Quasi-4-Connected Components

We introduce a new decomposition of a graphs into quasi-4-connected components, where we call a graph quasi-4-connected if it is 3-connected and it only has separations of order 3 that separate a single vertex from the rest of the graph. Moreover, we give a cubic time algorithm computing the decomposition of a given graph. Our decomposition into quasi-4-connected components refines the well-known decompositions of graphs into biconnected and triconnected components. We relate our decomposition to Robertson and Seymour\u27s theory of tangles by establishing a correspondence between the quasi-4-connected components of a graph and its tangles of order 4

Excluding a Weakly 4-connected Minor

A 3-connected graph $G$ is called weakly 4-connected if min $(|E(G_1)|, |E(G_2)|) \leq 4$ holds for all 3-separations $(G_1,G_2)$ of $G$. A 3-connected graph $G$ is called quasi 4-connected if min $(|V(G_1)|, |V(G_2)|) \leq 4$. We first discuss how to decompose a 3-connected graph into quasi 4-connected components. We will establish a chain theorem which will allow us to easily generate the set of all quasi 4-connected graphs. Finally, we will apply these results to characterizing all graphs which do not contain the Pyramid as a minor, where the Pyramid is the weakly 4-connected graph obtained by performing a $\Delta Y$ transformation to the octahedron. This result can be used to show an interesting characterization of quasi 4-connected, outer-projective graphs

Moduli Spaces of Higher Spin Klein Surfaces

We study the connected components of the space of higher spin bundles on hyperbolic Klein surfaces. A Klein surface is a generalisation of a Riemann surface to the case of non-orientable surfaces or surfaces with boundary. The category of Klein surfaces is isomorphic to the category of real algebraic curves. An m-spin bundle on a Klein surface is a complex line bundle whose m-th tensor power is the cotangent bundle. The spaces of higher spin bundles on Klein surfaces are important because of their applications in singularity theory and real algebraic geometry, in particular for the study of real forms of Gorenstein quasi-homogeneous surface singularities. In this paper we describe all connected components of the space of higher spin bundles on hyperbolic Klein surfaces in terms of their topological invariants and prove that any connected component is homeomorphic to a quotient of a Euclidean space by a discrete group. We also discuss applications to real forms of Brieskorn-Pham singularities.Comment: v3: 21 pages, shortened sec. 2 (summary of previous results), added an example in sec. 4, added sec. 5 (applications in sing. theory), added references; v2: 25 pages, minor corrections, added reference

From Anderson to anomalous localization in cold atomic gases with effective spin-orbit coupling

We study the dynamics of a one-dimensional spin-orbit coupled Schrodinger particle with two internal components moving in a random potential. We show that this model can be implemented by the interaction of cold atoms with external lasers and additional Zeeman and Stark shifts. By direct numerical simulations a crossover from an exponential Anderson-type localization to an anomalous power-law behavior of the intensity correlation is found when the spin-orbit coupling becomes large. The power-law behavior is connected to a Dyson singularity in the density of states emerging at zero energy when the system approaches the quasi-relativistic limit of the random mass Dirac model. We discuss conditions under which the crossover is observable in an experiment with ultracold atoms and construct explicitly the zero-energy state, thus proving its existence under proper conditions.Comment: 4 pages and 4 figure

Crystallizing the hypoplactic monoid: from quasi-Kashiwara operators to the Robinson--Schensted--Knuth-type correspondence for quasi-ribbon tableaux

Crystal graphs, in the sense of Kashiwara, carry a natural monoid structure given by identifying words labelling vertices that appear in the same position of isomorphic components of the crystal. In the particular case of the crystal graph for the $q$-analogue of the special linear Lie algebra $\mathfrak{sl}_{n}$, this monoid is the celebrated plactic monoid, whose elements can be identified with Young tableaux. The crystal graph and the so-called Kashiwara operators interact beautifully with the combinatorics of Young tableaux and with the Robinson--Schensted--Knuth correspondence and so provide powerful combinatorial tools to work with them. This paper constructs an analogous `quasi-crystal' structure for the hypoplactic monoid, whose elements can be identified with quasi-ribbon tableaux and whose connection with the theory of quasi-symmetric functions echoes the connection of the plactic monoid with the theory of symmetric functions. This quasi-crystal structure and the associated quasi-Kashiwara operators are shown to interact just as neatly with the combinatorics of quasi-ribbon tableaux and with the hypoplactic version of the Robinson--Schensted--Knuth correspondence. A study is then made of the interaction of the crystal graph of the plactic monoid and the quasi-crystal graph for the hypoplactic monoid. Finally, the quasi-crystal structure is applied to prove some new results about the hypoplactic monoid.Comment: 55 pages. Minor revision to fix typos, add references, and discuss an open questio

Real valued functions and metric spaces quasi-isometric to trees

We prove that if X is a complete geodesic metric space with uniformly generated first homology group and $f: X\to R$ is metrically proper on the connected components and bornologous, then X is quasi-isometric to a tree. Using this and adapting the definition of hyperbolic approximation we obtain an intrinsic sufficent condition for a metric space to be PQ-symmetric to an ultrametric space.Comment: 12 page