Conformal Field Theories in Fractional Dimensions

We study the conformal bootstrap in fractional space-time dimensions, obtaining rigorous bounds on operator dimensions. Our results show strong evidence that there is a family of unitary CFTs connecting the 2D Ising model, the 3D Ising model, and the free scalar theory in 4D. We give numerical predictions for the leading operator dimensions and central charge in this family at different values of D and compare these to calculations of phi^4 theory in the epsilon-expansion.Comment: 11 pages, 4 figures - references updated - one affiliation modifie

A new microvertebrate fauna from the Middle Hettangian (early Jurassic) of Fontenoille (Province of Luxembourg, south Belgium)

A Lower Jurassic horizon from Fontenoille yielding fossil fish remains can be dated to the Middle Hettangian Liasicus zone on the basis of the early belemnite Schwegleria and the ammonite Alsatites Iciqueus francus. Hybodontiform sharks are represented by Hybodus reticularis, Lissodus sp„ Polxacrodus sp, and Neoselachians by Synechodus paludinensis nov. sp. and Synechodus streitzi, nov. sp. Earlier reports of a scyliorhinid are not confirmed; teeth of similar morphology to scyliorhinids seem to be juvenile variants of 5. paludinensis. Chimaeriform remains include Squaloraja sp., the earliest occurrence of the genus. The Actinopterygian fauna is introduced, comprising a palaeonisciform cf. Ptxcholepis, a possible late perleidiform cf. Platysiagum, the dapediid semionotiforms Dapedium and cf. Tetragonolepis, the pycnodontiform Eomesodon, halecomorphs cf. Furidae or Ophiopsidae, pholidophoriforms and/or Leptolepididae, and actinistians. Lepidosaur remains are also present

Discrete complex analysis on planar quad-graphs

We develop a linear theory of discrete complex analysis on general quad-graphs, continuing and extending previous work of Duffin, Mercat, Kenyon, Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our approach based on the medial graph yields more instructive proofs of discrete analogs of several classical theorems and even new results. We provide discrete counterparts of fundamental concepts in complex analysis such as holomorphic functions, derivatives, the Laplacian, and exterior calculus. Also, we discuss discrete versions of important basic theorems such as Green's identities and Cauchy's integral formulae. For the first time, we discretize Green's first identity and Cauchy's integral formula for the derivative of a holomorphic function. In this paper, we focus on planar quad-graphs, but we would like to mention that many notions and theorems can be adapted to discrete Riemann surfaces in a straightforward way. In the case of planar parallelogram-graphs with bounded interior angles and bounded ratio of side lengths, we construct a discrete Green's function and discrete Cauchy's kernels with asymptotics comparable to the smooth case. Further restricting to the integer lattice of a two-dimensional skew coordinate system yields appropriate discrete Cauchy's integral formulae for higher order derivatives.Comment: 49 pages, 8 figure

Functional Maps Representation on Product Manifolds

We consider the tasks of representing, analyzing and manipulating maps between shapes. We model maps as densities over the product manifold of the input shapes; these densities can be treated as scalar functions and therefore are manipulable using the language of signal processing on manifolds. Being a manifold itself, the product space endows the set of maps with a geometry of its own, which we exploit to define map operations in the spectral domain; we also derive relationships with other existing representations (soft maps and functional maps). To apply these ideas in practice, we discretize product manifolds and their Laplace--Beltrami operators, and we introduce localized spectral analysis of the product manifold as a novel tool for map processing. Our framework applies to maps defined between and across 2D and 3D shapes without requiring special adjustment, and it can be implemented efficiently with simple operations on sparse matrices.Comment: Accepted to Computer Graphics Foru

A note on dimer models and McKay quivers

We give one formulation of an algorithm of Hanany and Vegh which takes a lattice polygon as an input and produces a set of isoradial dimer models. We study the case of lattice triangles in detail and discuss the relation with coamoebas following Feng, He, Kennaway and Vafa.Comment: 25 pages, 35 figures. v3:completely rewritte

On the truncation of the harmonic oscillator wavepacket

We present an interesting result regarding the implication of truncating the wavepacket of the harmonic oscillator. We show that disregarding the non-significant tails of a function which is the superposition of eigenfunctions of the harmonic oscillator has a remarkable consequence: namely, there exist infinitely many different superpositions giving rise to the same function on the interval. Uniqueness, in the case of a wavepacket, is restored by a postulate of quantum mechanics

Comment on ``the Klein-Gordon Oscillator''

The different ways of description of the $S=0$ particle with oscillator-like interaction are considered. The results are in conformity with the previous paper of S. Bruce and P. Minning.Comment: LaTeX file, 5p

Approximation of conformal mappings by circle patterns

A circle pattern is a configuration of circles in the plane whose combinatorics is given by a planar graph G such that to each vertex of G corresponds a circle. If two vertices are connected by an edge in G, the corresponding circles intersect with an intersection angle in $(0,\pi)$. Two sequences of circle patterns are employed to approximate a given conformal map $g$ and its first derivative. For the domain of $g$ we use embedded circle patterns where all circles have the same radius decreasing to 0 and which have uniformly bounded intersection angles. The image circle patterns have the same combinatorics and intersection angles and are determined from boundary conditions (radii or angles) according to the values of $g'$ ($|g'|$ or $\arg g'$). For quasicrystallic circle patterns the convergence result is strengthened to $C^\infty$-convergence on compact subsets.Comment: 36 pages, 7 figure

A Decomposed Fourier-Motzkin Elimination Framework to Derive Vessel Capacity Models

Accurate Vessel Capacity Models (VCMs) expressing thetrade-off between different container types that can be stowed on containervessels are required in core liner shipping functions such as uptake-,capacity-, and network management. Today, simple models based on volume,weight, and refrigerated container capacity are used for these tasks,which causes overestimations that hamper decision making. Though previouswork on stowage planning optimization in principle provide finegrainedlinear Vessel Stowage Models (VSMs), these are too complexto be used in the mentioned functions. As an alternative, this papercontributes a novel framework based on Fourier-Motzkin Eliminationthat automatically derives VCMs from VSMs by projecting unneededvariables. Our results show that the projected VCMs are reduced byan order of magnitude and can be solved 20–34 times faster than theircorresponding VSMs with only a negligible loss in accuracy. Our frameworkis applicable to LP models in general, but are particularly effectiveon block-angular structured problems such as VSMs. We show similarresults for a multi-commodity flow problem

Polyhedral Analysis using Parametric Objectives

The abstract domain of polyhedra lies at the heart of many program analysis techniques. However, its operations can be expensive, precluding their application to polyhedra that involve many variables. This paper describes a new approach to computing polyhedral domain operations. The core of this approach is an algorithm to calculate variable elimination (projection) based on parametric linear programming. The algorithm enumerates only non-redundant inequalities of the projection space, hence permits anytime approximation of the output