Taher Armaghani, Mojtaba Sepehrnia, Maysam Molana, Manasik M. Nour, Amir Safari
Investigating natural convection heat transfer of nanofluids in various geometries has garnered significant attention due to its potential applications across several disciplines. This study presents a numerical simulation of the natural convection heat transfer and entropy generation process in an E-shaped porous cavity filled with nanofluids, implementing Buongiorno’s simulation model. Analyzing the behavior of individual nanoparticles, or even the entire nanofluid system at the molecular level, can be extremely computationally intensive. Symmetry is a fundamental concept in science that can help reduce this computational burden considerably. In this study, nanofluids are frequently conceived of as a combination of water and Al2O3 nanoparticles at a concentration of up to 4% by volume. A unique correlation was proposed to model the effective thermal conductivity of nanofluids. The average Nusselt number, entropy production, and Rayleigh number have been illustrated to exhibit a decreasing trend when the volume concentration of nanoparticles inside the porous cavity rises; the 4% vol. water–alumina NFs yield 17.35% less average Nu number compared to the base water.
{"title":"Studying Alumina–Water Nanofluid Two-Phase Heat Transfer in a Novel E-Shaped Porous Cavity via Introducing New Thermal Conductivity Correlation","authors":"Taher Armaghani, Mojtaba Sepehrnia, Maysam Molana, Manasik M. Nour, Amir Safari","doi":"10.3390/sym15112057","DOIUrl":"https://doi.org/10.3390/sym15112057","url":null,"abstract":"Investigating natural convection heat transfer of nanofluids in various geometries has garnered significant attention due to its potential applications across several disciplines. This study presents a numerical simulation of the natural convection heat transfer and entropy generation process in an E-shaped porous cavity filled with nanofluids, implementing Buongiorno’s simulation model. Analyzing the behavior of individual nanoparticles, or even the entire nanofluid system at the molecular level, can be extremely computationally intensive. Symmetry is a fundamental concept in science that can help reduce this computational burden considerably. In this study, nanofluids are frequently conceived of as a combination of water and Al2O3 nanoparticles at a concentration of up to 4% by volume. A unique correlation was proposed to model the effective thermal conductivity of nanofluids. The average Nusselt number, entropy production, and Rayleigh number have been illustrated to exhibit a decreasing trend when the volume concentration of nanoparticles inside the porous cavity rises; the 4% vol. water–alumina NFs yield 17.35% less average Nu number compared to the base water.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136351390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Xuan Liu, Hongmin Lu, Yifeng He, Fulin Wu, Chengxi Zhang, Xiaoliang Wang
An oscillator is a key component of a satellite communication measurement and control system, performing symmetry precisely as a time frequency reference. At the same time, the phase noise index has a close coupling relationship with the overall performance of the entire system, while persistently breaking the symmetry property of the oscillator during work. It is very important to study and reasonably allocate the phase noise index. According to the theoretical formula of phase noise, this paper analyzes the power law spectral model in the frequency domain and the noise jitter characteristics in the time domain. Using the carrier tracking loop in the measurement system, the frequency domain transfer model of phase noise is established, and typical analysis results are given. A discrete fractional integration algorithm is proposed, which can generate the phase noise time domain sequence under the given power law spectral model coefficients. The proposed algorithm is more realistic compared with the previous numerical calculation method, and has sufficient accuracy compared with the results of the instrument. After frequency domain conversion, the RMS deviation between the simulated noise sequence in the frequency domain and the measured single sideband power spectral density is less than 2.5 dB, indicating that the phase noise sequence can reflect the frequency domain characteristics more completely. A communication measurement simulation system is built, and a discrete sequence simulation analysis method combining frequency domain and time domain is provided, and the coupling relationship of key indicators such as phase noise, thermal noise, communication data rate, modulation method and bit error rate is synthesized. The results show that the BER of the QPSK/BPSK communication system will not be significantly reduced if the phase jitter RMS caused by the phase noise is less than 5 degrees, so 5 degrees can be used as a reference for the decomposition of the carrier SSB phase noise index. The simulation results have been successfully applied to a satellite inter-satellite link system, which has universal practical application value.
