The Birkhoff systems are the generalization of the Hamiltonian systems. Generalized canonical transformations are studied. The symplectic algorithm of the Hamiltonian systems is extended into that of the Birkhofflan s...The Birkhoff systems are the generalization of the Hamiltonian systems. Generalized canonical transformations are studied. The symplectic algorithm of the Hamiltonian systems is extended into that of the Birkhofflan systems. Symplectic differential scheme of autonomous Birkhoffian systems vas structured and discussed by introducing the Kailey Transformation.展开更多
The quantum mechanical relationships between time-dependent oscillators, Hamilton-Jacobi theory and an invariant operator are clarified by making reference to a system with a generalized oscillator. We introduce a lin...The quantum mechanical relationships between time-dependent oscillators, Hamilton-Jacobi theory and an invariant operator are clarified by making reference to a system with a generalized oscillator. We introduce a linear transformation in position and momentum, and show that the correspondence between classical and quantum transformations is exactly one-to-one. We found that classical canonical transformations are constructed from quantum unitary transformations as long as we are concerned with linear transformations. We also show the relationship between the invariant operator and a linear transformation.展开更多
The mutual relationships between four generating functions F-1(q, Q), F-2(q, P), F-3(p, P), F-4(p, Q) and four kinds of canonical variables q, p, Q, P concerned in Hamilton's canonical transformations, can be got ...The mutual relationships between four generating functions F-1(q, Q), F-2(q, P), F-3(p, P), F-4(p, Q) and four kinds of canonical variables q, p, Q, P concerned in Hamilton's canonical transformations, can be got with linear transformations from seven basic formulae. All of them are Legendre's transformation, which are implemented by 32 matrices of 8 x 8 which are homomorphic to D-4 point group of 8 elements with correspondence of 4:1. Transformations and relationships of four state functions G(P, T), H(P, S), U(V, S), F(V, T) and four variables P, V, T, S in thermodynamics, are just the same Lagendre's transformations with the relationships of canonical transformations. The state functions of thermodynamics are summarily founded on experimental results of macroscope measurements, and Hamilton's canonical transformations are theoretical generalization of classical mechanics. Both group represents are the same, and it is to say, their mathematical frames are the same. This generality indicates the thermodynamical transformation is an example of one-dimensional Hamilton's canonical transformation.展开更多
In order to obtain with simplicity the known and new properties of linear canonical transformations (LCTs), we show that any relation between a couple of operators (A,B) having commutator identical to unity, called du...In order to obtain with simplicity the known and new properties of linear canonical transformations (LCTs), we show that any relation between a couple of operators (A,B) having commutator identical to unity, called dual couple in this work, is valuable for any other dual couple, so that from the known translation operator exp(a∂<sub>x</sub>) one may obtain the explicit form and properties of a category of linear and linear canonical transformations in 2N-phase spaces. Moreover, other forms of LCTs are also obtained in this work as so as the transforms by them of functions by integrations as so as by derivations. In this way, different kinds of LCTs such as Fast Fourier, Fourier, Laplace, Xin Ma and Rhodes, Baker-Campbell-Haussdorf, Bargman transforms are found again.展开更多
The canonical transformation and Poisson theory of dynamical systems with exponential,power-law,and logarithmic non-standard Lagrangians are studied,respectively.The criterion equations of canonical transformation are...The canonical transformation and Poisson theory of dynamical systems with exponential,power-law,and logarithmic non-standard Lagrangians are studied,respectively.The criterion equations of canonical transformation are established,and four basic forms of canonical transformations are given.The dynamic equations with non-standard Lagrangians admit Lie algebraic structure.From this,we es-tablish the Poisson theory,which makes it possible to find new conservation laws through known conserved quantities.Some examples are put forward to demonstrate the use of the theory and verify its effectiveness.展开更多
New objects characterizing the structure of complex linear transformations areintroduced. These new objects yield a new result for the decomposition of complexvector spaces relative to complex lrnear transformations a...New objects characterizing the structure of complex linear transformations areintroduced. These new objects yield a new result for the decomposition of complexvector spaces relative to complex lrnear transformations and provide all Jordan basesby which the Jordan canonical form is constructed. Accordingly, they can result in thecelebrated Jordan theorem and the third decomposition theorem of space directly. and,moreover, they can give a new deep insight into the exquisite and subtle structure ofthe Jordan form. The latter indicates that the Jordan canonical form of a complexlinear transformation is an invariant structure associated with double arbitrary. choices.展开更多
