 NSFPAR ID:
 10327714
 Date Published:
 Journal Name:
 Mathematics of Operations Research
 ISSN:
 0364765X
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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For generalized Korteweg–De Vries (KdV) models with polynomial nonlinearity, we establish a local smoothing property in [Formula: see text] for [Formula: see text]. Such smoothing effect persists globally, provided that the [Formula: see text] norm does not blow up in finite time. More specifically, we show that a translate of the nonlinear part of the solution gains [Formula: see text] derivatives for [Formula: see text]. Following a new simple method, which is of independent interest, we establish that, for [Formula: see text], [Formula: see text] norm of a solution grows at most by [Formula: see text] if [Formula: see text] norm is a priori controlled.more » « less

null (Ed.)Consider a fractional Brownian motion (fBM) [Formula: see text] with Hurst index [Formula: see text]. We construct a probability space supporting both B H and a fully simulatable process [Formula: see text] such that[Formula: see text] with probability one for any userspecified error bound [Formula: see text]. When [Formula: see text], we further enhance our error guarantee to the αHölder norm for any [Formula: see text]. This enables us to extend our algorithm to the simulation of fBMdriven stochastic differential equations [Formula: see text]. Under mild regularity conditions on the drift and diffusion coefficients of Y, we construct a probability space supporting both Y and a fully simulatable process [Formula: see text] such that[Formula: see text] with probability one. Our algorithms enjoy the toleranceenforcement feature, under which the error bounds can be updated sequentially in an efficient way. Thus, the algorithms can be readily combined with other advanced simulation techniques to estimate the expectations of functionals of fBMs efficiently.more » « less

We investigate the problem of recovering jointly [Formula: see text]rank and [Formula: see text]bisparse matrices from as few linear measurements as possible, considering arbitrary measurements as well as rankone measurements. In both cases, we show that [Formula: see text] measurements make the recovery possible in theory, meaning via a nonpractical algorithm. In case of arbitrary measurements, we investigate the possibility of achieving practical recovery via an iterativehardthresholding algorithm when [Formula: see text] for some exponent [Formula: see text]. We show that this is feasible for [Formula: see text], and that the proposed analysis cannot cover the case [Formula: see text]. The precise value of the optimal exponent [Formula: see text] is the object of a question, raised but unresolved in this paper, about head projections for the jointly lowrank and bisparse structure. Some related questions are partially answered in passing. For rankone measurements, we suggest on arcane grounds an iterativehardthresholding algorithm modified to exploit the nonstandard restricted isometry property obeyed by this type of measurements.more » « less

Auxetic foams exhibit novel mechanical properties due to their unique microstructure for improved energyabsorption and cavity expansion applications that have fascinated the scientific community since their inception. Given the advancements in material processing and performance of polymer open cell auxetic foams, there is a strong desire to fully understand the nonlinear ratedependent deformation of these materials. The influence of nonlinear compressibility is introduced here along with relaxation effects to improve model predictions for different stretch rates and finite deformation regimes. The viscoelastic behavior of the material is analyzed by comparing fractional order and integer order calculus models. All results are statistically validated using maximum entropy methods to obtain Bayesian posterior densities for the hyperelastic, auxetic, and viscoelastic parameters. It is shown that fractional order viscoelasticity provides [Formula: see text]–[Formula: see text] improvement in prediction over integer order viscoelastic models when the model is calibrated at higher stretch rates where viscoelasticity is more significant.

We consider a [Formula: see text] system of hyperbolic balance laws that is the converted form under inverse Hopf–Cole transformation of a Keller–Segel type chemotaxis model. We study Cauchy problem when Cauchy data connect two different endstates as [Formula: see text]. The background wave is a diffusive contact wave of the reduced system. We establish global existence of solution and study the time asymptotic behavior. In the special case where the cellular population initially approaches its stable equilibrium value as [Formula: see text], we obtain nonlinear stability of the diffusive contact wave under smallness assumption. In the general case where the population initially does not approach to its stable equilibrium value at least at one of the far fields, we use a correction function in the time asymptotic ansatz, and show that the population approaches logistically to its stable equilibrium value. Our result shows two significant differences when comparing to Euler equations with damping. The first one is the existence of a secondary wave in the time asymptotic ansatz. This implies that our solutions converge to the diffusive contact wave slower than those of Euler equations with damping. The second one is that the correction function logistically grows rather than exponentially decays.more » « less