By J. Gilbert, M. Murray

The purpose of this publication is to unite the probably disparate issues of Clifford algebras, research on manifolds, and harmonic research. The authors express how algebra, geometry, and differential equations play a extra primary position in Euclidean Fourier research. They then hyperlink their presentation of the Euclidean concept evidently to the illustration thought of semi-simple Lie teams.

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**Extra info for Clifford Algebras and Dirac Operators in Harmonic Analysis**

**Example text**

Observe that NAT has constants succ ◦ succ ◦ . . ◦ succ ◦ 0 where succ occurs zero or more times. In the exercises, n is the constant defined by induction by 1 = succ ◦ 0 and n + 1 = succ ◦ n. Composition in the category is composition of programs. Note that for composition to be well defined, if two composites of primitive operations are equal, then their composites with any other program must be equal. For example, we must have ord ◦ (chr ◦ succ ◦ ord) = ord ◦ (chr ◦ succ ◦ ord ◦ chr ◦ ord) as arrows from CHAR to NAT.

4 Under those conditions, a functional programming language L has a category structure C(L) for which: FPC–1 The types of L are the objects of C(L). FPC–2 The operations (primitive and derived) of L are the arrows of C(L). FPC–3 The source and target of an arrow are the input and output types of the corresponding operation. FPC–4 Composition is given by the composition constructor, written in the reverse order. FPC–5 The identity arrows are the do-nothing operations. The reader may wish to compare the discussion in [Pitt, 1986].

The function c is called composition, and if (g, f ) is a composable pair, c(g, f ) is written g ◦ f and is called the composite of g and f . If A is an object of C , u(A) is denoted idA , which is called the identity of the object A. C–1 The source of g ◦ f is the source of f and the target of g target of g. C–2 (h ◦ g) ◦ f = h ◦ (g ◦ f ) whenever either side is defined. C–3 The source and target of idA are both A. C–4 If f : A − → B, then f ◦ idA = idB ◦ f = f . ◦ f is the The significance of the fact that the composite c is defined on G2 is that g ◦ f is defined if and only if the source of g is the target of f .