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12 pages, 1909 KiB  
Article
A Multi-Mode Recognition Method for Broadband Oscillation Based on Compressed Sensing and EEMD
by Jinggeng Gao, Honglei Xu, Yong Yang, Haoming Niu, Jinping Liang and Haiying Dong
Appl. Sci. 2024, 14(24), 11484; https://doi.org/10.3390/app142411484 - 10 Dec 2024
Viewed by 242
Abstract
In power systems, the application of wind power generation equipment and power electronic devices leads to an increased frequency of broadband oscillation events, and the detection of oscillation information becomes extremely difficult, due to the limitations of communication bandwidth and the sampling theorem. [...] Read more.
In power systems, the application of wind power generation equipment and power electronic devices leads to an increased frequency of broadband oscillation events, and the detection of oscillation information becomes extremely difficult, due to the limitations of communication bandwidth and the sampling theorem. To ensure the safety and stability of a power system, this paper presents a new recognition method of broadband oscillation information, which combines compressed sensing (CS) technology and an ensemble empirical mode decomposition (EEMD) algorithm to solve the problem of wideband oscillation recognition. Firstly, the broadband oscillation signal data collected by the phasor measuring unit (PMU) is compressed and sampled by a Gaussian random matrix in the substation, then the low-dimensional data obtained is uploaded to the main station. Secondly, in the main station, the subspace pursuit (SP) algorithm is used to reconstruct the low-dimensional signal; the broadband oscillation signal is recovered without losing the main features of the signal. Finally, we use the EEMD algorithm to decompose the reconstructed signal; the intrinsic mode function (IMF) components containing wideband oscillation information are screened by the energy coefficient, and the wideband oscillation information is identified. Full article
(This article belongs to the Special Issue Power System Security and Stability)
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Figure 1

Figure 1
<p>Comparison between the reconstructed signal and original signal.</p>
Full article ">Figure 2
<p>EEMD decomposition results of the reconstructed signal.</p>
Full article ">Figure 3
<p>FFT results for the first three IMFs.</p>
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<p>The effect of compression and reconstruction for different SNRs.</p>
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<p>Simplified model for a power generation system composed of six wind turbines.</p>
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<p>Comparison of the broadband oscillation signal with a reconstructed signal.</p>
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<p>The identification results for the measurement signal.</p>
Full article ">Figure 7 Cont.
<p>The identification results for the measurement signal.</p>
Full article ">
15 pages, 5604 KiB  
Article
A Deep Evolution Policy-Based Approach for RIS-Enhanced Communication System
by Ke Zhao, Zhiqun Song, Yong Li, Xingjian Li, Lizhe Liu and Bin Wang
Entropy 2024, 26(12), 1056; https://doi.org/10.3390/e26121056 - 5 Dec 2024
Viewed by 324
Abstract
This paper investigates the design of active and passive beamforming in a reconfigurable intelligent surface (RIS)-aided multi-user multiple-input single-output (MU-MISO) system with the objective of maximizing the sum rate. We propose a deep evolution policy (DEP)-based algorithm to derive the optimal beamforming strategy [...] Read more.
This paper investigates the design of active and passive beamforming in a reconfigurable intelligent surface (RIS)-aided multi-user multiple-input single-output (MU-MISO) system with the objective of maximizing the sum rate. We propose a deep evolution policy (DEP)-based algorithm to derive the optimal beamforming strategy by generating multiple agents, each utilizing distinct deep neural networks (DNNs). Additionally, a random subspace selection (RSS) strategy is incorporated to effectively balance exploitation and exploration. The proposed DEP-based algorithm operates without the need for alternating iterations, gradient descent, or backpropagation, enabling simultaneous optimization of both active and passive beamforming. Simulation results indicate that the proposed algorithm can bring significant performance enhancements. Full article
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Figure 1
<p>The considered RIS-aided MU-MISO communication system comprised of an <span class="html-italic">M</span>-antenna BS simultaneously serving in the downlink <span class="html-italic">K</span> single-antenna users, and the transmit signal propagates to the users via the RIS assistance.</p>
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<p>M-P neuron model.</p>
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<p>DNN model with two hidden layers.</p>
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<p>The schematic diagram of the DEP-based algorithm, which includes the processes of interaction, selection, crossover and mutation.</p>
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<p>Crossover process. The boxes in gray and cyan represent the genetic information of <math display="inline"><semantics> <msub> <mi>Para</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>Para</mi> <mn>2</mn> </msub> </semantics></math>, respectively.</p>
Full article ">Figure 6
<p>Mutation process. The boxes in gray, cyan and yellow represent the genetic information of <math display="inline"><semantics> <msub> <mi>Para</mi> <mn>1</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>Para</mi> <mn>2</mn> </msub> </semantics></math>, and mutation respectively.</p>
Full article ">Figure 7
<p>Converged sum rate versus transmit power to show the proposed DEP-based algorithm in comparison with two benchmarks, <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mi>N</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mi>K</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 8
<p>Converged sum rate versus transmit power to show the proposed DEP-based algorithm in comparison with two benchmarks, <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mi>N</mi> <mo>=</mo> <mn>32</mn> <mo>,</mo> <mi>K</mi> <mo>=</mo> <mn>4</mn> <mo>.</mo> </mrow> </semantics></math></p>
Full article ">Figure 9
<p>Sum rate as a function of total time steps at <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>10</mn> <mspace width="3.33333pt"/> <mi>dBm</mi> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mi>N</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mi>K</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>. Shaded regions represent <math display="inline"><semantics> <mrow> <mn>95</mn> <mo>%</mo> </mrow> </semantics></math> confidence interval over 10 random seeds for each result.</p>
Full article ">Figure 10
<p>Sum rate versus total time steps under different mutation rates, i.e., {<math display="inline"><semantics> <mrow> <mn>0.1</mn> <mo>,</mo> <mn>0.01</mn> <mo>,</mo> <mn>0.001</mn> </mrow> </semantics></math>}, <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>5</mn> <mspace width="3.33333pt"/> <mi>dBm</mi> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mi>N</mi> <mo>=</mo> <mn>32</mn> <mo>,</mo> <mi>K</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, population size = 500, subspace proportion = 0.5. Shaded regions represent 95% confidence interval over 10 random seeds for each result.</p>
Full article ">Figure 11
<p>Sum rate versus total time steps under different population sizes, i.e., {<math display="inline"><semantics> <mrow> <mn>300</mn> <mo>,</mo> <mn>500</mn> <mo>,</mo> <mn>800</mn> </mrow> </semantics></math>}, <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>5</mn> <mspace width="3.33333pt"/> <mi>dBm</mi> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mi>N</mi> <mo>=</mo> <mn>32</mn> <mo>,</mo> <mi>K</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, mutation rate = <math display="inline"><semantics> <mrow> <mn>0.001</mn> </mrow> </semantics></math>, subspace proportion = <math display="inline"><semantics> <mrow> <mn>0.5</mn> </mrow> </semantics></math>. Shaded regions represent <math display="inline"><semantics> <mrow> <mn>95</mn> <mo>%</mo> </mrow> </semantics></math> confidence interval over 10 random seeds for each result.</p>
Full article ">Figure 12
<p>Sum rate versus total time steps under different subspace proportions, i.e., {<math display="inline"><semantics> <mrow> <mn>2</mn> <mo>/</mo> <mi>P</mi> <mo>,</mo> <mn>0.5</mn> <mo>,</mo> <mn>1</mn> </mrow> </semantics></math>}, <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>5</mn> <mspace width="3.33333pt"/> <mi>dBm</mi> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mi>N</mi> <mo>=</mo> <mn>32</mn> <mo>,</mo> <mi>K</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, mutation rate = 0.001, population size = 500. Shaded regions represent <math display="inline"><semantics> <mrow> <mn>95</mn> <mo>%</mo> </mrow> </semantics></math> confidence interval over 10 random seeds for each result.</p>
Full article ">
16 pages, 8249 KiB  
Technical Note
Impact Analysis of Orthogonal Circular-Polarized Interference on GNSS Spatial Anti-Jamming Array
by Ke Zhang, Xiangjun Li, Lei Chen, Zengjun Liu and Yuchen Xie
Remote Sens. 2024, 16(23), 4506; https://doi.org/10.3390/rs16234506 - 1 Dec 2024
Viewed by 391
Abstract
With the continuous advancement of electromagnetic countermeasures, new types of interference signals (e.g., multi-polarization suppression interference) pose a significant threat to conventional Global Navigation Satellite System (GNSS) services, even when the receiver employs a right-handed circularly polarized (RHCP) anti-jamming array. This paper proposes [...] Read more.
