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CONJUGATIONFREE GEOMETRIC PRESENTATIONS OF FUNDAMENTAL GROUPS OF ARRANGEMENTS
, 2008
"... We introduce the notion of a conjugationfree geometric presentation for a fundamental group of line arrangements’ complements, and we show that the fundamental group of following family of arrangements have a conjugationfree geometric presentation: An arrangement L, whose graph of multiple point ..."
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We introduce the notion of a conjugationfree geometric presentation for a fundamental group of line arrangements’ complements, and we show that the fundamental group of following family of arrangements have a conjugationfree geometric presentation: An arrangement L, whose graph of multiple points is a unique cycle of length n, and the multiplicities of the multiple points are arbitrary. We also compute the exact structure (by means of a semidirect product of groups) of the arrangement which consists of a cycle of length 3, where all the multiple points are of multiplicity 3.
LS303 Generation of Bright, Tunable, Polarized γRay Sources by Scattering Laser Pulses from APS Electron Beams
, 2003
"... We calculate the performance of possible Advanced Photon Source (APS) γray sources for applications in nuclear physics research. For the APS storage ring, it is possible to generate tagged γray photon fluxes of 10 8, 0.7×10 8, and 0.3×10 8 photons/s at photon energies of 1, 1.7, and 2.8 GeV, respe ..."
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We calculate the performance of possible Advanced Photon Source (APS) γray sources for applications in nuclear physics research. For the APS storage ring, it is possible to generate tagged γray photon fluxes of 10 8, 0.7×10 8, and 0.3×10 8 photons/s at photon energies of 1, 1.7, and 2.8 GeV, respectively. For untagged photons, fluxes higher than 10 8 photons/s are possible for those energies. For the injection booster, an untagged γray photon flux up to 10 8 photons/s at energy ranging from 5 MeV to 1 GeV is possible. This can be achieved using offtheshelf commercial Ti:Sa laser systems. The photon fluxes predicted here are in general one to two orders of magnitude higher than facilities with similar photon energies. 1.
Tangent unitvector fields: nonabelian homotopy invariants and the Dirichlet energy
, 2009
"... Let O be a closed geodesic polygon in S 2. Maps from O into S 2 are said to satisfy tangent boundary conditions if the edges of O are mapped into the geodesics which contain them. Taking O to be an octant of S 2, we compute the infimum Dirichlet energy, E(H), for continuous maps satisfying tangent b ..."
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Let O be a closed geodesic polygon in S 2. Maps from O into S 2 are said to satisfy tangent boundary conditions if the edges of O are mapped into the geodesics which contain them. Taking O to be an octant of S 2, we compute the infimum Dirichlet energy, E(H), for continuous maps satisfying tangent boundary conditions of arbitrary homotopy type H. The expression for E(H) involves a topological invariant – the spelling length – associated with the (nonabelian) fundamental group of the ntimes punctured twosphere, π1(S 2 − {s1,..., sn}, ∗). The lower bound for E(H) is obtained from combinatorial group theory arguments, while the upper bound is obtained by constructing explicit representatives which, on all but an arbitrarily small subset of O, are alternatively locally conformal or anticonformal. For conformal and anticonformal classes (classes containing wholly conformal and anticonformal representatives respectively), the expression for E(H) reduces to a previous result involving the degrees of a set of regular values s1,...,sn in the target S 2 space. These degrees may be viewed as invariants associated with the abelianization of π1(S 2 − {s1,..., sn}, ∗). For nonconformal classes, however, E(H) may be strictly greater than the abelian bound. This stems from the fact that, for nonconformal maps, the number of preimages of certain regular values may necessarily be strictly greater than the absolute value of their degrees. This work is motivated by the theoretical modelling of nematic liquid crystals in confined polyhedral geometries. The results imply new lower and upper bounds for the Dirichlet energy (oneconstant OseenFrank energy) of reflectionsymmetric tangent unitvector fields in a rectangular prism.