Isotropic Realizability Research Articles

How does an incomplete sky coverage affect the Hubble Constant variance?

We address the simeq 4.4sigma tension between local and the CMB measurements of the Hubble Constant using simulated Type Ia Supernova (SN) data-sets. We probe its directional dependence by means of a hemispherical comparison through the entire celestial sphere as an estimator of the H_0 cosmic variance. We perform Monte Carlo simulations assuming isotropic and non-uniform distributions of data points, the latter coinciding with the real data. This allows us to incorporate observational features, such as the sample incompleteness, in our estimation. We obtain that this tension can be alleviated to 3.4sigma for isotropic realizations, and 2.7sigma for non-uniform ones. We also find that the H_0 variance is largely reduced if the data-sets are augmented to 4 and 10 times the current size. Future surveys will be able to tell whether the Hubble Constant tension happens due to unaccounted cosmic variance, or whether it is an actual indication of physics beyond the standard cosmological model.

Isotropic Realizability of Fields and Reconstruction of Invariant Measures under Positivity Properties. Asymptotics of the Flow by a Non-Ergodic Approach

The paper is devoted to the isotropic realizability of a regular gradient field u or a more general vector field b, namely the existence of a continuous positive function σ such that σb is divergence free in R d or in an open set of R d. First, we prove that under some suitable positivity condition satisfied by u, the isotropic realizability of u holds either in R d if u does not vanish, or in the open sets {c j < u < c j+1 } if the c j are the critical values of u (including inf R d u and sup R d u) which are assumed to be in finite number. It turns out that this positivity condition is not sufficient to ensure the existence of a continuous positive invariant measure σ on the torus when u is periodic. Then, we establish a new criterium of the existence of an invariant measure for the flow associated with a regular periodic vector field b, which is based on the equality b · v = 1 in R d. We show that this gradient invertibility is not related to the classical ergodic assumption, but it actually appears as an alternative to get the asymptotics of the flow.

Reconstruction of isotropic conductivities from non smooth electric fields

In this paper we study the isotropic realizability of a given non smooth gradient field ∇u defined in ℝd, namely when one can reconstruct an isotropic conductivity σ &gt; 0 such that σ∇u is divergence free in ℝd. On the one hand, in the case where ∇u is non-vanishing, uniformly continuous in ℝd and Δu is a bounded function in ℝd, we prove the isotropic realizability of ∇u using the associated gradient flow combined with the DiPerna, Lions approach for solving ordinary differential equations in suitable Sobolev spaces. On the other hand, in the case where ∇u is piecewise regular, we prove roughly speaking that the isotropic realizability holds if and only if the normal derivatives of u on each side of the gradient discontinuity interfaces have the same sign. Some examples of conductivity reconstruction are given.

Picard group of isotropic realizations of twisted Poisson manifolds

Let $B$ be a twisted Poisson manifold with a fixed tropical affine structure given by a period bundle $P$. In this paper, we study the classification of almost symplectically complete isotropic realizations (ASCIRs) over $B$ in the spirit of [10]. We construct a product among ASCIRs in analogy with tensor product of line bundles, thereby introducing the notion of the Picard group of $B$. We give descriptions of the Picard group in terms of exact sequences involving certain sheaf cohomology groups, and find that the `Neron-Severi group' is isomorphic to $H^2(B, \underline{P})$. An example of an ASCIR over a certain open subset of a compact Lie group is discussed.

Isotropic realizability of a strain field for the two−dimensional incompressible elasticity system

In the paper we study the problem of the isotropic realizability in of a regular strain field for the incompressible elasticity system, namely the existence of a positive shear modulus solving the elasticity system in with the prescribed field e(U). We show that if e(U) does not vanish at some point, then the isotropic realizability holds in the neighborhood of that point. The global realizability in or in the torus is much more delicate, since it involves the global existence of a regular solution to a semilinear wave equation, the coefficients of which depend on the derivatives of U. Using this semilinear wave equation we prove a small perturbation result: if DU is periodic and close enough to its average value for the C4−norm, then the associated strain field is isotropically realizable in a given disk centered at the origin. On the other hand, a counterexample shows that the global realizability in may hold without the realizability in the torus, and it is discussed in connection with the associated semilinear wave equation. The case where the strain field vanishes is illustrated by an example. The singular case of a rank-one laminate field is also investigated.