With the continuous advancement of electromagnetic countermeasures, new types of interference signals (e.g., multi-polarization suppression interference) pose a significant threat to conventional Global Navigation Satellite System (GNSS) services, even when the receiver employs a right-handed circularly polarized (RHCP) anti-jamming array. This paper proposes a receiving signal model for orthogonal circularly polarized (OCP) interference signals based on conventional arrays, following an analysis of the non-ideal characteristics of actual arrays. Furthermore, the mechanism by which OCP interference signals affect anti-jamming performance is examined. Power inversion (PI) and linear constrained minimum variance (LCMV) techniques, applied to both uniform linear arrays and central circular arrays, are utilized to verify the impact of these interference signals. Simulation and physical testing demonstrate that OCP interference significantly affects the interference subspace of the conventional RHCP array, potentially leading to a reduction in the anti-jamming performance of the receiver. To effectively suppress multi-polarization interference, anti-jamming GNSS receivers must either ensure the consistency of cross-polarization among the elements of the array or adopt polarization-sensitive arrays. Full article
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Figure 1
<p>Array signal processing architecture.</p>
Full article ">Figure 2
<p>Patch antenna and its radiation characteristics: (<b>a</b>) patch antenna; (<b>b</b>) S11 parameter; (<b>c</b>) gain pattern; (<b>d</b>) cross-polarization gain pattern.</p>
Full article ">Figure 3
<p>Seven-element patch array: (<b>a</b>) uniform linear array; (<b>b</b>) central circular array.</p>
Full article ">Figure 4
<p>Characteristics of cross-polarization pattern: (<b>a</b>) uniform linear array with spacing 0.5 times of wave; (<b>b</b>) uniform linear array with spacing 0.4 times of wave. (<b>c</b>) central circular array with spacing 0.5 times of wave; (<b>d</b>) central circular array with spacing 0.4 times of wave.</p>
Full article ">Figure 4 Cont.
<p>Characteristics of cross-polarization pattern: (<b>a</b>) uniform linear array with spacing 0.5 times of wave; (<b>b</b>) uniform linear array with spacing 0.4 times of wave. (<b>c</b>) central circular array with spacing 0.5 times of wave; (<b>d</b>) central circular array with spacing 0.4 times of wave.</p>
Full article ">Figure 5
<p>The normalized RHCP pattern of 7-element linear array against OCP interference based on PI: (<b>a</b>) under unidirectional interference; (<b>b</b>) under multi-directional interference.</p>
Full article ">Figure 6
<p>The normalized LHCP pattern of 7-element linear array against OCP interference based on PI.</p>
Full article ">Figure 7
<p>The normalized RHCP pattern of 7-element linear array against OCP interference based on LCMV: (<b>a</b>) under unidirectional interference; (<b>b</b>) under multi-directional interference.</p>
Full article ">Figure 8
<p>The normalized LHCP pattern of 7-element linear array against OCP interference based on LCMV.</p>
Full article ">Figure 9
<p>Anti-jamming experiment of 7-element central circular array: (<b>a</b>) the 7-element patch array; (<b>b</b>) experimental scene.</p>
Full article ">Figure 10
<p>Eigenvalues of covariance matrix in different interference scenes.</p>
Full article ">Figure 11
<p>The normalized RHCP pattern of 7-element central circular array against OCP interference based on PI: (<b>a</b>) under one RHCP interference; (<b>b</b>) under one OCP interference.</p>
Full article ">
20 pages, 1795 KiB  
Article
Source Quantization and Coding over Noisy Channel Analysis
by Runfeng Wang, Dan Song, Jinkai Ren, Lin Wang and Zhiping Xu
Mathematics 2024, 12(23), 3798; https://doi.org/10.3390/math12233798 - 30 Nov 2024
Viewed by 255
Abstract
Recently, lossy source coding based on linear block code has been designed using the duality principle, i.e., the channel decoding algorithm is employed to realize the lossy source coding. However, the quantization structure has not been analyzed in this compression technique, and the [...] Read more.
Recently, lossy source coding based on linear block code has been designed using the duality principle, i.e., the channel decoding algorithm is employed to realize the lossy source coding. However, the quantization structure has not been analyzed in this compression technique, and the codebook design does not match the source characteristics well. Hence, the compression performance is not so good. To overcome this problem, the codebook design is correlated with the quantization structure in this work. It is found that the lossy source coding based on the linear block code can be defined as lattice vector quantization (VQ), which provides a new analytical perspective for the coding methodology. Then, the VQ scheme is generalized with the noisy channel to evaluate the transmission robustness of the continuous source compression. Finally, the codebook of the VQ scheme is optimally designed by uniforming the radiuses of the quantization subspace to reduce the quantization distortion. The proposed codebook outperforms existing codes in terms of its proximity to the rate–distortion limit, while also exhibiting enhanced robustness against channel noise. Full article
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Figure 1
<p>The coding relation between the quantization space and the linear block code.</p>
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<p>The proposed VQ scheme based on linear block codes over noisy channels.</p>
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<p>The VQ scheme for the Gaussian source over the noisy channels.</p>
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<p>The VQ scheme for the general source over the noisy channels.</p>
Full article ">Figure 5
<p>The distortion and total degree performance for code searching. The objective code is searched by the S-GDA algorithm at <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> when SNR = 7 dB and BER = <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </semantics></math>, and the initial degrees are given as (<b>a</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mn>1</mn> </msub> </semantics></math>: GDA<sub>9×10</sub><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mn>2</mn> </msub> </semantics></math>: GDA<sub>3×5</sub><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p>
Full article ">Figure 6
<p>The distortion and total degree performance for code searching. The objective code is searched by the S-GDA algorithm at <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> when SNR = 7 dB and BER = <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </semantics></math>, and the initial degrees are given as (<b>a</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mn>1</mn> </msub> </semantics></math>: GDA<sub>19×20</sub><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mn>21</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <msub> <mi>L</mi> <mn>2</mn> </msub> </semantics></math>: GDA<sub>4×5</sub><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mn>1</mn> </msub> </semantics></math>: GDA<sub>19×20</sub><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mn>21</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <msub> <mi>L</mi> <mn>3</mn> </msub> </semantics></math>: GDA<sub>2×8</sub><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mn>2</mn> </msub> </semantics></math>: GDA<sub>4×5</sub><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <msub> <mi>L</mi> <mn>3</mn> </msub> </semantics></math>: GDA<sub>2×8</sub><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p>
Full article ">Figure 7
<p>The distortion and total degree performance for code searching. The objective code is searched by the S-GDA algorithm at <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math> when SNR = 7 dB and BER = <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </semantics></math>, and the initial degrees are given as (<b>a</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mn>1</mn> </msub> </semantics></math>: GDA<sub>17×20</sub><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mn>51</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <msub> <mi>L</mi> <mn>2</mn> </msub> </semantics></math>: GDA<sub>3×5</sub><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mn>1</mn> </msub> </semantics></math>: GDA<sub>17×20</sub><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mn>51</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <msub> <mi>L</mi> <mn>3</mn> </msub> </semantics></math>: GDA<sub>2×40</sub><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mn>40</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mn>2</mn> </msub> </semantics></math>: GDA<sub>3×5</sub><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <msub> <mi>L</mi> <mn>3</mn> </msub> </semantics></math>: GDA<sub>2×40</sub><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mn>40</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p>
Full article ">Figure 8
<p>The distortion–rate performance is tested based on the Gaussian source compression using the MLC structure with the SP mapping over the AWGN channel with BPSK modulation when SNR = 7 dB and BER = <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </semantics></math>.</p>
Full article ">Figure 9
