The kth power of a graph is, by definition, the graph obtained by joining all vertices at distance less than or equal to k in the original graph. The size of the automorphism groups of the kth powers of the n-dimensional cube, 1 ::; k ::; n, arises naturally in an application of the second moment method for determining a probability threshold for bandwidth in hypercubes. In this paper we give a...

We present two short proofs of an identity found by Sylvester and rediscovered by Louck. The first proof is an elementary version of Knuth’s proof and is analogous to Macdonald’s proof of a related identity of Milne. The second is Sylvester’s own proof of his identity.

We determine in an elementary fashion the dimension of the space of innni-tesimal deformations of an arbitrary vector bundle on the complex projective line, showing its dependence on the precise form of the decomposition of the bundle under consideration.

We construct new monomorphisms between mapping class groups of surfaces. The first family of examples injects the mapping class group of a closed surface into that of a different closed surface. The second family of examples are defined on mapping class groups of oncepunctured surfaces and have quite curious behaviour. For instance, some pseudo-Anosov elements are mapped to multi-twists. Neithe...

Results from the rich and well-developed theory of well-quasi-ordering have often been rediscovered and republished. The purpose of this paper is to describe this intriguing subject. To illustrate the theory, here are two definitions and an elementary result. A partially ordered set is called well-partially-ordered if every subset has at least one, but only a finite number, of minimal elements....

A new method of normalization is used for the construction of the affine moment invariants. The affine transform is decomposed into translation, scaling, stretching, two rotations and mirror reflection. The object is successively normalized to these elementary transforms by means of low order moments. After normalization, other moments of normalized object can be used as affine invariant featur...

I f G induces a primitive permutation group on the set X of points (l-spaces) of the underlying vector space V, then well-known results of Burnside [I, p. 3411 and Schur [6] imply that G acts on X 2-transitively or as a regular or Frobenius group of prime degree; the theorem then follows from [2; 51. We must thus take a nontrivial block of imprimitivity d for G on X, and analyze the action of G...

We discuss an elementary, yet unsolved, problem of Niederreiter concerning the enumeration of a class of subspaces of finite dimensional vector spaces over finite fields. A short and self-contained account of some recent progress on this problem is included and some related problems are discussed.

introduction: students in elementary schools due to the physical and behavioral characteristics more than adults have exposed to environmental hazards. this study was conducted to examine the environmental health and safety indices in girls' elementary schools in ilam city. materials and methods: this cross-sectional study was carried out via census sampling method in girls' elementar...