Implementing machine learning algorithms in the non-conducive environment of the vehicular network requires some adaptations due to the high computational complexity of these algorithms.K-clustering algorithms are sim...Implementing machine learning algorithms in the non-conducive environment of the vehicular network requires some adaptations due to the high computational complexity of these algorithms.K-clustering algorithms are simplistic,with fast performance and relative accuracy.However,their implementation depends on the initial selection of clusters number(K),the initial clusters’centers,and the clustering metric.This paper investigated using Scott’s histogram formula to estimate the K number and the Link Expiration Time(LET)as a clustering metric.Realistic traffic flows were considered for three maps,namely Highway,Traffic Light junction,and Roundabout junction,to study the effect of road layout on estimating the K number.A fast version of the PAM algorithm was used for clustering with a modification to reduce time complexity.The Affinity propagation algorithm sets the baseline for the estimated K number,and the Medoid Silhouette method is used to quantify the clustering.OMNET++,Veins,and SUMO were used to simulate the traffic,while the related algorithms were implemented in Python.The Scott’s formula estimation of the K number only matched the baseline when the road layout was simple.Moreover,the clustering algorithm required one iteration on average to converge when used with LET.展开更多
We compare Newton’s force law of universal gravitation with a corrected simple approach based on Bhandari’s recently presented work, where the gravitation constant G is maintained. A reciprocity relation exists betw...We compare Newton’s force law of universal gravitation with a corrected simple approach based on Bhandari’s recently presented work, where the gravitation constant G is maintained. A reciprocity relation exists between both alternative gravity formulas with respect to the distances between mass centers. We conclude a one-to-one mapping of the two gravitational formulas. We don’t need Einstein’s construct of spacetime bending by matter.展开更多
While the Bushen Yizhi Formula can treat Alzheimer’s disease(AD),the yet to be ascertained specific mechanism of action was explored in this work.Methods:Different concentrations of the Bushen Yizhi Formula and amylo...While the Bushen Yizhi Formula can treat Alzheimer’s disease(AD),the yet to be ascertained specific mechanism of action was explored in this work.Methods:Different concentrations of the Bushen Yizhi Formula and amyloid-beta peptide(Aβ)were used to treat rat pheochromocytoma cells(P12)and human neuroblastoma cells(SH-SY5Y).Cell morphological changes were observed to determine the in vitro cell damage.Cell Counting Kit(CCK)-8 assay and flow cytometry were employed to identify cell viability and apoptosis/cell cycle,respectively.Western blotting and immunohistochemistry were employed to measure the expressions of endoplasmic reticulum stress(ERS)-related proteins(GRP78 and CHOP),p-IRE1α,IRE1α,ASK1,p-JNK,JNK,Bax,Bcl-2,XBP-1,and Bim.Fura 2-acetoxymethyl ester(Fura-2/AM)was used to determine the intracellular calcium(Ca^(2+))concentration.Also,an AD model was constructed by injecting Aβinto the CA1 area of the hippocampus in Sprague Dawley rats.AD model rats were gavaged with different concentrations of Bushen Yizhi Formula for 14 consecutive days.The Morris water maze experiment was conducted to test the learning and memory of rats.Hematoxylin&Eosin(H&E)and Terminal-deoxynucleotidyl Transferase(TdT)-mediated dUTP Nick-End Labeling(TUNEL)staining were done to determine histopathological changes in the brain.Results:Bushen Yizhi Formula relieved the Aβ-induced effects including cell injury,decreased viability,increased apoptosis,G0/G1 phase cell cycle arrest,upregulation of GRP78,CHOP,p-IRE1α,p-JNK,Bax,XBP-1 and Bim,as well as down-regulation of Bcl-2.These results were also seen with IRE1αsilencing.While Aβsuppressed the learning and memory abilities of rats,the Bushen Yizhi Formula alleviated these effects of Aβ.Brain nerve cell injury induced by Aβcould also be treated with Bushen Yizhi Formula.Conclusion:Bushen Yizhi Formula could influence ERS through the IRE1αsignaling pathway to achieve its therapeutic effects on AD.展开更多
