Two new methods, the generalized Levy method and the weighted iteration method, are presented for identification of non-integer order systems. The first method generalizes the Levy identification method from the integ...Two new methods, the generalized Levy method and the weighted iteration method, are presented for identification of non-integer order systems. The first method generalizes the Levy identification method from the integer order systems to the non-integer order systems. Then, the weighted iteration method is presented to overcome the shortcomings of the first method. Results show that the proposed methods have better performance compared with the integer order identification method. For the non-integer order systems, the proposed methods have the better fitting for the system frequency response. For the integer order system, if commensurate order scanning is applied, the proposed methods can also achieve the best integer order model which fits the system frequency response. At the same time, the proposed algorithms are more stable.展开更多
Polynomial functions containing terms with non-integer powers are studied to disclose possible approaches for obtaining their roots as well as employing them for curve-fitting purposes. Several special cases represent...Polynomial functions containing terms with non-integer powers are studied to disclose possible approaches for obtaining their roots as well as employing them for curve-fitting purposes. Several special cases representing equations from different categories are investigated for their roots. Curve-fitting applications to physically meaningful data by the use of fractional functions are worked out in detail. Relevance of this rarely worked subject to solutions of fractional differential equations is pointed out and existing potential in related future work is emphasized.展开更多
We examine variations of the harmonic series by grouping terms into “washings” that alternate sign with the number of terms in a washing growing exponentially with respect to a fixed base. The bases x = 1 and x = ∞...We examine variations of the harmonic series by grouping terms into “washings” that alternate sign with the number of terms in a washing growing exponentially with respect to a fixed base. The bases x = 1 and x = ∞ correspond to the alternating harmonic series and the usual harmonic series;we first consider other positive integral bases and further we consider positive real number bases with a unique way to make sense of adding a non-integral number of terms together. In both cases, we prove a remarkable result regarding the difference between the upper and lower convergent values of the series, and give some analysis of this behavior.展开更多
Many modern video encoders use the Lagrangian rate-distortion optimization （RDO） algorithm for mode deci- sions during the compression procedure. For each encoding stage, this approach involves minimizing a cost, wh...Many modern video encoders use the Lagrangian rate-distortion optimization （RDO） algorithm for mode deci- sions during the compression procedure. For each encoding stage, this approach involves minimizing a cost, which is a function of rate, distortion and a multiplier called Lambda. This paper proposes to improve the RDO process by applying two modifications. The first modification is to increase the ac- curacy of rate estimation, which is achieved by computing a non-integer number of bits for arithmetic coding of the syntax elements. This leads to a more accurate cost computation and therefore a better mode decision. The second modification is to search and adjust the value of Lambda based on the char- acteristics of each coding stage. For the encoder used, this paper proposes to search multiple values of Lambda for the intra-4x4 mode decision. Moreover, a simple shift in Lambda value is proposed for motion estimation. Each of these modi- fications offers a certain gain in RDO performance, and, when all are combined, an average bit-rate saving of up to 7.0% can be achieved for the H.264/AVC codec while the same concept is applicable to the H.265/HEVC codec as well. The extra added complexity is contained to a certain level, and is also adjustable according to the processing resources available.展开更多
Applications of a constitutive framework providing compound complexity analysis and indexing of coarse-grained self-similar time series representing behavioural data are presented. A notion of behavioural entropy and ...Applications of a constitutive framework providing compound complexity analysis and indexing of coarse-grained self-similar time series representing behavioural data are presented. A notion of behavioural entropy and hysteresis is introduced as two different forms of compound measures. These measures provide clinically applicable complexity analysis of behavioural patterns yielding scalar characterisation of time-varying behaviours registered over an extended period of time. The behavioural data are obtained using body attached sensors providing non-invasive readings of heart rate, skin blood perfusion, blood oxygenation, skin temperature, movement and steps frequency. The results using compound measures of behavioural patterns of fifteen healthy individuals are presented. The application of the compound measures is shown to correlate with complexity analysis. The correlation is demonstrated using two healthy subjects compared against a control group. This indicates a possibility to use these measures in place of fractional dimensions to provide a finer characterisation of behavioural patterns observed using sensory data acquired over a long period of time.展开更多
