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By Lupo D., Payne K.R.
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Extra info for A dual variational approach to a class of nonlocal semilinear Tricomi problems
To do this we must, if T = T(eu), insert a coefficient CT equal to the number of pairs (e\ w') for which T^u^ and T(e M) are isomorphic as rooted trees. As already remarked above we must have e' = e. Further if a: T(e,u') ~* T(e,u) i s a rooted tree isomorphism it must induce bijections Vj(T(euf)) -» Vj(T(eu)) for each y. The indexings then translate these bijections into permutations a- of ey such that the following diagram commutes. THE JACOBIAN CONJECTURE «A uh "* h-\ "* •*• u3 "* -> ••• -* e a Of, I eA e h-\i - eA_1 2 321 u2 "> °2^ Wh e l °\ i e2 -* «3 e!
Of integers from 1 to n with fx = /, and THE JACOBIAN CONJECTURE 325 The (a, b) entry of J(H) is Jab = DbHa, so we can write P Ldj = J fx,f2 ' ' ' Jfd-ufd ' Hfd 1 n ~ T 2 Jfx,f2 ' " Jfd_xJdJfd,aXa> where we have used Euler's formula for the cubic homogeneous polynomial Hu Finally oi(Ld) = 2PLd,f=ll(Jd),a-Xa9 f whence (6). " Suppose that we are given (T, v) and 5, where T G Td, v G V(T), and S G Td,. We write Tl 0S for the tree in Td+d, obtained by joining the root of S by a new edge to the vertex v of T, and rooting the resulting tree at the root of T.
Then [S:A] = e\/a(T) where e\ = e2\ • • -eh\ = Card S. With this information we can rewrite (10) as do "<** = ? 2(i) ;£)*•' where T varies over rooted trees (up to isomorphism) with d vertices, and ƒ varies over /-rooted labelings of T. ^'*,. This is the formula we sought. We recapitulate the result in the following theorem. 1) THEOREM. Let k be a commutative Q-algebra. , Xn — Hn) satisfies j(F) ( = d c t / ( F ) ) = l. Define Gt G *""" by G,(F) = X,. Then G, = ld>0GJd\ = Ht, and for d>2, < 12 > where Gj0) = X„ Gjl) ^-S^jï'w.
A dual variational approach to a class of nonlocal semilinear Tricomi problems by Lupo D., Payne K.R.