A new type of polarization sensitive interferometer is proposed,named the Delta interferometer,inspired by its geometry resembling the Greek letter Delta.The main difference between the Delta interferometer and other ...A new type of polarization sensitive interferometer is proposed,named the Delta interferometer,inspired by its geometry resembling the Greek letter Delta.The main difference between the Delta interferometer and other existing interferometers,such as Michelson,Mach-Zehnder and Young's double-slit interferometers,is that the two interfering paths are asymmetrical in the Delta interferometer.The visibility of the first-order interference pattern observed in the Delta interferometer is dependent on the polarization of the incidental light.Optical coherence theory is employed to interpret this phenomenon and single-mode continuous-wave laser light is employed to verify the theoretical predictions.The theoretical and experimental results are consistent.The Delta interferometer is a perfect tool to study the reflection of electromagnetic fields in different polarizations and may find applications in polarization-sensitive scenarios.展开更多
By using the two-mode Fresnel operator we derive a multiplication rule of two-dimensional (2D) Collins diffraction formula, the inverse of 2D Collins diffraction integration can also be conveniently derived in this wa...By using the two-mode Fresnel operator we derive a multiplication rule of two-dimensional (2D) Collins diffraction formula, the inverse of 2D Collins diffraction integration can also be conveniently derived in this way in the context of quantum optics theory.展开更多
Objective:Parkinson’s disease(PD)is the second largest neurodegenerative disease following Alzheimer’s disease(AD),which associated with aging.There are many similarities in pathology and pathogenesis,even in the TC...Objective:Parkinson’s disease(PD)is the second largest neurodegenerative disease following Alzheimer’s disease(AD),which associated with aging.There are many similarities in pathology and pathogenesis,even in the TCM theory understanding,so we can learn from each other in the process of drug discovery.The clinical results showed that Bushen-Yizhi formula(BSYZ)could effectively improve the neurological function score of senile dementia patients and had a better anti-dementia effect.Further pharmacological studies showed that BSYZ had neuroprotective effects,such as anti-inflammatory,anti-oxidation,anti-apoptosis and neurotrophic effects.In this study,the therapeutic effect of BSYZ on PD was evaluated in vivo and in vivo,and its molecular mechanism was discussed in order to expand the scope of application of BSYZ and to provide strategies for drug discovery of related neurodegenerative diseases.Methods:C57 BL/6 mice were injected intraperitoneally with MPTP to construct a PD mouse model.BSYZ(1.46,2.92,5.84 mg·kg-1)was administered for two weeks,and the positive control group was given a NSAID,piroxicam(12.5 mg·kg-1).After 1 week of pretreatment,MPTP was used to construct a PD mouse model.The mice were subjected to Rotation test on days 1,3 and 5,6th day.and the movement coordination and exercise ability of the drug on PD mice were observed on theThe number of TH-positive cells,Iba1 and CD68-labeled microglial cells in SNpc region were observed by immunofluorescence to observe the proliferation and activation of microglial cells and GFAP-labeled astrocytes.Western blotting was used to detect the nuclear transfer of NLRP3,Caspase-1,ASC,pro-IL-1β,IL-1βand NF-κB in the midbrain.Results:1.BSYZ could significantly improve the expression of MPTP model mice in the experiment of fatigue and Y-maze,increase the number of neurons in SNpc region and the positive expression of TH protein.2.BSYZ significantly inhibited the number of Iba1/CD68-positive microglial cells in MPTP-model mice and decreased the number of GFAP-positive astrocytes.3.BSYZ significantly inhibited the expression of NLRP3-associated protein in BV2 microglial cells induced by LPS+ATP and inhibited the nuclear transfer of NF-κB.Conclusion:BSYZ can effectively relieve the motor dysfunction of PD model mice,improve the damage of dopaminergic neurons,inhibit the proliferation and activation of microglial cells and astrocytes,and have good anti-MPTPinduced neuroinflammation and neuroinflammation mediated by nuclear transfer of NF-κB.The results show that BSYZ has a good prospect of anti-Parkinson’s disease and provides valuable drug discovery strategies for the related neurodegenerative diseases.展开更多
In this paper, Leibniz’ formula of generalized divided difference with respect to a class of differential operators whose basic sets of solutions have power form, is considered. The recurrence formula of Green functi...In this paper, Leibniz’ formula of generalized divided difference with respect to a class of differential operators whose basic sets of solutions have power form, is considered. The recurrence formula of Green function about the operators is also given.展开更多
