2,412 research outputs found
Dimensions of Imaginary Root Spaces of Hyperbolic Kac--Moody Algebras
We discuss the known results and methods for determining root multiplicities
for hyperbolic Kac--Moody algebras
Polynomial Invariants for Affine Programs
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
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
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
In the paper 'Mating quadratic maps with the modular group II' the current
authors proved that each member of the family of holomorphic
correspondences :
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 whenever the limit set of is connected.
Here we provide a dynamical description for the correspondences
which parallels the Douady and Hubbard description for
quadratic polynomials. We define a B\"ottcher map and a Green's function for
, 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 to be in the
connectedness locus of the family , the value of
the log-multiplier of an alpha fixed point which has combinatorial rotation
number lies in a strip whose width goes to zero at rate proportional to
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