Communications in Mathematical Physics - Volume 259 by M. Aizenman (Chief Editor)

By M. Aizenman (Chief Editor)

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Math. Phys. 227, 131 (2002) 3. : Funct. Anal. Appl. 22, 200–213 (1989) 4. : Commun. Pure. Appl. Math. 47, 437 (1994) 5. : Phys. Rev. Lett. 84, 5106 (2000) 6. : Commun. Math. Phys. 213, 523 (2000) 7. : Phys. Lett. B515, 483– 492 (2001) 8. : Funct. Anal. Appl. 21, No 2, 46–63 (1987) 9. :“Theta Functions on Riemann Surfaces”. Lect. Notes in Mathematics 352, Berlin-Heidelberg-New York: Springer-Verlag, 1973 10. : M´em. pr´esent´es par divers savants a` l’Acad. , 33, (1908) 11. : The Schwarz function and its applications.

That is to say, if we tune the B-field suitably we can always hit a bad conformal field theory as we pass from one phase to another. Thus we may associate singular conformal field theories with phase transitions. The discriminant itself is generically reducible. The combinatorial structure of this reduction has been studied in detail in [31]. In particular, any time an m-dimensional face of the convex hull of the set A contains more than m + 1 points, the resulting linear relationship between these points yields a component of .

More precisely, ρ0 (z) = 3 1 0 z∈D . z ∈ C \ D ≡ Dc One may have in mind the model of all complex or mutually hermitian conjugated or normal matrices. Integrable Structure of the Dirichlet Boundary Problem 33 Note that the saddle point condition is imposed only for z inside the support of eigenvalues. Writing (cf. 57)) p V (z) = τk Bk (z) , k=1 multiplying Eq. 13) by Ak (z) and integrating it over the boundary of the support of eigenvalues, one finds that the domain D is such that the coefficients τk are moments of its complement with respect to the basis functions Ak and the higher moments (with numbers greater than p) vanish.

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