an electromagnetic wave is considered. The problem of resonance capture of a particle with significantly varying energy is studied. The main result is the description of the capture area, i.e., the set of initial points in the phase plane from which the resonance trajectories issue. Such a description is obtained by the method of asymptotic approximation in a small parameter which corresponds to the rate of variation in the magnetic field. It is discovered that such an approximation is inapplicable in the case of small amplitudes of the electromagnetic wave.
Operator inclusions and quasi-variational inequalities
Abstract
The operator inclusion 0 ∈ A(x) + N(x) is studied. Themain results are concerned with the case where A is a bounded monotone-type operator from a reflexive space to its dual and N is a cone-valued operator. A criterion for this inclusion to have no solutions is obtained. Additive and homotopy-invariant integer characteristics of set-valued maps are introduced. Applications to the theory of quasi-variational inequalities with set-valued operators are given.
A regular differential operator with perturbed boundary condition
Abstract
The operator ℒ0 generated by a linear ordinary differential expression of nth order and regular boundary conditions of general form is considered on a closed interval. The characteristic determinant of the spectral problem for the operator ℒ1, where ℒ1 is an operator with the integral perturbation of one of its boundary conditions, is constructed, assuming that the unperturbed operator ℒ0 possesses a system of eigenfunctions and associated functions generating an unconditional basis in L2(0, 1). Using the obtained formula, we derive conclusions about the stability or instability of the unconditional basis properties of the system of eigenfunctions and associated functions of the problem under an integral perturbation of the boundary condition. The Samarskii–Ionkin problem with integral perturbation of its boundary condition is used as an example of the application of the formula.
An analog of Pólya’s theorem for multivalued analytic functions with finitely many branch points
Abstract
An analog of Pólya’s theorem on the estimate of the transfinite diameter for a class of multivalued analytic functions with finitely many branch points and of the corresponding class of admissible compact sets located on the associated (with this function) two-sheeted Stahl–Riemann surface is obtained.
Minimal two-spheres in G(2, 4; (C)) with parallel second fundamental form
Abstract
In this paper, we give a classification theorem of minimal two-spheres in G(2, 4; (C)) with parallel second fundamental form. Moreover, we also consider some special holomorphic two-spheres in G(2, n; (C)) and give the corresponding conditions of the parallel second fundamental form.
Short Communications
Spectral properties of the Sturm–Liouville operator with δ-potential and with spectral parameter in the boundary condition
On the number of transversals in n-ary quasigroups of order 4
Existence of the global solutions of the Cauchy problem for a system of semilinear pseudohyperbolic equations with structural dissipation
Punctured Lagrangian manifolds and asymptotic solutions of the linear water wave equations with localized initial conditions
Corrections and complements to my paper “On a class of operator monotone functions of several variables”
A note on the rationality of Sylow 2-subgroups of rational groups
Abstract
A finite group whose all irreducible characters are rational valued is called a rational group. Using the concept of transversal action, we get a sufficient condition on non Abelian rational groups that guarantees every Sylow 2-subgroup is also rational. This gives a partial answer to an old conjecture rationality of Sylow 2-subgroup of rational group.
On the number of singular points of terminal factorial Fano threefolds
Volume 101, Number 6, June, 2017
My dear Ludvig
On exact solutions of a Sobolev equation
Abstract
A nonlinear Sobolev-type equation that can be used to describe nonstationary processes in the semiconductor medium is studied. A number of families of exact solutions of this equation that can be expressed in terms of elementary functions and quadratures is obtained; some of these families contain arbitrary sufficiently smooth functions of one argument. The qualitative behavior of the resulting solutions is analyzed.
Automorphisms of graphs with intersection arrays {60, 45, 8; 1, 12, 50} and {49, 36, 8; 1, 6, 42}
Abstract
Automorphisms of distance-regular graphs are considered. It is proved that any graph with the intersection array {60, 45, 8; 1, 12, 50} is not vertex symmetric, and any graph with the intersection array {49, 36, 8; 1, 6, 42} is not edge symmetric.
