Algebraic Approaches to Partial Differential Equations

Partial differential equations are fundamental tools in mathematics,sciences and engineering. This book is mainly an exposition of the various algebraic techniques of solving partial differential equations for exact solutions developed by the author in recent years, with emphasis on physical equations such as: the Calogero-Sutherland model of quantum many-body system in one-dimension, the Maxwell equations, the free Dirac equations, the generalized acoustic system, the Kortweg and de Vries (KdV) equation, the Kadomtsev and Petviashvili (KP) equation, the equation of transonic gas flows, the short-wave equation, the Khokhlov and Zabolotskaya equation in nonlinear acoustics, the equation of geopotential forecast, the nonlinear Schrodinger equation and coupled nonlinear Schrodinger equations in optics, the Davey and Stewartson equations of three-dimensional packets of surface waves, the equation of the dynamic convection in a sea, the Boussinesq equations in geophysics, the incompressible Navier-Stokes equations and the classical boundary layer equations. In linear partial differential equations, we focus on finding all the polynomial solutions and solving the initial-value problems. Intuitive derivations of Lie symmetry of nonlinear partial differential equations are given. These symmetry transformations generate sophisticated solutions with more parameters from relatively simple ones. They are also used to simplify our process of finding exact solutions. We have extensively used moving frames, asymmetric conditions, stable ranges of nonlinear terms, special functions and linearizations in our approaches to nonlinear partial differential equations. The exact solutions we obtained usually contain multiple parameter functions and most of them are not of traveling-wave type.Comment: This is part of the monograph to be published by Springe

Miniaturized Transistors, Volume II

In this book, we aim to address the ever-advancing progress in microelectronic device scaling. Complementary Metal-Oxide-Semiconductor (CMOS) devices continue to endure miniaturization, irrespective of the seeming physical limitations, helped by advancing fabrication techniques. We observe that miniaturization does not always refer to the latest technology node for digital transistors. Rather, by applying novel materials and device geometries, a significant reduction in the size of microelectronic devices for a broad set of applications can be achieved. The achievements made in the scaling of devices for applications beyond digital logic (e.g., high power, optoelectronics, and sensors) are taking the forefront in microelectronic miniaturization. Furthermore, all these achievements are assisted by improvements in the simulation and modeling of the involved materials and device structures. In particular, process and device technology computer-aided design (TCAD) has become indispensable in the design cycle of novel devices and technologies. It is our sincere hope that the results provided in this Special Issue prove useful to scientists and engineers who find themselves at the forefront of this rapidly evolving and broadening field. Now, more than ever, it is essential to look for solutions to find the next disrupting technologies which will allow for transistor miniaturization well beyond silicon’s physical limits and the current state-of-the-art. This requires a broad attack, including studies of novel and innovative designs as well as emerging materials which are becoming more application-specific than ever before

ATMO 612: Atmospheric Physics II -Calculus Volume 3 (OpenStax Textbook)

Calculus Volume 3 covers parametric equations and polar coordinates, vectors, functions of several variables, multiple integration, and second-order differential equations

Computer simulation of a neurological model of learning

A number of problems in psychology and neurology are discussed to orient the reader to a theory of neural integration. The importance is stressed of the comprehensive temporal and spatial integration of sensory, motor and motivational aspects of brain function. It is argued that an extended neural template theory could provide such an integration. Contemporary solutions to the problem of neural integration are discussed. The available knowledge concerning the structure of neural tissue leads to the description of a theory of neural integration which might provide such neural templates. Integrating Neurons are suggested to be organised in columns or pools. Sub-sets of Neurons are formed as a result of excitation and can preferentially exchange excitation. These sub-sets or Linked Constellations would act as spatial templates to be matched with subsequent states of excitation. Inhibition acts to restrict spike emission to the most highly activated sub-sets. An initial computer simulation represented a simple learning or classical conditioning situation. In a variety of test computer runs the performance confirmed the main predictions of the theoretical model. The model was then extended to include representation of instrumental, consummatory, motivational and other aspects of behaviour. The intention of these further simulations was not to demonstrate the predictions of prior formulations but rather to use the computer to develop simulations progressively able to represent behaviour. Difficulties were encountered which were remedied by incorporating rhythmic mechanisms. A number of different versions of the model were explored. It was shown that the models could be trained to produced a different response to discriminative cues, when those cues had previously signalled different contingencies of obtaining the opportunity to perform consummatory behaviour. A published experiment on the Spiral Illusion is reported, which confirmed predictions suggested by the model.<p