<p>The distortion sensitivity is compared with each layer under the channel noise, and the codes are (<b>a</b>) GDA<sub>9×10</sub><math display="inline"><semantics> <mrow> <mspace width="3.33333pt"/> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, GDA<sub>3×5</sub><math display="inline"><semantics> <mrow> <mspace width="3.33333pt"/> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>; (<b>b</b>) GDA<sub>19×20</sub><math display="inline"><semantics> <mrow> <mspace width="3.33333pt"/> <mrow> <mo>(</mo> <mn>25</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, GDA<sub>4×5</sub><math display="inline"><semantics> <mrow> <mspace width="3.33333pt"/> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, GDA<sub>2×8</sub><math display="inline"><semantics> <mrow> <mspace width="3.33333pt"/> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>; (<b>c</b>) GDA<sub>17×20</sub><math display="inline"><semantics> <mrow> <mspace width="3.33333pt"/> <mrow> <mo>(</mo> <mn>29</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, GDA<sub>3×5</sub><math display="inline"><semantics> <mrow> <mspace width="3.33333pt"/> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, GDA<sub>2×40</sub><math display="inline"><semantics> <mrow> <mspace width="3.33333pt"/> <mrow> <mo>(</mo> <mn>40</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 10
<p>The distortion and total degree performance are tested for different code comparisons based on the Gaussian source compression over the AWGN channel with BPSK modulation when SNR = 7 dB and BER = <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </semantics></math>, and the codes are given as (<b>a</b>) GDA<sub>3×5</sub><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mi>d</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math>; (<b>b</b>) GDA<sub>3×6</sub><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mi>d</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>; (<b>c</b>) GDA<sub>3×7</sub><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mi>d</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.57</mn> </mrow> </semantics></math>; (<b>d</b>) GDA<sub>3×9</sub><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mi>d</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, at <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.66</mn> </mrow> </semantics></math>.</p>
Full article ">Figure 11
<p>The energy distribution of nodes is compared with the AR3A and S-GDA<sub>3×5</sub><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> codes at different lifting numbers.</p>
Full article ">Figure 12
<p>The distortion–rate performance is tested based on Gaussian source compression by using the NN structure over the AWGN channel with BPSK modulation when SNR = 7 dB and BER = <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </semantics></math>.</p>
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<p>The distortion and total degree performance are tested for different code comparisons based on Laplacian source compression over the AWGN channel with BPSK modulation when SNR = 7 dB and BER = <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </semantics></math>, and the codes are given as (<b>a</b>) GDA<sub>3×5</sub><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mi>d</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math>; (<b>b</b>) GDA<sub>3×7</sub><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mi>d</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.57</mn> </mrow> </semantics></math>; (<b>c</b>) GDA<sub>3×9</sub><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mi>d</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, at <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.66</mn> </mrow> </semantics></math>.</p>
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<p>The distortion–rate performance is tested based on Laplacian source compression using the NN structure over the AWGN channel with BPSK modulation when SNR = 7 dB and BER = <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </semantics></math>.</p>
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<p>The distortion–SNR performance is tested for different degree comparisons based on the Gaussian source with <math display="inline"><semantics> <mrow> <msup> <mi>σ</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. (<b>a</b>) S-GDA code: GDA<sub>3×5</sub><math display="inline"><semantics> <mrow> <mspace width="3.33333pt"/> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math>. (<b>b</b>) S-GDA code: GDA<sub>3×7</sub><math display="inline"><semantics> <mrow> <mspace width="3.33333pt"/> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.57</mn> </mrow> </semantics></math>. (<b>c</b>) S-GDA code: GDA<sub>3×9</sub><math display="inline"><semantics> <mrow> <mspace width="3.33333pt"/> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.66</mn> </mrow> </semantics></math>.</p>
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<p>The distortion–SNR performance is tested for different degree comparisons based on the Laplacian source with <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> </semantics></math>. (<b>a</b>) S-GDA code: GDA<sub>3×5</sub><math display="inline"><semantics> <mrow> <mspace width="3.33333pt"/> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math>. (<b>b</b>) S-GDA code: GDA<sub>3×7</sub><math display="inline"><semantics> <mrow> <mspace width="3.33333pt"/> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.57</mn> </mrow> </semantics></math>. (<b>c</b>) S-GDA code: GDA<sub>3×9</sub><math display="inline"><semantics> <mrow> <mspace width="3.33333pt"/> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>0.66</mn> </mrow> </semantics></math>.</p>
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26 pages, 4034 KiB  
Article
Semi-Supervised Deep Subspace Embedding for Binary Classification of Sella Turcica
by Kaushlesh Singh Shakya, Azadeh Alavi, Julie Porteous, Priti Khatri, Amit Laddi, Manojkumar Jaiswal and Vinay Kumar
Appl. Sci. 2024, 14(23), 11154; https://doi.org/10.3390/app142311154 - 29 Nov 2024
Viewed by 475
Abstract
In orthodontics, the manual tracing of cephalometric radiographs is a common practice, where the Sella Turcica (ST) serves as a reference point. The radiologist often manually traces the outline of the sella using manual tools (e.g., calipers on radiographs). Perhaps the inherent complexity [...] Read more.
In orthodontics, the manual tracing of cephalometric radiographs is a common practice, where the Sella Turcica (ST) serves as a reference point. The radiologist often manually traces the outline of the sella using manual tools (e.g., calipers on radiographs). Perhaps the inherent complexity and variability in the shapes of sella and the lack of advanced assessment tools make the classification of sella challenging, as it requires extensive training, skills, time, and manpower to detect subtle changes that often may not be apparent. Moreover, existing semi-supervised learning (SSL) methods face key limitations such as shift invariance, inadequate feature representation, overfitting on small datasets, and a lack of generalization to unseen variations in ST morphology. Medical imaging data are often unlabeled, limiting the training of automated classification systems for ST morphology. To address these limitations, a novel semi-supervised deep subspace embedding (SSLDSE) framework is proposed. This approach integrates real-time stochastic augmentation to significantly expand the training dataset and introduce natural variability in the ST morphology, overcoming the constraints of small and non-representative datasets. Non-linear features are extracted and mapped to a non-linear subspace using Kullback–Leibler divergence, which ensures that the model remains consistent despite image transformations, thus resolving issues related to shift invariance. Additionally, fine-tuning the Inception-ResNet-v2 network on these enriched features reduces retraining costs when new unlabeled data becomes available. t-distributed stochastic neighbor embedding (t-SNE) is employed for effective feature representation through manifold learning, capturing complex patterns that previous methods might miss. Finally, a zero-shot classifier is utilized to accurately categorize the ST, addressing the challenge of classifying new or unseen variations. Further, the proposed SSLDSE framework is evaluated through comparative analysis with the existing methods (Active SSL, GAN SSL, Contrastive SSL, Modified Inception-ResNet-v2) for ST classification using various evaluation metrics. The SSLDSE and the existing methods are trained on our dataset (sourced from PGI Chandigarh, India), and a blind test is conducted on the benchmark dataset (IEEE ISBI 2015). The proposed method improves classification accuracy by 15% compared to state-of-the-art models and reduces retraining costs. Full article
(This article belongs to the Special Issue Application of Artificial Intelligence in Biomedical Informatics)
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<p>Sample images of pre-defined Sella Turcica (ST) shapes: (<b>A</b>) Oval ST, (<b>B</b>) Circular ST, (<b>C</b>) Flat ST, and (<b>D</b>) Bridging ST. This study classified Circular ST as non-bridging, and Bridging ST was used for binary classification.</p>
