12,682 research outputs found

    Polytropic equation of state and primordial quantum fluctuations

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    We study the primordial Universe in a cosmological model where inflation is driven by a fluid with a polytropic equation of state p=αρ+kρ1+1/np = \alpha\rho + k\rho^{1 + 1/n}. We calculate the dynamics of the scalar factor and build a Universe with constant density at the origin. We also find the equivalent scalar field that could create such equation of state and calculate the corresponding slow-roll parameters. We calculate the scalar perturbations, the scalar power spectrum and the spectral index.Comment: 16 pages, 4 figure

    Bounds and extremal domains for Robin eigenvalues with negative boundary parameter

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    We present some new bounds for the first Robin eigenvalue with a negative boundary parameter. These include the constant volume problem, where the bounds are based on the shrinking coordinate method, and a proof that in the fixed perimeter case the disk maximises the first eigenvalue for all values of the parameter. This is in contrast with what happens in the constant area problem, where the disk is the maximiser only for small values of the boundary parameter. We also present sharp upper and lower bounds for the first eigenvalue of the ball and spherical shells. These results are complemented by the numerical optimisation of the first four and two eigenvalues in 2 and 3 dimensions, respectively, and an evaluation of the quality of the upper bounds obtained. We also study the bifurcations from the ball as the boundary parameter becomes large (negative).Comment: 26 pages, 20 figure

    On the behavior of clamped plates under large compression

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    We determine the asymptotic behavior of eigenvalues of clamped plates under large compression by relating this problem to eigenvalues of the Laplacian with Robin boundary conditions. Using the method of fundamental solutions, we then carry out a numerical study of the extremal domains for the first eigenvalue, from which we see that these depend on the value of the compression, and start developing a boundary structure as this parameter is increased. The corresponding number of nodal domains of the first eigenfunction of the extremal domain also increases with the compression.This work was partially supported by the Funda ̧c ̃ao para a Ciˆencia e a Tecnologia(Portugal) through the program “Investigador FCT” with reference IF/00177/2013 and the projectExtremal spectral quantities and related problems(PTDC/MAT-CAL/4334/2014).info:eu-repo/semantics/publishedVersio

    Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin Laplacian

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    We consider the problem of minimising the nthn^{th}-eigenvalue of the Robin Laplacian in RN\mathbb{R}^{N}. Although for n=1,2n=1,2 and a positive boundary parameter α\alpha it is known that the minimisers do not depend on α\alpha, we demonstrate numerically that this will not always be the case and illustrate how the optimiser will depend on α\alpha. We derive a Wolf-Keller type result for this problem and show that optimal eigenvalues grow at most with n1/Nn^{1/N}, which is in sharp contrast with the Weyl asymptotics for a fixed domain. We further show that the gap between consecutive eigenvalues does go to zero as nn goes to infinity. Numerical results then support the conjecture that for each nn there exists a positive value of αn\alpha_{n} such that the nthn^{\rm th} eigenvalue is minimised by nn disks for all 0<α<αn0<\alpha<\alpha_{n} and, combined with analytic estimates, that this value is expected to grow with n1/Nn^{1/N}
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