292 research outputs found
Some homogenization and corrector results for nonlinear monotone operators
This paper deals with the limit behaviour of the solutions of quasi-linear
equations of the form \ \ds -\limfunc{div}\left(a\left(x, x/{\varepsilon
_h},Du_h\right)\right)=f_h on with Dirichlet boundary conditions.
The sequence tends to and the map is
periodic in , monotone in and satisfies suitable continuity
conditions. It is proved that weakly in , where is the solution of a homogenized problem \
-\limfunc{div}(b(x,Du))=f on . We also prove some corrector results,
i.e. we find such that in
Correctors for some nonlinear monotone operators
In this paper we study homogenization of quasi-linear partial differential
equations of the form -\mbox{div}\left( a\left( x,x/\varepsilon _h,Du_h\right)
\right) =f_h on with Dirichlet boundary conditions. Here the
sequence tends to as
and the map is periodic in monotone in
and satisfies suitable continuity conditions. We prove that
weakly in as where
is the solution of a homogenized problem of the form -\mbox{div}\left(
b\left( x,Du\right) \right) =f on We also derive an explicit
expression for the homogenized operator and prove some corrector results,
i.e. we find such that in
Constraints on unroofing rates in the high Himalaya, eastern Nepal
Thermobarometric data for samples across the Main Central thrust zone in eastern Nepal show an inversion in temperature but not in pressure. These data have been interpreted to represent a portion of the paleogeotherm at the time of Main Central thrust deformation. A 40Ar/39Ar age on hornblende (closure temperature (Tc)=500±50°C) constrains the timing of this deformation to be ∼21±0.2 Ma. The 40Ar/39Ar ages of other minerals (muscovite, Tc=350°C, age (t)=12.0±0.2 Ma; K-feldspar, Tc=220°C, t=8.0±0.2 Ma) from the same location further constrain the cooling history of this region. Together the geochronologic and thermobarometric data yield an average unroofing rate of 1.2±0.6 mm/yr for the High Himalaya of eastern Nepal.
Simple thermal models show that these geochronologic and thermobarometric data are consistent with a wide range of different initial geotherms, applied boundary conditions and magnitude of radiogenic heat production. The variation through time of the unroofing rates can only be poorly constrained, however. The unroofing histories were found to be largely insensitive to the details of the assumed initial geotherm, fairly sensitive to the magnitude of radiogenic heat production, and extremely sensitive to the nature of the boundary conditions applied below the fault zone. This study underscores the difficulty in constraining uplift histories on the basis of cooling rates even when thermobarometric data are available to supplement geochronologic constraints on the cooling history of the region
Barycentric decomposition of quantum measurements in finite dimensions
We analyze the convex structure of the set of positive operator valued
measures (POVMs) representing quantum measurements on a given finite
dimensional quantum system, with outcomes in a given locally compact Hausdorff
space. The extreme points of the convex set are operator valued measures
concentrated on a finite set of k \le d^2 points of the outcome space, d<
\infty being the dimension of the Hilbert space. We prove that for second
countable outcome spaces any POVM admits a Choquet representation as the
barycenter of the set of extreme points with respect to a suitable probability
measure. In the general case, Krein-Milman theorem is invoked to represent
POVMs as barycenters of a certain set of POVMs concentrated on k \le d^2 points
of the outcome space.Comment: !5 pages, no figure
Geometric Approach to Pontryagin's Maximum Principle
Since the second half of the 20th century, Pontryagin's Maximum Principle has
been widely discussed and used as a method to solve optimal control problems in
medicine, robotics, finance, engineering, astronomy. Here, we focus on the
proof and on the understanding of this Principle, using as much geometric ideas
and geometric tools as possible. This approach provides a better and clearer
understanding of the Principle and, in particular, of the role of the abnormal
extremals. These extremals are interesting because they do not depend on the
cost function, but only on the control system. Moreover, they were discarded as
solutions until the nineties, when examples of strict abnormal optimal curves
were found. In order to give a detailed exposition of the proof, the paper is
mostly self\textendash{}contained, which forces us to consider different areas
in mathematics such as algebra, analysis, geometry.Comment: Final version. Minors changes have been made. 56 page
Theory of differential inclusions and its application in mechanics
The following chapter deals with systems of differential equations with
discontinuous right-hand sides. The key question is how to define the solutions
of such systems. The most adequate approach is to treat discontinuous systems
as systems with multivalued right-hand sides (differential inclusions). In this
work three well-known definitions of solution of discontinuous system are
considered. We will demonstrate the difference between these definitions and
their application to different mechanical problems. Mathematical models of
drilling systems with discontinuous friction torque characteristics are
considered. Here, opposite to classical Coulomb symmetric friction law, the
friction torque characteristic is asymmetrical. Problem of sudden load change
is studied. Analytical methods of investigation of systems with such
asymmetrical friction based on the use of Lyapunov functions are demonstrated.