An uncertainty principle(UP),which offers information about a signal and its Fourier transform in the time-frequency plane,is particularly powerful in mathematics,physics and signal processing community.Under the pola...An uncertainty principle(UP),which offers information about a signal and its Fourier transform in the time-frequency plane,is particularly powerful in mathematics,physics and signal processing community.Under the polar coordinate form of quaternion-valued signals,the UP of the two-sided quaternion linear canonical transform(QLCT)is strengthened in terms of covariance.The condition giving rise to the equal relation of the derived result is obtained as well.The novel UP with covariance can be regarded as one in a tighter form related to the QLCT.It states that the product of spreads of a quaternion-valued signal in the spatial domain and the QLCT domain is bounded by a larger lower bound.展开更多
Linear canonical transformation(LCT)is a generalization of the Fourier transform and fractional Fourier transform.The recent research has shown that the LCT is widely used in signal processing and applied mathematics,...Linear canonical transformation(LCT)is a generalization of the Fourier transform and fractional Fourier transform.The recent research has shown that the LCT is widely used in signal processing and applied mathematics,and the discretization of the LCT becomes vital for the applic-ations of LCT.Based on the development of discretization LCT,a review of important research progress and current situation is presented,which can help researchers to further understand the discretization of LCT and can promote its engineering application.Meanwhile,the connection among different discretization algorithms and the future research are given.展开更多
In order to transmit the speech information safely in the channel,a new speech encryp-tion algorithm in linear canonical transform(LCT)domain based on dynamic modulation of chaot-ic system is proposed.The algorithm fi...In order to transmit the speech information safely in the channel,a new speech encryp-tion algorithm in linear canonical transform(LCT)domain based on dynamic modulation of chaot-ic system is proposed.The algorithm first uses a chaotic system to obtain the number of sampling points of the grouped encrypted signal.Then three chaotic systems are used to modulate the corres-ponding parameters of the LCT,and each group of transform parameters corresponds to a group of encrypted signals.Thus,each group of signals is transformed by LCT with different parameters.Fi-nally,chaotic encryption is performed on the LCT domain spectrum of each group of signals,to realize the overall encryption of the speech signal.The experimental results show that the proposed algorithm is extremely sensitive to the keys and has a larger key space.Compared with the original signal,the waveform and LCT domain spectrum of obtained encrypted signal are distributed more uniformly and have less correlation,which can realize the safe transmission of speech signals.展开更多
The uncertainty principle proposed by German physicist Heisenberg in 1927 is a basic principle of quantum mechanics and signal processing.Since linear canonical transformation has been widely used in various fields of...The uncertainty principle proposed by German physicist Heisenberg in 1927 is a basic principle of quantum mechanics and signal processing.Since linear canonical transformation has been widely used in various fields of signal processing recently and Heisenberg uncertainty principle has been endowed with new expressive meaning in linear canonical transforms domain,in this manuscript,an improved Heisenberg uncertainty principle is obtained in linear canonical trans-forms domain.展开更多
In this paper, with the application of the Delauney variables, according to the Hamilton equations, the influence on the perturbation of a satellite exerted by the gravitational force of the earth through canonical tr...In this paper, with the application of the Delauney variables, according to the Hamilton equations, the influence on the perturbation of a satellite exerted by the gravitational force of the earth through canonical transformation has been found out. As a result, the equation about how the position and velocity of the satellite vary with time is deduced.展开更多
In this work, the canonical transformation method is applied to a general second order differential equation (DE) in order to trasform it into a Schr?dinger-like DE. Our proposal is based on an auxiliary function g(x)...In this work, the canonical transformation method is applied to a general second order differential equation (DE) in order to trasform it into a Schr?dinger-like DE. Our proposal is based on an auxiliary function g(x) which determines the transformation needed to find exactly-solvable potentials associated to a known DE. To show the usefulness of the proposed approach, we consider explicitly their application to the hypergeometric DE with the aim to find quantum potentials with hypergeometric wavefunctions. As a result, different potentials are obtained depending on the choice of the auxiliary function;the generalized Scarf, Posh-Teller, Eckart and Rosen-Morse trigonometric and hyperbolic potentials, are derived by selecting g(x) as constant and proportional to the P(x) hypergeometric coefficient. Similarly, the choices g(x)~P(x)/x2 and g(x)~x2/P(x) give rise to a class of exactly-solvable generalized multiparameter exponential-type potentials, which contain as particular cases the Hulthén, Manning-Rosen and Woods-Saxon models, among others. Our proposition is general and can be used with other important DE within the frame of applied matematics and physics.展开更多
