arXiv subjects

Xu Zhang

Publications and source records attributed to Xu Zhang.

At least 487 records · Page 27Linked to original sources

arXiv2016

Finite Codimensional Controllability for Evolution Equations

Motivated by infinite-dimensional optimal control problems with endpoint state constraints, in this Note, we introduce the notion of finite codimensional exact controllability for evolution equations. It is shown that this new controllability is equivalent to the finite codimensionality condition in the literatures to guarantee Pontryagin's maximum principle. As examples, LQ problems with fixed endpoint state constraints for a wave and a heat equation are analyzed, respectively.

math.OC↗

arXiv2016

A Mini-Course on Stochastic Control

This note is addressed to giving a short introduction to control theory of stochastic systems, governed by stochastic differential equations in both finite and infinite dimensions. We will mainly explain the new phenomenon and difficulties in the study of controllability and optimal control problems for these sort of equations. In particular, we will show by some examples that both the formulation of stochastic control problems and the tools to solve them may differ considerably from their deterministic counterpart.

math.OC↗

arXiv2016

Nonconforming Immersed Finite Element Spaces For Elliptic Interface Problems

In this paper, we use a unified framework introduced in [3] to study two classes of nonconforming immersed finite element (IFE) spaces with integral value degrees of freedom. The shape functions on interface elements are piecewise polynomials defined on sub-elements separated either by the actual interface or its line approximation. In this unified framework, we use the invertibility of the well known Sherman-Morison systems to prove the existence and uniqueness of shape functions on each interface element in either rectangular or triangular mesh. Furthermore, we develop a multi-edge expansion for piecewise functions and a group of identities for nonconforming IFE functions which enable us to show that these IFE spaces have the optimal approximation capability.

math.NA↗

arXiv2016

Characterization of Optimal Feedback for Stochastic Linear Quadratic Control Problems

One of the fundamental issues in Control Theory is to design feedback controls. It is well-known that, the purpose of introducing Riccati equations in the deterministic case is to provide the desired feedback controls for linear quadratic control problems. To date, the same problem in the stochastic setting is only partially well-understood. In this paper, we establish the equivalence between the existence of optimal feedback controls for the stochastic linear quadratic control problems with random coefficients and the solvability of the corresponding backward stochastic Riccati equations in a suitable sense. We also give a counterexample showing the nonexistence of feedback controls to a solvable stochastic linear quadratic control problem. This is a new phenomenon in the stochastic setting, significantly different from its deterministic counterpart.

math.OC↗

arXiv2016

Second Order Necessary Conditions for Optimal Control Problems on Riemannian Manifolds

This work is concerned with an optimal control problem on a Riemannian manifold, for which two typical cases are considered. The first case is when the endpoint is free. For this case, the control set is assumed to be a separable metric space. By introducing suitable dual equations, which depend on the curvature tensor of the manifold, we establish the second order necessary and sufficient optimality conditions of integral form. In particular, when the control set is a Polish space, the second order necessary condition is reduced to a pointwise form. As a key preliminary result and also an interesting byproduct, we derive a geometric lemma, which may have some independent interest. The second case is when the endpoint is fixed. For this more difficult case, the control set is assumed to be open in an Euclidian space. We obtain the second order necessary and sufficient optimality conditions, in which the curvature tensor also appears explicitly. Our optimality conditions can be used to recover the following famous geometry result: Any geodesic connecting two fixed points on a Riemannian manifold satisfies the second variation of energy; while the existing optimality conditions in control literatures fail to give the same result.

math.OC↗

arXiv2016

First and second order necessary conditions for stochastic optimal controls

The main purpose of this paper is to establish the first and second order necessary optimality conditions for stochastic optimal controls using the classical variational analysis approach. The control system is governed by a stochastic differential equation, in which both drift and diffusion terms may contain the control variable and the set of controls is allowed to be nonconvex. Only one adjoint equation is introduced to derive the first order necessary condition; while only two adjoint equations are needed to state the second order necessary conditions for stochastic optimal controls.

