4,977 research outputs found
Constructing Numerical Semigroups of a Given Genus
Let n_g denote the number of numerical semigroups of genus g. Bras-Amoros
conjectured that n_g possesses certain Fibonacci-like properties. Almost all
previous attempts at proving this conjecture were based on analyzing the
semigroup tree. We offer a new, simpler approach to counting numerical
semigroups of a given genus. Our method gives direct constructions of families
of numerical semigroups, without referring to the generators or the semigroup
tree. In particular, we give an improved asymptotic lower bound for n_g.Comment: 11 pages, 3 figures, 2 tables; accepted by Semigroup Foru
Images of Galois representations in mod Hecke algebras
Let denote the mod local Hecke algebra
attached to a normalised Hecke eigenform , which is a commutative algebra
over some finite field of characteristic and with residue
field . By a result of Carayol we know that, if the residual
Galois representation
is
absolutely irreducible, then one can attach to this algebra a Galois
representation that
is a lift of . We will show how one can determine the image
of under the assumptions that the image of the residual
representation contains , that
and that the coefficient ring is generated by the
traces. As an application we will see that the methods that we use allow us to
deduce the existence of certain -elementary abelian extensions of big
non-solvable number fields
Eliciting Socially Optimal Rankings from Unfair Jurors
A jury must provide a ranking of contestants (students applying for scholarships or Ph. D. programs, gymnasts in a competition, etc.). There exists a true ranking which is common knowledge among the jurors, but it is not verifiable. The socially optimal rule is that the contestants be ranked according to the true ranking. The jurors are not impartial and, for example, may have friends (contestants that they would like to benefit) and enemies (contestants that they would like to prejudice). We study necessary and sufficient conditions on the jury under which the socially optimal rule is Nash implementable. We also propose a simple mechanism that Nash implements the socially optimal rule under these conditions.Ranking of contestants; Implementation Theory; Nash Equilibrium
Nash Implementation and Uncertain Renegotiation
This paper studies Nash implementation when the outcomes of the mechanism can be renegotiated among the agents but the planner does not know the renegotiation function that they will use. We characterize the social objectives that can be implemented in Nash equilibrium when the same mechanism must work for every admissible renegotiation function. The constrained Walrasian correspondence, the core correspondence, and the Pareto-efficient and envy-free correspondence satisfy the necessary and sufficient conditions for this form of implementation if and only if freedisposal of the commodities is allowed. The uniform rule, on the other hand, is not Nash implementable for some admissible renegotiations functions.Implementation theory, Nash equilibrium, renegotiation function.
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