We present the problem of the time-dependent Harmonic oscillator with time-dependent mass and frequency in phase space and by using a canonical transformation and delta functional integration we could find the propaga...We present the problem of the time-dependent Harmonic oscillator with time-dependent mass and frequency in phase space and by using a canonical transformation and delta functional integration we could find the propagator related to the system. New examples of time-dependent frequencies are presented.展开更多
Uncertainty principle plays an important role in multiple fields such as physics,mathem-atics,signal processing,etc.The linear canonical transform(LCT)has been used widely in optics and information processing and so o...Uncertainty principle plays an important role in multiple fields such as physics,mathem-atics,signal processing,etc.The linear canonical transform(LCT)has been used widely in optics and information processing and so on.In this paper,a few novel uncertainty inequalities on Fisher information associated with linear canonical transform are deduced.These newly deduced uncer-tainty relations not only introduce new physical interpretation in signal processing,but also build the relations between the uncertainty lower bounds and the LCT transform parameters a,b,c and d for the first time,which give us the new ideas for the analysis and potential applications.In addi-tion,these new uncertainty inequalities have sharper and tighter bounds which are the generalized versions of the traditional counterparts.Furthermore,some numeric examples are given to demon-strate the efficiency of these newly deduced uncertainty inequalities.展开更多
As generalization of the fractional Fourier transform (FRFT), the linear canonical transform (LCT) has been used in several areas, including optics and signal processing. Many properties for this transform are alr...As generalization of the fractional Fourier transform (FRFT), the linear canonical transform (LCT) has been used in several areas, including optics and signal processing. Many properties for this transform are already known, but the convolution theorems, similar to the version of the Fourier transform, are still to be determined. In this paper, the authors derive the convolution theorems for the LCT, and explore the sampling theorem and multiplicative filter for the band limited signal in the linear canonical domain. Finally, the sampling and reconstruction formulas are deduced, together with the construction methodology for the above mentioned multiplicative filter in the time domain based on fast Fourier transform (FFT), which has much lower computational load than the construction method in the linear canonical domain.展开更多
The aim of this paper is to establish an extension of quantitative uncertainty principles and an algorithm for signal recovery about the essential supports related to a Bessel type of(LCT)so called canonical Fourier-B...The aim of this paper is to establish an extension of quantitative uncertainty principles and an algorithm for signal recovery about the essential supports related to a Bessel type of(LCT)so called canonical Fourier-Bessel transform.展开更多
The main purpose of this paper is to study different types of sampling formulas of quaternionic functions,which are bandlimited under various quaternion Fourier and linear canonical transforms.We show that the quatern...The main purpose of this paper is to study different types of sampling formulas of quaternionic functions,which are bandlimited under various quaternion Fourier and linear canonical transforms.We show that the quaternionic bandlimited functions can be reconstructed from their samples as well as the samples of their derivatives and Hilbert transforms.In addition,the relationships among different types of sampling formulas under various transforms are discussed.First,if the quaternionic function is bandlimited to a rectangle that is symmetric about the origin,then the sampling formulas under various quaternion Fourier transforms are identical.If this rectangle is not symmetric about the origin,then the sampling formulas under various quaternion Fourier transforms are different from each other.Second,using the relationship between the two-sided quaternion Fourier transform and the linear canonical transform,we derive sampling formulas under various quaternion linear canonical transforms.Third,truncation errors of these sampling formulas are estimated.Finally,some simulations are provided to show how the sampling formulas can be used in applications.展开更多
In this paper we consider the boundedness of all the solutions for the equation x'+ax^+ -bx^-=f(t). where f(t) is a smooth 2π-periodic function, a and b are positive constants (a≠b).