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<p>The schematic representation of a Hybrid Database (<span class="html-italic">L</span>) and Hybrid Case Base (<math display="inline"><semantics> <mrow> <mi>S</mi> <mi>L</mi> </mrow> </semantics></math>) from labeled (<math display="inline"><semantics> <mrow> <mi>S</mi> <msub> <mi>L</mi> <mi>i</mi> </msub> </mrow> </semantics></math>) and unlabeled (<math display="inline"><semantics> <mrow> <mi>S</mi> <msub> <mi>L</mi> <mover accent="true"> <mi>i</mi> <mo>^</mo> </mover> </msub> </mrow> </semantics></math>) case data. Feature extraction using KL divergence, mean (<math display="inline"><semantics> <mi>μ</mi> </semantics></math>), and standard deviation (<math display="inline"><semantics> <mi>σ</mi> </semantics></math>) is applied to both databases. Labeled data form a featured database with labels (<math display="inline"><semantics> <msub> <mi>L</mi> <mi>i</mi> </msub> </semantics></math>), while unlabeled data create a featured database without labels (<math display="inline"><semantics> <msub> <mi>L</mi> <mover accent="true"> <mi>i</mi> <mo>^</mo> </mover> </msub> </semantics></math>). A dynamic responsive data and label mechanism integrates both, resulting in (1) a Hybrid Database (<span class="html-italic">L</span>) and (2) a Hybrid Case Base (<math display="inline"><semantics> <mrow> <mi>S</mi> <mi>L</mi> </mrow> </semantics></math>) for further analysis.</p>
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<p>Process flow diagram of the proposed SSLDSE framework.</p>
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<p>The figure illustrates a comprehensive framework of the proposed SSLDSE that integrates labeled (<math display="inline"><semantics> <mrow> <mi>S</mi> <msub> <mi>L</mi> <mi>i</mi> </msub> </mrow> </semantics></math>) and unlabeled case databases (<math display="inline"><semantics> <mrow> <mi>S</mi> <msub> <mi>L</mi> <mover accent="true"> <mi>i</mi> <mo>^</mo> </mover> </msub> </mrow> </semantics></math>). Features are extracted using Kullback–Leibler divergence, mean (<math display="inline"><semantics> <mi>μ</mi> </semantics></math>), and standard deviation (<math display="inline"><semantics> <mi>σ</mi> </semantics></math>), forming a Hybrid Database. The data undergo stochastic augmentation and are processed through an Inception-ResNet-V2 model. A deep subspace descriptor with t-SNE refines the feature representations, and the outputs are classified by a zero-shot classifier (<math display="inline"><semantics> <mrow> <mi>Z</mi> <mi>s</mi> <mi>C</mi> </mrow> </semantics></math>) with KL divergence loss, enabling the model to handle unseen or unlabeled ST structures.</p>
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<p>The illustrated SSLDSE architectural framework processes labeled (L) and semi-labeled (<math display="inline"><semantics> <mrow> <mi>S</mi> <msub> <mi>L</mi> <mi>i</mi> </msub> </mrow> </semantics></math>) data using Inception-ResNet-V2 as the CNN backbone to extract features (P, Q) and estimate pairwise probability densities (<math display="inline"><semantics> <msub> <mi mathvariant="normal">P</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi mathvariant="normal">Q</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </semantics></math>). KL divergence (<math display="inline"><semantics> <mrow> <msub> <mi>D</mi> <mi>KL</mi> </msub> <mrow> <mo>(</mo> <mi>P</mi> <mo>‖</mo> <mi>Q</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>) minimizes divergence through the optimization of (Y). Manifold learning maps feature matrices (X) to t-SNE representations (Y) while preserving structural relationships. The SSL framework employs deep embedding and clustering (mean: <math display="inline"><semantics> <msub> <mi>μ</mi> <mi>j</mi> </msub> </semantics></math>, covariance: <math display="inline"><semantics> <msub> <mi>Σ</mi> <mi>j</mi> </msub> </semantics></math>) for feature representation. The zero-shot classifier constructs semantic vectors and applies KL divergence loss for output prediction (<math display="inline"><semantics> <msub> <mi>O</mi> <mi>i</mi> </msub> </semantics></math>).</p>
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<p>Confusion matrix and ROC curve showcasing the validation results of the binary classifier, highlighting the proposed model’s classification performance through true positive/negative rates and the AUC-ROC score.</p>
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<p>t-SNE plots visualizing the quantitative assessment of the proposed SSLDSE method. The plots illustrate the effective separation between bridging and non-bridging labels from our proprietary and IEEE ISBI 2015 datasets, demonstrating a clear class distinction.</p>
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<p>Boxplots illustrate a detailed comparison of the classification error rates for the proposed SSLDSE method, showing the distribution of error rates across (<b>a</b>) the proprietary dataset and (<b>b</b>) the IEEE ISBI 2015 dataset, highlighting the variability and consistency in classification accuracy.</p>
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<p>Boxplots comparing classification error rates across utilized SSL approaches and the proposed SSLDSE method, illustrating performance differences and the effectiveness of SSLDSE in reducing classification errors.</p>
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<p>Visual interpretation of errors in ST-binary classification predictions, illustrating misclassified instances and highlighting the areas where the model’s predictions diverge from the true labels.</p>
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21 pages, 5901 KiB  
Article
A Rapid Identification Method for Cottonseed Varieties Based on Near-Infrared Spectral and Generative Adversarial Networks
by Qingxu Li, Hao Li, Renhao Liu, Xiaofeng Dong, Hongzhou Zhang and Wanhuai Zhou
Agriculture 2024, 14(12), 2177; https://doi.org/10.3390/agriculture14122177 - 29 Nov 2024
Viewed by 356
Abstract
China is a major cotton-growing country with numerous cotton varieties, each exhibiting significant differences in yield and fiber quality. However, the current management of cottonseed varieties is disorganized, resulting in severe homogenization and the presence of counterfeit and mislabeled varieties. The detection of [...] Read more.
China is a major cotton-growing country with numerous cotton varieties, each exhibiting significant differences in yield and fiber quality. However, the current management of cottonseed varieties is disorganized, resulting in severe homogenization and the presence of counterfeit and mislabeled varieties. The detection of cottonseed variety information has become a critical issue for the Chinese cotton industry. In this study, we collected near-infrared (NIR) spectral data from six cottonseed varieties and constructed a GAN for cottonseed NIR data (GAN-CNIRD) model to generate additional cottonseed NIR data. The Euclidean distance method was used to label the generated NIR data according to the characteristics of the true NIR data. We then applied Standard Normal Variate (SNV), Multiplicative Scatter Correction (MSC), and Normalization algorithms to preprocess the combined dataset of generated and real cottonseed NIR data. Feature wavelengths were extracted using Bootstrap Soft Shrinkage (BOSS) and Competitive Adaptive Reweighted Sampling (CARS) algorithms. Subsequently, we developed Linear Discriminant Analysis (LDA), Random subspace method (RSM), and convolutional neural network (CNN) models to classify the cottonseed varieties. The results showed that for the LDA model, the use of feature wavelengths extracted after Normalization-BOSS processing achieved the best performance with an accuracy of 97.00%. For the RSM model, the use of feature wavelengths extracted after SNV-CARS processing achieved the best performance with an accuracy of 98.00%. For the CNN model, the use of feature wavelengths extracted after MSC-CARS processing achieved the best performance with an accuracy of 100.00%. Data augmentation using GAN-CNIRD-generated cottonseed data improved the accuracy of the three optimal models by 6%, 5%, and 6%, respectively. This study provides a crucial reference for the rapid detection of cottonseed variety information and has significant implications for the standardized management of cottonseed varieties. Full article
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<p>Six different cottonseed varieties.</p>
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<p>The NIR data acquisition system for cottonseeds.</p>
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<p>WGAN-GP.</p>
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<p>GAN-CNIRD.</p>
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<p>CNN.</p>
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<p>Loss of GAN-CNIRD.</p>
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<p>Generated and real cottonseed data.</p>
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<p>The preprocessed cottonseed data.</p>
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<p>Selection of cottonseed features using the CARS.</p>
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<p>Feature wavelengths selected by CARS.</p>
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<p>The variation of RMSECV with the number of iterations.</p>
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<p>The weight distribution at the 23rd iteration.</p>
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<p>Feature wavelengths selected by BOSS.</p>
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18 pages, 1427 KiB  
Article
Asymptotic and Probabilistic Perturbation Analysis of Controllable Subspaces
by Vera Angelova, Mihail Konstantinov and Petko Petkov
Computation 2024, 12(12), 236; https://doi.org/10.3390/computation12120236 - 28 Nov 2024
Viewed by 299
Abstract
In this paper, we consider the sensitivity of the controllable subspaces of single-input linear control systems to small perturbations of the system matrices. The analysis is based on the strict component-wise asymptotic bounds for the matrix of the orthogonal transformation to canonical form [...] Read more.