The Watt governor and Chua system are considered to show different aspects of
computer modeling of discontinuous systems
Statistical-Thermodynamic Model for Light Scattering from Eye Lens Protein Mixtures
We model light-scattering cross sections of concentrated aqueous mixtures of the bovine eye lens proteins γB- and α-crystallin by adapting a statistical-thermodynamic model of mixtures of spheres with short-range attractions. The model reproduces measured static light scattering cross sections, or Rayleigh ratios, of γB-α mixtures from dilute concentrations where light scattering intensity depends on molecular weights and virial coefficients, to realistically high concentration protein mixtures like those of the lens. The model relates γB-γB and γB-α attraction strengths and the γB-α size ratio to the free energy curvatures that set light scattering efficiency in tandem with protein refractive index increments. The model includes (i) hard-sphere α-α interactions, which create short-range order and transparency at high protein concentrations, (ii) short-range attractive plus hard-core γ-γ interactions, which produce intense light scattering and liquid-liquid phase separation in aqueous γ-crystallin solutions, and (iii) short-range attractive plus hard-core γ-α interactions, which strongly influence highly non-additive light scattering and phase separation in concentrated γ-α mixtures. The model reveals a new lens transparency mechanism, that prominent equilibrium composition fluctuations can be perpendicular to the refractive index gradient. The model reproduces the concave-up dependence of the Rayleigh ratio on α/γ composition at high concentrations, its concave-down nature at intermediate concentrations, non-monotonic dependence of light scattering on γ-α attraction strength, and more intricate, temperature-dependent features. We analytically compute the mixed virial series for light scattering efficiency through third order for the sticky-sphere mixture, and find that the full model represents the available light scattering data at concentrations several times those where the second and third mixed virial contributions fail. The model indicates that increased γ-γ attraction can raise γ-α mixture light scattering far more than it does for solutions of γ-crystallin alone, and can produce marked turbidity tens of degrees celsius above liquid-liquid separation
On Linear Programming Duality and Necessary and Sufficient Conditions in Minimax Theory
In this paper we discuss necessary and sufficient conditions for different minimax results to hold using only linear programming duality and the finite intersection property for compact sets. It turns out that these necessary and sufficient conditions have a clear interpretation within zero-sum game theory. We apply these results to derive necessary and sufficient conditions for strong duality for a general class of optimization problems
Haptic pop-out of movable stimuli
When, in visual and haptic search, a target is easily found among distractors, this is called a pop-out effect. The target feature is then believed to be salient, and the search is performed in a parallel way. We investigated this effect with movable stimuli in a haptic search task. The task was to find a movable ball among anchored distractors or the other way round. Results show that reaction times were independent of the number of distractors if the movable ball was the target but increased with the number of items if the anchored ball was the target. Analysis of hand movements revealed a parallel search strategy, shorter movement paths, a higher average movement speed, and a narrower direction distribution with the movable target, as compared with a more detailed search for an anchored target. Taken together, these results show that a movable object pops out between anchored objects and this indicates that movability is a salient object feature. Vibratory signals resulting from the movable ball were found to be a reasonable explanation regarding the sensation responsible for the pop-out of movability
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