When are Multiples of Polygonal Numbers again Polygonal Numbers?

Euler showed that there are infinitely many triangular numbers that are three times other triangular numbers. In general, it is an easy consequence of the Pell equation that for a given square-free m > 1, the relation P=mP' is satisfied by infinitely many pairs of triangular numbers P, P'. After recalling what is known about triangular numbers, we shall study this problem for higher polygonal numbers. Whereas there are always infinitely many triangular numbers which are fixed multiples of other triangular numbers, we give an example that this is false for higher polygonal numbers. However, as we will show, if there is one such solution, there are infinitely many. We will give conditions which conjecturally assure the existence of a solution. But due to the erratic behavior of the fundamental unit in quadratic number fields, finding such a solution is exceedingly difficult. Finally, we also show in this paper that, given m > n > 1 with obvious exceptions, the system of simultaneous relations P = mP', P = nP'' has only finitely many possibilities not just for triangular numbers, but for triplets P, P', P'' of polygonal numbers, and give examples of such solutions.Comment: 17 pages, 1 figure, 2 tables. New version added a table of solutions to the second proble

Insider Threats to Information Systems

There are few, if any, organizations immune to the adverse and costly effects of successful information system attacks. As reliance on information systems continues to increase, organizations must continue to implement effective computer security measures to maintain their operability. This paper focuses on internal attacks executed by those individuals within the organization who have authorized access to information systems and behave in an unethical manner. We examine categorization of insiders; the motives and psychological profiles behind their destructive behavior; and conclude with a discussion of several measures that organizations can implement in order to detect and defend against insider threats

Clinical and Forensic Outcomes from the Illinois Mental Health Juvenile Justice Initiative

To address the mental health needs of youths who are arrested and detained in Illinois, an initiative was designed and implemented that identified youths with psychotic or affective disorders, linked them to community services, and monitored their cases. This study assessed whether such linkage is possible and whether it improves clinical and forensic outcomes. METHODS: Under the initiative, court staff refer youths who may have a mental illness to a clinical liaison. If the youth is eligible for the program, the liaison works with the family to develop a community-based action plan. For the analysis presented here, the Child and Adolescent Needs and Strengths-Mental Health Scale (CANS-MH) and the Child and Adolescent Functional Assessment Scale (CAFAS) were used to assess outcomes among 314 youths who had completed the program at the time of the study. School and forensic outcomes were also monitored. RESULTS: Seventy-five percent of the youths were successfully linked to at least one mental health or community service. A comparison of average CANS-MH dimension scores at enrollment and program completion indicated that youths\u27 emotional problems decreased considerably within three months of referral. CAFAS scores six months after enrollment improved across nearly all dimensions. Home, community, and school functioning were significantly improved from baseline. Only 42 percent of the youths were rearrested, compared with a statewide rate of 72 percent of detained youths. CONCLUSIONS: By linking youths with significant mental health needs to existing community-based services, it appears possible both to ameliorate psychopathology and to reduce delinquency

Light-cone QCD on the lattice

Ideas and recent results for light-front Hamiltonian quantisation of lattice gauge theories.Comment: LATTICE99(Plenary), 5 pages, 3 figure

Nav1.7 expression is increased in painful human dental pulp

© 2008 Luo et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution Licens

Plenary Session: \u3cem\u3eWater Research Needs in Kentucky\u3c/em\u3e

Four panelists were invited to present their visions/predictions of current and future water research needs in Kentucky in an opening plenary session at the 2012 Kentucky Water Resources Annual Symposium. The requested overall general focus was ground water research needs

Reducing Plasmodium falciparum malaria transmission in Africa: a model-based evaluation of intervention strategies.

BACKGROUND: Over the past decade malaria intervention coverage has been scaled up across Africa. However, it remains unclear what overall reduction in transmission is achievable using currently available tools. METHODS AND FINDINGS: We developed an individual-based simulation model for Plasmodium falciparum transmission in an African context incorporating the three major vector species (Anopheles gambiae s.s., An. arabiensis, and An. funestus) with parameters obtained by fitting to parasite prevalence data from 34 transmission settings across Africa. We incorporated the effect of the switch to artemisinin-combination therapy (ACT) and increasing coverage of long-lasting insecticide treated nets (LLINs) from the year 2000 onwards. We then explored the impact on transmission of continued roll-out of LLINs, additional rounds of indoor residual spraying (IRS), mass screening and treatment (MSAT), and a future RTS,S/AS01 vaccine in six representative settings with varying transmission intensity (as summarized by the annual entomological inoculation rate, EIR: 1 setting with low, 3 with moderate, and 2 with high EIRs), vector-species combinations, and patterns of seasonality. In all settings we considered a realistic target of 80% coverage of interventions. In the low-transmission setting (EIR approximately 3 ibppy [infectious bites per person per year]), LLINs have the potential to reduce malaria transmission to low levels (90%) or novel tools and/or substantial social improvements will be required, although considerable reductions in prevalence can be achieved with existing tools and realistic coverage levels. CONCLUSIONS: Interventions using current tools can result in major reductions in P. falciparum malaria transmission and the associated disease burden in Africa. Reduction to the 1% parasite prevalence threshold is possible in low- to moderate-transmission settings when vectors are primarily endophilic (indoor-resting), provided a comprehensive and sustained intervention program is achieved through roll-out of interventions. In high-transmission settings and those in which vectors are mainly exophilic (outdoor-resting), additional new tools that target exophagic (outdoor-biting), exophilic, and partly zoophagic mosquitoes will be required