In this paper, we consider the sensitivity of the controllable subspaces of single-input linear control systems to small perturbations of the system matrices. The analysis is based on the strict component-wise asymptotic bounds for the matrix of the orthogonal transformation to canonical form derived by the method of the splitting operators. The asymptotic bounds are used to obtain probabilistic bounds on the angles between perturbed and unperturbed controllable subspaces implementing the Markoff inequality. It is demonstrated that the probability bounds allow us to obtain sensitivity estimates, which are much tighter than the usual deterministic bounds. The analysis is illustrated by a high-order example. Full article
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<p>The mean value of <math display="inline"><semantics> <mrow> <mo>[</mo> <mo>Δ</mo> <msub> <mi>a</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>/</mo> <mo>|</mo> <mi>δ</mi> <msub> <mi>a</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>|</mo> <mo>]</mo> </mrow> </semantics></math> as a function of <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msup> </semantics></math> (<b>left</b>) and the mean value of <math display="inline"><semantics> <mrow> <mrow> <mi>N</mi> <mo>{</mo> <mo>Δ</mo> </mrow> <msub> <mi>a</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow> <mo>&gt;</mo> <mo>|</mo> <mi>δ</mi> </mrow> <msub> <mi>a</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow> <mo>|</mo> <mo>}</mo> </mrow> <mo>/</mo> <msup> <mi>n</mi> <mn>2</mn> </msup> </mrow> </semantics></math> (<b>right</b>) as a function of <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msup> </semantics></math> for normal random distributions of the entries of the <math display="inline"><semantics> <mrow> <mn>100</mn> <mo>×</mo> <mn>100</mn> </mrow> </semantics></math> matrix <span class="html-italic">A</span>.</p>
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<p>The mean value of <math display="inline"><semantics> <mrow> <mo>[</mo> <mo>Δ</mo> <msub> <mi>b</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>/</mo> <mo>|</mo> <mi>δ</mi> <msub> <mi>b</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>|</mo> <mo>]</mo> </mrow> </semantics></math> as a function of <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msup> </semantics></math> (<b>left</b>) and the mean value of <math display="inline"><semantics> <mrow> <mrow> <mi>N</mi> <mo>{</mo> <mo>Δ</mo> </mrow> <msub> <mi>b</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow> <mo>&gt;</mo> <mo>|</mo> <mi>δ</mi> </mrow> <msub> <mi>b</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow> <mo>|</mo> <mo>}</mo> </mrow> <mo>/</mo> <msup> <mi>n</mi> <mn>2</mn> </msup> </mrow> </semantics></math> (<b>right</b>) as a function of <math display="inline"><semantics> <msup> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msup> </semantics></math> for normal random distributions of the entries of the <math display="inline"><semantics> <mrow> <mn>100</mn> <mo>×</mo> <mn>1</mn> </mrow> </semantics></math> matrix <span class="html-italic">B</span>.</p>
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<p>Perturbation bounds of the entries of <span class="html-italic">U</span> for different probabilities.</p>
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<p>The ratio <math display="inline"><semantics> <mrow> <mo>[</mo> <mi>δ</mi> <msubsup> <mi>U</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mrow> <mi>l</mi> <mi>i</mi> <mi>n</mi> </mrow> </msubsup> <mo>/</mo> <mo>|</mo> <mi>δ</mi> <msub> <mi>U</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>|</mo> <mo>]</mo> </mrow> </semantics></math> (<b>left</b>) and the ratio <math display="inline"><semantics> <mrow> <mo>[</mo> <mi>δ</mi> <msubsup> <mi>U</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mrow> <mi>p</mi> <mi>r</mi> <mi>o</mi> <mi>b</mi> </mrow> </msubsup> <mo>/</mo> <mo>|</mo> <mi>δ</mi> <msub> <mi>U</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>|</mo> <mo>]</mo> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <msup> <mrow> <mi mathvariant="script">P</mi> </mrow> <mrow> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msup> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math> (<b>right</b>) for normal random distributions of the entries of the matrices <span class="html-italic">A</span> and <span class="html-italic">B</span>.</p>
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<p>Angles between the perturbed and unperturbed controllable subspaces and their bounds.</p>
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24 pages, 28615 KiB  
Article
Modal Parameter Identification of Jacket-Type Offshore Wind Turbines Under Operating Conditions
by Chen Zhang, Xu Han, Chunhao Li, Bernt Johan Leira, Svein Sævik, Dongzhe Lu, Wei Shi and Xin Li
J. Mar. Sci. Eng. 2024, 12(11), 2083; https://doi.org/10.3390/jmse12112083 - 18 Nov 2024
Viewed by 746
Abstract
Operational modal analysis (OMA) is essential for long-term health monitoring of offshore wind turbines (OWTs), helping identifying changes in structural dynamic characteristics. OMA has been applied under parked or idle states for OWTs, assuming a linear and time-invariant dynamic system subjected to white [...] Read more.
Operational modal analysis (OMA) is essential for long-term health monitoring of offshore wind turbines (OWTs), helping identifying changes in structural dynamic characteristics. OMA has been applied under parked or idle states for OWTs, assuming a linear and time-invariant dynamic system subjected to white noise excitations. The impact of complex operating environmental conditions on structural modal identification therefore requires systematic investigation. This paper studies the applicability of OMA based on covariance-driven stochastic subspace identification (SSI-COV) under various non-white noise excitations, using a DTU 10 MW jacket OWT model as a basis for a case study. Then, a scaled (1:75) 10 MW jacket OWT model test is used for the verification. For pure wave conditions, it is found that accurate identification for the first and second FA/SS modes can be achieved with significant wave energy. Under pure wind excitations, the unsteady servo control behavior leads to significant identification errors. The combined wind and wave actions further complicate the picture, leading to more scattered identification errors. The SSI-COV based modal identification method is suggested to be reliably applied for wind speeds larger than the rated speed and with sufficient wave energy. In addition, this method is found to perform better with larger misalignment of wind and wave directions. This study provides valuable insights in relation to the engineering applications of in situ modal identification techniques under operating conditions in real OWT projects. Full article
(This article belongs to the Topic Wind, Wave and Tidal Energy Technologies in China)
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<p>Illustration of the considered jacket OWT.</p>
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<p>The 3D turbulent wind map at a turbulence level of 18.75% simulated by Turbsim.</p>
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<p>Rotor speed time history at a wind speed of 9.5 m/s.</p>
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<p>Average rotor speed and variance of the rotor speed over different wind speeds.</p>
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<p>Acceleration response signals in FA and SS directions excited by a white noise spectrum.</p>
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<p>Stabilization diagram using 10 min acceleration response in the FA direction of the OWT when excited by white noise.</p>
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<p>Clustering results of the stable points under a white noise condition.</p>
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<p>The first two reference mode shapes in FA/SS directions.</p>
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<p>The identified first FA and SS modes for irregular wave conditions: (<b>a</b>) the natural frequency, and (<b>b</b>) the MAC. Note that the gradient color bars represent variations in Tp from 6.5 s to 17 s in the irregular wave excitation conditions, the blue color indicates FA, and the red color is SS.</p>
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<p>The second FA/SS modes for irregular wave conditions: (<b>a</b>) the second natural frequency, (<b>b</b>) the second MAC, and (<b>c</b>) the relative frequency error. Note that the gradient bars represent variations in Tp from 6.5 s to 17 s for the pure wave excitation conditions, the blue color indicates FA, and the red color is SS.</p>
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<p>Stabilization diagram using the 10 min of a condition where Hs is 0.5 m, Tp is 10.5 s.</p>
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<p>Clustering results of the stable points under a condition where Hs is 0.5 m, Tp is 10.5 s.</p>
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<p>The first FA/SS modes for turbulent wind conditions: (<b>a</b>) the first natural frequency, (<b>b</b>) the first MAC.</p>
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<p>The identified harmonic frequencies for turbulent wind conditions.</p>
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<p>The identified second modes for different turbulent wind conditions: (<b>a</b>) second frequency, (<b>b</b>) second MAC.</p>
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<p>The first frequency at different Hs for combined wave and wind conditions. Note that the gradient bars represent variations in Tp from 6.5 s to 17 s, the blue color indicates FA, and the red color is SS.</p>
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<p>The first MAC at different Hs for combined wave and wind conditions. Note that the gradient bars represent variations in Tp from 6.5 s to 17 s, the blue color indicates FA, and the red color is SS.</p>
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<p>The second frequency at different Hs for combined wave and wind conditions. Note that the gradient bars represent variations in Tp from 6.5 s to 17 s, the blue color indicates FA, and the red color is SS.</p>
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<p>The second MAC at different Hs for combined wave and wind conditions. Note that the gradient bars represent variations in Tp from 6.5 s to 17 s, the blue color indicates FA, and the red color is SS.</p>
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<p>The first two modes of OWT by combined wave and wind conditions under different noise levels.</p>
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<p>The first frequency for variation in wave directions, while at a wind direction of 0°.</p>
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<p>The second frequency for variation in wave directions, for a constant wind direction of 0°.</p>
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<p>The test configurations.</p>
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<p>Modal parameter identification results for the scale test.</p>
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28 pages, 2411 KiB  
Review
Cosmological Models in Lovelock Gravity: An Overview of Recent Progress
by Sergey Pavluchenko
Universe 2024, 10(11), 429; https://doi.org/10.3390/universe10110429 - 18 Nov 2024
Viewed by 507
Abstract
In the current review, we provide a summary of the recent progress made in the cosmological aspect of extra-dimensional Lovelock gravity. Our review covers a wide variety of particular model/matter source combinations: Einstein–Gauss–Bonnet as well as cubic Lovelock gravities with vacuum, cosmological constant, [...] Read more.
