Relative asymptotic equivalence of dynamic equations on time scales

This paper aims to study the relative equivalence of the solutions of the following dynamic equations $y^{\Delta }(t)=A(t)y(t)$ y Δ ( t ) = A ( t ) y ( t ) and $x^{\Delta }(t)=A(t)x(t)+f(t,x(t))$ x Δ ( t ) = A ( t ) x ( t ) + f ( t , x ( t ) ) in the sense that if $y(t)$ y ( t ) is a given solution of the unperturbed system, we provide sufficient conditions to prove that there exists a family of solutions $x(t)$ x ( t ) for the perturbed system such that $\Vert y(t)-x(t) \Vert =o( \Vert y(t) \Vert )$ ∥ y ( t ) − x ( t ) ∥ = o ( ∥ y ( t ) ∥ ) , as $t\rightarrow \infty $ t → ∞ , and conversely, given a solution $x(t)$ x ( t ) of the perturbed system, we give sufficient conditions for the existence of a family of solutions $y(t)$ y ( t ) for the unperturbed system, and such that $\Vert y(t)-x(t) \Vert =o( \Vert x(t) \Vert )$ ∥ y ( t ) − x ( t ) ∥ = o ( ∥ x ( t ) ∥ ) , as $t\rightarrow \infty $ t → ∞ ; and in doing so, we have to extend Rodrigues inequality, the Lyapunov exponents, and the polynomial exponential trichotomy on time scales.

Comparison of ARIMA and LSTM for prediction of hemorrhagic fever at different time scales in China

Objectives This study intends to build and compare two kinds of forecasting models at different time scales for hemorrhagic fever incidence in China. Methods Autoregressive Integrated Moving Average (ARIMA) and Long Short-Term Memory Neural Network (LSTM) were adopted to fit monthly, weekly and daily incidence of hemorrhagic fever in China from 2013 to 2018. The two models, combined and uncombined with rolling forecasts, were used to predict the incidence in 2019 to examine their stability and applicability. Results ARIMA (2, 1, 1) (0, 1, 1)12, ARIMA (1, 1, 3) (1, 1, 1)52 and ARIMA (5, 0, 1) were selected as the best fitting ARIMA model for monthly, weekly and daily incidence series, respectively. The LSTM model with 64 neurons and Stochastic Gradient Descent (SGDM) for monthly incidence, 8 neurons and Adaptive Moment Estimation (Adam) for weekly incidence, and 64 neurons and Root Mean Square Prop (RMSprop) for daily incidence were selected as the best fitting LSTM models. The values of root mean square error (RMSE), mean absolute error (MAE) and mean absolute percentage error (MAPE) of the models combined with rolling forecasts in 2019 were lower than those of the direct forecasting models for both ARIMA and LSTM. It was shown from the forecasting performance in 2019 that ARIMA was better than LSTM for monthly and weekly forecasting while the LSTM was better than ARIMA for daily forecasting in rolling forecasting models. Conclusions Both ARIMA and LSTM could be used to build a prediction model for the incidence of hemorrhagic fever. Different models might be more suitable for the incidence prediction at different time scales. The findings can provide a good reference for future selection of prediction models and establishments of early warning systems for hemorrhagic fever.

Generalizations of the Jensen functional involving diamond integrals via Abel–Gontscharoff interpolation

In this paper we obtain several refinements of the Jensen inequality on time scales by generalizing Jensen’s functional for n-convex functions. We also investigate the bounds for the identities related to the new improvements obtained.

Direct Determination of Reduced Models of a Class of Singularly Perturbed Nonlinear Systems on Three Time Scales in a Bond Graph Approach

One of the most important features in the analysis of the singular perturbation methods is the reduction of models. Likewise, the bond graph methodology in dynamic system modeling has been widely used. In this paper, the bond graph modeling of nonlinear systems with singular perturbations is presented. The class of nonlinear systems is the product of state variables on three time scales (fast, medium, and slow). Through this paper, the symmetry of mathematical modeling and graphical modeling can be established. A main characteristic of the bond graph is the application of causality to its elements. When an integral causality is assigned to the storage elements that determine the state variables, the dynamic model is obtained. If the storage elements of the fast dynamics have a derivative causality and the storage elements of the medium and slow dynamics an integral causality is assigned, a reduced model is obtained, which consists of a dynamic model for the medium and slow time scales and a stationary model of the fast time scale. By applying derivative causality to the storage elements of the fast and medium dynamics and an integral causality to the storage elements of the slow dynamics, the quasi-steady-state model for the slow dynamics is obtained and stationary models for the fast and medium dynamics are defined. The exact and reduced models of singularly perturbed systems can be interpreted as another symmetry in the development of this paper. Finally, the proposed methodology was applied to a system with three time scales in a bond graph approach, and simulation results are shown in order to indicate the effectiveness of the proposed methodology.