{"title":"Analysis on the Effect of Phase Noise on the Performance of Satellite Communication and Measurement System","authors":"Xuan Liu, Hongmin Lu, Yifeng He, Fulin Wu, Chengxi Zhang, Xiaoliang Wang","doi":"10.3390/sym15112053","DOIUrl":"https://doi.org/10.3390/sym15112053","url":null,"abstract":"An oscillator is a key component of a satellite communication measurement and control system, performing symmetry precisely as a time frequency reference. At the same time, the phase noise index has a close coupling relationship with the overall performance of the entire system, while persistently breaking the symmetry property of the oscillator during work. It is very important to study and reasonably allocate the phase noise index. According to the theoretical formula of phase noise, this paper analyzes the power law spectral model in the frequency domain and the noise jitter characteristics in the time domain. Using the carrier tracking loop in the measurement system, the frequency domain transfer model of phase noise is established, and typical analysis results are given. A discrete fractional integration algorithm is proposed, which can generate the phase noise time domain sequence under the given power law spectral model coefficients. The proposed algorithm is more realistic compared with the previous numerical calculation method, and has sufficient accuracy compared with the results of the instrument. After frequency domain conversion, the RMS deviation between the simulated noise sequence in the frequency domain and the measured single sideband power spectral density is less than 2.5 dB, indicating that the phase noise sequence can reflect the frequency domain characteristics more completely. A communication measurement simulation system is built, and a discrete sequence simulation analysis method combining frequency domain and time domain is provided, and the coupling relationship of key indicators such as phase noise, thermal noise, communication data rate, modulation method and bit error rate is synthesized. The results show that the BER of the QPSK/BPSK communication system will not be significantly reduced if the phase jitter RMS caused by the phase noise is less than 5 degrees, so 5 degrees can be used as a reference for the decomposition of the carrier SSB phase noise index. The simulation results have been successfully applied to a satellite inter-satellite link system, which has universal practical application value.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135036811","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Anna A. Rybkina, Alevtina A. Gogina, Artem V. Tarasov, Ye Xin, Vladimir Yu. Voroshnin, Dmitrii A. Pudikov, Ilya I. Klimovskikh, Anatoly E. Petukhov, Kirill A. Bokai, Chengxun Yuan, Zhongxiang Zhou, Alexander M. Shikin, Artem G. Rybkin
Intercalation of noble metals can produce giant Rashba-type spin–orbit splittings in graphene. The spin–orbit splitting of more than 100 meV has yet to be achieved in graphene on metal or semiconductor substrates. Here, we report the p-type graphene obtained by Pt intercalation of zero-layer graphene on SiC substrate. The spin splitting of ∼200 meV was observed at a wide range of binding energies. Comparing the results of theoretical studies of different models with the experimental ones measured by spin-ARPES, XPS and STM methods, we concluded that inducing giant spin–orbit splitting requires not only a relatively close distance between graphene and Pt layer but also the presence of graphene corrugation caused by a non-flat Pt layer. This makes it possible to find a compromise between strong hybridization and increased spin–orbit interaction. In our case, the Pt submonolayer possesses nanometer-scale lateral ordering under graphene.
{"title":"Origin of Giant Rashba Effect in Graphene on Pt/SiC","authors":"Anna A. Rybkina, Alevtina A. Gogina, Artem V. Tarasov, Ye Xin, Vladimir Yu. Voroshnin, Dmitrii A. Pudikov, Ilya I. Klimovskikh, Anatoly E. Petukhov, Kirill A. Bokai, Chengxun Yuan, Zhongxiang Zhou, Alexander M. Shikin, Artem G. Rybkin","doi":"10.3390/sym15112052","DOIUrl":"https://doi.org/10.3390/sym15112052","url":null,"abstract":"Intercalation of noble metals can produce giant Rashba-type spin–orbit splittings in graphene. The spin–orbit splitting of more than 100 meV has yet to be achieved in graphene on metal or semiconductor substrates. Here, we report the p-type graphene obtained by Pt intercalation of zero-layer graphene on SiC substrate. The spin splitting of ∼200 meV was observed at a wide range of binding energies. Comparing the results of theoretical studies of different models with the experimental ones measured by spin-ARPES, XPS and STM methods, we concluded that inducing giant spin–orbit splitting requires not only a relatively close distance between graphene and Pt layer but also the presence of graphene corrugation caused by a non-flat Pt layer. This makes it possible to find a compromise between strong hybridization and increased spin–orbit interaction. In our case, the Pt submonolayer possesses nanometer-scale lateral ordering under graphene.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135038486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Aleksey Lutsenko, Alla Sidorova, Denis Shpigun, Ekaterina Belova, Vsevolod Tverdislov
Chirality plays an important role in studies of natural protein structures. Therefore, much attention is paid to solving the problems associated with the development of criteria and methods for assessing the chirality of biomolecules. In this paper, a new method for calculating the sign and degree of chirality of superhelices is proposed. The method makes it possible to characterize the chirality sign and to quantify coiled-coils and collagen superhelices. The degree of chirality is understood as a value indicating the intensity of twisting of individual helices around the axis of the superhelix. The calculation requires information about the relative spatial arrangement of the alpha carbon of the amino acid residues of the helices that make up the superhelix. The use of a small amount of raw data makes the method easy to apply, and the validity of the results of this study is confirmed through the analysis of real protein structures.