Generalizations of the geometric construction that repeatedly attaches rectangles to a square, originally given by Myerson, are presented. The initial square is replaced with a rectangle, and also the dimensionality o...Generalizations of the geometric construction that repeatedly attaches rectangles to a square, originally given by Myerson, are presented. The initial square is replaced with a rectangle, and also the dimensionality of the construction is increased. By selecting values for the various parameters, such as the lengths of the sides of the original rectangle or rectangular box in dimensions more than two and their relationships to the size of the attached rectangles or rectangular boxes, some interesting formulas are found. Examples are Wallis-type infinite-product formulas for the areas of p-circles with p > 1.展开更多
By introducing the s-parameterized generalized Wigner operator into phase-space quantum mechanics we invent the technique of integration within s-ordered product of operators(which considers normally ordered,antinorma...By introducing the s-parameterized generalized Wigner operator into phase-space quantum mechanics we invent the technique of integration within s-ordered product of operators(which considers normally ordered,antinormally ordered and Weyl ordered product of operators as its special cases).The s-ordered operator expansion(denoted by...) formula of density operators is derived,which is ρ = 2 1 s ∫ d2βπβ|ρ |β exp { 2 s 1(s|β|2 β a + βa a a) }.The s-parameterized quantization scheme is thus completely established.展开更多
Based on the displacement-squeezing related squeezed coherent state representation |z>g and using the technique of integration within an ordered product of operators,this paper finds a generalized Fresnel operator,...Based on the displacement-squeezing related squeezed coherent state representation |z>g and using the technique of integration within an ordered product of operators,this paper finds a generalized Fresnel operator,whose matrix element in the coordinate representation leads to a generalized Collins formula (Huygens-Fresnel integration transformation describing optical diffraction).The generalized Fresnel operator is derived by a quantum mechanical mapping from z to sz-rz* in the |z>g representation,while |z>g in phase space is graphically denoted by an ellipse.展开更多
WT5”HZ] A simple method for solving Cauchy’s problem of wave equations in higher space dimensions with initial condition of separated variables, has been given by using D’Alembert’s formula and some examples have ...WT5”HZ] A simple method for solving Cauchy’s problem of wave equations in higher space dimensions with initial condition of separated variables, has been given by using D’Alembert’s formula and some examples have been shown.展开更多
As is well known,the definitions of fractional sum and fractional difference of f(z)on non-uniform lattices x(z)=c1z^(2)+c2z+c3 or x(z)=c1q^(z)+c2q^(-z)+c3 are more difficult and complicated.In this article,for the fi...As is well known,the definitions of fractional sum and fractional difference of f(z)on non-uniform lattices x(z)=c1z^(2)+c2z+c3 or x(z)=c1q^(z)+c2q^(-z)+c3 are more difficult and complicated.In this article,for the first time we propose the definitions of the fractional sum and fractional difference on non-uniform lattices by two different ways.The analogue of Euler’s Beta formula,Cauchy’Beta formula on non-uniform lattices are established,and some fundamental theorems of fractional calculas,the solution of the generalized Abel equation on non-uniform lattices are obtained etc.展开更多
A generalized Liouville’s formula is established for linear matrix differential equations involving left and right multiplications.Its special cases are used to determine the localness of characteristics of symmetrie...A generalized Liouville’s formula is established for linear matrix differential equations involving left and right multiplications.Its special cases are used to determine the localness of characteristics of symmetries and solutions to Riemann-Hilbert problems in soltion theory.展开更多
