Simple formulas for lattice paths avoiding certain periodic staircase boundaries

There is a strikingly simple classical formula for the number of lattice paths avoiding the line x = ky when k is a positive integer. We show that the natural generalization of this simple formula continues to hold when the line x = ky is replaced by certain periodic staircase boundaries--but only under special conditions. The simple formula fails in general, and it remains an open question to what extent our results can be further generalized.Comment: Accepted version (JCTA); proof of Corollary 7 expanded, and 2 new refs adde

Assessing housing quality and its impact on health, safety and sustainability

Background The adverse health and environmental effects of poor housing quality are well established. A central requirement for evidence-based policies and programmes to improve housing standards is a valid, reliable and practical way of measuring housing quality that is supported by policy agencies, the housing sector, researchers and the public. Methods This paper provides guidance on the development of housing quality-assessment tools that link practical measures of housing conditions to their effects on health, safety and sustainability, with particular reference to tools developed in New Zealand and England. Results The authors describe how information on housing quality can support individuals, agencies and the private sector to make worthwhile improvements to the health, safety and sustainability of housing. The information gathered and the resultant tools developed should be guided by the multiple purposes and end users of this information. Other important issues outlined include deciding on the scope, detailed content, practical administration issues and how the information will be analysed and summarised for its intended end users. There are likely to be considerable benefits from increased international collaboration and standardisation of approaches to measuring housing hazards. At the same time, these assessment approaches need to consider local factors such as climate, geography, culture, predominating building practices, important housing-related health issues and existing building codes. Conclusions An effective housing quality-assessment tool has a central role in supporting improvements to housing. The issues discussed in this paper are designed to motivate and assist the development of such tools

Adaptive gait changes due to spectacle magnification and dioptric blur in older people

YesPurpose. A recent study suggested that updated spectacles could increase falls rate in older people. We hypothesized that this may be due to changes in spectacle magnification and this study assessed the effects of spectacle magnification on adaptive gait. Methods. Adaptive gait and visual function was measured in 10 older adults (mean age 77.1 ¿ 4.3 years) with the participants¿ optimal refractive correction and when blurred with +1.00DS, +2.00DS, -1.00DS and -2.00DS lenses. Adaptive gait measurements for the lead and trail foot included foot position before the step, toe clearance of the step edge and foot position on the step. Vision measurements included visual acuity, contrast sensitivity and stereoacuity. Results. The blur lenses led to equal decrements in visual acuity and stereoacuity for the +1.00DS and -1.00DS and the +2.00DS and -2.00DS lenses. However, they had very different effects on adaptive gait compared to the optimal correction: Positive blur lenses led to an increased distance of the feet from the step, increased vertical toe clearance and reduced distance of the lead heel position on the step. Negative lenses led to the opposite of these changes. Conclusion. The adaptive gait changes did not mirror the effects of blur on vision, but were driven by the magnification changes of the lenses. Steps appear closer and larger with positive lenses and further away and smaller with negative ones. Magnification likely explains the mobility problems some older adults have with updated spectacles and after cataract surgery.Federation of Ophthalmic and Dispensing Optician

Thermal analysis of the southern Powder River Basin, Wyoming

Journal ArticleTemperature and geologic data from over 3000 oil and gas wells within a 180 km x 30 km area that transect across the southern Powder River Basin in Wyoming, U.S.A., were used to determine the present thermal regime of the basin. Three-dimensional temperature fields within the transect, based on corrected bottom-hole temperatures (BHTs) and other geologic information, were assessed using: (1) A laterally constant temperature gradient model in conjunction with an L1 norm inversion method, and (2) a laterally variable temperature gradient model in conjunction with a stochastic inversion technique

Microwave-assisted synthesis and electrochemical evaluation of VO2 (B) nanostructures

Understanding how intercalation materials change during electrochemical operation is paramount to optimizing their behaviour and function and in situ characterization methods allow us to observe these changes without sample destruction. Here we first report the improved intercalation properties of bronze phase vanadium dioxide VO2 (B) prepared by a microwave-assisted route which exhibits a larger electrochemical capacity (232 mAh g-1) compared with VO2 (B) prepared by a solvothermal route (197 mAh g-1). These electrochemical differences have also been followed using in situ X-ray absorption spectroscopy allowing us to follow oxidation state changes as they occur during battery operation