On the neutrix composition of the distributions x −s ln m |x| and x r

Let F be a distribution and let f be a locally summable function. The distribution F(f) is defined as the neutrix limit of the sequence {F _n (f)}, where F _n (x) = F(x) * δ _n (x) and {δ _n (x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function δ(x). The composition of the distributions x ^?s ln ^m |x| and x ^r is proved to exist and be equal to r ^m x ^?rs ln ^m |x| for r, s, m = 2, 3….     

Braid groups and Hodge theory

This paper gives an account of the unitary representations of the braid group that arise via the Hodge theory of cyclic branched coverings of ${\mathbb{P}^1}$ , highlighting their connections with ergodic theory, complex reflection groups, moduli spaces of 1-forms and open problems in surface topology.     

Stepping into iSMART: Understanding Science–Mathematics Integration for Middle School Science and Mathematics Teachers

This paper is based on an online graduate program for middle school science and mathematics teachers in Texas titled Integration of Science, Mathematics and Reflective Teaching (iSMART). Launching the program for its first cohort in fall 2010, the authors attempted to answer the following two questions in this paper: (a) How do the members of the iSMART design team and the first cohort of teacher participants define science and mathematics integration with similar and different emphases? and (b) How would these definitions and concerns impact the ongoing design of the program? The iSMART design team members and the participating cohort teachers had a shared view regarding the importance of integration and its possible impact on student motivation. The findings also revealed that the two groups showed some different points of emphasis in their definitions of integration. These issues will be addressed in the ongoing design of the program in the following three areas: (a) design of the second summer meeting activities, (b) greater emphasis on teacher as researcher and action research, and (c) administrative support for teacher collaboration.     

Nonparametric spatial models for extremes: application to extreme temperature data

Estimating the probability of extreme temperature events is difficult because of limited records across time and the need to extrapolate the distributions of these events, as opposed to just the mean, to locations where observations are not available. Another related issue is the need to characterize the uncertainty in the estimated probability of extreme events at different locations. Although the tools for statistical modeling of univariate extremes are well-developed, extending these tools to model spatial extreme data is an active area of research. In this paper, in order to make inference about spatial extreme events, we introduce a new nonparametric model for extremes. We present a Dirichlet-based copula model that is a flexible alternative to parametric copula models such as the normal and t-copula. The proposed modelling approach is fitted using a Bayesian framework that allow us to take into account different sources of uncertainty in the data and models. We apply our methods to annual maximum temperature values in the east-south-central United States.     

Variations on a Theme of Cline and Donkin

Let N be a normal subgroup of a group G. An N-module Q is called G-stable provided that Q is equivalent to the twist Q ^g of Q by g, for every g?∈?G. If the action of N on Q extends to an action of G on Q, then Q is obviously G-stable, but the converse need not hold. A famous conjecture in the modular representation theory of reductive algebraic groups G asserts that the (obviously G-stable) projective indecomposable modules (PIMs) Q for the Frobenius kernels G _r (r?≥?1) of G have a G-module structure. It is sometimes just as useful (for a general module Q) to know that a finite direct sum Q ^?⊕?n of Q has a compatible G-module structure. In this paper, this property is called numerical stability. In recent work (Parshall and Scott, Adv Math 226:2065–2088, 2011), the authors established numerical stability in the special case of PIMs. We provide in this paper a more general context for that result, working in the context of k-group schemes and a suitable version of G-stability, called strong G-stability. Among our results here is the determination of necessary and sufficient conditions for the existence of a compatible G-module structure on a strongly G-stable N-module, in the form of a cohomological obstruction which must be trivial precisely when the G-module structure exists. Our main result is achieved by giving an approach to killing the obstruction by tensoring with certain finite dimensional G/N-modules.     

Ascending Subgraph Decompositions of Tournaments of Order 6n + 3

In 1987, Alavi, Boals, Chartrand, Erdös, and Oellermann conjectured that all graphs have an ascending subgraph decomposition (ASD). Though several classes of graphs have been shown to have an ASD, the conjecture remains open. In this paper, we investigate the similar problem for tournaments. In particular, using Kirkman Triple Systems, we will show that all tournaments of order 6n + 3 have an ASD.     

Global stability in the 2D Ricker equation

We improve a previous result for the 2D Ricker equation by reducing an infinite number of topological conditions to a finite number. We also give sufficient conditions in terms of the parameters where many of these topological conditions are satisfied. We also discuss the various pathologies that occur for other parameter choices.     

Size-dependent chemical transformation,structural phase change,and optical properties of nanowires

Nanowires offer a unique approach for the bottom-up assembly of electronic and photonic devices with the potential of integrating photonics with existing technologies. The anisotropic geometry and mesoscopic length scales of nanowires also make them very interesting systems to study a variety of size-dependent phenomenon where finite-size effects become important. We will discuss the intriguing size-dependent properties of nanowire systems with diameters in the 5–300?nm range, where finite-size and interfacial phenomena become more important than quantum mechanical effects. The ability to synthesize and manipulate nanostructures by chemical methods allows tremendous versatility in creating new systems with well-controlled geometries, dimensions, and functionality, which can then be used for understanding novel processes in finite-sized systems and devices.     

Feasibility Study on Quantitative Measurements of Singlet Oxygen Generation Using Singlet Oxygen Sensor Green

The purpose of this study is to investigate the feasibility for quantitative measurement of singlet oxygen (¹O₂) generation by using a newly developed ¹O₂-specific fluorescence probe Singlet Oxygen Sensor Green reagent (SOSG). ¹O₂ generation from photoirradiation of a model photosensitizer Rose Bengal (RB), in initially air-statured phosphate buffered saline (PBS) was indirectly monitored with SOSG. In the presence of ¹O₂, SOSG can react with ¹O₂ to produce SOSG endoperoxides (SOSG-EP) that emit strong green fluorescence with the maximum at 531 nm. The green fluorescence of SOSG-EP is mainly dependent on the initial concentrations of RB and SOSG, and the photoirradiation time for ¹O₂ generation. Furthermore, kinetic analysis of the RB-sensitized photooxidation of SOSG is performed that, for the first time, allows quantitative measurement of ¹O₂ generation directly from the determination of reaction rate. In addition, the obtained ¹O₂ quantum yield of porphyrin-based photosensitizer hematoporphyrin monomethyl ether (HMME) in PBS by using SOSG is in good agreement with the value that independently determined by using direct measurement of ¹O₂ luminescence. The results of this study clearly demonstrate that the quantitative measurement of ¹O₂ generation using SOSG can be achieved by determining the reaction rate with an appropriate measurement protocol.     

Comparison of the discriminating power of Raman and surface‐enhanced Raman spectroscopy with established techniques for the examination of liquid and gel inks

The ability to discriminate between inks is important for forensic document analysis. Here, Raman spectroscopy (RS) and surface‐enhanced RS have been compared to the traditional document examination techniques of video spectral comparison and thin layer chromatography on a population of blue and black‐coloured liquid and gel inks. It was found that in most cases, the Raman techniques provided a similar or better discriminating power than the conventional methods. Importantly, this study allowed us to determine whether the same underlying changes in composition were being exploited by the different methods to discriminate between samples. It was found that there was indeed a high degree of commonality in the sample pairs being discriminated by the various techniques. This work can therefore underpin introduction of Raman methods into standard operating procedures for ink analysis since it not only measures the extent of discrimination between samples but can also explain the origin of the spectral changes that are used to distinguish between them. © 2013 John Wiley & Sons Ltd and Crown copyright