Research problems

This section appears from time to time. Contributions are invited, and should be submitted to R. C. Thompson, Mathematics Department, University of California, Santa Barbara, CA 93106.     

Singularities and noncommutative Frobenius manifolds

We prove that the quaternionic miniversal deformations of an A _n singularity have the structure of a noncommutative Frobenius manifold in the sense of the extended cohomological field theory.     

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.     

A note on 4-dimensional locally conformally flat Walker manifolds

A 4-dimensional Walker metric on a semi Riemanian manifold M, for the canonical metric with c = 0, have been investigated by M. Chaichi, E. García—Río and Y. Matsushita. The paper generalizes these notions to the case of constant c ≠ 0. Specially the form of defining functions of this metric in locally conformally flat 4-dimensional Walker manifolds is found.     

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.     

Generic polynomials for transitive subgroups of the group S 4 over fields of characteristic 2

Generic polynomials for the symmetric group S₄, the Klein group V₄, the cyclic group of order 4, the dihedral group D₄, and the alternating group A₄ over fields of characteristic 2 are described explicitly. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 330, 2006, pp. 247–258.     

A mathematical model of the response of the semicircular canal and otolith to vestibular system rotation under gravity

A mathematical model of the system composed of two sensors, the semicircular canal and the sacculus, is suggested. The model is described by three lines of blocks, each line of which has the following structure: a biomechanical block, a mechanoelectrical transduction mechanism, and a block describing the hair cell ionic currents and membrane potential dynamics. The response of this system to various stimuli (head rotation under gravity and falling) is investigated. Identification of the model parameters was done with the experimental data obtained for the axolotl (Ambystoma tigrinum) at the Institute of Physiology, Autonomous University of Puebla, Mexico. Comparative analysis of the semicircular canal and sacculus membrane potentials is presented. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 7, pp. 207–220, 2005.