A note on communicating between information systems based on including degrees

In order to study the communication between information systems, Gong and Xiao [Z. Gong and Z. Xiao, Communicating between information systems based on including degrees, International Journal of General Systems 39 (2010) 189--206] proposed the concept of general relation mappings based on including degrees. Some properties and the extension for fuzzy information systems of the general relation mappings have been investigated there. In this paper, we point out by counterexamples that several assertions (Lemma 3.1, Lemma 3.2, Theorem 4.1, and Theorem 4.3) in the aforementioned work are not true in general.Comment: 4 page

Cluster automorphism groups of cluster algebras with coefficients

We study the cluster automorphism group of a skew-symmetric cluster algebra with geometric coefficients. For this, we introduce the notion of gluing free cluster algebra, and show that under a weak condition the cluster automorphism group of a gluing free cluster algebra is a subgroup of the cluster automorphism group of its principal part cluster algebra (i.e. the corresponding cluster algebra without coefficients). We show that several classes of cluster algebras with coefficients are gluing free, for example, cluster algebras with principal coefficients, cluster algebras with universal geometric coefficients, and cluster algebras from surfaces (except a 4-gon) with coefficients from boundaries. Moreover, except four kinds of surfaces, the cluster automorphism group of a cluster algebra from a surface with coefficients from boundaries is isomorphic to the cluster automorphism group of its principal part cluster algebra; for a cluster algebra with principal coefficients, its cluster automorphism group is isomorphic to the automorphism group of its initial quiver.Comment: 23 pages,3 figure

Global Classical Large Solutions to Navier-Stokes Equations for Viscous Compressible and Heat Conducting Fluids with Vacuum

In this paper, we consider the 1D Navier-Stokes equations for viscous compressible and heat conducting fluids (i.e., the full Navier-Stokes equations). We get a unique global classical solution to the equations with large initial data and vacuum. Because of the strong nonlinearity and degeneration of the equations brought by the temperature equation and by vanishing of density (i.e., appearance of vacuum) respectively, to our best knowledge, there are only two results until now about global existence of solutions to the full Navier-Stokes equations with special pressure, viscosity and heat conductivity when vacuum appears (see \cite{Feireisl-book} where the viscosity $\mu=$const and the so-called {\em variational} solutions were obtained, and see \cite{Bresch-Desjardins} where the viscosity $\mu=\mu(\rho)$ degenerated when the density vanishes and the global weak solutions were got). It is open whether the global strong or classical solutions exist. By applying our ideas which were used in our former paper \cite{Ding-Wen-Zhu} to get $H^3-$estimates of $u$ and $\theta$ (see Lemma \ref{non-le:3.14}, Lemma \ref{non-le:3.15}, Lemma \ref{non-rle:3.12} and the corresponding corollaries), we get the existence and uniqueness of the global classical solutions (see Theorem \ref{non-rth:1.1})

Some improved results on communication between information systems

To study the communication between information systems, Wang et al. [C. Wang, C. Wu, D. Chen, Q. Hu, and C. Wu, Communicating between information systems, Information Sciences 178 (2008) 3228-3239] proposed two concepts of type-1 and type-2 consistent functions. Some properties of such functions and induced relation mappings have been investigated there. In this paper, we provide an improvement of the aforementioned work by disclosing the symmetric relationship between type-1 and type-2 consistent functions. We present more properties of consistent functions and induced relation mappings and improve upon several deficient assertions in the original work. In particular, we unify and extend type-1 and type-2 consistent functions into the so-called neighborhood-consistent functions. This provides a convenient means for studying the communication between information systems based on various neighborhoods.Comment: 12 page

Global symmetric classical and strong solutions of the full compressible Navier-Stokes equations with vacuum and large initial data

First of all, we get the global existence of classical and strong solutions of the full compressible Navier-Stokes equations in three space dimensions with initial data which is large and spherically or cylindrically symmetric. The appearance of vacuum is allowed. In particular, if the initial data is spherically symmetric, the space dimension can be taken not less than two. The analysis is based on some delicate {\it a priori} estimates globally in time which depend on the assumption $\kappa=O(1+\theta^q)$ where $q>r$ ($r$ can be zero), which relaxes the condition $q\ge2+2r$ in [14,29,42]. This could be viewed as an extensive work of [18] where the equations hold in the sense of distributions in the set where the density is positive with initial data which is large, discontinuous, and spherically or cylindrically symmetric in three space dimension. Finally, with the assumptions that vacuum may appear and that the solutions are not necessarily symmetric, we establish a blow-up criterion in terms of $\|\rho\|_{L^\infty_tL_x^\infty}$ and $\|\rho\theta\|_{L^4_tL^(12/5)_x}$ for strong solutions.Comment: arXiv admin note: text overlap with arXiv:1103.142

