By Carl B. Boyer, Uta C. Merzbach, Isaac Asimov

Boyer and Merzbach distill millions of years of arithmetic into this attention-grabbing chronicle. From the Greeks to Godel, the maths is exceptional; the forged of characters is individual; the ebb and stream of rules is in all places glaring. And, whereas tracing the improvement of ecu arithmetic, the authors don't put out of your mind the contributions of chinese language, Indian, and Arabic civilizations. surely, this is—and will lengthy remain—a vintage one-volume heritage of arithmetic and mathematicians who create it.

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REMARK. 1. See § 3 of [17] for a proof of the fact that £ is a twisting cochain, and and that C i F{C). F{C). The The statement that £ is a twisting cochain is synonymous with the statement that the differential of ^(C) T(C) is self-annihilating. 4. Let A be a chain-complex. Then A will be called an A (oo) -algebra if it is equipped with maps //;: A11 — —•» A A of of degree i — — 22satisfying satisfying the the identity identity n n—k +A+fc Y. /i* ® ® i"in-fck-Ax) == oo E £(-i)* £ ( - i ) * + A + f c Vw*+i - * + i o (iA ®fik A == l Jb = l A where jli fli = d: A -+ —• A; A (oo) -algebra, (A, (A, {fii}), {//;}), the associated bar construction is defined to be the the Given an A .

A(oo)-coalgebras In this chapter we obtain formulas for the coproduct structure (and higher coproduct structure) of the cobar construction of a space in terms (essentially) of chain-level descriptions of Steenrod operations on the space. These are applied to determining the coproduct structure of the total space of fibrations. In this section we will define and develop some of the properties of yi(oo)-coalgebras and A (oo) -algebras. A(oo)-coalgebras represent a homotopy-invariant version of coassociative coalgebras.

If A is a DGA-algebra, the transposition map defines a canonical isomorphism of chain-complexes T: A^x,{jil}C = C x A, where C is the DGA coalgebra with the same underlying chain-complex as C, but whose coproduct A c = T o Ac and A c is the coproduct of C. Here is the twisted tensor product formed with respect to the coalgebra structure of C and the A (oo) -algebra structure of A. Note that a:(Co) = 0 so that the O-dimensional components of the coproduct, A, have no effect on the formula for b(x) or the definition of a twisting cochain.

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