Number Of Discrete Modes Research Articles

A hybrid short-term load forecasting model based on variational mode decomposition and long short-term memory networks considering relevant factors with Bayesian optimization algorithm

Short-term load forecasting plays an essential role in the safe and stable operation of power systems and has always been a vital research issue of energy management. In this research, a hybrid short-load forecasting method with Variational Mode Decomposition (VMD) and Long Short-Term Memory (LSTM) networks considering relevant factors which optimized by the Bayesian Optimization Algorithm (BOA) is studied. This method firstly decomposition with VMD which is a non-recursive signal processing technology that can decompose a signal into a discrete number of modes, then, consider the relevant factors and extend to the sequence according to the coefficient of association. Specifically, for the day type and higher or lower temperature, the nonlinear mapping is used and optimized by the BOA. Finally, the subsequences are predicted by LSTM which is a special Recurrent Neural Network with memory cells and reconstructed. To validate the performance of the proposed method, two categories of contrast methods including individual methods and decomposition-based methods are demonstrated in this study. The individual methods which without decomposition processes including LSTM, Support Vector Regression, Multi-Layered Perceptron Regressor, Linear Regression, and Random Forest Regressor, and the decomposition based methods including Empirical Mode Decomposition-Long Short-Term Memory, and Ensemble Empirical Mode Decomposition-Long Short-Term Memory. The simulation results, which developed in four periods of Hubei Province, China, show that the prediction accuracy of the proposed model is significantly improved compared with the contrast methods.

Dynamical error bounds for continuum discretisation via Gauss quadrature rules—A Lieb-Robinson bound approach

Instances of discrete quantum systems coupled to a continuum of oscillators are ubiquitous in physics. Often the continua are approximated by a discrete set of modes. We derive error bounds on expectation values of system observables that have been time evolved under such discretised Hamiltonians. These bounds take on the form of a function of time and the number of discrete modes, where the discrete modes are chosen according to Gauss quadrature rules. The derivation makes use of tools from the field of Lieb-Robinson bounds and the theory of orthonormal polynomials.

A study of VLF signals variations associated with the changes of ionization level in the D-region in consequence of solar conditions

In this paper we confine our attention to the analysis of amplitude and phase data acquired by monitoring VLF/LF radio signals emitted by four European transmitters during a seven-year period (2008–2014). All the data were recorded at a Belgrade site (44.85°N, 20.38°E) by the Stanford University ELF/VLF receiver AWESOME. Propagation of VLF/LF radio signal takes place in the Earth–ionosphere waveguide and strongly depends on ionization level of the D-region, which means that it is mainly controlled by solar conditions. Some results of amplitude and phase variations on GQD/22.10kHz, DHO/23.40kHz, ICV/20.27kHz and NSC/45.90kHz radio signals measurements at short distances (D<2Mm) over Central Europe and their interpretation are summarized in this paper. Attention is restricted to regular diurnal, seasonal and solar variations including sunrise and sunset effects on propagation characteristics of four VLF/LF radio signals. We study VLF/LF propagation over short path as a superposition of different number of discrete modes which depends on the variations of the path parameters. Although the solar X-ray flare effects on propagation of VLF/LF radio signals are well recognized on all paths, similarities and differences between them are defined under existing conditions over the paths. Statistical results show that the size of amplitude and phase perturbations on VLF/LF radio signal is in correlation with the intensity of X-ray flux. We present the calculations of electron density enhancements in the D-region caused by different classes of solar X-ray flares during the period of ascending phase and maximum of the solar cycle 24.

A clustering-based approach for the identification of a class of temporally switched linear systems

The behaviours of hybrid dynamic systems (HDS) are determined by combining continuous variables with discrete switching logic. The identification of a HDS aims to find an accurate model of the system’s dynamics based on its past inputs and outputs. In pattern recognition (PR) methods, each mode is represented by a set of similar patterns that form restricted regions in the feature space. These sets of patterns are called classes. A pattern is a vector built from past inputs and outputs. HDS identification is a challenging problem since it involves the estimation of different sets of parameters without knowing in advance which sections of the measured data correspond to the different modes of the system. Therefore, HDS identification can be achieved by combining two steps: clustering and parameter estimation. In the clustering step, the number of discrete modes (i.e., the classes that input–output data points belong) is estimated. The parameter estimation step finds the parameters of the models that govern the continuous dynamics in each mode. In this paper, an unsupervised PR method is proposed to achieve the clustering step of the identification of temporally switched linear HDS. The determination of the number of modes does not require prior information about the modes or their number.

