Inversion Problem, Legendre Transform and Inviscid Burgers' Equations

Let $F(z)=z-H(z)$ with order $o(H(z))\geq 1$ be a formal map from \bC^n to \bC^n and $G(z)$ the formal inverse map of $F(z)$. We first study the deformation $F_t(z)=z-tH(z)$ of $F(z)$ and its formal inverse $G_t(z)=z+tN_t(z)$. (Note that $G_{t=1}(z)=G(z)$ when $o(H(z))\geq 2$.) We show that $N_t(z)$ is the unique power series solution of a Cauchy problem of a PDE, from which we derive a recurrent formula for $G_t(z)$. Secondly, motivated by the gradient reduction obtained by M. de Bondt, A. van den Essen \cite{BE1} and G. Meng \cite{M} for the Jacobian conjecture, we consider the formal maps $F(z)=z-H(z)$ satisfying the gradient condition, i.e. $H(z)=\nabla P(z)$ for some P(z)\in \bC[[z]] of order $o(P(z))\geq 2$. We show that, under the gradient condition, $N_t(z)=\nabla Q_t(z)$ for some Q_t(z)\in \bC[[z, t]] and the PDE satisfied by $N_t(z)$ becomes the $n$-dimensional inviscid Burgers' equation, from which a recurrent formula for $Q_t(z)$ also follows. Furthermore, we clarify some close relationships among the inversion problem, Legendre transform and the inviscid Burgers' equations. In particular the Jacobian conjecture is reduced to a problem on the inviscid Burgers' equations. Finally, under the gradient condition, we derive a binary rooted tree expansion inversion formula for $Q_t(z)$. The recurrent inversion formula and the binary rooted tree expansion inversion formula derived in this paper can also be used as computational algorithms for solutions of certain Cauchy problems of the inviscid Burgers' equations and Legendre transforms of the power series $f(z)$ of $o(f(z))\geq 2$.Comment: Latex, 21 pages. Some misprints have been correcte

Uniqueness of entire functions concerning difference polynomials

summary:In this paper, we investigate the uniqueness problem of difference polynomials sharing a small function. With the notions of weakly weighted sharing and relaxed weighted sharing we prove the following: Let $f(z)$ and $g(z)$ be two transcendental entire functions of finite order, and $\alpha (z)$ a small function with respect to both $f(z)$ and $g(z)$. Suppose that $c$ is a non-zero complex constant and $n\geq 7$ (or $n\geq 10$) is an integer. If $f^{n}(z)(f(z)-1)f(z+c)$ and $g^{n}(z)(g(z)-1)g(z+c)$ share “$(\alpha (z),2)$” (or $(\alpha (z),2)^{*}$), then $f(z)\equiv g(z)$. Our results extend and generalize some well known previous results

Development of a Collaborative, Distance Learning Based route to BEng/MEng qualification for Engineers in Employment.

This report presents the development of an integrated programme entitled MEng Engineering that will provide an opportunity for engineers already in the workforce to engage with upskilling that can lead to their gaining professional recognition. Progression will be provided directly from level HNC/HND/FdSc qualifications for engineers already in employment to gain BEng and MEng qualifications by part time distance learning mode. The programme will be jointly delivered by BU and BPC, the award will be a BU award. The programme will be delivered in 2-6 years depending upon entry and exit points and amount of credits studied in an academic year. The programme has been developed through close discussion with various employer representatives. It is the intention that the programme will be submitted to the Institution of Engineering Designers (IED) and the Institution of Mechanical Engineers (IMechE) for Engineering Council accreditation for the academic requirements of IEng/CEng as appropriate to level

Is SN 2006X from a WD + MS system with optically thick wind?

The single-degenerate channel is widely accepted as the progenitors of type Ia supernovae (SNe Ia). Following the work of Meng, Chen and Han (2009), we reproduced the birth rate and age of supernovae like SN 2006X by the single-degenerate model (WD + MS) with an optically thick wind, which may imply that the progenitor of SN 2006X is a WD + MS system.Comment: 10 pages, 2 figures, accepted for publication in New

Nuclear charge-exchange excitations in localized covariant density functional theory

The recent progress in the studies of nuclear charge-exchange excitations with localized covariant density functional theory is briefly presented, by taking the fine structure of spin-dipole excitations in 16O as an example. It is shown that the constraints introduced by the Fock terms of the relativistic Hartree-Fock scheme into the particle-hole residual interactions are straightforward and robust.Comment: 4 pages, 1 figure, Proceedings of INPC2013, Florence, Italy, 2-7 June 201

Testing cosmic opacity from SNe Ia and Hubble parameter through three cosmological-model-independent methods

We use the newly published 28 observational Hubble parameter data ($H(z)$) and current largest SNe Ia samples (Union2.1) to test whether the universe is transparent. Three cosmological-model-independent methods (nearby SNe Ia method, interpolation method and smoothing method) are proposed through comparing opacity-free distance modulus from Hubble parameter data and opacity-dependent distance modulus from SNe Ia . Two parameterizations, $\tau(z)=2\epsilon z$ and $\tau(z)=(1+z)^{2\epsilon}-1$ are adopted for the optical depth associated to the cosmic absorption. We find that the results are not sensitive to the methods and parameterizations. Our results support a transparent universe.Comment: 11 pages, 8 figures, 1 table, PLB(in press