Loss of regularity for p-evolution type models

The goal of the paper is to study the loss of regularity for special p-evolution type models with bounded coefficients in the principal part. The obtained loss of regularity is related in an optimal way to some unboundedness conditions for the derivatives of coefficients up to the second-order with respect to t.     

On the global regularity of generalized Leray-alpha type models

We generalize Leray-alpha type models studied in Cheskidov et al. (2005) [1] and Linshiz and Titi (2007) [4] via fractional Laplacians and employ Besov space techniques to obtain global regularity results with the logarithmically supercritical dissipation.     

Nonlinear models of suspension bridges

Nonlinear variational equations describing one type of suspension bridges are proposed and studied. The variational equations describe the behaviour of road bed, main cables and cable stays. The road bed is described by two functions connected with vertical and horizontal deformation of any cross section. The main cable is considered to be perfectly flexible and inextensible. The cable stays only resist tensile forces. The variational equations are derived from the principle of minimum potential energy. The existence of solution is based on the Brouwer Fixed Point Theorem. The local uniqueness and continuous dependence on the data represented by gravitational forces acting on the road bed are studied. The local results are based on the Implicit Function Theorem for Banach spaces. A certain stability criterion for suspension bridges is formulated and this criterion indicates how to influence the stability of suspension bridges.     

Nonlinear semigroups and age-dependent population models

Summary This paper treats the nonlinear age-dependent population problem (1)(0,a)=(a), a I; (2)(t, 0)=F((t, ·)), t0; (3) ,t0,where I is the age range of the population, (t, ·) is the unknown age density at time t, is the known initial age distribution, and the functionals F and G are nonlinear. The problems of existence, uniqueness, continuous dependence upon initial values, and the positivity of solutions are investigated using the method of nonlinear semigroups.Supported in part by the National Science Foundation Grant NSF 75-06332A01.     

Nonlinear reaction-diffusion models for interacting populations

This paper proves that several initial-boundary value problems for a wide class of nonlinear reaction-diffusion equations have solutions c_i(x, t), 1 ? i ? N (with c_i(x, t) representing the concentration of the ith species at position x in a set $\text{Ω}$ at time t ? 0), which exist for all t ? 0 and are unique, smooth, nonnegative, and strictly positive for t > 0. The Volterra-Lotka predator-prey model with diffusion (to which the results above are proved to apply) is then studied in more detail. It is proved that any bounded solution of this model loses its spatial dependence and behaves like a periodic function of time alone as t → ∞. It is proved that if the spatial dimension is one or if the diffusion coefficients of the two species are equal, then all solutions are bounded.     

Nonlinear instability in advection-diffusion numerical models

The phenomenon of nonlinear instability, which affects all propagation equations, is simply presented and explained; its origin lies in the numerical dispersion of fundamental solutions by the discrete schemes. Methods to avoid it or to minimize its effects on solutions are proposed and discussed.     

Nonlinear autoregressive models with heavy-tailed innovation

In this paper, we discuss the relationship between the stationary marginal tail probability and the innovation's tail probability of nonlinear autoregressive models. We show that under certain conditions that ensure the stationarity and ergodicity, one dimension stationary marginal distribution has the heavy-tailed probability property with the same index as that of the innovation's tail probability.     

Nonlinear time-domain models of human controllers

Data from a compensatory tracking task are analyzed by using time-domain models. The linear time-domain results are transformed and compared with frequency-domain results. The nonlinear time-domain model of the same data reduced the remnant or residual power by only a small amount. The need for testing models on independent data is discussed. A novel, but attractive, method of generating functions for an efficient functional expansion of time-domain models is offered.Notation c Pilot output (control deflection), inches - E Error matrix - e Error, radians - F[·] Fourier transform - h Time interval, seconds - h _i Sample of impulse response of pilot - h _p Impulse response of pilot, inches/radian or inches/degree - i Input (external disturbance function), radians - M Maximum value ofm, M=T _M/ - m Index for the argument ofh _p - N Maximum value ofn - n Index for time - o Linear output of pilot model (control deflection), inches - r Remnant signal of pilot model (control deflection), inches - S Matrix - s Laplace variable - T _M Maximum memory time of the pilot model, seconds - t Time, seconds - Y _c Transfer function of controlled element - Y _c j) Controlled-element transfer function, radians/inch - Y _p j) Pilot-describing function, inches/radian - Argument ofh _p, seconds - Incremental value of, seconds - Frequency, radians/second - ^ Estimate - Absolute value - Phase angle     

Nonlinear parametric models from volterra kernels measurements

Methods for nonlinear system identification are often classified, based on the employed model form, into parametric (nonlinear differential or difference equations) and nonparametric (functional expansions). These methods exhibit distinct sets of advantages and disadvantages that have motivated comparative studies and point to potential benefits from combined use. Fundamental to these studies are the mathematical relations between nonlinear differential (or difference, in discrete time) equations (NDE) and Volterra functional expansions (VFE) of the class of nonlinear systems for which both model forms exist, in continuous or discrete time. Considerable work has been done in obtaining the VFE's of a broad class of NDE's, which can be used to make the transition from nonparametric models (obtained from experimental input-output data) to more compact parametric models. This paper presents a methodology by which this transition can be made in discrete time. Specifically, a method is proposed for obtaining a parametric NARMAX (Nonlinear Auto-Regressive Moving-Average with exogenous input) model from Volterra kernels estimated by use of input-output data.     