{"title":"A Method for Calculating the Sign and Degree of Chirality of Supercoiled Protein Structures","authors":"Aleksey Lutsenko, Alla Sidorova, Denis Shpigun, Ekaterina Belova, Vsevolod Tverdislov","doi":"10.3390/sym15112051","DOIUrl":"https://doi.org/10.3390/sym15112051","url":null,"abstract":"Chirality plays an important role in studies of natural protein structures. Therefore, much attention is paid to solving the problems associated with the development of criteria and methods for assessing the chirality of biomolecules. In this paper, a new method for calculating the sign and degree of chirality of superhelices is proposed. The method makes it possible to characterize the chirality sign and to quantify coiled-coils and collagen superhelices. The degree of chirality is understood as a value indicating the intensity of twisting of individual helices around the axis of the superhelix. The calculation requires information about the relative spatial arrangement of the alpha carbon of the amino acid residues of the helices that make up the superhelix. The use of a small amount of raw data makes the method easy to apply, and the validity of the results of this study is confirmed through the analysis of real protein structures.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135036678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jelena Mišić, Aleksandar Kemiveš, Milan Ranđelović, Dragan Ranđelović
This study proposes an innovative model that determines the importance of selected factors of a univariate problem. The proposed model has been developed based on the example of determining the impact of non-medical factors on the quality of inpatient treatment, but it is generally applicable to any process of binary classification. In addition, an ensemble stacking model that involves the asymmetric use of two different well-known algorithms is proposed to determine the importance of individual factors. This model is constructed so that the standard logistic regression is first applied as mandatory. Further, the classification algorithms are implemented if the defined conditions are met. Finally, feature selection algorithms, which belong to the optimization group of algorithms, are applied as a combinatorial algorithm. The proposed model is verified through a case study conducted using real data obtained from health institutions in the region connected to the city of Nis, Republic of Serbia. The obtained results show that the proposed model can achieve better results than each of the methods included in it and surpasses several state-of-the-art ensemble algorithms in the field of machine learning. The proposed solution has been implemented in the form of a modern mobile application.
