Homological Domination in Large Random Simplicial Complexes

In this paper we state the homological domination principle for random multi-parameter simplicial complexes, claiming that the Betti number in one specific dimension (which is explicitly determined by the probability multi-parameter) significantly dominates the Betti numbers in all other dimensions. We also state and discuss evidence for two interesting conjectures which would imply a stronger version of the homological domination principle, namely that generically homology of a random simplicial complex coincides with that of a wedges of k-dimensional spheres. These two conjectures imply that under an additional assumption (specified in the paper) a random simplicial complex collapses to a k-dimensional complex homotopy equivalent to a wedge of spheres of dimension k.Comment: 8 pages, 1 figur

Solar energy, its conversion and utilization

The work being carried out at the University of Florida Solar Energy and Energy Conversion Laboratory in converting solar energy, our only income, into other needed and useful forms of energy is described. A treatment such as this demonstrates, in proper perspective, how solar energy can benefit mankind with its many problems of shortages and pollution. Descriptions were given of the conversion processes, equipment, and performance. The testing of materials, solar water heating, space heating, cooking and baking, solar distillation, refrigeration and air-conditioning, work with the solar furnace, conversion to mechanical power, hot air engines, solar-heated sewage digestion, conversion to electricity, and other devices will be discussed

California Climate Law---Model or Object Lesson?

In the invitation to this Symposium on Reconceptualizing the Future of Environmental Law, the organizers explained that the Symposium “focuses on the continued expansion of environmental law into distinct areas of the law, requiring an increasingly multidisciplinary approach beyond that of traditional federal regulation.” In short, the question posed is about the future proliferation of environmental measures outside the previous domains of federal environmental statutes. At the risk of being guilty of local parochialism, I would like to discuss how the future described by the organizers has already arrived in California--both in the sense that a great deal is happening outside the purview of “federal statutes,” and that much of it involves “distinct areas of law” other than traditional environmental regulation. My focus will be on the issue of climate change, where California has been particularly active. Not all of California\u27s efforts have been met with approval, even from observers who are highly sympathetic to the goals. Some influential environmental scholars have debated whether California might have done better to simply set a price on carbon and avoid further regulatory apparatus, either by traditional regulators or elsewhere. I will use this debate to discuss some of the costs and benefits of mainstreaming environmental law into areas outside of the traditional environmental statutes. Part I will address California\u27s broad portfolio of climate measures. These measures certainly fit the organizer\u27s description: none of them are federal, but many are implemented by parts of the state government other than environmental agencies, and some reach forms of conduct well outside traditional environmental regulation. Part II will ask whether the breadth of this regulatory portfolio is really desirable: would we be better off to stick to a simple direct attack on carbon emissions? Using so many different tools may simply be an unnecessary complication, if not counterproductive. But a broad portfolio might also be more effective in some ways