Maps on Ultrametric Spaces,Hensel's Lemma,and Differential Equations Over Valued Fields

We give a criterion for maps on ultrametric spaces to be surjective and to preserve spherical completeness. We show how Hensel's Lemma and the multidimensional Hensel's Lemma follow from our result. We give an easy proof that the latter holds in every henselian field. We also prove a basic infinite-dimensional Implicit Function Theorem. Further, we apply the criterion to deduce various versions of Hensel's Lemma for polynomials in several additive operators, and to give a criterion for the existence of integration and solutions of certain differential equations on spherically complete valued differential fields, for both valued D-fields in the sense of Scanlon, and differentially valued fields in the sense of Rosenlicht. We modify the approach so that it also covers logarithmic-exponential power series fields. Finally, we give a criterion for a sum of spherically complete subgroups of a valued abelian group to be spherically complete. This in turn can be used to determine elementary properties of power series fields in positive characteristic.     

The Dedekind-Mertens lemma and the contents of polynomials

We prove a sharpening of the Dedekind-Mertens Lemma relating the contents of two polynomials to the content of their product. We show that for a polynomial the integer in the Dedekind-Mertens Lemma may be replaced by the number of local generators of the content of . We also raise a question concerning the converse.

     

On Lelong-Bremermann Lemma

The main theorem of this note is the following refinement of the well-known Lelong-Bremermann Lemma:

Let be a continuous plurisubharmonic function on a Stein manifold of dimension Then there exists an integer , natural numbers , and analytic mappings such that the sequence of functions

converges to uniformly on each compact subset of .

In the case when is a domain in the complex plane, it is shown that one can take in the theorem above (Section 3); on the other hand, for -circular plurisubharmonic functions in the statement of this theorem is true with (Section 4). The last section contains some remarks and open questions.

     

A generalization of Baer’s Lemma

There is a classical result known as Baer’s Lemma that states that an R-module E is injective if it is injective for R. This means that if a map from a submodule of R, that is, from a left ideal L of R to E can always be extended to R, then a map to E from a submodule A of any R-module B can be extended to B; in other words, E is injective. In this paper, we generalize this result to the category q _ω consisting of the representations of an infinite line quiver. This generalization of Baer’s Lemma is useful in proving that torsion free covers exist for q _ω.      

Eppstein's bound on intersecting triangles revisited

Let S be a set of n points in the plane, and let T be a set of m triangles with vertices in S. Then there exists a point in the plane contained in Ω(m³/(n⁶log²n)) triangles of T. Eppstein [D. Eppstein, Improved bounds for intersecting triangles and halving planes, J. Combin. Theory Ser. A 62 (1993) 176-182] gave a proof of this claim, but there is a problem with his proof. Here we provide a correct proof by slightly modifying Eppstein's argument.     

A combinatorial proof of the Removal Lemma for Groups

Green [B. Green, A Szemerédi-type regularity lemma in abelian groups, with applications, Geom. Funct. Anal. 15 (2005) 340-376] established a version of the Szemerédi Regularity Lemma for abelian groups and derived the Removal Lemma for abelian groups as its corollary. We provide another proof of his Removal Lemma that allows us to extend its statement to all finite groups. We also discuss possible extensions of the Removal Lemma to systems of equations.     

On a condition for α-starlikeness

In this paper we give a sufficient condition for function to be α-starlike function and some of its applications. We use the techniques of convolution and differential subordinations.     

图的邻点可区别Ⅵ-全色数和邻点可区别E-全色数

利用穷染、递推的方法讨论了路、圈、完全图、轮和扇的邻点可区别Ⅵ-全染色.并用概率方法研究了一般图的邻点可区别E-全染色,给出了图的邻点可区别E-全色数的一个上界.即δ≥7且△≥28,则有x_(at)~e(G)≤10△,其中δ是图G的最小度,△是图G的最大度.