Fredholm theory and transversality for noncompact pseudoholomorphic mapsin symplectizations

We study pseudoholomorphic maps from a punctured Riemann surface into the symplectization of a contact manifold. A Fredholm theory yields the virtual dimension of the moduli spaces of such maps in terms of the Euler characteristic of the Riemann surface and the asymptotics data given by the periodic solutions of the Reeb vector field associated to the contact form. The transversality results establish the existence of additional structure for these spaces. To be more precise, we prove that these spaces are generically smooth manifolds, and therefore their virtual dimension coincides with their actual dimension. © 2004 Wiley Periodicals, Inc.     

Moduli of Polarised Abelian Surfaces

The moduli variety parametrising abelian surfaces over ? with a polarisation of type (1, p) may be compactified by toroidal methods. We show that if p is a prime and p ≥ 173 then the compactified moduli variety is of general type.     

Space-filling curves for 2-simplicial meshes created with bisections and reflections

Numerical experiments in J. Maubach: Local bisection refinement and optimal order algebraic multilevel preconditioners, PRISM-97 conference Proceedings, 1977, 121–136 indicated that the refinement with the use of local bisections presented in J. Maubach: Local bisection refinement for n-simplicial grids generated by reflections, SIAM J. Sci. Comput. 16 (1995), 210–227 leads to highly locally refined computational 2-meshes which can be very efficiently load-balanced with the use of a space-filling curve. This paper introduces the construction of this curve which can be produced at almost no costs, proofs that all its properties are invariant under local bisection, and comments on the 3-dimensional case.With the use of a space-filling curve (which passes through all triangular elements), load balancing over several processors is trivial: The load can be distributed over N processors by cutting the curve into N almost equilength parts. Each processor then operates on the triangles which are passed by its part of the curve.     

Clifford's inequality for real algebraic curves

We improve Clifford's Inequality for real algebraic curves. As an application we improve Harnack's Inequality for real space curves having a certain number of pseudo-lines. Another application involves the number of ovals that a real space curve can have.     

Modulation spaces with scaling symmetry

We indicate how to construct a family of modulation spaces that have a scaling symmetry. We also illustrate the behavior of the Schrödinger multiplier on such function spaces.     

An isomorphism theorem for Teichmüller curves

It is proved that for any Fuchsian group Γ such that ℍ/Γ is a hyperbolic Riemann surface, the Teichmüller curve V(Γ) has a unique complex manifold structure so that the natural projection of the Bers fiber space F(Γ) onto V(Γ) is holomorphic with local holomorphic sections. An isomorphism theorem for Teichmüller curves is deduced, which generalizes a classical result that the Teichmüller curve V(Γ) depends only on the type of Γ and not on the orders of the elliptic elements of Γ when ℍ/Γ is a compact hyperbolic Riemann surface.     

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     

On nonparametric tests for symmetry

Summary This paper is concerned with an extension of the problem of testing symmetry about zero of a distribution function. In order to obtain the asymptotic null distribution of test statistics for the problem, a limit theorem is proved, which indeed plays an essential role in the asymptotic theory of testing, problem for symmetry. The Institute of Statistical Mathematics     

Analytic regularity for an operator with Treves curves

We study a partial differential operator with analytic coefficients, which is of the form “sum of squares”. is hypoelliptic on any open subset of , yet possesses the following properties: (1) is not analytic hypoelliptic on any open subset of that contains 0. (2) If u is any distribution defined near with the property that is analytic near 0, then u must be analytic near 0. (3) The point 0 lies on the projection of an infinite number of Treves curves (bicharacteristics).These results are consistent with the Treves conjectures. However, it follows that the analog of Treves conjecture, in the sense of germs, is false.As far as we know, is the first example of a “sum of squares” operator which is not analytic hypoelliptic in the usual sense, yet is analytic hypoelliptic in the sense of germs.     

Solutions of the spatially-dependent mass Dirac equation with the spin and pseudospin symmetry for the Coulomb-like potential

We study the effect of spatially-dependent mass function over the solution of the Dirac equation with the Coulomb potential in the (3+1)-dimensions for any arbitrary spin-orbit κ state. In the framework of the spin and pseudospin symmetry concept, the analytic bound state energy eigenvalues and the corresponding upper and lower two-component spinors of the two Dirac particles are obtained by means of the Nikiforov-Uvarov method, in closed form. This physical choice of the mass function leads to an exact analytical solution for the pseudospin part of the Dirac equation. The special cases , the constant mass and the non-relativistic limits are briefly investigated.     

A symmetry exploiting Lanczos method for symmetric Toeplitz matrices

Several methods for computing the smallest eigenvalues of a symmetric and positive definite Toeplitz matrix T have been studied in the literature. Most of them share the disadvantage that they do not reflect symmetry properties of the corresponding eigenvector. In this note we present a Lanczos method which approximates simultaneously the odd and the even spectrum of T at the same cost as the classical Lanczos approach.     

Gonality for stable curves and their maps with a smooth curve as their target

Here we study the deformation theory of some maps f: X → ℙ ^r , r = 1, 2, where X is a nodal curve and f|T is not constant for every irreducible component T of X. For r = 1 we show that the “stratification by gonality” for any subset of with fixed topological type behaves like the stratification by gonality of M _g.      

Optimal control of differential-difference systems with intragroup symmetry

A decomposition method of finding the optimal control of differential-difference systems with intragroup symmetry is presented. The introduction of two forms of motions of coordinates of separate groups allows us to reduce the original optimal control problem to the solution of optimal control problems of order less than the order of the initial problem.     

Nonclassical symmetry solutions for reaction–diffusion equations with explicit spatial dependence

Nonclassical symmetry methods are used to study the linear diffusion equation with a nonlinear source term which includes explicit spatial dependence. Mathematical forms for the spatial dependence are found which enable strictly nonclassical symmetries to be admitted when the nonlinearity is cubic. A number of new exact solutions are constructed, and an application of one of these solutions to diploid population genetics is discussed.     

A birational embedding with two Galois points for quotient curves

A criterion for the existence of a birational embedding with two Galois points for quotient curves is presented. We apply our criterion to several curves, for example, some cyclic subcovers of the Giulietti–Korchmáros curve or of the curves constructed by Skabelund. New examples of plane curves with two Galois points are described, as plane models of such quotient curves.     

Galois points for double-Frobenius nonclassical curves

We determine the distribution of Galois points for plane curves over a finite field of q elements, which are Frobenius nonclassical for different powers of q. This family is an important class of plane curves with many remarkable properties. It contains the Dickson–Guralnick–Zieve curve, which has been recently studied by Giulietti, Korchmáros, and Timpanella from several points of view. A problem posed by the second author in the theory of Galois points is modified.