Ground state periodic solutions for difference equations with discrete p-Laplacian

We use the critical point theory to establish existence results for periodic solutions of some nonlinear boundary value problems involving the discrete p-Laplacian operator. As an application we give an alternative proof to the upper and lower solutions theorem.     

Flag graphs and their applications in mathematical chemistry for benzenoids

Flag graphs have been used in the past for describing maps on closed surfaces. In this paper we use them for the first time in mathematical chemistry for describing benzenoids and some other similar structures. Examples include catacondensed and pericondensed benzenoids. Several theorems are included. Symmetries of benzenoid systems, flag graphs, and symmetry type graphs are briefly discussed.     

How visual representations participate in algebra classes’ mathematical activity

Our aim is to discuss how a visual display introduced in a classroom activity to represent a specific algebraic procedure is transformed, taking a central role and modifying the ongoing activity. To discuss how visualization comes about in this activity, we describe an illustrative example selected from observations carried out in a 9th grade classroom and analyze the class interaction from a cultural-historical perspective. Our analysis illuminates the tensions that emerge from a difference between the teacher’s way of signifying the algebraic procedure and the students’ overuse of a visual display they associate with it, and how these tensions impel changes in the activity. We further discuss some pros and cons of using visual displays in algebra classes, and we argue that it is very important for the teacher to be aware of them in order to realize the benefits of using such displays.     

Dilation coefficient, plane-width, and resolution coefficient of graphs

In this paper we investigate the relations between three graph invariants that are related to the ‘compactness’ of graph drawing in the plane: the dilation coefficient, defined as the smallest possible quotient between the longest and the shortest edge length; the plane-width, which is the smallest possible quotient between the largest distance between any two points and the shortest length of an edge; and the resolution coefficient, the smallest possible quotient between the longest edge length and the smallest distance between any two points. These three invariants coincide for complete graphs. More specifically, we show that the plane-width and the dilation coefficient are equivalent graph parameters in the sense that they are bounded on the same sets of graphs, while bounded resolution coefficient implies bounded plane-width (or dilation coefficient) but not conversely. It is known that the one-dimensional analogues of the plane-width and the resolution coefficient are closely related to the chromatic number and the bandwidth respectively. We complete this picture by showing that a graph of one-dimensional dilation coefficient $d$ is exactly the same as a graph of circular chromatic number  $d+1$ . Finally, we complement the result that a graph of resolution coefficient less than $\sqrt{2}$ is planar by constructing a family of graphs of resolution coefficient $\sqrt{2}$ not contained in any nontrivial minor-closed graph family.     

Hamilton cycles in graph bundles over a cycle with tree as a fibre

A sufficient and necessary condition for the existence of a Hamilton cycle in a graph bundle with a cycle as a base and a tree as a fibre is obtained.     

Rapid Screening and Identification of Polar Constituents from Brazilian Arnica Lychnophora sp. by LC-UV/DAD-ESI-MS and LC-UV/DAD-ESI-MS/MS Analysis

This paper describes an analytical method for the rapid screening and identification of the phenolic constituents present in the polar extracts of different Lychnophora spp. using LC-UV/DAD-ESI-MS and LC-UV/DAD-ESI-MS/MS. Compounds were identified based on UV, retention time, MS experiments and MS/MS of precursor ion or standard. On-line phytochemical investigation of Lychnophora spp. allowed for the identification of flavonoids, chlorogenic acid derivatives and lactones. Some of the observed compounds were for the first time identified in Lychnophora species in a fast analytical procedure. The data obtained here may be helpful to the investigation of polar constituents from other Lychnophora species.     

Nonorientable genus of cartesian products of regular graphs

A special type of surgery developed by A. T. White and later used by the author to construct orientable quadrilateral embeddings of Cartesian products of graphs is here expanded to cover the nonorientable case as well. This enables the nonorientable genus of many families of Cartesian products of triangle-free graphs to be computed.     

Dual imbeddings and wrapped quasi-coverings of graphs

This paper gives a new method for constructing imbeddings of graphs which are “nearly” coverings of given imbedded graphs. The method is based on the dual theories of current graphs and voltage graphs. Some applications are given, in particular the following theorem: Let G be any graph which has a triangular imbedding in the sphere. Then there are infinitely many integers n for which the composition G[nK₁] has a triangular (orientable surface) imbedding.