Journal Description
Symmetry
Symmetry
is an international, peer-reviewed, open access journal covering research on symmetry/asymmetry phenomena wherever they occur in all aspects of natural sciences. Symmetry is published monthly online by MDPI.
- Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
- High Visibility: indexed within Scopus, SCIE (Web of Science), CAPlus / SciFinder, Inspec, Astrophysics Data System, and other databases.
- Journal Rank: JCR - Q2 (Multidisciplinary Sciences) / CiteScore - Q1 (General Mathematics); Q1 (Physics and Astronomy); Q1 (Computer Science)
- Rapid Publication: manuscripts are peer-reviewed and a first decision is provided to authors approximately 16.2 days after submission; acceptance to publication is undertaken in 3.5 days (median values for papers published in this journal in the second half of 2023).
- Recognition of Reviewers: reviewers who provide timely, thorough peer-review reports receive vouchers entitling them to a discount on the APC of their next publication in any MDPI journal, in appreciation of the work done.
- Testimonials: See what our editors and authors say about Symmetry.
Impact Factor:
2.7 (2022);
5-Year Impact Factor:
2.7 (2022)
Latest Articles
The Expansion Methods of Inception and Its Application
Symmetry 2024, 16(4), 494; https://doi.org/10.3390/sym16040494 - 18 Apr 2024
Abstract
In recent years, with the rapid development of deep learning technology, a large number of excellent convolutional neural networks (CNNs) have been proposed, many of which are based on improvements to classical methods. Based on the Inception family of methods, depthwise separable convolution
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In recent years, with the rapid development of deep learning technology, a large number of excellent convolutional neural networks (CNNs) have been proposed, many of which are based on improvements to classical methods. Based on the Inception family of methods, depthwise separable convolution was applied to Xception to achieve lightweighting, and Inception-ResNet introduces residual connections to accelerate model convergence. However, existing improvements for the Inception module often neglect further enhancement of its receptive field, while increasing the receptive field of CNNs has been widely studied and proven to be effective in improving classification performance. Motivated by this fact, three effective expansion modules are proposed in this paper. The first expansion module, Inception expand (Inception-e) module, is proposed to improve the classification accuracy by concatenating more and deeper convolutional branches. To reduce the number of parameters for Inception e, this paper proposes a second expansion module—Equivalent Inception-e (Eception) module, which is equivalent to Inception-e in terms of feature extraction capability, but which suppresses the growth of the parameter quantity brought by the expansion by effectively reducing the redundant convolutional layers; on the basis of Eception, this paper proposes a third expansion module—Lightweight Eception (Lception) module, which crosses depthwise convolution with ordinary convolution to further effectively reduce the number of parameters. The three proposed modules have been validated on the Cifar10 dataset. The experimental results show that all these extensions are effective in improving the classification accuracy of the models, and the most significant effect is the Lception module, where Lception (rank = 4) on the Cifar10 dataset improves the accuracy by 1.5% compared to the baseline model (Inception module A) by using only 0.15 M more parameters.
