Discrete Complex Analysis on Planar Quad-Graphs

In 1993 the 200th anniversary of George Green, a great physicist and mathematician's, birth was celebrated. His contribution to world's science is beyond the question. This can best be seen in the frequency of mentioning his name and quoting his works in other physicists' and mathematicians' works. Some of historians of science and researchers are deeply convinced that George Green together with Maxwell originated modern electromagnetism. George Green is also famous for his inventions as far as light, stress and accoustic theories are concerned but electromagnetism ows him most of all. Indeed, none of those who have ever dealt with mathematical electromagnetism will question George Green's part and position. In nearly each paper referring to it, Green's function or Green's identities are the terms that are mentioned or quoted. Without these notions contemporary numerical techniques such as finite element method (first Green's identity) or boundary element method (second Green's identity) or integral methods (Green's function) are hard to imagine.

A new TLM‐based concept analogous to the Green's function approach in classical electromagnetic theory is presented. It employs the procedure known as time domain diakoptics. The response of a TLM mesh to a unit impulse excitation at selected input points is interpreted as a discrete Green's function in the time domain, and the term ‘Johns matrix’ is proposed for this characteristic response. As in classical theory, the response of the mesh to an arbitrary excitation is found by convolving the excitation with its Johns matrix. This concept extends the generalized scattering parameter concept into the time dimension, opens unprecedented possibilities for partioning time domain problems at the field level, and permits large‐scale preprocessing of substructures for computer‐aided design. It also represents and elegant way of modelling broadband absorbing boundary conditions, imperfectly conducting walls and general frequency dispersive boundaries in the time domain.

This is the second paper in a set that defines the discrete Green's function (DGF). This paper focuses first on the turbulent boundary layer and presents two different methods to estimate the DGF. The long-element formulation defines the DGF with just two simple algebraic equations, but it is not quantitatively accurate for short element lengths. A short element correction is derived, but must be recalculated for each selection of flow parameters and element lengths. A similarity solution is derived that allows accurate estimates of the DGF diagonal elements for laminar boundary layers and for turbulent boundary layers discretized with short element lengths. To illustrate other methods to derive DGFs in more complex flows, a low-resolution DGF for laminar stagnation line boundary layers is determined using the skin-friction formulation combined with similarity solutions for two different thermal boundary conditions. Stagnation line flow is shown to be highly sensitive to the thermal boundary condition, and this can be analyzed effectively using the DGF.

A modal absorbing boundary condition (ABC) based on the discrete Green's function (DGF) is introduced and applied for termination of waveguides simulated by means of the finite-difference time-domain (FDTD) method. The differences between the developed approach and implementations already demonstrated in the literature are presented. By applying DGF, a consistent theoretical approach to modal ABC in the FDTD method is obtained. Developed ABC is applicable to propagating and evanescent modes yielding the reflection error below -250 dB. Because the perfect matching may be computationally expensive, results of investigations of the tradeoff between computational efficiency and accuracy of the developed method are presented.

In this article, we first introduce interpolation operator of projection type in three dimensions, from which we then derive weak estimates for tensor‐product block finite elements of degree m ≥ 1. Finally, using estimates for the discrete Green's function and the discrete derivative Green's function, we prove that both of the gradient and the function value of the finite element solution uh and the corresponding interpolant Πmu of projection type and degree m are superclose in the pointwise sense of the L∞‐norm. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

The heat fluxes across the turbine tip gap are characterized by large unsteady pressure gradients and shear from the viscous effects. The classical Newton heat convection equation, based on the turbine inlet total temperature, is inadequate. Previous research from our team relied on the use of the adiabatic wall temperature. In this paper, we propose an alternative approach to predict the convective heat transfer problem across the turbine rotor tip using discrete Green's functions (DGF). The linearity of the energy equation in the solid domain with constant thermal properties can be applied with a superposition technique to measure the data extracted from flow simulations to determine the Green's function distribution. The DGF is a matrix of coefficients that relate the temperature spatial (GF) distribution with the heat flux. This methodology is first applied to a backward facing step, validated using experimental data. The final aim of this paper is to demonstrate the method in the rotor turbine tip. A turbine stage at engine-like conditions was assessed using cfd software. The heat flux pulses were applied at different locations in the rotor tip geometry, and the increment of temperature in this zone was evaluated for different clearances, with a consequent variation of the DGF coefficients. Ultimately, a detailed uncertainty analysis of the methodology was included based on the magnitude of the heat flux pulses used in the DGF coefficients calculation and the uncertainty in the experimental measurements of the wall temperature.

