18,795 research outputs found
Bessel Integrals and Fundamental Solutions for a Generalized Tricomi Operator
Partial Fourier transforms are used to find explicit formulas for two
remarkable fundamental solutions for a generalized Tricomi operator. These
fundamental solutions reflect clearly the mixed type of the operator. In order
to prove these results, we establish explicit formulas for Fourier transforms
of some type of Bessel functions
Cities: Continuity, transformation and emergence
Book synopsis: This book applies ideas and methods from the complexity perspective to key concerns in the social sciences, exploring co-evolutionary processes that have not yet been addressed in the technical or popular literature on complexity. \ud
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Authorities in a variety of fields – including evolutionary economics, innovation and regeneration studies, urban modelling and history – re-evaluate their disciplines within this framework. The book explores the complex dynamic processes that give rise to socio-economic change over space and time, with reference to empirical cases including the emergence of knowledge-intensive industries and decline of mature regions, the operation of innovative networks and the evolution of localities and cities. Sustainability is a persistent theme and the practicability of intervention is examined in the light of these perspectives. \ud
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Specialists in disciplines that include economics, evolutionary theory, innovation, industrial manufacturing, technology change, and archaeology will find much to interest them in this book. In addition, the strong interdisciplinary emphasis of the book will attract a non-specialist audience interested in keeping abreast of current theoretical and methodological approaches through evidence-based and practical examples
The Surface Laplacian Technique in EEG: Theory and Methods
This paper reviews the method of surface Laplacian differentiation to study
EEG. We focus on topics that are helpful for a clear understanding of the
underlying concepts and its efficient implementation, which is especially
important for EEG researchers unfamiliar with the technique. The popular
methods of finite difference and splines are reviewed in detail. The former has
the advantage of simplicity and low computational cost, but its estimates are
prone to a variety of errors due to discretization. The latter eliminates all
issues related to discretization and incorporates a regularization mechanism to
reduce spatial noise, but at the cost of increasing mathematical and
computational complexity. These and several others issues deserving further
development are highlighted, some of which we address to the extent possible.
Here we develop a set of discrete approximations for Laplacian estimates at
peripheral electrodes and a possible solution to the problem of multiple-frame
regularization. We also provide the mathematical details of finite difference
approximations that are missing in the literature, and discuss the problem of
computational performance, which is particularly important in the context of
EEG splines where data sets can be very large. Along this line, the matrix
representation of the surface Laplacian operator is carefully discussed and
some figures are given illustrating the advantages of this approach. In the
final remarks, we briefly sketch a possible way to incorporate finite-size
electrodes into Laplacian estimates that could guide further developments.Comment: 43 pages, 8 figure
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