An E-Learning Investigation into Learning Style Adaptivity

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Spectrum of one dimensional p-Laplacian operator with indefinite weight

This paper is concerned with the nonlinear boundary eigenvalue problem $$-(|u'|^{p-2}u')'=\lambda m|u|^{p-2}u\qquad u \in I=]a,b[,\quad u(a)=u(b)=0,$$ where $p>1$, $\lambda$ is a real parameter, $m$ is an indefinite weight, and $a$, $b$ are real numbers. We prove there exists a unique sequence of eigenvalues for this problem. Each eigenvalue is simple and verifies the strict monotonicity property with respect to the weight $m$ and the domain $I$, the k-th eigenfunction, corresponding to the $k$-th eigenvalue, has exactly $k-1$ zeros in $(a,b)$. At the end, we give a simple variational formulation of eigenvalues

Measurement of the intrinsic damping constant in individual nanodisks of YIG and YIG{\textbar}Pt

We report on an experimental study on the spin-waves relaxation rate in two series of nanodisks of diameter $\phi=$300, 500 and 700~nm, patterned out of two systems: a 20~nm thick yttrium iron garnet (YIG) film grown by pulsed laser deposition either bare or covered by 13~nm of Pt. Using a magnetic resonance force microscope, we measure precisely the ferromagnetic resonance linewidth of each individual YIG and YIG{\textbar}Pt nanodisks. We find that the linewidth in the nanostructure is sensibly smaller than the one measured in the extended film. Analysis of the frequency dependence of the spectral linewidth indicates that the improvement is principally due to the suppression of the inhomogeneous part of the broadening due to geometrical confinement, suggesting that only the homogeneous broadening contributes to the linewidth of the nanostructure. For the bare YIG nano-disks, the broadening is associated to a damping constant $\alpha = 4 \cdot 10^{-4}$. A 3 fold increase of the linewidth is observed for the series with Pt cap layer, attributed to the spin pumping effect. The measured enhancement allows to extract the spin mixing conductance found to be $G_{\uparrow \downarrow}= 1.55 \cdot 10^{14}~ \Omega^{-1}\text{m}^{-2}$ for our YIG(20nm){\textbar}Pt interface, thus opening large opportunities for the design of YIG based nanostructures with optimized magnetic losses.Comment: 4 pages, 3 figure

Nearly total spin polarization in La2/3Sr1/3MnO3 from tunneling experiments

We have performed magnetotransport measurements on La2/3Sr1/3MnO3 / SrTiO3 / La2/3Sr1/3MnO3 magnetic tunnel junctions. A magnetoresistance ratio of more than 1800 % is obtained at 4K, from which we infer an electrode spin polarization of at least 95 %. This result strongly underscores the half-metallic nature of mixed-valence manganites and demonstrates its capability as a spin analyzer. The magnetoresistance extends up to temperatures of more than 270K. We argue that these improvements over most previous works may result from optimizing the patterning process for oxide heterostructures.Comment: to appear in Applied Physics Letter

Spectrum of the Ap-Laplacian Operator

This work deals with the nonlinear boundary eigenvalue problem(V:P(Gammaho;I)):-A_p u = lambda ho(x)|u|^{p-2}u in I =], b[,u(a) = u(b) = 0,where A_p is called the A_p-Laplacian operator and defined by A_p u = (Gamma(x) |u'|^{p-2}u'),p > 1, lambda is a real parameter, ho is an indefinite weight, a, b are real numbers and Gamma in C^1(I) cap C^0(overline{I}) and it is nonnegative on overline{I}.We prove in this paper that the spectrum of the A_p-Laplacian operator is given by a sequence of eigenvalues. Moreover, each eigenvalue is simple, isolated andverifies the strict monotonicity property with respect to the weight ho and the domain I. The k¡th eigenfunction corresponding to the k-th eigenvalue has exactly k-1 zeros in (a,b). Finally, we give a simple variational formulation of eigenvalues