arXiv subjects
Publications and source records attributed to Rui Han.
We give a simple proof of absence of point spectrum for the self-dual extended Harper's model. We get a sharp result which improves that of Avila-Jitomirskaya-Marx in the isotropic self-dual regime.
To meet the women appearance needs, we present a novel virtual experience approach of facial makeup transfer, developed into windows platform application software. The makeup effects could present on the user's input image in real time, with an only single reference image. The input image and reference image are divided into three layers by facial feature points landmarked: facial structure layer, facial color layer, and facial detail layer. Except for the above layers are processed by different algorithms to generate output image, we also add illumination transfer, so that the illumination effect of the reference image is automatically transferred to the input image. Our approach has the following three advantages: (1) Black or dark and white facial makeup could be effectively transferred by introducing illumination transfer; (2) Efficiently transfer facial makeup within seconds compared to those methods based on deep learning frameworks; (3) Reference images with the air-bangs could transfer makeup perfectly.
Let $ Tf =\sum_{ I} \varepsilon_I \langle f,h_{I^+}\rangle h_{I^-}$. Here, $ \lvert \varepsilon _I\rvert=1 $, and $ h_J$ is the Haar function defined on dyadic interval $ J$. We show that, for instance, \begin{equation*} \lVert T \rVert _{L ^{2} (w) \to L ^{2} (w)} \lesssim [w] _{A_2 ^{+}} . \end{equation*} Above, we use the one sided $ A_2$ characteristic for the weight $ w$. This is an instance of a one sided $A_2$ conjecture. Our proof of this fact is difficult, as the very quick known proofs of the $A_2$ theorem do not seem to apply in the one sided setting.
We extend some basic results known for finite range operators to long range operators with off-diagonal decay. Namely, we prove an analogy of Sch'nol's theorem. We also establish the connection between the almost sure spectrum of long range random operators and the spectra of deterministic periodic operators.
We study the one-dimensional discrete Schrödinger operator with the skew-shift potential $2λ\cos\left(2π\left(\binom{j}{2} ω+jy+x\right)\right)$. This potential is long conjectured to behave like a random one, i.e., it is expected to produce Anderson localization for arbitrarily small coupling constants $λ>0$. In this paper, we introduce a novel perturbative approach for studying the zero-energy Lyapunov exponent $L(λ)$ at small $λ$. Our main results establish that, to second order in perturbation theory, a natural upper bound on $L(λ)$ is fully consistent with $L(λ)$ being positive and satisfying the usual Figotin-Pastur type asymptotics $L(λ)\sim Cλ^2$ as $λ\to 0$. The analogous quantity behaves completely differently in the Almost-Mathieu model, whose zero-energy Lyapunov exponent vanishes for $λ<1$. The main technical work consists in establishing good lower bounds on the exponential sums (quadratic Weyl sums) that appear in our perturbation series.
We study discrete Schrödinger operators on the graphs corresponding to the triangular lattice, the hexagonal lattice, and the square lattice with next-nearest neighbor interactions. For each of these lattice geometries, we analyze the behavior of small periodic potentials. In particular, we provide sharp bounds on the number of gaps that may perturbatively open, we describe sharp arithmetic criteria on the periods that ensure that no gaps open, and we characterize those energies at which gaps may open in the perturbative regime. In all three cases, we provide examples that open the maximal number of gaps and estimate the scaling behavior of the gap lengths as the coupling constant goes to zero.
Four-dimensional variational data assimilation (4DVar) has become an increasingly important tool in data science with wide applications in many engineering and scientific fields such as geoscience1-12, biology13 and the financial industry14. The 4DVar seeks a solution that minimizes the departure from the background field and the mismatch between the forecast trajectory and the observations within an assimilation window. The current state-of-the-art 4DVar offers only two choices by using different forms of the forecast model: the strong- and weak-constrained 4DVar approaches15-16. The former ignores the model error and only corrects the initial condition error at the expense of reduced accuracy; while the latter accounts for both the initial and model errors and corrects them separately, which increases computational costs and uncertainty. To overcome these limitations, here we develop an integral correcting 4DVar (i4DVar) approach by treating all errors as a whole and correcting them simultaneously and indiscriminately. To achieve that, a novel exponentially decaying function is proposed to characterize the error evolution and correct it at each time step in the i4DVar. As a result, the i4DVar greatly enhances the capability of the strong-constrained 4DVar for correcting the model error while also overcomes the limitation of the weak-constrained 4DVar for being prohibitively expensive with added uncertainty. Numerical experiments with the Lorenz model show that the i4DVar significantly outperforms the existing 4DVar approaches. It has the potential to be applied in many scientific and engineering fields and industrial sectors in the big data era because of its ease of implementation and superior performance.
We establish a version of the fractal uncertainty principle, obtained by Bourgain and Dyatlov in 2016, in higher dimensions. The Fourier support is limited to sets $Y\subset \mathbb{R}^d$ which can be covered by finitely many products of $δ$-regular sets in one dimension, but relative to arbitrary axes. Our results remain true if $Y$ is distorted by diffeomorphisms. Our method combines the original approach by Bourgain and Dyatlov, in the more quantitative 2017 rendition by Jin and Zhang, with Cartan set techniques.
