2,412 research outputs found

    Dimensions of Imaginary Root Spaces of Hyperbolic Kac--Moody Algebras

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    We discuss the known results and methods for determining root multiplicities for hyperbolic Kac--Moody algebras

    Polynomial Invariants for Affine Programs

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    We exhibit an algorithm to compute the strongest polynomial (or algebraic) invariants that hold at each location of a given affine program (i.e., a program having only non-deterministic (as opposed to conditional) branching and all of whose assignments are given by affine expressions). Our main tool is an algebraic result of independent interest: given a finite set of rational square matrices of the same dimension, we show how to compute the Zariski closure of the semigroup that they generate

    Fivebranes and 4-manifolds

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    We describe rules for building 2d theories labeled by 4-manifolds. Using the proposed dictionary between building blocks of 4-manifolds and 2d N=(0,2) theories, we obtain a number of results, which include new 3d N=2 theories T[M_3] associated with rational homology spheres and new results for Vafa-Witten partition functions on 4-manifolds. In particular, we point out that the gluing measure for the latter is precisely the superconformal index of 2d (0,2) vector multiplet and relate the basic building blocks with coset branching functions. We also offer a new look at the fusion of defect lines / walls, and a physical interpretation of the 4d and 3d Kirby calculus as dualities of 2d N=(0,2) theories and 3d N=2 theories, respectivelyComment: 81 pages, 18 figures. v2: misprints corrected, clarifications and references added. v3: additions and corrections about lens space theory, 4-manifold gluing, smooth structure

    Information Ranking and Power Laws on Trees

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    We study the situations when the solution to a weighted stochastic recursion has a power law tail. To this end, we develop two complementary approaches, the first one extends Goldie's (1991) implicit renewal theorem to cover recursions on trees; and the second one is based on a direct sample path large deviations analysis of weighted recursive random sums. We believe that these methods may be of independent interest in the analysis of more general weighted branching processes as well as in the analysis of algorithms

    Dynamics of Modular Matings

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    In the paper 'Mating quadratic maps with the modular group II' the current authors proved that each member of the family of holomorphic (2:2)(2:2) correspondences Fa\mathcal{F}_a: (az+1z+1)2+(az+1z+1)(aw1w1)+(aw1w1)2=3,\left(\frac{az+1}{z+1}\right)^2+\left(\frac{az+1}{z+1}\right)\left(\frac{aw-1}{w-1}\right) +\left(\frac{aw-1}{w-1}\right)^2=3, introduced by the first author and C. Penrose in 'Mating quadratic maps with the modular group', is a mating between the modular group and a member of the parabolic family of quadratic rational maps PA:zz+1/z+AP_A:z\to z+1/z+A whenever the limit set of Fa\mathcal{F}_a is connected. Here we provide a dynamical description for the correspondences Fa\mathcal{F}_a which parallels the Douady and Hubbard description for quadratic polynomials. We define a B\"ottcher map and a Green's function for Fa\mathcal{F}_a, and we show how in this setting periodic geodesics play the role played by external rays for quadratic polynomials. Finally, we prove a Yoccoz inequality which implies that for the parameter aa to be in the connectedness locus MΓM_{\Gamma} of the family Fa\mathcal{F}_a, the value of the log-multiplier of an alpha fixed point which has combinatorial rotation number 1/q1/q lies in a strip whose width goes to zero at rate proportional to (logq)/q2(\log q)/q^2
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