In this paper, we analyze and design a new type of servo system with noninteger voltage controlled crystal oscillator （VCXO） for rubidium atomic frequency standard （RAFS）, which does not require fractional frequen...In this paper, we analyze and design a new type of servo system with noninteger voltage controlled crystal oscillator （VCXO） for rubidium atomic frequency standard （RAFS）, which does not require fractional frequency synthesizer. By the estab- lishment of the loop equations with noises and drifts, we prove that all the components of the loop can affect its performance in- dex, and in which, RAFS long-term frequency stability is mainly determined by frequency multiplier, quantum system, and servo amplifier; the short-term one is mostly decided by VCXO. Owing to the elimination of the frequency synthesizer and its additive mixing unit, we can reduce phase noise and stray of the servo sys- tem, and it is favorable for miniaturizing the RAFS system. In addition, we adopt some targeted optimization measures to im- prove the frequency stability index. The good short-term fre- quency stability index is also validated by the test results.展开更多
Tracking load changes in a pressurized water reactor(PWR)with the help of an efficient core power control scheme in a nuclear power station is very important.The reason is that it is challenging to maintain a stable c...Tracking load changes in a pressurized water reactor(PWR)with the help of an efficient core power control scheme in a nuclear power station is very important.The reason is that it is challenging to maintain a stable core power according to the reference value within an acceptable tolerance for the safety of PWR.To overcome the uncertainties,a non-integer-based fractional order control method is demonstrated to control the core power of PWR.The available dynamic model of the reactor core is used in this analysis.Core power is controlled using a modified state feedback approach with a non-integer integral scheme through two different approximations,CRONE(Commande Robuste d’Ordre Non Entier,meaning Non-integer orderRobust Control)and FOMCON(non-integer order modeling and control).Simulation results are produced using MATLAB■program.Both non-integer results are compared with an integer order PI(Proportional Integral)algorithm to justify the effectiveness of the proposed scheme.Sate-spacemodel Core power control Non-integer control Pressurized water reactor PI controller CRONE FOMCON.展开更多
Triphenylmethane(tpm) derivatives(e.g. tpm CV) have threatened the safety of the aquatic environment due to the potential toxicity and carcinogenicity. In this study, the novel ultrasonic/persulfate/chlorite(US/S_(2)O...Triphenylmethane(tpm) derivatives(e.g. tpm CV) have threatened the safety of the aquatic environment due to the potential toxicity and carcinogenicity. In this study, the novel ultrasonic/persulfate/chlorite(US/S_(2)O_(8)^(2-)/ClO_(2)^(-)) oxidation process was developed for the effective removal of tpm CV in wastewater. The apparent non-integer kinetics( n around 1.20) of tpm CV degradation under different factors( R^(2)_(Adj) > 0.990) were investigated, respectively. Inhibiting effects of anions were greater than those of cations(except Fe(Ⅱ/Ⅲ)). The adding of micromolecule organic acids could regulate degradation towards positive direction. The double response surface methodology(RSM) was designed to optimize tpm CV removal process, and the acoustic-piezoelectric interaction was simulated to determine the propagation process of acoustic wave in the reactor. The possible degradation pathway was explored to mainly include carbonylation, carboxylation, and demethylation. The estimated effectivemean temperature at the bubble-water interface was calculated from 721 to 566 K after introducing the ClO_(2)^(-), however, the adsorption or partitioning capacity of tpm CV in the reactive zone was widened from 0.0218 to 0.0982. The proposed co-catalysis of US/S_(2)O_(8)^(2-)/ClO_(2)^(-)was based on the determined active species mainly including ClO_(2), SO_(4)·^(-), and ·OH. Compared with other US-based processes, the operating cost(3.97 $/m^(3)) of US/S_(2)O_(8)^(2-)/ClO_(2)^(-)with the EE/O value(16.8 k Wh/m^(3)) was relatively reduced.展开更多
High-order harmonics from helium atom in the orthogonally two-color(OTC) laser field are investigated by solving the two-dimensional time-dependent Schrodinger equation.Non-integer high-order harmonics are obtained in...High-order harmonics from helium atom in the orthogonally two-color(OTC) laser field are investigated by solving the two-dimensional time-dependent Schrodinger equation.Non-integer high-order harmonics are obtained in some ratio of frequencies of two components.Pure odd and even harmonics from atoms could be separated in two components by adjusting the ratio of frequencies in OTC scheme,and the resolution of harmonics is improved at the same time.The physical mechanism is explained by the periodicity of dipole.With the same intensity of the incident laser,the intensity of the high-order harmonics from the OTC field scheme is improved by three orders of magnitude compared to the monochromatic laser field scheme.A theoretical scheme is provided for experimentally achieving improving energy resolution and separation of pure odd and even harmonics in atoms.Also,we provide a means for improving harmonic intensity.展开更多
