7,132 research outputs found

    Abstract Algebra : An Introductory Course

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    This book is intended for students encountering the beautiful subject of abstract algebra for the first time. My goal here is to provide a text that is suitable for you, whether you plan to take only a single course in abstract algebra, or to carry on to more advanced courses at the senior undergraduate and graduate levels. Naturally, I wish to encourage you to study the subject further and to ensure that you are prepared if you do so. At many universities, including my own, abstract algebra is the first serious proof-based course taken by mathematics majors. While it is quite possible to get through, let us say, a course in calculus simply by memorizing a list of rules and applying them correctly, without really understanding why anything works, such an approach would be disastrous here. To be sure, you must carefully learn the definitions and the statements of theorems, but that is nowhere near sufficient. In order to master the material, you need to understand the proofs and then be able to prove things yourself. This book contains hundreds of problems, and I cannot stress strongly enough the need to solve as many of them as you can. Do not be discouraged if you cannot get all of them! Some are very difficult. But try to figure out as many as you can. You will only learn by getting your hands dirty

    Group rings whose skew elements are bounded Lie Engel

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    Let FG be the group ring of a group G over a field F of characteristic different from 2, and let FG have an involution induced from one on G. Assuming that G has no elements of order 2 and no dihedral group involved, we determine the conditions under which the set of skew elements of FG is bounded Lie Engel. Furthermore, we make the determination with no restrictions upon G when the involution on FG is classical

    Torsion matrices over commutative integral group rings

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    Let ZA be the integral group ring of a finite abelian group A, and n a positive integer greater than 5. We provide conditions on n and A under which every torsion matrix U, with identity augmentation, in GLn(ZA) is conjugate in GLn(QA) to a diagonal matrix with group elements on the diagonal. When A is infinite, we show that under similar conditions, U has a group trace and is stably conjugate to such a diagonal matrix

    Testing improved staggered fermions with msm_s and BKB_K

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    We study the improvement of staggered fermions using hypercubically smeared (HYP) links. We calculate the strange quark mass and the kaon B-parameter, BKB_K, in quenched QCD on a 163×6416^3 \times 64 lattice at β=6.0\beta=6.0. We find ms(MSˉ,2GeV)=101.2±1.3±4m_s(\bar{\rm MS},2 {\rm GeV})=101.2\pm1.3\pm4 MeV and BK(MSˉ,2GeV)=0.578±0.018±0.042B_K(\bar{\rm MS},2 {\rm GeV}) = 0.578 \pm 0.018\pm 0.042, where the first error is from statistics and fitting, and the second from using one-loop matching factors. The scale (1/a=1.951/a=1.95GeV) is set by MρM_\rho, and msm_s is determined using the kaon mass. Comparing to quenched results obtained using unimproved staggered fermions and other discretizations, we argue that the size of discretization errors in BKB_K is substantially reduced by improvement.Comment: 9 pages, 12 figure, referee's comments are incorporate

    Zerobrane Matrix Mechanics, Monopoles and Membrane Approach in QCD

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    We conjecture that a T-dual form of pure QCD describes dynamics of point-like monopoles. T-duality transforms the QCD Lagrangian into a matrix quantum mechanics of zerobranes which we identify with monopoles. At generic points of the monopole moduli space the SU(N) gauge group is broken down to U(1)N1U(1)^{N-1} reproducing the key feature of 't Hooft's Abelian projection. There are certain points in the moduli space where monopole positions coincide, gauge symmetry is enhanced and gluons emerge as massless excitations. We show that there is a linearly rising potential between zerobranes. This indicates the presence of a stretched flux tube between monopoles. The lowest energy state is achieved when monopoles are sitting on top of each other and gauge symmetry is enhanced. In this case they behave as free massive particles and can condense. In fact, we find a constant eigenfunction of the corresponding Hamiltonian which describes condensation of monopoles. Using the monopole quantum mechanics, we argue that large NN QCD in this T-dual picture is a theory of a closed bosonic membrane propagating in {\em five} dimensional space-time. QCD point-like monopoles can be regarded in this approach as constituents of the membrane.Comment: 16 pages, new section and references adde

    Concurrent Image Processing Executive (CIPE)

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    The design and implementation of a Concurrent Image Processing Executive (CIPE), which is intended to become the support system software for a prototype high performance science analysis workstation are discussed. The target machine for this software is a JPL/Caltech Mark IIIfp Hypercube hosted by either a MASSCOMP 5600 or a Sun-3, Sun-4 workstation; however, the design will accommodate other concurrent machines of similar architecture, i.e., local memory, multiple-instruction-multiple-data (MIMD) machines. The CIPE system provides both a multimode user interface and an applications programmer interface, and has been designed around four loosely coupled modules; (1) user interface, (2) host-resident executive, (3) hypercube-resident executive, and (4) application functions. The loose coupling between modules allows modification of a particular module without significantly affecting the other modules in the system. In order to enhance hypercube memory utilization and to allow expansion of image processing capabilities, a specialized program management method, incremental loading, was devised. To minimize data transfer between host and hypercube a data management method which distributes, redistributes, and tracks data set information was implemented
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