On the limit cycles of a quintic planar vector field

This paper concerns the number and distributions of limit cycles in a Z_2-equivariant quintic planar vector field.25 limit cycles are found in this special planar polynomial system and four different configurations of these limit cycles are also given by using the methods of the bifurcation theory and the qualitative analysis of the differential equation.It can be concluded that H(5)≥25=5~2, where H(5)is the Hilbert number for quintic polynomial systems.The results obtained are useful to study the weakened 16th Hilbert problem.     

On polynomial symbols for subdivision schemes

Given a dilation matrix A :ℤ^d→ℤ^d, and G a complete set of coset representatives of 2π(A ^−Tℤ^d/ℤ^d), we consider polynomial solutions M to the equation ∑ _g∈G M(ξ+g)=1 with the constraints that M≥0 and M(0)=1. We prove that the full class of such functions can be generated using polynomial convolution kernels. Trigonometric polynomials of this type play an important role as symbols for interpolatory subdivision schemes. For isotropic dilation matrices, we use the method introduced to construct symbols for interpolatory subdivision schemes satisfying Strang–Fix conditions of arbitrary order. Research partially supported by the Danish Technical Science Foundation, Grant No. 9701481, and by the Danish SNF-PDE network.     

Solid codes and the uniform density of fd-domains

By means of further investigation of solid codes,the problem“Is every fd-domain uni- formly dense”proposed by Yuqi Guo,C.M.Reis and G.Thierrin in 1988 is solved in this paper.     

Covalent functionalization of multi-walled carbon nanotubes by lipase

Lipase from Candida rugosa was covalently anchored onto acid-treated multi-walled carbon nanotubes (MWNTs) through a self-catalytic mechanism. A variety of characterization techniques including FTIR, Raman spectroscopy, and XPS were employed to demonstrate the formation of the ester linkage between lipase and MWNTs. The MWNTs-lipase biocomposites showed significantly increased solubility in some common-used organic solvents, such as THF, DMF and chloroform. This study may offer a novel and facile route for covalent modification of carbon nanotubes, and expand the potential utilization of both lipases and MWNTs in the fields of biocatalyst and biosensor.     

Nonnegative curvature and cobordism type

We show that in each dimension n = 4k, k≥ 2, there exist infinite sequences of closed simply connected Riemannian n-manifolds with nonnegative sectional curvature and mutually distinct oriented cobordism type. W. Tuschmann’s research was supported in part by a DFG Heisenberg Fellowship.     

Hyers–Ulam stability of linear differential operator with constant coefficients

Let P(z) be a polynomial of degree n with complex coefficients and consider the n–th order linear differential operator P(D). We show that the equation P(D)f = 0 has the Hyers–Ulam stability, if and only if the equation P(z) = 0 has no pure imaginary solution. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Quantitative measurements of micro- and macromixing in a stirred vessel using Planar Laser-Induced Fluorescence

The Planar Laser-Induced Fluorescence (PLIF) technique enables measurement of the local degree of deviation in an arbitrary plane inside the mixing vessel. This is achieved by injecting a mixture of an inert dye and a reacting fluorescent into the vessel. The inert dye serves as a tracer for the macromixing. The fluorescent characteristics of the reacting dye change while undergoing a fast chemical reaction with the vessel content and it therefore shows the micromixing indirectly. The concentration fields of the dyes are measured simultaneously. For that a laser beam is expanded into a thin light sheet which illuminates an arbitrary plane in the mixing vessel, exciting the fluorescent dye in this area. The emitted light is detected by a CCD-camera which is positioned vertical to the measurement plane. The fluorescent light passes through two optical filters which are suitable for separating the fluorescent light of the two dyes. The light is then projected on half of the camera chip each so that the same display window is detected twice, and thus the local concentration of the two dyes can be measured simultaneously. Low Reynolds number measurements are performed in a mixing vessel equipped with a Rushton turbine. The lamellar structure is clearly resolved. Areas of micromixing are detected by calculating the local degree of deviation from the concentration fields. These areas are mainly found in the boundary layer of the lamellas.     

Dihedral crossed products of exponent 2 are abelian

We prove that assuming enough roots of unity in the base field, a central simple algebra of exponent 2 which is split by a dihedral group, is also split by certain abelian groups.     

A fictitious domain approach to the numerical solution of PDEs in stochastic domains

We present an efficient method for the numerical realization of elliptic PDEs in domains depending on random variables. Domains are bounded, and have finite fluctuations. The key feature is the combination of a fictitious domain approach and a polynomial chaos expansion. The PDE is solved in a larger, fixed domain (the fictitious domain), with the original boundary condition enforced via a Lagrange multiplier acting on a random manifold inside the new domain. A (generalized) Wiener expansion is invoked to convert such a stochastic problem into a deterministic one, depending on an extra set of real variables (the stochastic variables). Discretization is accomplished by standard mixed finite elements in the physical variables and a Galerkin projection method with numerical integration (which coincides with a collocation scheme) in the stochastic variables. A stability and convergence analysis of the method, as well as numerical results, are provided. The convergence is “spectral” in the polynomial chaos order, in any subdomain which does not contain the random boundaries.     

Irregular Behavior of Solutions for Fisher’s Equation

This paper is concerned with the irregular behavior of solutions for Fisher’s equation when initial data do not decay in a regular way at the spatial infinity. In the one-dimensional case, we show the existence of a solution whose profile and average speed are not convergent. In the higher-dimensional case, we show the existence of expanding fronts with arbitrarily prescribed profiles. We also show the existence of irregularly expanding fronts whose profile varies in time. Proofs are based on some estimate of the difference of two distinct solutions and a comparison technique. Dedicated to Professor Pavol Brunovsky on his 70th birthday.