179,653 research outputs found

### 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

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