Nonlinear nonparametric mixed-effects models for unsupervised classification

In this work we propose a novel EM method for the estimation of nonlinear nonparametric mixed-effects models, aimed at unsupervised classification. We perform simulation studies in order to evaluate the algorithm performance and we apply this new procedure to a real dataset.     

Global regularity for a class of 2D generalized tropical climate models

This paper establishes the global existence and regularity of solutions to a two-dimensional (2D) tropical climate model (TCM) with fractional dissipation. The inviscid counterpart of this model was derived by Frierson, Majda and Pauluis [8] as a model for tropical geophysical flows. This model reflects the interaction and coupling among the barotropic mode u, the first baroclinic mode v of the velocity and the temperature θ. The systems with fractional dissipation studied here may arise in the modeling of geophysical circumstances. Mathematically these systems allow simultaneous examination of a family of systems with various levels of regularization. The aim here is the global regularity with the least dissipation. We prove two main results: first, the global regularity of the system with ${(?\mathrm{\Delta })}^{\beta }v$ and ${(?\mathrm{\Delta })}^{\gamma }\theta$ for $\beta >1$ and $\beta +\gamma >\frac{3}{2}$; and second, the global regularity of the system with ${(?\mathrm{\Delta })}^{\beta }v$ for $\beta >\frac{3}{2}$. The proofs of these results are not trivial and the requirements on the fractional indices appear to be optimal. The key tools employed here include the maximal regularity for general fractional heat operators, the Littlewood–Paley decomposition and Besov space techniques, lower bounds involving fractional Laplacian and simultaneous estimates of several coupled quantities.     

Nonlinear evolution inclusions arising from phase change models

The paper is devoted to the analysis of an abstract evolution inclusion with a non-invertible operator, motivated by problems arising in nonlocal phase separation modeling. Existence, uniqueness, and long-time behaviour of the solution to the related Cauchy problem are discussed in detail. The Italian authors would like to point out financial support from theMIUR-COFIN 2002 research program on “Free boundary problems in applied sciences”. The second author was supported by GA ČR under Grant No. 201/02/1058, and by GNAMPA of INDAM during his stay at Pavia in May 2003: in this respect, the kind hospitality of the Department of Mathematics in Pavia is gratefully acknowledged as well. The work also benefited from partial support of the IMATI of CNR in Pavia, Italy.     

Nonlinear and bilinear models for chemical reactor control

The dynamic behavior of a continuously stirred tank reactor (CSTR) with an exothermic reversible reaction is studied. The balance equations of the reaction lead to a set of highly nonlinear differential equations. For system analysis and control synthesis the dynamic equation are rewritten as state space model. From this nonlinear model a bilinear model is derived. Then, two optimization problems are solved: The time optimal problem for the nonlinear model and the quadratic problem for the bilinear model. In case of the finite time bilinear-quadratic problem a modified Riccati approximation algorithm for a stabilizing feedback controller is presented.     

Nonlinear models of diffusion on a finite space

Summary The basic convergence theorems for finite state Markov chains are extended to the nonlinear case. An operatorT inl ₁ of a finite space with counting measure is called nonexpansive if Tf-Tg₁f-g₁ holds for allf, g. It is shown that, for anyf, there exists an integer p>=1 such thatT ^pnf converges. Sufficient conditions forp=1 are given. In the case of continuous parameter nonexpansive semigrous {T _t, t>=0},T _tf converges fort.The main tool is a geometric theorem on isometriesS of compact subsets of the abovel ₁: It is shown that any orbit underS is finite.The exponential speed of convergence does not extend from the Markov chain case to nonlinearT.This research has been done during a visit of M.A.A. to the University of Göttingen. The principal results were announced in C.R. Math. Rep. Acad. Sci. Canada Vol. VIII, 1. Feb. 1986The research of this author is supported in part by an N.S.E.R.C.-Grant     

A note on the regularity of reduced models obtained by nonlocal quasi-continuum-like approaches

The paper investigates model reduction techniques that are based on a nonlocal quasi-continuum-like approach. These techniques reduce a large optimization problem to either a system of nonlinear equations or another optimization problem that are expressed in a smaller number of degrees of freedom. The reduction is based on the observation that many of the components of the solution of the original optimization problem are well approximated by certain interpolation operators with respect to a restricted set of representative components. Under certain assumptions, the “optimize and interpolate” and the “interpolate and optimize” approaches result in a regular nonlinear equation and an optimization problem whose solutions are close to the solution of the original problem, respectively. The validity of these assumptions is investigated by using examples from potential-based and electronic structure-based calculations in Materials Science models. A methodology is presented for using quasi-continuum-like model reduction for real-space DFT computations in the absence of periodic boundary conditions. The methodology is illustrated using a basic Thomas–Fermi–Dirac case study.     

Energy analysis and improved regularity estimates for multiscale deconvolution models of incompressible flows

This paper presents new analytical results and the first numerical results for a recently proposed multiscale deconvolution model (MDM) recently proposed. The model involves a large‐eddy simulation closure that uses a novel deconvolution approach based on the introduction of two distinct filtering length scales. We establish connections between the MDM and two other models, and, on the basis of one of these connections, we establish an improved regularity estimate for MDM solutions. We also prove that the MDM preserves Taylor‐eddy solutions of the Navier–Stokes equations and therefore does not distort this particular vortex structure. Simulations of the MDM are performed to examine the accuracy of the MDM and the effect of the filtering length scales on energy spectra for three‐dimensional homogeneous and isotropic flows. Numerical evidence for all tests clearly indicates that the MDM gives very accurate coarse‐mesh solutions and that this multiscale approach to deconvolution is effective. Copyright © 2015 John Wiley & Sons, Ltd.