Entire solutions of delay differential equations of Malmquist type

In this paper, we investigate the delay differential equations of Malmquist type of form \begin{equation*} w(z+1)-w(z-1)+a(z)\frac{w'(z)}{w(z)}=R(z, w(z)),~~~~~~~~~~~~~~(*) \end{equation*} where $R(z, w(z))$ is an irreducible rational function in $w(z)$ with rational coefficients and $a(z)$ is a rational function. We characterize all reduced forms when the equation $(*)$ admits a transcendental entire solutions with hyper-order less than one. When we compare with the results obtained by Halburd and Korhonen[Proc.Amer.Math.Soc., forcoming],we obtain the reduced forms without the assumptions that the denominator of rational function $R(z,w(z))$ has roots that are nonzero rational functions in $z$. The growth order and value distribution of transcendental entire solutions for the reduced forms are also investigated

Improved distance sensitivity oracles via tree partitioning

We introduce an improved structure of distance sensitivity oracle (DSO). The task is to pre-process a non-negatively weighted graph so that a data structure can quickly answer replacement path length for every triple of source, terminal and failed vertex. The previous best algorithm constructs in time $\tilde{O}(mn)$ a distance sensitivity oracle of size $O(n^2\log n)$ that processes queries in $O(1)$ time. As an improvement, our oracle takes up $O(n^2)$ space, while preserving $O(1)$ query efficiency and $\tilde{O}(mn)$ preprocessing time. One should notice that space complexity and query time of our novel data structure are asymptotically optimal

Angle-of-Attack Modulation in Trajectory Tracking for a Reusable Launch Vehicle

This paper deals with the problem of angle-of-attack modulation with the aim of enhancing transient performance of entry guidance during bank reversals, while compensating adverse effects of fast time-varying transient disturbances. An extended single-input/single-output system is developed in the velocity domain by means of a dynamic extension technique, and explicitly captures the trajectory dynamics of angle-of-attack modulation. A normal form for this extended system is derived for the sake of employing a feedback linearization controller. Further, the control characteristics of angle-of-attack modulation is found to be a non-minimum phase behavior under two common conditions in a near- equilibrium glide flight. Therefore, the issue of angle-of-attack modulation is formulated as robust output stabilization of the non-minimum phase system. A disturbance observer-based feedback linearization technique is used to design a robustly dynamical output-feedback controller for angle-of-attack modulation, and an internal-state feedback controller for bank-angle modulation is used to stabilize the unstable internal dynamics. Numerical simulations are conducted to demonstrate that the performance of the proposed method of angle-of-attack modulation is enhanced compared to the existing shuttle method.Comment: 29 pages, 12 figure

A Weak Galerkin Finite Element Scheme for the Biharmonic Equations by Using Polynomials of Reduced Order

A new weak Galerkin (WG) finite element method for solving the biharmonic equation in two or three dimensional spaces by using polynomials of reduced order is introduced and analyzed. The WG method is on the use of weak functions and their weak derivatives defined as distributions. Weak functions and weak derivatives can be approximated by polynomials with various degrees. Different combination of polynomial spaces leads to different WG finite element methods, which makes WG methods highly flexible and efficient in practical computation. This paper explores the possibility of optimal combination of polynomial spaces that minimize the number of unknowns in the numerical scheme, yet without compromising the accuracy of the numerical approximation. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete $H^2$ norm and the standard $L^2$ norm. In addition, the paper also presents some numerical experiments to demonstrate the power of the WG method. The numerical results show a great promise of the robustness, reliability, flexibility and accuracy of the WG method.Comment: 28 pages. arXiv admin note: substantial text overlap with arXiv:1303.0927, arXiv:1309.5560, arXiv:1510.06001 by other author

Quantum Metrological Bounds for Vector Parameter

Precise measurement is crucial to science and technology. However, the rule of nature imposes various restrictions on the precision that can be achieved depending on specific methods of measurement. In particular, quantum mechanics poses the ultimate limit on precision which can only be approached but never be violated. Depending on analytic techniques, these bounds may not be unique. Here, in view of prior information, we investigate systematically the precision bounds of the total mean-square error of vector parameter estimation which contains $d$ independent parameters. From quantum Ziv-Zakai error bounds, we derive two kinds of quantum metrological bounds for vector parameter estimation, both of which should be satisfied. By these bounds, we show that a constant advantage can be expected via simultaneous estimation strategy over the optimal individual estimation strategy, which solves a long-standing problem. A general framework for obtaining the lower bounds in a noisy system is also proposed.Comment: 8 pages, 4 figure

Zeno dynamics in quantum open systems

Quantum Zeno effect shows that frequent observations can slow down or even stop the unitary time evolution of an unstable quantum system. This effect can also be regarded as a physical consequence of the the statistical indistinguishability of neighboring quantum states. The accessibility of quantum Zeno dynamics under unitary time evolution can be quantitatively estimated by quantum Zeno time in terms of Fisher information. In this work, we investigate the accessibility of quantum Zeno dynamics in quantum open systems by calculating noisy Fisher information, in which a trace preserving and completely positive map is assumed. We firstly study the consequences of non-Markovian noise on quantum Zeno effect and give the exact forms of the dissipative Fisher information and the quantum Zeno time. Then, for the operator-sum representation, an achievable upper bound of the quantum Zeno time is given with the help of the results in noisy quantum metrology. It is of significance that the noise affecting the accuracy in the entanglement-enhanced parameter estimation can conversely be favorable for the accessibility of quantum Zeno dynamics of entangled states.Comment: 6 pages, 2 figure

Acceleration of weak Galerkin methods for the Laplacian eigenvalue problem

Recently, we proposed a weak Galerkin finite element method for the Laplace eigenvalue problem. In this paper, we present two-grid and two-space skills to accelerate the weak Galerkin method. By choosing parameters properly, the two-grid and two-space weak Galerkin method not only doubles the convergence rate, but also maintains the asymptotic lower bounds property of the weak Galerkin method. Some numerical examples are provided to validate our theoretical analysis.Comment: 4 figure2, 20 page

The Shifted-inverse Power Weak Galerkin Method for Eigenvalue Problems

This paper proposes and analyzes a new weak Galerkin method for the eigenvalue problem by using the shifted-inverse power technique. A high order lower bound can be obtained at a relatively low cost via the proposed method. The error estimates for both eigenvalue and eigenfunction are provided and asymptotic lower bounds are shown as well under some conditions. Numerical examples are presented to validate the theoretical analysis.Comment: 19 pages, 3 table

Quantization on Generalized Heisenberg-Virasoro Algebra

In a recent paper by the authors, Lie bialgebra structures on generalized Heisenberg- Virasoro algebra L are considered. In this paper, the explicit formula of the quantization on generalized Heisenberg-Virasoro algebra is presented.Comment: 11 pages. arXiv admin note: text overlap with arXiv:1004.3646, arXiv:1004.3645 by other author

Lanthanum-Cerium Based Bulk Metallic Glasses with Superior Glass-Forming Ability

A quinary (La0.5Ce0.5)65Al10(Co0.6Cu0.4)25 alloy with superior glass-forming ability (GFA), identified by the formation of fully glassy rod of 32 mm in diameter by tilt-pour casting, was reported. By comparing with the GFA of quarternary (La0.5Ce0.5)65Al10TM25 and ternary Ln65Al10TM25 alloys (Ln = La or Ce; TM = Co or Cu), we suggest that the strong frustration of crystallization by utilizing the coexistence of La-Ce and Co-Cu to complicate competing crystalline phases is helpful to construct BMG component with superior GFA.Comment: 15 pages, 4 figures, 1 tabl