Difference analogue of the Lemma on the Logarithmic Derivative with applications to difference equations

The Lemma on the Logarithmic Derivative of a meromorphic function has many applications in the study of meromorphic functions and ordinary differential equations. In this paper, a difference analogue of the Logarithmic Derivative Lemma is presented and then applied to prove a number of results on meromorphic solutions of complex difference equations. These results include a difference analogue of the Clunie Lemma, as well as other results on the value distribution of solutions.Comment: 12 pages. To appear in the Journal of Mathematical Analysis and Application

Finite-order meromorphic solutions and the discrete Painleve equations

Let w(z) be a finite-order meromorphic solution of the second-order difference equation w(z+1)+w(z-1) = R(z,w(z)) (eqn 1) where R(z,w(z)) is rational in w(z) and meromorphic in z. Then either w(z) satisfies a difference linear or Riccati equation or else equation (1) can be transformed to one of a list of canonical difference equations. This list consists of all known difference Painleve equation of the form (1), together with their autonomous versions. This suggests that the existence of finite-order meromorphic solutions is a good detector of integrable difference equations.Comment: 34 page

Nevanlinna theory for the difference operator

Certain estimates involving the derivative $f\mapsto f'$ of a meromorphic function play key roles in the construction and applications of classical Nevanlinna theory. The purpose of this study is to extend the usual Nevanlinna theory to a theory for the exact difference $f\mapsto \Delta f=f(z+c)-f(z)$. An $a$-point of a meromorphic function $f$ is said to be $c$-paired at z\in\C if $f(z)=a=f(z+c)$ for a fixed constant c\in\C. In this paper the distribution of paired points of finite-order meromorphic functions is studied. One of the main results is an analogue of the second main theorem of Nevanlinna theory, where the usual ramification term is replaced by a quantity expressed in terms of the number of paired points of $f$. Corollaries of the theorem include analogues of the Nevanlinna defect relation, Picard's theorem and Nevanlinna's five value theorem. Applications to difference equations are discussed, and a number of examples illustrating the use and sharpness of the results are given.Comment: 19 page

Emergent leadership in student-led collaborative activity in a school-based makerspace

The aim of this dissertation study is to analyze how leadership emerges in a school-based makerspace, and analyze the ways in which emergent leadership mediates student-led making activities. The study also aims to investigate how teachers can support productive leadership during student-led collaboration. In the study, leadership is defined as a reciprocal social process that develops in students’ interactions. I argue that school-based makerspaces provide an important platform for the development of students’ emergent leadership. © ISLS.Peer reviewe

Magnetic Structure of Rapidly Rotating FK Comae-Type Coronae

We present a three-dimensional simulation of the corona of an FK Com-type rapidly rotating G giant using a magnetohydrodynamic model that was originally developed for the solar corona in order to capture the more realistic, non-potential coronal structure. We drive the simulation with surface maps for the radial magnetic field obtained from a stellar dynamo model of the FK Com system. This enables us to obtain the coronal structure for different field topologies representing different periods of time. We find that the corona of such an FK Com-like star, including the large scale coronal loops, is dominated by a strong toroidal component of the magnetic field. This is a result of part of the field being dragged by the radial outflow, while the other part remains attached to the rapidly rotating stellar surface. This tangling of the magnetic field,in addition to a reduction in the radial flow component, leads to a flattening of the gas density profile with distance in the inner part of the corona. The three-dimensional simulation provides a global view of the coronal structure. Some aspects of the results, such as the toroidal wrapping of the magnetic field, should also be applicable to coronae on fast rotators in general, which our study shows can be considerably different from the well-studied and well-observed solar corona. Studying the global structure of such coronae should also lead to a better understanding of their related stellar processes, such as flares and coronal mass ejections, and in particular, should lead to an improved understanding of mass and angular momentum loss from such systems.Comment: Accepted to ApJ, 10 pages, 6 figure

Piracy prevention methods in software business

Abstract. There are various forms of piracy in software business, and many prevention techniques have been developed against them. Forms of software piracy are, for example, cracks and serials, softlifting and hard disk loading, internet piracy and software counterfeiting, mischanneling, reverse engineering, and tampering. There are various prevention methods that target these types of piracy, although all of these methods have been broken. The piracy prevention measures can be divided into ethical, legal, and technical measures. Technical measures include measures like obfuscation and tamper-proofing, for example. However, relying on a single method does not provide complete protection from attacks against intellectual property, so companies wishing to protect their product should consider combining multiple methods of piracy prevention

Quality and Access in Lifelong Learning

Learner-directed choice is central to the process of educational achievemen

Consumer’s right to be informed:communicating sustainability issues through fashion marketing

Abstract. The purpose of this study is to explore fashion companies’ current marketing efforts in the context of sustainability. More specifically, this study is focusing on the level of leading and slow fashion companies’ sustainability communication and their possible socially responsible marketing strategies applied. A qualitative content analysis was conducted where the data was collected through the selected sets of samples of leading and slow fashion companies’ websites. The theoretic framework for this study is based on the previous research about the level of sustainability communication and different socially responsible marketing strategies. Based on these studies, two coding schemes were formed. The first coding scheme includes the sustainability topics that can be identified in the fashion companies’ communication while the second coding scheme consists of the different sustainability messages communicated based on the socially responsible marketing approaches. One of the main findings was that the current level of fashion companies’ sustainability communication is wide addressing several different sustainability issues. The difference found between the leading and slow fashion companies was in the nature of this communication. Another main finding was the differences in the socially responsible marketing messages; all slow fashion companies were promoting one of the socially responsible marketing messages while half of the leading fashion companies did not have any specific message included. This study contributes to give an outlook in the current state of fashion companies’ sustainability communication. Additionally, some generalized differences between leading and slow fashion companies’ sustainability communication were identified. As for the managerial implications, the results of this study can be used especially by fashion marketers to identify their company’s level of sustainability communication. The results also indicate how different socially responsible marketing strategies can be applied in order to promote more sustainable consumption. This study had some limitations. Since sustainability is a wide topic, it could have limited the possibility to construct a comprehensive coding schemes around the sustainability topics. Other factors that can be considered as limitations were using the websites as the unit of analysis and having only one researcher doing the coding for the content analysis. For the future research, more research efforts are needed to enable categorizing sustainability topics of fashion and to better identify different socially responsible marketing strategies

An Algorithm for Projecting a Reference Direction onto the Nondominated Set of Given Points

In this paper, we consider the problem of searching nondominated alternatives in a discrete multiple criteria problem. The search procedure is based on the use of a reference direction. A reference direction reflects the desire of the decision maker (DM) to specify a search direction. To find a set of given alternatives related somehow to the reference direction specified by the DM, the reference direction has to be projected onto the set of nondominated alternatives. Our purpose is to develop an efficient algorithm for making this projection. The projection of each given reference direction determines a nondominated ordered subset. The set is provided to a decision maker for evaluation. The decision maker will choose the most preferred alternative from this subset and continues the search from this alternative with a new reference direction. The search will end when no direction of of improvement is found. A critical point in the procedure is the efficiency of the projection operation. This efficiency of our algorithm is considered theoretically and numerically. The projection is made by parametrizing an achievement scalarizing function originally proposed by Wierzbicki (1980) to project any single point onto the nondominated set