Kinematic cosmology in conformally flat spacetime

In a recent series of papers Endean examines the properties of spatially homogeneous and isotropic (FLRW) cosmological models filled with dust in the ``conformally flat spacetime presentation of cosmology'' (CFS cosmology). This author claims it is possible to resolve a certain number of the difficulties the standard model exhibits when confronted to observations, if the theoretical predictions are obtained in the special framework of CFS cosmology. As a by-product of his analysis Endean claims that no initial (big-bang) nor final (big-crunch) singularities occur in the closed FLRW model. In this paper we show up the fallacious arguments leading to Endean's conclusions and we consistently reject his CFS cosmology.Comment: 5 pages, LaTeX2e, aas2pp4.sty, cases.sty, to be published in the Astrophysical Journa

Lattice symmetry breaking perturbations for spiral waves

Spiral waves in two-dimensional excitable media have been observed experimentally and studied extensively. It is now well-known that the symmetry properties of the medium of propagation drives many of the dynamics and bifurcations which are experimentally observed for these waves. Also, symmetry-breaking induced by boundaries, inhomogeneities and anisotropy have all been shown to lead to different dynamical regimes as to that which is predicted for mathematical models which assume infinite homogeneous and isotropic planar geometry. Recent mathematical analyses incorporating the concept of forced symmetry-breaking from the Euclidean group of all planar translations and rotations have given model-independent descriptions of the effects of media imperfections on spiral wave dynamics. In this paper, we continue this program by considering rotating waves in dynamical systems which are small perturbations of a Euclidean-equivariant dynamical system, but for which the perturbation preserves only the symmetry of a regular square lattice

Orange Peels and Fresnel Integrals

There are two standard ways of peeling an orange: either cut the skin along meridians, or cut it along a spiral. We consider here the second method, and study the shape of the spiral strip, when unfolded on a table. We derive a formula that describes the corresponding flattened-out spiral. Cutting the peel with progressively thinner strip widths, we obtain a sequence of increasingly long spirals. We show that, after rescaling, these spirals tends to a definite shape, known as the Euler spiral. The Euler spiral has applications in many fields of science. In optics, the illumination intensity at a point behind a slit is computed from the distance between two points on the Euler spiral. The Euler spiral also provides optimal curvature for train tracks between a straight run and an upcoming bend. It is striking that it can be also obtained with an orange and a kitchen knife

Distortion of imbeddings of groups of intermediate growth into metric spaces

For every metric space $\mathcal X$ in which there exists a sequence of finite groups of bounded-size generating set that does not embed coarsely, and for every unbounded, increasing function $\rho$, we produce a group of subexponential word growth all of whose imbeddings in $\mathcal X$ have distortion worse than $\rho$. This applies in particular to any B-convex Banach space $\mathcal X$, such as Hilbert space.Comment: Used to appear as first half of arXiv:1403.558

Growth Series and Random Walks on Some Hyperbolic Graphs

Consider the tesselation of the hyperbolic plane by m-gons, l per vertex. In its 1-skeleton, we compute the growth series of vertices, geodesics, tuples of geodesics with common extremities. We also introduce and enumerate "holly trees", a family of reduced loops in these graphs. We then apply Grigorchuk's result relating cogrowth and random walks to obtain lower estimates on the spectral radius of the Markov operator associated with a symmetric random walk on these graphs.Comment: 21 pages. to appear in monash. mat

Born Reciprocity in String Theory and the Nature of Spacetime

After many years, the deep nature of spacetime in string theory remains an enigma. In this letter we incorporate the concept of Born reciprocity in order to provide a new point of view on string theory in which spacetime is a derived dynamical concept. This viewpoint may be thought of as a dynamical chiral phase space formulation of string theory, in which Born reciprocity is implemented as a choice of a Lagrangian submanifold of the phase space, and amounts to a generalization of T-duality. In this approach the fundamental symmetry of string theory contains phase space diffeomorphism invariance and the underlying string geometry should be understood in terms of dynamical bi-Lagrangian manifolds and an apparently new geometric structure, somewhat reminiscent of para-quaternionic geometry, which we call Born geometry