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(This article belongs to the Section Computer)
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Accelerated Stability Testing in Food Supplements Underestimates Shelf Life Prediction of Resveratrol with Super-Arrhenius Behavior
by
Andrea Biagini, Nicola Refrigeri, Concetta Caglioti, Paola Sabbatini, Silvia Ticconi, Giada Ceccarelli, Rossana Giulietta Iannitti, Federico Palazzetti and Bernard Fioretti
Symmetry 2024, 16(4), 493; https://doi.org/10.3390/sym16040493 - 18 Apr 2024
Abstract
Thermo-oxidative stability testing plays a critical role in accurately predicting shelf life. These tests are performed in real time and under stress conditions, where degradation processes are accelerated by increasing storage conditions. In this study, high-performance liquid chromatography (HPLC) analyses were performed to
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Thermo-oxidative stability testing plays a critical role in accurately predicting shelf life. These tests are performed in real time and under stress conditions, where degradation processes are accelerated by increasing storage conditions. In this study, high-performance liquid chromatography (HPLC) analyses were performed to evaluate the degradation of resveratrol in nutraceutical tablets as a function of time under different storage conditions in terms of temperature and relative humidity (RH), namely 25 °C/60% RH, 30 °C/65% RH, and 40 °C/75% RH. The latter is an accelerated test and is used to estimate shelf life for long-term storage. Resveratrol is present in both pure form and as a solid dispersion on magnesium dihydroxide microparticles (Resv@MDH). Degradation kinetic constants were determined at 25 °C, 30 °C, and 40 °C, and the Arrhenius behavior of the kinetic constants as a function of temperature was verified. The main results of this work are as follows: (i) the stability of resveratrol in nutraceutical tablets is affected by temperature; (ii) the dependence of the kinetic constants on temperature does not follow the Arrhenius equation, determining an overestimation of the degradation rate at 25 °C; in this regard a modified version of the Arrhenius equation that takes into account the deviation from linearity has been used to estimate the dependence of the kinetic constant on the temperature. These results suggest that accelerated testing does not provide a general model for predicting the shelf life of foods and dietary supplements. The reason may be due to possible matrix effects that result in different degradation mechanisms depending on the temperature. In this regard, symmetry relationships in the kinetics of chemical reactions resulting from microscopic reversibility and their relationship to the deviation from the Arrhenius equation are discussed. However, further research is needed to characterize the degradation mechanisms at different temperatures. The results of these studies would allow accurate prediction of food degradation to improve food safety and risk management and reduce food waste. In addition, knowledge of stability processes is necessary to ensure the maintenance of physiological processes by dietary supplements.
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(This article belongs to the Special Issue From Nanoclusters to Nanoparticles: Symmetry, Theory, Experiments, and Applications)
Open AccessArticle
Generalized Neuromorphism and Artificial Intelligence: Dynamics in Memory Space
by
Said Mikki
Symmetry 2024, 16(4), 492; https://doi.org/10.3390/sym16040492 - 18 Apr 2024
Abstract
This paper introduces a multidisciplinary conceptual perspective encompassing artificial intelligence (AI), artificial general intelligence (AGI), and cybernetics, framed within what we call the formalism of generalized neuromorphism. Drawing from recent advancements in computing, such as neuromorphic computing and spiking neural networks, as well
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This paper introduces a multidisciplinary conceptual perspective encompassing artificial intelligence (AI), artificial general intelligence (AGI), and cybernetics, framed within what we call the formalism of generalized neuromorphism. Drawing from recent advancements in computing, such as neuromorphic computing and spiking neural networks, as well as principles from the theory of open dynamical systems and stochastic classical and quantum dynamics, this formalism is tailored to model generic networks comprising abstract processing events. A pivotal aspect of our approach is the incorporation of the memory space and the intrinsic non-Markovian nature of the abstract generalized neuromorphic system. We envision future computations taking place within an expanded space (memory space) and leveraging memory states. Positioned at a high abstract level, generalized neuromorphism facilitates multidisciplinary applications across various approaches within the AI community.
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(This article belongs to the Section Mathematics)
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The General Solution to a Classical Matrix Equation AXB = C over the Dual Split Quaternion Algebra
by
Kai-Wen Si and Qing-Wen Wang
Symmetry 2024, 16(4), 491; https://doi.org/10.3390/sym16040491 - 18 Apr 2024
Abstract
In this paper, we investigate the necessary and sufficient conditions for solving a dual split quaternion matrix equation , and present the general solution expression when the solvability conditions are met. As an application, we delve
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In this paper, we investigate the necessary and sufficient conditions for solving a dual split quaternion matrix equation , and present the general solution expression when the solvability conditions are met. As an application, we delve into the necessary and sufficient conditions for the existence of a Hermitian solution to this equation by using a newly defined real representation method. Furthermore, we obtain the solutions for the dual split quaternion matrix equations and . Finally, we provide a numerical example to demonstrate the findings of this paper.
