Q2Q^2 evolution of chiral-odd twist-3 distribution e(x,Q2)e(x,Q^2)

We study the $Q^2$ dependence of the chiral-odd twist-3 distribution $e(x,Q^2)$.The anomalous dimension matrix for the corresponding twist-3 operators is calculated in the one-loop level. This study completes the calculation of the anomalous dimension matrices for all the twist-3 distributions together with the known results for the other twist-3 distributions $g_2(x,Q^2)$ and $h_L(x,Q^2)$. We also have confirmed that in the large $N_c$ limit the $Q^2$-evolution of $e(x,Q^2)$ is wholely governed by the lowest eigenvalue of the anomalous dimension matrix which takes a very simple analytic form as in the case of $g_2$ and $h_L$.Comment: 16 pages LaTeX, 4 postscript figure

Rational approximation of xnx^n

Let $E_{kk}^{(n)}$ denote the minimax (i.e., best supremum norm) error in approximation of $x^n$ on $[\kern .3pt 0,1]$ by rational functions of type $(k,k)$ with $k<n$. We show that in an appropriate limit $E_{kk}^{(n)} \sim 2\kern .3pt H^{k+1/2}$ independently of $n$, where $H \approx 1/9.28903$ is Halphen's constant. This is the same formula as for minimax approximation of $e^x$ on $(-\infty,0\kern .3pt]$.Comment: 5 page

Liouville/Toda central charges from M5-branes

We show that the central charge of the Liouville and ADE Toda theories can be reproduced by equivariantly integrating the anomaly eight-form of the corresponding six-dimensional N=(0,2) theories, which describe the low-energy dynamics of M5-branes.Comment: 9 page

Rational minimax approximation via adaptive barycentric representations

Computing rational minimax approximations can be very challenging when there are singularities on or near the interval of approximation - precisely the case where rational functions outperform polynomials by a landslide. We show that far more robust algorithms than previously available can be developed by making use of rational barycentric representations whose support points are chosen in an adaptive fashion as the approximant is computed. Three variants of this barycentric strategy are all shown to be powerful: (1) a classical Remez algorithm, (2) a "AAA-Lawson" method of iteratively reweighted least-squares, and (3) a differential correction algorithm. Our preferred combination, implemented in the Chebfun MINIMAX code, is to use (2) in an initial phase and then switch to (1) for generically quadratic convergence. By such methods we can calculate approximations up to type (80, 80) of $|x|$ on $[-1, 1]$ in standard 16-digit floating point arithmetic, a problem for which Varga, Ruttan, and Carpenter required 200-digit extended precision.Comment: 29 pages, 11 figure

An algorithm for real and complex rational minimax approximation

Rational minimax approximation of real functions on real intervals is an established topic, but when it comes to complex functions or domains, there appear to be no algorithms currently in use. Such a method is introduced here, the {\em AAA-Lawson algorithm,} available in Chebfun. The new algorithm solves a wide range of problems on arbitrary domains in a fraction of a second of laptop time by a procedure consisting of two steps. First, the standard AAA algorithm is run to obtain a near-best approximation and a set of support points for a barycentric representation of the rational approximant. Then a "Lawson phase" of iteratively reweighted least-squares adjustment of the barycentric coefficients is carried out to improve the approximation to minimax

Reciprocal-log approximation and planar PDE solvers

This article is about both approximation theory and the numerical solution of partial differential equations (PDEs). First we introduce the notion of {\em reciprocal-log} or {\em log-lightning approximation} of analytic functions with branch point singularities at points $\{z_k\}$ by functions of the form $g(z) = \sum_k c_k /(\log(z-z_k) - s_k)$, which have $N$ poles potentially distributed along a Riemann surface. We prove that the errors of best reciprocal-log approximations decrease exponentially with respect to $N$ and that exponential or near-exponential convergence (i.e., at a rate $O(\exp(-C N / \log N))$) also holds for near-best approximations with preassigned singularities constructed by linear least-squares fitting on the boundary. We then apply these results to derive a "log-lightning method" for numerical solution of Laplace and related PDEs in two-dimensional domains with corner singularities. The convergence is near-exponential, in contrast to the root-exponential convergence for the original lightning methods based on rational functions.Comment: 20 pages, 11 figure

Classical and Quantum Correlations of Scalar Field in the Inflationary Universe

We investigate classical and quantum correlations of a quantum field in the inflationary universe using a particle detector model. By considering the entanglement and correlations between two comoving detectors interacting with a scalar field, we find that the entanglement between the detectors becomes zero after their physical separation exceeds the Hubble horizon. Furthermore, the quantum discord, which is defined as the quantum part of total correlation, approaches zero on super horizon scale. These behaviors support appearance of classical nature of the quantum fluctuation generated during the inflationary era.Comment: 21 pages, accepted for publication in Phys. Rev.

Glaucomatous vertical vessel density asymmetry of the temporal raphe detected with optical coherence tomography angiography.

Changes in retinal vasculature and ocular circulation may play an important role in the glaucoma development and progression. We evaluated the vertical asymmetry across the temporal raphe of the deep retinal layer vessel density, using swept-source optical coherence tomography angiography (SS-OCTA), and its relationship with the central visual field (VF) loss. Thirty-four eyes of 27 patients with open-angle glaucoma were included. SS-OCTA macular scanning was performed within a 3 × 3 mm (300 × 300 pixels) volume, centred on the fovea. The relationships between the vertical asymmetrical deep retinal vessel density reduction (ADRVD) across the temporal raphe and various ocular parameters were analysed. Twenty-two glaucomatous eyes with ADRVDs had central VF loss. Contrarily, ADRVDs were not found in any of the 12 eyes without central VF loss. Thirteen eyes (59.1%) with central VF loss had ADRVDs topographically corresponding to the central VF loss and macular ganglion cell complex thinning. The glaucomatous eyes with ADRVDs exhibited inferior rather than superior central VF loss (P = 0.032). Thus, ADRVD specifically indicates the glaucomatous central visual loss. Further analysis of ADRVD may improve our understanding on glaucoma pathogenesis, offering new treatment insights

Rigid Limit in N=2 Supergravity and Weak-Gravity Conjecture

We analyze the coupled N=2 supergravity and Yang-Mills system using holomorphy, near the rigid limit where the former decouples from the latter. We find that there appears generically a new mass scale around g M_{pl} where g is the gauge coupling constant and M_{pl} is the Planck scale. This is in accord with the weak-gravity conjecture proposed recently. We also study the scale dependence of the gauge theory prepotential from its embedding into supergravity.Comment: 17 pages, minor correction