Unexpected local extrema for the Sendov conjecture

Let S(n) be the set of all polynomials of degree n with all roots in the unit disk, and define d(P) to be the maximum of the distances from each of the roots of a polynomial P to that root's nearest critical point. In this notation, Sendov's conjecture asserts that d(P)<=1 for every P in S(n). Define P in S(n) to be locally extremal if d(P)>=d(Q) for all nearby Q in S(n), and note that maximizing d(P) over all locally extremal polynomials P would settle the Sendov conjecture. Prior to now, the only polynomials known to be locally extremal were of the form P(z)=c(z^n+a) with |a|=1. In this paper, we determine sufficient conditions for real polynomials of a different form to be locally extremal, and we use these conditions to find locally extremal polynomials of this form of degrees 8, 9, 12, 13, 14, 15, 19, 20, and 26.Comment: 10 pages, AMS-LaTeX, no figures. The Maple code and results used in this paper are included in the source files. We constructed an unexpected locally extremal polynomial of degree 8 in version 1, then added degrees 12, 14, 20 and 26 in version 2, and degrees 9, 13, 15 and 19 in version

Sustainable construction for the future - the role of government in energy efficiency and sustainability in buildings

This paper will summarise the findings from a study that explored the link between dwelling design, or type, and energy efficiencies in sub-tropical climates. An increasing number of government and private sector development companies are initiating projects that aim to deliver enhanced environmental outcomes at both sub-divisional and dwelling levels. The study used AccuRate, a new thermal modelling tool developed by CSIRO that responds to the need to improve ventilation modelling. The study found that dwellings developed in conjunction with the Departments of Housing and Public Works have set the benchmark. It provides a snapshot of the energy efficiency of a range of dwelling types found in recent subdivisions. However, the trend toward increasing urban densities may reduce the likelihood that cooling breezes will be available to cool dwellings. The findings are relevant to regulators, designers and industry in all states interested in reducing the energy used to cool dwellings in summer

Energy efficient multi storey residential developments

Worldwide, the current pattern of urban development is unsustainable and metropolitan planning and development strategies deliver poor environmental outcomes in relation to energy production. As a result, an increasing number of governments and private sector development companies are initiating projects that aim to deliver enhanced environmental outcomes rather than a ‘business as usual’ approach. This paper will summarise the findings from a study that explored the link between building orientation and energy efficiencies in sub-tropical and tropical climates. The study used a new thermal modelling software tool developed by CSIRO that responds more accurately to residential heating and cooling energy performance in those climate zones. This software tool responds to industry criticisms regarding cold climate modelling systems that do not make sufficient allowance for natural ventilation. The study examined a range of low, medium and high-density dwelling types and investigated the impact of orientation, insulation, ventilation and shading devices on energy efficiencies. This paper will examine the findings from the medium and high-density case study developments as these are relevant to residential developments in many South East Asian countries, such as Singapore, Hong Kong and Malaysia. Finally, the paper will explore the potential benefits that medium and high-density residential developments have in the development of ‘solar cities’ and ‘solar suburbs’

How to achieve sustainability : regulatory challenges

The importance of designing sustainable buildings is gaining greater acceptance worldwide. Evidence of this is how regulators are incorporating sustainable design principles into building regulations and requirements. The aim being to increase the number of sustainable buildings and move from a traditional voluntary compliance to one that is mandatory. However, developing regulations that actually achieve these aims can be a difficult exercise. Several countries in South East Asia, such as Singapore and Malaysia, have performance based building regulations that are supplemented by prescriptive measures for achieving the desired performance. Australia too has similar building regulations and has had energy efficiency regulations within the Building Code of Australia for over a decade. This paper explores some of the difficulties and problems that Australian regulators have experienced with the performance-based method and the prescriptive or “deemed-to-comply” method and measures that have been taken to try and overcome these problems. These experiences act as a useful guide to all regulators considering the incorporation of sustainable design measures into their countries building regulations. The paper also speculates on future environmental requirements being incorporated into regulations, including the possibility of non-residential buildings being required to meet minimum energy efficiency requirements, and the possible systems that would need to be in place before such requirements were included. Finally, the paper looks at a possible way forward using direct assessment from electronic designs and introduces several software tools that are currently being developed that move towards achieving this goal. Keywords: Sustainable buildings, Performance-based, Regulations, Energy efficiency, Assessment tools