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(This article belongs to the Section Mathematics)
Open AccessArticle
Characteristics of Midface Asymmetry in Skeletal Class III Malocclusion Using Three-Dimensional Analysis
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Chia-Yi (Jessica) Wang, Chen-Jung Chang, Meng-Yen Chen, Tung-Yiu Wong and Jing-Jing Fang
Symmetry 2024, 16(4), 490; https://doi.org/10.3390/sym16040490 - 18 Apr 2024
Abstract
Background: The midface plays an important role in the judgment of symmetry. However, studies on three-dimensional analyses of midface asymmetry are limited. This study investigated the characteristics of midface asymmetry in skeletal Class III malocclusion patients through three-dimensional analysis. Methods: Sixty-eight adult subjects
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Background: The midface plays an important role in the judgment of symmetry. However, studies on three-dimensional analyses of midface asymmetry are limited. This study investigated the characteristics of midface asymmetry in skeletal Class III malocclusion patients through three-dimensional analysis. Methods: Sixty-eight adult subjects with skeletal Class III malocclusion were included and divided into mandible symmetry and asymmetry groups. The prevalence of recognizable malar asymmetry and the deviation of anterior nasal spine (ANS) were examined. The relation between midface and mandible asymmetry were investigated with Spearman correlation. The difference in distance of landmarks to reference planes were compared between the two groups using Mann–Whitney U test (p < 0.05). Results: The overall prevalence of malar asymmetry was 7.35% and of ANS deviation was 38.24%. In subjects with chin deviated to the right, there was a moderate negative correlation between chin deviation and difference of zygion and zygomatic process to mid-sagittal plane. The absolute value of difference in the glenoid fossa was significantly greater in female asymmetry subjects. Conclusions: The prevalence of midface asymmetry is not low. The more severely the chin is shifted, the greater asymmetrical position of the zygoma and glenoid fossa was found. Therefore, pre-surgical case-by-case evaluation of the midface region is essential for understanding the midface skeletal characteristics of Class III patients with chin deviation, thereby providing patients with realistic expectations and optimizing surgical outcomes and patient satisfaction.
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(This article belongs to the Special Issue Advances in Imaging Evaluation of Head and Neck Spaces with Symmetry)
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Controlled State Transfer in Central Spin Models
by
Martiros Khurshudyan
Symmetry 2024, 16(4), 489; https://doi.org/10.3390/sym16040489 - 17 Apr 2024
Abstract
In the recent literature, various aspects of the transfer of quantum states by spin chains have been thoroughly investigated. Part of the existing study is devoted to the problem of optimal control, with the goal of achieving a highly reliable information/state transfer for
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In the recent literature, various aspects of the transfer of quantum states by spin chains have been thoroughly investigated. Part of the existing study is devoted to the problem of optimal control, with the goal of achieving a highly reliable information/state transfer for a given time T. In general, achieving this goal is not an easy task in the case of (open) quantum systems. Various approaches have been developed and applied, including Krotov’s method to study the problem. It is a gradient-based method used here to study the problem of state transfer control in central spin models. Our results show that with Krotov’s method, it is possible to find an optimal control form that allows for very-high-fidelity state transfer in the central spin models we have developed. Our results will be of interest for a better understanding of the non-trivial effects of the classical world on the quantum world, which have been discussed in the form of various new effects, including the Epstein effect, in the recent literature.
