8,082 research outputs found
Discrete Midpoint Convexity
For a function defined on a convex set in a Euclidean space, midpoint
convexity is the property requiring that the value of the function at the
midpoint of any line segment is not greater than the average of its values at
the endpoints of the line segment. Midpoint convexity is a well-known
characterization of ordinary convexity under very mild assumptions. For a
function defined on the integer lattice, we consider the analogous notion of
discrete midpoint convexity, a discrete version of midpoint convexity where the
value of the function at the (possibly noninteger) midpoint is replaced by the
average of the function values at the integer round-up and round-down of the
midpoint. It is known that discrete midpoint convexity on all line segments
with integer endpoints characterizes L-convexity, and that it
characterizes submodularity if we restrict the endpoints of the line segments
to be at -distance one. By considering discrete midpoint convexity
for all pairs at -distance equal to two or not smaller than two,
we identify new classes of discrete convex functions, called local and global
discrete midpoint convex functions, which are strictly between the classes of
L-convex and integrally convex functions, and are shown to be
stable under scaling and addition. Furthermore, a proximity theorem, with the
same small proximity bound as that for L-convex functions, is
established for discrete midpoint convex functions. Relevant examples of
classes of local and global discrete midpoint convex functions are provided.Comment: 39 pages, 6 figures, to appear in Mathematics of Operations Researc
Landau-De Gennes theory of nematic liquid\ud crystals: the Oseen-Frank limit and beyond
We study global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals, in three-dimensional domains, subject to uniaxial boundary conditions. We analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly, in W 1,2 , to a global minimizer predicted by the Oseen-Frank theory for uniaxial nematic liquid crystals with constant order parameter. Moreover, the convergence is uniform in the interior of the domain, away from the singularities of the limiting Oseen-Frank global minimizer. We obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the Landau-De Gennes global minimizer.\ud
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We also study the interplay between biaxiality and uniaxiality in Landau-De Gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions
Kicked Burgers Turbulence
Burgers turbulence subject to a force ,
where the 's are ``kicking times'' and the ``impulses'' have
arbitrary space dependence, combines features of the purely decaying and the
continuously forced cases. With large-scale forcing this ``kicked'' Burgers
turbulence presents many of the regimes proposed by E, Khanin, Mazel and Sinai
(1997) for the case of random white-in-time forcing. It is also amenable to
efficient numerical simulations in the inviscid limit, using a modification of
the Fast Legendre Transform method developed for decaying Burgers turbulence by
Noullez and Vergassola (1994). For the kicked case, concepts such as
``minimizers'' and ``main shock'', which play crucial roles in recent
developments for forced Burgers turbulence, become elementary since everything
can be constructed from simple two-dimensional area-preserving Euler--Lagrange
maps.
One key result is for the case of identical deterministic kicks which are
periodic and analytic in space and are applied periodically in time: the
probability densities of large negative velocity gradients and of
(not-too-large) negative velocity increments follow the power law with -7/2
exponent proposed by E {\it et al}. (1997) in the inviscid limit, whose
existence is still controversial in the case of white-in-time forcing. (More in
the full-length abstract at the beginning of the paper.)Comment: LATEX 30 pages, 11 figures, J. Fluid Mech, in pres
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