There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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Table Of Integrals Series And Products

Real and Complex Analysis

Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

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Methods Of Modern Mathematical Physics

Noncommutative Functional Calculus

* Preface * Background Material * Computational Algebraic Analysis for Systems of Linear Constant Coefficients Differential Equations * The Cauchy-Fueter Systems and its Variations * Special First Order Systems in Clifford Analysis * Some First Order Linear Operators in Physics * Open Problems and Avenues for Further Research * References * Index

Analysis of Dirac Systems and Computational Algebra

1 Introduction.- 2 Slice monogenic functions.- 3 Functional calculus for n-tuples of operators.- 4 Quaternionic Functional Calculus.- 5 Appendix: The Riesz-Dunford functional calculus.- Bibliography.- Index.

Noncommutative Functional Calculus: Theory and Applications of Slice Hyperholomorphic Functions

Clifford algebras will be the setting in which we will work throughout this book. They were introduced under the name of geometric algebras by Clifford in 1878. Since then, several people have extensively studied them and nowadays there are, in the literature, several possible ways to introduce Clifford algebras: for example one can use exterior algebras, or present them as a quotient of a tensor algebra or by means of a universal property (see [23], [31], [34], or [75] for a survey on the various possible definitions). In this book, we will adopt an equivalent but more direct approach, using generators and relations.

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Irene Sabadini

Papers

Noncommutative Functional Calculus

Analysis of Dirac Systems and Computational Algebra

Noncommutative Functional Calculus: Theory and Applications of Slice Hyperholomorphic Functions

Slice monogenic functions

Extension results for slice regular functions of a quaternionic variable