265 research outputs found
Variations on a theorem of Davenport concerning abundant numbers
Full text linkLet \sigma(n) = \sum_{d \mid n}d be the usual sum-of-divisors function. In
1933, Davenport showed that that n/\sigma(n) possesses a continuous
distribution function. In other words, the limit D(u):= \lim_{x\to\infty}
\frac{1}{x}\sum_{n \leq x,~n/\sigma(n) \leq u} 1 exists for all u \in [0,1] and
varies continuously with u. We study the behavior of the sums \sum_{n \leq
x,~n/\sigma(n) \leq u} f(n) for certain complex-valued multiplicative functions
f. Our results cover many of the more frequently encountered functions,
including \varphi(n), \tau(n), and \mu(n). They also apply to the
representation function for sums of two squares, yielding the following
analogue of Davenport's result: For all u \in [0,1], the limit exists, and
\tilde{D}(u) is both continuous and strictly increasing on [0,1]
Let\u27s All Agree to Disagree, and Move On: Analyzing \u3cem\u3eSlaughter-House\u3c/em\u3e and the Fourteenth Amendment\u27s Privileges or Immunities Clause Under Sunk Cost Principles
Get PDFThe Privileges or Immunities Clause of the Fourteenth Amendment has lain nearly dormant since the U.S. Supreme Court’s 1872 decision in the Slaughter-House Cases. Although legal historians have fought to overturn Slaughter-House for decades to restore the Privileges or Immunities Clause to its intended preeminence in American jurisprudence, these historians cannot agree on the correct meaning and scope of the clause. Each historical interpretation of the clause would affect the scope and power of the Privileges or Immunities Clause in the modern era; however, American jurisprudence has already found the clause’s intended powers in alternative constitutional provisions post-Slaughter-House. Accordingly, the Supreme Court’s likely reliance on “sunk cost” principles to justify its modern refusal to revive the clause is the most rational resolution to this long-debated issue of American Constitutional law
Witness History as Juries Become History: How the Eleventh Circuit Allowed the Opinions of Lay Witnesses to Overtake the Duty of the Jury in \u3cem\u3eUnited States v. Jayyousi\u3c/em\u3e
Get PDFOn September 19, 2011, in United States v. Jayyousi, the U.S. Court of Appeals for the Eleventh Circuit held that an FBI agent’s testimony regarding his post-hoc review of investigation materials was admissible as lay opinion testimony under Rule 701 of the Federal Rules of Evidence. In so holding, the Eleventh Circuit joined a minority of courts in adopting the most liberal interpretation of the Rule 701 perception and helpfulness requirements. This Comment argues that the Eleventh Circuit erred in adopting the most liberal interpretation of Rule 701 and that only by adopting the strictest interpretation can the courts continue to show due reverence for our jurors
Extension Operators and Finite Elements for Fractal Boundary Value Problems
Get PDFThe dissertation is organized into two main parts. The first part considers fractal extension operators. Although extension operators are available for general subsets of Euclidean domains or metric spaces, our extension operator is unique in that it utilizes both the iterative nature of the fractal and finite element approximations to construct the operator. The resulting operator is especially well suited for future numerical work on domains with prefractal boundaries. In the dissertation we prove the existence of a linear extension operator, Π from the space of Hölder continuous functions on a fractal set S to the space of Hölder continuous functions on a larger domain Ω. Moreover this same extension operator maps functions of finite energy on the fractal to H1 functions on the larger domain Ω.
In the second part, we consider boundary value problems in domains with fractal boundaries. First we consider the Sierpinski prefractal and how we might apply the technique of singular homogenization to thin layers constructed on the prefractal. We will also discuss numerical approximation in domains with fractal boundaries and introduce a finite element mesh developed for studying problems in domains with prefractal Koch boundaries. This mesh exploits the self-similarity of the Koch curve for arbitrary rational values of α and its construction is crucial for future numerical study of problems in domains with prefractal Koch curve boundaries. We also show a technique for mesh refinement so that singularities in the domain can be handled and present sample numerical results for the transmission problem
Involving Students as Partners in a Course Redesign
Get PDFAs a team composed of student partners, a course instructor, and a distance learning program development specialist, we share our experiences of working together on a course development project. We used a collaborative autoethnographic approach to document and reflect on our experiences. Although our individual reflections reveal that our experiences of working together varied, we all valued working with each other and were engaged in the project. A change that we would make for future collaborations of this nature would be to invest more effort into team building and cohesion at the start of the project. We also found that individually, we all experienced our collaboration differently, which we believe is important to keep in mind when we think about inclusivity with respect to course instruction and design
Quality assessment of work recovery activities: Guidance for recovering from work-related demands
Get PDFThe proposed study is designed to test a revised work recovery process model and gather data to provide guidance for work recovery activities based on their recovery quality value. Using an integrated and modified model of the stress-recovery process, recovery quality will be measured in terms of potential for psychological detachment, mastery, and control, with relaxation serving as an outcome state associated with the proposed three core recovery mechanisms. Underlying theoretical frameworks such as the Conservation of Resources Theory, the Effort-Recovery Model, and the Job-Demands Resource model served as the foundation to describe the importance of recovering depleted resources. Past research suggests active forms of recovery in natural environments hold the greatest potential for work recovery, but research has been limited to broad activity category classifications. In this study we take a more holistic approach to identifying specific recovery activities and their associated recovery experience quality by asking participants to list, rank order, and provide quality-related details regarding their three most common recovery activities. A variety of analyses will be used to compare average ratings of recovery quality elements and identify common recovery themes
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