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Register a new userAn experimental investigation of Neumanns conjecture
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We use a large census of hyperbolic 3-manifolds to experimentally investigate a conjecture of Neumann regarding the Bloch Group. We present an augmented census including, for feasible invariant trace fields, explicit manifolds (associated to that field) that appear to generate the Bloch group of that field. We also make use of Ptolemy coordinates to compute exotic volumes of representations, and attempt to realize these volumes as linear combinations of generator volumes. We thus present a large body of empirical support for Neumanns conjecture.
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Columnar jointing is a fracture pattern common in igneous rocks in which cracks self-organize into a roughly hexagonal arrangement, leaving behind an ordered colonnade. We report observations of columnar jointing in a laboratory analog system, desiccated corn starch slurries. Using measurements of moisture density, evaporation rates, and fracture advance rates as evidence, we suggest an advective-diffusive system is responsible for the rough scaling behavior of columnar joints. This theory explains the order of magnitude difference in scales between jointing in lavas and in starches. We investigated the scaling of average columnar cross-sectional areas due to the evaporation rate, the analog of the cooling rate of igneous columnar joints. We measured column areas in experiments where the evaporation rate depended on lamp height and time, in experiments where the evaporation rate was fixed using feedback methods, and in experiments where gelatin was added to vary the rheology of the starch. Our results suggest that the column area at a particular depth is related to both the current conditions, and hysteretically to the geometry of the pattern at previous depths. We argue that there exists a range of stable column scales allowed for any particular evaporation rate.
We generate a pair of entangled beams from the interference of two amplitude squeezed beams. The entanglement is quantified in terms of EPR-paradox [Reid88] and inseparability [Duan00] criteria, with observed results of $Delta^{2} X_{x|y}^{+} Delta^{2} X_{x|y}^{-} = 0.58 pm 0.02$ and $sqrt{Delta^{2} X_{x pm y}^{+} Delta^{2} X_{x pm y}^{-}} = 0.44 pm 0.01$, respectively. Both results clearly beat the standard quantum limit of unity. We experimentally analyze the effect of decoherence on each criterion and demonstrate qualitative differences. We also characterize the number of required and excess photons present in the entangled beams and provide contour plots of the efficacy of quantum information protocols in terms of these variables.
The Hilbert-Smith Conjecture states that if G is a locally compact group which acts effectively on a connected manifold as a topological transformation group, then G is a Lie group. A rather straightforward proof of this conjecture is given. The motivation is work of Cernavskii (``Finite-to-one mappings of manifolds, Trans. of Math. Sk. 65 (107), 1964.) His work is generalized to the orbit map of an effective action of a p-adic group on compact connected n-manifolds with the aid of some new ideas. There is no attempt to use Smith Theory even though there may be similarities. It is well known that if a locally compact group acts effectively on a connected n-manifold M and G is not a Lie group, then there is a subgroup H of G isomorphic to a p-adic group A_p which acts effectively on M. It can be shown that A_p can not act effectively on an n-manifold and, hence, The Hilbert Smith Conjecture is true. The existence of a non empty fixed point set adds some complexity to the proof. In this paper, it is shown that A_p can not act freely on a compact connected n-manifold. The basic ideas for the general case are more clearly seen in this case. The general proof will be given in another paper.
An emerging long obstacle placed in a boundary layer developing under a free-surface generates a complex horseshoe vortex (HSV) system, which is composed of a set of vortices exhibiting a rich variety of dynamics. The present experimental study examines such flow structure and characterizes precisely, using PIV measurements, the evolution of the HSV geometrical and dynamical properties over a wide range of dimensionless parameters (Reynolds number $Re_h in [750, 8300]$, boundary layer development ratio $h/delta in [1.25, 4.25]$ and obstacle aspect ratio $W/h in [0.67, 2.33]$). The dynamical study of the HSV is based on the categorization of the HSV vortices motion into an enhanced specific bi-dimensional typology, separating a coherent (due to vortex-vortex interactions) and an irregular evolution (due to appearance of small-scale instabilities). This precise categorization is made possible thanks to the use of vortex tracking methods applied on PIV measurements, A semi-empirical model for the HSV vortices motion is then proposed to highlight some important mechanisms of the HSV dynamics, as (i) the influence of the surrounding vortices on a vortex motion and (ii) the presence of a phase shift between the motion of all vortices. The study of the HSV geometrical properties (vortex position and characteristic lengths and frequencies) evolution with the flow parameters shows that strong dependencies exist between the streamwise extension of the HSV and the obstacle width, and between the HSV vortex number and its elongation. Comparison of these data with prior studies for immersed obstacles reveals that emerging obstacles lead to greater adverse pressure gradients and down-flows in front of the obstacle.
The Hilbert-Smith Conjecture states that if G is a locally compact group which acts effectively on a connected manifold as a topological transformation group, then G is a Lie group. A rather straightforward proof of this conjecture is given. The motivation is work of Cernavskii (``Finite-to-one mappings of manifolds, Trans. of Math. Sk. 65 (107), 1964.) His work is generalized to the orbit map of an effective action of a p-adic group on compact connected n-manifolds with the aid of some new ideas. There is no attempt to use Smith Theory even though there may be similarities.
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