A New Algorithm for Solving the Word Problem in Braid Groups

One of the most interesting questions about a group is if its word problem can be solved and how. The word problem in the braid group is of particular interest to topologists, algebraists and geometers, and is the target of intensive current research. We look at the braid group from a topological point of view (rather than a geometrical one). The braid group is defined by the action of diffeomorphisms on the fundamental group of a punctured disk. We exploit the topological definition of the braid group in order to give a new approach for solving its word problem. Our algorithm is faster, in comparison with known algorithms, for short braid words with respect to the number of generators combining the braid, and it is almost independent of the number of strings in the braids. Moreover, the algorithm is based on a new computer presentation of the elements of the fundamental group of a punctured disk. This presentation can be used also for other algorithms.Comment: 24 pages, 13 figure

Asymptotic Behavior of the Wave Packet Propagation through a Barrier: the Green's Function Approach Revisited

To model the decay of a quasibound state we use the modified two-potential approach introduced by Gurvitz and Kalbermann. This method has proved itself useful in the past for calculating the decay width and the energy shift of an isolated quasistationary state. We follow the same approach in order to propagate the wave-packet in time with the ultimate goal of extracting the momentum-distribution of emitted particles. The advantage of the method is that it provides the time-dependent wave function in a simple semi-analytic form. We intend to apply this method to the modeling of metastable states for which no direct integration of the time-dependent Schroedinger equation is available today.Comment: 7 page

D-Branes in the Background of NS Fivebranes

We study the dynamics of $D$-branes in the near-horizon geometry of $NS$ fivebranes. This leads to a holographically dual description of the physics of $D$-branes ending on and/or intersecting $NS5$-branes. We use it to verify some properties of such $D$-branes which were deduced indirectly in the past, and discuss some instabilities of non-supersymmetric brane configurations. Our construction also describes vacua of Little String Theory which are dual to open plus closed string theory in asymptotically linear dilaton spacetimes.Comment: 44 pages, 16 figures, harvma

String Theory on AdS_3 x S^3 x S^3 x S^1

Spacetime properties of superstrings on AdS_3 x S^3 x S^3 x S^1 are studied. The boundary theory is a two dimensional superconformal field theory with a large N=(4,4) supersymmetry.Comment: 12 pages, harvma

Skating on a Film of Air: Drops Impacting on a Surface

Drops impacting on a surface are ubiquitous in our everyday experience. This impact is understood within a commonly accepted hydrodynamic picture: it is initiated by a rapid shock and a subsequent ejection of a sheet leading to beautiful splashing patterns. However, this picture ignores the essential role of the air that is trapped between the impacting drop and the surface. Here we describe a new imaging modality that is sensitive to the behavior right at the surface. We show that a very thin film of air, only a few tens of nanometers thick, remains trapped between the falling drop and the surface as the drop spreads. The thin film of air serves to lubricate the drop enabling the fluid to skate on the air film laterally outward at surprisingly high velocities, consistent with theoretical predictions. Eventually this thin film of air must break down as the fluid wets the surface. We suggest that this occurs in a spinodal-like fashion, and causes a very rapid spreading of a wetting front outwards; simultaneously the wetting fluid spreads inward much more slowly, trapping a bubble of air within the drop. Our results show that the dynamics of impacting drops are much more complex than previously thought and exhibit a rich array of unexpected phenomena that require rethinking classical paradigms.Comment: 4 pages, 4 figure

Probabilistic Solutions of Equations in the Braid Group

Given a system of equations in a "random" finitely generated subgroup of the braid group, we show how to find a small ordered list of elements in the subgroup, which contains a solution to the equations with a significant probability. Moreover, with a significant probability, the solution will be the first in the list. This gives a probabilistic solution to: The conjugacy problem, the group membership problem, the shortest representation of an element, and other combinatorial group-theoretic problems in random subgroups of the braid group. We use a memory-based extension of the standard length-based approach, which in principle can be applied to any group admitting an efficient, reasonably behaving length function.Comment: Small update

Melanocytes and Langerhans Cells in Aged Versus Young Skin Before and After Transplantation onto Nude Mice

Previous studies have demonstrated decreased numbers of melanocytes and Langerhans cells (LC) in aged skin. In the present study, we employed dopa and indirect immunoperoxidase techniques in epidermal sheets to determine the fate of melanocytes and LC of aged versus young donors after skin transplantations onto nude mice. The detection of positive homologous leucocytic antibody reaction of degeneration (HLA-DR) of LC indicates an age-associated reduction in sun-protected thigh skin in aged versus young subjects (263 ± 63 versus 589.25 ± 142.643, p < 0.001). The mean number of LC four weeks after transplanation remained almost constant. Prior to skin engraftment, a decreased number of melanocytes was found in aged versus young epidermis (160.77 ± 51.7 versus 255.83 ± 81.2, respectively, p < 0.05). A significantly increased number of melanocytes was noted four weeks following engraftment in epidermis from aged (307.44 ± 174, p <0.05) and young human donors (402.16 ± l39,p < 0.02). The marked increase in density of dopa-positive melanocytes following engraftment onto nude mice may indicate the existence of circulating factors in nude mice that perhaps both stimulates and enhances proliferation and activity of these cells