Russian Journal of Mathematical Physics最新文献

We study an initial boundary value problem of axially symmetric one-dimensional unsteady shear in the viscoplastic space (a Bingham solid) initiated by a rectilinear vortex thread located along the symmetry axis. The force intensity of the thread is represented by a given monotone piecewise continuous function of time. The density and the dynamical viscosity of the medium are constant, and the yield point is a given piecewise continuous function of radius. We find similar and quasisimilar expressions for the tangent stress and for the rotating component of the velocity both in viscoplastic shear domains and in rigid zones. We show that the vortex thread with time-bounded force intensity may generate a viscoplastic shear only inside a cylinder of certain radius. If the thread intensity growth linearly with time, then the radius of the shear domain grows proportionally to (sqrt t).

We study the operator acting in (L_2(mathbb{R})) by the formula (( mathcal{A} psi)(x)=psi(x+omega)+psi(x-omega)+ lambda e^{-2pi i x} psi(x)), where (xinmathbb R) is a variable, and (lambda>0) and (omegain(0,1)) are parameters. It is related to the simplest quasi-periodic operator introduced by P. Sarnak in 1982. We investigate ( mathcal{A} ) using the monodromization method, the Buslaev–Fedotov renormalization approach, which arose when trying to extend the Bloch–Floquet theory to difference equations on ( mathbb{R} ). Within this approach, the analysis of ( mathcal{A} ) turns out to be very natural and transparent. We describe the geometry of the spectrum and calculate the Lyapunov exponent.

We consider real submanifolds of (mathbf{C}^2) with singularities of three types: (RC)-singular 2 - dimensional surfaces, real quadratic cones, and hypersurfaces with degeneration of the Levi form. The holomorphic automorphisms of singular germs are evaluated. We also discuss resolution of singularities in the context of (mathit{CR}) geometry.

We study the existence of localized waves that can propagate in an acoustic medium bounded by two thin semi-infinite elastic membranes along their common edge. The membranes terminate an infinite wedge that is filled by the medium, and are rigidly connected at the points of their common edge. The acoustic pressure of the medium in the wedge satisfies the Helmholtz equation and the third-order boundary conditions on the bounding membranes as well as the other appropriate conditions like contact conditions at the edge. The existence of such localized waves is equivalent to existence of the discrete spectrum of a semi-bounded self-adjoint operator attributed to this problem. In order to compute the eigenvalues and eigenfunctions, we make use of an integral representation (of the Sommerfeld type) for the solutions and reduce the problem to functional equations. Their nontrivial solutions from a relevant class of functions exist only for some values of the spectral parameter. The asymptotics of the solutions (eigenfunctions) is also addressed. The far-zone asymptotics contains exponentially vanishing terms. The corresponding solutions exist only for some specific range of physical and geometrical parameters of the problem at hand.

Let (M) be a smooth manifold, smoothly triangulated by a simplicial complex (K), and ( {cal A} ) a transitive Lie algebroid on (M). A piecewise smooth form on ( {cal A} ) is a family (omega=(omega_{Delta})_{Deltain K}) such that (omega_{Delta}) is a smooth form on the Lie algebroid restriction of ( {cal A} ) to (Delta), satisfying the compatibility condition concerning the restrictions of (omega_{Delta}) to the faces of (Delta), that is, if (Delta') is a face of (Delta), the restriction of the form (omega_{Delta}) to the simplex (Delta') coincides with the form (omega_{Delta'}). The set (Omega^{ast}( {cal A} ;K)) of all piecewise smooth forms on ( {cal A} ) is a cochain algebra. There exists a natural morphism

We consider a boundary-value problem for singularly perturbed integro-differential equation describing stationary reaction–diffusion processes with due account of nonlocal interactions. The principal feature of the problem is the presence of a singularly perturbed Neumann condition describing intense flows on the boundary. We prove that there exists a boundary-layer solution, construct its asymptotic approximation, and establish its asymptotic Lyapunov stability. Illustrative examples are given.

We study a discrete Miura-type transformation between the hierarcies of non-Abelian semi-infinite Volterra (Kac–van Moerbeke) and Toda lattices with operator coefficients in terms of the inverse spectral problem for three-diagonal band operators, which appear in the Lax representation for such systems. This inverse problem method, which amounts to reconstruction of the operator from the moments of its Weyl operator-valued function, can be used in solving initial-boundary value problem for the systems of both these hierarchies. It is shown that the Miura transformation can be easily described in terms of these moments. Using this description we establish a bijection between the Volterra hierarchy and the Toda sub-hierarchy which can be characterized via Lax operators corresponding to its lattices.

An algorithmic concept of the physical world is proposed in which the main ideas of the Darwinian evolution act as algorithms of becoming.

We solve the inverse problem for Jacobi operators on the half lattice with finitely supported perturbations, in particular, in terms of resonances. Our proof is based on the results for the inverse eigenvalue problem for specific finite Jacobi matrices and theory of polynomials. We determine forbidden domains for resonances and maximal possible multiplicities of real and complex resonances.

An analogue of reflexivity in asymmetric cone spaces is introduced and studied. Some classical results known for ordinary normalized spaces are carried over to the case of essentially asymmetric spaces.