3,698 research outputs found
Renormalization : A number theoretical model
We analyse the Dirichlet convolution ring of arithmetic number theoretic
functions. It turns out to fail to be a Hopf algebra on the diagonal, due to
the lack of complete multiplicativity of the product and coproduct. A related
Hopf algebra can be established, which however overcounts the diagonal. We
argue that the mechanism of renormalization in quantum field theory is modelled
after the same principle. Singularities hence arise as a (now continuously
indexed) overcounting on the diagonals. Renormalization is given by the map
from the auxiliary Hopf algebra to the weaker multiplicative structure, called
Hopf gebra, rescaling the diagonals.Comment: 15 pages, extended version of talks delivered at SLC55 Bertinoro,Sep
2005, and the Bob Delbourgo QFT Fest in Hobart, Dec 200
On the shape factor of interaction laws for a non-local approximation of the Sobolev norm and the total variation
We consider the family of non-local and non-convex functionals introduced by
H. Brezis and H.-M. Nguyen in a recent paper. These functionals Gamma-converge
to a multiple of the Sobolev norm or the total variation, depending on a
summability exponent, but the exact values of the constants are unknown in many
cases.
We describe a new approach to the Gamma-convergence result that leads in some
special cases to the exact value of the constants, and to the existence of
smooth recovery families.Comment: Compte-rendu that summarizes the strategy developed in
ArXiv:1708.01231 and ArXiv:1712.04413. This version extends the previous one
keeping into account the changes in the above papers. 9 page
Twofold Optimality of the Relative Utilitarian Bargaining Solution
Given a bargaining problem, the `relative utilitarian' (RU) solution maximizes the sum total of the bargainer's utilities, after having first renormalized each utility function to range from zero to one. We show that RU is `optimal' in two very different senses. First, RU is the maximal element (over the set of all bargaining solutions) under any partial ordering which satisfies certain axioms of fairness and consistency; this result is closely analogous to the result of Segal (2000). Second, RU offers each person the maximum expected utility amongst all rescaling-invariant solutions, when it is applied to a random sequence of future bargaining problems which are generated using a certain class of distributions; this is somewhat reminiscent of the results of Harsanyi (1953) and Karni (1998).relative utilitarian; bargaining solution; impartial observer
Extremality conditions for isolated and dynamical horizons
A maximally rotating Kerr black hole is said to be extremal. In this paper we
introduce the corresponding restrictions for isolated and dynamical horizons.
These reduce to the standard notions for Kerr but in general do not require the
horizon to be either stationary or rotationally symmetric. We consider physical
implications and applications of these results. In particular we introduce a
parameter e which characterizes how close a horizon is to extremality and
should be calculable in numerical simulations.Comment: 13 pages, 4 figures, added reference; v3 appendix added with proof of
result from section IIID, some discussion and references added. Version to
appear in PR
On the Derivative Imbalance and Ambiguity of Functions
In 2007, Carlet and Ding introduced two parameters, denoted by and
, quantifying respectively the balancedness of general functions
between finite Abelian groups and the (global) balancedness of their
derivatives , (providing an
indicator of the nonlinearity of the functions). These authors studied the
properties and cryptographic significance of these two measures. They provided
for S-boxes inequalities relating the nonlinearity to ,
and obtained in particular an upper bound on the nonlinearity which unifies
Sidelnikov-Chabaud-Vaudenay's bound and the covering radius bound. At the
Workshop WCC 2009 and in its postproceedings in 2011, a further study of these
parameters was made; in particular, the first parameter was applied to the
functions where is affine, providing more nonlinearity parameters.
In 2010, motivated by the study of Costas arrays, two parameters called
ambiguity and deficiency were introduced by Panario \emph{et al.} for
permutations over finite Abelian groups to measure the injectivity and
surjectivity of the derivatives respectively. These authors also studied some
fundamental properties and cryptographic significance of these two measures.
Further studies followed without that the second pair of parameters be compared
to the first one.
In the present paper, we observe that ambiguity is the same parameter as
, up to additive and multiplicative constants (i.e. up to rescaling). We
make the necessary work of comparison and unification of the results on ,
respectively on ambiguity, which have been obtained in the five papers devoted
to these parameters. We generalize some known results to any Abelian groups and
we more importantly derive many new results on these parameters
An axiomatization of the Euclidean compromise solution
The utopia point of a multicriteria optimization problem is the vector that specifies for each criterion the most favourable among the feasible values. The Euclidean compromise solution in multicriteria optimization is a solution concept that assigns to a feasible set the alternative with minimal Euclidean distance to the utopia point. The purpose of this paper is to provide a characterization of the Euclidean compromise solution. Consistency plays a crucial role in our approach.Consistency; Euclidean compromise solution; Multicriteria optimization
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