Rook theory and cycle-counting permutation statistics

A statistic is found to combinatorially generate the cycle-counting q-hit numbers, defined algebraically by Haglund [Adv. in Appl. Math. 17 (1996) 408–459]. We then define the notion of a cycle-Mahonian pair of statistics (generalizing that of a Mahonian statistic), and show that our newly discovered statistic is part of such a pair. Finally, we note a second example of a cycle-Mahonian pair of statistics which leads us to define the stronger property of being a cycle-Euler–Mahonian pair.     

Ordinary and strong density continuity of complex analytic functions

In the paper we prove that the complex analytic functions are (ordinarily) density continuous. This stays in contrast with the fact that even such a simple function asG:²²,G(x,y)=(x,y ³ ), is not density continuous [1]. We will also characterize those analytic functions which are strongly density continuous at the given pointa . From this we conclude that a complex analytic functionf is strongly density continuous if and only iff(z)=a+bz, wherea, b andb is either real or imaginary.     

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.

     

On cardinal interpolation by Gaussian radial-basis functions: Properties of fundamental functions and estimates for Lebesgue constants

Suppose λ is a positive number. Basic theory of cardinal interpolation ensures the existence of the Gaussian cardinal functionL _λ(x)

Shallow arch model. An asymptotic approach

We examine an asymptotic expansion of the time dependent displacement field of a solid, three-dimensional elastic arch. We show that it is possible to derive from the principles of virtual work four different models of arches. We also show that the approximate equations of motion are universal within a class of homogeneous, hyperelastic materials provided that the deformations of the solid arch are not too large.     

A result about a selection problem of Michael

It is shown that a continuum that is an space in the sense of Michael must be hereditarily decomposable. This improves known results, thereby providing more evidence that such continua must be dendrites.

     

The harmonic and geometric mean of vectors

New vector operations are defined that generalize the classical harmonic and geometric means of scalars. The definition of the harmonic mean involves an explicit vector formula, whereas the geometric mean of two vectors is defined by an iterative procedure. Two special cases are described in some detail, the case of 2-vectors which are equivalent to the complex numbers, and the case of 4-vectors which are a representation of the quaternions.     

Graph family operations

In previous papers, Catlin introduced four functions, denoted , , , and , between sets of finite graphs. These functions proved to be very useful in establishing properties of several classes of graphs, including supereulerian graphs and graphs with nowhere zero k-flows for a fixed integer k3. Unfortunately, a subtle error caused several theorems previously published in Catlin (Discrete Math. 160 (1996) 67–80) to be incorrect. In this paper we correct those errors and further explore the relations between these functions, showing that there is a sort of duality between them and that they act as inverses of one another on certain sets of graphs.     

A note on class one graphs with maximum degree six

It is known that there are class two graphs with Δ=6 which can be embedded in a surface Σ with Euler characteristic χ(Σ)?0. However, it is unknown whether there are class two graphs on the projective plane or on the plane with Δ=6. In this paper, we prove that every graph with Δ=6 is class one if it can be embedded in a surface with Euler characteristic at least -3 and is C₃-free, or C₄-free or if it can be embedded in a surface with Euler characteristic at least -1 and is C₅-free. This generalizes Zhou's results in [G. Zhou, A note on graphs of class I, Discrete Math. 263 (2003) 339-345] on planar graphs.