In this paper we prove the following Hajek Renyi inequality:Let 0<p≤1 ,then for any Banach space B , any L p integrable B valued random variable sequence {D n,n≥1} ,any real number sequence {b...In this paper we prove the following Hajek Renyi inequality:Let 0<p≤1 ,then for any Banach space B , any L p integrable B valued random variable sequence {D n,n≥1} ,any real number sequence {b n,n≥1} with 0<b n↑∞ ,any integer n≥1 ,there exits a constant C=C p>0 (only depending on p ) such thatP( sup j≥nji=1D ib j≥ε)≤Cε -p (∞j=n+1E‖D j‖ pb p j+nj=1E‖D j‖ pb p n) In the other direction,we prove some strong laws of large numbers and the integrability of the maximal functions for B valued random variable sequences by using this inequality and the Hajeck Renyi inequality we have obtained recently.Some known results are extended and improved.展开更多
A stochastic averaging method is proposed for quasi-Hamiltonian systems (Hamiltonian systems with light dampings subject to weakly stochastic excitations). Various versions of the method, depending on whether the asso...A stochastic averaging method is proposed for quasi-Hamiltonian systems (Hamiltonian systems with light dampings subject to weakly stochastic excitations). Various versions of the method, depending on whether the associated Hamiltonian systems are integrable or nonintegrable, resonant or nonresonant, are discussed. It is pointed out that the standard stochastic averaging method and the stochastic averaging method of energy envelope are special cases of the stochastic averaging method of quasi-Hamiltonian systems and that the results obtained by this method for several examples prove its effectiveness.展开更多
The Liouville's integrability of the second order autonomous system is studied. It isproved that a second order polynomial system is Liouville integrable if and only if thereis an integral factor μ(x, y), such th...The Liouville's integrability of the second order autonomous system is studied. It isproved that a second order polynomial system is Liouville integrable if and only if thereis an integral factor μ(x, y), such that or a rational function in x and y.展开更多
The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability.The well-known modified Kd V equation is a prototypical...The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability.The well-known modified Kd V equation is a prototypical example of an integrable evolution equation in one spatial dimension.Do there exist integrable analogs of the modified Kd V equation in higher spatial dimensions?In what follows,we present a positive answer to this question.In particular,rewriting the(1+1)-dimensional integrable modified Kd V equation in conservation forms and adding deformation mappings during the process allows one to construct higher-dimensional integrable equations.Further,we illustrate this idea with examples from the modified Kd V hierarchy and also present the Lax pairs of these higher-dimensional integrable evolution equations.展开更多
We present an introduction to the Darboux integrability theory of planar complex and real polynomial differential systems containing some improvements to the classical theory.
Let x （xn）≥1 be a martingale on a noncommutative probability space n （M, r） and （wn）n≥1 a sequence of positive numbers such that Wn = ∑ k=1^n wk →∞ as n →∞ We prove that x = （x.）n≥1 converges in E（M...Let x （xn）≥1 be a martingale on a noncommutative probability space n （M, r） and （wn）n≥1 a sequence of positive numbers such that Wn = ∑ k=1^n wk →∞ as n →∞ We prove that x = （x.）n≥1 converges in E（M） if and only if （σn（x）n≥1 converges in E（.hd）, where E（A//） is a noncommutative rearrangement invariant Banach function space with the Fatou property and σn（x） is given by σn（x） = 1/Wn ∑k=1^n wkxk, n=1, 2, .If in addition, E（Ad） has absolutely continuous norm, then, （an（x））≥1 converges in E（.M） if and only if x = （Xn）n≥1 is uniformly integrable and its limit in measure topology x∞∈ E（M）.展开更多
