Occupation times of intervals until last passage times for spectrally negative Levy processes

In this paper, we derive the joint Laplace transforms of occupation times until its last passage times as well as its positions. Motivated by Baurdoux [2], the last times before an independent exponential variable are studied. By applying dual arguments, explicit formulas are derived in terms of new analytical identities from Loeffen et al. [12]

Fast Multipole Method For 3-D Helmholtz Equation In Layered Media

In this paper, a fast multipole method (FMM) is proposed to compute long-range interactions of wave sources embedded in 3-D layered media. The layered media Green's function for the Helmholtz equation, which satisfies the transmission conditions at material interfaces, is decomposed into a free space component and four types of reaction field components arising from wave reflections and transmissions through the layered media. The proposed algorithm is a combination of the classic FMM for the free space component and FMMs specifically designed for the four types reaction components, made possible by new multipole expansions (MEs) and local expansions (LEs) as well as the multipole-to-local translation (M2L) operators for the reaction field components. { Moreover, equivalent polarization source can be defined for each reaction component based on the convergence analysis of its ME. The FMMs for the reaction components, implemented with the target particles and equivalent polarization sources, are found to be much more efficient than the classic FMM for the free space component due to the fact that the equivalent polarization sources and the target particles are always separated by a material interface.} As a result, the FMM algorithm developed for layered media has a similar computational cost as that for the free space. Numerical results validate the fast convergence of the MEs and the $O(N)$ complexity of the FMM for interactions of low-frequency wave sources in 3-D layered media

Exponential convergence for multipole and local expansions and their translations for sources in layered media: 2-D acoustic wave

In this paper, we will first give a derivation of the multipole expansion (ME) and local expansion (LE) for the far field from sources in general 2-D layered media and the multipole-to-local translation (M2L) operator by using the generating function for Bessel functions. Then, we present a rigorous proof of the exponential convergence of the ME, LE, and M2L for 2-D Helmholtz equations in layered media. It is shown that the convergence of ME, LE, and M2L for the reaction field component of the Green's function depends on a polarized distance between the target and a polarized image of the source

Fast multipole method for 3-D Laplace equation in layered media

In this paper, a fast multipole method (FMM) is proposed for 3-D Laplace equation in layered media. The potential due to charges embedded in layered media is decomposed into a free space component and four types of reaction field components, and the latter can be associated with the potential of a polarization source defined for each type. New multipole expansions (MEs) and local expansions (LEs), as well as the multipole to local (M2L) translation operators are derived for the reaction components, based on which the FMMs for reaction components are then proposed. The resulting FMM for charge interactions in layered media is a combination of using the classic FMM for the free space components and the new FMMs for the reaction field components. With the help of a recurrence formula for the run-time computation of the Sommerfeld-type integrals used in M2L translation operators, pre-computations of a large number of tables are avoided. The new FMMs for the reaction components are found to be much faster than the classic FMM for the free space components due to the separation of equivalent polarization charges and the associated target charges by a material interface. As a result, the FMM for potential in layered media costs almost the same as the classic FMM in the free space case. Numerical results validate the fast convergence of the MEs for the reaction components, and the O(N) complexity of the FMM with a given truncation number p for charge interactions in 3-D layered media.Comment: arXiv admin note: text overlap with arXiv:1902.0513

Constraints on the Λ\LambdaCDM model with redshift tomography

Recently released Planck data favor a lower value of the Hubble constant and a higher value of the fraction matter density in the standard $\Lambda$CDM model, which are discrepant with some of the low-redshift measurements. Within the context of this cosmology, we examine the consistency of the estimated values for the Hubble constant and fraction matter density with redshift tomography. Using the SNe Ia, Hubble parameter, BAO and CMB data, which are divided into three bins, we find no statistical evidence for any tension in the three redshift bins, although there exists a 1.4$\sigma$ deviation of the Hubble constant in the middle redshift from the one in the high redshift bin.Comment: 18 pages,3 figure

