Degeneracy second main theorem for meromorphic mappings and moving hypersurfaces with truncated counting functions and applications

In this paper, we establish a new second main theorem for meromorphic mappings of [Formula: see text] into [Formula: see text] and moving hypersurfaces with truncated counting functions in the case, where the meromorphic mappings may be algebraically degenerate. A version of the second main theorem with weighted counting functions is also given. Our results improve the recent results on this topic. As an application, an algebraic dependence theorem for meromorphic mappings sharing moving hypersurfaces is given.

Degenerated second main theorem for holomorphic curves into algebraic varieties

In this paper, under the refinement of the subgeneral position, we give an improvement for the Second Main Theorem with truncated counting functions of algebraically non-degenerate holomorphic curves into algebraic varieties [Formula: see text] intersecting divisors in subgeneral position with some index.

Difference analogues of the second main theorem of zero-order meromorphic mappings for slowly moving targets

In this paper, we show some Second Main Theorems for zero-order meromorphic mappings intersecting slowly moving targets in [Formula: see text] by considering their [Formula: see text]-Casorati determinant. Our results are [Formula: see text]-difference analogues of Cartan’s Second Main Theorem for moving targets. As an application, we give an unicity theorem for meromorphic mappings of [Formula: see text] into [Formula: see text] under the growth condition “order [Formula: see text]”.

PERFECT SUBSETS OF GENERALIZED BAIRE SPACES AND LONG GAMES

We extend Solovay’s theorem about definable subsets of the Baire space to the generalized Baire spaceλλ, whereλis an uncountable cardinal withλ<λ= λ. In the first main theorem, we show that the perfect set property for all subsets ofλλthat are definable from elements ofλOrd is consistent relative to the existence of an inaccessible cardinal aboveλ. In the second main theorem, we introduce a Banach–Mazur type game of lengthλand show that the determinacy of this game, for all subsets ofλλthat are definable from elements ofλOrd as winning conditions, is consistent relative to the existence of an inaccessible cardinal aboveλ. We further obtain some related results about definable functions onλλand consequences of resurrection axioms for definable subsets ofλλ.