Closed 2-cell embeddings of graphs with no V8-minors

A closed 2-cell embedding of a graph embedded in some surface is an embedding such that each face is bounded by a cycle in the graph. The strong embedding conjecture says that every 2-connected graph has a closed 2-cell embedding in some surface. In this paper, we prove that any 2-connected graph without V₈ (the Möbius 4-ladder) as a minor has a closed 2-cell embedding in some surface. As a corollary, such a graph has a cycle double cover. The proof uses a classification of internally-4-connected graphs with no V₈-minor (due to Kelmans and independently Robertson), and the proof depends heavily on such a characterization.     

What Matters Most: A Comparison of Expert and Novice Teachers' Noticing of Mathematics Classroom Events

In this study, we examined 10 expert and 10 novice teachers' noticing of classroom events in China. It was found that both expert and novice teachers, who were selected from two cities in China, highly attended to developing students' mathematics knowledge coherently and developing students' mathematical thinking and ability; they also paid attention to students' self‐exploratory learning, students' participation, and teachers' instructional skills. Furthermore, compared with novice teachers, expert teachers paid greater attention to developing mathematical and high‐order thinking, and developing mathematics knowledge coherently, but paid less attention to teachers' guidance. Moreover, we further illustrated the qualitative differences and similarities in their noticing of classroom events. Finally, we discussed the findings and relevant implications.     

Electrohydrocyclization of alkoxy tether substituted mixed enone/enoate and bisenone systems: retention versus elimination

The electrohydrocyclization of tethered enone/enoate or bisenone systems which contain an alkoxyl substituent in the tether was studied. Rather surprisingly, the cyclization of these compounds afforded very different products depending upon the alkoxy substituent. This behavior and its potential applications will be discussed.     

A sharp lower bound of the Randi? index of cacti with r pendants

The Randi? index of a simple connected graph G is defined as ∑_uv∈E(G)(d(u)d(v))^-1/2. In this paper, we present a sharp lower bound on the Randi? index of cacti with r pendants.     

Time-resolved light scattering studies of Spin-coated titanium dioxide

Spin coating of titanium alkoxides is a fast and straightforward method of forming compact and durable titanium dioxide thin films. In this report, an optical interference technique is used to understand the dynamics of titanium dioxide thin film formation during the spin coating process of titanium (IV) ethoxide sols, and a 4-stage thinning model is proposed. Optical monitoring of interference fringes as a function of time enables determination of the time-dependent film thickness through measurement of the interference fringe separation while the sol to gel transition can be monitored by analyzing the fringe visibility. For films that are spun at 2,000 rpm or faster, a spin duration of 5 s is sufficient for the film to reach its final thickness. At a given spin speed, spin times longer than 5 s do not affect the final film thickness. However, thinner final film thicknesses can be achieved by increasing the spin speed. All films were found to densify on annealing with an annealed thickness approximately 0.4 times that of the pre-anneal thickness regardless of spin speed. Spin-coated titanium dioxide films are expected to play an important role in low-cost and scalable next generation solar cells employing dyes and quantum dots as the light harvesting agents.     

High order smoothing schemes for inhomogeneous parabolic problems with applications in option pricing

A new family of numerical schemes for inhomogeneous parabolic partial differential equations is developed utilizing diagonal Padé schemes combined with positivity–preserving Padé schemes as damping devices. We also develop a split version of the algorithm using partial fraction decomposition to address difficulties with accuracy and computational efficiency in solving and to implement the algorithms in parallel. Numerical experiments are presented for several inhomogeneous parabolic problems, including pricing of financial options with nonsmooth payoffs.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Non-bipartite Graphs with Third Largest Laplacian Eigenvalue Less Than Three

All bipartite graphs whose third largest Laplacian eigenvalue is less than 3 have been characterized by Zhang. In this paper, all connected non-bipartite graphs with third largest Laplacian eigenvalue less than three are determined.     

An application of Vizing and Vizing-like adjacency lemmas to Vizing’s Independence Number Conjecture of edge chromatic critical graphs

In 1968, Vizing conjectured that, if G is a Δ-critical graph with n vertices, then , where α(G) is the independence number of G. In this note, we apply Vizing and Vizing-like adjacency lemmas to this problem and obtain better bounds for Δ∈{7,…,19}.     

The orientable genus of some joins of complete graphs with large edgeless graphs

In an earlier paper the authors showed that with one exception the nonorientable genus of the graph with m≥n−1, the join of a complete graph with a large edgeless graph, is the same as the nonorientable genus of the spanning subgraph . The orientable genus problem for with m≥n−1 seems to be more difficult, but in this paper we find the orientable genus of some of these graphs. In particular, we determine the genus of when n is even and m≥n, the genus of when n=2^p+2 for p≥3 and m≥n−1, and the genus of when n=2^p+1 for p≥3 and m≥n+1. In all of these cases the genus is the same as the genus of K_m,n, namely ⌈(m−2)(n−2)/4⌉.