Time-delay embedding and dimensionality reduction are powerful techniques for discovering effective coordinate systems to represent the dynamics of physical systems. Recently, it has been shown that ...models identified by dynamic mode decomposition on time-delay coordinates provide linear representations of strongly nonlinear systems, in the so-called Hankel alternative view of Koopman (HAVOK) approach. Curiously, the resulting linear model has a matrix representation that is approximately antisymmetric and tridiagonal; for chaotic systems, there is an additional forcing term in the last component. In this paper, we establish a new theoretical connection between HAVOK and the Frenet-Serret frame from differential geometry, and also develop an improved algorithm to identify more stable and accurate models from less data. In particular, we show that the sub- and super-diagonal entries of the linear model correspond to the intrinsic curvatures in the Frenet-Serret frame. Based on this connection, we modify the algorithm to promote this antisymmetric structure, even in the noisy, low-data limit. We demonstrate this improved modelling procedure on data from several nonlinear synthetic and real-world examples.
The use of Singular Value Decomposition (SVD) under the Hankel matrix has emerged as a powerful technique for denoising non-stationary signals. The efficacy of the denoising process is significantly ...influenced by the structure of the Hankel matrix and the selection of subsignals. This paper systematically investigates these factors and introduces an Analytical Signal-based SVD (A-SVD) method. Initially, the analytical signal is introduced. This is based on the observed correlation between subsignals, aiming to reduce this correlation. Subsequently, a parameter unit energy change index (ECI) is introduced for assessing the decomposition's stability across different Hankel matrices, aiming to optimize the structure of the Hankel matrix. Moreover, the Group Gini index (GGI) of the reconstructed signal is utilized to select the optimal denoised signal. Lastly, the envelope spectrum is utilized for the analysis and extraction of relevant fault features. The effectiveness and superiority of the A-SVD method are confirmed through its application to both simulated bearing fault signals and two actual bearing fault cases.
Since the concept of deep learning (DL) was formally proposed in 2006, it has had a major impact on academic research and industry. Nowadays, DL provides an unprecedented way to analyze and process ...data with demonstrated great results in computer vision, medical imaging, natural language processing, and so forth. Herein, applications of DL in NMR spectroscopy are summarized, and a perspective for DL as an entirely new approach that is likely to transform NMR spectroscopy into a much more efficient and powerful technique in chemistry and life sciences is outlined.
Applying automation: The applications of deep learning in NMR spectroscopy are summarized, and a perspective for deep learning as an entirely new approach that is likely to transform NMR spectroscopy into a much more efficient and powerful technique in chemistry and life sciences is outlined.
Bessel orbits of normal operators Philipp, Friedrich
Journal of mathematical analysis and applications,
04/2017, Letnik:
448, Številka:
2
Journal Article
Recenzirano
Odprti dostop
Given a bounded normal operator A in a Hilbert space and a fixed vector x, we elaborate on the problem of finding necessary and sufficient conditions under which (Akx)k∈N constitutes a Bessel ...sequence. We provide a characterization in terms of the measure ‖E(⋅)x‖2, where E is the spectral measure of the operator A. In the separately treated special cases where A is unitary or selfadjoint we obtain more explicit characterizations. Finally, we apply our results to a sequence (Akx)k∈N, where A arises from the heat equation. The problem is motivated by and related to the new field of Dynamical Sampling which was recently initiated by Aldroubi et al. in 3.
Aiming at solving the existing sharp problems by using singular value decomposition (SVD) in the fault diagnosis of rolling bearings, such as the determination of the delay step k for creating the ...Hankel matrix and selection of effective singular values, the present study proposes a novel adaptive SVD method for fault feature detection based on the correlation coefficient by analyzing the principles of the SVD method. This proposed method achieves not only the optimal determination of the delay step k by means of the absolute value of the autocorrelation function sequence of the collected vibration signal, but also the adaptive selection of effective singular values using the index corresponding to useful component signals including weak fault information to detect weak fault signals for rolling bearings, especially weak impulse signals. The effectiveness of this method has been verified by contrastive results between the proposed method and traditional SVD, even using the wavelet-based method through simulated experiments. Finally, the proposed method has been applied to fault diagnosis for a deep-groove ball bearing in which a single point fault located on either the inner or outer race of rolling bearings is obtained successfully. Therefore, it can be stated that the proposed method is of great practical value in engineering applications.