{"title":"An Asymmetric Ensemble Method for Determining the Importance of Individual Factors of a Univariate Problem","authors":"Jelena Mišić, Aleksandar Kemiveš, Milan Ranđelović, Dragan Ranđelović","doi":"10.3390/sym15112050","DOIUrl":"https://doi.org/10.3390/sym15112050","url":null,"abstract":"This study proposes an innovative model that determines the importance of selected factors of a univariate problem. The proposed model has been developed based on the example of determining the impact of non-medical factors on the quality of inpatient treatment, but it is generally applicable to any process of binary classification. In addition, an ensemble stacking model that involves the asymmetric use of two different well-known algorithms is proposed to determine the importance of individual factors. This model is constructed so that the standard logistic regression is first applied as mandatory. Further, the classification algorithms are implemented if the defined conditions are met. Finally, feature selection algorithms, which belong to the optimization group of algorithms, are applied as a combinatorial algorithm. The proposed model is verified through a case study conducted using real data obtained from health institutions in the region connected to the city of Nis, Republic of Serbia. The obtained results show that the proposed model can achieve better results than each of the methods included in it and surpasses several state-of-the-art ensemble algorithms in the field of machine learning. The proposed solution has been implemented in the form of a modern mobile application.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135087296","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Zhiqiang Wu, Yebo Yin , Jie Liu, De Zhang, Jie Chen, Wei Jiang
The path planning problem of nuclear environment robots refers to finding a collision-free path under the constraints of path length and an accumulated radiation dose. To solve this problem, the Improved Dueling Deep Double Q Network algorithm (ID3QN) based on asymmetric neural network structure was proposed. To address the issues of overestimation and low sample utilization in the traditional Deep Q Network (DQN) algorithm, we optimized the neural network structure and used the double network to estimate action values. We also improved the action selection mechanism, adopted a priority experience replay mechanism, and redesigned the reward function. To evaluate the efficiency of the proposed algorithm, we designed simple and complex radioactive grid environments for comparison. We compared the ID3QN algorithm with traditional algorithms and some deep reinforcement learning algorithms. The simulation results indicate that in the simple radioactive grid environment, the ID3QN algorithm outperforms traditional algorithms such as A*, GA, and ACO in terms of path length and accumulated radiation dosage. Compared to other deep reinforcement learning algorithms, including DQN and some improved DQN algorithms, the ID3QN algorithm reduced the path length by 15.6%, decreased the accumulated radiation dose by 23.5%, and converged approximately 2300 episodes faster. In the complex radioactive grid environment, the ID3QN algorithm also outperformed the A*, GA, ACO, and other deep reinforcement learning algorithms in terms of path length and an accumulated radiation dose. Furthermore, the ID3QN algorithm can plan an obstacle-free optimal path with a low radiation dose even in complex environments. These results demonstrate that the ID3QN algorithm is an effective approach for solving robot path planning problems in nuclear environments, thereby enhancing the safety and reliability of robots in such environments.
{"title":"A Novel Path Planning Approach for Mobile Robot in Radioactive Environment Based on Improved Deep Q Network Algorithm","authors":"Zhiqiang Wu, Yebo Yin , Jie Liu, De Zhang, Jie Chen, Wei Jiang","doi":"10.3390/sym15112048","DOIUrl":"https://doi.org/10.3390/sym15112048","url":null,"abstract":"The path planning problem of nuclear environment robots refers to finding a collision-free path under the constraints of path length and an accumulated radiation dose. To solve this problem, the Improved Dueling Deep Double Q Network algorithm (ID3QN) based on asymmetric neural network structure was proposed. To address the issues of overestimation and low sample utilization in the traditional Deep Q Network (DQN) algorithm, we optimized the neural network structure and used the double network to estimate action values. We also improved the action selection mechanism, adopted a priority experience replay mechanism, and redesigned the reward function. To evaluate the efficiency of the proposed algorithm, we designed simple and complex radioactive grid environments for comparison. We compared the ID3QN algorithm with traditional algorithms and some deep reinforcement learning algorithms. The simulation results indicate that in the simple radioactive grid environment, the ID3QN algorithm outperforms traditional algorithms such as A*, GA, and ACO in terms of path length and accumulated radiation dosage. Compared to other deep reinforcement learning algorithms, including DQN and some improved DQN algorithms, the ID3QN algorithm reduced the path length by 15.6%, decreased the accumulated radiation dose by 23.5%, and converged approximately 2300 episodes faster. In the complex radioactive grid environment, the ID3QN algorithm also outperformed the A*, GA, ACO, and other deep reinforcement learning algorithms in terms of path length and an accumulated radiation dose. Furthermore, the ID3QN algorithm can plan an obstacle-free optimal path with a low radiation dose even in complex environments. These results demonstrate that the ID3QN algorithm is an effective approach for solving robot path planning problems in nuclear environments, thereby enhancing the safety and reliability of robots in such environments.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135086744","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, the non-scaling backstepping approach is used to examine the controller design process and stability analysis of a class of leader-type stochastic nonlinear systems. By utilizing the non-scaling backstepping design method and Lyapunov method, the controller of the leader-type stochastic nonlinear system is derived. Different from the previous literature on controller design, we develop a more computationally efficient way for designing controllers because the scaling function in the coordinate transformation is not included. Meanwhile, the prescribed-time mean-square stabilization on the equilibrium and two important estimates are derived by combining the Lyapunov method with the matrix norm. Compared to the finite-time stabilization in other studies, the prescribed-time stabilization can determine the convergence time without relying on the initial value and has more real-world applicability. To illustrate the effectiveness of the controller derived in this paper, numerical examples are provided finally.
