To what extent do the Classical Equations of Motion Determine the Quantization Scheme?

A simple example of one particle moving in a (1+1) space-time is considered. As an example we take the harmonic oscillator. We confirm the statement that the classical Equations of Motion do not determine at all the quantization scheme. To this aim we use two inequivalent Lagrange functions, yielding Euler-Lagrange Equations, having the same set of solutions. We present in detail the calculations of both cases to emphasize the differences occuring between them.Comment: LaTeX 20 page

Torsion zero-cycles and the Abel-Jacobi map over the real numbers

This is a study of the torsion in the Chow group of zero-cycles on a variety over the real numbers. The first section recalls important results from the literature. The rest of the paper is devoted to the study of the AbelJacobi map a: A0XAlbXR restricted to torsion subgroups. Using Roitmans theorem over the complex numbers and a version of Blochs cohomological AbelJacobi map over the real numbers, it is shown that this map can be described completely in terms of ´etale cohomology. For some examples (products of curves, abelian varieties, certain fibre bundles) the torsion in the kernel and cokernel of the AbelJacobi map a is computed explicitly

Lichtenbaum-Tate duality for varieties over p-adic fields

S. Lichtenbaum has proved in [L1] that there is a nondegenerate pairing Pic(C)x Br(C)->Br(K) =Q/Z (1) between the Picard group and the Brauer group of a nonsingular projective curve C over a p-adic field K (a finite extension of the p-adic numbers Qp). His proof consists of a reduction via explicit cocycle calculations in Galois cohomology to a combination of Tate duality for group schemes over p-adic fields and the autoduality of the Jacobian of a smooth curve. In this paper we will reconstruct the above duality as a purely formal combination of a generalized form of Tate duality over p-adic fields and a form of Poincar´ e duality for curves over arbitrary fields of characteristic zero. This gives a more conceptual proof of Lichtenbaum's result and an analogue in higher dimensions

IMPROVING THE RIGOR OF ONLINE EDUCATION: EXPLORING CHARACTERISTICS OF FACULTY AND SYLLABI WITHIN AN ONLINE PROGRAM ASSESSMENT PROCESS

As online course enrollments grow, overall perception of rigor still lags compared to that of traditional face-to-face education. The purpose of this research was to tie faculty and syllabi characteristics to the rigor of online courses. This study explored the relationship between faculty and syllabi characteristics and performance on an online entry-level course design quality assurance assessment (pass, pass with concern, or fail). A decision tree analysis was used to predict the relationship of the independent (faculty and syllabi characteristics) and the dependent (entry-level course design assessment) variables. Findings suggest that faculty rank and writing intensive are key characteristics predictive of the rigor of design for online courses. Knowing what characteristics are likely to fail the course design assessments, extra resources and support can be shifted to elevate the rigor of online courses

Children\u27s Understanding and Experience of Anger Within Their Peer Groups

The purpose of this study was to describe how children understand, experience and express their anger with others in their peer group, from their perspectives. Semi-structured focus groups were conducted with third grade students in an elementary school. Children were asked three questions: 1) What did you learn about dealing with anger this week? 2) What are some good/positive things that happened between you and your classmates this week? 3) What are some not so good or negative things that happened between you and your classmates this week? Six themes emerged from analysis of the transcripts of the semi-structured focus groups: Managing Anger, Roots of Anger, Expression of Anger, Consequences of Anger, Avoidance Strategies, and Impact of Anger on Relationships. Children think that anger is bad, and they view anger and aggression as the same. Implications for counseling and research are offered

Safavid Trade During the 17th Century: Iran\u27s Transit Economy

The seventeenth century ushered in a plethora of changes in global trade patterns. These fluctuating trade patterns began to generate nascent economic, political, and social trends the likes of which had never been seen before. Ultimately, the product of these trends points towards the emergence of a truly global economy. Evidence of this phenomenon is fairly well documented, as it served as the catalyst which transformed civilizations in nearly every corner of the world. It is in this critical time period where the foundation for modern terms such as “globalization” and “interconnected global economy” were first laid. Essentially, a select group of industrializing Western powers experienced an increase in demand and consumption, thereby triggering an increase in the production and transport of goods and resources from the East. Obviously, there were major consequences, intended and unintended, which stemmed from this inherently unbalanced system—e.g., colonialism, capitalism, modern imperialism, and nationalism. Despite the global scope of these developments, there are very few publications which properly document the role that Safavid Iran’s economy played within this complex system. As a result, conventional wisdom has incorrectly bred the notion that Safavid Iran’s economic impact was minimal. The purpose of this paper is to disprove this theory in three stages. Firstly, it will analyze the role that the Safavid economy played within the rapidly developing economic system itself. Secondly, this paper will consult sources which focus on the economic relationship between the Mughal Empire, Europe, and the Safavid Empire during this time period. Finally, the paper will examine the trade routes used to transport goods and services in and out of Iran