We study out-of-time ordered four-point functions in two dimensional conformal field theories by suitably analytically continuing the Euclidean correlator. For large central charge theories with a sparse spectrum, chaotic dynamics is revealed in an exponential decay; this is seen directly in the contribution of the vacuum block to the correlation function. However, contributions from individual non-vacuum blocks with large spin and small twist dominate over the vacuum block. We argue, based on holographic intuition, that suitable summations over such intermediate states in the block decomposition of the correlator should be sub-dominant, and attempt to use this criterion to constrain the OPE data with partial success. Along the way we also discuss the relation between the spinning Virasoro blocks and the on-shell worldline action of spinning particles in an asymptotically AdS spacetime.

Let <i>T</i> be a compact disjointness preserving linear operator from <i>C</i>₀(<i>X</i>) into <i>C</i>₀(<i>Y</i>), where <i>X</i> and <i>Y</i> are locally compact Hausdorff spaces. We show that <i>T</i> can be represented as a norm convergent countable sum of disjoint rank one operators. More precisely, <i>T</i> = _<i>n</i> <i></i> <i>h_n</i> for a (possibly finite) sequence {<i>x_n</i> }_<i>n</i> of distinct points in <i>X</i> and a norm null sequence {<i>h_n</i> }_<i>n</i> of mutually disjoint functions in <i>C</i>₀(<i>Y</i>). Moreover, we develop a graph theoretic method to describe the spectrum of such an operator ( 2009 WILEYVCH Verlag GmbH &amp; Co. KGaA, Weinheim)

We study tree level scattering amplitudes of four massless states in the double scaled little string theory, and compare them to perturbative loop amplitudes in six-dimensional super-Yang-Mills theory. The little string amplitudes are computed from correlators in the cigar coset CFT and in N = 2 minimal models. The results are expressed in terms of integrals of conformal blocks and evaluated numerically in the α ′ expansion. We find striking agreements with up to 2-loop scattering amplitudes of massless gluons in 6D SU(k) SYM at a ℤ_k invariant point on the Coulomb branch. We comment on the issue of UV divergence at higher loop orders in the gauge theory and discuss the implication of our results.

In this paper, we solve a problem of Kobayashi posed in (Complex Finsler vector bundles, American Mathematical Society, Providence, 1996) by introducing a Donaldson type functional on the space F + ( E ) of strongly pseudo-convex complex Finsler metrics on <i>E</i>a holomorphic vector bundle over a closed Khler manifold <i>M</i>. This Donaldson type functional is a generalization in the complex Finsler geometry setting of the original Donaldson functional and has FinslerEinstein metrics on <i>E</i> as its only critical points, at which this functional attains the absolute minimum.

The advance of technology facilitates the collection of statistical data. Flexible and refined statistical models are widely sought in a large array of statistical problems. The question arises frequently whether or not a family of parametric or nonparametric models fit adequately the given data. In this paper we give a selective overview on nonparametric inferences using generalized likelihood ratio (<i>GLR</i>) statistics. We introduce generalized likelihood ratio statistics to test various null hypotheses against nonparametric alternatives. The trade-off between the flexibility of alternative models and the power of the statistical tests is emphasized. Well-established Wilks phenomena are discussed for a variety of semi- and non-parametric models, which sheds light on other research using <i>GLR</i> tests. Anumber of open topics worthy of further study are given in a discussion section.