Re-embedding a 1-Plane Graph into a Straight-line Drawing in Linear Time

Thomassen characterized some 1-plane embedding as the forbidden configuration such that a given 1-plane embedding of a graph is drawable in straight-lines if and only if it does not contain the configuration [C. Thomassen, Rectilinear drawings of graphs, J. Graph Theory, 10(3), 335-341, 1988]. In this paper, we characterize some 1-plane embedding as the forbidden configuration such that a given 1-plane embedding of a graph can be re-embedded into a straight-line drawable 1-plane embedding of the same graph if and only if it does not contain the configuration. Re-embedding of a 1-plane embedding preserves the same set of pairs of crossing edges. We give a linear-time algorithm for finding a straight-line drawable 1-plane re-embedding or the forbidden configuration.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016). This is an extended abstract. For a full version of this paper, see Hong S-H, Nagamochi H.: Re-embedding a 1-Plane Graph into a Straight-line Drawing in Linear Time, Technical Report TR 2016-002, Department of Applied Mathematics and Physics, Kyoto University (2016

Tame concealed algebras and cluster quivers of minimal infinite type

In this paper we explain how and why the list of Happel-Vossieck of tame concealed algebras is closely related to the list of A. Seven of minimal infinite cluster quivers.Comment: 16 pages, new version with an additional section on cluster-tilted algebras of minimal infinite typ

Alterations in composition and diversity of the intestinal microbiota in patients with diarrhea-predominant irritable bowel syndrome: Alterations in composition and diversity of the intestinal microbiota in D-IBS

The intestinal microbiota has been implicated in the pathophysiology of Irritable Bowel Syndrome (IBS). Due to the variable resolutions of techniques used to characterize the intestinal microbiota, and the heterogeneity of IBS, the defined alterations of the IBS intestinal microbiota are inconsistent. We analyzed the composition of the intestinal microbiota in a defined subgroup of IBS patients (diarrhea-predominant IBS, D-IBS) using a technique that provides the deepest characterization available for complex microbial communities

Semi-invariants of symmetric quivers of tame type

A symmetric quiver $(Q,\sigma)$ is a finite quiver without oriented cycles $Q=(Q_0,Q_1)$ equipped with a contravariant involution $\sigma$ on $Q_0\sqcup Q_1$. The involution allows us to define a nondegenerate bilinear form on a representation $V$ of $Q$. We shall say that $V$ is orthogonal if is symmetric and symplectic if $$ is skew-symmetric. Moreover, we define an action of products of classical groups on the space of orthogonal representations and on the space of symplectic representations. So we prove that if $(Q,\sigma)$ is a symmetric quiver of tame type then the rings of semi-invariants for this action are spanned by the semi-invariants of determinantal type $c^V$ and, when matrix defining $c^V$ is skew-symmetric, by the Pfaffians $pf^V$. To prove it, moreover, we describe the symplectic and orthogonal generic decomposition of a symmetric dimension vector

Scattering of massless particles in one-dimensional chiral channel

We present a general formalism describing a propagation of an arbitrary multiparticle wave packet in a one-dimensional multimode chiral channel coupled to an ensemble of emitters which are distributed at arbitrary positions. The formalism is based on a direct and exact resummation of diagrammatic series for the multiparticle scattering matrix. It is complimentary to the Bethe Ansatz and to approaches based on equations of motion, and it reveals a simple and transparent structure of scattering states. In particular, we demonstrate how this formalism works on various examples, including scattering of one- and two-photon states off two- and three-level emitters, off an array of emitters as well as scattering of coherent light. We argue that this formalism can be constructively used for study of scattering of an arbitrary initial photonic state off emitters with arbitrary degree of complexity.Comment: 25 pages, 5 figure

Applications of BGP-reflection functors: isomorphisms of cluster algebras

Given a symmetrizable generalized Cartan matrix $A$, for any index $k$, one can define an automorphism associated with $A,$ of the field $\mathbf{Q}(u_1, >..., u_n)$ of rational functions of $n$ independent indeterminates $u_1,..., u_n.$ It is an isomorphism between two cluster algebras associated to the matrix $A$ (see section 4 for precise meaning). When $A$ is of finite type, these isomorphisms behave nicely, they are compatible with the BGP-reflection functors of cluster categories defined in [Z1, Z2] if we identify the indecomposable objects in the categories with cluster variables of the corresponding cluster algebras, and they are also compatible with the "truncated simple reflections" defined in [FZ2, FZ3]. Using the construction of preprojective or preinjective modules of hereditary algebras by Dlab-Ringel [DR] and the Coxeter automorphisms (i.e., a product of these isomorphisms), we construct infinitely many cluster variables for cluster algebras of infinite type and all cluster variables for finite types.Comment: revised versio

Contact Representations of Graphs in 3D

We study contact representations of graphs in which vertices are represented by axis-aligned polyhedra in 3D and edges are realized by non-zero area common boundaries between corresponding polyhedra. We show that for every 3-connected planar graph, there exists a simultaneous representation of the graph and its dual with 3D boxes. We give a linear-time algorithm for constructing such a representation. This result extends the existing primal-dual contact representations of planar graphs in 2D using circles and triangles. While contact graphs in 2D directly correspond to planar graphs, we next study representations of non-planar graphs in 3D. In particular we consider representations of optimal 1-planar graphs. A graph is 1-planar if there exists a drawing in the plane where each edge is crossed at most once, and an optimal n-vertex 1-planar graph has the maximum (4n - 8) number of edges. We describe a linear-time algorithm for representing optimal 1-planar graphs without separating 4-cycles with 3D boxes. However, not every optimal 1-planar graph admits a representation with boxes. Hence, we consider contact representations with the next simplest axis-aligned 3D object, L-shaped polyhedra. We provide a quadratic-time algorithm for representing optimal 1-planar graph with L-shaped polyhedra

Out-of-Equilibrium Admittance of Single Electron Box Under Strong Coulomb Blockade

We study admittance and energy dissipation in an out-of-equlibrium single electron box. The system consists of a small metallic island coupled to a massive reservoir via single tunneling junction. The potential of electrons in the island is controlled by an additional gate electrode. The energy dissipation is caused by an AC gate voltage. The case of a strong Coulomb blockade is considered. We focus on the regime when electron coherence can be neglected but quantum fluctuations of charge are strong due to Coulomb interaction. We obtain the admittance under the specified conditions. It turns out that the energy dissipation rate can be expressed via charge relaxation resistance and renormalized gate capacitance even out of equilibrium. We suggest the admittance as a tool for a measurement of the bosonic distribution corresponding collective excitations in the system