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By H. Fine, H. Thompson

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37, 180-184 (1951). L. OnigSik: ON THE CLASSIFICATION OF FIBER SPACES, Soy. Math. Dokl. 2, 1561-1564 (1961). L. Oni~ik: CONNECTIONS WITH ZERO CURVATURE AND THE DE RHAM THEOREM, Soy. Math. Dokl. 5, 1654-1657 (1964). L. Onig6ik: SOME CONCEPTS AND APPLICATIONS OF NON-ABELIAN COHOMOLOGY THEORY,Trans. Moscow Math. Soc. 17, 49-98 (1967). L. Oni~ik: ON COMPLETELY INTEGRABLE EQUATIONS ON HOMOGENOUS SPACES, Mat. Zametld 9 (14), 365-373 (1970) (russian). [R] H. R6hrl, Das Riemann-Hilbertsche Problem der Theorie der linearen Differentialgleichungen.

Stanton: THE HEAT EQUATION IN SEVERAL COMPLEX VARIABLES, Bull. Am. Mat. Soc. 11, 65-84 (1984). K. Tolpygo: ON TWO-DIMENSIONAL COHOMOLOGY THEORIES, Usp. Mat. Nauk. 27 (5), 251-252, (1972) (russian). K. Tolpygo: UNIVERSALITY OF NON-ABELIAN COHOMOLOGY THEORIES Mat. Sbomik 91 (2), 267-278 (1973) (russian). Tolpygo: TWO-DIMENSIONAL COHOMOLOGY AND SPECTRAL SEQUENCES IN NON-ABELIAN THEORY. in "Questions in group theory and homological algebra". Jaroslavl 1977 p. 156-197. (russian). [Y] K. Yosida: AN ERGODIC THEOREM ASSOCIATED WITH HARMONIC INTEGRALS, Proc.

Ii). PS(~) is linear in~ . (iii). The following formula holds 44 (7) PO(~) (iv). I6(P~(~)) - = P8 (~ - I8 (~) = ~. The following estimates hold. I IPr,£)(~)(x) ] (and ] IP ,8(~)(x) I IPad,~(T)(x)ll I) ! ~ (x)emcp~(x)Llxll <= ~(x)e 2mcpe(x)l xll Proof. (i),(ii) and (iii) follow from the definition. (iv follows from lemma i. 3. Lemma 3. Let T be a p-form. Then we have (8)r dPr,o(~) : Pr,0(~)/k~ + Pr,_0 (d~-(-l)pI(Pr,e (~)Ade'P0 (8) dPx, e(~) = ~AP /J,~ (~) + P X,8 (d~-I( d e ~/~ P ~/,8(~))) (8) ad dPad,9(~) = [8'Pad,8 (T)~ + Pad,[~(d~-I[de~'@'Pad,8 (~)3 Since we have dIr, o 0(~)=dO and dI r,e (~)=~0-I((-I)P~Ad0+ Proof.

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Coordinate Geometry by H. Fine, H. Thompson

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