In 1673, Yoshimasu Murase made a cubic equation to obtain the thickness of a hearth. He introduced two kinds of recurrence formulas of square and the deformation (Ref.[1]). We find that the three formulas lead to the ...In 1673, Yoshimasu Murase made a cubic equation to obtain the thickness of a hearth. He introduced two kinds of recurrence formulas of square and the deformation (Ref.[1]). We find that the three formulas lead to the extension of Newton-Raphson’s method and Horner’s method at the same time. This shows originality of Japanese native mathematics (Wasan) in the Edo era (1600- 1867). Suzuki (Ref.[2]) estimates Murase to be a rare mathematician in not only the history of Wasan but also the history of mathematics in the world. Section 1 introduces Murase’s three solutions of the cubic equation of the hearth. Section 2 explains the Horner’s method. We give the generalization of three formulas and the relation between these formulas and Horner’s method. Section 3 gives definitions of Murase-Newton’s method (Tsuchikura-Horiguchi’s method), general recurrence formula of Murase-Newton’s method (Tsuchikura-Horiguchi’s method), and general recurrence formula of the extension of Murase-Newton’s method (the extension of Tsuchikura-Horiguchi’s method) concerning n-degree polynomial equation. Section 4 is contents of the title of this paper.展开更多
This paper presents a new nonstationary iterative method for solving non linear algebraic equations that does not require the use of any derivative. The study uses only the Newton’s divided differences of first and s...This paper presents a new nonstationary iterative method for solving non linear algebraic equations that does not require the use of any derivative. The study uses only the Newton’s divided differences of first and second orders instead of the derivatives of (1).展开更多
The purpose of this paper is that we give an extension of Halley’s method (Section 2), and the formulas to compare the convergences of the Halley’s method and extended one (Section 3). For extension of Halley’s met...The purpose of this paper is that we give an extension of Halley’s method (Section 2), and the formulas to compare the convergences of the Halley’s method and extended one (Section 3). For extension of Halley’s method we give definition of function by variable transformation in Section 1. In Section 4 we do the numerical calculations of Halley’s method and extended one for elementary functions, compare these convergences, and confirm the theory. Under certain conditions we can confirm that the extended Halley’s method has better convergence or better approximation than Halley’s method.展开更多
Contrary to the opinion about approximation nature of a simple-iteration method, the exact solution of a system of linear algebraic equations (SLAE) in a finite number of iterations with a stationary matrix is demonst...Contrary to the opinion about approximation nature of a simple-iteration method, the exact solution of a system of linear algebraic equations (SLAE) in a finite number of iterations with a stationary matrix is demonstrated. We present a theorem and its proof that confirms the possibility to obtain the finite process and imposes the requirement for the matrix of SLAE. This matrix must be unipotent, i.e. all its eigenvalues to be equal to 1. An example of transformation of SLAE given analytically to the form with a unipotent matrix is presented. It is shown that splitting the unipotent matrix into identity and nilpotent ones results in Cramer’s analytical formulas in a finite number of iterations.展开更多
文摘Implementing machine learning algorithms in the non-conducive environment of the vehicular network requires some adaptations due to the high computational complexity of these algorithms.K-clustering algorithms are simplistic,with fast performance and relative accuracy.However,their implementation depends on the initial selection of clusters number(K),the initial clusters’centers,and the clustering metric.This paper investigated using Scott’s histogram formula to estimate the K number and the Link Expiration Time(LET)as a clustering metric.Realistic traffic flows were considered for three maps,namely Highway,Traffic Light junction,and Roundabout junction,to study the effect of road layout on estimating the K number.A fast version of the PAM algorithm was used for clustering with a modification to reduce time complexity.The Affinity propagation algorithm sets the baseline for the estimated K number,and the Medoid Silhouette method is used to quantify the clustering.OMNET++,Veins,and SUMO were used to simulate the traffic,while the related algorithms were implemented in Python.The Scott’s formula estimation of the K number only matched the baseline when the road layout was simple.Moreover,the clustering algorithm required one iteration on average to converge when used with LET.
