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The modelling of vibration problems is of great importance in engineering. A popular method of analysing such problems is the variational method. The simplest vibration model is represented using the example of a long rod. Two kinds of eigenfunctions orthogonality are proved and the corresponding norms are used to derive Green's function that gives rise to an analytical solution of the problem. The method can be easily generalized to a broad class of hyperbolic problems. [ABSTRACT FROM AUTHOR]

Chapter nineteen of the book "Green's Function Estimates for Lattice Schrödinger Operators and Applications," by J. Bourgain is presented. It explores the application of Green's function estimates to the construction of quasi-periodic solutions of non-linear Schrödinger operators. It also discusses induction hypothesis of the estimates.

Chapter thirteen of the book "Green's Function Estimates for Lattice Schrödinger Operators and Applications," by J. Bourgain is presented. It explores the point spectrum of lattice Schrödinger operators in multifrequency mathematical models. It also discusses the localization results of the operators.

Chapter fourteen of the book "Green's Function Estimates for Lattice Schrödinger Operators and Applications," by J. Bourgain is presented. It explores a matrix-valued cartan-type theorem with regards to the Green's function estimates of Schrödinger operators. It also discusses the perturbative quasiperiodic localization of the operators, resulting from the theorem.

Chapter nine of the book "Green's Function Estimates for Lattice Schrödinger Operators and Applications," by J. Bourgain is presented. It explores the facts regarding semialgebraic sets, as well as its polynomial equalities and inequalities. It also discusses the assumed diophantine and standard equidistribution considerations of semialgebraic sets.

Chapter sixteen of the book "Green's Function Estimates for Lattice Schrödinger Operators and Applications," by J. Bourgain is presented. It explores the application of green's function estimates to the kicked rotor problem. It also discusses a kicked rotor model considering the periodic sequence of time-dependent Schrödinger equations.

Chapter fifteen of the book "Green's Function Estimates for Lattice Schrödinger Operators and Applications," by J. Bourgain is presented. It explores the application of Green's function estimates to Jacobi matrices in Schrödinger operators with skew shift transformation. It also discusses the Anderson localization of the operators.

Chapter seven of the book "Green's Function Estimates for Lattice Schrödinger Operators and Applications," by J. Bourgain is presented. It explores the consequences for Lyapounov exponent and Green's function estimates of lattice Schrödinger operators. It also provides mathematical proofs and propositions of the subject matter.

Chapter eight of the book "Green's Function Estimates for Lattice Schrödinger Operators and Applications," by J. Bourgain is presented. It explores the details of the estimates of Schrödinger operators for small Lyapounov exponents. It also looks into the 1-frequency shift model for the estimates and their analytic extension.

Chapter eleven of the book "Green's Function Estimates for Lattice Schrödinger Operators and Applications," by J. Bourgain is presented. It explores the long-range models of the localized lattice Schrödinger operators, as well their extension to Hamiltonian forms. It also discusses the application of Green's function estimates to localization.

Chapter one of the book "Green's Function Estimates for Lattice Schrödinger Operators and Applications," by J. Bourgain is presented. It explores some of the spectral problems for lattice Schrödinger operators and their localization. It also discusses the long-time behavior of time-dependent Schrödinger operators.

In Chapter 3, the authors discuss scattering and emission by periodic rough surfaces. Two solution methods are used to solve this problem. The first is the T-matrix method, which makes use of Floquet mode expansions and the extended boundary condition. The second method uses a surface integral equation approach with MoM. The authors also describe Ewald's method for speeding up calculations of the Green's function in periodic medium. [ABSTRACT FROM PUBLISHER]

Many numerical methods exist to date for analyzing microwave and millimeter-wave passive structures. Among them, the spectral-domain analysis (SDA) is one of the most popular ones. It was developed in 1974. A SDA version for quasi-static analysis was also presented. SDA is basically a Fourier-transformed version of the integral equation method. However, as compared to the conventional space-domain integral equation method, the SDA has several advantages. Its formulation results in a system of coupled algebraic equations instead of coupled integral equations. Closed-form expressions can easily be obtained for the Green's functions. In addition, incorporation of physical conditions of analyzed structures via the so-called basis functions is achieved, and the obtained solutions are stationary. These features make the SDA numerically simpler and more efficient than the conventional integral equation method. SDA has been used extensively in analyzing planar transmission lines. In this chapter, we present a detailed formulation of the SDA for planar transmission lines in both quasi-static and dynamic domains. Other applications of SDA, to resonators and antenna and scattering problems, can be found in the references cited. [ABSTRACT FROM PUBLISHER]

Green's function is one of the most commonly used functions in solving microwave boundary-value problems. It represents a response (e.g., an electric field) due to a source of unit amplitude (e.g., a unit current). Green's function has been used in finding solutions for many microwave problems such as scattering and transmission line analysis. Its particular use in analyzing transmission lines is described in Chapters 6 (Variational Methods) and 7 (Spectral-Domain Method). [ABSTRACT FROM PUBLISHER]

This article focuses on the integral equations and wire modeling. The derivation of an integral equation for a perfect electric conductor is approached in various ways, but perhaps the most physically appealing is to begin with Maxwell's equations written in their source-integral form whose kernel is a Green's function for an electric source in an infinite medium. The source integral gives the secondary field also called induced, scattered, or radiated field caused by a current on the body flowing in response to a primary field. Two broad general classes of integral equations are encountered, depending on whether the unknown source occurs only under the integral called a first-kind Fredholm integral equation or also outside of a second-kind integral equation. In electromagnetics, a first-kind integral equation arises when the forcing function is an electric field, and a second-kind integral equation arises when the forcing function is a magnetic field.