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(This article belongs to the Special Issue Feature Papers in 'Physics' Section 2024)
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Cascading Failure Modeling for Circuit Systems Considering Continuous Degradation and Random Shocks Using an Impedance Network
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Yi Jin and Qingyuan Zhang
Symmetry 2024, 16(4), 488; https://doi.org/10.3390/sym16040488 - 17 Apr 2024
Abstract
The reliability of circuit systems is primarily affected by cascading failures due to their complex structural and functional coupling. Causes of cascading failure during circuit operation include the continuous degradation process of components and external random shocks. Circuit systems can exhibit asymmetric structural
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The reliability of circuit systems is primarily affected by cascading failures due to their complex structural and functional coupling. Causes of cascading failure during circuit operation include the continuous degradation process of components and external random shocks. Circuit systems can exhibit asymmetric structural changes and functional loss during cascading failure propagation due to the coupling of degradation and shock and their uncertainty effects. To tackle this issue, this paper abstracts the circuit into an impedance network and constructs a component failure behavior model that considers the correlation between degradation and shock. The interactions between soft and hard failure processes among different components are discussed. Two types of cascading failure propagation processes are described: slow propagation associated with continuous degradation and damage shock, and fast propagation due to fatal shock. Based on this, a cascading failure simulation algorithm is developed. This article presents a case study to demonstrate the proposed models and to analyze the reliability of a typical circuit system.
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(This article belongs to the Special Issue Advances and Applications of Uncertainty Theory in Reliability and Systems Engineering)
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Research on Target Ranging Method for Live-Line Working Robots
by
Guoxiang Hua, Guo Chen, Qingxin Luo and Jiyuan Yan
Symmetry 2024, 16(4), 487; https://doi.org/10.3390/sym16040487 - 17 Apr 2024
Abstract
Due to the operation of live-line working robots at elevated heights for precision tasks, a suitable visual assistance system is essential to determine the position and distance of the robotic arm or gripper relative to the target object. In this study, we propose
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Due to the operation of live-line working robots at elevated heights for precision tasks, a suitable visual assistance system is essential to determine the position and distance of the robotic arm or gripper relative to the target object. In this study, we propose a method for distance measurement in live-line working robots by integrating the YOLOv5 algorithm with binocular stereo vision. The camera’s intrinsic and extrinsic parameters, as well as distortion coefficients, are obtained using the Zhang Zhengyou calibration method. Subsequently, stereo rectification is performed on the images to establish a standardized binocular stereovision model. The Census and Sum of Absolute Differences (SAD) fused stereo matching algorithm is applied to compute the disparity map. We train a dataset of transmission line bolts within the YOLO framework to derive the optimal model. The identified bolts are framed, and the depth distance of the target is ultimately calculated. And through the experimental verification of the bolt positioning, the results show that the method can achieve a relative error of 1% in the proximity of positioning. This approach provides real-time and accurate environmental perception for symmetrical structural live-line working robots, enhancing the stability of these robots.
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(This article belongs to the Section Computer)
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Removable Singularities of Harmonic Functions on Stratified Sets
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Nurlan S. Dairbekov, Oleg M. Penkin and Denis V. Savasteev
Symmetry 2024, 16(4), 486; https://doi.org/10.3390/sym16040486 - 17 Apr 2024
Abstract
There are deep historical connections between symmetry, harmonic functions, and stratified sets. In this article, we prove an analog of the removable singularity theorem for bounded harmonic functions on stratified sets. The harmonic functions are understood in the sense of the soft Laplacian.
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There are deep historical connections between symmetry, harmonic functions, and stratified sets. In this article, we prove an analog of the removable singularity theorem for bounded harmonic functions on stratified sets. The harmonic functions are understood in the sense of the soft Laplacian. The result can become one of the main technical components for extending the well-known Poincaré–Perron’s method of proving the solvability of the Dirichlet problem for the soft Laplacian.
