Maximal Subrings and Covering Numbers of Finite Semisimple Rings

We classify the maximal subrings of the ring of n×n matrices over a finite field, and show that these subrings may be divided into three types. We also describe all of the maximal subrings of a finite semisimple ring, and categorize them into two classes. As an application of these results, we calculate the covering number of a finite semisimple ring.     

Path integral quantization corresponding to the deformed Heisenberg algebra

In this paper, the deformation of the Heisenberg algebra, consistent with both the generalized uncertainty principle and doubly special relativity, has been analyzed. It has been observed that, though this algebra can give rise to fractional derivative terms in the corresponding quantum mechanical Hamiltonian, a formal meaning can be given to them by using the theory of harmonic extensions of function. Depending on this argument, the expression of the propagator of the path integral corresponding to the deformed Heisenberg algebra, has been obtained. In particular, the consistent expression of the one dimensional free particle propagator has been evaluated explicitly. With this propagator in hand, it has been shown that, even in free particle case, normal generalized uncertainty principle and doubly special relativity show very much different result.     

Numerical integration of ordinary differential equations on manifolds

Summary This paper is concerned with the problem of developing numerical integration algorithms for differential equations that, when viewed as equations in some Euclidean space, naturally evolve on some embedded submanifold. It is desired to construct algorithms whose iterates also evolve on the same manifold. These algorithms can therefore be viewed as integrating ordinary differential equations on manifolds. The basic method “decouples” the computation of flows on the submanifold from the numerical integration process. It is shown that two classes of single-step and multistep algorithms can be posed and analyzed theoretically, using the concept of “freezing” the coefficients of differential operators obtained from the defining vector field. Explicit third-order algorithms are derived, with additional equations augmenting those of their classical counterparts, obtained from “obstructions” defined by nonvanishing Lie brackets.     

Algebra matrix and similarity classification of operators

In this paper, by the Gelfand representation theory and the Silov idempotents theorem, we first obtain a central decomposition theorem related to a unital semi-simple n-homogeneous Banach algebra, and then give a similarity classification of two strongly irreducible Cowen-Douglas operators using this theorem.     

Duality for Finite Hopf Algebras Explained by Corings

We give a coring version for the duality theorem for actions and coactions of a finitely generated projective Hopf algebra. We also provide a coring analogue for a theorem of H.-J. Schneider, which generalizes and unifies the duality theorem for finite Hopf algebras and its refinements. This paper was written while the first author visited the Mathematics Departments of Syracuse University and California State University Dominguez Hills. He would like to thank both departments for their hospitality.     

Stationary turbulent closure via the Hopf functional equation

Exact closed-form solutions are exhibited for the Hopf equation for stationary incompressible 3D Navier-Stokes flow, for the cases of homogeneous forced flow (including a solution with depleted nonlinearity) and inhomogeneous flow with arbitrary boundary conditions. This provides an exact method for computing two- and higher-point moments, given the mean flow.     

On a bifurcation governed by hysteresis nonlinearity

We consider autonomous systems with a nonlinear part depending on a parameter and study Hopf bifurcations at infinity. The nonlinear part consists of the nonlinear functional term and the Prandtl--Ishlinskii hysteresis term. The linear part of the system has a special form such that the close-loop system can be considered as a hysteresis perturbation of a quasilinear Hamiltonian system. The Hamiltonian system has a continuum of arbitrarily large cycles for each value of the parameter. We present sufficient conditions for the existence of bifurcation points for the non-Hamiltonian system with hysteresis. These bifurcation points are determined by simple characteristics of the hysteresis nonlinearity.