We prove the boundedness of all solutions for the equation x" + V'(x) = DxG(x,t), where V(x) is of singular potential, i.e., limx→-1 Y(x) = ∞, and G(x, t) is bounded and periodic in t. We give sufficien...We prove the boundedness of all solutions for the equation x" + V'(x) = DxG(x,t), where V(x) is of singular potential, i.e., limx→-1 Y(x) = ∞, and G(x, t) is bounded and periodic in t. We give sufficient conditions on V(x) and G(x, t) to ensure that all solutions are bounded.展开更多
基金Foundation items:the National Natural Science Foundation of Cina(19990510)Planed item for distinguished teacher invested by Ministry of Education PRC
文摘The Birkhoff systems are the generalization of the Hamiltonian systems. Generalized canonical transformations are studied. The symplectic algorithm of the Hamiltonian systems is extended into that of the Birkhofflan systems. Symplectic differential scheme of autonomous Birkhoffian systems vas structured and discussed by introducing the Kailey Transformation.
文摘The quantum mechanical relationships between time-dependent oscillators, Hamilton-Jacobi theory and an invariant operator are clarified by making reference to a system with a generalized oscillator. We introduce a linear transformation in position and momentum, and show that the correspondence between classical and quantum transformations is exactly one-to-one. We found that classical canonical transformations are constructed from quantum unitary transformations as long as we are concerned with linear transformations. We also show the relationship between the invariant operator and a linear transformation.
文摘The mutual relationships between four generating functions F-1(q, Q), F-2(q, P), F-3(p, P), F-4(p, Q) and four kinds of canonical variables q, p, Q, P concerned in Hamilton's canonical transformations, can be got with linear transformations from seven basic formulae. All of them are Legendre's transformation, which are implemented by 32 matrices of 8 x 8 which are homomorphic to D-4 point group of 8 elements with correspondence of 4:1. Transformations and relationships of four state functions G(P, T), H(P, S), U(V, S), F(V, T) and four variables P, V, T, S in thermodynamics, are just the same Lagendre's transformations with the relationships of canonical transformations. The state functions of thermodynamics are summarily founded on experimental results of macroscope measurements, and Hamilton's canonical transformations are theoretical generalization of classical mechanics. Both group represents are the same, and it is to say, their mathematical frames are the same. This generality indicates the thermodynamical transformation is an example of one-dimensional Hamilton's canonical transformation.
文摘In order to obtain with simplicity the known and new properties of linear canonical transformations (LCTs), we show that any relation between a couple of operators (A,B) having commutator identical to unity, called dual couple in this work, is valuable for any other dual couple, so that from the known translation operator exp(a∂<sub>x</sub>) one may obtain the explicit form and properties of a category of linear and linear canonical transformations in 2N-phase spaces. Moreover, other forms of LCTs are also obtained in this work as so as the transforms by them of functions by integrations as so as by derivations. In this way, different kinds of LCTs such as Fast Fourier, Fourier, Laplace, Xin Ma and Rhodes, Baker-Campbell-Haussdorf, Bargman transforms are found again.
基金Supported by the National Natural Science Foundation of China(12272248,11972241)。
文摘The canonical transformation and Poisson theory of dynamical systems with exponential,power-law,and logarithmic non-standard Lagrangians are studied,respectively.The criterion equations of canonical transformation are established,and four basic forms of canonical transformations are given.The dynamic equations with non-standard Lagrangians admit Lie algebraic structure.From this,we es-tablish the Poisson theory,which makes it possible to find new conservation laws through known conserved quantities.Some examples are put forward to demonstrate the use of the theory and verify its effectiveness.
