Improved critical eigenfunction restriction estimates on Riemannian manifolds with constant negative curvature

We show that one can obtain logarithmic improvements of $L^2$ geodesic restriction estimates for eigenfunctions on 3-dimensional compact Riemannian manifolds with constant negative curvature. We obtain a $(\log\lambda)^{-\frac12}$ gain for the $L^2$-restriction bounds, which improves the corresponding bounds of Burq, G\'erard and Tzvetkov, Hu, Chen and Sogge. We achieve this by adapting the approaches developed by Chen and Sogge, Blair and Sogge, Xi and the author. We derive an explicit formula for the wave kernel on 3D hyperbolic space, which improves the kernel estimates from the Hadamard parametrix in Chen and Sogge. We prove detailed oscillatory integral estimates with fold singularities by Phong and Stein and use the Poincar\'e half-space model to establish bounds for various derivatives of the distance function restricted to geodesic segments on the universal cover $\mathbb{H}^3$.Comment: 28 pages, 8 figure

Cycles of the logistic map

The onset and bifurcation points of the $n$-cycles of a polynomial map are located through a characteristic equation connecting cyclic polynomials formed by periodic orbit points. The minimal polynomials of the critical parameters of the logistic, H\'enon, and cubic maps are obtained for $n$ up to 13, 9, and 8, respectively.Comment: 37 pages, 4 figure

An asymptotic formula for the zeros of the deformed exponential function

We study the asymptotic representation for the zeros of the deformed exponential function $\sum\nolimits_{n = 0}^\infty {\frac1{n!}{q^{n(n - 1)/2}{x^n}}}$, $q\in (0,1)$. Indeed, we obtain an asymptotic formula for these zeros: $$x_n=- nq^{1-n}(1 + g(q)n^{-2}+o(n^{-2})),n\ge1,$$ where $g(q)=\sum\nolimits_{k = 1}^\infty {\sigma (k){q^k}}$ is the generating function of the sum-of-divisors function $\sigma(k)$. This improves earlier results by Langley and Liu. The proof of this formula is reduced to estimating the sum of an alternating series, where the Jacobi's triple product identity plays a key role.Comment: 10 pages. To appear in Journal of Mathematical Analysis and Application

Dressing the boundary: on soliton solutions of the nonlinear Schr\"odinger equation on the half-line

Based on the theory of integrable boundary conditions (BCs) developed by Sklyanin, we provide a direct method for computing soliton solutions of the focusing nonlinear Schr\"odinger (NLS) equation on the half-line. The integrable BCs at the origin are represented by constraints of the Lax pair, and our method lies on dressing the Lax pair by preserving those constraints in the Darboux-dressing process. The method is applied to two classes of solutions: solitons vanishing at infinity and self-modulated solitons on a constant background. Half-line solitons in both cases are explicitly computed. In particular, the boundary-bound solitons, that are static solitons bounded at the origin, are also constructed. We give a natural inverse scattering transform interpretation of the method as evolution of the scattering data determined by the integrable BCs in space.Comment: 21 pages, 10 figures, correcting typos of the previous uploa

Temperature measurement from perturbations

The notion of configuration temperature is extended to discontinuous systems by identifying the temperature as the nontrivial root of several integral equations regarding the distribution of the energy change upon configuration perturbations. The relations are generalized to pressure and a distribution mean force.Comment: 8 pages, 2 figure

Topological insulators from the Perspective of first-principles calculations

Topological insulators are new quantum states with helical gapless edge or surface states inside the bulk band gap.These topological surface states are robust against the weak time-reversal invariant perturbations, such as lattice distortions and non-magnetic impurities. Recently a variety of topological insulators have been predicted by theories, and observed by experiments. First-principles calculations have been widely used to predict topological insulators with great success. In this review, we summarize the current progress in this field from the perspective of first-principles calculations. First of all, the basic concepts of topological insulators and the frequently-used techniques within first-principles calculations are briefly introduced. Secondly, we summarize general methodologies to search for new topological insulators. In the last part, based on the band inversion picture first introduced in the context of HgTe, we classify topological insulators into three types with s-p, p-p and d-f, and discuss some representative examples for each type.Comment: 10 pages, 7 figure

SO(5) Quantum Nonlinear sigma Model Theory of the High Tc Superconductivity

We show that the complex phase diagram of high $T_c$ superconductors can be deduced from a simple symmetry principle, a $SO(5)$ symmetry which unifies antiferromagnetism with $d$ wave superconductivity. We derive the approximate $SO(5)$ symmetry from the microscopic Hamiltonian and show furthermore that this symmetry becomes exact under the renormalization group flow towards a bicritical point. With the help of this symmetry, we construct a $SO(5)$ quantum nonlinear $\sigma$ model to describe the effective low energy degrees of freedom of the high $T_c$ superconductors, and use it to deduce the phase diagram and the nature of the low lying collective excitations of the system. We argue that this model naturally explains the basic phenomenology of the high $T_c$ superconductors from the insulating to the underdoped and the optimally doped region.Comment: 36 pages, 1 Postscript figur

Exact microscopic wave function for a topological quantum membrane

The higher dimensional quantum Hall liquid constructed recently supports stable topological membrane excitations. Here we introduce a microscopic interacting Hamiltonian and present its exact ground state wave function. We show that this microscopic ground state wave function describes a topological quantum membrane. We also construct variational wave functions for excited states using the non-commutative algebra on the four sphere. Our approach introduces a non-perturbative method to quantize topological membranes

Modeling Market Mechanism with Evolutionary Games

This is an essay solicited by Europhysics News, published in its March/April 1998 issue with slight modifications. We outline some highlights of the econophysics models, especially the so-called Minority model of competition and evolution. Even without the usual math, this essay offers an analytical solution to the Minority model, revealing some key features of the solution.Comment: 4 pages, no figure

Why Financial Markets Will Remain Marginally Inefficient?

I summarize the recent work on market (in)efficiency, highlighting key elements why financial markets will never be made efficient. My approach is not by adding more empirical evidence, but giving plausible reasons as to where inefficiency arises and why it's not rational to arbitrage it away.Comment: 5 pages, 1 figure. based on a speech at Tokyo Econophysics Meeting, Nov 14th 200