Structural Parameters for 10 Halo Globular Clusters in M33

In this paper, we present the properties of 10 halo globular clusters with luminosities $L\simeq 5-7\times 10^5{L_\odot}$ in the Local Group galaxy M33 using the images of {\it Hubble Space Telescope} Wide Field Planetary Camera 2 in the F555W and F814W bands. We obtained ellipticities, position angles and surface brightness profiles for them. In general, the ellipticities of M33 sample clusters are similar to those of M31 clusters. The structural and dynamical parameters are derived by fitting the profiles to three different models combined with mass-to-light ratios ($M/L$ values) from population-synthesis models. The structural parameters include core radii, concentration, half-light radii {\bf and} central surface brightness. The dynamical parameters include the integrated cluster mass, integrated binding energy, central surface mass density {\bf and} predicted line-of-sight velocity dispersion at the cluster center. The velocity dispersions of four clusters predicted here agree well with the observed dispersions by Larsen et al. The results here showed that the majority of the sample halo globular clusters are well fitted by King model as well as by Wilson model, and better than by S\'ersic model. In general, the properties of clusters in M33, M31 and the Milky Way fall in the same regions of parameter spaces. The tight correlations of cluster properties indicate a "fundamental plane" for clusters, which reflects some universal physical conditions and processes operating at the epoch of cluster formation.Comment: Accepted for Publication in AJ, 27 pages, 23 figures and 6 table

New ubvriubvri photometry of 234 M33 star clusters

This is the second paper of our series. In this paper, we present $UBVRI$ photometry for 234 star clusters in the field of M33. For most of these star clusters, there is photometry in only two bands in previous studies. The photometry of these star clusters is performed using archival images from the Local Group Galaxies Survey, which covers 0.8 deg$^2$ along the major axis of M33. Detailed comparisons show that, in general, our photometry is consistent with previous measurements, especially, our photometry is in good agreement with Zloczewski & Kaluzny. Combined with the star clusters' photometry in previous studies, we present some results: none of the M33 youngest clusters ($\sim 10^7$ yr) have masses approaching $10^5$ $M_{\odot}$; comparisons with models of simple stellar populations suggest a large range of ages of M33 star clusters, and some as old as the Galactic globular clusters.Comment: Accepted for Publication in AJ, 23 pages, 9 figures and 3 tables. arXiv admin note: text overlap with arXiv:1205.482

Asymptotically exact a posteriori error estimates of eigenvalues by the Crouzeix-Raviart element and enriched Crouzeix-Raviart element

Two asymptotically exact a posteriori error estimates are proposed for eigenvalues by the nonconforming Crouzeix--Raviart and enriched Crouzeix-- Raviart elements. The main challenge in the design of such error estimators comes from the nonconformity of the finite element spaces used. Such nonconformity causes two difficulties, the first one is the construction of high accuracy gradient recovery algorithms, the second one is a computable high accuracy approximation of a consistency error term. The first difficulty was solved for both nonconforming elements in a previous paper. Two methods are proposed to solve the second difficulty in the present paper. In particular, this allows the use of high accuracy gradient recovery techniques. Further, a post-processing algorithm is designed by utilizing asymptotically exact a posteriori error estimators to construct the weights of a combination of two approximate eigenvalues. This algorithm requires to solve only one eigenvalue problem and admits high accuracy eigenvalue approximations both theoretically and numerically.Comment: arXiv admin note: text overlap with arXiv:1802.0189

Negative Magneto-Resistance Beyond Weak Localization in Three-Dimensional Billiards: Effect of Arnold Diffusion

We investigate a semiclassical conductance for ballistic open three-dimensional (3-d) billiards. For partially or completely broken-ergodic 3-d billiards such as SO(2) symmetric billiards, the dependence of the conductance on the Fermi wavenumber is dramatically changed by the lead orientation. Application of a symmetry-breaking weak magnetic field brings about mixed phase-space structures of 3-d billiards which ensures a novel Arnold diffusion that cannot be seen in 2-d billiards. In contrast to the 2-d case, the anomalous increment of the conductance should inevitably include a contribution arising from Arnold diffusion as well as a weak localization correction. Discussions are devoted to the physical condition for observing this phenomenon.Comment: 14 pages, 3 figure