The text may contain information such as personal privacy and national security, and it is vital to protect these sensitive information. The traditional text encryption methods only encrypt text into ...the garbled code. Although the encrypted text appears to be garbled, there may be text length exposure and semantic preservation. Attackers can use these information leakages to analyze and attack. Therefore, in order to confuse the attackers, we propose a technique to encrypt the text into the image in which it can hide the information that may be exposed in the ciphertext such as text length, language structure, frequency distribution, etc. It mainly uses coordinate substitution encryption and image encryption to encrypt text into image. First, the text is encrypted using coordinate substitution to initially disrupt the correlation between characters. Second, the encrypted text is converted to ASCII values and presented as the image. It is then multiplied with the Hankel matrix to adjust its elements values close to the true range of pixel values. Finally, the two-dimensional polynomial chaotic system and the image encryption algorithm are combined to obfuscate and diffusion encrypt images. It is experimentally verified that the scheme has good security and robustness for protecting text data.
In this paper, we study the numerical ranges of (finite) Hankel matrices and Hankel operators (on an infinite-dimensional space). The main concern is which nonempty bounded convex set △ in the plane ...is the numerical range W(A) of a Hankel matrix or a Hankel operator A. In Section 1 below, we prove results for △ a line segment, an elliptic disc, or a polygonal region. For example, we show that if △ is a closed elliptic disc in the plane, then a necessary and sufficient condition for the existence of an n-by-n Hankel matrix An with W(An) equal to △ for all n≥2 is that 0 is in △. In Section 2, we use the Megretskiĭ–Peller–Treil characterization of Hermitian Hankel operators to obtain an analogous condition for △ a (finite) line segment in the plane.
The two-point Padé approximation problem is to find a ratio of two coprime polynomials with some constraints on their degrees to approximate a function whose power series expansions at the origin and ...at infinity are given. In this paper, we introduce the Hankel vector for the two-point Padé approximation problem and establish the intrinsic connections between the two-point Padé approximation problem and a certain Padé approximation problem at infinity determined by the Hankel vector of the former. These connections provide us with a new way to study the structural characteristics of the two-point Padé table and to deduce the three-term recursive relations for the numerators and denominators of three adjacent entries in the two-point Padé table.
Abstract
Simultaneous-source acquisition helps greatly boost an economic saving, while it brings an unprecedented challenge of removing the crosstalk interference in the recorded seismic data. In ...this paper, we propose a novel iterative method to separate the simultaneous source data based on a coherency-pass shaping operator. The coherency-pass filter is used to constrain the model, that is, the unblended data to be estimated, in the shaping regularization framework. In the simultaneous source survey, the incoherent interference from adjacent shots greatly increases the rank of the frequency domain Hankel matrix that is formed from the blended record. Thus, the method based on rank reduction is capable of separating the blended record to some extent. However, the shortcoming is that it may cause residual noise when there is strong blending interference. We propose to cascade the rank reduction and thresholding operators to deal with this issue. In the initial iterations, we adopt a small rank to severely separate the blended interference and a large thresholding value as strong constraints to remove the residual noise in the time domain. In the later iterations, since more and more events have been recovered, we weaken the constraint by increasing the rank and shrinking the threshold to recover weak events and to guarantee the convergence. In this way, the combined rank reduction and thresholding strategy acts as a coherency-pass filter, which only passes the coherent high-amplitude component after rank reduction instead of passing both signal and noise in traditional rank reduction based approaches. Two synthetic examples are tested to demonstrate the performance of the proposed method. In addition, the application on two field data sets (common receiver gathers and stacked profiles) further validate the effectiveness of the proposed method.
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