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(This article belongs to the Section Mathematics)
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Influence of the Symmetry Neural Network Morphology on the Mine Detection Metric
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Roman Mykhailovych Peleshchak, Vasyl Volodymyrovych Lytvyn, Mariia Andriivna Nazarkevych, Ivan Romanovych Peleshchak and Hanna Yaroslavivna Nazarkevych
Symmetry 2024, 16(4), 485; https://doi.org/10.3390/sym16040485 - 17 Apr 2024
Abstract
Presently, active detectors are widely used to detect mines, providing high accuracy. However, the principle of the operation of active detectors can lead to the explosion of hidden mines. The novelty of this work is the development of the morphology of a neural
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Presently, active detectors are widely used to detect mines, providing high accuracy. However, the principle of the operation of active detectors can lead to the explosion of hidden mines. The novelty of this work is the development of the morphology of a neural network for the classification of mines made of different materials (metallic, semi-metallic, plastic) with high accuracy (99.23%), based on a vector of input features with the following components: the value of the output voltage of the FLC-100 magnetic field sensor, which measures magnetic field anomalies in the vicinity of mines with an accuracy of 10−10–10−4 Tesla; six different soil types, depending on the humidity; and the height at which the magnetic field sensor is located above the mine. Due to the fact that mines, when made of different materials (metallic, semi-metallic, plastic), have different magnetic properties, the neural network method of mine classification, based on the sensor data regarding anomalies of the magnetic field in the vicinity of mines, allows the classification of mines made of different materials. The accuracy of mine classification was assessed with two-layer and three-layer neural networks on various metrics (confusion matrix, ROC curves, accuracy–loss curves), using ADAM, RMSprop, and SGD optimisers, and analyses and comparisons were then carried out. The impact of asymmetry in the neuron number and the types of activation functions in the first and second hidden layers on the values of the accuracy and loss metrics was studied. In particular, it was established that the asymmetry of the number of neurons in the first and second hidden layers relative to the plane of symmetry between the hidden layers has a significant effect on the accuracy of the model (decrease in accuracy by 25%), while the loss function, when the symmetry of the neurons number in the hidden layers is violated, increases to a maximum of 50%.
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(This article belongs to the Section Computer)
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Evaluation of Symmetrical Face Pressure of EPB
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Hasan Eray Yaman and Cemalettin Okay Aksoy
Symmetry 2024, 16(4), 484; https://doi.org/10.3390/sym16040484 - 16 Apr 2024
Abstract
The content of this study combines city safety, optimum excavation situation, mining, geology, and civil engineering principles. Tunnel boring machines (TBM) are the most commonly used machines in the excavation of urban tunnels. These machines prevent the inward movement of the tunnel face
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The content of this study combines city safety, optimum excavation situation, mining, geology, and civil engineering principles. Tunnel boring machines (TBM) are the most commonly used machines in the excavation of urban tunnels. These machines prevent the inward movement of the tunnel face and control the amount of settlement formed on the ground by applying pressure to the tunnel face. The most important question here is to determine the amount of pressure to be applied to the tunnel face. There are many widely accepted formulas used in the calculation of the face pressure and these formulas generally attempt to limit the settlements on the ground by using parameters such as groundwater level, overburden thickness, physical and mechanical properties of the surrounding rocks, etc. In this study, a new formula was developed. This new formula calculates the face pressure required to be applied by EPB to the tunnel face in order to prevent damage to a structure located on the route and within the area to be affected by tunnel excavation, instead of only preventing settlements on the surface. In the formula, produced within the scope of this study, in addition to other studies, 3D distances of the structure to which the deformation limitation will be made to prevent damage is also one of the parameters.
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(This article belongs to the Section Engineering and Materials)
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Common Fixed-Point Theorem and Projection Method on a Hadamard Space
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Yasunori Kimura
Symmetry 2024, 16(4), 483; https://doi.org/10.3390/sym16040483 - 16 Apr 2024
Abstract
In this paper, we obtain an equivalent condition to the existence of a common fixed point of a given family of nonexpansive mappings defined on a Hadamard space. Moreover, if the space is bounded, we show that the generating process of the approximate
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In this paper, we obtain an equivalent condition to the existence of a common fixed point of a given family of nonexpansive mappings defined on a Hadamard space. Moreover, if the space is bounded, we show that the generating process of the approximate sequence by a specific projection method will stop in finite steps if there is no common fixed point. It is a significant advantage to reveal the nonexistence of a common fixed point in a finite time.