Structural parameters for globular clusters in M31

In this paper, we present surface brightness profiles for 79 globular clusters in M31, using images observed with {\it Hubble Space Telescope}, some of which are from new observations. The structural and dynamical parameters are derived from fitting the profiles to several different models for the first time. The results show that in the majority of cases, King models fit the M31 clusters as well as Wilson models, and better than S\'{e}rsic models. However, there are 11 clusters best fitted by S\'{e}rsic models with the S\'{e}rsic index $n>2$, meaning that they have cuspy central density profiles. These clusters may be the well-known core-collapsed candidates. There is a bimodality in the size distribution of M31 clusters at large radii, which is different from their Galactic counterparts. In general, the properties of clusters in M31 and the Milky Way fall in the same regions of parameter spaces. The tight correlations of cluster properties indicate a "fundamental plane" for clusters, which reflects some universal physical conditions and processes operating at the epoch of cluster formation.Comment: Accepted for Publication in AJ, 17 pages, 15 figures and 7 table

Superconvergence of both the Crouzeix-Raviart and Morley elements

In this paper, a new method is proposed to prove the superconvergence of both the Crouzeix-Raviart and Morley elements. The main idea is to fully employ equivalences with the first order Raviart-Thomas element and the first order Hellan-Herrmann-Johnson element, respectively. In this way, some special conformity of discrete stresses is explored and superconvergence of mixed elements can be used to analyze superconvergence of nonconforming elements. Finally, a half order superconvergence by postprocessing is proved for both nonconforming elements.Comment: 16 pages, 6 figure

Coupling motion of colloidal particles in quasi-two-dimensional confinement

Brownian motion of colloidal particles in the quasi-two-dimensional (qTD) confinement displays distinct kinetic characters from that in bulk. Here we experimentally report a dynamic evolution of Brownian particles in the qTD system. The dynamic system displays a quasi-equilibrium state of colloidal particles performing Brownian motion. In the quasi-equilibrium process, the qTD confinement results in the coupling of particle motions, which slowly dampens the motion and interaction of particles until the final equilibrium state reaches. The theory is developed to explain coupling motions of Brownian particles in the qTD confinement.Comment: 7 pages, 4 figure

High accuracy methods for eigenvalues of elliptic operators by nonconforming elements

In this paper, three high-accuracy methods for eigenvalues of second order elliptic operators are proposed by using the nonconforming Crouzeix-Raviart(CR for short) element and the nonconforming enriched Crouzeix-Raviart(ECR for short) element. They are based on a crucial full one order superconvergence of the first order mixed Raviart-Thomas(RT for short) element. The main ingredient of such a superconvergence analysis is to employ a discrete Helmholtz decomposition of the difference between the canonical interpolation and the finite element solution of the RT element. In particular, it allows for some vital cancellation between terms in one key sum of boundary terms. Consequently, a full one order superconvergence follows from a special relation between the CR element and the RT element, and the equivalence between the ECR element and the RT element for these two nonconforming elements. These superconvergence results improve those in literature from a half order to a full one order for the RT element, the CR element and the ECR element. Based on the aforementioned superconvergence of the RT element, asymptotic expansions of eigenvalues are established and employed to achieve high accuracy extrapolation methods for these two nonconforming elements. In contrast to a classic analysis in literature, the novelty herein is to use not only the canonical interpolations of these nonconforming elements but also that of the RT element to analyze such asymptotic expansions. Based on the superconvergence of these nonconforming elements, asymptotically exact a posteriori error estimators of eigenvalues are constructed and analyzed for them. Finally, two post-processing methods are proposed to improve accuracy of approximate eigenvalues by employing these a posteriori error estimators.Numerical tests are provided to justify and compare the performance of the aforementioned methods

An Equivalence of Fully Connected Layer and Convolutional Layer

This article demonstrates that convolutional operation can be converted to matrix multiplication, which has the same calculation way with fully connected layer. The article is helpful for the beginners of the neural network to understand how fully connected layer and the convolutional layer work in the backend. To be concise and to make the article more readable, we only consider the linear case. It can be extended to the non-linear case easily through plugging in a non-linear encapsulation to the values like this $\sigma(x)$ denoted as $x^{\prime}$.Comment: 9 page

Conforming mixed triangular prism and nonconforming mixed tetrahedral elements for the linear elasticity problem

We propose two families of mixed finite elements for solving the classical Hellinger-Reissner mixed problem of the linear elasticity equations in three dimensions. First, a family of conforming mixed triangular prism elements is constructed by product of elements on triangular meshes and elements in one dimension. The well-posedness is established for all elements with $k\geq1$, which are of $k+1$ order convergence for both the stress and displacement. Besides, a family of reduced stress spaces is proposed by dropping the degrees of polynomial functions associated with faces. As a result, the lowest order conforming mixed triangular prism element has 93 plus 33 degrees of freedom on each element. Second, we construct a new family of nonconforming mixed tetrahedral elements. The shape function spaces of our stress spaces are different from those of the elements in literature