A Hopf bifurcation with spherical symmetry

Summary We provide a theoretical analysis of a Hopf bifurcation that can occur in systems with spherical geometry, based on the general theory of Hopf bifurcation in the presence of symmetry. In this particular bifurcation the imaginary eigenspace is a direct sum of two copies of the 5-dimensional irreducible representation of the groupSO(3). The same bifurcation has been studied by looss and Rossi (1988), using extensive computer-assisted calculations. Here we describe a simpler and more conceptual approach in which the representation ofSO(3) is realised as its conjugation action on the space of symmetric traceless 3 × 3 matrices. We prove the generic existence of five types of symmetry-breaking oscillation: two rotating waves and three standing waves. We analyse the stabilities of the bifurcating branches, describe the restrictions of the dynamics to various fixed-point spaces of subgroups ofSO(3), and discuss possible degeneracies in the stability conditions.     

Gravitation collapse of spherical symmetry with non-uniform density

A star with sufficiently large mass will collapse in its senectitude because of gravitation. Many researchers have tried to describe this collapse process. Investigations have been conducted in the case that the star has spherical symmetry and the initial density is uniform. In this article, the case that the initial density is not uniform will be considered. When the density function is high in the center and decreasing with the radius, the collapse process will be described, and in this case, the singularity will only come out in the center because of collision. If the density is not monotonic and there is a crust with high density around the star, it is proved that the non-central collision singularity may come out either in the Schwarzschild sphere or outside of it, i.e., the naked singularize may come out. __________ Translated from Journal of Fudan University (Natural Science), 1973, 1: 73–78     

Self-similar solutions of the Euler equations with spherical symmetry

We consider self-similar flows arising from the uniform expansion of a spherical piston and preceded by a shock wave front. With appropriate boundary conditions imposed on the piston surface and the spherical shock, the isentropic compressible Euler system is transformed into a nonlinear ODE system. We formulate the problem in a simple form in order to present the analytic proof of the global existence of positive smooth solutions.     

Some necessary uniform tests for spherical symmetry

While spherical distributions have been used in many statistical models for high-dimensional data analysis, there are few easily implemented statistics for testing spherical symmetry for the underlying distribution of high-dimensional data. Many existing statistics for this purpose were constructed by the theory of empirical processes and turn out to converge slowly to their limiting distributions. Some existing statistics for the same purpose were given in the form of high-dimensional integrals that are not easily evaluated in numerical computation. In this paper, we develop some necessary tests for spherical symmetry based on both univariate and multivariate uniform statistics. These statistics are easily evaluated numerically and have simple limiting distributions. A Monte Carlo study is carried out to demonstrate the performance of the statistics on controlling type I error rates and power.     

Model Dirac-Maxwell equations with pseudounitary symmetry

We introduce a new system of equations called a model system of Dirac-Maxwell equations, reproducing the main properties of the standard system. At the same time, the model system of equations differs from the standard system in several ways; in particular, it is a tensor system and has a new symmetry with respect to the pseudounitary group. We also propose a version of the model system of Dirac-Maxwell equations with local (gauge) pseudounitary symmetry. We show that any spinor solution of the standard system of Dirac-Maxwell equations can be obtained from the corresponding tensor solution of the model system. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 3, pp. 425–435, December, 2008.     

A note on the radial symmetry of solutions with spherical nodal sets

We will investigate the radial symmetry of solutions with spherical nodal sets of semilinear elliptic equations.     

Large time behavior of Euler-Poisson equations for isothermal fluids with spherical symmetry

In this paper, the large time behavior of spherically symmetric weak solutions to the multi-dimensional isothermal Euler-Poisson system in an annulus is considered. When space dimension N=2, it is shown that the weak solutions converge to the unique stationary solution exponentially in time. No smallness and regularity conditions are assumed.     

Robust estimation of common regression coefficients under spherical symmetry

Consider the problem of estimating the common regression coefficients of two linear regression models where the two distributions of the errors may be different and unknown. Under the spherical symmetry assumption, the paper proves the superiority of a Graybill-Deal type combined estimator and the further improvement by the Stein effect which were exhibited by Shinozaki (1978, Comm. Statist. Theory Methods, 7, 1421–1432) in the normal case. This shows the robustness of the dominations since the conditions for the dominations are independent of the errors distributions.Research supported by NSERC Grant No. A3088 and GR-5 Grant from Faculty of Graduate Studies, Carleton University, Ottawa, Canada.     

Single step methods for general second order singular initial value problems with spherical symmetry

A one parameter family of self-starting explicit Runge-Kutta-Nyström methods has been obtained for the solution of the general second order singular initial value problem with spherical symmetry of the formu+2r ^–1 u=f(r, u, u),u(0)=A, u(0)=0. The methods are exact foru(r)=r ^–1, l,r, r ²,r ³ andr ⁴.     

Asymptotic spherical symmetry of the free boundary in degenerate diffusion equations

Summary A general class of nonlinear degenerate parabolic equations in many space dimensions is considered and two main results concerning the free boundary are proved: (i) the «eventual» Lipschitz continuity in the space variable, (ii) the asymptotic spherical symmetry in a stronger sense than the «almost radiality» proved by Aronson & Caffarelli [2] for the porous medium equation. The proofs make use of geometric ideas based on the comparison principle and the method of moving planes.     

A measure of symmetry for the moduli of spherical minimal immersions

A Grünbaum type of measure of symmetry is calculated and estimated for the DoCarmo-Wallach moduli spaces for eigenmaps and spherical minimal immersions. The DeTurck-Ziller classification of minimal imbeddings of 3-dimensional space forms is used to obtain exact determination of the measure for the SU(2)-equivariant moduli.     

Inverse problem for the diffusion equation in the case of spherical symmetry

The initial boundary value problem for the diffusion equation is considered in the case of spherical symmetry and an unknown initial condition. Additional information used for determining the unknown initial condition is an external volume potential whose density is the Laplace operator applied to the solution of the initial boundary value problem. The uniqueness of the solution of the inverse problem is studied depending on the parameters entering into the boundary conditions. It is shown that the solution of the inverse problem is either unique or not unique up to a one-dimensional linear subspace.     

Non-circular,non-well-founded set universes

We show that there are universes of sets which contain descending ?-sequences of length α for every ordinal α, though they do not contain any ?-cycle. It is also shown that there is no set universe containing a descending ?-sequence of length On. MSC: 03E30; 03E65.     

Fuzzy sets and Boolean-valued universes

A connection between fuzzy sets and Boolean-valued universes is made by developing a formal axiomatic system for fuzzy sets among whose models are suitably interpreted Boolean-valued universes. The formal system is an independent first-order axiomatization of fuzzy set theory which parallels the Zermelo-Fraenkel development of classical set theory.     

Finite difference schemes for the singularly perturbed reaction-diffusion equation in the case of spherical symmetry

The boundary value problem for the singularly perturbed reaction-diffusion parabolic equation in a ball in the case of spherical symmetry is considered. The derivatives with respect to the radial variable appearing in the equation are written in divergent form. The third kind boundary condition, which admits the Dirichlet and Neumann conditions, is specified on the boundary of the domain. The Laplace operator in the differential equation involves a perturbation parameter ?², where ? takes arbitrary values in the half-open interval (0, 1]. When ? → 0, the solution of such a problem has a parabolic boundary layer in a neighborhood of the boundary. Using the integro-interpolational method and the condensing grid technique, conservative finite difference schemes on flux grids are constructed that converge ?-uniformly at a rate of O(N ^?2ln² N + N ₀ ^?1 ), where N + 1 and N ₀ + 1 are the numbers of the mesh points in the radial and time variables, respectively.