In the current review, we provide a summary of the recent progress made in the cosmological aspect of extra-dimensional Lovelock gravity. Our review covers a wide variety of particular model/matter source combinations: Einstein–Gauss–Bonnet as well as cubic Lovelock gravities with vacuum, cosmological constant, perfect fluid, spatial curvature, and some of their combinations. Our analysis suggests that it is possible to set constraints on the parameters of the above-mentioned models from the simple requirement of the existence of a smooth transition from the initial singularity to a realistic low-energy regime. Initially, anisotropic space naturally evolves into a configuration with two isotropic subspaces, and if one of these subspaces is three-dimensional and is expanding while another is contracting, we call it realistic compactification. Of course, the process is not devoid of obstacles, and in our paper, we review the results of the compactification occurrence investigation for the above-mentioned models. In particular, for vacuum and Λ-term EGB models, compactification is not suppressed (but is not the only possible outcome either) if the number of extra dimensions is D2; for vacuum cubic Lovelock gravities it is always present (however, cubic Lovelock gravity is defined only for D3 number of extra dimensions); for the EGB model with perfect fluid it is present for D=2 (we have not considered this model in higher dimensions yet), and in the presence of spatial curvature, the realistic stabilization of extra dimensions is always present (however, such a model is well-defined only in D4 number of extra dimensions). Full article
(This article belongs to the Special Issue Cosmological Models of the Universe)
Show Figures

Figure 1

Figure 1
<p>Resulting regimes for the <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> vacuum case: <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math> on (<b>a</b>) and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math> on (<b>b</b>) (see the text for more details).</p>
Full article ">Figure 2
<p>Resulting regimes for EGB vacuum cases: <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> (panel (<b>a</b>) for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math> and panel (<b>b</b>) for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math>), <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> (panel (<b>c</b>) for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math> and panel (<b>d</b>) for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math>), and general <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>⩾</mo> <mn>4</mn> </mrow> </semantics></math> case (panel (<b>e</b>) for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, and panel (<b>f</b>) for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math>) (see the text for more details).</p>
Full article ">Figure 3
<p>Viable regimes for <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>⩾</mo> <mn>4</mn> </mrow> </semantics></math> EGB cosmology with <math display="inline"><semantics> <mi mathvariant="normal">Λ</mi> </semantics></math>-term: (<b>a</b>) panel: <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi mathvariant="normal">Λ</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math> featuring <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mn>3</mn> </msub> <mo>→</mo> <msub> <mi>K</mi> <mn>3</mn> </msub> </mrow> </semantics></math> regime; (<b>b</b>) panel: <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi mathvariant="normal">Λ</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mi mathvariant="normal">Λ</mi> <mo>⩽</mo> <msub> <mi>ζ</mi> <mn>1</mn> </msub> </mrow> </semantics></math> featuring <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mn>3</mn> </msub> <mo>→</mo> <msubsup> <mi>E</mi> <mrow> <mn>3</mn> <mo>+</mo> <mi>D</mi> </mrow> <mn>1</mn> </msubsup> <mo>←</mo> <msub> <mi>P</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </msub> </mrow> </semantics></math> transitions; (<b>c</b>) panel: <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi mathvariant="normal">Λ</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math> featuring the same <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mn>3</mn> </msub> <mo>→</mo> <msubsup> <mi>E</mi> <mrow> <mn>3</mn> <mo>+</mo> <mi>D</mi> </mrow> <mn>1</mn> </msubsup> <mo>←</mo> <msub> <mi>P</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </msub> </mrow> </semantics></math> transitions; (<b>d</b>) panel: <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi mathvariant="normal">Λ</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mi mathvariant="normal">Λ</mi> <mo>&lt;</mo> <msub> <mi>ζ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> still featuring the same <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mn>3</mn> </msub> <mo>→</mo> <msubsup> <mi>E</mi> <mrow> <mn>3</mn> <mo>+</mo> <mi>D</mi> </mrow> <mn>1</mn> </msubsup> <mo>←</mo> <msub> <mi>P</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </msub> </mrow> </semantics></math> transitions; (<b>e</b>) panel: <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi mathvariant="normal">Λ</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mi mathvariant="normal">Λ</mi> <mo>=</mo> <msub> <mi>ζ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mn>3</mn> </msub> <mo>→</mo> <msubsup> <mi>E</mi> <mrow> <mn>3</mn> <mo>+</mo> <mi>D</mi> </mrow> <mn>1</mn> </msubsup> <mo>←</mo> <msub> <mi>P</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </msub> </mrow> </semantics></math> transitions; (<b>f</b>) panel: <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi mathvariant="normal">Λ</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>ζ</mi> <mn>2</mn> </msub> <mo>&lt;</mo> <mi>α</mi> <mi mathvariant="normal">Λ</mi> <mo>&lt;</mo> <msub> <mi>ζ</mi> <mn>3</mn> </msub> </mrow> </semantics></math> (see the text for more details).</p>
Full article ">Figure 4
<p>Viable regimes for vacuum cubic Lovelock cosmology (all viable regimes are located in the second quadrant): (<b>a</b>) panel: <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </msub> <mo>→</mo> <msub> <mi>E</mi> <mrow> <mn>3</mn> <mo>+</mo> <mi>D</mi> </mrow> </msub> </mrow> </semantics></math> transition on the green branch for <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>=</mo> <mn>3</mn> <mo>÷</mo> <mn>7</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>⩽</mo> <msub> <mi>μ</mi> <mn>1</mn> </msub> </mrow> </semantics></math>; (<b>b</b>) panel: <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </msub> <mo>→</mo> <msub> <mi>K</mi> <mn>1</mn> </msub> </mrow> </semantics></math> transition on the green-blue branch for <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>=</mo> <mn>3</mn> <mo>÷</mo> <mn>7</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>&gt;</mo> <msub> <mi>μ</mi> <mn>1</mn> </msub> </mrow> </semantics></math>; (<b>c</b>) panel: <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </msub> <mo>→</mo> <msub> <mi>E</mi> <mrow> <mn>3</mn> <mo>+</mo> <mi>D</mi> </mrow> </msub> </mrow> </semantics></math> transition on the green branch as well as <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mn>5</mn> </msub> <mo>→</mo> <msub> <mi>E</mi> <mrow> <mn>3</mn> <mo>+</mo> <mi>D</mi> </mrow> </msub> </mrow> </semantics></math> transition on the red branch for <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>⩾</mo> <mn>8</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>⩽</mo> <msub> <mi>μ</mi> <mn>2</mn> </msub> </mrow> </semantics></math>; (<b>d</b>) panel: <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </msub> <mo>→</mo> <msub> <mi>E</mi> <mrow> <mn>3</mn> <mo>+</mo> <mi>D</mi> </mrow> </msub> <mo>←</mo> <msub> <mi>K</mi> <mn>5</mn> </msub> </mrow> </semantics></math> double transition on the green branch for <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>⩾</mo> <mn>8</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>⩽</mo> <msub> <mi>μ</mi> <mn>3</mn> </msub> </mrow> </semantics></math>; (<b>e</b>) panel: <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </msub> <mo>→</mo> <msub> <mi>K</mi> <mn>1</mn> </msub> </mrow> </semantics></math> transition on the right green-blue branch and <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mn>5</mn> </msub> <mo>→</mo> <msub> <mi>E</mi> <mrow> <mn>3</mn> <mo>+</mo> <mi>D</mi> </mrow> </msub> </mrow> </semantics></math> transition on the left green-blue branch for <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>⩾</mo> <mn>8</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>&gt;</mo> <msub> <mi>μ</mi> <mn>3</mn> </msub> </mrow> </semantics></math> (see the text for more details).</p>
Full article ">Figure 5
<p>Viable regimes for EGB model with a perfect fluid as a source: (<b>a</b>) panel: large-scale structure of the <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math> quadrant for <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>b</b>) panel: vicinity of <math display="inline"><semantics> <msub> <mi>E</mi> <mrow> <mn>3</mn> <mo>+</mo> <mn>2</mn> </mrow> </msub> </semantics></math> stable point for <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math>, initial conditions leading to <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mn>3</mn> </msub> <mo>→</mo> <msub> <mi>E</mi> <mrow> <mn>3</mn> <mo>+</mo> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> transition are bounded by light-blue lines; (<b>c</b>) panel: large-scale structure of the <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math> quadrant for <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>/</mo> <mn>3</mn> <mo>&gt;</mo> <mi>ω</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>; (<b>d</b>) panel: vicinity of <math display="inline"><semantics> <msub> <mi>E</mi> <mrow> <mn>3</mn> <mo>+</mo> <mn>2</mn> </mrow> </msub> </semantics></math> stable point for <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>/</mo> <mn>3</mn> <mo>&gt;</mo> <mi>ω</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, initial conditions leading to <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mn>3</mn> </msub> <mo>→</mo> <msub> <mi>E</mi> <mrow> <mn>3</mn> <mo>+</mo> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> transition are bounded by light-blue lines; (<b>e</b>) panel: large-scale structure of the <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math> quadrant for <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>&gt;</mo> <mn>1</mn> <mo>/</mo> <mn>3</mn> </mrow> </semantics></math>, only <math display="inline"><semantics> <mrow> <mi>n</mi> <mi>S</mi> <mo>→</mo> <msub> <mi>E</mi> <mrow> <mn>3</mn> <mo>+</mo> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> transition remains; (<b>f</b>) vicinity of the exponential constant volume solution (<math display="inline"><semantics> <msub> <mi>E</mi> <mrow> <mi>C</mi> <mi>V</mi> <mi>S</mi> </mrow> </msub> </semantics></math>) (see the text for more details).</p>
Full article ">Figure 6
<p>Illustrations for the dynamics of the EGB case with spatial curvature: regime with stabilization of extra dimensions (<math display="inline"><semantics> <mrow> <mi>H</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>→</mo> <mn>0</mn> </mrow> </semantics></math>) on (<b>a</b>) panel, areas on the parameters space where stabilization of extra dimensions with positive spatial curvature is possible and stable, as a function of <span class="html-italic">D</span> (<b>b</b>,<b>c</b>) panels (see the text for more details).</p>
Full article ">Figure 7
<p>Illustrations for the dynamics of the EGB case initial total anisotropy (Bianchi-I-type): different initial conditions for the model with 5 spatial dimensions could lead to either <math display="inline"><semantics> <mrow> <mo>[</mo> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mo>]</mo> </mrow> </semantics></math> spatial splitting ((<b>a</b>) panel) or isotropization ((<b>b</b>) panel); distribution of the initial conditions for the model with 6 spatial dimensions leading to either <math display="inline"><semantics> <mrow> <mo>[</mo> <mn>4</mn> <mo>+</mo> <mn>2</mn> <mo>]</mo> </mrow> </semantics></math> or <math display="inline"><semantics> <mrow> <mo>[</mo> <mn>3</mn> <mo>+</mo> <mn>3</mn> <mo>]</mo> </mrow> </semantics></math> spatial splittings ((<b>c</b>) panel) (see the text for more details).</p>
Full article ">Figure 8
<p>Summary of the bounds on <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>α</mi> <mo>,</mo> <mi mathvariant="normal">Λ</mi> <mo>)</mo> </mrow> </semantics></math> from this paper alone on (<b>a</b>) panel; from other considerations found in the literature on (<b>b</b>) panel; and the intersection between them on (<b>c</b>) panel (see the text for more details).</p>
Full article ">
21 pages, 5789 KiB  
Article
Joint Sparse Local Linear Discriminant Analysis for Feature Dimensionality Reduction of Hyperspectral Images
by Cong-Yin Cao, Meng-Ting Li, Yang-Jun Deng, Longfei Ren, Yi Liu and Xing-Hui Zhu
Remote Sens. 2024, 16(22), 4287; https://doi.org/10.3390/rs16224287 - 17 Nov 2024
Viewed by 516
Abstract
Although linear discriminant analysis (LDA)-based subspace learning has been widely applied to hyperspectral image (HSI) classification, the existing LDA-based subspace learning methods exhibit several limitations: (1) They are often sensitive to noise and demonstrate weak robustness; (2) these methods ignore the local information [...] Read more.
