662,617 research outputs found
Liberal approaches to ranking infinite utility streams: When can we avoid interferences?
In this work we analyse social welfare relations on sets of infinite utility streams that verify various types of liberal non-interference principles. Earlier contributions have established that (finitely) anonymous and strongly Paretian quasiorderings exist that agree with axioms of that kind together with weak preference continuity and further consistency. Nevertheless Mariotti and Veneziani prove that a fully liberal non-interfering view of a finite society leads to dictatorship if weak Pareto optimality is imposed. We first prove that extending the horizon to infinity produces a reversal of such impossibility result. Then we investigate a related problem: namely, the possibility of combining âstandardâ semicontinuity with efficiency in the presence of non-interference. We provide several impossibility results that prove that there is a generalised incompatibility between continuity and non-interference principles, both under ordinal and cardinal views of the problem. Our analysis ends with some insights on the property of representability in the presence of non-interference assumptions. In particular we prove that all social welfare functions that verify a very mild efficiency property must exert some interference (penalising both adverse and favorable changes) on the affairs of particular generations.Pareto axiom; Intergenerational justice; Social welfare relation; Non-interference; Continuity
Stabilized Schemes for the Hydrostatic Stokes Equations
Some new stable finite element (FE) schemes are presented for the hydrostatic Stokes
system or primitive equations of the ocean. It is known that the stability of the mixed formulation ap-
proximation for primitive equations requires the well-known LadyzhenskayaâBabuËskaâBrezzi condi-
tion related to the Stokes problem and an extra inf-sup condition relating the pressure and the vertical
velocity.
The main goal of this paper is to avoid this extra condition by adding a residual stabilizing term to the
vertical momentum equation. Then, the stability for Stokes-stable FE combinations is extended to
the primitive equations and some error estimates are provided using TaylorâHood P2 âP1 or miniele-
ment (P1 +bubble)âP1 FE approximations, showing the optimal convergence rate in the P2 âP1 case.
These results are also extended to the anisotropic (nonhydrostatic) problem. On the other hand,
by adding another residual term to the continuity equation, a better approximation of the vertical
derivative of pressure is obtained. In this case, stability and error estimates including this better
approximation are deduced, where optimal convergence rate is deduced in the (P 1 +bubble)âP1 case.
Finally, some numerical experiments are presented supporting previous results
Exponential self-similar mixing by incompressible flows
We study the problem of the optimal mixing of a passive scalar under the
action of an incompressible flow in two space dimensions. The scalar solves the
continuity equation with a divergence-free velocity field, which satisfies a
bound in the Sobolev space , where and . The mixing properties are given in terms of a characteristic length
scale, called the mixing scale. We consider two notions of mixing scale, one
functional, expressed in terms of the homogeneous Sobolev norm ,
the other geometric, related to rearrangements of sets. We study rates of decay
in time of both scales under self-similar mixing. For the case and (including the case of Lipschitz continuous velocities, and
the case of physical interest of enstrophy-constrained flows), we present
examples of velocity fields and initial configurations for the scalar that
saturate the exponential lower bound, established in previous works, on the
time decay of both scales. We also present several consequences for the
geometry of regular Lagrangian flows associated to Sobolev velocity fields.Comment: To appear in Journal of the American Mathematical Society. Some
results were announced in G. Alberti, G. Crippa, A. L. Mazzucato,
"Exponential self-similar mixing and loss of regularity for continuity
equations", C. R. Math. Acad. Sci. Paris, 352(11):901--906, 2014,
arXiv:1407.2631v
BSDEs with weak terminal condition
We introduce a new class of Backward Stochastic Differential Equations in
which the -terminal value of the solution is not fixed as a
random variable, but only satisfies a weak constraint of the form
, for some (possibly random) non-decreasing map and
some threshold . We name them \textit{BSDEs with weak terminal condition}
and obtain a representation of the minimal time -values such that
is a supersolution of the BSDE with weak terminal condition. It
provides a non-Markovian BSDE formulation of the PDE characterization obtained
for Markovian stochastic target problems under controlled loss in Bouchard,
Elie and Touzi \cite{BoElTo09}. We then study the main properties of this
minimal value. In particular, we analyze its continuity and convexity with
respect to the -parameter appearing in the weak terminal condition, and show
how it can be related to a dual optimal control problem in Meyer form. These
last properties generalize to a non Markovian framework previous results on
quantile hedging and hedging under loss constraints obtained in F\"{o}llmer and
Leukert \cite{FoLe99,FoLe00}, and in Bouchard, Elie and Touzi \cite{BoElTo09}
Frechet Differentiable Norm and Locally Uniformly Rotund Renormings
In this paper, we study briefly the role played by the locally uniformly rotund (LUR) norm and Frechet differentiability of a norm on the Banach space theory. Our old outstanding open Problem 3.8 mentioned below is the main object of this paper. We study nearly about it and find some additional assumptions on the space attached with this problem to obtain its positive or negative answer. We investigate different results related to these norms and their duals on different settings. In particular, we introduce reflexive spaces, Banach spaces with unconditional basis, weakly locally uniformly rotund (WLUR) norm, Almost locally uniformly rotund (ALUR) norm, strongly exposed point, sub-differentiability and Ï”-sub-differentiability, Ïâslicely continuity, weakly compactly generated (WCG) Banach spaces with ck âsmooth norms, Symulianâs Theorem, and some technical lemmas
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