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

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

Uranus evolution models with simple thermal boundary layers

The strikingly low luminosity of Uranus (Teff ~ Teq) constitutes a long-standing challenge to our understanding of Ice Giant planets. Here we present the first Uranus structure and evolution models that are constructed to agree with both the observed low luminosity and the gravity field data. Our models make use of modern ab initio equations of state at high pressures for the icy components water, methane, and ammonia. Proceeding step by step, we confirm that adiabatic models yield cooling times that are too long, even when uncertainties in the ice:rock ratio (I:R) are taken into account. We then argue that the transition between the ice/rock-rich interior and the H/He-rich outer envelope should be stably stratified. Therefore, we introduce a simple thermal boundary and adjust it to reproduce the low luminosity. Due to this thermal boundary, the deep interior of the Uranus models are up to 2--3 warmer than adiabatic models, necessitating the presence of rocks in the deep interior with a possible I:R of $1\times$ solar. Finally, we allow for an equilibrium evolution (Teff ~ Teq) that begun prior to the present day, which would therefore no longer require the current era to be a "special time" in Uranus' evolution. In this scenario, the thermal boundary leads to more rapid cooling of the outer envelope. When Teff ~ Teq is reached, a shallow, subadiabatic zone in the atmosphere begins to develop. Its depth is adjusted to meet the luminosity constraint. This work provides a simple foundation for future Ice Giant structure and evolution models, that can be improved by properly treating the heat and particle fluxes in the diffusive zones.Comment: 13 pages, Accepted to Icaru