Homological properties of Banach and C*-algebras of continuous fields

PhD ThesisOne concern in the homological theory of Banach algebras is the identification of projective algebras and projective closed ideals of algebras. Besides being of independent interest, this question is closely connected to the continuous Hochschild cohomology. In this thesis we give necessary and sufficient conditions for the left projectivity and biprojectivity of Banach algebras defined by locally trivial continuous fields of Banach algebras. We identify projective C*-algebras A defined by locally trivial continuous fields U = fW, (At)t2W,Qg such that each C*-algebra At has a strictly positive element. We also identify projective Banach algebras A defined by locally trivial continuous fields U = fW, (K(Et))t2W,Qg such that each Banach space Et has an extended unconditional basis. In particular, for a left projective Banach algebra A defined by locally trivial continuous fields U = fW, (At)t2W,Qg we prove that W is paracompact. We also show that the biprojectivity of A implies that W is discrete. In the case U is a continuous field of elementary C*-algebras satisfying Fell’s condition (not nessecarily a locally trivial field) we show that the left projectivity of A defined by U, under some additional conditions on U, implies paracompactness of W. For the above Banach algebras A we give applications to the second continuous Hochschild cohomology group H2(A, X) of A and to the strong splittability of singular extensions of A

The fluctuation of year-classes and the regulation of fisheries

There are three sources of variation of recruitment; first that due to random fluctuations alone, secondly that, due to variations in stock and thirdly variations which are extreme at low and high levels of stock and if these occur sequentially can cause catastrophic declines or sharp increases in stock density. From a study of stock/recruitment curves of a number of fish species, it appears that the curve for herring-like fishes differs markedly from those, for example, of cod-like fishes or in more general terms that the shape of the stock/recruitment curve is really a function of its fecundity. That for the herring-like fishes is a near linear curve whereas that for the cod-like fishes is markedly dome-shaped. It follows that the herring-like fishes are very vulnerable to long-term climatic changes and that the cod-like fishes are able to stabilize their populations through long-term climatic changes merely because the increased recruitment when stock is slightly reduced is very much greater than that for the herring-like fishes. If stocks of fish are to be managed then the stock/recruitment relatioilship has to be taken into account; a form of self regenerating yield curve combining stock/recruitment and yield-per-recruit was in fact put forward by BEVERTON and HOLT. A development of such a curve is needed and one of the uses of such a curve would he that management need not take too much notice of environmental change, because if it occurred the stock would have been exploited at its best rate before any environmental collapse took place

The fluctuation of year-classes and the regulation of fisheries

There are three sources of variation of recruitment; first that due to random fluctuations alone, secondly that, due to variations in stock and thirdly variations which are extreme at low and high levels of stock and if these occur sequentially can cause catastrophic declines or sharp increases in stock density. From a study of stock/recruitment curves of a number of fish species, it appears that the curve for herring-like fishes differs markedly from those, for example, of cod-like fishes or in more general terms that the shape of the stock/recruitment curve is really a function of its fecundity. That for the herring-like fishes is a near linear curve whereas that for the cod-like fishes is markedly dome-shaped. It follows that the herring-like fishes are very vulnerable to long-term climatic changes and that the cod-like fishes are able to stabilize their populations through long-term climatic changes merely because the increased recruitment when stock is slightly reduced is very much greater than that for the herring-like fishes. If stocks of fish are to be managed then the stock/recruitment relatioilship has to be taken into account; a form of self regenerating yield curve combining stock/recruitment and yield-per-recruit was in fact put forward by BEVERTON and HOLT. A development of such a curve is needed and one of the uses of such a curve would he that management need not take too much notice of environmental change, because if it occurred the stock would have been exploited at its best rate before any environmental collapse took place

You need 27 tickets to guarantee a win on the UK National Lottery

In the UK National Lottery, players purchase tickets comprising their choices of six different numbers between 1 and 59. During the draw, six balls are randomly selected without replacement from a set numbered from 1 to 59. A prize is awarded to any player who matches at least two of the six drawn numbers. We identify 27 tickets that guarantee a prize, regardless of which of the 45,057,474 possible draws occurs. Moreover, we determine that 27 is the optimal number of tickets required, as achieving the same guarantee with 26 tickets is not possible.Comment: 23 page