Edge-Fault Tolerance of Hypercube-like Networks

This paper considers a kind of generalized measure $\lambda_s^{(h)}$ of fault tolerance in a hypercube-like graph $G_n$ which contain several well-known interconnection networks such as hypercubes, varietal hypercubes, twisted cubes, crossed cubes and M\"obius cubes, and proves $\lambda_s^{(h)}(G_n)= 2^h(n-h)$ for any $h$ with $0\leqslant h\leqslant n-1$ by the induction on $n$ and a new technique. This result shows that at least $2^h(n-h)$ edges of $G_n$ have to be removed to get a disconnected graph that contains no vertices of degree less than $h$. Compared with previous results, this result enhances fault-tolerant ability of the above-mentioned networks theoretically

Gabor Frames for Quasicrystals, KK-theory, and Twisted Gap Labeling

We study the connection between Gabor frames for quasicrystals, the topology of the hull of a quasicrystal $\Lambda,$ and the $K$-theory of the twisted groupoid $C^*$-algebra $\mathcal{A}_\sigma$ arising from a quasicrystal. In particular, we construct a finitely generated projective module \mathcal{H}_\L over $\mathcal{A}_\sigma$ related to time-frequency analysis, and any multiwindow Gabor frame for $\Lambda$ can be used to construct an idempotent in $M_N(\mathcal{A}_\sigma)$ representing \mathcal{H}_\L in $K_0(\mathcal{A}_\sigma).$ We show for lattice subsets in dimension two, this element corresponds to the Bott element in $K_0(\mathcal{A}_\sigma),$ allowing us to prove a twisted version of Bellissard's gap labeling theorem

Partial hyperbolicity far from homoclinic bifurcations

We prove that any diffeomorphism of a compact manifold can be C^1-approximated by a diffeomorphism which exhibits a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by a diffeomorphism which is partially hyperbolic (its chain-recurrent set splits into partially hyperbolic pieces whose centre bundles have dimensions less or equal to two). We also study in a more systematic way the central models introduced in arXiv:math/0605387

Homology for higher-rank graphs and twisted C*-algebras

We introduce a homology theory for k-graphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homology of a k-graph coincides with the homology of its topological realisation as described by Kaliszewski et al. We exhibit combinatorial versions of a number of standard topological constructions, and show that they are compatible, from a homological point of view, with their topological counterparts. We show how to twist the C*-algebra of a k-graph by a T-valued 2-cocycle and demonstrate that examples include all noncommutative tori. In the appendices, we construct a cubical set \tilde{Q}(\Lambda) from a k-graph {\Lambda} and demonstrate that the homology and topological realisation of {\Lambda} coincide with those of \tilde{Q}(\Lambda) as defined by Grandis.Comment: 33 pages, 9 pictures and one diagram prepared in TiK

Temporal Evolution of the Magnetic Topology of the NOAA Active Region 11158

We studied the temporal evolution of the magnetic topology of the active region (AR) 11158 based on the reconstructed three-dimensional magnetic fields in the corona. The \nlfff\ extrapolation method was applied to the 12 minutes cadence data obtained with the \hmi\ (HMI) onboard the \sdo\ (SDO) during five days. By calculating the squashing degree factor Q in the volume, the derived quasi-separatrix layers (QSLs) show that this AR has an overall topology, resulting from a magnetic quadrupole, including an hyperbolic flux tube (HFT) configuration which is relatively stable at the time scale of the flare ($\sim 1-2$ hours). A strong QSL, which corresponds to some highly sheared arcades that might be related to the formation of a flux rope, is prominent just before the M6.6 and X2.2 flares, respectively. These facts indicate the close relationship between the strong QSL and the high flare productivity of AR 11158. In addition, with a close inspection of the topology, we found a small-scale HFT which has an inverse tear-drop structure above the aforementioned QSL before the X2.2 flare. It indicates the existence of magnetic flux rope at this place. Even though a global configuration (HFT) is recognized in this AR, it turns out that the large-scale HFT only plays a secondary role during the eruption. In final, we dismiss a trigger based on the breakout model and highlight the central role of the flux rope in the related eruption.Comment: Accepted by Ap

Singular Instantons Made Regular

The singularity present in cosmological instantons of the Hawking-Turok type is resolved by a conformal transformation, where the conformal factor has a linear zero of codimension one. We show that if the underlying regular manifold is taken to have the topology of $RP^4$, and the conformal factor is taken to be a twisted field so that the zero is enforced, then one obtains a one-parameter family of solutions of the classical field equations, where the minimal action solution has the conformal zero located on a minimal volume noncontractible $RP^3$ submanifold. For instantons with two singularities, the corresponding topology is that of a cylinder $S^3\times [0,1]$ with D=4 analogues of `cross-caps' at each of the endpoints.Comment: 23 pages, compressed and RevTex file, including nine postscript figure files. Submitted versio