1,273 research outputs found
A Holder Continuous Nowhere Improvable Function with Derivative Singular Distribution
We present a class of functions in which is variant
of the Knopp class of nowhere differentiable functions. We derive estimates
which establish \mathcal{K} \sub C^{0,\al}(\R) for 0<\al<1 but no is pointwise anywhere improvable to C^{0,\be} for any \be>\al.
In particular, all 's are nowhere differentiable with derivatives singular
distributions. furnishes explicit realizations of the functional
analytic result of Berezhnoi.
Recently, the author and simulteously others laid the foundations of
Vector-Valued Calculus of Variations in (Katzourakis), of
-Extremal Quasiconformal maps (Capogna and Raich, Katzourakis) and of
Optimal Lipschitz Extensions of maps (Sheffield and Smart). The "Euler-Lagrange
PDE" of Calculus of Variations in is the nonlinear nondivergence
form Aronsson PDE with as special case the -Laplacian.
Using , we construct singular solutions for these PDEs. In the
scalar case, we partially answered the open regularity problem of
Viscosity Solutions to Aronsson's PDE (Katzourakis). In the vector case, the
solutions can not be rigorously interpreted by existing PDE theories and
justify our new theory of Contact solutions for fully nonlinear systems
(Katzourakis). Validity of arguments of our new theory and failure of classical
approaches both rely on the properties of .Comment: 5 figures, accepted to SeMA Journal (2012), to appea
Existence and uniqueness of global solutions to fully nonlinear second order elliptic systems
We consider the problem of existence and uniqueness of strong a.e. solutions u:Rnâ¶RNu:Rnâ¶RN to the fully nonlinear PDE system
F(â
,D2u)=f, a.e. on Rn,(1)
F(â
,D2u)=f, a.e. on Rn,(1)
when fâL2(Rn)NfâL2(Rn)N and F is a CarathĂ©odory map. (1) has not been considered before. The case of bounded domains has been studied by several authors, firstly by Campanato and under Campanatoâs ellipticity condition on F. By introducing a new much weaker notion of ellipticity, we prove solvability of (1) in a tailored Sobolev âenergyâ space and a uniqueness estimate. The proof is based on the solvability of the linearised problem by Fourier transform methods, together with a âperturbation deviceâ which allows to use Campanatoâs near operators. We also discuss our hypothesis via counterexamples and give a stability theorem of strong global solutions for systems of the form (1)
Quasivariational solutions for first order quasilinear equations with gradient constraint
We prove the existence of solutions for an evolution quasi-variational
inequality with a first order quasilinear operator and a variable convex set,
which is characterized by a constraint on the absolute value of the gradient
that depends on the solution itself. The only required assumption on the
nonlinearity of this constraint is its continuity and positivity. The method
relies on an appropriate parabolic regularization and suitable {\em a priori}
estimates. We obtain also the existence of stationary solutions, by studying
the asymptotic behaviour in time. In the variational case, corresponding to a
constraint independent of the solution, we also give uniqueness results
Nonlinear Dynamics of Aeolian Sand Ripples
We study the initial instability of flat sand surface and further nonlinear
dynamics of wind ripples. The proposed continuous model of ripple formation
allowed us to simulate the development of a typical asymmetric ripple shape and
the evolution of sand ripple pattern. We suggest that this evolution occurs via
ripple merger preceded by several soliton-like interaction of ripples.Comment: 6 pages, 3 figures, corrected 2 typo
Interpolating orientation fields : an axiomatic approach
International audienc
First search for gravitational waves from the youngest known neutron star
We present a search for periodic gravitational waves from the neutron star in the supernova remnant Cassiopeia
A. The search coherently analyzes data in a 12 day interval taken from the fifth science run of the Laser
Interferometer Gravitational-Wave Observatory. It searches gravitational-wave frequencies from 100 to 300 Hz
and covers a wide range of first and second frequency derivatives appropriate for the age of the remnant and
for different spin-down mechanisms. No gravitational-wave signal was detected. Within the range of search
frequencies, we set 95% confidence upper limits of (0.7â1.2) Ă 10^(â24) on the intrinsic gravitational-wave
strain, (0.4â4) Ă 10^(â4) on the equatorial ellipticity of the neutron star, and 0.005â0.14 on the amplitude of
r-mode oscillations of the neutron star. These direct upper limits beat indirect limits derived from energy
conservation and enter the range of theoretical predictions involving crystalline exotic matter or runaway r-modes.
This paper is also the first gravitational-wave search to present upper limits on the r-mode amplitude
Mapping trait versus species turnover reveals spatiotemporal variation in functional redundancy and network robustness in a plant-pollinator community
Functional overlap among species (redundancy) is considered important in shaping competitive and mutualistic interactions that determine how communities respond to environmental change. Most studies view functional redundancy as static, yet traits within speciesâwhich ultimately shape functional redundancyâcan vary over seasonal or spatial gradients. We therefore have limited understanding of how trait turnover within and between species could lead to changes in functional redundancy or how loss of traits could differentially impact mutualistic interactions depending on where and when the interactions occur in space and time. Using an Arctic bumblebee community as a case study, and 1277 individual measures from 14 species over three annual seasons, we quantified how inter- and intraspecific body-size turnover compared to species turnover with elevation and over the season. Coupling every individual and their trait with a plant visitation, we investigated how grouping individuals by a morphological trait or by species identity altered our assessment of network structure and how this differed in space and time. Finally, we tested how the sensitivity of the network in space and time differed when simulating extinction of nodes representing either morphological trait similarity or traditional species groups. This allowed us to explore the degree to which trait-based groups increase or decrease interaction redundancy relative to species-based nodes. We found that (i) groups of taxonomically and morphologically similar bees turn over in space and time independently from each other, with trait turnover being larger over the season; (ii) networks composed of nodes representing species versus morphologically similar bees were structured differently; and (iii) simulated loss of bee trait groups caused faster coextinction of bumblebee species and flowering plants than when bee taxonomic groups were lost. Crucially, the magnitude of these effects varied in space and time, highlighting the importance of considering spatiotemporal context when studying the relative importance of taxonomic and trait contributions to interaction network architecture. Our finding that functional redundancy varies spatiotemporally demonstrates how considering the traits of individuals within networks is needed to understand the impacts of environmental variation and extinction on ecosystem functioning and resilience
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