The Empirical Adaptive Wavelet Decomposition (EAWD): An adaptive decomposition for the variability analysis of observation time series in atmospheric science

Most observational data sequences in geophysics can be interpreted as resulting from the interaction of several physical processes at several time and space scales. As a consequence, measurements time series have often characteristics of non-linearity and non-stationarity and thereby exhibit strong fluctuations at different time-scales. The variability analysis of a time series consists in decomposing it into several mode of variability, each mode representing the fluctuations of the original time series at a specific time-scale. Such a decomposition enables to obtain a time-frequency representation of the original time series and turns out to be very useful to estimate the dimensionality of the underlying dynamics. Decomposition techniques very well suited to non-linear and non-stationary time series have recently been developed in the literature. Among the most widely used of these technics are the empirical mode decomposition (EMD) and the empirical wavelet transformation (EWT). The purpose of this paper is to present a new adaptive filtering method that combines the advantages of the EMD and EWT technics, while remaining close to the dynamics of the original signal made of atmospheric observations, which means reconstructing as close as possible to the original time series, while preserving its variability at different time scales.

Impact of Solar Activity on Snow Cover Variation Over the Tibetan Plateau and Linkage to the Summer Precipitation in China

Solar activity is one of the main external forcing factors driving the Earth’s climate system to change. The snow cover over the Tibetan Plateau is an important physical factor affecting the East Asian climate. At present, insufficient research on the connection between solar activity and snow cover over the Tibetan Plateau has been carried out. Using Solar Radio Flux (SRF), Solar Sunspot Number (SSN), and Total Solar Irradiance (TSI) data, this paper calculated the correlation coefficients with snow indices over the Tibetan Plateau, such as winter and spring snow depth (WSD/SSD) and snow day number (WSDN/SSDN). These snow indices are obtained from the daily gauge snow data in the Tibetan Plateau. Through correlation analyses, it is found that there are significant synchronous or lag correlations between snow indices and solar parameters on multi-time scales. In particular, the Spring Snow Day Number (SSDN) is of significant synchronous or lag correlation with SRF, SSN, and TSI on multi-time scales. It is further found that SSDN over the Tibetan Plateau has more stable positive correlations with SRF by using the 21-year running mean and cross spectrum analyses. Therefore, SSDN can be ascertained to be the most sensitive snow index to the solar activity compared with other snow indices. Moreover, its influence on summer precipitation of China is strongly regulated by solar activity. In high solar activity years (HSAY), the significant correlated area of summer precipitation in China to SSDN is located further north than that in low solar activity years (LSAY). Such impact by solar activity is also remarkable after excluding the impact of ENSO (i.e., El Niño–Southern Oscillation) events. These results provide support for the application of snow indices in summer rainfall prediction in China.

Dynamic Wetting of Photo-Responsive Arylazopyrazole Monolayers is Controlled by the Molecular Kinetics of the Monolayer

Smart surfaces that can change their wetting behavior on demand are interesting for applications such as self-cleaning surfaces or lab-on-a-chip devices. In order to functionalize aluminum oxide surfaces, we have synthesized arylazopyrazole phosphonic acids (butyl-AAP-C18PA) that represent a new class of photoswitchable molecules for these oxide surfaces. Butyl-AAP-C18PA monolayers were deposited on alpha-Al2O3(0001) and show reversible E/Z photo-switching with UV (Z) and green (E) light that can trigger contact angle changes of up to ~10°. We monitored these changes on the macroscopic level by recording the dynamic contact angle while the monolayer was switched in situ from the E to the Z state. On the molecular level, time-resolved vibrational sum-frequency generation (SFG) spectroscopy provided information on the kinetic changes within the AAP monolayer and the relevant characteristic time scales for E to Z switching and vice versa. In addition, vibrational SFG at different relative humidity indicates that the thermal stability of the Z configuration is largely influenced by the presence of water and that water can stabilize the Z state and, thus, hinder the AAP monolayer to switch into the E state when it is immersed in H2O. Having established the characteristic times for switching on the molecular scale from SFG spectroscopy, we additional measure the dynamic contact angle. Further, we reveal the time scales of the coupled substrate and droplet dynamics which we have extracted individually. For that, we report on a relaxation model, that can be solved analytically and which is verified via comparison with simulations of a Lennard Jones system and a comparison with experimental data. Indeed, our modelling of these coupled relaxation processes allows us to predict the non-trivial variation of the time-dependence of the contact angle when changing the size of the droplet. The observed slowing-down for E to Z switching upon the presence of the droplet is rationalized in terms of specific interactions of water with the exposed AAP moieties.