Linial arrangements and local binary search trees

We study the set of NBC sets (no broken circuit sets) of the Linial arrangement and deduce a constructive bijection to the set of local binary search trees. We then generalize this construction to two families of Linial type arrangements for which the bijections are with some $k$-ary labelled trees that we introduce for this purpose.Comment: 13 pages, 1 figure. arXiv admin note: text overlap with arXiv:1403.257

Compartmentalized and signal-selective gap junctional coupling in the hearing cochlea

Gap junctional intercellular communication (GJIC) plays a major role in cochlear function. Recent evidence suggests that connexin 26 (Cx26) and Cx30 are the major constituent proteins of cochlear gap junction channels, possibly in a unique heteromeric configuration. We investigated the functional and structural properties of native cochlear gap junctions in rats, from birth to the onset of hearing [ postnatal day 12 (P12)]. Confocal immunofluorescence revealed increasing Cx26 and Cx30 expression from P0 to P12. Functional GJIC was assessed by coinjection of Lucifer yellow (LY) and Neurobiotin (NBN) during whole-cell recordings in cochlear slices. At P0, there was restricted dye transfer between supporting cells around outer hair cells. Transfer was more extensive between supporting cells around inner hair cells. At P8, there was extensive transfer of both dyes between all supporting cell types. By P12, LY no longer transferred between the supporting cells immediately adjacent to hair cells but still transferred between more peripheral cells. NBN transferred freely, but it did not transfer between inner and outer pillar cells. Freeze fracture further demonstrated decreasing GJIC between inner and outer pillar cells around the onset of hearing. These data are supportive of the appearance of signal-selective gap junctions around the onset of hearing, with specific properties required to support auditory function. Furthermore, they suggest that separate medial and lateral buffering compartments exist in the hearing cochlea, which are individually dedicated to the homeostasis of inner hair cells and outer hair cells

Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements

Hyperplanes of the form x_j = x_i + c are called affinographic. For an affinographic hyperplane arrangement in R^n, such as the Shi arrangement, we study the function f(M) that counts integral points in [1,M]^n that do not lie in any hyperplane of the arrangement. We show that f(M) is a piecewise polynomial function of positive integers M, composed of terms that appear gradually as M increases. Our approach is to convert the problem to one of counting integral proper colorations of a rooted integral gain graph. An application is to interval coloring in which the interval of available colors for vertex v_i has the form [(h_i)+1,M]. A related problem takes colors modulo M; the number of proper modular colorations is a different piecewise polynomial that for large M becomes the characteristic polynomial of the arrangement (by which means Athanasiadis previously obtained that polynomial). We also study this function for all positive moduli.Comment: 13 p

The Interest is Not Mutual: Effect of the Personal Property Securities Act 2009 (CTH) on Contractual Rights of Set-Off

In Hamersley Iron Pty Ltd v Forge Group Power Pty Ltd (In Liquidation) (Receivers and Managers Appointed), the Supreme Court of Western Australia held that the rights of ANZ, a secured creditor of Forge Group Power Pty Ltd (Forge) holding a security interest under the Personal Property Securities Act 2009 (Cth) (PPSA), trumped Hamersley Iron Pty Ltd’s rights of contractual and equitable set-off. Forge is in receivership and in liquidation. In answering the preliminary issues in dispute between the parties, the Supreme Court examined the complex interaction between contractual and equitable rights, the PPSA and section 553C of the Corporations Act 2001 (Cth) (the CA)

Lattice Points in Orthotopes and a Huge Polynomial Tutte Invariant of Weighted Gain Graphs

A gain graph is a graph whose edges are orientably labelled from a group. A weighted gain graph is a gain graph with vertex weights from an abelian semigroup, where the gain group is lattice ordered and acts on the weight semigroup. For weighted gain graphs we establish basic properties and we present general dichromatic and forest-expansion polynomials that are Tutte invariants (they satisfy Tutte's deletion-contraction and multiplicative identities). Our dichromatic polynomial includes the classical graph one by Tutte, Zaslavsky's two for gain graphs, Noble and Welsh's for graphs with positive integer weights, and that of rooted integral gain graphs by Forge and Zaslavsky. It is not a universal Tutte invariant of weighted gain graphs; that remains to be found. An evaluation of one example of our polynomial counts proper list colorations of the gain graph from a color set with a gain-group action. When the gain group is Z^d, the lists are order ideals in the integer lattice Z^d, and there are specified upper bounds on the colors, then there is a formula for the number of bounded proper colorations that is a piecewise polynomial function of the upper bounds, of degree nd where n is the order of the graph. This example leads to graph-theoretical formulas for the number of integer lattice points in an orthotope but outside a finite number of affinographic hyperplanes, and for the number of n x d integral matrices that lie between two specified matrices but not in any of certain subspaces defined by simple row equations.Comment: 32 pp. Submitted in 2007, extensive revisions in 2013 (!). V3: Added references, clarified examples. 35 p

An elementary chromatic reduction for gain graphs and special hyperplane arrangements

A gain graph is a graph whose edges are labelled invertibly by "gains" from a group. "Switching" is a transformation of gain graphs that generalizes conjugation in a group. A "weak chromatic function" of gain graphs with gains in a fixed group satisfies three laws: deletion-contraction for links with neutral gain, invariance under switching, and nullity on graphs with a neutral loop. The laws lead to the "weak chromatic group" of gain graphs, which is the universal domain for weak chromatic functions. We find expressions, valid in that group, for a gain graph in terms of minors without neutral-gain edges, or with added complete neutral-gain subgraphs, that generalize the expression of an ordinary chromatic polynomial in terms of monomials or falling factorials. These expressions imply relations for chromatic functions of gain graphs. We apply our relations to some special integral gain graphs including those that correspond to the Shi, Linial, and Catalan arrangements, thereby obtaining new evaluations of and new ways to calculate the zero-free chromatic polynomial and the integral and modular chromatic functions of these gain graphs, hence the characteristic polynomials and hypercubical lattice-point counting functions of the arrangements. We also calculate the total chromatic polynomial of any gain graph and especially of the Catalan, Shi, and Linial gain graphs.Comment: 31 page

Spectral Graph Forge: Graph Generation Targeting Modularity

Community structure is an important property that captures inhomogeneities common in large networks, and modularity is one of the most widely used metrics for such community structure. In this paper, we introduce a principled methodology, the Spectral Graph Forge, for generating random graphs that preserves community structure from a real network of interest, in terms of modularity. Our approach leverages the fact that the spectral structure of matrix representations of a graph encodes global information about community structure. The Spectral Graph Forge uses a low-rank approximation of the modularity matrix to generate synthetic graphs that match a target modularity within user-selectable degree of accuracy, while allowing other aspects of structure to vary. We show that the Spectral Graph Forge outperforms state-of-the-art techniques in terms of accuracy in targeting the modularity and randomness of the realizations, while also preserving other local structural properties and node attributes. We discuss extensions of the Spectral Graph Forge to target other properties beyond modularity, and its applications to anonymization