Although linear discriminant analysis (LDA)-based subspace learning has been widely applied to hyperspectral image (HSI) classification, the existing LDA-based subspace learning methods exhibit several limitations: (1) They are often sensitive to noise and demonstrate weak robustness; (2) these methods ignore the local information inherent in data; and (3) the number of extracted features is restricted by the number of classes. To address these drawbacks, this paper proposes a novel joint sparse local linear discriminant analysis (JSLLDA) method by integrating embedding regression and locality-preserving regularization into the LDA model for feature dimensionality reduction of HSIs. In JSLLDA, a row-sparse projection matrix can be learned, to uncover the joint sparse structure information of data by imposing a L2,1-norm constraint. The L2,1-norm is also employed to measure the embedding regression reconstruction error, thereby mitigating the effects of noise and occlusions. A locality preservation term is incorporated to fully leverage the local geometric structural information of the data, enhancing the discriminability of the learned projection. Furthermore, an orthogonal matrix is introduced to alleviate the limitation on the number of acquired features. Finally, extensive experiments conducted on three hyperspectral image (HSI) datasets demonstrated that the performance of JSLLDA surpassed that of some related state-of-the-art dimensionality reduction methods. Full article
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Figure 1

Figure 1
<p>For the Salinas dataset, the OA varied with <math display="inline"><semantics> <msub> <mi>λ</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>λ</mi> <mn>2</mn> </msub> </semantics></math> when <math display="inline"><semantics> <msub> <mi>λ</mi> <mn>3</mn> </msub> </semantics></math> was set to (<b>a</b>) 0.1, (<b>b</b>) 0.01, (<b>c</b>) 0.001, (<b>d</b>) 0.0001, and (<b>e</b>) 0.00001.</p>
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<p>For the University of Pavia dataset, the OA varied with <math display="inline"><semantics> <msub> <mi>λ</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>λ</mi> <mn>2</mn> </msub> </semantics></math> when <math display="inline"><semantics> <msub> <mi>λ</mi> <mn>3</mn> </msub> </semantics></math> was set to (<b>a</b>) 0.1, (<b>b</b>) 0.01, (<b>c</b>) 0.001, (<b>d</b>) 0.0001, and (<b>e</b>) 0.00001.</p>
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<p>For the Heihe dataset, the OA varied with <math display="inline"><semantics> <msub> <mi>λ</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>λ</mi> <mn>2</mn> </msub> </semantics></math> when <math display="inline"><semantics> <msub> <mi>λ</mi> <mn>3</mn> </msub> </semantics></math> was set to (<b>a</b>) 0.1, (<b>b</b>) 0.01, (<b>c</b>) 0.001, (<b>d</b>) 0.0001, and (<b>e</b>) 0.00001.</p>
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<p>The average recognition rates for the different datasets versus the different dimensions of the features extracted by different methods. (<b>a</b>) The Salinas dataset. (<b>b</b>) The University of Pavia dataset. (<b>c</b>) The Heihe dataset.</p>
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<p>Classification maps of the different methods for the Salinas dataset. (<b>a</b>) Ground truth, (<b>b</b>) LDA, (<b>c</b>) LPP, (<b>d</b>) RSLDA, (<b>e</b>) TRLDA, (<b>f</b>) LRPER, (<b>g</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mn>2</mn> <mo>,</mo> <mi>p</mi> </mrow> </msub> </semantics></math>-RER, (<b>h</b>) JSLLDA.</p>
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<p>Classification maps of different methods for the University of Pavia dataset. (<b>a</b>) Ground truth, (<b>b</b>) LDA, (<b>c</b>) LPP, (<b>d</b>) RSLDA, (<b>e</b>) TRLDA, (<b>f</b>) LRPER, (<b>g</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mn>2</mn> <mo>,</mo> <mi>p</mi> </mrow> </msub> </semantics></math>-RER, (<b>h</b>) JSLLDA.</p>
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<p>Classification maps of the different methods for the Heihe dataset. (<b>a</b>) Ground truth, (<b>b</b>) LDA, (<b>c</b>) LPP, (<b>d</b>) RSLDA, (<b>e</b>) TRLDA, (<b>f</b>) LRPER, (<b>g</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mn>2</mn> <mo>,</mo> <mi>p</mi> </mrow> </msub> </semantics></math>-RER, (<b>h</b>) JSLLDA.</p>
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33 pages, 394 KiB  
Article
On the Problem of the Uniqueness of Fixed Points and Solutions for Quadratic Fractional-Integral Equations on Banach Algebras
by Kinga Cichoń, Mieczysław Cichoń and Maciej Ciesielski
Symmetry 2024, 16(11), 1535; https://doi.org/10.3390/sym16111535 - 16 Nov 2024
Viewed by 464
Abstract
In this paper, we study the problem of the uniqueness of fixed points for operators defined on subspaces of the space of continuous functions C[a,b] equipped with norms stronger than the supremum norm. We are particularly interested in [...] Read more.
In this paper, we study the problem of the uniqueness of fixed points for operators defined on subspaces of the space of continuous functions C[a,b] equipped with norms stronger than the supremum norm. We are particularly interested in Hölder spaces since they are natural ranges of integral operators of fractional order. Our goal is to preserve the expected regularity of the fixed points (solutions of the equations) when investigating their uniqueness, without assuming a contraction condition on the space under study. We claim some symmetry between the case of the obtained results and the case of the classical Banach fixed-point theorem in such spaces, even for operators which are not necessarily contractions in the sense of the norm of these subspaces. This result is of particular interest for the study of quadratic integral equations, and as an application example we prove the uniqueness theorem for such a kind equations with general fractional order integral operators, which are not necessarily contractions, in a suitably constructed generalized Hölder space. Full article
(This article belongs to the Special Issue New Trends in Fixed Point Theory with Emphasis on Symmetry)
11 pages, 280 KiB  
Article
Several Constructions of Cyclic Subspace Codes Utilizing (k + 1)-Dimensional Sidon Spaces
by Yongfeng Niu and Yu Li
Electronics 2024, 13(22), 4457; https://doi.org/10.3390/electronics13224457 - 14 Nov 2024
Viewed by 400
Abstract
In recent years, network coding has attracted widespread attention due to its important role in digital communication and network security. Among these, cyclic subspace codes have very important applications in error correction and rectification in random network coding, attracting many scholars to conduct [...] Read more.