Isotropic Realizability of Current Fields in $\mathbb{R}^3$

This paper deals with the isotropic realizability of a given regular divergence free field $j$ in $\mathbb{R}^3$ as a current field, namely, to know when $j$ can be written as $\sigma\nabla u$ for some isotropic conductivity $\sigma>0$ and some gradient field $\nabla u$. The local isotropic realizability in $\mathbb{R}^3$ is obtained by Frobenius' theorem provided that $j$ and $\mbox{curl}\,j$ are orthogonal in $\mathbb{R}^3$. A counterexample shows that Frobenius' condition is not sufficient to derive the global isotropic realizability in $\mathbb{R}^3$. However, assuming that $(j,\mbox{curl}\,j,j\times\mbox{curl}\,j)$ is an orthogonal basis of $\mathbb{R}^3$, an admissible conductivity $\sigma$ is constructed from a combination of the three dynamical flows along the directions $j/|j|$, $\mbox{curl}\,j/|\mbox{curl}\,j|,$ and $(j/|j|^2)\times\mbox{curl}\,j$. When the field $j$ is periodic, the isotropic realizability in the torus needs in addition a boundedness assumption satisfied by the flow along the t...

Which electric fields are realizable in conducting materials?

In this paper we study the realizability of a given smooth periodic gradient field $\nabla u$ defined in $\RR^d$, in the sense of finding when one can obtain a matrix conductivity $\si$ such that $\si\nabla u$ is a divergence free current field. The construction is shown to be always possible locally in $\RR^d$ provided that $\nabla u$ is non-vanishing. This condition is also necessary in dimension two but not in dimension three. In fact the realizability may fail for non-regular gradient fields, and in general the conductivity cannot be both periodic and isotropic. However, using a dynamical systems approach the isotropic realizability is proved to hold in the whole space (without periodicity) under the assumption that the gradient does not vanish anywhere. Moreover, a sharp condition is obtained to ensure the isotropic realizability in the torus. The realizability of a matrix field is also investigated both in the periodic case and in the laminate case. In this context the sign of the matrix field determinant plays an essential role according to the space dimension.

Isotropic realizability of electric fields around critical points

In this paper we study the isotropic realizability of a given regular gradient field $\nabla u$ as an electric field, namely when $u$ is solution to the equation div$\left(\sigma\nabla u\right)=0$ for some isotropic conductivity $\sigma>0$. The case of a smooth function $u$ without critical point was investigated in [7] thanks to a dynamical system approach which yields a global isotropic realizability result in $\mathbb{R}^d$. The presence of a critical point $x^*$ needs a specific treatment according to the behavior of the gradient flow in the neighborhood of $x^*$. The case where the hessian matrix $\nabla^2 u(x^*)$ is invertible with both positive and negative eigenvalues is the most favorable: the anisotropic realizability is a consequence of Morse's lemma, while the Hadamard-Perron theorem leads us to a characterization of the isotropic realizability around $x^*$ through some boundedness condition involving the laplacian of $u$ along the gradient flow. When the matrix $\nabla^2 u(x^*)$ has $d$ positive eigenvalues or $d$ negative eigenvalues, we get a strong maximum principle under the same boundedness condition. However, when the matrix $\nabla^2 u(x^*)$ is not invertible, the derivation of the isotropic realizability is much more intricate: the Hartman-Wintner theorem gives necessary conditions for the isotropic realizability in dimension two, while the dynamical system approach provides a criterion of non realizability in any dimension. The two methods are illustrated by a two-dimensional and a three-dimensional example.

Simulation of lognormal random fields with varying resolution scale and local average for Darcy flow

The paper presents a new method for generating isotropic realizations of a random field with varying resolution scales and local block averages in one, two, or three dimensions. The random field has a lognormal distribution as found for the hydraulic conductivity in Darcy's flow in hydrology. We also present a fast method for generating anisotropic realizations of aGaussian random field using superposition of harmonic modes. Both methods are implemented in a software package which is presented. The realizations of the rescaled random field are made up ofupscaled local averages of the underlying random field which are consistentwith the level of resolution and which can be conditioned. The approach is motivated by the need to represent engineering properties aslocal averages and to be able to easily condition the realizations to incorporate known data or to change the resolution within sub-regions. The numerical results show that the method is very efficient computationally.

Random field models of heterogeneous materials

One of the main challenges in solid mechanics lies in the passage from a heterogeneous microstructure to an approximating continuum model. In many cases (e.g. stochastic finite elements, statistical fracture mechanics), the interest lies in resolution of stress and other dependent fields over scales not infinitely larger than the typical microscale. This may be accomplished with the help of a meso-scale window which becomes the classical representative volume element (RVE) in the infinite limit. It turns out that the material properties at such a mesoscale cannot be uniquely approximated by a random field of stiffness/compliance with locally isotropic realizations, but rather two random continuum fields with locally anisotropic realizations, corresponding, respectively, to essential and natural boundary conditions on the meso-scale, need to be introduced to bound the material response from above and from below. We study the first- and second-order characteristics of these two meso-scale random fields for anti-plane elastic response of random matrix-inclusion composites over a wide range of contrasts and aspect ratios. Special attention is given to the convergence of effective responses obtained from the essential and natural boundary conditions, which sheds light on the minimum size of an RVE. Additionally, the spatial correlation structure of the crack density tensor with the meso-scale moduli is studied.