Hexagon OPE Resummation and Multi-Regge Kinematics

We analyse the OPE contribution of gluon bound states in the double scaling limit of the hexagonal Wilson loop in planar N=4 super Yang-Mills theory. We provide a systematic procedure for perturbatively resumming the contributions from single-particle bound states of gluons and expressing the result order by order in terms of two-variable polylogarithms. We also analyse certain contributions from two-particle gluon bound states and find that, after analytic continuation to the $2\to 4$ Mandelstam region and passing to multi-Regge kinematics (MRK), only the single-particle gluon bound states contribute. From this double-scaled version of MRK we are able to reconstruct the full hexagon remainder function in MRK up to five loops by invoking single-valuedness of the results.Comment: 29 pages, 3 figures, 4 ancillary file

An extended Stein-type covariance identity for the Pearson family with applications to lower variance bounds

For an absolutely continuous (integer-valued) r.v. $X$ of the Pearson (Ord) family, we show that, under natural moment conditions, a Stein-type covariance identity of order $k$ holds (cf. [Goldstein and Reinert, J. Theoret. Probab. 18 (2005) 237--260]). This identity is closely related to the corresponding sequence of orthogonal polynomials, obtained by a Rodrigues-type formula, and provides convenient expressions for the Fourier coefficients of an arbitrary function. Application of the covariance identity yields some novel expressions for the corresponding lower variance bounds for a function of the r.v. $X$, expressions that seem to be known only in particular cases (for the Normal, see [Houdr\'{e} and Kagan, J. Theoret. Probab. 8 (1995) 23--30]; see also [Houdr\'{e} and P\'{e}rez-Abreu, Ann. Probab. 23 (1995) 400--419] for corresponding results related to the Wiener and Poisson processes). Some applications are also given.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ282 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

A Hybrid Boundary Element Method for Elliptic Problems with Singularities

The singularities that arise in elliptic boundary value problems are treated locally by a singular function boundary integral method. This method extracts the leading singular coefficients from a series expansion that describes the local behavior of the singularity. The method is fitted into the framework of the widely used boundary element method (BEM), forming a hybrid technique, with the BEM computing the solution away from the singularity. Results of the hybrid technique are reported for the Motz problem and compared with the results of the standalone BEM and Galerkin/finite element method (GFEM). The comparison is made in terms of the total flux (i.e. the capacitance in the case of electrostatic problems) on the Dirichlet boundary adjacent to the singularity, which is essentially the integral of the normal derivative of the solution. The hybrid method manages to reduce the error in the computed capacitance by a factor of 10, with respect to the BEM and GFEM

Optimization of Patterned Surfaces for Improved Superhydrophobicity Through Cost-Effective Large-Scale Computations

The growing need for creating surfaces with specific wetting properties, such as superhyrdophobic behavior, asks for novel methods for their efficient design. In this work, a fast computational method for the evaluation of patterned superhyrdophobic surfaces is introduced. The hydrophobicity of a surface is quantified in energy terms through an objective function. The increased computational cost led to the parallelization of the method with the Message Passing Interface (MPI) communication protocol that enables calculations on distributed memory systems allowing for parametric investigations at acceptable time frames. The method is demonstrated for a surface consisting of an array of pillars with inverted conical (frustum) geometry. The parallel speedup achieved allows for low cost parametric investigations on the effect of the fine features (curvature and slopes) of the pillars on the superhydophobicity of the surface and consequently for the optimization of superhyrdophobic surfaces.Comment: 18 pages, 18 figure