math.OC↗

arXiv2016

Comment on "Linear Scaling of the Exciton Binding Energy versus the Band Gap of Two-Dimensional Materials"

In a recent letter [J. -H. Choi et al. Phys. Rev. Lett. 115, 066403 (2015)], a universal linear relation between the binding energy Eb of exciton and the band gap Eg is found in different quasi-2D semiconductors. However, when one extrapolates the straight line to Eg=0, a positive Eb is obtained. This means that a stable exciton exists in a metal (Eg=0), or in other words, the excitation energy is negative. Therefore, for quasi-2D semiconductors with small band gap, Eb vs. Eg must deviate from the straight line suggested in the Letter. We performed similar GW+BSE calculation as the Letter for compressed phosphorene and stretched graphene. At larger band gap, our data fall in the same straight lien as the Letter. But for the quasi-2D semiconductors with small band gap, the Eb vs. Eg curve approaches to zero. A simple model based on local instantaneous screened interaction gives reasonable agreement with the more rigorous calculations. For the quasi-2D semiconductors with diminishing gap Eg<0.3eV, due to the nonlocal retarded effect induced by the strong screening in the small gap materials, one has to use 2-time transition amplitude (Bethe-Salpeter wave function) to describe exciton.

cond-mat.mtrl-sci↗

arXiv2016

Spatio-temporal Aware Non-negative Component Representation for Action Recognition

This paper presents a novel mid-level representation for action recognition, named spatio-temporal aware non-negative component representation (STANNCR). The proposed STANNCR is based on action component and incorporates the spatial-temporal information. We first introduce a spatial-temporal distribution vector (STDV) to model the distributions of local feature locations in a compact and discriminative manner. Then we employ non-negative matrix factorization (NMF) to learn the action components and encode the video samples. The action component considers the correlations of visual words, which effectively bridge the sematic gap in action recognition. To incorporate the spatial-temporal cues for final representation, the STDV is used as the part of graph regularization for NMF. The fusion of spatial-temporal information makes the STANNCR more discriminative, and our fusion manner is more compact than traditional method of concatenating vectors. The proposed approach is extensively evaluated on three public datasets. The experimental results demonstrate the effectiveness of STANNCR for action recognition.

cs.CV↗

arXiv2016

Enhancing the resolution of hyperlens by the compensation of losses without gain media

We present a method to improve the resolution of available hyperlenses in the literature. In this method, we combine the operation of hyperlens with the recently proposed plasmon injection scheme for loss compensation in metamaterials. Image of an object, which is otherwise not resolvable by the hyperlens alone, was reconstructed up to the minimum feature size of one seventh of the free-space wavelength.

physics.optics↗

arXiv2016

Bringing the 'perfect lens' into focus by near-perfect compensation of losses without gain media

In this paper, the optical properties and imaging performance of a non-ideal Pendry's negative index flat lens with a practical value for loss are studied. Analytical calculations of the optical properties of the lens are performed, and those results are used to further study the lens and corresponding imaging system numerically. The plasmon injection scheme for loss compensation in negative index metamaterials is applied to the results from the imaging system, resulting in a perfect reconstruction of a previously unresolved image that demonstrates sub-diffraction-limit resolution.

physics.optics↗

arXiv2016

Long time well-posdness of the Prandtl equations in Sobolev space

In this paper, we study the long time well-posedness for the nonlinear Prandtl boundary layer equation on the half plane. While the initial data are small perturbations of some monotonic shear profile, we prove the existence, uniqueness and stability of solutions in weighted Sobolev space by energy methods. The key point is that the life span of the solution could be any large $T$ as long as its initial date is a perturbation around the monotonic shear profile of small size like $e^{-T}$. The nonlinear cancellation properties of Prandtl equations under the monotonic assumption are the main ingredients to establish a new energy estimate.

math.AP↗

arXiv2016

Transport anomalies and quantum criticality in electron-doped cuprate superconductors