We give a method to estimate non-integer power function|u|~ku in modulation space which is an open question in the study of modulation space.As an application,we can study Cauchy problem for the nonlinear Klein-Gordon...We give a method to estimate non-integer power function|u|~ku in modulation space which is an open question in the study of modulation space.As an application,we can study Cauchy problem for the nonlinear Klein-Gordon equation with nonlinear term|u|~ku in modulation space,where k is not an integer.Moreover,we also study the global solution with small initial value for the Klein-Gordon-Hartree equation.The results show some advantages of modulation space both in high and low regularity cases.展开更多
A nonlinear impact damping model of single-degree-of-freedom spur cylindrical gear with backlash and time-varying stiffness was established. Systematic analyses of the dynamic responses were performed. First, the nonl...A nonlinear impact damping model of single-degree-of-freedom spur cylindrical gear with backlash and time-varying stiffness was established. Systematic analyses of the dynamic responses were performed. First, the nonlinear damping coefficient was considered as a constant parameter with two types of compliance exponent, meanwhile, dynamic factors were adopted to depict the dynamic characteristics. Second, the bifurcation graphs were plotted, where the damping coefficient was obtained along with the impact velocity and coefficient of restitution. The results show that light and heavy load conditions have an effect on the responses when the compliance exponent is integer. On the contrary, when the compliance exponent is non-integer, the dynamic responses are slightly affected, namely the system is more stable than the former situation.展开更多
Unique expansions in non-integer bases have been investigated in many papers during the last thirty years.They are often conveniently generated by labeled directed graphs.We give a precise description of the set of se...Unique expansions in non-integer bases have been investigated in many papers during the last thirty years.They are often conveniently generated by labeled directed graphs.We give a precise description of the set of sequences generated by these graphs.This provides a geometric explanation of many former abstract results in this domain.Our results are illustrated by many examples.展开更多
The conformable fractional derivative method has been utilized in order to examine the logistic model with constant harvesting.Such method introduces a generalization to the classical analysis of Logistic model,and he...The conformable fractional derivative method has been utilized in order to examine the logistic model with constant harvesting.Such method introduces a generalization to the classical analysis of Logistic model,and hence the features of the Logistic model,such as subcritical and supercritical harvesting,have been investigated in a view of fractional calculus.The positive auxiliary parameter,σ,with dimension of time is implemented to maintain the dimensionality of the system.The significant information of such parameter to the population has been discussed.The population expressions,obtained by conformable description,are compared with the expressions of the classical derivative.This comparison shows that the non-integer expressions are in a parallel line with that of the classical one.展开更多
文摘Two new methods, the generalized Levy method and the weighted iteration method, are presented for identification of non-integer order systems. The first method generalizes the Levy identification method from the integer order systems to the non-integer order systems. Then, the weighted iteration method is presented to overcome the shortcomings of the first method. Results show that the proposed methods have better performance compared with the integer order identification method. For the non-integer order systems, the proposed methods have the better fitting for the system frequency response. For the integer order system, if commensurate order scanning is applied, the proposed methods can also achieve the best integer order model which fits the system frequency response. At the same time, the proposed algorithms are more stable.
文摘Polynomial functions containing terms with non-integer powers are studied to disclose possible approaches for obtaining their roots as well as employing them for curve-fitting purposes. Several special cases representing equations from different categories are investigated for their roots. Curve-fitting applications to physically meaningful data by the use of fractional functions are worked out in detail. Relevance of this rarely worked subject to solutions of fractional differential equations is pointed out and existing potential in related future work is emphasized.
文摘We examine variations of the harmonic series by grouping terms into “washings” that alternate sign with the number of terms in a washing growing exponentially with respect to a fixed base. The bases x = 1 and x = ∞ correspond to the alternating harmonic series and the usual harmonic series;we first consider other positive integral bases and further we consider positive real number bases with a unique way to make sense of adding a non-integral number of terms together. In both cases, we prove a remarkable result regarding the difference between the upper and lower convergent values of the series, and give some analysis of this behavior.
