Pre-Symmetry Sets of 3D shapes

The investigation of 3D euclidean symmetry sets (SS) and medial axis is an important area, due in particular to their various important applications. The pre-symmetry set of a surface M in 3-space (resp. smooth closed curve in 2D) is the set of pairs of points which contribute to the symmetry set, that is, the closure of the set of pairs of distinct points p and q in M, for which there exists a sphere (resp. a circle) tangent to M at p and at q. The aim of this paper is to address problems related to the smoothness and the singularities of the pre-symmetry sets of 3D shapes. We show that the pre-symmetry set of a smooth surface in 3-space has locally the structure of the graph of a function from R^2 to R^2, in many cases of interest.Comment: ACM-class: I.2; I.5; I.4; J.2. Latex, 3 grouped figures. The final version will appear in the proceedings of the First International Workshop on Deep Structure, Singularities and Computer Vision, Maastricht June 200

The "Coulomb phase" in frustrated systems

The "Coulomb phase" is an emergent state for lattice models (particularly highly frustrated antiferromagnets) which have local constraints that can be mapped to a divergence-free "flux". The coarse-grained version of this flux or polarization behave analogously to electric or magnetic fields; in particular, defects at which the local constraint is violated behave as effective charges with Coulomb interactions. I survey the derivation of the characteristic power-law correlation functions and the pinch-points in reciprocal space plots of diffuse scattering, as well as applications to magnetic relaxation, quantum-mechanical generalizations, phase transitions to long-range-ordered states, and the effects of disorder.Comment: 30 pp, 5 figures (Sub. to Annual Reviews of Condensed Matter Physics

Cell motility driving mediolateral intercalation in explants of Xenopus laevis

In Xenopus, convergence and extension are produced by active intercalation of the deep mesodermal cells between one another along the mediolateral axis (mediolateral cell intercalation), to form a narrower, longer array. The cell motility driving this intercalation is poorly understood. A companion paper shows that the endodermal epithelium organizes the outermost mesodermal cells immediately beneath it to undergo convergence and extension, and other evidence suggests that these deep cells are the most active participants in mediolateral intercalation (Shih, J. and Keller, R. (1992) Development 116, 887–899). In this paper, we shave off the deeper layers of mesodermal cells, which allows us to observe the protrusive activity of the mesodermal cells next to the organizing epithelium with high resolution video microscopy. These mesodermal cells divide in the early gastrula and show rapid, randomly directed protrusive activity. At the early midgastrula stage, they begin to express a characteristic sequence of behaviors, called mediolateral intercalation behavior (MIB): (1) large, stable, filiform and lamelliform protrusions form in the lateral and medial directions, thus making the cells bipolar; (2) these protrusions are applied directly to adjacent cell surfaces and exert traction on them, without contact inhibition; (3) as a result, the cells elongate and align parallel to the mediolateral axis and perpendicular to the axis of extension; (4) the elongate, aligned cells intercalate between one another along the mediolateral axis, thus producing a longer, narrower array. Explants of essentially a single layer of deep mesodermal cells, made at stage 10.5, converge and extend by mediolateral intercalation. Thus by stage 10.5 (early midgastrula), expression of MIB among deep mesodermal cells is physiologically and mechanically independent of the organizing influence of the endodermal epithelium, described previously (Shih, J. and Keller, R. (1992) Development 116 887–899), and is the fundamental cell motility underlying mediolateral intercalation and convergence and extension of the body axis

From segment to somite: segmentation to epithelialization analyzed within quantitative frameworks

One of the most visually striking patterns in the early developing embryo is somite segmentation. Somites form as repeated, periodic structures in pairs along nearly the entire caudal vertebrate axis. The morphological process involves short- and long-range signals that drive cell rearrangements and cell shaping to create discrete, epithelialized segments. Key to developing novel strategies to prevent somite birth defects that involve axial bone and skeletal muscle development is understanding how the molecular choreography is coordinated across multiple spatial scales and in a repeating temporal manner. Mathematical models have emerged as useful tools to integrate spatiotemporal data and simulate model mechanisms to provide unique insights into somite pattern formation. In this short review, we present two quantitative frameworks that address the morphogenesis from segment to somite and discuss recent data of segmentation and epithelialization

Boundary chromatic polynomial

We consider proper colorings of planar graphs embedded in the annulus, such that vertices on one rim can take Q_s colors, while all remaining vertices can take Q colors. The corresponding chromatic polynomial is related to the partition function of a boundary loop model. Using results for the latter, the phase diagram of the coloring problem (with real Q and Q_s) is inferred, in the limits of two-dimensional or quasi one-dimensional infinite graphs. We find in particular that the special role played by Beraha numbers Q=4 cos^2(pi/n) for the usual chromatic polynomial does not extend to the case Q different from Q_s. The agreement with (scarce) existing numerical results is perfect; further numerical checks are presented here.Comment: 20 pages, 7 figure

Farthest-Polygon Voronoi Diagrams

Given a family of k disjoint connected polygonal sites in general position and of total complexity n, we consider the farthest-site Voronoi diagram of these sites, where the distance to a site is the distance to a closest point on it. We show that the complexity of this diagram is O(n), and give an O(n log^3 n) time algorithm to compute it. We also prove a number of structural properties of this diagram. In particular, a Voronoi region may consist of k-1 connected components, but if one component is bounded, then it is equal to the entire region

An information theoretic characterisation of auditory encoding.

The entropy metric derived from information theory provides a means to quantify the amount of information transmitted in acoustic streams like speech or music. By systematically varying the entropy of pitch sequences, we sought brain areas where neural activity and energetic demands increase as a function of entropy. Such a relationship is predicted to occur in an efficient encoding mechanism that uses less computational resource when less information is present in the signal: we specifically tested the hypothesis that such a relationship is present in the planum temporale (PT). In two convergent functional MRI studies, we demonstrated this relationship in PT for encoding, while furthermore showing that a distributed fronto-parietal network for retrieval of acoustic information is independent of entropy. The results establish PT as an efficient neural engine that demands less computational resource to encode redundant signals than those with high information content