Elementary Maps on Triangular Algebras

In this note we prove that elementary maps on triangular algebras are automically additive.Comment: 8 page

On Jordan Derivations of Triangular Algebras

In this short note we prove that every Jordan derivation of triangular algebras is a derivation.Comment: 5 page

On Pointed Hopf Algebras with Sporadic Simple Groups HS{\rm HS} and Co3{\rm Co3}

Every non quasi- -1-type Nichols algebra is infinite dimensional. All quasi- -1-type Nichols algebra over sporadic simple groups ${\rm HS}$ and ${\rm Co3}$ are found.Comment: 9page

A Novel Recursive Construction for Coded Caching Schemes

As a strategy to further reduce the transmission pressure during the peak traffic times in wireless network, coded caching has been widely studied recently. And several coded caching schemes are constructed focusing on the two core problems in practice, i.e., the rate transmitted during the peak traffic times and the packet number of each file divided during the off peak traffic times. It is well known that there exits a tradeoff between the rate and the packet number. In this paper, a novel recursive construction is proposed. As an application, several new schemes are obtained. Comparing with previously known schemes, new schemes could further reduce packet number by increasing little rate. And for some parameters in coded caching systems, the packet number of our new schemes are smaller than that of schemes generated by memory sharing method which is widely used in the field of caching. By the way our new schemes include all the results constructed by Tang et al., (IEEE ISIT, 2790-2794, 2017) as special cases.Comment: 10 page

On the classification of fractal squares

In \cite{LaLuRa13}, the authors completely classified the topological structure of so called {\it fractal square} $F$ defined by $F=(F+{\mathcal D})/n$, where ${\mathcal{D}}\subsetneq\{0,1,\dots,n-1\}^2, n\ge 2$. In this paper, we further provide simple criteria for the $F$ to be totally disconnected, then we discuss the Lipschitz classification of $F$ in the case $n=3$, which is an attempt to consider non-totally disconnected sets.Comment: 16 pages, 12 figure

Multipartite entanglement, quantum coherence, and quantum criticality in triangular and Sierpi\'nski fractal lattices

We investigate the quantum phase transitions of the transverse-field quantum Ising model on the triangular lattice and Sierpi\'nski fractal lattices by employing multipartite entanglement and quantum coherence along with the quantum renormalization group method. It is shown that the quantum criticalities of these high-dimensional models closely relate to the behaviors of the multipartite entanglement and quantum coherence. As the thermodynamic limit is approached, the first derivatives of multipartite entanglement and quantum coherence exhibit singular behaviors and the consistent finite-size scaling behaviors for each lattice are also obtained from the first derivatives. The multipartite entanglement and quantum coherence are demonstrated to be good indicators for detecting the quantum phase transitions in the triangular lattice and Sierpi\'nski fractal lattices. Furthermore, the factors that determine the relations between the critical exponents and the correlation length exponents for these models are diverse. For the triangular lattice, the decisive factor is the spatial dimension, while for the Sierpi\'nski fractal lattices, it is the Hausdorff dimension.Comment: 12 pages; 12 figure

Strongly Separable Codes

Binary $t$-frameproof codes ($t$-FPCs) are used in multimedia fingerprinting schemes where the identification of authorized users taking part in the averaging collusion attack is required. In this paper, a binary strongly $\bar{t}$-separable code ($\bar{t}$-SSC) is introduced to improve such a scheme based on a binary $t$-FPC. A binary $\bar{t}$-SSC has the same traceability as a binary $t$-FPC but has more codewords than a binary $t$-FPC. A composition construction for binary $\bar{t}$-SSCs from $q$-ary $\bar{t}$-SSCs is described, which stimulates the research on $q$-ary $\bar{t}$-SSCs with short length. Several infinite series of optimal $q$-ary $\bar{2}$-SSCs of length $2$ are derived from the fact that a $q$-ary $\bar{2}$-SSC of length $2$ is equivalent to a $q$-ary $\bar{2}$-separable code of length $2$. Combinatorial properties of $q$-ary $\bar{2}$-SSCs of length $3$ are investigated, and a construction for $q$-ary $\bar{2}$-SSCs of length $3$ is provided. These $\bar{2}$-SSCs of length $3$ have more than $12.5\%$ codewords than $2$-FPCs of length $3$ could have.Comment: 11 pages, submitted to Designs, Codes and Cryptography. arXiv admin note: text overlap with arXiv:1411.684

The distribution of faint satellites around central galaxies in the CFHT Legacy Survey

We investigate the radial number density profile and the abundance distribution of faint satellites around central galaxies in the low redshift universe using the CFHT Legacy Survey. We consider three samples of central galaxies with magnitudes of M_r=-21, -22, and -23 selected from the Sloan Digital Sky Survey (SDSS) group catalog of Yang et al.. The satellite distribution around these central galaxies is obtained by cross-correlating these galaxies with the photometric catalogue of the CFHT Legacy Survey. The projected radial number density of the satellites obeys a power law form with the best-fit logarithmic slope of -1.05, independent of both the central galaxy luminosity and the satellite luminosity. The projected cross correlation function between central and satellite galaxies exhibits a non-monotonic trend with satellite luminosity. It is most pronounced for central galaxies with M_r=-21, where the decreasing trend of clustering amplitude with satellite luminosity is reversed when satellites are fainter than central galaxies by more than 2 magnitudes. A comparison with the satellite luminosity functions in the Milky Way and M31 shows that the Milky Way/M31 system has about twice as many satellites as around a typical central galaxy of similar luminosity. The implications for theoretical models are briefly discussed.Comment: 14 pages, 6 figures, accepted for publication in Ap

Coherent bimolecular reactions with quantum-degenerate matter waves

We demonstrate theoretically that the abstraction reaction $A+B_2 \to AB+B$ can be driven coherently and efficiently with quantum-degenerate bosonic or fermionic matter waves. We show that the initial stages of the reaction are dominated by quantum fluctuations, resulting in the appearance of macroscopic non-classical correlations in the final atomic and molecular fields. The dynamics associated with the creation of bosonic and of fermionic dimer-atom pairs are also compared. This study opens up a promising new regime of quantum degenerate matter-wave chemistry.Comment: 9 pages, 5 figues, 1 tabl

The character tables of centralizers in Sporadic Simple Groups of HS{\rm HS} and Co3{\rm Co_3}

To classify the finite dimensional pointed Hopf algebras with $G= {\rm HS}$ or ${\rm Co3}$ we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software {\rm GAP}.Comment: 409 page