Live imaging of whole mouse embryos during gastrulation : migration analyses of epiblast and mesodermal cells

During gastrulation in the mouse embryo, dynamic cell movements including epiblast invagination and mesodermal layer expansion lead to the establishment of the three-layered body plan. The precise details of these movements, however, are sometimes elusive, because of the limitations in live imaging. To overcome this problem, we developed techniques to enable observation of living mouse embryos with digital scanned light sheet microscope (DSLM). The achieved deep and high time-resolution images of GFP-expressing nuclei and following 3D tracking analysis revealed the following findings: (i) Interkinetic nuclear migration (INM) occurs in the epiblast at embryonic day (E)6 and 6.5. (ii) INM-like migration occurs in the E5.5 embryo, when the epiblast is a monolayer and not yet pseudostratified. (iii) Primary driving force for INM at E6.5 is not pressure from neighboring nuclei. (iv) Mesodermal cells migrate not as a sheet but as individual cells without coordination

Inflationary Scenarios from Branes at Angles

We describe a simple mechanism that can lead to inflation within string-based brane-world scenarios. The idea is to start from a supersymmetric configuration with two parallel static Dp-branes, and slightly break the supersymmetry conditions to produce a very flat potential for the field that parametrises the distance between the branes, i.e. the inflaton field. This breaking can be achieved in various ways: by slight relative rotations of the branes with small angles, by considering small relative velocities between the branes, etc. If the breaking parameter is sufficiently small, a large number of e-folds can be produced within the D-brane, for small changes of the configuration in the compactified directions. Such a process is local, i.e. it does not depend very strongly on the compactification space nor on the initial conditions. Moreover, the breaking induces a very small velocity and acceleration, which ensures very small slow-roll parameters and thus an almost scale invariant spectrum of metric fluctuations, responsible for the observed temperature anisotropies in the microwave background. Inflation ends as in hybrid inflation, triggered by the negative curvature of the string tachyon potential. In this paper we elaborate on one of the simplest examples: two almost parallel D4-branes in a flat compactified space.Comment: 29 pages, 9 eps figures, using JHEP3.cls, published in JHE

Knots with small rational genus

If K is a rationally null-homologous knot in a 3-manifold M, the rational genus of K is the infimum of -\chi(S)/2p over all embedded orientable surfaces S in the complement of K whose boundary wraps p times around K for some p (hereafter: S is a p-Seifert surface for K). Knots with very small rational genus can be constructed by "generic" Dehn filling, and are therefore extremely plentiful. In this paper we show that knots with rational genus less than 1/402 are all geometric -- i.e. they may be isotoped into a special form with respect to the geometric decomposition of M -- and give a complete classification. Our arguments are a mixture of hyperbolic geometry, combinatorics, and a careful study of the interaction of small p-Seifert surfaces with essential subsurfaces in M of non-negative Euler characteristic.Comment: 38 pages, 3 figures; version 3 corrects minor typos; keywords: knots, rational genu

Divergence in right-angled Coxeter groups

Let W be a 2-dimensional right-angled Coxeter group. We characterise such W with linear and quadratic divergence, and construct right-angled Coxeter groups with divergence polynomial of arbitrary degree. Our proofs use the structure of walls in the Davis complex.Comment: This version incorporates the referee's comments. It contains the complete appendix (which will be abbreviated in the journal version). To appear in Transactions of the AM

A new approach to upscaling fracture network models while preserving geostatistical and geomechanical characteristics

A new approach to upscaling two-dimensional fracture network models is proposed for preserving geostatistical and geomechanical characteristics of a smaller-scale “source” fracture pattern. First, the scaling properties of an outcrop system are examined in terms of spatial organization, lengths, connectivity, and normal/shear displacements using fractal geometry and power law relations. The fracture pattern is observed to be nonfractal with the fractal dimension D ≈ 2, while its length distribution tends to follow a power law with the exponent 2 < a < 3. To introduce a realistic distribution of fracture aperture and shear displacement, a geomechanical model using the combined finite-discrete element method captures the response of a fractured rock sample with a domain size L = 2 m under in situ stresses. Next, a novel scheme accommodating discrete-time random walks in recursive self-referencing lattices is developed to nucleate and propagate fractures together with their stress- and scale-dependent attributes into larger domains of up to 54 m × 54 m. The advantages of this approach include preserving the nonplanarity of natural cracks, capturing the existence of long fractures, retaining the realism of variable apertures, and respecting the stress dependency of displacement-length correlations. Hydraulic behavior of multiscale growth realizations is modeled by single-phase flow simulation, where distinct permeability scaling trends are observed for different geomechanical scenarios. A transition zone is identified where flow structure shifts from extremely channeled to distributed as the network scale increases. The results of this paper have implications for upscaling network characteristics for reservoir simulation

Approach to a rational rotation number in a piecewise isometric system

We study a parametric family of piecewise rotations of the torus, in the limit in which the rotation number approaches the rational value 1/4. There is a region of positive measure where the discontinuity set becomes dense in the limit; we prove that in this region the area occupied by stable periodic orbits remains positive. The main device is the construction of an induced map on a domain with vanishing measure; this map is the product of two involutions, and each involution preserves all its atoms. Dynamically, the composition of these involutions represents linking together two sector maps; this dynamical system features an orderly array of stable periodic orbits having a smooth parameter dependence, plus irregular contributions which become negligible in the limit.Comment: LaTeX, 57 pages with 13 figure

An introduction to finite type invariants of knots and 3-manifolds defined by counting graph configurations

These introductory lectures show how to define finite type invariants of links and 3-manifolds by counting graph configurations in 3-manifolds, following ideas of Witten and Kontsevich. The linking number is the simplest finite type invariant for 2-component links. It is defined in many equivalent ways in the first section. As an important example, we present it as the algebraic intersection of a torus and a 4-chain called a propagator in a configuration space. In the second section, we introduce the simplest finite type 3-manifold invariant, which is the Casson invariant (or the Theta-invariant) of integer homology 3-spheres. It is defined as the algebraic intersection of three propagators in the same two-point configuration space. In the third section, we explain the general notion of finite type invariants and introduce relevant spaces of Feynman Jacobi diagrams. In Sections 4 and 5, we sketch an original construction based on configuration space integrals of universal finite type invariants for links in rational homology 3-spheres and we state open problems. Our construction generalizes the known constructions for links in the ambient space, and it makes them more flexible. In Section 6, we present the needed properties of parallelizations of 3-manifolds and associated Pontrjagin classes, in details.Comment: 68 pages. Change of title, updates and minor reorganization of notes of five lectures presented in the ICPAM-ICTP research school of Mekn{\`e}s in May 2012. To appear in the Proceedings of the conference "Quantum topology" organized by Chelyabinsk State University in July 2014 (Vestnik ChelGU