Runge–Kutta convolution quadrature methods for well-posed equations with memory

Runge–Kutta based convolution quadrature methods for abstract, well-posed, linear, and homogeneous Volterra equations, non necessarily of sectorial type, are developed. A general representation of the numerical solution in terms of the continuous one is given. The error and stability analysis is based on this representation, which, for the particular case of the backward Euler method, also shows that the numerical solution inherits some interesting qualitative properties, such as positivity, of the exact solution. Numerical illustrations are provided.     

Zeroth-order phase transitions and Zipf law quantization

From the standpoint of thermodynamic averaging of fission microprocesses, we investigate the origin of radioactive release in an NPP after an accident or after resource depletion. The genesis of the NPP release is interpreted as a new thermodynamic phenomenon, a zeroth-order phase transition. This problem setting results in a problem in probabilistic number theory. We prove the corresponding theorem leading to quantization of the Zipf law for the frequency of a zeroth-order phase transition with different values of the jump of the Gibbs thermodynamic potential. We introduce the notion of hole dimension. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 1, pp. 118–142, January, 2007.     

Cauchy problem in the class of monotone increasing functions

The paper establishes some solvability conditions of the Cauchy problem for linear differential equation in the class of monotone increasing functions. The results are applied for clarifying the possibility of flight along a given trajectory under existence of braking forces.     

The centralizer algebra of the diagonal action of the group <Emphasis Type="Italic">GL</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(ℂ) in a mixed tensor space

We consider the walled Brauer algebra Br _{k, l}(n) introduced by V. Turaev and K. Koike. We prove that it is a subalgebra of the Brauer algebra and that it is isomorphic, for sufficiently large n ∈ ℕ, to the centralizer algebra of the diagonal action of the group GL_n(ℂ) in a mixed tensor space. We also give the presentation of the algebra Br _{k, l}(n) by generators and relations. For a generic value of the parameter, the algebra is semisimple, and in this case we describe the Bratteli diagram for this family of algebras and give realizations for the irreducible representations. We also give a new, more natural proof of the formulas for the characters of the walled Brauer algebras. Bibliography: 29 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 331, 2006, pp. 170–198.     

Steinberg unitary Lie conformal algebras

In this paper, we study Steinberg unitary Lie conformal algebras, which are universal central extensions of unitary Lie conformal algebras. We describe the kernels of these extensions by means of skew-dihedral homology. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 135–155, 2005.     

On the bidirectional vortex and other similarity solutions in spherical coordinates

The bidirectional vortex refers to the bipolar, coaxial swirling motion that can be triggered, for example, in cyclone separators and some liquid rocket engines with tangential aft-end injectors. In this study, we present an exact solution to describe the corresponding bulk motion in spherical coordinates. To do so, we examine both linear and nonlinear solutions of the momentum and vorticity transport equations in spherical coordinates. The assumption will be that of steady, incompressible, inviscid, rotational, and axisymmetric flow. We further relate the vorticity to some power of the stream function. At the outset, three possible types of similarity solutions are shown to fulfill the momentum equation. While the first type is incapable of satisfying the conditions for the bidirectional vortex, it can be used to accommodate other physical settings such as Hill’s vortex. This case is illustrated in the context of inviscid flow over a sphere. The second leads to a closed-form analytical expression that satisfies the boundary conditions for the bidirectional vortex in a straight cylinder. The third type is more general and provides multiple solutions. The spherical bidirectional vortex is derived using separation of variables and the method of variation of parameters. The three-pronged analysis presented here increases our repertoire of general mean flow solutions that rarely appear in spherical geometry. It is hoped that these special forms will permit extending the current approach to other complex fluid motions that are easier to capture using spherical coordinates.     

Integrable models for vicious and friendly walkers

We consider random walks of two essentially different classes of random walkers, namely, of vicious and friendly ones, on one-dimensional lattices with periodic boundary conditions. The walkers are called vicious since, arriving at a lattice site, they annihilate not only one another but all the remaining walkers as well. On the contrary, an arbitrary number of friendly walkers can share the same lattice sites. It is shown that a natural model describing the behavior of friendly walkers is an integrable model of the boson type. A representation of the generating function for the number of the lattice paths performed by a fixed number of friendly walkers for a certain number of steps is obtained. Bibliography: 22 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 59–74.     

Realization of graphs in a subspace of bounded height

A realization of graphs with vertices of bounded branching in a subspace of bounded depth is considered. A volume order inside of which an arbitrary graph can be realized is determined.     

Surfaces of revolution in the Heisenberg group and the spectral generalization of the Willmore functional

We study the generalization of the Willmore functional for surfaces in the three-dimensional Heisenberg group. Its construction is based on the spectral theory of the Dirac operator entering into theWeierstrass representation of surfaces in this group. Using the surfaces of revolution we demonstrate that the generalization resembles the Willmore functional for the surfaces in the Euclidean space in many geometrical aspects. We also observe the relation of these functionals to the isoperimetric problem.