To this end, we propose Time-Aware Multi-Scale Recurrent Neural Networks (TAMS-RNNs), which disentangle representations of different scales and adaptively select the most important scale for each sample at each time step. First, the hidden state of the RNN is disentangled into multiple independently updated small hidden states, which use different update frequencies to model time-series multi-scale information. Then, at each time step, the temporal context information is used to modulate the features of different scales, selecting the most important time-series scale. Therefore, the proposed model can capture the multi-scale information for each time series at each time step adaptively. Extensive experiments demonstrate that the model outperforms state-of-the-art methods on multivariate time series classification and human motion prediction tasks. Furthermore, visualized analysis on music genre recognition verifies the effectiveness of the model.

• We apply the proposed procedure to improve the multiple-scales method to obtain the optimum solution of the forced oscillator with strongly nonlinear restoring and inertial forces.
• This is done by introducing fast-scale and slow-scale variables for an independent variable, and subsequently treating these variables, fast and slow, as if they are independent.
• The scale() function is specified with either one or two values, which represent the amount of scaling to be applied in each direction.
• We propose a novel procedure to improve the solution obtained by perturbation methods for analyzing the solutions of strongly nonlinear systems.
• The solutions obtained from multiple-scales method and the proposed method are examined by the numerical solution obtained from 4th-order Runge-Kutta method.

This agrees with the nonlinear frequency changes found by employing the Lindstedt–Poincaré method. This term is O and has the same order of magnitude as the leading-order term. Because the terms have become disordered, the series is no longer an asymptotic expansion of the solution.

Multiple

We propose a novel procedure to improve the solution obtained by perturbation methods for analyzing the solutions of strongly nonlinear systems. The multiple-scales method, one of the perturbation method, is widely used in many areas. However, Multiple-scales method fails in analyzing the solutions of oscillators if the oscillator nonlinearity is strong. We apply the proposed procedure to improve the multiple-scales method to obtain the optimum solution of the forced oscillator with strongly nonlinear restoring and inertial forces. The solutions obtained from multiple-scales method and the proposed method are examined by the numerical solution obtained from 4th-order Runge-Kutta method.

Also, consider making use of the prefers-reduced-motion media feature — use it to write a media query that will turn off animations if the user has reduced animation specified in their system preferences. Scaling/zooming animations are problematic for accessibility, as they are a common trigger https://wizardsdev.com/ for certain types of migraine. If you need to include such animations on your website, you should provide a control to allow users to turn off animations, preferably site-wide. When a coordinate value is outside the [-1, 1] range, the element grows along that dimension; when inside, it shrinks.

References

The results show that the proposed method is effective for the oscillators with nonlinear restoring force as well as nonlinear inertial force even if the nonlinearities are strong. Numerical results and comparison show that the proposed method can improve the solution lot in comparison to the solution obtained by conventional multiple-scales method. This is done by introducing fast-scale and slow-scale variables for an independent variable, and subsequently treating these variables, fast and slow, as if they are independent.

In the solution process of the perturbation problem thereafter, the resulting additional freedom – introduced by the new independent variables – is used to remove secular terms. The latter puts constraints on the approximate solution, which are called solvability conditions. This scaling transformation is characterized by a two-dimensional vector. If both coordinates are equal, the scaling is uniform and the aspect ratio of the element is preserved . Journals.sagepub.com needs to review the security of your connection before proceeding.

Scaling The X And Y Dimensions Together

Onlinelibrary.wiley.com needs to review the security of your connection before proceeding. More difficult examples are better treated using a time-dependent coordinate transform involving complex exponentials (as also invoked in the previous multiple time-scale approach). A web service will perform the analysis for a wide range of examples. However, this introduces possible ambiguities in the perturbation multi-scale analysis series solution, which require a careful treatment (see Kevorkian & Cole 1996; Bender & Orszag 1999). The scale() function is specified with either one or two values, which represent the amount of scaling to be applied in each direction. Alternatively, modern approaches derive these sorts of models using coordinate transforms, like in the method of normal forms, as described next.