Differential Geometrical Formulation of Gauge Theory of Gravity

Differential geometric formulation of quantum gauge theory of gravity is studied in this paper. The quantum gauge theory of gravity which is proposed in the references hep-th/0109145 and hep-th/0112062 is formulated completely in the framework of traditional quantum field theory. In order to study the relationship between quantum gauge theory of gravity and traditional quantum gravity which is formulated in curved space, it is important to find the differential geometric formulation of quantum gauge theory of gravity. We first give out the correspondence between quantum gauge theory of gravity and differential geometry. Then we give out differential geometric formulation of quantum gauge theory of gravity.Comment: 10 pages, no figur

The Sylvester equation and integrable equations: I. The Korteweg-de Vries system and sine-Gordon equation

The paper is to reveal the direct links between the well known Sylvester equation in matrix theory and some integrable systems. Using the Sylvester equation $\boldsymbol{K} \boldsymbol{M}+\boldsymbol{M} \boldsymbol{K}=\boldsymbol{r}\, \boldsymbol{s}^{T}$ we introduce a scalar function $S^{(i,j)}=\boldsymbol{s}^{T}\, \boldsymbol{K}^j(\boldsymbol{I}+\boldsymbol{M})^{-1}\boldsymbol{K}^i\boldsymbol{r}$ which is defined as same as in discrete case. $S^{(i,j)}$ satisfy some recurrence relations which can be viewed as discrete equations and play indispensable roles in deriving continuous integrable equations. By imposing dispersion relations on $\boldsymbol{r}$ and $\boldsymbol{s}$, we find the Korteweg-de Vries equation, modified Korteweg-de Vries equation, Schwarzian Korteweg-de Vries equation and sine-Gordon equation can be expressed by some discrete equations of $S^{(i,j)}$ defined on certain points. Some special matrices are used to solve the Sylvester equation and prove symmetry property $S^{(i,j)}=S^{(i,j)}$. The solution $\boldsymbol{M}$ provides $\tau$ function by $\tau=|\boldsymbol{I}+\boldsymbol{M}|$. We hope our results can not only unify the Cauchy matrix approach in both continuous and discrete cases, but also bring more links for integrable systems and variety of areas where the Sylvester equation appears frequently.Comment: 23 page

Kinetic behavior of the general modifier mechanism of Botts and Morales with non-equilibrium binding

In this paper, we perform a complete analysis of the kinetic behavior of the general modifier mechanism of Botts and Morales in both equilibrium steady states and non-equilibrium steady states (NESS). Enlightened by the non-equilibrium theory of Markov chains, we introduce the net flux into discussion and acquire an expression of product rate in NESS, which has clear biophysical significance. Up till now, it is a general belief that being an activator or an inhibitor is an intrinsic property of the modifier. However, we reveal that this traditional point of view is based on the equilibrium assumption. A modifier may no longer be an overall activator or inhibitor when the reaction system is not in equilibrium. Based on the regulation of enzyme activity by the modifier concentration, we classify the kinetic behavior of the modifier into three categories, which are named hyperbolic behavior, bell-shaped behavior, and switching behavior, respectively. We show that the switching phenomenon, in which a modifier may convert between an activator and an inhibitor when the modifier concentration varies, occurs only in NESS. Effects of drugs on the Pgp ATPase activity, where drugs may convert from activators to inhibitors with the increase of the drug concentration, are taken as a typical example to demonstrate the occurrence of the switching phenomenon.Comment: 19 pages, 10 figure

Uniform L1L^1 error bounds for semi-discrete finite element solutions of evolutionary integral equations

summary:In this paper, we consider the second-order continuous time Galerkin approximation of the solution to the initial problem $u_{t}+\int_{0}^{t}\beta (t-s) Au(s)ds=0,u(0)=v,t>0,$ where A is an elliptic partial-differential operator and $\beta(t)$ is positive, nonincreasing and log-convex on $(0,\infty)$ with $0\leq\beta(\infty)<\beta(0^{+})\leq\infty$. Error estimates are derived in the norm of $L^{1}_{t}(0,\infty;L^{2}_{x})$, and some estimates for the first order time derivatives of the errors are also given