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(This article belongs to the Special Issue Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk)
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Relativistic Formulation in Dual Minkowski Spacetime
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Timothy Ganesan
Symmetry 2024, 16(4), 482; https://doi.org/10.3390/sym16040482 - 16 Apr 2024
Abstract
The objective of this work is to derive the structure of Minkowski spacetime using a Hermitian spin basis. This Hermitian spin basis is analogous to the Pauli spin basis. The derived Minkowski metric is then employed to obtain the corresponding Lorentz factors, potential
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The objective of this work is to derive the structure of Minkowski spacetime using a Hermitian spin basis. This Hermitian spin basis is analogous to the Pauli spin basis. The derived Minkowski metric is then employed to obtain the corresponding Lorentz factors, potential Lie algebra, effects on gamma matrices and complex representations of relativistic time dilation and length contraction. The main results, a discussion of the potential applications and future research directions are provided.
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(This article belongs to the Special Issue Symmetry and Asymmetry in Nature: From Quantum Physics to the Universe)
Open AccessArticle
A Globally Convergent Iterative Method for Matrix Sign Function and Its Application for Determining the Eigenvalues of a Matrix Pencil
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Munish Kansal, Vanita Sharma, Pallvi Sharma and Lorentz Jäntschi
Symmetry 2024, 16(4), 481; https://doi.org/10.3390/sym16040481 - 16 Apr 2024
Abstract
In this research article, we propose a new matrix iterative method with a convergence order of five for computing the sign of a complex matrix by examining the different patterns and symmetry of existing methods. Analysis of the convergence of the method was
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In this research article, we propose a new matrix iterative method with a convergence order of five for computing the sign of a complex matrix by examining the different patterns and symmetry of existing methods. Analysis of the convergence of the method was explored on a global scale, and attraction basins were demonstrated. In addition to this, the asymptotic stability of the scheme was explored.Then, an algorithm for determing thegeneralized eigenvalues for the case of regular matrix pencils was investigated using the matrix sign computation. We performed a series of numerical experiments using numerous matrices to confirm the usefulness and superiority of the proposed method.
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(This article belongs to the Special Issue The Advances of Nonlinear Equations: Mathematical Models, Symmetry and Applications)
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Systemic Financial Risk Forecasting with Decomposition–Clustering-Ensemble Learning Approach: Evidence from China
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Zhongzhe Ouyang and Min Lu
Symmetry 2024, 16(4), 480; https://doi.org/10.3390/sym16040480 - 15 Apr 2024
Abstract
Establishing a scientifically effective systemic financial risk early warning model is of great significance for prudently mitigating systemic financial risks and enhancing the efficiency of financial supervision. Based on the measurement of systemic financial risk and the network sentiment index of 47 financial
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Establishing a scientifically effective systemic financial risk early warning model is of great significance for prudently mitigating systemic financial risks and enhancing the efficiency of financial supervision. Based on the measurement of systemic financial risk and the network sentiment index of 47 financial institutions, this study adopted the “decomposition–reconstruction–integration” approach, utilizing techniques such as extreme-point symmetric empirical mode decomposition (ESMD), empirical mode decomposition (EMD), variational mode decomposition (VMD), hierarchical clustering, fast independent component analysis (FastICA), attention mechanism, bidirectional long short-term memory neural network (BiLSTM), support vector regression (SVR), and their combination, to construct a systemic financial risk prediction model. The empirical results demonstrate that decomposing and reconstructing relevant indicators before predicting systemic financial risks can enhance prediction accuracy. Among the proposed models, the ESMD-HFastICA-BiLSTM-Attention model exhibits superior performance in systemic financial risk early warning.