A hybrid fixed-point theorem for set-valued maps
Abstract
In 1955, M. A. Krasnosel’skii proved a fixed-point theorem for a single-valued map which is a completely continuous contraction (a hybrid theorem). Subsequently, his work was continued in various directions. In particular, it has stimulated the development of the theory of condensing maps (both single-valued and set-valued); the images of such maps are always compact. Various versions of hybrid theorems for set-valued maps with noncompact images have also been proved. The set-valued contraction in these versions was assumed to have closed images and the completely continuous perturbation, to be lower semicontinuous (in a certain sense). In this paper, a new hybrid fixed-point theorem is proved for any set-valued map which is the sum of a set-valued contraction and a compact set-valued map in the case where the compact set-valued perturbation is upper semicontinuous and pseudoacyclic. In conclusion, this hybrid theorem is used to study the solvability of operator inclusions for a new class of operators containing all surjective operators. The obtained result is applied to solve the solvability problem for a certain class of control systems determined by a singular differential equation with feedback.
Saddle-type solenoidal basis sets
Abstract
An example of a diffeomorphism of the 3-sphere with positive topological entropy which has a one-dimensional solenoidal basis set with a two-dimensional unstable and a one-dimensional stable invariant manifold at each point (in particular, the basis set is neither an attractor nor a repeller) is given. On the basis of this diffeomorphism, a nondissipative fast kinematic dynamo with a one-dimensional invariant solenoidal set is constructed.
Checking the congruence between accretive matrices
Abstract
We call a finite computational process using only arithmetic operations a rational algorithm. A rational algorithm that is able to check the congruence between arbitrary complex matrices A and B is currently not known. The situation may be different if A and B belong to a certain class of specialmatrices. For instance, there exist rational algorithms for the case where both matrices are Hermitian or unitary. In this paper, rational algorithms for checking the congruence between accretive or dissipative A and B are proposed.
On the stabilization to zero of the solutions of the inverse problem for a degenerate parabolic equation with two independent variables
Abstract
Theorems on the stabilization to zero as t → +∞ of solutions of the inverse problem of determining the unknown right-hand side of a degenerate parabolic equation with one space variable belonging to a bounded closed interval are proved under an integral observation condition. Further, it is assumed that the coefficients of the equation are unbounded. These theorems can be used to obtain an upper bound for the stabilization rate. The paper contains examples of equations for which the assumptions of the theorems hold, as well as an example showing that such assumptions are essential.
Asymptotics of multipoint Hermite–Padé approximants of the first type for two beta functions
Abstract
The asymptotic behavior of the Hermite–Padé approximants of the first type for two beta functions are studied. The results are expressed in terms of equilibrium problems of logarithmic potential theory and in terms of meromorphic functions on Riemann surfaces.
Linear boundary-value problems described by Drazin invertible operators
Abstract
The main subject of this paper is the study of a general linear boundary-value problem with Drazin or right Drazin (respectively, left Drazin) invertible operators corresponding to initial boundary operators. The obtained results are then employed to solve a Schro¨ dinger equation.
Semiclassical asymptotics of the spectrum near the lower boundary of spectral clusters for a Hartree-type operator
Abstract
The eigenvalue problem for a perturbed two-dimensional resonant oscillator is considered. The exciting potential is given by a nonlocal nonlinearity of Hartree type with smooth self-action potential. To each representation of the rotation algebra corresponds the spectral cluster around an energy level of the unperturbed operator. Asymptotic eigenvalues and asymptotic eigenfunctions close to the lower boundary of spectral clusters are obtained. For their calculation, asymptotic formulas for quantum means are used.
On the Hamiltonian property of linear dynamical systems in Hilbert space
Abstract
Conditions for the operator differential equation \(\dot x = Ax\) possessing a quadratic first integral (1/2)(Bx, x) to be Hamiltonian are obtained. In the finite-dimensional case, it suffices to require that ker B ⊂ ker A*. For a bounded linear mapping x → Ωx possessing a first integral, sufficient conditions for the preservation of the (possibly degenerate) Poisson bracket are obtained.