Side Lengths of a Right Triangle Reproduced as Perimeters of Other Right Triangles: A Sketch and Proof

In this paper, the author builds upon the work of DiDomenico (1995) and Alsina and Nelson (2011). Given an arbitrary right triangle with side lengths a, b, and c, the author illustrates a method for constructing three smaller right triangles with perimeters a, b, and c. The author also provides a link to a dynamic sketch of the construction that can be used to help students informally explore geometric relationships related to the proof

Area and perimeter relationship for different shapes with an inradius

In this paper, we show that an interesting relationship for a triangle, which is Area/Perimeter = r/2, extends to other shapes with an inradius r (i.e., shapes with an inscribed circle). These shapes include squares, circles, rhombuses, regular polygons, and even irregular polygons and more

Physiological Responses and Thermal Comfort of Subjects in a Tractor Cab

P.O. Fanger of Denmark was the first scientist to generalize the physiological basis for predicting human thermal comfort based on activity level, clothing type, air temperature, air velocity, mean radiant temperature, and air humidity. Thus any activity level and clothing type, it is possible to determine combinations of air temperature, mean radiant temperature, air humidity and relative air velocity which would produce optimum thermal comfort. The Fanger approach has been applied successfully to offices and buildings. It would be quite useful to demonstrate that Fanger’s concepts could be applied to tractor cabs. The objectives of this study were as follows: 1. Determine the effect of cab air temperature an velocity on physiological parameters within a tractor cab for summer conditions. 2. Determine in Fanger’s criteria for thermal comfort is applicable to tractor cabs for summer conditions

Homotaurine, a safe blood-brain barrier permeable GABAA-R-specific agonist, ameliorates disease in mouse models of multiple sclerosis.

There is a need for treatments that can safely promote regulatory lymphocyte responses. T cells express GABA receptors (GABAA-Rs) and GABA administration can inhibit Th1-mediated processes such as type 1 diabetes and rheumatoid arthritis in mouse models. Whether GABAA-R agonists can also inhibit Th17-driven processes such as experimental autoimmune encephalomyelitis (EAE), a model of multiple sclerosis (MS), is an open question. GABA does not pass through the blood-brain barrier (BBB) making it ill-suited to inhibit the spreading of autoreactivity within the CNS. Homotaurine is a BBB-permeable amino acid that antagonizes amyloid fibril formation and was found to be safe but ineffective in long-term Alzheimer's disease clinical trials. Homotaurine also acts as GABAA-R agonist with better pharmacokinetics than that of GABA. Working with both monophasic and relapsing-remitting mouse models of EAE, we show that oral administration of homotaurine can (1) enhance CD8+CD122+PD-1+ and CD4+Foxp3+ Treg, but not Breg, responses, (2) inhibit autoreactive Th17 and Th1 responses, and (3) effectively ameliorate ongoing disease. These observations demonstrate the potential of BBB-permeable GABAA-R agonists as a new class of treatment to enhance CD8+ and CD4+ Treg responses and limit Th17 and Th1-medaited inflammation in the CNS

A further critique of the analytic strategy of adjusting for covariates to identify biologic mediation

BACKGROUND: Epidemiologic research is often devoted to etiologic investigation, and so techniques that may facilitate mechanistic inferences are attractive. Some of these techniques rely on rigid and/or unrealistic assumptions, making the biologic inferences tenuous. The methodology investigated here is effect decomposition: the contrast between effect measures estimated with and without adjustment for one or more variables hypothesized to lie on the pathway through which the exposure exerts its effect. This contrast is typically used to distinguish the exposure's indirect effect, through the specified intermediate variables, from its direct effect, transmitted via pathways that do not involve the specified intermediates. METHODS: We apply a causal framework based on latent potential response types to describe the limitations inherent in effect decomposition analysis. For simplicity, we assume three measured binary variables with monotonic effects and randomized exposure, and use difference contrasts as measures of causal effect. Previous authors showed that confounding between intermediate and the outcome threatens the validity of the decomposition strategy, even if exposure is randomized. We define exchangeability conditions for absence of confounding of causal effects of exposure and intermediate, and generate two example populations in which the no-confounding conditions are satisfied. In one population we impose an additional prohibition against unit-level interaction (synergism). We evaluate the performance of the decomposition strategy against true values of the causal effects, as defined by the proportions of latent potential response types in the two populations. RESULTS: We demonstrate that even when there is no confounding, partition of the total effect into direct and indirect effects is not reliably valid. Decomposition is valid only with the additional restriction that the population contain no units in which exposure and intermediate interact to cause the outcome. This restriction implies homogeneity of causal effects across strata of the intermediate. CONCLUSIONS: Reliable effect decomposition requires not only absence of confounding, but also absence of unit-level interaction and use of linear contrasts as measures of causal effect. Epidemiologists should be wary of etiologic inference based on adjusting for intermediates, especially when using ratio effect measures or when absence of interacting potential response types cannot be confidently asserted