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(This article belongs to the Special Issue Symmetry and Asymmetry in Machine Learning)
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A Coordinated Control Strategy of Multi-Type Flexible Resources and Under-Frequency Load Shedding for Active Power Balance
by
Jian Zhang, Jiaying Wang, Yongji Cao, Baoliang Li and Changgang Li
Symmetry 2024, 16(4), 479; https://doi.org/10.3390/sym16040479 - 15 Apr 2024
Abstract
With the increasing expansion of power systems, there is a growing trend towards active distribution networks for decentralized power generation and energy management. However, the instability of distributed renewable energy introduces complexity to power system operation. The active symmetry and balance of power
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With the increasing expansion of power systems, there is a growing trend towards active distribution networks for decentralized power generation and energy management. However, the instability of distributed renewable energy introduces complexity to power system operation. The active symmetry and balance of power systems are becoming increasingly important. This paper focuses on the characteristics of distributed resources and under-frequency load shedding, and a coordinated operation and control strategy based on the rapid adjustment of energy storage power is proposed. The characteristics of various controllable resources are analyzed to explore the rapid response capabilities of energy storage. The energy storage types are categorized based on the support time, and the final decision is achieved with power allocation and adjustment control of the energy storage system. Additionally, a comprehensive control strategy for under-frequency load shedding and hierarchical systems is provided for scenarios with insufficient active support. The feasibility of the proposed model and methods is verified via a multi-energy system case.
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(This article belongs to the Special Issue Symmetry/Asymmetry Studies in Modern Power Systems)
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Floating-Point Embedding: Enhancing the Mathematical Comprehension of Large Language Models
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Xiaoxiao Jin, Chenyang Mao, Dengfeng Yue and Tuo Leng
Symmetry 2024, 16(4), 478; https://doi.org/10.3390/sym16040478 - 15 Apr 2024
Abstract
The processing and comprehension of numerical information in natural language represent pivotal focal points of scholarly inquiry. Across diverse applications spanning text analysis to information retrieval, the adept management and understanding of the numerical content within natural language are indispensable in achieving task
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The processing and comprehension of numerical information in natural language represent pivotal focal points of scholarly inquiry. Across diverse applications spanning text analysis to information retrieval, the adept management and understanding of the numerical content within natural language are indispensable in achieving task success. Specialized encoding and embedding techniques tailored to numerical data offer an avenue toward improved performance in tasks involving masked prediction and numerical reasoning, inherently characterized by numerical values. Consequently, treating numbers in text merely as words is inadequate; their numerical semantics must be underscored. Recent years have witnessed the emergence of a range of specific encoding methodologies designed explicitly for numerical content, demonstrating promising outcomes. We observe similarities between the Transformer architecture and CPU architecture, with symmetry playing a crucial role. In light of this observation and drawing inspiration from computer system theory, we introduce a floating-point representation and devise a corresponding embedding module. The numerical representations correspond one-to-one with their semantic vector values, rendering both symmetric regarding intermediate transformation methods. Our proposed methodology facilitates the more comprehensive encoding and embedding of numerical information within a predefined precision range, thereby ensuring a distinctive encoding representation for each numerical entity. Rigorous testing on multiple encoder-only models and datasets yielded results that stand out in terms of competitiveness. In comparison to the default embedding methods employed by models, our approach achieved an improvement of approximately in Top-1 accuracy and a reduction in perplexity of approximately . These outcomes affirm the efficacy of our proposed method. Furthermore, the enrichment of numerical semantics through a more comprehensive embedding contributes to the augmentation of the model’s capacity for semantic understanding.
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(This article belongs to the Special Issue Applications Based on AI in Mathematics and Asymmetry/Symmetry)
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Determination of Na+ Cation Locations in Nanozeolite ECR-1 Using a 3D ED Method
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Taylan Örs, Irena Deroche, Corentin Chatelard, Mathias Dodin, Raquel Martinez-Franco, Alain Tuel and Jean-Louis Paillaud
Symmetry 2024, 16(4), 477; https://doi.org/10.3390/sym16040477 - 15 Apr 2024
Abstract
Until now, the comprehensive structural analysis of single crystals of zeolite ECR-1, an aluminosilicate with the EON topology, has been hindered owing to the submicron dimensions of the obtained crystals. Additionally, this zeolite, which is characterized by a topology comprising alternating periodic building
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Until now, the comprehensive structural analysis of single crystals of zeolite ECR-1, an aluminosilicate with the EON topology, has been hindered owing to the submicron dimensions of the obtained crystals. Additionally, this zeolite, which is characterized by a topology comprising alternating periodic building units of MAZ and MOR layers, exhibits stacking faults that impede accurate refinement through the Rietveld method. In this report, we present, for the first time, the structure of ECR-1 elucidated by studying a nanocrystal with a significantly reduced number of stacking faults. The sample used was synthesized hydrothermally using trioxane as the organic structure-directing agent. The structure determination was conducted using precession electron diffraction (PED) at 103 K. Partial dehydration occurred owing to the high vacuum conditions in the TEM sample chamber. From the dynamical refinement (Robs = 0.097), 8.16 Na+ compensating cations were localized on six distinct crystallographic sites, along with approximately four water molecules per unit cell. Furthermore, a canonical Monte Carlo computational study was conducted to compare the experimental cationic distribution and location of water molecules with the simulation.