文摘Many modern video encoders use the Lagrangian rate-distortion optimization （RDO） algorithm for mode deci- sions during the compression procedure. For each encoding stage, this approach involves minimizing a cost, which is a function of rate, distortion and a multiplier called Lambda. This paper proposes to improve the RDO process by applying two modifications. The first modification is to increase the ac- curacy of rate estimation, which is achieved by computing a non-integer number of bits for arithmetic coding of the syntax elements. This leads to a more accurate cost computation and therefore a better mode decision. The second modification is to search and adjust the value of Lambda based on the char- acteristics of each coding stage. For the encoder used, this paper proposes to search multiple values of Lambda for the intra-4x4 mode decision. Moreover, a simple shift in Lambda value is proposed for motion estimation. Each of these modi- fications offers a certain gain in RDO performance, and, when all are combined, an average bit-rate saving of up to 7.0% can be achieved for the H.264/AVC codec while the same concept is applicable to the H.265/HEVC codec as well. The extra added complexity is contained to a certain level, and is also adjustable according to the processing resources available.
文摘Applications of a constitutive framework providing compound complexity analysis and indexing of coarse-grained self-similar time series representing behavioural data are presented. A notion of behavioural entropy and hysteresis is introduced as two different forms of compound measures. These measures provide clinically applicable complexity analysis of behavioural patterns yielding scalar characterisation of time-varying behaviours registered over an extended period of time. The behavioural data are obtained using body attached sensors providing non-invasive readings of heart rate, skin blood perfusion, blood oxygenation, skin temperature, movement and steps frequency. The results using compound measures of behavioural patterns of fifteen healthy individuals are presented. The application of the compound measures is shown to correlate with complexity analysis. The correlation is demonstrated using two healthy subjects compared against a control group. This indicates a possibility to use these measures in place of fractional dimensions to provide a finer characterisation of behavioural patterns observed using sensory data acquired over a long period of time.
文摘In this paper, we analyze and design a new type of servo system with noninteger voltage controlled crystal oscillator （VCXO） for rubidium atomic frequency standard （RAFS）, which does not require fractional frequency synthesizer. By the estab- lishment of the loop equations with noises and drifts, we prove that all the components of the loop can affect its performance in- dex, and in which, RAFS long-term frequency stability is mainly determined by frequency multiplier, quantum system, and servo amplifier; the short-term one is mostly decided by VCXO. Owing to the elimination of the frequency synthesizer and its additive mixing unit, we can reduce phase noise and stray of the servo sys- tem, and it is favorable for miniaturizing the RAFS system. In addition, we adopt some targeted optimization measures to im- prove the frequency stability index. The good short-term fre- quency stability index is also validated by the test results.
基金This project was funded by the Deanship of Scientific Research(DSR),King Abdulaziz University,Jeddah,Saudi Arabia under grant no.(KEP-Msc-36-135-38).
文摘Tracking load changes in a pressurized water reactor(PWR)with the help of an efficient core power control scheme in a nuclear power station is very important.The reason is that it is challenging to maintain a stable core power according to the reference value within an acceptable tolerance for the safety of PWR.To overcome the uncertainties,a non-integer-based fractional order control method is demonstrated to control the core power of PWR.The available dynamic model of the reactor core is used in this analysis.Core power is controlled using a modified state feedback approach with a non-integer integral scheme through two different approximations,CRONE(Commande Robuste d’Ordre Non Entier,meaning Non-integer orderRobust Control)and FOMCON(non-integer order modeling and control).Simulation results are produced using MATLAB■program.Both non-integer results are compared with an integer order PI(Proportional Integral)algorithm to justify the effectiveness of the proposed scheme.Sate-spacemodel Core power control Non-integer control Pressurized water reactor PI controller CRONE FOMCON.
