Primitive divisors of Lucas and Lehmer sequences

Stewart reduced the problem of determining all Lucas and Lehmer sequences whose $n$-th element does not have a primitive divisor to solving certain Thue equations. Using the method of Tzanakis and de Weger for solving Thue equations, we determine such sequences for $n \leq 30$. Further computations lead us to conjecture that, for $n > 30$, the $n$-th element of such sequences always has a primitive divisor

Decline of Gomphus \u3ci\u3eFraternus Fraternus\u3c/i\u3e (Odonata: Gomphidae) in Lake Erie

Collections and literature reports indicate that Gomphus fraternus fraternus was abundant on the shoreline of Lake Erie prior to 1960, and tens of thousands were reported at Long Point Bay. After 1960 there were no reports from the shoreline, although there have been a number of comprehensive studies that have included the shoreline area and a number of Odonata specialists have also visited the Lake Erie shoreline regularly. A survey of portions of the Lake Erie shoreline, including the Long Point Bay area in 1999 and 2000, during the established peak and late peak flight period in southwestern Ontario, did not result in any observations. It is concluded that G. fraternus has declined substantially in Lake Erie and is possibly extirpated from the lake. The decline appears to have occurred between 1950 and 1960, and thus approximates the mid-1950s decline of burrowing mayflies in Lake Erie, which has been associated with warm weather oxygen depletion and pollution. Although it may never be possible to precisely deter- mine the cause of the decline of G. fraternus, it is likely that a number of factors are involved including climatic warming, pollution, changes to the shore- line, other effects of shoreline development, and introduced species

At the Crossroads of Title IX and a New “Idea”: Why Bullying Need Not Be “A Normal Part of Growing Up” for Special Education Children

In 21st century America, bullying of children by other children at school continues at epidemic levels. Organizations as diverse as the Department of Health and Human Services (through the National Institutes of Health and Health Resources and Services Administration (HRSA)), the National Crime Prevention Council, the National Youth Violence Prevention Resource Center, and the National Education Association, have all targeted school bullying and its consequences in various initiatives and campaigns over the last several years. These organizations argue that if schools and parents do not properly intervene to prevent bullying, the long-term ramifications for both the bully and the bullied could be disastrous. In fact, there appears to be a new momentum among parents and educators to take the problem of bullying more seriously. Recent violence at schools across the country, including the now-infamous Columbine school shooting, has made educators, parents, and children more reluctant to accept that bullying is just a normal part of growing up. Even so, legal remedies for victims of bullying continue to be woefully inadequate. Although victims of student-on-student sexual harassment have a claim for compensatory damages under the federal gender discrimination in education law, Title IX of the Education Amendments of 1972, the Supreme Court of the United States has yet to endorse the idea of a same-sex harassment cause of action for more common forms of bullying under Title IX (i.e., boys bullying boys or girls bullying girls). That being said, there is some hope that same-sex harassment causes of action for bullying behavior may become more common under Title IX as an increasing number of courts and the United States Department of Education have adopted the reasoning of the Supreme Court\u27s Title VII decision of Oncale v. Sundowner Offshore Services, Inc. In these instances, decisions have embraced Oncale\u27s central teaching that same-sex harassment need not depend upon sexual attraction or desire, but may also derive from nonsexual animus based on the failure of the harassed individual to live up to stereotypical gender norms. Indeed, research studies and anecdotal evidence suggest that those victimized by bullying are typically students who do not fit stereotypical notions of what it is to be masculine or feminine, athletic, cool, or in at school. Classic examples include not only children who are smaller, younger, gay or effeminate, obese, or from different countries, but also children who look and/or act differently from other children as a result of physical and/or mental impairments (i.e., special education children). It is the legal ramifications for verbally and physically abused special education children, mostly by members of their own gender, upon which this article focuses. Presently, most observers, and even the Supreme Court in its seminal student-on-student sexual harassment decision, Davis v. Monroe County Bd. of Educ., have yet to undertake a sufficient evaluation of the complex legal issues surrounding the bullying of special education children. To date, commentators and lower courts have primarily focused on the Title IX implications for students subject to same-sex harassment by other students and then, mostly in the context of harassment based on the actual, or perceived, sexual orientation of the student. Yet, when another student bullies a special education child based on that child\u27s appearance, behavior, or failure to live up to stereotyped notions of gender, it is necessary to consider the intersection between Title IX and the primary, federal special education law, the Individuals with Disabilities in Education Act (IDEA). Under IDEA, children with disabilities are entitled to a free and appropriate education (FAPE) in the least restrictive environment practicable. In turn, a FAPE includes special education and related services which are reasonably calculated to permit a child with a disability to benefit educationally. Consequently, actions taken by school districts to alleviate an unpleasant bullying situation for a special education child in order to comply with Title IX\u27s dictates concerning peer sexual harassment may inadvertently also violate a child\u27s right to a FAPE by altering that child\u27s placement and/or programs. In order to prevent placing school officials in this legal Catch-22, a legal model needs to be developed which ties the overlapping statutory frameworks of Title IX and IDEA together in one hybrid legal cause of action. Alternatively, if Title IX\u27s stringent legal standards for peer sexual harassment cannot be met in a given bullying case even after incorporating IDEA concepts, IDEA may also provide legal bases for special education children to obtain monetary damages against school officials who have failed to protect them from bullying, which, in turn, has violated that child\u27s right to a FAPE under IDEA. With these issues at the forefront, this article advocates two IDEA-based legal models to increase the legal protections available for special education children who are the subject of same-sex harassment/bullying at school. The first proposal attempts to provide this additional protection by strengthening the Davis Title IX framework for peer sexual harassment at school by incorporating IDEA concepts directly into that framework. Under this hybrid model, a school is liable for the same-sex harassment of a special education child where (1) the school had actual notice of the harassment; (2) the character of the harassment was severe, pervasive, and objectively offensive; (3) the school\u27s response to the known harassment was clearly unreasonable in light of its obligations under Title IX and IDEA; and (4) the student was denied a free and appropriate education in the least restrictive environment practicable or otherwise denied access to appropriate educational opportunities and benefits as a result of the harassment. Next, after addressing the lack of effective legal theories of recovery outside the Title IX context because of various procedural and substantive limitations on such claims, this article sets forth a second legal model. The second legal model involves an IDEA-based Section 1983 action for money damages, while at the same time recognizing that special education plaintiffs should normally have to first exhaust their administrative remedies under IDEA before bringing such a claim

Solid-state Isotopic Power Source for Computer Memory Chips

Recent developments in materials technology now make it possible to fabricate nonthermal thin-film radioisotopic energy converters (REC) with a specific power of 24 W/kg and a 10 year working life at 5 to 10 watts. This creates applications never before possible, such as placing the power supply directly on integrated circuit chips. The efficiency of the REC is about 25 percent which is two to three times greater than the 6 to 8 percent capabilities of current thermoelectric systems. Radio isotopic energy converters have the potential to meet many future space power requirements for a wide variety of applications with less mass, better efficiency, and less total area than other power conversion options. These benefits result in significant dollar savings over the projected mission lifetime