Superconductivity research is like running a marathon. Three decades after the discovery of high-Tc cuprates, there have been mass data generated from transport measurements, which bring fruitful information. In this review, we give a brief summary of the intriguing phenomena reported in electron-doped cuprates from the aspect of electrical transport as well as the complementary thermal transport. We attempt to sort out common features of the electron-doped family, e.g. the strange metal, negative magnetoresistance, multiple sign reversals of Hall in mixed state, abnormal Nernst signal, complex quantum criticality. Most of them have been challenging the existing theories, nevertheless, a unified diagram certainly helps to approach the nature of electron-doped cuprates.

cond-mat.supr-con↗

arXiv2016

An Internal Observability Estimate for Stochastic Hyperbolic Equations

This paper is addressed to establishing an internal observability estimate for some linear stochastic hyperbolic equations. The key is to establish a new global Carleman estimate for forward stochastic hyperbolic equations in the $L^2$-space. Different from the deterministic case, a delicate analysis of the adaptedness for some stochastic processes is required in the stochastic setting.

math.OC↗

arXiv2015

Parallel Stitching of Two-Dimensional Materials

Diverse parallel stitched two-dimensional heterostructures are synthesized, including metal-semiconductor (graphene-MoS2), semiconductor-semiconductor (WS2-MoS2), and insulator-semiconductor (hBN-MoS2), directly through selective sowing of aromatic molecules as the seeds in chemical vapor deposition (CVD) method. Our methodology enables the large-scale fabrication of lateral heterostructures with arbitrary patterns, and clean and precisely aligned interfaces, which offers tremendous potential for its application in integrated circuits.

cond-mat.mtrl-sci↗

arXiv2015

Pointwise second-order necessary conditions for stochastic optimal controls, Part II: The general case

This paper is the second part of our series of work to establish pointwise second-order necessary conditions for stochastic optimal controls. In this part, we consider the general cases, i.e., the control region is allowed to be nonconvex, and the control variable enters into both the drift and the diffusion terms of the control systems. By introducing four variational equations and four adjoint equations, we obtain the desired necessary conditions for stochastic singular optimal controls in the sense of Pontryagin-type maximum principle.

math.OC↗

arXiv2015

Discontinuous Galerkin Immersed Finite Element Methods for Parabolic Interface Problems

In this article, interior penalty discontinuous Galerkin methods using immersed finite element functions are employed to solve parabolic interface problems. Typical semi-discrete and fully discrete schemes are presented and analyzed. Optimal convergence for both semi-discrete and fully discrete schemes are proved. Some numerical experiments are provided to validate our theoretical results.

math.NA↗

arXiv2015

Extend the Kompaneets Equation to Down-Comptonization Process in a Novel Way

Comptonization is a very important phenomenon in astrophysics. Kompaneets equation describes up-Comptonization process in nonrelativistic astrophysics, while it fails for down-Comptonization Scattering which is the most important radiative transfer process in hard X-rays and $γ$-rays astronomy. In this study, we explore both up-Comptonization and down-Comptonization processes. A new relativistic corrections is introduced to the Kompaneets equation which is valid in both nonrelativistic energy regime with the photon energy $hν<< m_ec^2$ and the electron temperature $KT_e << m_ec^2$. Numerical comparisons are presented which show excellent agreement between the Kompaneets equation and the new equation.

astro-ph.HE↗

arXiv2015

Quantum Electrodynamics is Crucial for Plasmonic Resonance of Metallic Nanostructures

Plasmonic resonance of a metallic nanostructure results from coherent motion of its conduction electrons driven by incident light. At the resonance, the induced dipole in the nanostructure is proportional to the number of the conduction electrons, hence $10^{7}$ times larger than that in an atom. The interaction energy between the induced dipole and fluctuating virtual field of the incident light can reach a few tenths of an eV. Therefore, the classical electromagnetism dominating the field becomes inadequate. We argue that quantum electrodynamics (QED) should be used instead as the fundamental theory to describe the virtual field and its interaction with the electrons. Based on QED, we derive analytic expressions for the plasmonic resonant frequency, which depends on three easily accessible material parameters. The analytic theory reproduces very well the experimental data, and can be used for rational design of materials.

cond-mat.mtrl-sci↗