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.     

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.     

Noncovalent Substrate‐Directed Enantioselective Heck Reactions: Synthesis of S‐ and P‐Stereogenic Heterocycles

S‐ and P‐Stereogenic heterocycles were synthesized by a remarkably simple enantioselective Heck desymmetrization reaction based on the unprecedented noncovalent directing effect of S=O and P=O functionalities. Selected prochiral symmetric substrates were efficiently arylated using the recently disclosed chiral PyraBOx ligand under mild and open‐flask reaction conditions. Several five‐membered aryl‐ sulfones, sulfoxides, and phosphine oxides were synthesized in good to excellent yields, in good to high diastereoselectivity, and enantiomeric ratios up to 98:2. Theoretical calculations also support the noncovalent directing effect of the S=O and P=O functionalities during the arylation process.     

Exotic n-universal graphs

An n-universal graph is a graph that contains as an induced subgraph a copy of every graph on n vertices. It is shown that for each positive integer n > 1 there exists an n-universal graph G on 4ⁿ - 1 vertices such that G is a (v, k, λ)-graph, and both G and its complement G¯ are 1-transitive in the sense of W. T. Tutte and are of diameter 2. The automorphism group of G is a transitive rank 3 permutation group, i.e., it acts transitively on (1) the vertices of G, (2) the ordered pairs uv of adjacent vertices of G, and (3) the ordered pairs xy of nonadjacent vertices of G.     

Transmission electron microscopic study of reduced Ca2UO5

Structural changes of Ca₂UO₅ during reduction in hydrogen were studied by transmission electron microscopy. It was shown that monoclinic Ca₂UO₅ changes into triclinic Ca₄U₂O₉. They are related, respectively, to the fluorite and the CM₂O₃structures, so that the product is a superstructure of the latter. Reduction occurs along the (100)_t planes originating from the (006)_m planes of the parent structure by diminishing the coordination number of the Ca cation from 7 to 6.     

Minimal ordered triangulations of surfaces

A finite simplicial complex is orderable if its simplices are the chains of a poset. For each closed surface an orderable triangulation is given that is minimal with respect to the number of vertices. The construction of minimal ordered triangulations implies that for each surface S the minimal number of vertices of a bipartite graph, which has a quadrilateral embedding into S, is equal to b(S) = ?4 + (16 – 8χ)^1/2?, where χ is the Euler characteristic of S.     

Field cooling-induced magnetic anisotropy in exchange biased CoO/Fe bilayer studied by ferromagneticresonance

Exchange-biased CoO/Fe bilayer grown on MgO (0 0 1) substrate by sputtering, studied by variable angle and temperature ferromagnetic resonance. Room temperature in-plane measurements reveal that the Fe layer was epitaxially grown on MgO substrate with a fourfold cubic symmetry. The data also show that the easy axis of magnetization is in the film plane and makes an angle of 45° with the [1 0 0] crystallographic direction of MgO substrate. The low temperature data exhibit a sudden onset of a field cooling-induced and shifted cubic anisotropy below the Néel temperature of CoO. This results in a twofold uniaxial or fourfold cubic symmetry for in-plane magnetic anisotropy depending on a field cooling direction. Low temperature measurements also present a reduction in the resonance fields due to the antiferromagnetic/ferromagnetic coupling. The developed theoretical model perfectly simulates the experimental data of coupled CoO/Fe bilayer.     

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.     

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.     

A Taylor method for numerical solution of generalized pantograph equations with linear functional argument

This paper is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument. In this paper, we introduce a numerical method based on the Taylor polynomials for the approximate solution of the pantograph equation with retarded case or advanced case. The method is illustrated by studying the initial value problems. The results obtained are compared by the known results.