{"title":"Controller Design and Stability Analysis for a Class of Leader-Type Stochastic Nonlinear Systems","authors":"Haiying Zhang","doi":"10.3390/sym15112049","DOIUrl":"https://doi.org/10.3390/sym15112049","url":null,"abstract":"In this paper, the non-scaling backstepping approach is used to examine the controller design process and stability analysis of a class of leader-type stochastic nonlinear systems. By utilizing the non-scaling backstepping design method and Lyapunov method, the controller of the leader-type stochastic nonlinear system is derived. Different from the previous literature on controller design, we develop a more computationally efficient way for designing controllers because the scaling function in the coordinate transformation is not included. Meanwhile, the prescribed-time mean-square stabilization on the equilibrium and two important estimates are derived by combining the Lyapunov method with the matrix norm. Compared to the finite-time stabilization in other studies, the prescribed-time stabilization can determine the convergence time without relying on the initial value and has more real-world applicability. To illustrate the effectiveness of the controller derived in this paper, numerical examples are provided finally.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135086745","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Some might say that the eigenproblem is one of the examples people discovered by looking at the sky and wondering. Even though it was formulated to explain the movement of the planets, today it has become the ansatz of solving many linear and nonlinear problems. Formulation in the terms of the eigenproblem is one of the key tools to solve complex problems, especially in the area of molecular geometry. However, the basic concept is difficult without proper preparation. A review paper covering basic concepts and algorithms is very useful. This review covers the basics of the topic. Definitions are provided for defective, Hermitian, Hessenberg, modal, singular, spectral, symmetric, skew-symmetric, skew-Hermitian, triangular, and Wishart matrices. Then, concepts of characteristic polynomial, eigendecomposition, eigenpair, eigenproblem, eigenspace, eigenvalue, and eigenvector are subsequently introduced. Faddeev–LeVerrier, von Mises, Gauss–Jordan, Pohlhausen, Lanczos–Arnoldi, Rayleigh–Ritz, Jacobi–Davidson, and Gauss–Seidel fundamental algorithms are given, while others (Francis–Kublanovskaya, Gram–Schmidt, Householder, Givens, Broyden–Fletcher–Goldfarb–Shanno, Davidon–Fletcher–Powell, and Saad–Schultz) are merely discussed. The eigenproblem has thus found its use in many topics. The applications discussed include solving Bessel’s, Helmholtz’s, Laplace’s, Legendre’s, Poisson’s, and Schrödinger’s equations. The algorithm extracting the first principal component is also provided.
{"title":"Eigenproblem Basics and Algorithms","authors":"Lorentz Jäntschi","doi":"10.3390/sym15112046","DOIUrl":"https://doi.org/10.3390/sym15112046","url":null,"abstract":"Some might say that the eigenproblem is one of the examples people discovered by looking at the sky and wondering. Even though it was formulated to explain the movement of the planets, today it has become the ansatz of solving many linear and nonlinear problems. Formulation in the terms of the eigenproblem is one of the key tools to solve complex problems, especially in the area of molecular geometry. However, the basic concept is difficult without proper preparation. A review paper covering basic concepts and algorithms is very useful. This review covers the basics of the topic. Definitions are provided for defective, Hermitian, Hessenberg, modal, singular, spectral, symmetric, skew-symmetric, skew-Hermitian, triangular, and Wishart matrices. Then, concepts of characteristic polynomial, eigendecomposition, eigenpair, eigenproblem, eigenspace, eigenvalue, and eigenvector are subsequently introduced. Faddeev–LeVerrier, von Mises, Gauss–Jordan, Pohlhausen, Lanczos–Arnoldi, Rayleigh–Ritz, Jacobi–Davidson, and Gauss–Seidel fundamental algorithms are given, while others (Francis–Kublanovskaya, Gram–Schmidt, Householder, Givens, Broyden–Fletcher–Goldfarb–Shanno, Davidon–Fletcher–Powell, and Saad–Schultz) are merely discussed. The eigenproblem has thus found its use in many topics. The applications discussed include solving Bessel’s, Helmholtz’s, Laplace’s, Legendre’s, Poisson’s, and Schrödinger’s equations. The algorithm extracting the first principal component is also provided.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135186938","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The expectation that the physical expansion of space occurs smoothly may be expressed mathematically as a requirement for continuity in the time derivative of the metric scale factor of the Friedmann–Robertson–Walker cosmology. We explore the consequences of imposing such a smoothness requirement, examining the forms of possible interpolating functions between the end of inflation and subsequent radiation- or matter-dominated eras, using a straightforward geometric model of the interpolating behavior. We quantify the magnitude of the cusp found in a direct transition from the end of slow-roll inflation to the subsequent era, analyze the validity of several smooth interpolator candidates, and investigate equation-of-state and thermodynamic constraints. We find an order-of-magnitude increase in the size of the universe at the end of the transition to a single-component radiation or matter era. We also evaluate the interpolating functions in terms of the standard theory of preheating and determine the effect on the number of bosons produced.