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(This article belongs to the Special Issue Electron Diffraction and Structural Imaging II)
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A 3D Descriptive Geometry Problem-Solving Methodology Using CAD and Orthographic Projection
by
Eduardo Gutiérrez de Ravé and Francisco J. Jiménez-Hornero
Symmetry 2024, 16(4), 476; https://doi.org/10.3390/sym16040476 - 14 Apr 2024
Abstract
In solving descriptive geometry (DG) problems, board (i.e., hand drawing) methods are frequently used, despite this discipline is still very important to enhance spatial vision. These methods are very different from CAD tools which are used in the field of design. CAD facilitates
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In solving descriptive geometry (DG) problems, board (i.e., hand drawing) methods are frequently used, despite this discipline is still very important to enhance spatial vision. These methods are very different from CAD tools which are used in the field of design. CAD facilitates the realization of geometric constructions and transformations (i.e., rotation, translation, copying, scaling, alignment, and symmetry, among others) are performed analytically. For this reason, a 3D DG problem-solving methodology using CAD and orthographic projection (CADOP) is introduced. Once the principles of DG, orthographic projection, and CAD fundamentals and tools are described, CADOP is applied to obtain (i) orthogonal views; (ii) principal lines of a plane; (iii) true-size view of a plane; (iv) parallelism, perpendicularity, and distance, and (v) angles. Considering the user coordinate system in CADOP allows one to place the horizontal plane in the suitable position to solve DG problems directly in one step. In the traditional methods, the use of auxiliary views must be carried out in several steps instead. The dynamic management of the 3D view of the scene is facilitated in CADOP, improving its understanding, and achieving the precision inherent in analytical calculations.
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(This article belongs to the Special Issue Challenges and Trends in the Application of Descriptive Geometry Field under the Current Paradigm of Computer – Assisted Design (CAD))
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Noether Symmetry of Multi-Time-Delay Non-Conservative Mechanical System and Its Conserved Quantity
by
Xingyu Ji, Zhengwei Yang and Xianghua Zhai
Symmetry 2024, 16(4), 475; https://doi.org/10.3390/sym16040475 - 14 Apr 2024
Abstract
The study of multi-time-delay dynamical systems has highlighted many challenges, especially regarding the solution and analysis of multi-time-delay equations. The symmetry and conserved quantity are two important and effective essential properties for understanding complex dynamical behavior. In this study, a multi-time-delay non-conservative mechanical
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The study of multi-time-delay dynamical systems has highlighted many challenges, especially regarding the solution and analysis of multi-time-delay equations. The symmetry and conserved quantity are two important and effective essential properties for understanding complex dynamical behavior. In this study, a multi-time-delay non-conservative mechanical system is investigated. Firstly, the multi-time-delay Hamilton principle is proposed. Then, multi-time-delay non-conservative dynamical equations are deduced. Secondly, depending on the infinitesimal group transformations, the invariance of the multi-time-delay Hamilton action is studied, and Noether symmetry, Noether quasi-symmetry, and generalized Noether quasi-symmetry are discussed. Finally, Noether-type conserved quantities for a multi-time-delay Lagrangian system and a multi-time-delay non-conservative mechanical system are obtained. Two examples in terms of a multi-time-delay non-conservative mechanical system and a multi-time-delay Lagrangian system are given.
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