Applied Functional Analysis: Numerical Methods, Wavelet by Abul Hasan Siddiqi

By Abul Hasan Siddiqi

Advisor covers the most up-tp-date analytical and numerical equipment in infinite-dimensional areas, introducing fresh ends up in wavelet research as utilized in partial differential equations and sign and photo processing. For researchers and practitioners. contains index and references.

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Example text

Part I is mostly devoted to the preliminaries. In Section 1 we introduce the notion of a quasi-map (in rather general circumstances) and prove some of its basic properties. The reader familiar with any of the works [FFKM], [BG1], or [BFGM] may skip Section 1 and return for proofs of statements referred to in the subsequent sections. In Section 2 we collect some facts about Kashiwara’s flag schemes Gg ,p for a general Kac–Moody Lie algebra g , and study the quasi-map spaces from a curve C to Gg ,p .

In other words, perhaps we have not yet found the right language or framework to see the ultimate simplicity of nature. To get a better idea of what I am trying to say, let us consider GR as a description of gravity. To a mathematician this theory is beautifully simple but yet subtle. Moreover, it is highly nonlinear so that it is extremely complicated in its detailed implications. This is no doubt why it appeals to both Einstein and Penrose as a model theory. Is it not possible that something having the same inherent simplicity (and nonlinearity) can explain all of nature?

In special finite-dimensional cases these are now understood mathematically, and are related to classical ideas of integral geometry. These include the Penrose Transform, the Mukai Transform, the Nahm Transform, and the inverse scattering transform in soliton theory. In fact, solitons are a prominent part of all these dualities. However, the full dualities of string theory (or QFT) are infinite dimensional and nonlinear. All this suggests that a prominent theme of 21st century mathematics might be the development of a fully-fledged nonlinear Fourier Transform theory for function spaces.

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