基金supported by the State Key Laboratory of Urban Water Resource and Environment, Harbin Institute of Technology (No. 2019DX08)the Natural Science Foundation of Shandong Province of China (No. ZR2019QEE012)。
文摘Triphenylmethane(tpm) derivatives(e.g. tpm CV) have threatened the safety of the aquatic environment due to the potential toxicity and carcinogenicity. In this study, the novel ultrasonic/persulfate/chlorite(US/S_(2)O_(8)^(2-)/ClO_(2)^(-)) oxidation process was developed for the effective removal of tpm CV in wastewater. The apparent non-integer kinetics( n around 1.20) of tpm CV degradation under different factors( R^(2)_(Adj) > 0.990) were investigated, respectively. Inhibiting effects of anions were greater than those of cations(except Fe(Ⅱ/Ⅲ)). The adding of micromolecule organic acids could regulate degradation towards positive direction. The double response surface methodology(RSM) was designed to optimize tpm CV removal process, and the acoustic-piezoelectric interaction was simulated to determine the propagation process of acoustic wave in the reactor. The possible degradation pathway was explored to mainly include carbonylation, carboxylation, and demethylation. The estimated effectivemean temperature at the bubble-water interface was calculated from 721 to 566 K after introducing the ClO_(2)^(-), however, the adsorption or partitioning capacity of tpm CV in the reactive zone was widened from 0.0218 to 0.0982. The proposed co-catalysis of US/S_(2)O_(8)^(2-)/ClO_(2)^(-)was based on the determined active species mainly including ClO_(2), SO_(4)·^(-), and ·OH. Compared with other US-based processes, the operating cost(3.97 $/m^(3)) of US/S_(2)O_(8)^(2-)/ClO_(2)^(-)with the EE/O value(16.8 k Wh/m^(3)) was relatively reduced.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11974229 and 11504221)the Natural Science Foundation of Shanxi Province+4 种基金China(Grant No.201901D111288)the Scientific and Technological Innovation Programs of Higher Education Institutions in ShanxiChina(Grant No.2019L0452)the Program for the Top Young Academic Leaders of Higher Learning Institutions of Shanxi ProvinceChina。
文摘High-order harmonics from helium atom in the orthogonally two-color(OTC) laser field are investigated by solving the two-dimensional time-dependent Schrodinger equation.Non-integer high-order harmonics are obtained in some ratio of frequencies of two components.Pure odd and even harmonics from atoms could be separated in two components by adjusting the ratio of frequencies in OTC scheme,and the resolution of harmonics is improved at the same time.The physical mechanism is explained by the periodicity of dipole.With the same intensity of the incident laser,the intensity of the high-order harmonics from the OTC field scheme is improved by three orders of magnitude compared to the monochromatic laser field scheme.A theoretical scheme is provided for experimentally achieving improving energy resolution and separation of pure odd and even harmonics in atoms.Also,we provide a means for improving harmonic intensity.
基金supported by National Natural Science Foundation of China (Grant Nos. 11671363 and 11471288)Natural Science Foundation of Zhejiang Province (Grant No. LQ15A010003)
文摘We give a method to estimate non-integer power function|u|~ku in modulation space which is an open question in the study of modulation space.As an application,we can study Cauchy problem for the nonlinear Klein-Gordon equation with nonlinear term|u|~ku in modulation space,where k is not an integer.Moreover,we also study the global solution with small initial value for the Klein-Gordon-Hartree equation.The results show some advantages of modulation space both in high and low regularity cases.
基金Project(51275530)supported by the National Natural Science Foundation of ChinaProject(2011CB706800)supported by the National Basic Research Program of ChinaProject(2013zzts198)supported by the Fundamental Research Founds of Central South University,China
文摘A nonlinear impact damping model of single-degree-of-freedom spur cylindrical gear with backlash and time-varying stiffness was established. Systematic analyses of the dynamic responses were performed. First, the nonlinear damping coefficient was considered as a constant parameter with two types of compliance exponent, meanwhile, dynamic factors were adopted to depict the dynamic characteristics. Second, the bifurcation graphs were plotted, where the damping coefficient was obtained along with the impact velocity and coefficient of restitution. The results show that light and heavy load conditions have an effect on the responses when the compliance exponent is integer. On the contrary, when the compliance exponent is non-integer, the dynamic responses are slightly affected, namely the system is more stable than the former situation.
基金supported by National Natural Science Foundation of China(Grant Nos.11871348 and 61972265)Natural Science Foundation of Guangdong Province of China(Grant No.2020B1515310008)+1 种基金Project of Educational Commission of Guangdong Province of China(Grant No.2019KZDZX1007)Shenzhen Key Laboratory of Advanced Machine Learning and Applications.
文摘Unique expansions in non-integer bases have been investigated in many papers during the last thirty years.They are often conveniently generated by labeled directed graphs.We give a precise description of the set of sequences generated by these graphs.This provides a geometric explanation of many former abstract results in this domain.Our results are illustrated by many examples.
文摘The conformable fractional derivative method has been utilized in order to examine the logistic model with constant harvesting.Such method introduces a generalization to the classical analysis of Logistic model,and hence the features of the Logistic model,such as subcritical and supercritical harvesting,have been investigated in a view of fractional calculus.The positive auxiliary parameter,σ,with dimension of time is implemented to maintain the dimensionality of the system.The significant information of such parameter to the population has been discussed.The population expressions,obtained by conformable description,are compared with the expressions of the classical derivative.This comparison shows that the non-integer expressions are in a parallel line with that of the classical one.