A way to a single frequency precession of an inhomogeneous ensemble of electron spins in InGaAs quantum dots

AbstractWe show a possible way to the realization of a robust macroscopical quantum bit on the basis of an inhomogeneous ensemble of electron spins. In our case, electrons are confined in a self‐assembled (In,Ga)As/GaAs quantum dots which are excited by a periodical laser pulses. The precession of electron spins in a transversemagnetic field is monitored by the time‐resolved Faraday rotation technique. Due to mode‐locking effect and a focusing of the electron spins precession on a discrete number of modes by the nuclei it should be possible to drive all the electron spins into a single precession mode. (© 2009 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Surface elastic waves in granular media under gravity and their relation to booming avalanches

Due to the nonlinearity of Hertzian contacts, the speed of sound c in granular matter is expected to increase with pressure as P1/6. A static layer of grains under gravity is thus stratified so that the bulk waves are refracted toward the surface. The reflection at the surface being total, there is a discrete number of modes (both sagittal plane and transverse ones) localized close to the free surface. The shape of these modes and the corresponding dispersion relation are investigated in the framework of an elastic description taking into account the main features of granular matter: the nonlinearity between stress and strain and the existence of a yield transition. We show in this context that the surface modes localized at the free surface exhibit a waveguide effect related to the nonlinear Hertz contact. Sound propagation measurements in a laboratory scale channel as well as in the field (slip face of a singing dune), will be presented and compared to the theoretical predictions. Finally, these results will be used to constrain the possible dynamical mechanisms at the origin of booming avalanches.

Quasigeostrophic planetary waves in a two-layer ocean with one-dimensional periodic bottom topography

Although the study of topographic effects on the Rossby waves in a stratified ocean has a long history, the wave property over a periodic bottom topography whose lateral scale is comparable to the wavelength is still not clear. The present paper treats this problem in a two-layer ocean with one-dimensional periodic bottom topography by a simple numerical method, in which no restriction on the wavelength and/or the horizontal scale of the topography is required. The dispersion diagram is obtained for a wavenumber range of [−π/L b , π/L b ], where L b is the periodic length of the topography. When the topographic β is not negligible compared to the planetary β, the Rossby wave solutions around the wavenumbers which satisfy the resonant condition among the waves and topography disappear and separate into an infinite number of discrete modes. For convenience, each mode is numbered in order of frequency. As topographic height is increased, the high frequency barotropic Rossby wave (mode 1) becomes a topographic mode which can exist even on the f plane, and the highfrequency baroclinic mode (mode 2) becomes a surface intensified mode. Behaviors of low frequency modes are somewhat complicated. When the topographic amplitude is small, the low frequency baroclinic modes tend to be bottom trapped and the low frequency barotropic modes tend to be surface intensified. As topographic amplitude further increases, the relation between the mode number and vertical structure changes. This change can be attributed to the increase of the frequency of the topographic mode with the topographic amplitude.

Surface elastic waves in granular media under gravity and their relation to booming avalanches

Due to the nonlinearity of Hertzian contacts, the speed of sound c in granular matter is expected to increase with pressure as P(1/6). A static layer of grains under gravity is thus stratified so that the bulk waves are refracted toward the surface. The reflection at the surface being total, there is a discrete number of modes (both in the sagittal plane and transverse to it) localized close to the free surface. The shape of these modes and the corresponding dispersion relation are investigated in the framework of an elastic description taking into account the main features of granular matter: Nonlinearity between stress and strain and the existence of a yield transition. We show in this context that the surface modes localized at the free surface exhibit a waveguide effect related to the nonlinear Hertz contact. Recent results about the song of dunes are reinterpreted in light of the theoretical results. The predicted propagation speed is compared with measurements performed in the field. Taking into account the finite depth effects, we show that the booming instability threshold can be explained quantitatively by a waveguide cutoff frequency below which no sound can propagate. Therefore, we propose another look at a recent controversy, confirming that the song of dunes can well originate from a coupling between avalanching grains and surface elastic waves once the specificity of surface waves (we baptized Rayleigh-Hertz) is correctly taken into account.

The influence of finite extensibility on the eigenspectrum of dilute polymeric solutions

Linear stability of plane Couette flow of dilute polymeric solutions modeled using the FENE-P constitutive equation has been examined. Specifically, the spurious, discrete and continuous spectra are identified as a function of the ratio of the solvent to the total viscosity β, maximum chain length L, disturbance wavenumber α and the Weissenberg number ( We). It is observed that reducing L shifts the entire spectrum to the left (in the eigenspectrum plane) by a factor directly related to the steady-state value of the Peterlin function f. Additionally, decreasing the value of L causes the splitting of the regular continuous spectrum, initially located at −1/ We, into two branches. Overall, the plane Couette flow of a FENE-P model is found to be unconditionally stable. Specifically, decreasing L increases the decay rate of the most dangerous disturbance while increasing the number of discrete and spurious modes. However, increasing both β and α decreases the number of discrete and spurious modes whereas increasing We has no influence on the number of spurious modes but results in an increase in the number of discrete modes.

Recasting developmental evolution in terms of genetic pathway and network evolution … and the implications for comparative biology

The morphological features of complex organisms are the outcomes of developmental processes. Developmental processes, in turn, reflect the genetic networks that underlie them. Differences in morphology must ultimately, therefore, reflect differences in the underlying genetic networks. A mutation that affects a developmental process does so by affecting either a gene whose product acts as an upstream controlling element, an intermediary connecting link, or as a downstream output of the network that governs the trait's development. Although the immense diversity of gene networks in the animal and plant kingdoms would seem to preclude any general “rules” of network evolution, the material discussed here suggests that the patterns of genetic pathway and network evolution actually fall into a number of discrete modes. The potential utility of this conceptual framework in reconstructing instances of developmental evolution and for comparative neurobiology will be discussed.

