18 research outputs found
Adjoint bi-continuous semigroups and semigroups on the space of measures
For a given bi-continuous semigroup T on a Banach space X we define its
adjoint on an appropriate closed subspace X^o of the norm dual X'. Under some
abstract conditions this adjoint semigroup is again bi-continuous with respect
to the weak topology (X^o,X). An application is the following: For K a Polish
space we consider operator semigroups on the space C(K) of bounded, continuous
functions (endowed with the compact-open topology) and on the space M(K) of
bounded Baire measures (endowed with the weak*-topology). We show that
bi-continuous semigroups on M(K) are precisely those that are adjoints of a
bi-continuous semigroups on C(K). We also prove that the class of bi-continuous
semigroups on C(K) with respect to the compact-open topology coincides with the
class of equicontinuous semigroups with respect to the strict topology. In
general, if K is not Polish space this is not the case
Rights in conflict? The rationale and implications of using human rights in conflict prevention strategies
Online auction is an essence of many modern markets, particularly networked markets, in which information about goods, agents, and outcomes is revealed over a period of time, and the agents must make irrevocable decisions without knowing future information. Optimal stopping theory, especially the classic "secretary problem", is a powerful tool for analyzing such online scenarios which generally require optimizing an objective function over the input. The secretary problem and its generalization the "multiple-choice secretary problem" were under a thorough study in the literature. In this paper, we consider a very general setting of the latter problem called the "submodular secretary problem", in which the goal is to select k secretaries so as to maximize the expectation of a (not necessarily monotone) submodular function which defines efficiency of the selected secretarial group based on their overlapping skills. We present the first constant-competitive algorithm for this case. In a more general setting in which selected secretaries should form an independent (feasible) set in each of l given matroids as well, we obtain an O(l log^2 r)-competitive algorithm generalizing several previous results, where r is the maximum rank of the matroids. Another generalization is to consider l knapsack constraints instead of the matroid constraints, for which we present an O(l)-competitive algorithm. In a sharp contrast, we show for a more general setting of "subadditive secretary problem, there is no o~(sqrt(n))-competitive algorithm and thus submodular functions are the most general functions to consider for constant competitiveness in our setting. We complement this result by giving a matching O(sqrt(n))-competitive algorithm for the subadditive case. At the end, we consider some special cases of our general setting as well