The First and Second Variation of the Total Energy of Closed Duplex DNA in Planar Case

DNA's shape mostly lies on its total energy $F$. Its corresponding equilibrium shape equations can be got by classical variation method: letting the first energy variation $\delta^{(1)}F=0$. Here we not only provide the first variation $\delta^{(1)}F$ but also give the second variation $\delta^{(2)}F$ in planar case. Moreover, the general shape equations of DNA are obtained and a mistake in [Zhang, \emph{et al}. {\it Phys. Rev. E} {\bf 70} 051902 (2004)] is pointed out.Comment: 9 pages, 1 figure,. Accepted by International Journal of Modern Physics

Stability analysis of kinked DNA in F(K,τ)\mathcal{F}(K,\tau) model

We phenomenologically analyze short DNA rings' stability by discussing the second variation of its elastic free energy. Through expanding the perturbation functions as Fourier series, we obtain DNA rings' stability condition in a general case. By reviewing the relationship between the Kirchhoff model and the worm-like road chain (WLRC) model, we insert a spontaneous curvature term which can partly reflect the twist angle's contribution to free energy in the WLRC model and name this extended model the EWLRC model. By choosing suitable spontaneous curvature, stability analysis in this model provide us with some useful results which are consistent with the experimental observations.Comment: 13 pages, 1 figure, 2 table

Discontinuity of curvature on axisymmetric Willmore surfaces

The equilibrium shapes of vesicles are governed by the general shape equation which is derived from the minimization of the Helfrich elastic free energy and can be reduced to the Willmore equation in a special case. The general shape equation is a high order nonlinear partial differential equation and it is very difficult to find analytical solution even in axisymmetric case, which is reduced to a seconder ordinary differential equation. Traditional axisymmetric shape equation is with the turning radius as the variable. Here we study the shape equation with the tangential angle as the variable. In this case, the Willmore equation is reduced to the Bernoulli differential equation and the general solution is obtained conveniently. We find that the curvature in this solution is discontinuous in some cases, which was ignored by previous researchers. This solution can satisfy the boundary conditions for open vesicle with free edges.Comment: 5 pages, 3 figure

Periodic-cylinder vesicle with minimal energy

We give some details about the periodic cylindrical solution found by Zhang and Ou-Yang in [Phys. Rev. E 53, 4206(1996)] for the general shape equation of vesicle. Three different kinds of periodic cylindrical surfaces and a special closed cylindrical surface are obtained. Using the elliptic functions contained in \emph{mathematic}, we find that this periodic shape has the minimal total energy for one period when the period-amplitude ratio $\beta\simeq1.477$, and point out that it is a discontinuous deformation between plane and this periodic shape. Our results also are suitable for DNA and multi-walled carbon nanotubes (MWNTs)Comment: 10 pages, 6 figures, accepted by Chinese Physics

Quantum discord for the general two-qubit case

Recently, Girolami and Adesso have demonstrated that the calculation of quantum discord for two-qubit case can be viewed as to solve a pair of transcendental equation (Phys. Rev. A, {\bf 83}, 052108(2011)). In present work, we introduce the generalized Choi-Jamiolkowski isomorphism and apply it as a convenient tool for constructing transcendental equations. For the general two-qubit case, we show that the transcendental equations always have a finite set of universal solutions, this result can be viewed as a generalization of the one get by Ali, Rau, and Alber (Phys. Rev. A, {\bf 81}, 042105 (2010)). For a subclass of $X$ state, we find the analytical solutions by solving the transcendental equations.Comment: 6 pages, no figure, accepted by QIN

A note on the KK-stability on toric manifolds

In this note, we prove that on polarized toric manifolds the relative $K$-stability with respect to Donaldson's toric degenerations is a necessary condition for the existence of Calabi's extremal metrics, and also we show that the modified $K$-energy is proper in the space of $G_0$-invariant K\"ahler metrics for the case of toric surfaces which admit the extremal metrics.Comment: 8 page

Geometric discord: A resource for increments of quantum key distribution through twirling

In the present work, we consider a scenario where an arbitrary two-qubit pure state is applied to generate a randomly distributed key via the generalized EPR protocol. Using the twirling procedure to convert the pure state into a Werner state, the error rate of the key can be reduced by a factor of $2/3$. This effect indicates that entanglement is not the sufficient resource of the generalized EPR protocol since it is not increased in the twirling procedure. Instead of entanglement, the geometric discord is suggested to be the general quantum resource for this task.Comment: 4 pages, no figure, comments are welcom

Relative KK-stability and modified KK-energy on toric manifolds

In this paper, we discuss the relative $K$-stability and the modified $K$-energy associated to the Calabi's extremal metric on toric manifolds. We give a sufficient condition in the sense of convex polytopes associated to toric manifolds for both the relative $K$-stability and the properness of modified $K$-energy. In particular, our results hold for toric Fano manifolds with vanishing Futaki-invariant. We also verify our results on the toric Fano surfaces

Minimizing weak solutions for calabi's extremal metrics on toric manifolds

In this paper, we discuss a Donaldson's version of the modified $K$-energy associated to the Calabi's extremal metrics on toric manifolds and prove the existence of the weak solution for extremal metrics in the sense of convex functions which minimizes the modified $K$-energy.Comment: 28 pages, 3 figure

Modified Futaki invariant and equivariant Riemann-Roch formula

In this paper, we give a new version of the modified Futaki invariant for a test configuration associated to the soliton action on a Fano manifold. Our version will naturally come from toric test configurations defined by Donaldson for toric manifolds. As an application, we show that the modified $K$-energy is proper for toric invariant K\"ahler potentials on a toric Fano manifold.Comment: welcome comment