Triangular bases of integral closures

In this work, we consider the problem of computing triangular bases of integral closures of one-dimensional local rings. Let $(K, v)$ be a discrete valued field with valuation ring $\mathcal{O}$ and let $\mathfrak{m}$ be the maximal ideal. We take $f \in \mathcal{O}[x]$, a monic irreducible polynomial of degree $n$ and consider the extension $L = K[x]/(f(x))$ as well as $\mathcal{O}_{L}$ the integral closure of $\mathcal{O}$ in $L$, which we suppose to be finitely generated as an $\mathcal{O}$-module. The algorithm $\operatorname{MaxMin}$, presented in this paper, computes triangular bases of fractional ideals of $\mathcal{O}_{L}$. The theoretical complexity is equivalent to current state of the art methods and in practice is almost always faster. It is also considerably faster than the routines found in standard computer algebra systems, excepting some cases involving very small field extensions

Spectral Analysis Program (SAP)

Program eliminates or reduces time-consuming aspects of computation of power spectrum for high-frequency communication system. This program was written in FORTRAN IV for UNIVAC 1230 or 1108 computer

Optimality of programmable quantum measurements

We prove that for a programmable measurement device that approximates every POVM with an error $\le \delta$, the dimension of the program space has to grow at least polynomially with $\frac{1}{\delta}$. In the case of qubits we can improve the general result by showing a linear growth. This proves the optimality of the programmable measurement devices recently designed in [G. M. D'Ariano and P. Perinotti, Phys. Rev. Lett. \textbf{94}, 090401 (2005)]

Prioritizing Interdictions on a Shortest Path Network

We consider a variant to the shortest path network interdiction problem with symmetric information from Israeli and Wood (Networks 40, 97-111,2002) which arises in the context of nuclear smuggling prevention. In the basic shortest path interdiction problem, an interdictor has a limited number of interdictions with which he can lengthen arcs in a network in order to maximize the length of the network’s shortest path. This thesis considers the case in which the interdictor does not make all of the interdictions at once. Rather, the interdictor must make the interdictions over a set number of periods. Each period has a budget for the number of interdictions that can be placed during the period. The interdictor must prioritize the interdictions and decide the order in which the interdictions should take place. This problem is formulated as an integer program with an objective to maximize the average of the shortest paths across all periods

Random subspaces for encryption based on a private shared Cartesian frame

A private shared Cartesian frame is a novel form of private shared correlation that allows for both private classical and quantum communication. Cryptography using a private shared Cartesian frame has the remarkable property that asymptotically, if perfect privacy is demanded, the private classical capacity is three times the private quantum capacity. We demonstrate that if the requirement for perfect privacy is relaxed, then it is possible to use the properties of random subspaces to nearly triple the private quantum capacity, almost closing the gap between the private classical and quantum capacities.Comment: 9 pages, published versio

Blessed Are the Poor (in Spirit): Wealth and Poverty in the Writings of the Greek Christian Fathers of the Second Century

This paper examines how Greek Christian authors engaged with the topics of poverty and wealth during the second and third centuries CE - a period of major transition for the Christian Church. Beginning with the latest documents in the Greek New Testament (c. 90 - 120), this study traces these themes through the works of the Apostolic Fathers, including Clement of Rome (c. 35 - 99), Ignatius of Antioch (c. 34 - 108), and Polycarp of Smyrna (c. 69 - 155). It then addresses to the apologetic authors Justin Martyr (c. 100 - 165) and Irenaeus (c. 130 - 202). In the works of all these authors, “the poor” were considered to be a privileged class in the eyes of God. However, this status came to be questioned by Clement of Alexandria (c. 150 - 215), whose work The Rich Man’s Salvation is demonstrated to have fundamentally deviated from the traditional understanding of wealth and poverty. Clement eschewed the notion of an inherently “blessed” class of the poor in order to make Christian doctrine welcoming to more wealthy individuals. Clement’s work altered part of the character of Christian theology, but in doing so, he helped the movement expand throughout the Roman Empire

Not to Teach Any Different Doctrine: Examining the Doctrines of the Early Latter-Day Saints Movement and the Church Fathers

Explanations of the nature of humanity, God, and the purpose of life have a direct influence on the daily lives of the adherents of a given religious tradition. In the early third century, Origen of Alexandria proposed doctrines of preexistence, subordinationism, and theosis, which were dismissed in the early Church to various degrees. Some sixteen hundred years later, members of the upstart Latter-day Saints movement, such as Orson Pratt, would maintain strikingly similar positions about the nature of the soul, the godhead, and the final cause of humanity. These concepts represent essential aspects of the worldviews of these traditions; so why and how do they arise in such different times and contexts? Both Origen and Pratt expounded the idea that each human soul existed before its physical birth, and that during its preexistent life, it acted with free will; these actions have an influence on the physical life of the incarnate soul. The two men also argued that, within the godhead, the person of the Son derives his authority and power from the Father in such a way that the Son is not coequal with the Father. Finally, both men maintained that the ultimate goal of human existence is to become like God, through a process of learning and purification, in this life and the next. This essay, however, through examination of the immediate contexts of the two authors, makes the case that the opinions expressed by Origen and Pratt are, in some ways, demonstrably distinct. In studying how these ideas differ, underlying values of ancient Christians and early Mormons can be discussed, allowing the observer to gain a greater understanding of the major concepts which motivate adherents of these traditions. These include the underlying concept of material-naturalism in Mormonism, and the importance of reason and knowledge within some forms of Christianity. As regards the similarities, the study argues that there is no need for a direct connection between these two thinkers, either natural or supernatural. Rather, these common ideas appear throughout Christian history due to common themes found in Christian scripture and to constant problems of the human condition, with which all religious thinkers must contend. Additionally, these doctrines serve functional purposes, such as the legitimizing concept of the preexistent soul, which are useful for these traditions to maintain

A Universal Problem: The Universal Injunction

The article discusses the appropriateness of and alternatives to the imposition of universal injunction in cases and controversies involving executive orders in the U.S