文摘We compare Newton’s force law of universal gravitation with a corrected simple approach based on Bhandari’s recently presented work, where the gravitation constant G is maintained. A reciprocity relation exists between both alternative gravity formulas with respect to the distances between mass centers. We conclude a one-to-one mapping of the two gravitational formulas. We don’t need Einstein’s construct of spacetime bending by matter.
基金supported by the National Natural Science Foundation of China[81904266,82004309].
文摘While the Bushen Yizhi Formula can treat Alzheimer’s disease(AD),the yet to be ascertained specific mechanism of action was explored in this work.Methods:Different concentrations of the Bushen Yizhi Formula and amyloid-beta peptide(Aβ)were used to treat rat pheochromocytoma cells(P12)and human neuroblastoma cells(SH-SY5Y).Cell morphological changes were observed to determine the in vitro cell damage.Cell Counting Kit(CCK)-8 assay and flow cytometry were employed to identify cell viability and apoptosis/cell cycle,respectively.Western blotting and immunohistochemistry were employed to measure the expressions of endoplasmic reticulum stress(ERS)-related proteins(GRP78 and CHOP),p-IRE1α,IRE1α,ASK1,p-JNK,JNK,Bax,Bcl-2,XBP-1,and Bim.Fura 2-acetoxymethyl ester(Fura-2/AM)was used to determine the intracellular calcium(Ca^(2+))concentration.Also,an AD model was constructed by injecting Aβinto the CA1 area of the hippocampus in Sprague Dawley rats.AD model rats were gavaged with different concentrations of Bushen Yizhi Formula for 14 consecutive days.The Morris water maze experiment was conducted to test the learning and memory of rats.Hematoxylin&Eosin(H&E)and Terminal-deoxynucleotidyl Transferase(TdT)-mediated dUTP Nick-End Labeling(TUNEL)staining were done to determine histopathological changes in the brain.Results:Bushen Yizhi Formula relieved the Aβ-induced effects including cell injury,decreased viability,increased apoptosis,G0/G1 phase cell cycle arrest,upregulation of GRP78,CHOP,p-IRE1α,p-JNK,Bax,XBP-1 and Bim,as well as down-regulation of Bcl-2.These results were also seen with IRE1αsilencing.While Aβsuppressed the learning and memory abilities of rats,the Bushen Yizhi Formula alleviated these effects of Aβ.Brain nerve cell injury induced by Aβcould also be treated with Bushen Yizhi Formula.Conclusion:Bushen Yizhi Formula could influence ERS through the IRE1αsignaling pathway to achieve its therapeutic effects on AD.
基金Project supported by the Shanxi Key Research and Development Project(Grant No.2019ZDLGY09-08)Shanxi Nature and Science Basic Research Project(Grant No.2019JLP-18).
文摘A new type of polarization sensitive interferometer is proposed,named the Delta interferometer,inspired by its geometry resembling the Greek letter Delta.The main difference between the Delta interferometer and other existing interferometers,such as Michelson,Mach-Zehnder and Young's double-slit interferometers,is that the two interfering paths are asymmetrical in the Delta interferometer.The visibility of the first-order interference pattern observed in the Delta interferometer is dependent on the polarization of the incidental light.Optical coherence theory is employed to interpret this phenomenon and single-mode continuous-wave laser light is employed to verify the theoretical predictions.The theoretical and experimental results are consistent.The Delta interferometer is a perfect tool to study the reflection of electromagnetic fields in different polarizations and may find applications in polarization-sensitive scenarios.
基金supported by the Doctoral Scientific Research Startup Fund of Anhui University, China (Grant No. 33190059)the National Natural Science Foundation of China (Grant No. 10874174)+1 种基金the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20113401120004)the Open Funds from National Laboratory for Infrared Physics, Chinese Academy of Sciences (Grant No. 201117)
文摘By using the two-mode Fresnel operator we derive a multiplication rule of two-dimensional (2D) Collins diffraction formula, the inverse of 2D Collins diffraction integration can also be conveniently derived in this way in the context of quantum optics theory.
