Combinatorics and geometry of finite and infinite squaregraphs

Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar dual of a finite squaregraph is determined by a triangle-free chord diagram of the unit disk, which could alternatively be viewed as a triangle-free line arrangement in the hyperbolic plane. This representation carries over to infinite plane graphs with finite vertex degrees in which the balls are finite squaregraphs. Algebraically, finite squaregraphs are median graphs for which the duals are finite circular split systems. Hence squaregraphs are at the crosspoint of two dualities, an algebraic and a geometric one, and thus lend themselves to several combinatorial interpretations and structural characterizations. With these and the 5-colorability theorem for circle graphs at hand, we prove that every squaregraph can be isometrically embedded into the Cartesian product of five trees. This embedding result can also be extended to the infinite case without reference to an embedding in the plane and without any cardinality restriction when formulated for median graphs free of cubes and further finite obstructions. Further, we exhibit a class of squaregraphs that can be embedded into the product of three trees and we characterize those squaregraphs that are embeddable into the product of just two trees. Finally, finite squaregraphs enjoy a number of algorithmic features that do not extend to arbitrary median graphs. For instance, we show that median-generating sets of finite squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure

On the zone of the boundary of a convex body

We consider an arrangement \A of $n$ hyperplanes in $\R^d$ and the zone $\Z$ in \A of the boundary of an arbitrary convex set in $\R^d$ in such an arrangement. We show that, whereas the combinatorial complexity of $\Z$ is known only to be $O$ \cite{APS}, the outer part of the zone has complexity $O$ (without the logarithmic factor). Whether this bound also holds for the complexity of the inner part of the zone is still an open question (even for $d=2$)

The Complexity of Order Type Isomorphism

The order type of a point set in $R^d$ maps each $(d{+}1)$-tuple of points to its orientation (e.g., clockwise or counterclockwise in $R^2$). Two point sets $X$ and $Y$ have the same order type if there exists a mapping $f$ from $X$ to $Y$ for which every $(d{+}1)$-tuple $(a_1,a_2,\ldots,a_{d+1})$ of $X$ and the corresponding tuple $(f(a_1),f(a_2),\ldots,f(a_{d+1}))$ in $Y$ have the same orientation. In this paper we investigate the complexity of determining whether two point sets have the same order type. We provide an $O(n^d)$ algorithm for this task, thereby improving upon the $O(n^{\lfloor{3d/2}\rfloor})$ algorithm of Goodman and Pollack (1983). The algorithm uses only order type queries and also works for abstract order types (or acyclic oriented matroids). Our algorithm is optimal, both in the abstract setting and for realizable points sets if the algorithm only uses order type queries.Comment: Preliminary version of paper to appear at ACM-SIAM Symposium on Discrete Algorithms (SODA14

Three-Dimensional Myoarchitecture of the Lower Esophageal Sphincter and Esophageal Hiatus Using Optical Sectioning Microscopy.

Studies to date have failed to reveal the anatomical counterpart of the lower esophageal sphincter (LES). We assessed the LES and esophageal hiatus morphology using a block containing the human LES and crural diaphragm, serially sectioned at 50 μm intervals and imaged at 8.2 μm/pixel resolution. A 3D reconstruction of the tissue block was reconstructed in which each of the 652 cross sectional images were also segmented to identify the boundaries of longitudinal (LM) and circular muscle (CM) layers. The CM fascicles on the ventral surface of LES are arranged in a helical/spiral fashion. On the other hand, the CM fascicles from the two sides cross midline on dorsal surface and continue as sling/oblique muscle on the stomach. Some of the LM fascicles of the esophagus leave the esophagus to enter into the crural diaphragm and the remainder terminate into the sling fibers of the stomach. The muscle fascicles of the right crus of diaphragm which form the esophageal hiatus are arranged like a "noose" around the esophagus. We propose that circumferential squeeze of the LES and crural diaphragm is generated by a unique myo-architectural design, each of which forms a "noose" around the esophagus

On-chip spectro-detection for fully integrated coherent beam combiners

This paper presents how photonics associated with new arising detection technologies is able to provide fully integrated instrument for coherent beam combination applied to astrophysical interferometry. The feasibility and operation of on-chip coherent beam combiners has been already demonstrated using various interferometric combination schemes. More recently we proposed a new detection principle aimed at directly sampling and extracting the spectral information of an input signal together with its flux level measurement. The so-called SWIFTS demonstrated concept that stands for Stationary-Wave Integrated Fourier Transform Spectrometer, provides full spectral and spatial information recorded simultaneously thanks to a motionless detecting device. Due to some newly available detection principles considered for the implementation of the SWIFTS concept, some technologies can even provide photo-counting operation that brought a significant extension of the interferometry domain of investigation in astrophysics . The proposed concept is applicable to most of the interferometric instrumental modes including fringe tracking, fast and sensitive detection, Fourier spectral reconstruction and also to manage a large number of incoming beams. The paper presents three practical implementations, two dealing with pair-wise integrated optics beam combinations and the third one with an all-in-one 8 beam combination. In all cases the principles turned into a pair wise baseline coding after proper data processing.Comment: 12 pages, 6 figures, part of the Optics Express special issue dedicated to Astrophotonic

Eulerian method for multiphase interactions of soft solid bodies in fluids

We introduce an Eulerian approach for problems involving one or more soft solids immersed in a fluid, which permits mechanical interactions between all phases. The reference map variable is exploited to simulate finite-deformation constitutive relations in the solid(s) on the same fixed grid as the fluid phase, which greatly simplifies the coupling between phases. Our coupling procedure, a key contribution in the current work, is shown to be computationally faster and more stable than an earlier approach, and admits the ability to simulate both fluid--solid and solid--solid interaction between submerged bodies. The interface treatment is demonstrated with multiple examples involving a weakly compressible Navier--Stokes fluid interacting with a neo-Hookean solid, and we verify the method's convergence. The solid contact method, which exploits distance-measures already existing on the grid, is demonstrated with two examples. A new, general routine for cross-interface extrapolation is introduced and used as part of the new interfacial treatment