In this paper the theory of jets based on Weil's near points is applied to Lie equations and pseudogroups. Linear systems of partial differential equations are interpreted, in a canonical way, as distributions on the fibre bundles of invertible jets invariant under translations. We prove the two fundamental theorems for Lie equations and generalize the results of Rodrigues; a geometric correspondence between linear and nonlinear Lie equations is given, and the symbols of a linear Lie equation and its prolongations are canonically identified with the symbols of their attached nonlinear equations. From this fact we deduce that a linear Lie equation verifies the conditions of Goldsmichmidt's criterion on formal integrability if and only if its attached nonlinear Lie equation satisfies them locally. Finally, we define the Cartan 1-form on the fibre bundle of invertible jets and give a global form to the equivalence between the Lie and Cartan definitions of continuous groups. ᮊ
Background: Bridging strategies are necessary for patients with HCC, who are on a liver transplant waiting list. Few centers have started using TARE as a bridging strategy and Yttrium 90 (Y90) has been approved for use in HCC, however, it is very expensive and not practical to use as standard of care in countries, where there is no universal insurance coverage. 188 Rhenium (188Re) radio labeled lipiodol for the treatment of HCC is an alternative isotope with very similar properties to Y90. Methods: We did a retrospective review of 3 patients who were treated with 188Re radio labeled lipiodol as a bridging modality and then later on underwent cadaveric orthotopic liver transplant (OLT). Radiological, histopathological (HPE), alpha-feto protein (AFP) levels, survival and recurrence outcomes were analyzed. Results: First patient had 2 tumors and was transplanted 1 month after TARE. He developed recurrence 8 months post transplant. Second patient had single HCC and had a complete radiological response post TARE. He underwent OLT, 2 months post TARE and HPE also showed complete tumor necrosis. Third patient had 3 lesions and underwent TARE and had a partial radiological response. He underwent OLT, 9 months post TARE. There was a partial response on HPE. Second and third patients are doing well 15 and 11 months post transplant respectively. Conclusion: 188Re radio labeled lipiodol is a very effective bridging strategy for maintaining or down staging HCC in appropriately selected liver transplant waiting list patients.
In this paper we apply Weil bundle techniques to the study of formal integrability for systems of nonlinear partial differential equations. We clarify the role of the curvature and show that both the statement and proof of Goldschmidt's criterion on formal integrability are of algebraic nature. This fact allows us to obtain a version of this theorem which holds for smooth, algebraic, or analytic manifolds. Finally, we give a definition for the characteristic co-vectors of a system of partial differential equations and how their relationship with the Cauchy᎐Kowalevski normal form. ᮊ
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