570 research outputs found
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
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