A new discrete Green's function formulation of the finite difference time domain (DGF-FDTD) method has been developed which expresses the field response as a convolution of the current sources and the impulse response (which is referred to as the discrete Green's function) of the FDTD equation system. DGF-FDTD avoids the need for computation of free space nodes and absorbing boundary conditions. The method has been demonstrated by its application to a simple Yagi-Uda antenna with an efficiency comparable to the method of moments, which most of the time domain methods lack.

Written by a master of the subject, this text will be appreciated by students and experts for the way it develops the classical theory of functions of a complex variable in a clear and straightforward manner. In general, the approach taken here emphasises geometrical aspects of the theory in order to avoid some of the topological pitfalls associated with this subject. Thus, Cauchy's integral formula is first proved in a topologically simple case from which the author deduces the basic properties of holomorphic functions. Starting from the basics, students are led on to the study of conformal mappings, Riemann's mapping theorem, analytic functions on a Riemann surface, and ultimately the Riemann-Roch and Abel theorems. Profusely illustrated, and with plenty of examples, and problems (solutions to many of which are included), this book should be a stimulating text for advanced courses in complex analysis.

For matrix A, vector b and function f, the computation of vector f(A)b arises in many scientific computing applications. We consider the problem of obtaining quantum state |f> corresponding to vector f(A)b. There is a quantum algorithm to compute state |f> using eigenvalue estimation that uses phase estimation and Hamiltonian simulation e^{\im A t}. However, the algorithm based on eigenvalue estimation needs \poly(1/\epsilon) runtime, where \epsilon is the desired accuracy of the output state. Moreover, if matrix A is not Hermitian, \e^{\im A t} is not unitary and we cannot run eigenvalue estimation. In this paper, we propose a quantum algorithm that uses Cauchy's integral formula and the trapezoidal rule as an approach that avoids eigenvalue estimation. We show that the runtime of the algorithm is \poly(\log(1/\epsilon)) and the algorithm outputs state |f> even if A is not Hermitian.

The Cauchy integral formula is a fundamental and critical formula in the complex function, and it has a variety of applications in many aspects such as physics, engineering, and mathematics. This article focuses on the applications in the functions and two main theorems in the Cauchy integral formula. It explores the formula and provides corresponding proofs of the formula. To this end, the aim of the article is to use the Cauchy integral formula to cope with some difficult and complex integrals and functions. As a result, this article denotes to use the formula to simplify the complex and difficult problems and provides a practical and fast method. Hence, the article lists some examples to demonstrate the functions’ use of the Cauchy integral formula. At last, the article may serve as a reference for relevant research and survey in the field of the complex functions and provide insights into the Cauchy integral formula’s uses.

Holomorphic functions differ fundamentally from real differentiable functions: they are infinitely often (real and complex) differentiable (II.3, II.7), they even admit power series expansions (II.4), their entire behaviour is determined by their values on arbitrarily small open sets (II.4, II.7), and they satisfy powerful convergence theorems and estimates (II.5). All of these properties are consequences of the Cauchy integral theorem and the integral representations that arise from it (II.1–3). Meromorphic functions extend the class of holomorphic functions (II.6); their study leads to the notion of isolated singularities and to generalizations of power series obtained by allowing negative powers (Laurent series). In addition to the phenomena that occur in the theory of functions of one complex variable, a fundamentally new phenomenon enters the picture in higher dimensions: the simultaneous holomorphic continuation of all holomorphic functions from a given domain to a larger one (II.7). Here the Cauchy integral formula (in one variable!) is again the decisive tool.

Multiscale modeling of failure in metal matrix composites

This user-friendly textbook introduces complex analysis at the beginning graduate or advanced undergraduate level. Unlike other textbooks, it follows Weierstrass' approach, stressing the importance of power series expansions instead of starting with the Cauchy integral formula, an approach that illuminates many important concepts. This view allows readers to quickly obtain and understand many fundamental results of complex analysis, such as the maximum principle, Liouville's theorem, and Schwarz's lemma. The book covers all the essential material on complex analysis, and includes several elegant proofs that were recently discovered. It includes the zipper algorithm for computing conformal maps, as well as a constructive proof of the Riemann mapping theorem, and culminates in a complete proof of the uniformization theorem. Aimed at students with some undergraduate background in real analysis, though not Lebesgue integration, this classroom-tested textbook will teach the skills and intuition necessary to understand this important area of mathematics.

Graph simplification and matching using commute times

A lattice random walk model based on walkers wandering on discrete lattice of scattering space by discrete spatial and temporal step is presented. The discrete Green's function for infinite homogeneous turbid medium deduced by inductive method from the lattice random walk model is used with the method of images to describe the photon density distribution in two dimensional semi-infinite and finite slab homogenous turbid media such as tissue. The scattering matrix and linking matrix of the lattice random walk are obtained to describe the scattering and absorption processes in homogeneous and inhomogeneous turbid media. All the results obey the principle of causality.