Stability of the surface area preserving mean curvature flow in Euclidean space

We show that the surface area preserving mean curvature flow in Euclidean space exists for all time and converges exponentially to a round sphere, if initially the L^2-norm of the traceless second fundamental form is small (but the initial hypersurface is not necessarily convex).Comment: 17 page

Achieving an Efficient and Fair Equilibrium Through Taxation

It is well known that a game equilibrium can be far from efficient or fair, due to the misalignment between individual and social objectives. The focus of this paper is to design a new mechanism framework that induces an efficient and fair equilibrium in a general class of games. To achieve this goal, we propose a taxation framework, which first imposes a tax on each player based on the perceived payoff (income), and then redistributes the collected tax to other players properly. By turning the tax rate, this framework spans the continuum space between strategic interactions (of selfish players) and altruistic interactions (of unselfish players), hence provides rich modeling possibilities. The key challenge in the design of this framework is the proper taxing rule (i.e., the tax exemption and tax rate) that induces the desired equilibrium in a wide range of games. First, we propose a flat tax rate (i.e., a single tax rate for all players), which is necessary and sufficient for achieving an efficient equilibrium in any static strategic game with common knowledge. Then, we provide several tax exemption rules that achieve some typical fairness criterions (such as the Max-min fairness) at the equilibrium. We further illustrate the implementation of the framework in the game of Prisoners' Dilemma.Comment: This manuscript serves as the technical report for the paper with the same title published in APCC 201

Scale dependences of local form non-Gaussianity parameters from a DBI isocurvature field

We derive the spectral indices and their runnings of local form $f_{NL}$ and $g_{NL}$ from a DBI isocurvature field and we find that the indices are suppressed by the sound speed $c_s$. This effect can be interpreted by the Lorentz boost from the viewpoint in the frame where brane is moving.Comment: 12 pages, 1 figur

Entanglement renormalization and integral geometry

We revisit the applications of integral geometry in AdS$_3$ and argue that the metric of the kinematic space can be realized as the entanglement contour, which is defined as the additive entanglement density. From the renormalization of the entanglement contour, we can holographically understand the operations of disentangler and isometry in multi-scale entanglement renormalization ansatz. Furthermore, a renormalization group equation of the long-distance entanglement contour is then derived. We then generalize this integral geometric construction to higher dimensions and in particular demonstrate how it works in bulk space of homogeneity and isotropy.Comment: 40 pages, 7 figures. v2: discussions on the general measure added, typos fixed; v3: sections reorganized, various points clarified, to appear in JHE

Quasi-Quantum Planes and Quasi-Quantum Groups of Dimension p3p^3 and p4p^4

The aim of this paper is to contribute more examples and classification results of finite pointed quasi-quantum groups within the quiver framework initiated in \cite{qha1, qha2}. The focus is put on finite dimensional graded Majid algebras generated by group-like elements and two skew-primitive elements which are mutually skew-commutative. Such quasi-quantum groups are associated to quasi-quantum planes in the sense of nonassociative geomertry \cite{m1, m2}. As an application, we obtain an explicit classification of graded pointed Majid algebras with abelian coradical of dimension $p^3$ and $p^4$ for any prime number $p.$Comment: 12 pages; Minor revision according to the referee's suggestio