In recent years, network coding has attracted widespread attention due to its important role in digital communication and network security. Among these, cyclic subspace codes have very important applications in error correction and rectification in random network coding, attracting many scholars to conduct research on them. As an important tool in constructing cyclic subspace codes, the present study focuses on the construction of Sidon spaces by leveraging the roots of irreducible polynomials and primitive elements over finite fields. In this paper, we construct some Sidon spaces with new parameters. Specifically, we let k,m1,m2 be three positive integers and define ρ1=m12k1,ρ2=m22m11. On the basis of these newly constructed Sidon spaces, we obtain new cyclic subspace codes with size ρ1ρ2qkqn1q1 and minimum distance 2k. Full article
33 pages, 504 KiB  
Article
Gelfand–Phillips Type Properties of Locally Convex Spaces
by Saak Gabriyelyan
Mathematics 2024, 12(22), 3537; https://doi.org/10.3390/math12223537 - 12 Nov 2024
Viewed by 426
Abstract
We let 1pq. Being motivated by the classical notions of the Gelfand–Phillips property and the (coarse) Gelfand–Phillips property of order p of Banach spaces, we introduce and study different types of the Gelfand–Phillips property of order [...] Read more.
We let 1pq. Being motivated by the classical notions of the Gelfand–Phillips property and the (coarse) Gelfand–Phillips property of order p of Banach spaces, we introduce and study different types of the Gelfand–Phillips property of order (p,q) (the GP(p,q) property) and the coarse Gelfand–Phillips property of order p in the realm of all locally convex spaces. We compare these classes and show that they are stable under taking direct product, direct sums and closed subspaces. It is shown that any locally convex space is a quotient space of a locally convex space with the GP(p,q) property. Characterizations of locally convex spaces with the introduced Gelfand–Phillips type properties are given. Full article
17 pages, 2899 KiB  
Article
Mangrove Extraction Algorithm Based on Orthogonal Matching Filter-Weighted Least Squares
by Yongze Li, Jin Ma, Dongyang Fu, Jiajun Yuan and Dazhao Liu
Sensors 2024, 24(22), 7224; https://doi.org/10.3390/s24227224 - 12 Nov 2024
Viewed by 484
Abstract
High-precision extraction of mangrove areas is a crucial prerequisite for estimating mangrove area as well as for regional planning and ecological protection. However, mangroves typically grow in coastal and near-shore areas with complex water colors, where traditional mangrove extraction algorithms face challenges such [...] Read more.
High-precision extraction of mangrove areas is a crucial prerequisite for estimating mangrove area as well as for regional planning and ecological protection. However, mangroves typically grow in coastal and near-shore areas with complex water colors, where traditional mangrove extraction algorithms face challenges such as unclear region segmentation and insufficient accuracy. To address this issue, in this paper we propose a new algorithm for mangrove identification and extraction based on Orthogonal Matching Filter–Weighted Least Squares (OMF-WLS) target spectral information. This method first selects GF-6 remote sensing images with less cloud cover, then enhances mangrove feature information through preprocessing and band extension, combining whitened orthogonal subspace projection with the whitened matching filter algorithm. Notably, this paper innovatively introduces Weighted Least Squares (WLS) filtering technology. WLS filtering precisely processes high-frequency noise and edge details in images using an adaptive weighting matrix, significantly improving the edge clarity and overall quality of mangrove images. This innovative approach overcomes the bottleneck of traditional methods in effectively extracting edge information against complex water color backgrounds. Finally, Otsu’s method is used for adaptive threshold segmentation of GF-6 remote sensing images to achieve target extraction of mangrove areas. Our experimental results show that OMF-WLS improves extraction accuracy compared to traditional methods, with overall precision increasing from 0.95702 to 0.99366 and the Kappa coefficient rising from 0.88436 to 0.98233. In addition, our proposed method provides significant improvements in other metrics, demonstrating better overall performance. These findings can provide more reliable technical support for the monitoring and protection of mangrove resources. Full article
(This article belongs to the Section Sensing and Imaging)
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<p>Study area (Yingluo Port).</p>
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<p>Extraction process of mangrove remote sensing imagery based on OMF-WLS.</p>
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<p>Extraction areas of mangrove samples selected in the Yingluo Port area: (<b>a</b>) the locations of the three selected areas in the GF-6 image, indicated by the rectangular boxes labeled A, B, and C; (<b>b</b>) area A; (<b>c</b>) area B; (<b>d</b>) area C.</p>
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<p>Comparison of extraction results from different algorithms for the three areas: (<b>a</b>–<b>g</b>) the original image, ground truth map, OMF-WLS extraction map, MF extraction map, CEM extraction map, Adaptive Coherence Estimator (ACE) extraction map, and Maximum Likelihood Estimation (MLE) extraction map for area A; (<b>h</b>–<b>n</b>) and (<b>o</b>–<b>u</b>) similarly represent the extraction results of the different algorithms for areas B and C, respectively, in the same order as for area A.</p>
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23 pages, 6761 KiB  
Article
Enhanced Subspace Iteration Technique for Probabilistic Modal Analysis of Statically Indeterminate Structures
by Hongfei Cao, Xi Peng, Bin Xu, Fengjiang Qin and Qiuwei Yang
Mathematics 2024, 12(22), 3486; https://doi.org/10.3390/math12223486 - 7 Nov 2024
Viewed by 512
Abstract
In structural stochastic dynamic analysis, the consideration of the randomness in the physical parameters of the structure necessitates the establishment of numerous stochastic finite element models and the subsequent computation of their corresponding vibration modes. When the complete analysis is applied to calculate [...] Read more.
In structural stochastic dynamic analysis, the consideration of the randomness in the physical parameters of the structure necessitates the establishment of numerous stochastic finite element models and the subsequent computation of their corresponding vibration modes. When the complete analysis is applied to calculate the vibration modes for each sample of the stochastic finite element model, a substantial computational expense is incurred. To enhance computational efficiency, this work presents an extended subspace iteration method aimed at rapidly determining the vibration modal parameters of statically indeterminate structures. The essence of this proposed method revolves around efficiently constructing reduced basis vectors during the subspace iteration process, utilizing flexibility disassembly perturbation and the Krylov subspace. This extended subspace iteration method proves particularly advantageous for the modal analysis of finite element models that incorporate a multitude of random variables. The proposed modal random analysis method has been validated using both a truss structure and a beam structure. The results demonstrate that the proposed method achieves substantial savings in computational time. Specifically, for the truss structure, the calculation time of the proposed method is approximately 1.2% and 65% of that required by the comprehensive analysis method and the combined approximation method, respectively. For the beam structure, on average, the computational time of the proposed method is roughly 2.1% of a full analysis and approximately 48.2% of the Ritz vector method’s time requirement. Compared to existing stochastic modal analysis algorithms, the proposed method offers improved computational accuracy and efficiency, particularly in scenarios involving high-discreteness random analyses. Full article
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<p>The operation flow of the proposed algorithm for random modal analysis of statically indeterminate structures.</p>
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<p>A truss structure with 58 rods.</p>
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<p>Calculation result of the first eigenvalue when the standard deviation is 0.1 times the mean value.</p>
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<p>Calculation errors of the first eigen-pairs when the standard deviation is 0.1 times the mean value.</p>
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<p>Calculation result of the second eigenvalue when the standard deviation is 0.1 times the mean value.</p>
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<p>Calculation errors of the second eigen-pairs when the standard deviation is 0.1 times the mean value.</p>
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<p>Calculation result of the third eigenvalue when the standard deviation is 0.1 times the mean value.</p>
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<p>Calculation errors of the third eigen-pairs when the standard deviation is 0.1 times the mean value.</p>
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<p>Calculation result of the first eigenvalue when the standard deviation is 0.15 times the mean value.</p>
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<p>Calculation errors of the first eigen-pairs when the standard deviation is 0.15 times the mean value.</p>
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<p>Calculation result of the second eigenvalue when the standard deviation is 0.15 times the mean value.</p>
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<p>Calculation errors of the second eigen-pairs when the standard deviation is 0.15 times the mean value.</p>
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<p>Calculation result of the third eigenvalue when the standard deviation is 0.15 times the mean value.</p>
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<p>Calculation errors of the third eigen-pairs when the standard deviation is 0.15 times the mean value.</p>
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<p>Calculation result of the first eigenvalue when the standard deviation is 0.2 times the mean value.</p>
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<p>Calculation errors of the first eigen-pairs when the standard deviation is 0.2 times the mean value.</p>
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<p>Calculation result of the second eigenvalue when the standard deviation is 0.2 times the mean value.</p>
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<p>Calculation errors of the second eigen-pairs when the standard deviation is 0.2 times the mean value.</p>
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<p>Calculation result of the third eigenvalue when the standard deviation is 0.2 times the mean value.</p>
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<p>Calculation errors of the third eigen-pairs when the standard deviation is 0.2 times the mean value.</p>
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<p>A beam structure.</p>
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