High-order compact schemes for parabolic problems with mixed derivatives in multiple space dimensions

We present a high-order compact finite difference approach for a rather general class of parabolic partial differential equations with time and space dependent coefficients as well as with mixed second-order derivative terms in n spatial dimensions. Problems of this type arise frequently in computational fluid dynamics and computational finance. We derive general conditions on the coefficients which allow us to obtain a high-order compact scheme which is fourth-order accurate in space and second-order accurate in time. Moreover, we perform a thorough von Neumann stability analysis of the Cauchy problem in two and three spatial dimensions for vanishing mixed derivative terms, and also give partial results for the general case. The results suggest unconditional stability of the scheme. As an application example we consider the pricing of European Power Put Options in the multidimensional Black-Scholes model for two and three underlying assets. Due to the low regularity of typical initial conditions we employ the smoothing operators of Kreiss et al. to ensure high-order convergence of the approximations of the smoothed problem to the true solution

Algebraic Structures and Stochastic Differential Equations driven by Levy processes

We construct an efficient integrator for stochastic differential systems driven by Levy processes. An efficient integrator is a strong approximation that is more accurate than the corresponding stochastic Taylor approximation, to all orders and independent of the governing vector fields. This holds provided the driving processes possess moments of all orders and the vector fields are sufficiently smooth. Moreover the efficient integrator in question is optimal within a broad class of perturbations for half-integer global root mean-square orders of convergence. We obtain these results using the quasi-shuffle algebra of multiple iterated integrals of independent Levy processes.Comment: 41 pages, 11 figure

RG-improved single-particle inclusive cross sections and forward-backward asymmetry in ttˉt\bar t production at hadron colliders

We use techniques from soft-collinear effective theory (SCET) to derive renormalization-group improved predictions for single-particle inclusive (1PI) observables in top-quark pair production at hadron colliders. In particular, we study the top-quark transverse-momentum and rapidity distributions, the forward-backward asymmetry at the Tevatron, and the total cross section at NLO+NNLL order in resummed perturbation theory and at approximate NNLO in fixed order. We also perform a detailed analysis of power corrections to the leading terms in the threshold expansion of the partonic hard-scattering kernels. We conclude that, although the threshold expansion in 1PI kinematics is susceptible to numerically significant power corrections, its predictions for the total cross section are in good agreement with those obtained by integrating the top-pair invariant-mass distribution in pair invariant-mass kinematics, as long as a certain set of subleading terms appearing naturally within the SCET formalism is included.Comment: 55 pages, 14 figures, 6 table

Rank-based attachment leads to power law graphs

We investigate the degree distribution resulting from graph generation models based on rank-based attachment. In rank-based attachment, all vertices are ranked according to a ranking scheme. The link probability of a given vertex is proportional to its rank raised to the power -a, for some a in (0,1). Through a rigorous analysis, we show that rank-based attachment models lead to graphs with a power law degree distribution with exponent 1+1/a whenever vertices are ranked according to their degree, their age, or a randomly chosen fitness value. We also investigate the case where the ranking is based on the initial rank of each vertex; the rank of existing vertices only changes to accommodate the new vertex. Here, we obtain a sharp threshold for power law behaviour. Only if initial ranks are biased towards lower ranks, or chosen uniformly at random, we obtain a power law degree distribution with exponent 1+1/a. This indicates that the power law degree distribution often observed in nature can be explained by a rank-based attachment scheme, based on a ranking scheme that can be derived from a number of different factors; the exponent of the power law can be seen as a measure of the strength of the attachment

A General Mathematical Formulation for the Determination of Differential Leakage Factors in Electrical Machines with Symmetrical and Asymmetrical Full or Dead-Coil Multiphase Windings

This paper presents a simple and general mathematical formulation for the determination of the differential leakage factor for both symmetrical and asymmetrical full and dead-coil windings of electrical machines. The method can be applied to all multiphase windings and considers Görges polygons in conjunction with masses geometry in order to find an easy and affordable way to compute the differential leakage factor, avoiding the adoption of traditional methods that refer to the Ossanna's infinite series, which has to be obviously truncated under the bound of a predetermined accuracy. Moreover, the method described in this paper allows the easy determination of both the minimum and maximum values of the differential leakage factor, as well as its average value and the time trend. The proposed method, which does not require infinite series, is validated by means of several examples in order to practically demonstrate the effectiveness and the easiness of application of this procedure

The contribution of Fermi-2LAC blazars to the diffuse TeV-PeV neutrino flux

The recent discovery of a diffuse cosmic neutrino flux extending up to PeV energies raises the question of which astrophysical sources generate this signal. One class of extragalactic sources which may produce such high-energy neutrinos are blazars. We present a likelihood analysis searching for cumulative neutrino emission from blazars in the 2nd Fermi-LAT AGN catalogue (2LAC) using an IceCube neutrino dataset 2009-12 which was optimised for the detection of individual sources. In contrast to previous searches with IceCube, the populations investigated contain up to hundreds of sources, the largest one being the entire blazar sample in the 2LAC catalogue. No significant excess is observed and upper limits for the cumulative flux from these populations are obtained. These constrain the maximum contribution of the 2LAC blazars to the observed astrophysical neutrino flux to be $27 \%$ or less between around 10 TeV and 2 PeV, assuming equipartition of flavours at Earth and a single power-law spectrum with a spectral index of $-2.5$. We can still exclude that the 2LAC blazars (and sub-populations) emit more than $50 \%$ of the observed neutrinos up to a spectral index as hard as $-2.2$ in the same energy range. Our result takes into account that the neutrino source count distribution is unknown, and it does not assume strict proportionality of the neutrino flux to the measured 2LAC $\gamma$-ray signal for each source. Additionally, we constrain recent models for neutrino emission by blazars.Comment: 18 pages, 22 figure