Second main theorem and uniqueness theorem with moving targets on parabolic manifolds

On the Second Main Theorem of Cartan

The possibility of reversion of the inequality in the Second Main Theorem of Cartan in the theory of holomorphic curves in projective space is discussed. A new version of this theorem is proved that becomes an asymptotic equality for a class of holomorphic curves defined by solutions of linear differential equations. 2010 MSC: 30D35, 32A22.

Kamke's Uniqueness Theorem

A generalization of Kamke's uniqueness theorem in ordinary differential equations is obtained for the limit Cauchy problem, viz x'{t) = f(t, x(t)), x{t) -> x0 as 1J10, where / and x take values in an arbitrary normed linear space X and the initial point {t0, x0) is permitted to be on the boundary of the domain of/. Kamke's hypothesis that \\f(t,x)-f{t,y)\\ < <(>(\t-to\, ||x-,y||) is replaced by...

Picard’s Existence and Uniqueness Theorem

One of the most important theorems in Ordinary Di↵erential Equations is Picard’s Existence and Uniqueness Theorem for first-order ordinary di↵erential equations. Why is Picard’s Theorem so important? One reason is it can be generalized to establish existence and uniqueness results for higher-order ordinary di↵erential equations and for systems of di↵erential equations. Another is that it is a g...

The Second Main Theorem of Hypersurfaces in the Projective Space

A Remark on Dobrushin's Uniqueness Theorem

Ten years ago, Dobrushin [1] proved a beautiful result showing that under suitable hypotheses, a statistical mechanical lattice system interaction has a unique equilibrium state. In particular, there is no long range order, etc. see [6,7] for related material, Israel [4] for analyticity results and Gross [3] for falloff of correlations. There does not appear to have been systematic attempts to ...