Cluster automorphism groups of cluster algebras of finite type

We study the cluster automorphism group $Aut(\mathcal{A})$ of a coefficient free cluster algebra $\mathcal{A}$ of finite type. A cluster automorphism of $\mathcal{A}$ is a permutation of the cluster variable set $\mathscr{X}$ that is compatible with cluster mutations. We show that, on the one hand, by the well-known correspondence between $\mathscr{X}$ and the almost positive root system $\Phi_{\geq -1}$ of the corresponding Dynkin type, the piecewise-linear transformations $\tau_+$ and $\tau_-$ on $\Phi_{\geq -1}$ induce cluster automorphisms $f_+$ and $f_-$ of $\mathcal{A}$ respectively; on the other hand, excepting type $D_{2n},(n\geqslant 2)$, all the cluster automorphisms of $\mathcal{A}$ are compositions of $f_+$ and $f_-$. For a cluster algebra of type $D_{2n},(n\geqslant 2)$, there exists exceptional cluster automorphism induced by a permutation of negative simple roots in $\Phi_{\geq -1}$, which is not a composition of $\tau_+$ and $\tau_-$. By using these results and folding a simply laced cluster algebra, we compute the cluster automorphism group for a non-simply laced finite type cluster algebra. As an application, we show that $Aut(\mathcal{A})$ is isomorphic to the cluster automorphism group of the $FZ$-universal cluster algebra of $\mathcal{A}$.Comment: 24 pages, 2 figures in Journal of Algebra 201

Homomorphisms between fuzzy information systems revisited

Recently, Wang et al. discussed the properties of fuzzy information systems under homomorphisms in the paper [C. Wang, D. Chen, L. Zhu, Homomorphisms between fuzzy information systems, Applied Mathematics Letters 22 (2009) 1045-1050], where homomorphisms are based upon the concepts of consistent functions and fuzzy relation mappings. In this paper, we classify consistent functions as predecessor-consistent and successor-consistent, and then proceed to present more properties of consistent functions. In addition, we improve some characterizations of fuzzy relation mappings provided by Wang et al.Comment: 10 page

Cluster automorphism groups and automorphism groups of exchange graphs

For a coefficient free cluster algebra $\mathcal{A}$, we study the cluster automorphism group $Aut(\mathcal{A})$ and the automorphism group $Aut(E_{\mathcal{A}})$ of its exchange graph $E_{\mathcal{A}}$. We show that these two groups are isomorphic with each other, if $\mathcal{A}$ is of finite type excepting types of rank two and type $F_4$, or if $\mathcal{A}$ is of skew-symmetric finite mutation type.Comment: 18 pages. to appear at Pacific Journal of Mathematic

Comment on "A Random Quantum Key Distribution Achieved by Using Bell States"

In a recent paper [J. Opt. B: Quantum Semiclass. Opt. 5 (2003) 155-157], a quantum key distribution scheme based on entanglement swapping was proposed, which exhibited two improvements over the previous protocols. In this Comment, it is shown that the scheme has no properties as been discussed.Comment: 1 page, no figure

Stable Recovery of Sparse Signals via lp−l_p-Minimization

In this paper, we show that, under the assumption that \|\e\|_2\leq \epsilon, every $k-$sparse signal \x\in \mathbb{R}^n can be stably ($\epsilon\neq0$) or exactly recovered ($\epsilon=0$) from \y=\A\x+\e via $l_p-$mnimization with $p\in(0, \bar{p}]$, where \beqnn \bar{p}= \begin{cases} \frac{50}{31}(1-\delta_{2k}), &\delta_{2k}\in[\frac{\sqrt{2}}{2}, 0.7183)\cr 0.4541, &\delta_{2k}\in[0.7183,0.7729)\cr 2(1-\delta_{2k}), &\delta_{2k}\in[0.7729,1) \end{cases}, \eeqnn even if the restricted isometry constant of \A satisfies $\delta_{2k}\in[\frac{\sqrt{2}}{2}, 1)$. Furthermore, under the assumption that $n\leq 4k$, we show that the range of $p$ can be further improved to $p\in(0,\frac{3+2\sqrt{2}}{2}(1-\delta_{2k})]$. This not only extends some discussions of only the noiseless recovery (Lai et al. and Wu et al.) to the noise recovery, but also greatly improves the best existing results where $p\in(0,\min\{1, 1.0873(1-\delta_{2k}) \})$ (Wu et al.)