An expansion of the type (@(&ij &(&n~&() = (fr(' (& pl&(&2l&0(@(&p~' @(&n~&p (x, ) Xi is derived, where yi -fl, ci] are labels for infinite-dimensional symmetric tensor representations of the Euclidean conformal group 0 (2@+1, 1), X, =[ l, , -ci], the constants C (y i) are real, and Q)( and u)x have the properties of vacuum expectation values of field products. The starting point is an infinite set of coupled nonlinear integral equations for Euclidean Green's functions in 2h space-time dimensions of the type written some 15 years ago by Fradkin and Symanzik. The Green's functions of the corresponding Gell-Mann-Low limit theory are expanded in conformal partial waves. The dynamical equations imply the existence of poles and factorization of residues in the partial waves as functions of the representation parameters. In proving the validity of the expansion we use some differential relations between partially equivalent exceptional representations of 0& (2 h + 1, 1), established in an earlier paper. This work completes the group-theoretical derivation of the vacuum operator-product expansion undertaken by Mack in 1973.

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ABSTRACT:Lakova, I. and Petrowa, S. 2013. Towards a standard Tithonian to Valanginian calpionellid zonation of the Tethyan Realm. Acta Geologica Polonica, 63 (2), 201-221. Warszawa.A calpionellid zonal scheme is proposed for the Tithonian through Valanginian pelagic carbonates of the Western Balkan Unit, based on the vertical ranges of 57 chitinoidellid and calpionellid species recognized. This zonal scheme consists of calpionellid zones that are widely accepted in the Mediterranean Realm, such as the zones of Chitinoidella, Praetintinnopsella, Crassicollaria, Calpionella, Calpionellopsis, Calpionellites and Tintinnopsella. Subzonal divisions are comparable to those in the Carpathians. Direct correlations between ammonite and calpionellid ranges suggest that the base of the Upper Tithonian corresponds to the FO of Chitinoidella boneti; that of the Upper Berriasian to the FO of the genus Calpionellopsis; and the base and top of the Lower Valanginian to the FO and LO of the genus Calpionellites respectively. Correlations of the calpionellid zonation in the study area with zonations in other areas are discussed.

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