The Trapezoidal Rule for Computing Cauchy Principal Value Integral on Circle

The composite trapezoidal rule for the computation of Cauchy principal value integral with the singular kernel cot((x-s)/2) is discussed. Our study is based on the investigation of the pointwise superconvergence phenomenon; that is, when the singular point coincides with some a priori known point, the convergence rate of the trapezoidal rule is higher than what is globally possible. We show that the superconvergence rate of the composite trapezoidal rule occurs at middle of each subinterval and obtain the corresponding superconvergence error estimate. Some numerical examples are provided to validate the theoretical analysis

Numerical Methods for Partial Differential Equations

These lecture notes are devoted to the numerical solution of partial differential equations (PDEs). PDEs arise in many fields and are extremely important in modeling of technical processes with applications in physics, biology, chemisty, economics, mechanical engineering, and so forth. In these notes, not only classical topics for linear PDEs such as finite differences, finite elements, error estimation, and numerical solution schemes are addressed, but also schemes for nonlinear PDEs and coupled problems up to current state-of-the-art techniques are covered. In the Winter 2020/2021 an International Class with additional funding from DAAD (German Academic Exchange Service) and local funding from the Leibniz University Hannover, has led to additional online materials such as links to youtube videos, which complement these lecture notes. This is the updated and extended Version 2. The first version was published under the DOI: https://doi.org/10.15488/9248

Seventh International Workshop on Simulation, 21-25 May, 2013, Department of Statistical Sciences, Unit of Rimini, University of Bologna, Italy. Book of Abstracts

Seventh International Workshop on Simulation, 21-25 May, 2013, Department of Statistical Sciences, Unit of Rimini, University of Bologna, Italy. Book of Abstract

Seventh International Workshop on Simulation, 21-25 May, 2013, Department of Statistical Sciences, Unit of Rimini, University of Bologna, Italy. Book of Abstracts

Seventh International Workshop on Simulation, 21-25 May, 2013, Department of Statistical Sciences, Unit of Rimini, University of Bologna, Italy. Book of Abstract

A Geometric Framework for Analyzing the Performance of Multiple-Antenna Systems under Finite-Rate Feedback

We study the performance of multiple-antenna systems under finite-rate feedback of some function of the current channel realization from a channel-aware receiver to the transmitter. Our analysis is based on a novel geometric paradigm whereby the feedback information is modeled as a source distributed over a Riemannian manifold. While the right singular vectors of the channel matrix and the subspace spanned by them are located on the traditional Stiefel and Grassmann surfaces, the optimal input covariance matrix is located on a new manifold of positive semi-definite matrices - specified by rank and trace constraints - called the Pn manifold. The geometry of these three manifolds is studied in detail; in particular, the precise series expansion for the volume of geodesic balls over the Grassmann and Stiefel manifolds is obtained. Using these geometric results, the distortion incurred in quantizing sources using either a sphere-packing or a random code over an arbitrary manifold is quantified. Perturbative expansions are used to evaluate the susceptibility of the ergodic information rate to the quality of feedback information, and thereby to obtain the tradeoff of the achievable rate with the number of feedback bits employed. For a given system strategy, the gap between the achievable rates in the infinite and finite-rate feedback cases is shown to be $O(2^{-\frac{2N_f}{N}})$ for Grassmann feedback and $O(2^{-\frac{N_f}{N}})$ for other cases, where $N$ is the dimension of the manifold used for quantization and $N_f$ is the number of bits used by the receiver per block for feedback. The geometric framework developed enables the results to hold for arbitrary distributions of the channel matrix and extends to all covariance computation strategies including, waterfilling in the short-term/long-term power constraint case, antenna selection and other rank-limited scenarios that could not be analyzed using previous probabilistic approaches

Multiphoton Quantum Optics and Quantum State Engineering

We present a review of theoretical and experimental aspects of multiphoton quantum optics. Multiphoton processes occur and are important for many aspects of matter-radiation interactions that include the efficient ionization of atoms and molecules, and, more generally, atomic transition mechanisms; system-environment couplings and dissipative quantum dynamics; laser physics, optical parametric processes, and interferometry. A single review cannot account for all aspects of such an enormously vast subject. Here we choose to concentrate our attention on parametric processes in nonlinear media, with special emphasis on the engineering of nonclassical states of photons and atoms. We present a detailed analysis of the methods and techniques for the production of genuinely quantum multiphoton processes in nonlinear media, and the corresponding models of multiphoton effective interactions. We review existing proposals for the classification, engineering, and manipulation of nonclassical states, including Fock states, macroscopic superposition states, and multiphoton generalized coherent states. We introduce and discuss the structure of canonical multiphoton quantum optics and the associated one- and two-mode canonical multiphoton squeezed states. This framework provides a consistent multiphoton generalization of two-photon quantum optics and a consistent Hamiltonian description of multiphoton processes associated to higher-order nonlinearities. Finally, we discuss very recent advances that by combining linear and nonlinear optical devices allow to realize multiphoton entangled states of the electromnagnetic field, that are relevant for applications to efficient quantum computation, quantum teleportation, and related problems in quantum communication and information.Comment: 198 pages, 36 eps figure