Lagrangian Data-Driven Reduced Order Modeling of Finite Time Lyapunov Exponents

There are two main strategies for improving the projection-based reduced order model (ROM) accuracy: (i) improving the ROM, i.e., adding new terms to the standard ROM; and (ii) improving the ROM basis, i.e., constructing ROM bases that yield more accurate ROMs. In this paper, we use the latter. We propose new Lagrangian inner products that we use together with Eulerian and Lagrangian data to construct new Lagrangian ROMs. We show that the new Lagrangian ROMs are orders of magnitude more accurate than the standard Eulerian ROMs, i.e., ROMs that use standard Eulerian inner product and data to construct the ROM basis. Specifically, for the quasi-geostrophic equations, we show that the new Lagrangian ROMs are more accurate than the standard Eulerian ROMs in approximating not only Lagrangian fields (e.g., the finite time Lyapunov exponent (FTLE)), but also Eulerian fields (e.g., the streamfunction). We emphasize that the new Lagrangian ROMs do not employ any closure modeling to model the effect of discarded modes (which is standard procedure for low-dimensional ROMs of complex nonlinear systems). Thus, the dramatic increase in the new Lagrangian ROMs' accuracy is entirely due to the novel Lagrangian inner products used to build the Lagrangian ROM basis

Polarized Deep Inelastic Scattering Off the "Neutron" From Gauge/String Duality

We investigate deep inelastic scattering off the polarized "neutron" using gauge/string duality. The "neutron" corresponds to a supergravity mode of the neutral dilatino. Through introducing the Pauli interaction term into the action in $\textrm{AdS}_{5}$ space, we calculate the polarized deep inelastic structure functions of the "neutron" in supergravity approximation at large t' Hooft coupling $\lambda$ and finite $x$ with $\lambda^{-1/2}\ll x<1$. In comparison with the charged dilatino "proton," which has been obtained in the previous work by Gao and Xiao, we find the structure functions of the "neutron" are power suppressed at the same order as the ones of the "proton." Especially, we find the Burkhardt-Cottingham-like sum rule, which is satisfied in the work by Gao and Xiao, is broken due to the Pauli interaction term. We also illustrate how such a Pauli interaction term can arise naturally from higher dimensional fermion-graviton coupling through the usual Kaluza-Klein reduction.Comment: 21pages,5figures, published versio

Drifting diffusion on a circle as continuous limit of a multiurn Ehrenfest model

We study the continuous limit of a multibox Erhenfest urn model proposed before by the authors. The evolution of the resulting continuous system is governed by a differential equation, which describes a diffusion process on a circle with a nonzero drifting velocity. The short time behavior of this diffusion process is obtained directly by solving the equation, while the long time behavior is derived using the Poisson summation formula. They reproduce the previous results in the large $M$ (number of boxes) limit. We also discuss the connection between this diffusion equation and the Schr$\ddot{\rm o}$dinger equation of some quantum mechanical problems.Comment: 4 pages prevtex4 file, 1 eps figur