文摘Objective:Parkinson’s disease(PD)is the second largest neurodegenerative disease following Alzheimer’s disease(AD),which associated with aging.There are many similarities in pathology and pathogenesis,even in the TCM theory understanding,so we can learn from each other in the process of drug discovery.The clinical results showed that Bushen-Yizhi formula(BSYZ)could effectively improve the neurological function score of senile dementia patients and had a better anti-dementia effect.Further pharmacological studies showed that BSYZ had neuroprotective effects,such as anti-inflammatory,anti-oxidation,anti-apoptosis and neurotrophic effects.In this study,the therapeutic effect of BSYZ on PD was evaluated in vivo and in vivo,and its molecular mechanism was discussed in order to expand the scope of application of BSYZ and to provide strategies for drug discovery of related neurodegenerative diseases.Methods:C57 BL/6 mice were injected intraperitoneally with MPTP to construct a PD mouse model.BSYZ(1.46,2.92,5.84 mg·kg-1)was administered for two weeks,and the positive control group was given a NSAID,piroxicam(12.5 mg·kg-1).After 1 week of pretreatment,MPTP was used to construct a PD mouse model.The mice were subjected to Rotation test on days 1,3 and 5,6th day.and the movement coordination and exercise ability of the drug on PD mice were observed on theThe number of TH-positive cells,Iba1 and CD68-labeled microglial cells in SNpc region were observed by immunofluorescence to observe the proliferation and activation of microglial cells and GFAP-labeled astrocytes.Western blotting was used to detect the nuclear transfer of NLRP3,Caspase-1,ASC,pro-IL-1β,IL-1βand NF-κB in the midbrain.Results:1.BSYZ could significantly improve the expression of MPTP model mice in the experiment of fatigue and Y-maze,increase the number of neurons in SNpc region and the positive expression of TH protein.2.BSYZ significantly inhibited the number of Iba1/CD68-positive microglial cells in MPTP-model mice and decreased the number of GFAP-positive astrocytes.3.BSYZ significantly inhibited the expression of NLRP3-associated protein in BV2 microglial cells induced by LPS+ATP and inhibited the nuclear transfer of NF-κB.Conclusion:BSYZ can effectively relieve the motor dysfunction of PD model mice,improve the damage of dopaminergic neurons,inhibit the proliferation and activation of microglial cells and astrocytes,and have good anti-MPTPinduced neuroinflammation and neuroinflammation mediated by nuclear transfer of NF-κB.The results show that BSYZ has a good prospect of anti-Parkinson’s disease and provides valuable drug discovery strategies for the related neurodegenerative diseases.
文摘In this paper, Leibniz’ formula of generalized divided difference with respect to a class of differential operators whose basic sets of solutions have power form, is considered. The recurrence formula of Green function about the operators is also given.
文摘Generalizations of the geometric construction that repeatedly attaches rectangles to a square, originally given by Myerson, are presented. The initial square is replaced with a rectangle, and also the dimensionality of the construction is increased. By selecting values for the various parameters, such as the lengths of the sides of the original rectangle or rectangular box in dimensions more than two and their relationships to the size of the attached rectangles or rectangular boxes, some interesting formulas are found. Examples are Wallis-type infinite-product formulas for the areas of p-circles with p > 1.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 10775097 and 10874174)
文摘By introducing the s-parameterized generalized Wigner operator into phase-space quantum mechanics we invent the technique of integration within s-ordered product of operators(which considers normally ordered,antinormally ordered and Weyl ordered product of operators as its special cases).The s-ordered operator expansion(denoted by...) formula of density operators is derived,which is ρ = 2 1 s ∫ d2βπβ|ρ |β exp { 2 s 1(s|β|2 β a + βa a a) }.The s-parameterized quantization scheme is thus completely established.
