Velocity-stack inversion is the process of creating a model in velocity space that can correctly reconstruct the measured data. This is usually implemented by minimizing the L2 norm of the data misfit and the L2 norm of the model. Superior velocity space images, with a better separation of primaries and multiples, can be created by minimizing a different norm of the data misfit and a different ...

In the last lecture we defined metric spaces, normed spaces, and considered the distortion resulting from certain embeddings. In particular, we proved that l1 norms cannot always be embedded isometrically into l2 by considering a specific four-point l1 norm and showing that it requires at least √ 2 distortion. Today’s lecture further explores the 1 norm. We see a couple of interesting examples ...

We study the Cauchy problem for a strictly hyperbolic n×n system of conservation laws in one space dimension: ut + f(u)x = 0, u(0, x) = ū(x). The initial data ū is a small BV perturbation of a single rarefaction wave with an arbitrary strength. All characteristic fields are assumed to be genuinely nonlinear or linearly degenerate in the vicinity of the reference rarefaction curve. We prove that...

In this paper, we develop an analytic solution for the best one-sided approximation of polynomials under L1 norm, that is, we 0nd two polynomials with lower degree which bound the given polynomial such that the areas between the bounding polynomials and the given polynomial attain minimum. The key ingredient of our technique is a characterization for one-sided approximations based on orthogonal...

Recently, In [Phys. Rev. Lett. 104, 018701 (2010)] the authors studied a spatial network which is constructed from a regular lattice by adding long-range edges (shortcuts) with probability Pij ∼ r −α ij , where rij is the Manhattan length of the long-range edges. The total length of the additional edges is subject to a cost constraint ( ∑ r = C). These networks have fixed optimal exponent α for...

The Canberra distance is the sum of absolute values of the differences between ranks divided by their sum, thus it is a weighted version of the L1 distance. As a metric on permutation groups, the Canberra distance is a measure of disarray for ranked lists, where rank differences in top positions need to pay higher penalties than movements in the bottom part of the lists. Here we describe the di...

For d ∈ N and Ω 6= ∅ an open set in R, we consider the eigenfunctions Φ of the Dirichlet Laplacian −∆Ω of Ω. If Φ is associated with an eigenvalue below the essential spectrum of −∆Ω we provide estimates for the L1-norm of Φ in terms of its L2-norm and spectral data. These L1estimates are then used in the comparison of the heat content of Ω at time t > 0 and the heat trace at times t′ > 0, wher...