{"title":"Exiting Inflation with a Smooth Scale Factor","authors":"Harry Oslislo, Brett Altschul","doi":"10.3390/sym15112042","DOIUrl":"https://doi.org/10.3390/sym15112042","url":null,"abstract":"The expectation that the physical expansion of space occurs smoothly may be expressed mathematically as a requirement for continuity in the time derivative of the metric scale factor of the Friedmann–Robertson–Walker cosmology. We explore the consequences of imposing such a smoothness requirement, examining the forms of possible interpolating functions between the end of inflation and subsequent radiation- or matter-dominated eras, using a straightforward geometric model of the interpolating behavior. We quantify the magnitude of the cusp found in a direct transition from the end of slow-roll inflation to the subsequent era, analyze the validity of several smooth interpolator candidates, and investigate equation-of-state and thermodynamic constraints. We find an order-of-magnitude increase in the size of the universe at the end of the transition to a single-component radiation or matter era. We also evaluate the interpolating functions in terms of the standard theory of preheating and determine the effect on the number of bosons produced.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135186927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Gudrun Albrecht, Esmeralda Mainar, Juan Manuel Peña, Beatriz Rubio
This paper proposes a new approach to define two frequency trigonometric spline curves with interesting shape preserving properties. This construction requires the normalized B-basis of the space U4(Iα)=span{1,cost,sint,cos2t,sin2t} defined on compact intervals Iα=[0,α], where α is a global shape parameter. It will be shown that the normalized B-basis can be regarded as the equivalent in the trigonometric space U4(Iα) to the Bernstein polynomial basis and shares its well-known symmetry properties. In fact, the normalized B-basis functions converge to the Bernstein polynomials as α→0. As a consequence, the convergence of the obtained piecewise trigonometric curves to uniform quartic B-Spline curves will be also shown. The proposed trigonometric spline curves can be used for CAM design, trajectory-generation, data fitting on the sphere and even to define new algebraic-trigonometric Pythagorean-Hodograph curves and their piecewise counterparts allowing the resolution of C(3 Hermite interpolation problems.
{"title":"A Shape Preserving Class of Two-Frequency Trigonometric B-Spline Curves","authors":"Gudrun Albrecht, Esmeralda Mainar, Juan Manuel Peña, Beatriz Rubio","doi":"10.3390/sym15112041","DOIUrl":"https://doi.org/10.3390/sym15112041","url":null,"abstract":"This paper proposes a new approach to define two frequency trigonometric spline curves with interesting shape preserving properties. This construction requires the normalized B-basis of the space U4(Iα)=span{1,cost,sint,cos2t,sin2t} defined on compact intervals Iα=[0,α], where α is a global shape parameter. It will be shown that the normalized B-basis can be regarded as the equivalent in the trigonometric space U4(Iα) to the Bernstein polynomial basis and shares its well-known symmetry properties. In fact, the normalized B-basis functions converge to the Bernstein polynomials as α→0. As a consequence, the convergence of the obtained piecewise trigonometric curves to uniform quartic B-Spline curves will be also shown. The proposed trigonometric spline curves can be used for CAM design, trajectory-generation, data fitting on the sphere and even to define new algebraic-trigonometric Pythagorean-Hodograph curves and their piecewise counterparts allowing the resolution of C(3 Hermite interpolation problems.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135136531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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