By Alejandro Adem, Jack Morava, Yongbin Ruan
This booklet publishes papers initially awarded at a convention at the Mathematical elements of Orbifold String thought, hosted via the college of Wisconsin-Madison. It encompasses a good deal of knowledge now not totally coated within the released literature and showcases the present cutting-edge in orbital string concept. the topic of orbifolds has a protracted prehistory, going again to the paintings of Thurston and Haefliger, with roots within the thought of manifolds, team activities, and foliations. the hot explosion of job at the subject has been powered via functions of orbifolds to moduli difficulties and quantum box theory.The current quantity offers an interdisciplinary examine orbifold difficulties. subject matters resembling stacks, vertex operator algebras, branes, groupoids, K-theory and quantum cohomology are mentioned. The ebook displays the contemplating extraordinary investigators operating within the components of mathematical physics, algebraic geometry, algebraic topology, symplectic geometry and illustration conception. by way of providing the paintings of a wide variety of mathematicians and physicists who use and examine orbifolds, it familiarizes readers with a few of the issues of view and kinds of effects the researchers convey to the topic
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Extra info for Orbifolds in Mathematics and Physics: Proceedings of a Conference on Mathematical Aspects of Orbifold String Theory, May 4-8, 2001, University of ... Madison, Wisconsin
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E. n 1. profile such that ∇⊥ Within these assumptions, Eq. 35) which is a nonlinear Schrödinger equation (NLSE), one of the most recurrent and studied equations in nonlinear physics (Sulem and Sulem 1999). 36) where ∂μ = (∂x , ∇ ⊥ ) and L is the functional (also named the Lagrangian density) i 1 1 4 L = − (a˜ ∗ ∂x a˜ − a∂ ˜ . 37) The rule is that a, ˜ ∂t a, ˜ ∇ ⊥ a˜ and their complex conjugates must be considered as independent variables. 35). This formulation generalizes the Lagrangian approach to classical mechanics that should be known to most readers, and is strongly used in the quantum theory of fields.
31) by numerical methods (Sun et al. 1987) shows that in the limit of uniform density n = 1 for which the threshold is evaluated (see next Sect. 3), also η → 1, making the calculation consistent. 3 Nonlinear Schrödinger Equation. Self-focusing Threshold Following the discussion at the end of the preceding Sect. e. n 1. profile such that ∇⊥ Within these assumptions, Eq. 35) which is a nonlinear Schrödinger equation (NLSE), one of the most recurrent and studied equations in nonlinear physics (Sulem and Sulem 1999).
As a case which will be of interest in Chap. 5, let us consider an electrostatic regime with no magnetic field, so that E = −∇Φ, and where we are interested on phenomena occurring on the time scale of ion motion. 71) yields en e ∇Φ − ∇Pe = 0. We further assume that the electrons have an isothermal, perfect gas-like EoS Pe = n e Te with constant and uniform temperature Te . g. Feynman et al. 1963, Sect. 4). 73) and Eqs. 71) for ions yield a consistent model which, when the equations are linearized, provides the dispersion relation for ion-acoustic waves.