A linear finite element procedure for the Naghdi shell model

We prove the accuracy of a mixed finite element method for bending dominated shells in which a major part of the membrane/shear strain is reduced, to free up membrane/shear locking. When no part of the membrane/shear strain is reduced, the method becomes a consistent discontinuous Galerkin method that is proven accurate for membrane/shear dominated shells and intermediate shells. The two methods can be coded in a single program by using a parameter. We propose a procedure of numerically detecting the asymptotic behavior of a shell, choosing the parameter value in the method, and producing accurate approximation for a given shell problem. The method uses piecewise linear functions to approximate all the variables. The analysis is carried out for shells whose middle surfaces have the most general geometries, which shows that the method has the optimal order of accuracy for general shells and the accuracy is robust with respect to the shell thickness. In the particular case that the geometrical coefficients of the shell middle surface are piecewise constants the accuracy is uniform with respect to the shell thickness

A discontinuous Galerkin method for the Naghdi shell model

We propose a mixed discontinuous Galerkin method for the bending problem of Naghdi shell, and present an analysis for its accuracy. The error estimate shows that when components of the curvature tensor and Christoffel symbols are piecewise linear functions, the finite element method has the optimal order of accuracy, which is uniform with respect to the shell thickness. Generally, the error estimate shows how the accuracy is affected by the shell geometry and thickness. It suggests that to achieve optimal rate of convergence, the triangulation should be properly refined in regions where the shell geometry changes dramatically. These are the results for a balanced method in which the primary displacement components and rotation components are approximated by discontinuous piecewise quadratic polynomials, while components of the scaled membrane stress tensor and shear stress vector are approximated by continuous piecewise linear functions. On elements that have edges on the free boundary of the shell, finite element space for displacement components needs to be enriched slightly, for stability purpose. Results on higher order finite elements are also included.Comment: arXiv admin note: substantial text overlap with arXiv:1403.705

Coarse Quotient Mappings between Metric Spaces

We give a definition of coarse quotient mapping and show that several results for uniform quotient mapping also hold in the coarse setting. In particular, we prove that any Banach space that is a coarse quotient of $L_p\equiv L_p[0,1]$, $1<p<\infty$, is isomorphic to a linear quotient of $L_p$. It is also proved that $\ell_q$ is not a coarse quotient of $\ell_p$ for $1<p<q<\infty$ using Rolewicz's property ($\beta$)

Asymptotic properties of Banach spaces and coarse quotient maps

We give a quantitative result about asymptotic moduli of Banach spaces under coarse quotient maps. More precisely, we prove that if a Banach space $Y$ is a coarse quotient of a subset of a Banach space $X$, where the coarse quotient map is coarse Lipschitz, then the ($\beta$)-modulus of $X$ is bounded by the modulus of asymptotic uniform smoothness of $Y$ up to some constants. In particular, if the coarse quotient map is a coarse homeomorphism, then the modulus of asymptotic uniform convexity of $X$ is bounded by the modulus of asymptotic uniform smoothness of $Y$ up to some constants

Analysis of a discontinuous Galerkin method for Koiter shell

We present an analysis for a mixed finite element method for the bending problem of Koiter shell. We derive an error estimate showing that when the geometrical coefficients of the shell mid-surface satisfy certain conditions the finite element method has the optimal order of accuracy, which is uniform with respect to the shell thickness. Generally, the error estimate shows how the accuracy is affected by the shell geometry and thickness. It suggests that to achieve optimal rate of convergence, the triangulation should be properly refined in regions where the shell geometry changes dramatically. The analysis is carried out for a balanced method in which the normal component of displacement is approximated by discontinuous piecewise cubic polynomials, while the tangential components are approximated by discontinuous piecewise quadratic polynomials, with some enrichment on elements that have edges on the free boundary. Components of the membrane stress are approximated by continuous piecewise linear functions

Quantum coin flipping secure against channel noises

So far, most of existed single-shot quantum coin flipping(QCF) protocols failed in a noisy quantum channel. Here, we present a nested-structured framework that makes it possible to achieve partially noise-tolerant QCF, due to that there is a trade-off between the security and the justice correctness. It is showed that noise-tolerant single-shot QCF protocols can be produced by filling the presented framework up with existed or even future protocols. We also proved a lower bound of 0.25, with which a cheating Alice or Bob could bias the outcome.Comment: 6 pages, 2 figure

Novel ansatz for charge radii in density functional theories

Charge radii are one of the most fundamental properties of atomic nuclei characterizing their charge distributions. Though the general trend as a function of the mass number is well described by the $A^{1/3}$ rule, some fine structures, such as the evolution along the calcium isotopic chain and the corresponding odd-even staggerings, are notoriously difficult to describe both in density functional theories and ab initio methods. In this letter, we propose a novel ansatz to describe the charge radii of calcium isotopes, by adding a correction term, proportional to the number of Cooper pairs, and determined by the BCS amplitudes and a single parameter, to the charge radii calculated in the relativistic mean field model with the pairing interaction treated with the BCS method. The new ansatz yields results consistent with data not only for calcium isotopes, but also for ten other isotopic chains, including oxygen, neon, magnesium, chromium, nickel, germanium, zirconium, cadmium, tin, and lead. It is remarkable that this ansatz with a single parameter can describe nuclear charge radii throughout the periodic table, particularly the odd-even staggerings and parabolic behavior. We hope that the present study can stimulate more discussions about its nature and relation with other effects proposed to explain the odd-even staggerings of charge radii.Comment: 6 pages, 5 figure

Random Euler Complex-Valued Nonlinear Filters

Over the last decade, both the neural network and kernel adaptive filter have successfully been used for nonlinear signal processing. However, they suffer from high computational cost caused by their complex/growing network structures. In this paper, we propose two random Euler filters for complex-valued nonlinear filtering problem, i.e., linear random Euler complex-valued filter (LRECF) and its widely-linear version (WLRECF), which possess a simple and fixed network structure. The transient and steady-state performances are studied in a non-stationary environment. The analytical minimum mean square error (MSE) and optimum step-size are derived. Finally, numerical simulations on complex-valued nonlinear system identification and nonlinear channel equalization are presented to show the effectiveness of the proposed methods

Quench Dynamics in a Trapped Bose-Einstein Condensate with Spin-Orbit Coupling

We consider the phase transition dynamics of a trapped Bose-Einstein condensate subject to Raman-type spin-orbit coupling (SOC). By tuning the coupling strength the condensate is taken through a second order phase transition into an immiscible phase. We observe the domain wall defects produced by a finite speed quench is described by the Kibble-Zurek mechanism (KZM), and quantify a power law behavior for the scaling of domain number and formation time with the quench speed.Comment: 6 pages, 4 figure

A Survey of directed graphs invariants

In this paper, various kinds of invariants of directed graphs are summarized. In the first topic, the invariant w(G) for a directed graph G is introduced, which is primarily defined by S. Chen and X.M. Chen to solve a problem of weak connectedness of tensor product of two directed graphs. Further, we present our recent studies on the invariant w(G) in categorical view. In the second topic, Homology theory on directed graph is introduced, and we also cast on categorical view of the definition. The third topic mainly focuses on Laplacians on graphs, including traditional work and latest result of 1-laplacian by K.C.Chang. Finally, Zeta functions and Graded graphs are introduced, inclduing Bratteli-Vershik diagram, dual graded graphs and differential posets, with some applications in dynamic system.Comment: This survey is written by Y.L.Zhang, supervised by Pro. S.Chen, containing 20 pages and 5 figure