Sentences with phrase «of nonlinear equations»

Nor is there much understanding of the nonlinear equations at the core of climate models — and why that curtails climate prediction.

Each of the members of the ensemble is a non-unique solution of a set of nonlinear equations.

Lorenz was able to show that even for a simple set of nonlinear equations (1.1), the evolution of the solution could be changed by minute perturbations to the initial conditions, in other words, beyond a certain forecast lead time, there is no longer a single, deterministic solution and hence all forecasts must be treated as probabilistic.

Despite growing up in grinding poverty on a Hong Kong farm, Yau made his way to the University of California at Berkeley, where he studied with Chinese geometer Shiing - Shen Chern and the master of nonlinear equations, Charles Morrey.

Not exact matches

His formula called for summing up all the gravitational forces of the universe, whereas Einstein employed a nonlinear equation.

In cases for which reactant and product (A, B, P, Q) concentrations are not held constant but are allowed to follow the usual (nonlinear) equations which govern ordinary diffusion, development of spatial order is to be expected, if the chemical equations of the Brusselator apply.

I had always assumed I was incapable of understanding it, but as Burnham introduced us to fractals, complexity and nonlinear equations I had what I can only call a religious experience.

This relation of the Schrödinger equation to classical waves is already revealed in the way that a variant called the nonlinear Schrödinger equation is commonly used to describe other classical wave systems — for example in optics and even in ocean waves, where it provides a mathematical picture of unusually large and robust «rogue waves.»

In a discipline where one can spend a lifetime working on a single problem, Tao has made major contributions in a number of categories ranging from nonlinear equations to number theory — which explains why colleagues continually seek his guidance.

They converted those relationships into a coupled pair of nonlinear differential equations.

The Einstein equation described the curvature of the universe, and it was nonlinear.

For example, the classical equations used to predict the performance of such materials have indicated that the logarithm of the reaction rate should vary linearly as voltage is increased — but experiments have shown a nonlinear response, with the uptake of lithium flattening out at high voltage.

Demetrios Christodoulou of ETH Zurich in Switzerland and Richard Hamilton of Columbia University in New York City won the mathematics prize for their work on nonlinear partial differential equations in Lorentzian and Riemannian geometry and their applications to general relativity and topology.

In mathematical terms, this phenomenon can be described through exact solutions of the nonlinear Schrödinger equation, also referred to as «breathers.»

To do this, they combined ocean wave data available from measurements taken by ocean buoys, with nonlinear analysis of the underlying water wave equations.

The framework is based on solving nonlinear coupled ordinary and partial differential equations that model the kinetics of the following phenomena: (1) mass transport in the electrolyte and electrode using the Nernst - Planck equation; (2) electrical potential distribution using the Poisson equation; (3) interfacial reactions that determine the boundary conditions or source terms (using the Butler - Volmer equation or constant - flux conditions); and (4) evolution of the electrode / electrolyte interface using the Allen - Cahn equation within the phase - field modeling (PFM) approach.

With Prof. Karlsen has organized a special year on Nonlinear Partial Differential Equations (2008 - 09) at the Centre for Advanced Study of the Norwegian Academy in Science and Letters, Oslo.

Worse, the basic hydrodynamical equations are themselves nonlinear, which makes the description of their interactions and influences even more difficult.

«A dynamical system such as the climate system, governed by nonlinear deterministic equations (see Nonlinearity), may exhibit erratic or chaotic behaviour in the sense that very small changes in the initial state of the system in time lead to large and apparently unpredictable changes in its temporal evolution.

Mind you — I should not encourage the idea that model runs do anything but diverge exponentially from each other due to the nature of the core nonlinear equations.

If one tried to actually write «the» partial differential equation for the global climate system, it would be a set of coupled Navier - Stokes equations with unbelievably nasty nonlinear coupling terms — if one can actually include the physics of the water and carbon cycles in the N - S equations at all.

The equations are nonlinear (this means they involve products or powers of the things we don't know and are trying to solve for).

This spread results because the model equations provide a deterministic set of results that each can be different since the climate is a chaotic nonlinear system both in the model, and even more so in the real world.

Complex nonlinear multivariate systems often exhibit «strange attractors» — local fixed points in a set of coupled nonlinear ordinary differential equations — that function as foci for Poincare cycles in the multivariate phase space.

This, plus the fact that remarkable close simulations of the time series are obtained with a model consisting of a few nonlinear differential equations suggest the intriguing possibility that there are simple rules governing the complex behavior of global paleoclimate.»

It shows thousands of diverging solutions that is the defining property of these chaotic models that have at their core nonlinear equations of fluid transport.

In a system such as the climate, we can never include enough variables to describe the actual system on all relevant length scales (e.g. the butterfly effect — MICROSCOPIC perturbations grow exponentially in time to drive the system to completely different states over macroscopic time) so the best that we can often do is model it as a complex nonlinear set of ordinary differential equations with stochastic noise terms — a generalized Langevin equation or generalized Master equation, as it were — and average behaviors over what one hopes is a spanning set of butterfly - wing perturbations to assess whether or not the resulting system trajectories fill the available phase space uniformly or perhaps are restricted or constrained in some way.

For that, we require that A˜mOrt should be higher than the sum of the nonlinear terms in the original nonlinear barotropic vorticity equation on a sphere (see, e.g., ref.

In this case in particular, the correct formulae are the full nonlinear Navier - Stokes equations with external forcings, implemented in a full thermal model of the Earth.

Any change in a model can produce divergent solutions that are not predictable beforehand — it is the nature of the nonlinear Navier - Stokes equations — this extends to the range of uncertainty in climate data and to the number and breadth of couplings.

Our ability to solve the equation for turbulence fails first, but the failure of the equation itself is not far behind due to its nonlinear structure that amplifies the consequences of the small errors.

Any change in a model can produce divergent solutions that are not predictable beforehand — it is the nature of the nonlinear Navier - Stokes equations.

They have at their core nonlinear equations of fluid transport.

Although the Earth system does seem to share behaviours with these nonlinear sets of equations — is this merely coincidental?

We can not solve the many body atomic state problem in quantum theory exactly any more than we can solve the many body problem exactly in classical theory or the set of open, nonlinear, coupled, damped, driven chaotic Navier - Stokes equations in a non-inertial reference frame that represent the climate system.

You don't really know whether the process is this or that nonlinear differential equation, and you aren't really sure what the distribution of forcing should be, etc..

The time scale of variability of the patterns is longer than the decorrelation time scale of the stochastic forcing, because of the temporal integration of the forcing by the equations of motion limited by the effects of nonlinear dynamics and friction.

The study involved administering all 3 sets of scales to a general population sample who were then interviewed by clinical interviewers blinded to screening scales scores and classified as having or not having SMI based on 12 - month prevalences of DSM - IV disorders, as assessed by the Structured Clinical Interview (SCID) for DSM - IV16 and scores on the GAF.1 Logistic regression analyses were then carried out to estimate the strength of associations between the screening scales and SMI using linear and nonlinear prediction equations that assumed either additive or multiplicative associations among the different screening scales.

On Universality of Transition to Chaos Scenario in Nonlinear Systems of Ordinary Differential Equations of Shilnikov's Type