A dynamic nonlinear regression method for the determination of thediscrete relaxation spectrum

The relaxation spectrum is animportant tool for studying the behaviour of viscoelasticmaterials. The most popular procedure is to use data froma small-amplitude oscillatory shear experiment to determinethe parameters in a multi-mode Maxwell model. However,the discrete relaxation times appear nonlinearly inthe mathematical model for the relaxation modulus.The indirect calculation of the relaxationtimes is an ill-posed problem and its numerical solution isfraught with difficulties. The ill-posedness of the linearregression approach, in which the relaxation times arespecified a priori and the minimization is performed withrespect to the elastic moduli, is well documented.A nonlinear regression technique is described in this paperin which the minimization is performedwith respect to both the discrete relaxationtimes and the elastic moduli. In this technique the number ofdiscrete modes is increased dynamically andthe procedure is terminated whenthe calculated values of the model parameters aredominated by a measure of their expected values.The sequence of nonlinear least-squares problems,solved using the Marquardt-Levenberg procedure,is shown to be robust and efficient. Numericalcalculations on model and experimental data are presentedand discussed.

Nonlinear response of driven systems in weak turbulence theory

A method is presented for predicting the saturation levels and particle transport in weakly unstable systems with a discrete number of modes. Conditions are established for either steady-state or pulsating responses when several modes are excited for cases where there is and there is not resonance overlap. The conditions for saturation and the associated transport are discussed. Depending on parameters, the saturation level can be low, with only a small fraction of the available free energy released to waves and with no global transport, or the saturation level can be quite high, with almost complete conversion of free energy to wave energy coupled with rapid transport.

The Rate of Convergence of Spectral-Viscosity Methods for Periodic Scalar Conservation Laws

A rate of convergence is proven for spectral-viscosity methods for periodic scalar conservation laws. This rate is obtained by showing the discretization error to be small enough that the difference between the solutions of the spectral-viscosity method and the ordinary viscosity method tends to zero in $L^1 $ as the number of discrete modes tends to infinity and the viscosity simultaneously tends to zero at the appropriate rate. The method is also used to obtain an $L^\infty $ bound for the spectral-viscosity approximations to the elasticity equations; convergence, although without a rate, then follows from compensated compactness theory.

Mass of the sine-Gordon soliton in the Hartree approximation.

We derive the quantum corrections to the mass of the one-soliton solution of the sine-Gordon system in the Hartree approximation. In the weak-coupling limit we reproduce the semiclassical correction to the soliton mass. This happens only after a nontrivial cancellation of contributions related to the deformation of the soliton due to quantum fluctuations. Numerical results are obtained up to the critical value of the coupling constant as given by Coleman. In approaching the critical point we find an increasing number of discrete modes which seem to build up a new continuum with a lower mass.

Normal mode model for electromagnetic propagation in the earth crust waveguide

Propagation of modes in an idealized model of the earth-crust waveguide is considered. The two-layer region is bounded above and below by a perfectly conducting plane. For this structure, there are an infinite number of discrete modes without any contribution from a continuous spectrum. The excitation factor for the structure is derived for voltage-type and electric, current line sources. Specific numerical results for 1 to 1000 Hz are given for the case where the lower conductive region is effectively of infinite depth. It is found that the conductivity of the upper region is the main controlling factor on the attenuation rate.

SURFACE WAVES ALONG AN AXIALLY MAGNETIZED PLASMA COLUMN: I. SYMMETRIC MODES

The characteristics of surface waves excited by an axially oriented point electric dipole situated along the axis of an infinitely long; and axially magnetized column of uniform cold plasma are investigated. The surface waves are found to be slow waves and exist only below the upper hybrid resonant frequency. In the gyromagnetic and plasma resonance regions, the existence of an infinite number of discrete modes is noted. The dispersion curves are presented for different values of the strength of the applied magnetic field and the column radius. Also, the power transported by the surface waves is evaluated and is found to become infinite in the two resonance frequency regions. A technique for the removal of this singularity in the surface wave power is indicated.

Quasi-static solution for diffraction of a plane electromagnetic wave by a small oblate spheroid

The problem of the diffraction of a plane electromagnetic wave by a small perfectly conducting oblate spheroid for the case of normal incidence has been investigated by an expansion method. By retaining three terms of the exponential function contained in the incident field, it is possible to describe both the incident field and the near-zone scattered field in terms of a finite number of discrete modes which satisfy the third order vector equation ? × (?2 A) = 0, where A denotes a solenoidal vector representing either the electric field or the magnetic field. When the eccentricity of the spheroid approaches zero, the general solution reduces to the exact solution for a small sphere. When the spheroid degenerates into a disk, the expressions for the charge and the current agree with the leading terms of the corresponding expressions obtained by Meixner, Andrejewski, and Bouwkamp.