We study fractal dimension properties of singular Jacobi operators. We prove quantitative lower spectral/quantum dynamical bounds for general operators with strong repetition properties and controlled singularities. For analytic quasiperiodic Jacobi operators in the positive Lyapunov exponent regime, we obtain a sharp arithmetic criterion of full spectral dimensionality. The applications include the extended Harper's model where we obtain arithmetic results on spectral dimensions and quantum dynamical exponents.
We study the discrimination of weak coherent states of light with significant overlaps by nondestructive measurements on the light states through measuring atomic states that are entangled to the coherent states via dipole coupling. In this way, the problem of measuring and discriminating coherent light states is shifted to finding the appropriate atom-light interaction and atomic measurements. We show that this scheme allows us to attain a probability of error extremely close to the Helstrom bound, the ultimate quantum limit for discriminating binary quantum states, through the simple Jaynes-Cummings interaction between the field and ancilla with optimized light-atom coupling and projective measurements on the atomic states. Moreover, since the measurement is nondestructive on the light state, information that is not detected by one measurement can be extracted from the post-measurement light states through subsequent measurements.
When quantum states are used to send classical information, the receiver performs a measurement on the signal states. The amount of information extracted is often not optimal due to the receiver's measurement scheme and experimental apparatus. For quantum non-demolition measurements, there is potentially some residual information in the post-measurement state, while part of the information has been extracted and the rest is destroyed. Here, we propose a framework to characterize a quantum measurement by how much information it extracts and destroys, and how much information it leaves in the residual post-measurement state. The concept is illustrated for several receivers discriminating coherent states.
We prove a conditional theorem on the positivity of the Lyapunov exponent for a Schrödinger cocycle over a skew shift base with a cosine potential and the golden ratio as frequency. For coupling below 1, which is the threshold for Herman's subharmonicity trick, we formulate three conditions on the Lyapunov exponent in a finite but large volume and on the associated large deviation estimates at that scale. Our main results demonstrate that these finite-size conditions imply the positivity of the infinite volume Lyapunov exponent. This paper shows that it is possible to make the techniques developed for the study of Schrödinger operators with deterministic potentials, based on large deviation estimates and the avalanche principle, effective.
We consider a quantum graph as a model of graphene in magnetic fields and give a complete analysis of the spectrum, for all constant fluxes. In particular, we show that if the reduced magnetic flux $Φ/2π$ through a honeycomb is irrational, the continuous spectrum is an unbounded Cantor set of Lebesgue measure zero.
We study the multi-dimensional operator $(H_x u)_n=\sum_{|m-n|=1}u_{m}+f(T^n(x))u_n$, where $T$ is the shift of the torus $\T^d$. When $d=2$, we show the spectrum of $H_x$ is almost surely purely continuous for a.e. $α$ and generic continuous potentials. When $d\geq 3$, the same result holds for frequencies under an explicit arithmetic criterion. We also show that general multi-dimensional operators with measurable potentials do not have eigenvalue for generic $α$.
The CFETR baseline scenario is based on a H-mode equilibrium with high pedestal and highly peaked edge bootstrap current, along with strong reverse shear in safety factor profile. The stability of ideal MHD modes for the CFETR baseline scenario has been evaluated using NIMROD and AEGIS codes. The toroidal mode numbers (n=1-10) are considered in this analysis for different positions of perfectly conducting wall in order to estimate the ideal wall effect on the stability of ideal MHD modes for physics and engineering designs of CFETR. Although, the modes (n=1-10) are found to be unstable in ideal MHD, the structure of all modes is edge localized. Growth rates of all modes are found to be increasing initially with wall position before they reach ideal wall saturation limit (no wall limit). No global core modes are found to be dominantly unstable in our analysis. The design of $q_{min}>2$ and strong reverse shear in $q$ profile is expected to prevent the excitation of global modes. Therefore, this baseline scenario is considered to be suitable for supporting long time steady state discharge in context of ideal MHD physics, if ELMs could be controlled.
We quantify the coupling asymptotics for the Lyapunov-exponent of a one-frequency quasi-periodic Schrödinger operator with analytic potential sampling function. The result refines the well-known lower bound of the Lyapunov-exponent by Sorets and Spencer.
We discuss a novel implementation of the minimum error state discrimination measurement, originally introduced by Helstrom. In this implementation, instead of performing the optimal projective measurement directly on the system, it is first entangled to an ancillary system and the measurement is performed on the ancilla. We show that, by an appropriate choice of the entanglement transformation, the Helstrom bound can be attained. The advantage of this approach is twofold. First, it provides a novel implementation when the optimal projective measurement cannot be directly performed. For example, in the case of continuous variable states (binary and N phase-shifted coherent signals), the available detection methods, photon counting and homodyning, are insufficient to perform the required cat-state projection. In the case of symmetric states, the square-root measurement is optimal, but it is not easy to perform directly for more than two states. Our approach provides a feasible alternative in both cases. Second, the measurement is non-destructive from the point of view of the original system and one has a certain amount of freedom in designing the post-measurement state, which can then be processed further.
We establish a topological criterion for connection between reducibility to constant rotations and dual localization, for the general family of analytic quasiperiodic Jacobi operators. As a corollary, we obtain the sharp arithmetic phase transition for the extended Harper's model in the positive Lyapunov exponent region.