文摘New objects characterizing the structure of complex linear transformations areintroduced. These new objects yield a new result for the decomposition of complexvector spaces relative to complex lrnear transformations and provide all Jordan basesby which the Jordan canonical form is constructed. Accordingly, they can result in thecelebrated Jordan theorem and the third decomposition theorem of space directly. and,moreover, they can give a new deep insight into the exquisite and subtle structure ofthe Jordan form. The latter indicates that the Jordan canonical form of a complexlinear transformation is an invariant structure associated with double arbitrary. choices.
基金supported by Startup Foundation for Phd Research of Henan Normal University(No.5101119170155).
文摘An uncertainty principle(UP),which offers information about a signal and its Fourier transform in the time-frequency plane,is particularly powerful in mathematics,physics and signal processing community.Under the polar coordinate form of quaternion-valued signals,the UP of the two-sided quaternion linear canonical transform(QLCT)is strengthened in terms of covariance.The condition giving rise to the equal relation of the derived result is obtained as well.The novel UP with covariance can be regarded as one in a tighter form related to the QLCT.It states that the product of spreads of a quaternion-valued signal in the spatial domain and the QLCT domain is bounded by a larger lower bound.
基金supported by the National Natural Science Found-ation of China(No.62001193).
文摘Linear canonical transformation(LCT)is a generalization of the Fourier transform and fractional Fourier transform.The recent research has shown that the LCT is widely used in signal processing and applied mathematics,and the discretization of the LCT becomes vital for the applic-ations of LCT.Based on the development of discretization LCT,a review of important research progress and current situation is presented,which can help researchers to further understand the discretization of LCT and can promote its engineering application.Meanwhile,the connection among different discretization algorithms and the future research are given.
基金supported by the National Natural Science Found-ation of China(No.61901248)the Scientific and Tech-nological Innovation Programs of Higher Education Institu-tions in Shanxi(No.2019L0029).
文摘In order to transmit the speech information safely in the channel,a new speech encryp-tion algorithm in linear canonical transform(LCT)domain based on dynamic modulation of chaot-ic system is proposed.The algorithm first uses a chaotic system to obtain the number of sampling points of the grouped encrypted signal.Then three chaotic systems are used to modulate the corres-ponding parameters of the LCT,and each group of transform parameters corresponds to a group of encrypted signals.Thus,each group of signals is transformed by LCT with different parameters.Fi-nally,chaotic encryption is performed on the LCT domain spectrum of each group of signals,to realize the overall encryption of the speech signal.The experimental results show that the proposed algorithm is extremely sensitive to the keys and has a larger key space.Compared with the original signal,the waveform and LCT domain spectrum of obtained encrypted signal are distributed more uniformly and have less correlation,which can realize the safe transmission of speech signals.
文摘The uncertainty principle proposed by German physicist Heisenberg in 1927 is a basic principle of quantum mechanics and signal processing.Since linear canonical transformation has been widely used in various fields of signal processing recently and Heisenberg uncertainty principle has been endowed with new expressive meaning in linear canonical transforms domain,in this manuscript,an improved Heisenberg uncertainty principle is obtained in linear canonical trans-forms domain.
文摘In this paper, with the application of the Delauney variables, according to the Hamilton equations, the influence on the perturbation of a satellite exerted by the gravitational force of the earth through canonical transformation has been found out. As a result, the equation about how the position and velocity of the satellite vary with time is deduced.
基金supported by the projects UAM-A-CBI-2232004 and 009.JGR thanks to the Instituto Politécnico Nacional for the financial support given through the COFAA-IPN project SIP-200150019.
文摘In this work, the canonical transformation method is applied to a general second order differential equation (DE) in order to trasform it into a Schr?dinger-like DE. Our proposal is based on an auxiliary function g(x) which determines the transformation needed to find exactly-solvable potentials associated to a known DE. To show the usefulness of the proposed approach, we consider explicitly their application to the hypergeometric DE with the aim to find quantum potentials with hypergeometric wavefunctions. As a result, different potentials are obtained depending on the choice of the auxiliary function;the generalized Scarf, Posh-Teller, Eckart and Rosen-Morse trigonometric and hyperbolic potentials, are derived by selecting g(x) as constant and proportional to the P(x) hypergeometric coefficient. Similarly, the choices g(x)~P(x)/x2 and g(x)~x2/P(x) give rise to a class of exactly-solvable generalized multiparameter exponential-type potentials, which contain as particular cases the Hulthén, Manning-Rosen and Woods-Saxon models, among others. Our proposition is general and can be used with other important DE within the frame of applied matematics and physics.
