Proof. It would be easy to obtain these properties from the matrix form of U constructed in Sec. I.5.2, but we prefer to give a direct proof, independent of the particular realization of U . Part (i): To prove that L and L∗ are wandering subspaces, it suffices to show that UL0 ⊥L0 and UL0 ⊥ L0 for n = 1,2, . . .; by reason of symmetry it even suffices to consider one of these cases, say that of...

Proof. It would be easy to obtain these properties from the matrix form of U constructed in Sec. I.5.2, but we prefer to give a direct proof, independent of the particular realization of U . Part (i): To prove that L and L∗ are wandering subspaces, it suffices to show that UL0 ⊥L0 and UL0 ⊥ L0 for n = 1,2, . . .; by reason of symmetry it even suffices to consider one of these cases, say that of...

Given a compact n-dimensional immersed Riemannian manifold Mn in some Euclidean space we prove that if the Hausdorff dimension of the singular set of the Gauss map is small, then Mn is homeomorphic to the sphere Sn. Also, we define a concept of finite geometrical type and prove that finite geometrical type hypersurfaces with small set of points of zero GaussKronecker curvature are topologically...

We present measurements of the fractal dimensions associated to the spin clusters for Z4 and Z5 spin models. We also attempted to measure similar fractal dimensions for the generalised Fortuin Kastelyn (FK) clusters in these models but we discovered that these clusters do not percolate at the critical point of the model under consideration. These results clearly mark a difference in the behavio...

In a wide range of industrial applications there appear systems of hard particles of different shapes and sizes, known as “packings”. In this work, the force-biased algorithm, primarily designed to model close packings of equal spheres, is adapted to simulate mixtures of spherocylindrical particles of different radii and aspect ratios. The packing densities of simulated mono and polydisperse sy...

Three algorithms for determining the bulk geometric properties of cellular structures from a stationary digital image are presented. The geometrical properties of interest include cell orientation and cell aspect ratio. With future test data expected to possess high luminance variability, poor contrast near element boundaries and irregular discolouration, this problem is poorly suited for the s...

We study the benefits of the explicit formulas of a parametrized family of bijections and the explicit formula of a certain metric. These formulas are induced by a family of compactifications of C that ”account for all arguments of infinity”. This family of bijections map the union of C and a continuum of ideal points onto a family of spherical bowls. This family of bijections are shown to give...

We calculate the form factor H[q], i.e. the generating function of the gyration radii, and the generating function G [q] of the end-to-end radii, to first order in the dimensionless second viral coefficient g, or in 03B5 = 4 2014 d, d being the space dimension. These functions are calculated in the asymptotic limit of Kuhnian chains, i.e. very long chains in a good solvent. They are used to det...

Proof. It would be easy to obtain these properties from the matrix form of U constructed in Sec. I.5.2, but we prefer to give a direct proof, independent of the particular realization of U . Part (i): To prove that L and L∗ are wandering subspaces, it suffices to show that UL0 ⊥L0 and UL0 ⊥ L0 for n = 1,2, . . .; by reason of symmetry it even suffices to consider one of these cases, say that of...

We study the synchronization of N nearest neighbors coupled oscillators in a ring. We derive an analytic form for the phase difference among neighboring oscillators which shows the dependency on the periodic boundary conditions. At synchronization, we find two distinct quantities which characterize four of the oscillators, two pairs of nearest neighbors, which are at the border of the clusters ...