基金Project supported by the National Natural Science Foundation of China (Grant Nos.10874174 and 10675108)the President Foundation of the Chinese Academy of Sciencesthe Specilized Research Fund for the Doctorial Program of the Higher Education of China (Grant No.20070358009)
文摘Based on the displacement-squeezing related squeezed coherent state representation |z>g and using the technique of integration within an ordered product of operators,this paper finds a generalized Fresnel operator,whose matrix element in the coordinate representation leads to a generalized Collins formula (Huygens-Fresnel integration transformation describing optical diffraction).The generalized Fresnel operator is derived by a quantum mechanical mapping from z to sz-rz* in the |z>g representation,while |z>g in phase space is graphically denoted by an ellipse.
基金Supported by the Natural Science Foundation of Hubei Province!(992P0 30 7) the National Natural Science Foun-dation of Chi
文摘WT5”HZ] A simple method for solving Cauchy’s problem of wave equations in higher space dimensions with initial condition of separated variables, has been given by using D’Alembert’s formula and some examples have been shown.
基金Supported by the National Natural Science Foundation Fujian province of China(2016J01032).
文摘As is well known,the definitions of fractional sum and fractional difference of f(z)on non-uniform lattices x(z)=c1z^(2)+c2z+c3 or x(z)=c1q^(z)+c2q^(-z)+c3 are more difficult and complicated.In this article,for the first time we propose the definitions of the fractional sum and fractional difference on non-uniform lattices by two different ways.The analogue of Euler’s Beta formula,Cauchy’Beta formula on non-uniform lattices are established,and some fundamental theorems of fractional calculas,the solution of the generalized Abel equation on non-uniform lattices are obtained etc.
基金Supported by the National Natural Science Foundation of China(11975145,11972291)The National Science Foundation(DMS-1664561)。
文摘A generalized Liouville’s formula is established for linear matrix differential equations involving left and right multiplications.Its special cases are used to determine the localness of characteristics of symmetries and solutions to Riemann-Hilbert problems in soltion theory.
文摘In 1673, Yoshimasu Murase made a cubic equation to obtain the thickness of a hearth. He introduced two kinds of recurrence formulas of square and the deformation (Ref.[1]). We find that the three formulas lead to the extension of Newton-Raphson’s method and Horner’s method at the same time. This shows originality of Japanese native mathematics (Wasan) in the Edo era (1600- 1867). Suzuki (Ref.[2]) estimates Murase to be a rare mathematician in not only the history of Wasan but also the history of mathematics in the world. Section 1 introduces Murase’s three solutions of the cubic equation of the hearth. Section 2 explains the Horner’s method. We give the generalization of three formulas and the relation between these formulas and Horner’s method. Section 3 gives definitions of Murase-Newton’s method (Tsuchikura-Horiguchi’s method), general recurrence formula of Murase-Newton’s method (Tsuchikura-Horiguchi’s method), and general recurrence formula of the extension of Murase-Newton’s method (the extension of Tsuchikura-Horiguchi’s method) concerning n-degree polynomial equation. Section 4 is contents of the title of this paper.
文摘This paper presents a new nonstationary iterative method for solving non linear algebraic equations that does not require the use of any derivative. The study uses only the Newton’s divided differences of first and second orders instead of the derivatives of (1).
文摘The purpose of this paper is that we give an extension of Halley’s method (Section 2), and the formulas to compare the convergences of the Halley’s method and extended one (Section 3). For extension of Halley’s method we give definition of function by variable transformation in Section 1. In Section 4 we do the numerical calculations of Halley’s method and extended one for elementary functions, compare these convergences, and confirm the theory. Under certain conditions we can confirm that the extended Halley’s method has better convergence or better approximation than Halley’s method.
文摘Contrary to the opinion about approximation nature of a simple-iteration method, the exact solution of a system of linear algebraic equations (SLAE) in a finite number of iterations with a stationary matrix is demonstrated. We present a theorem and its proof that confirms the possibility to obtain the finite process and imposes the requirement for the matrix of SLAE. This matrix must be unipotent, i.e. all its eigenvalues to be equal to 1. An example of transformation of SLAE given analytically to the form with a unipotent matrix is presented. It is shown that splitting the unipotent matrix into identity and nilpotent ones results in Cramer’s analytical formulas in a finite number of iterations.