Recursive Representations of Arbitrary Virasoro Conformal Blocks

We derive recursive representations in the internal weights of N-point Virasoro conformal blocks in the sphere linear channel and the torus necklace channel, and recursive representations in the central charge of arbitrary Virasoro conformal blocks on the sphere, the torus, and higher genus Riemann surfaces in the plumbing frame.Comment: 39 pages, 8 figures, v2: comments on references added, reference added, typos corrected, v3: comments on the relation between the plumbing and the Schottky parameters added, v4: typos correcte

A Study of Different Modeling Choices For Simulating Platelets Within the Immersed Boundary Method

The Immersed Boundary (IB) method is a widely-used numerical methodology for the simulation of fluid-structure interaction problems. The IB method utilizes an Eulerian discretization for the fluid equations of motion while maintaining a Lagrangian representation of structural objects. Operators are defined for transmitting information (forces and velocities) between these two representations. Most IB simulations represent their structures with piecewise-linear approximations and utilize Hookean spring models to approximate structural forces. Our specific motivation is the modeling of platelets in hemodynamic flows. In this paper, we study two alternative representations - radial basis functions (RBFs) and Fourier-based (trigonometric polynomials and spherical harmonics) representations - for the modeling of platelets in two and three dimensions within the IB framework, and compare our results with the traditional piecewise-linear approximation methodology. For different representative shapes, we examine the geometric modeling errors (position and normal vectors), force computation errors, and computational cost and provide an engineering trade-off strategy for when and why one might select to employ these different representations.Comment: 33 pages, 17 figures, Accepted (in press) by APNU

An Image Morphing Technique Based on Optimal Mass Preserving Mapping

©2007 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or distribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.DOI: 10.1109/TIP.2007.896637Image morphing, or image interpolation in the time domain, deals with the metamorphosis of one image into another. In this paper, a new class of image morphing algorithms is proposed based on the theory of optimal mass transport. The 2 mass moving energy functional is modified by adding an intensity penalizing term, in order to reduce the undesired double exposure effect. It is an intensity-based approach and, thus, is parameter free. The optimal warping function is computed using an iterative gradient descent approach. This proposed morphing method is also extended to doubly connected domains using a harmonic parameterization technique, along with finite-element methods

The Euler anomaly and scale factors in Liouville/Toda CFTs

The role played by the Euler anomaly in the dictionary relating sphere partition functions of four dimensional theories of class $\mathcal{S}$ and two dimensional nonrational CFTs is clarified. On the two dimensional side, this involves a careful treatment of scale factors in Liouville/Toda correlators. Using ideas from tinkertoy constructions for Gaiotto duality, a framework is proposed for evaluating these scale factors. The representation theory of Weyl groups plays a critical role in this framework.Comment: 55 pages, 16 figures; v2:fixed referencing & typos ; v3: argument about scale factors in Liouville/Toda now phrased in terms of stripped correlators, leading to a sharper conjecture (earlier version had some inaccurate statements). Presentation improved, typos fixed, refs added. I thank the anonymous referee for comments. Version accepted for publication in JHE

Feynman integrals and motives

This article gives an overview of recent results on the relation between quantum field theory and motives, with an emphasis on two different approaches: a "bottom-up" approach based on the algebraic geometry of varieties associated to Feynman graphs, and a "top-down" approach based on the comparison of the properties of associated categorical structures. This survey is mostly based on joint work of the author with Paolo Aluffi, along the lines of the first approach, and on previous work of the author with Alain Connes on the second approach.Comment: 32 pages LaTeX, 3 figures, to appear in the Proceedings of the 5th European Congress of Mathematic

Singular Liouville fields and spiky strings in \rr^{1,2} and SL(2,\rr)

The closed string dynamics in \rr^{1,2} and SL(2,\rr) is studied within the scheme of Pohlmeyer reduction. In both spaces two different classes of string surfaces are specified by the structure of the fundamental quadratic forms. The first class in \rr^{1,2} is associated with the standard lightcone gauge strings and the second class describes spiky strings and their conformal deformations on the Virasoro coadjoint orbits. These orbits correspond to singular Liouville fields with the monodromy matrixes $\pm I$. The first class in SL(2,\rr) is parameterized by the Liouville fields with vanishing chiral energy functional. Similarly to \rr^{1,2}, the second class in SL(2,\rr) describes spiky strings, related to the vacuum configurations of the SL(2,\rr)/U(1) coset model.Comment: 37 p. 6 fi