The Mean Field Equation with Critical Parameter in a Plane Domain

Consider the mean field equation with critical parameter $8\pi$ in a bounded smooth domain $\Omega$. Denote by $E_{8\pi}(\Omega)$ the infimum of the associated functional $I_{8\pi}(\Omega)$. We call $E_{8\pi}(\Omega)$ the "energy" of the domain $\Omega$. We prove that if the area of $\Omega$ is equal to $\pi$, then the energy of $\Omega$ is always greater or equal to the energy of the unit disk and equality holds if and only if $\Omega$ is the unit disk. We also give a sufficient condition for the existence of a minimizer for $I_{8\pi}(\Omega)$.Comment: 13 pages, to appear in Differential and Integral Equation

The Ultraproducts of Quasirandom Groups

In this paper, we shall prove that an ultraproduct of non-abelian finite simple groups is either finite simple, or has no finite dimensional unitary representation other than the trivial one. Then we shall generalize this result for other kinds of quasirandom groups. A group is called D- quasirandom if all of its nontrivial representations over the complex numbers have dimensions at least D. We shall study the question of whether a non-principal ultraproduct of a given sequence of quasirandom groups remains quasirandom, and whether an ultraproduct of increasingly quasirandom groups becomes minimally almost periodic (i.e. no non-trivial finite-dimensional unitary representation at all). We answer this question in the affirmative when the groups in question are simple, quasisimple, semisimple, or when the groups in question have bounded number of conjugacy classes in their cosocles (the intersection of all maximal normal subgroups), or when the groups are arbitrary products (not necessarily finite) of the groups just listed. We shall also present with an ultraproduct of increasingly quasirandom groups with a non-trivial one-dimensional representation. We also obtain some results in the case of semi-direct products and short exact sequences of quasirandom groups. Finally, two applications of our results are given, one in triangle patterns of quasirandom groups and one in self-Bohrifying groups. Our main tools are some variations of the covering number for groups, different kinds of length functions on groups, and the classification of finite simple groups

One Dimensional Conformal Metric Flows

This is the second paper of our series of papers on one dimensional conformal metric flows. In this paper we continue our studies of the one dimensional conformal metric flows, which were introduced in math.AP/0611254. We prove the global existence and convergence of the one dimensional Yamabe and affine flows. Furthermore, we obtain exponential convergence of the metrics under these flows.Comment: 22 page

A Survey of directed graphs invariants

In this paper, various kinds of invariants of directed graphs are summarized. In the first topic, the invariant w(G) for a directed graph G is introduced, which is primarily defined by S. Chen and X.M. Chen to solve a problem of weak connectedness of tensor product of two directed graphs. Further, we present our recent studies on the invariant w(G) in categorical view. In the second topic, Homology theory on directed graph is introduced, and we also cast on categorical view of the definition. The third topic mainly focuses on Laplacians on graphs, including traditional work and latest result of 1-laplacian by K.C.Chang. Finally, Zeta functions and Graded graphs are introduced, inclduing Bratteli-Vershik diagram, dual graded graphs and differential posets, with some applications in dynamic system.Comment: This survey is written by Y.L.Zhang, supervised by Pro. S.Chen, containing 20 pages and 5 figure

Steady States for One Dimensional Conformal Metric Flows

We define two conformal structures on $S^1$ which give rise to a different view of the affine curvature flow and a new curvature flow, the ``$Q$-curvature flow". The steady state of these flows are studied. More specifically, we prove four sharp inequalities, which state the existences of the corresponding extremal metrics.Comment: 22 pages, typos correcte

Sharp form for improved Moser-Trudinger inequality

We prove that the improved Moser-Trudinger inequality with optimal coefficient $\alpha =1/2$ holds for all functions on $S^2$ with zero moments.Comment: 9 page

On well-posedness of Ericksen-Leslie's paraboloc-hyperbolic liquid crystal model

We establish the following well-posedness results on Ericksen-Leslie's parabolic-hyperbolic liquid crystal model: 1, if the dissipation coefficients \beta = \mu_4 - 4 \mu_6 > 0, and the size of the initial energy E^{in} is small enough, then the life span of the solution is at least -O(\ln E^{in}); 2, for the special case that the coefficients \mu_1 = \mu_2 = \mu_3 = \mu_5 = \mu_6 = 0, for which the model is the Navier-Stokes equations coupled with the wave map from \mathbb{R}^n to \mathbb{S}^2, the same existence result holds but without the smallness restriction on the size of the initial data; 3, with further constraints on the coefficients, namely \alpha = \mu_4 - 4 \mu_6 - \tfrac{ (|\lambda_1| - 7 \lambda_2)^2 }{\eta} - \tfrac{ 2 ( 7 |\lambda_1| - 2\lambda_2 )^2 }{ |\lambda_1| } > 0 and \mu_2 < \mu_3, the global classical solution with small initial data can be established. A relation between the Lagrangian multiplier and the geometric constraint |d|=1 plays a key role in the proof.Comment: 34 pages. arXiv admin note: text overlap with arXiv:1105.2180 by other author

Configurational temperature of charge-stabilized colloidal monolayers

Recent theoretical advances show that the temperature of a system in equilibriumcan be measured from static snapshots of its constituents' instantaneous configurations, withoutregard to their dynamics. We report the first measurements of the configurational temperature in an experimental system. In particular, we introduce a hierarchy of hyperconfigurational temperature definitions, which we use to analyze monolayers of charge-stabilized colloidal spheres. Equality of the hyperconfigurational and bulk thermodynamic temperatures provides previously lacking thermodynamic self-consistency checks for the measured colloidal pair potentials, and thereby casts new light on anomalous like-charge colloidal attractions induced by geometric confinement.Comment: 4 pages, 3 figure

2DR: Towards Fine-Grained 2-D RFID Touch Sensing

In this paper, we introduce 2DR, a single RFID tag which can seamlessly sense two-dimensional human touch using off-the-shelf RFID readers. Instead of using a two-dimensional tag array to sense human finger touch on a surface, 2DR only uses one or two RFID chip(s), which reduces the manufacturing cost and makes the tag more suitable for printing on flexible materials. The key idea behind 2DR is to design a custom-shape antenna and classify human finger touch based on unique phase information using statistical learning. We printed 2DR tag on FR-4 substrate and use off-the-shelf UHF-RFID readers (FCC frequency band) to sense different touch activities. Experiments show great potential of our design. Moreover, 2DR can be further extended to 3D by building stereoscopic model.Comment: RFID Touch Sensin

Colloidal electroconvection in a thin horizontal cell

Applying an electric field to an aqueous colloidal dispersion establishes a complex interplay of forces among the highly mobile simple ions, the more highly charged but less mobile colloidal spheres, and the surrounding water. This interplay can induce a wide variety of visually striking dynamical instabilities, even when the applied field is constant. This Article reports on the highly organized patterns that emerge when electrohydrodynamic forces compete with gravity in thin layers of charge-stabilized colloidal spheres subjected to low voltages between parallel plate electrodes. Depending on the conditions, these spheres can form into levitating clusters with morphologies ranging from tumbling clouds, to toroidal vortex rings, to writhing labyrinths.Comment: 12 pages, 17 figure