文摘We present the problem of the time-dependent Harmonic oscillator with time-dependent mass and frequency in phase space and by using a canonical transformation and delta functional integration we could find the propagator related to the system. New examples of time-dependent frequencies are presented.
基金supported by the National Natural Science Foundation of China(Nos.61771020,61471412)Project of Zhijiang Lab(No.2019KD0AC02).
文摘Uncertainty principle plays an important role in multiple fields such as physics,mathem-atics,signal processing,etc.The linear canonical transform(LCT)has been used widely in optics and information processing and so on.In this paper,a few novel uncertainty inequalities on Fisher information associated with linear canonical transform are deduced.These newly deduced uncer-tainty relations not only introduce new physical interpretation in signal processing,but also build the relations between the uncertainty lower bounds and the LCT transform parameters a,b,c and d for the first time,which give us the new ideas for the analysis and potential applications.In addi-tion,these new uncertainty inequalities have sharper and tighter bounds which are the generalized versions of the traditional counterparts.Furthermore,some numeric examples are given to demon-strate the efficiency of these newly deduced uncertainty inequalities.
基金supported by the National Natural Science Foundation of China(Grant Nos.60232010 and 60572094)the Ministerial Foundation of China(Grant No.6140445).
文摘As generalization of the fractional Fourier transform (FRFT), the linear canonical transform (LCT) has been used in several areas, including optics and signal processing. Many properties for this transform are already known, but the convolution theorems, similar to the version of the Fourier transform, are still to be determined. In this paper, the authors derive the convolution theorems for the LCT, and explore the sampling theorem and multiplicative filter for the band limited signal in the linear canonical domain. Finally, the sampling and reconstruction formulas are deduced, together with the construction methodology for the above mentioned multiplicative filter in the time domain based on fast Fourier transform (FFT), which has much lower computational load than the construction method in the linear canonical domain.
文摘The aim of this paper is to establish an extension of quantitative uncertainty principles and an algorithm for signal recovery about the essential supports related to a Bessel type of(LCT)so called canonical Fourier-Bessel transform.
基金the Research Development Foundation of Wenzhou Medical UniversityChina(No.QTJ18012)+6 种基金the Wenzhou Science and Technology Bureau of China(No.G2020031)the Guangdong Basic and Applied Basic Research Foundation of China(No.2019A1515111185)the Science and Technology Development FundMacao Special Administrative RegionChina(No.FDCT/085/2018/A2)the University of MacaoChina(No.MYRG2019-00039-FST)。
文摘The main purpose of this paper is to study different types of sampling formulas of quaternionic functions,which are bandlimited under various quaternion Fourier and linear canonical transforms.We show that the quaternionic bandlimited functions can be reconstructed from their samples as well as the samples of their derivatives and Hilbert transforms.In addition,the relationships among different types of sampling formulas under various transforms are discussed.First,if the quaternionic function is bandlimited to a rectangle that is symmetric about the origin,then the sampling formulas under various quaternion Fourier transforms are identical.If this rectangle is not symmetric about the origin,then the sampling formulas under various quaternion Fourier transforms are different from each other.Second,using the relationship between the two-sided quaternion Fourier transform and the linear canonical transform,we derive sampling formulas under various quaternion linear canonical transforms.Third,truncation errors of these sampling formulas are estimated.Finally,some simulations are provided to show how the sampling formulas can be used in applications.
文摘In this paper we consider the boundedness of all the solutions for the equation x'+ax^+ -bx^-=f(t). where f(t) is a smooth 2π-periodic function, a and b are positive constants (a≠b).
文摘We prove the boundedness of all solutions for the equation x" + V'(x) = DxG(x,t), where V(x) is of singular potential, i.e., limx→-1 Y(x) = ∞, and G(x, t) is bounded and periodic in t. We give sufficient conditions on V(x) and G(x, t) to ensure that all solutions are bounded.