广义Sierpinski集

本文考察了包括平面上的各种广义 Cantor集 ,Sierpinski集和包括某些连续不可微曲线在内的广义 Sierpinski集 .由相似变换 ,导出了它们的级数表达式 ,并利用它和字符串空间的对应关系 ,计算出它们的Hausdorff维数     

Nowhere zero 4‐flow in regular matroids

Jensen and Toft 8 conjectured that every 2‐edge‐connected graph without a K₅‐minor has a nowhere zero 4‐flow. Walton and Welsh 19 proved that if a coloopless regular matroid M does not have a minor in {M(K_3,3), M*(K₅)}, then M admits a nowhere zero 4‐flow. In this note, we prove that if a coloopless regular matroid M does not have a minor in {M(K₅), M*(K₅)}, then M admits a nowhere zero 4‐flow. Our result implies the Jensen and Toft conjecture. © 2005 Wiley Periodicals, Inc. J Graph Theory     

RECURSIVE BAIRE CLASSIFICATION AND SPEEDABLE FUNCTIONS

Using recursive variants of Baire notions of nowhere dense and meagre sets we study the topological size of speedable and infinitely often speedable functions in a machine-independent framework. We show that the set of speedable functions is not “small” whereas the set of infinitely often speedable functions is “large”. In this way we offer partial answers to a question in [4].     

On sums of Darboux and nowhere constant continuous functions

We show that the property

(P)

for every Darboux function there exists a continuous nowhere constant function such that is Darboux

follows from the following two propositions:

(A)

for every subset of of cardinality there exists a uniformly continuous function such that ,

(B)

for an arbitrary function whose image contains a non-trivial interval there exists an of cardinality such that the restriction of to is uniformly continuous,

which hold in the iterated perfect set model.

     

Porous sets that are Haar null, and nowhere approximately differentiable functions

We define a new notion of ``HP-small' set which implies that is both -porous and Haar null in the sense of Christensen. We show that the set of all continuous functions on which have finite unilateral approximate derivative at a point is HP-small, as well as its projections onto hyperplanes. As a corollary, the same is true for the set of all Besicovitch functions. Also, the set of continuous functions on which are Hölder at a point is HP-small.

     

A probabilistic property of Katsuura's continuous nowhere differentiable function

Katsuura's example of a continuous nowhere differentiable function is based on a fixed-point method. In this paper we use probability theory to study the increments of Katsuura's function.     

Nowhere‐Zero 5‐Flows On Cubic Graphs with Oddness 4

Tutte's 5‐flow conjecture from 1954 states that every bridgeless graph has a nowhere‐zero 5‐flow. It suffices to prove the conjecture for cyclically 6‐edge‐connected cubic graphs. We prove that every cyclically 6‐edge‐connected cubic graph with oddness at most 4 has a nowhere‐zero 5‐flow. This implies that every minimum counterexample to the 5‐flow conjecture has oddness at least 6.     

On differentiability properties of typical continuous functions and Haar null sets

Let () be the set of all continuous functions on which have a derivative ( , respectively) at least at one point . B. R. Hunt (1994) proved that is Haar null (in Christensen's sense) in .

In the present article it is proved that neither nor its complement is Haar null in . Moreover, the same assertion holds if we consider the approximate derivative (or the ``strong' preponderant derivative) instead of the ordinary derivative; these results are proved using a new result on typical (in the sense of category) continuous functions, which is of interest in its own right.

     

Dense-lineability in spaces of continuous functions

In this paper we provide a general method to prove that certain nonlinear families of continuous functions contain dense linear manifolds. An application is furnished.