This article discusses the Green's function methods. The Green's function method is a powerful technique for solving boundary-value problems. Green's function was named after George Green, who developed a general method to obtain solutions of Poisson's equation in potential theory. The Green's function method treats an arbitrary source for the potential as a linear superposition of weighted point sources. It then finds the potential as the corresponding linear superposition of the potentials produced by the point sources. Once the Green's function corresponding to the potential due to a point source is found, the potential produced by an arbitrary distribution of sources can he obtained easily.

The Standard Model is surely one of the major intellectual achievements of the twentieth century. In the late 1960s and early 1970s, decades of path-breaking experiments culminated in the emergence of a comprehensive theory of particle physics. This theory identifies the basic fundamental constituents of matter and describes all the forces of nature relevant at accessible energies – the strong, weak, and electromagnetic interactions. Science progresses in a much more muddled fashion than is often pictured in history books. This is especially true of theoretical physics, partly because history is written by the victorious. Consequently, historians of science often ignore the many alternate paths that people wandered down, the many false clues they followed, the many misconceptions they had. These alternate points of view are less clearly developed than the final theories, harder to understand and easier to forget, especially as these are viewed years later, when it all really does make sense. Thus reading history one rarely gets the feeling of the true nature of scientific development, in which the element of farce is as great as the element of triumph. The emergence of quantum chromodynamics, or QCD, is a wonderful example of the evolution from farce to triumph. During a very short period, a transition occurred from experimental discovery and theoretical confusion to theoretical triumph and experimental confirmation. We were lucky to have been young then, when we could stroll along the newly opened beaches and pick up the many beautiful shells that experiment had revealed. [ABSTRACT FROM AUTHOR]

In the spring of 1955 in Moscow there was a small conference on QED and elementary particle theory that took place at the Lebedev Physical Institute from March 31 through April 7. Among the participants were a few foreigners, including Ning Hu and Gunnar Källen. I remember it quite well as it was my first conference on quantum field theory (QFT) problems with scientists from abroad. My short contribution concerned finite Dyson transformations for renormalized Green's functions and matrix elements in QED. The central point of the conference was Lev Davydovich Landau's review talk “Basic Problems of Quantum Field Theory,” devoted to the ultraviolet behavior in local QFT. The point is that a few months earlier, the problem of short-distance behavior in QED was successfully attacked by Landau and his brilliant pupils Alesha Abrikosov and Isaak Khalatnikov. They managed to find a closed approximation to the Schwinger–Dyson equations for two propagators and the three-vertex function that was compatible with renormalizability and gauge invariance. Besides, this so-called three-gammas approximation admitted a solution in the massless limit that, in modern terms, was equivalent to the summation of leading ultraviolet logarithms. This solution had a peculiar feature that was controversial from the physical point of view (the “ghost-pole” in the renormalized photon propagator amplitude or Moscow-zero puzzle in the formal expression for the “physical electron charge”) that attracted attention and excited one's imagination. [ABSTRACT FROM AUTHOR]

The “island” in the title of this article means two things – the Soviet Union and my own mind. Partial isolation from the larger physics community had considerable effect on my work, both positive and negative. While the negative aspects of it are obvious, the good thing about isolation is that it gives independence, reduces the danger of being swept up by the intellectual “mass culture.” The beginning of modern field theory in Russia I would associate with the great work by Lev Davidovich Landau, Alexey Alexeevich Abrikosov, and Isak Markovich Khalatnikov. They studied the structure of the logarithmic divergences in QED and introduced the notion of the scale-dependent coupling. This scale dependence comes from the fact that the bare charge is screened by the cloud of virtual particles, and the larger this cloud is, the stronger the screening. They showed that at the scale r, the coupling has the form where a is the minimal cutoff scale. Similar, and in some respects stronger, results have been obtained by Gell-Mann and Low, who discovered the “renormalization group” equation: The catastrophic consequence of these results was that as a → 0 (no artificial cutoff) one obtains “Moscow zero” – total vanishing of the interaction. Immediately after that the search for different renormalizable theories was started in an attempt to find antiscreening (or as we would say today – asymptotic freedom). The only finding at that time had been the four-fermion interaction in two dimensions. [ABSTRACT FROM AUTHOR]

This is the history of the proof of renormalizability of gauge theories as I perceive it. It is a personal account. The importance of the proof of renormalizability is well known to all. Personally I have always felt that the proof was much more important than the actual construction of a model, the Standard Model. I felt that, once you knew the recipe, the road to a realistic description of Nature would be a matter of time and experiment. There some may disagree with me; I think, however, that a careful study of the recent history of high-energy physics will lead to this conclusion. Seldom has there been such a clear watershed. Old models, truly “dormant” (as Steven Weinberg put it), became credible and popular. Quantum chromodynamics came into being almost overnight. The proof of renormalizability also provided detailed technical methods such as, for example, suitable regularization methods, next to indispensable for any practical application of the theory. In longer perspective, the developments in supersymmetry and supergravity have been stimulated and enhanced by the renewed respectability of renormalizable field theory (including the absence of anomalies). If anything “turned the wheel,” as SLAC people have put it, it is this proof of renormalizability. Of course, the theory needs experimental verification, and whether people were convinced after the discovery of neutral currents, or after the discovery of charm, or W and Z, is another matter. [ABSTRACT FROM AUTHOR]

The table of contents for the book "Green's Function Estimates for Lattice Schrödinger Operators and Applications," by J. Bourgain is presented.