Probing the dynamical behavior of dark energy

We investigate dynamical behavior of the equation of state of dark energy $w_{de}$ by employing the linear-spline method in the region of low redshifts from observational data (SnIa, BAO, CMB and 12 $H(z)$ data). The redshift is binned and $w_{de}$ is approximated by a linear expansion of redshift in each bin. We leave the divided points of redshift bins as free parameters of the model, the best-fitted values of divided points will represent the turning positions of $w_{de}$ where $w_{de}$ changes its evolving direction significantly (if there exist such turnings in our considered region). These turning points are natural divided points of redshift bins, and $w_{de}$ between two nearby divided points can be well approximated by a linear expansion of redshift. We find two turning points of $w_{de}$ in $z\in(0,1.8)$ and one turning point in $z\in (0,0.9)$, and $w_{de}(z)$ could be oscillating around $w=-1$. Moreover, we find that there is a $2\sigma$ deviation of $w_{de}$ from -1 around $z=0.9$ in both correlated and uncorrelated estimates.Comment: Accepted by JCAP; 16 pages, 3 figure

Updated reduced CMB data and constraints on cosmological parameters

We obtain the reduced CMB data $\{l_A, R, z_*\}$ from WMAP9, WMAP9+BICEP2, Planck+WP and Planck+WP+BICEP2 for the $\Lambda$CDM and $w$CDM models with or without spatial curvature. We then use these reduced CMB data in combination with low-redshift observations to put constraints on cosmological parameters. We find that including BICEP2 results in a higher value of the Hubble constant especially when the equation of state of dark energy and curvature are allowed to vary. For the $\Lambda$CDM model with curvature, the estimate of the Hubble constant with Planck+WP+Lensing is inconsistent with the one derived from Planck+WP+BICEP at about 1.3 $\sigma$ confidence level.Comment: arXiv admin note: text overlap with arXiv:1312.430

Designing Markets for Daily Deals

Daily deals platforms such as Amazon Local, Google Offers, GroupOn, and LivingSocial have provided a new channel for merchants to directly market to consumers. In order to maximize consumer acquisition and retention, these platforms would like to offer deals that give good value to users. Currently, selecting such deals is done manually; however, the large number of submarkets and localities necessitates an automatic approach to selecting good deals and determining merchant payments. We approach this challenge as a market design problem. We postulate that merchants already have a good idea of the attractiveness of their deal to consumers as well as the amount they are willing to pay to offer their deal. The goal is to design an auction that maximizes a combination of the revenue of the auctioneer (platform), welfare of the bidders (merchants), and the positive externality on a third party (the consumer), despite the asymmetry of information about this consumer benefit. We design auctions that truthfully elicit this information from the merchants and maximize the social welfare objective, and we characterize the consumer welfare functions for which this objective is truthfully implementable. We generalize this characterization to a very broad mechanism-design setting and give examples of other applications.Comment: This is the full version of the paper appearing at WINE 2013 Proceedings of the Ninth Conference on Web and Internet Economics (WINE-13). 201

On the Secrecy Unicast Throughput Performance of NOMA Assisted Multicast-Unicast Streaming With Partial Channel Information

This paper considers a downlink single-cell non-orthogonal multiple access (NOMA) network with uniformly deployed users, while a mixed multicast and unicast traffic scenario is taken into account. By guaranteeing the quality of service (QoS) of the multicast traffic, the multicast outage and secrecy unicast throughput performance is evaluated. In particular, two types of partial channel state information (CSI), namely the imperfect CSI and CSI based on second order statistics (SOS), are investigated in the analytical framework. For both two cases, the closed-form approximations for the multicast outage probability and secrecy unicast throughput are derived except that the approximation for the secrecy unicast throughput in SOS-based CSI only concerns two users. As to the multicast outage probability, the simulation results demonstrate that the NOMA scheme considered in both two cases achieves superior performance compared to the traditional orthogonal multiple access (OMA) scheme, while for the secrecy unicast throughput, the NOMA scheme shows great advantages over the OMA scheme in good condition (high signal-to-noise ratio (SNR)), but is inferior to the OMA scheme in poor condition (low SNR). Moreover, both two cases achieve similar performance except that the NOMA scheme with imperfect CSI obtains larger secrecy unicast throughput than that based on SOS under high SNR condition. Finally, the provided numerical results also confirm that the derived approximations of both two cases for the multicast outage probability and secrecy unicast throughput match well with the Monte Carlo simulations.Comment: 14 pages, 5 figure

A Note On Intrinsic Regularization Method

There exist certain intrinsic relations between the ultraviolet divergent graphs and the convergent ones at the same loop order in renormalizable quantum field theories. Whereupon we may establish a new method, the intrinsic regularization method, to regulate those divergent graphs. In this note, we present a proposal, the inserter proposal, to the method. The $\phi^4$ theory and QED at the one loop order are dealt with in some detail. Inserters in the standard model are given. Some applications to SUSY-models are also made at the one loop order.Comment: 15 pages, LaTex, no figure