In this article,we explore the famous Selkov–Schnakenberg(SS)system of coupled nonlinear partial differential equations(PDEs)for Lie symmetry analysis,self-adjointness,and conservation laws.Moreover,miscellaneous sol...In this article,we explore the famous Selkov–Schnakenberg(SS)system of coupled nonlinear partial differential equations(PDEs)for Lie symmetry analysis,self-adjointness,and conservation laws.Moreover,miscellaneous soliton solutions like dark,bright,periodic,rational,Jacobian elliptic function,Weierstrass elliptic function,and hyperbolic solutions of the SS system will be achieved by a well-known technique called sub-ordinary differential equations.All these results are displayed graphically by 3D,2D,and contour plots.展开更多
In the present paper,we mostly focus on P_(p)^(2)-statistical convergence.We will look into the uniform integrability via the power series method and its characterizations for double sequences.Also,the notions of P_(p...In the present paper,we mostly focus on P_(p)^(2)-statistical convergence.We will look into the uniform integrability via the power series method and its characterizations for double sequences.Also,the notions of P_(p)^(2)-statistically Cauchy sequence,P_(p)^(2)-statistical boundedness and core for double sequences will be described in addition to these findings.展开更多
This paper studies the properties of Nambu-Poisson geometry from the(n-l,k)-Dirac structure on a smooth manifold M.Firstly,we examine the automorphism group and infinitesimal on higher order Courant algebroid,to prove...This paper studies the properties of Nambu-Poisson geometry from the(n-l,k)-Dirac structure on a smooth manifold M.Firstly,we examine the automorphism group and infinitesimal on higher order Courant algebroid,to prove the integrability of infinitesimal Courant automorphism.Under the transversal smooth morphismΦ:N-→M and anchor mapping of M on(n-1,k)-Dirac structure,it's holds that the pullback(n-1,k)-Dirac structure on M turns out an(n-1,k)-Dirac structure on N.Then,given that the graph of Nambu-Poisson structure takes the form of(n-1,n-2)-Dirac structure,it follows that the single parameter variety of Nambu-Poisson structure is related to one variety closed n-symplectic form under gauge transformation.WhenΦ:N-→M is taken as the immersion mapping of(n-1)-cosymplectic submanifold,the pullback Nambu-Poisson structure on M turns out the Nambu-Poisson structure on N.Finally,we discuss the(n-1,O)-Dirac structure on M can be integrated into a problem of(n-1)-presymplectic groupoid.Under the mapping II:M-→M/H,the corresponding(n-1,O)-Dirac structure is F and E respectively.If E can be integrated into(n-1)-presymplectic groupoid(g,2),then there exists the only,such that the corresponding integral of F is(n-1)-presymplectic groupoid(g,).展开更多
In this paper,we consider the derivatives of intersection local time for two independent d-dimensional symmetricα-stable processes X^(α) and X^(α)with respective indices α and α.We first study the sufficient cond...In this paper,we consider the derivatives of intersection local time for two independent d-dimensional symmetricα-stable processes X^(α) and X^(α)with respective indices α and α.We first study the sufficient condition for the existence of the derivatives,which makes us obtain the exponential integrability and H?lder continuity.Then we show that this condition is also necessary for the existence of derivatives of intersection local time at the origin.Moreover,we also study the power variation of the derivatives.展开更多
A discrete matrix spectral problem and the associated hierarchy of Lax integrable lattice equations are presented, and it is shown that the resulting Lax integrable lattice equations are all Liouville integrable discr...A discrete matrix spectral problem and the associated hierarchy of Lax integrable lattice equations are presented, and it is shown that the resulting Lax integrable lattice equations are all Liouville integrable discrete Hamiltonian systems. A new integrable symplectic map is given by binary Bargmann constraint of the resulting hierarchy. Finally, an infinite set of conservation laws is given for the resulting hierarchy.展开更多
文摘In this paper we prove the following Hajek Renyi inequality:Let 0<p≤1 ,then for any Banach space B , any L p integrable B valued random variable sequence {D n,n≥1} ,any real number sequence {b n,n≥1} with 0<b n↑∞ ,any integer n≥1 ,there exits a constant C=C p>0 (only depending on p ) such thatP( sup j≥nji=1D ib j≥ε)≤Cε -p (∞j=n+1E‖D j‖ pb p j+nj=1E‖D j‖ pb p n) In the other direction,we prove some strong laws of large numbers and the integrability of the maximal functions for B valued random variable sequences by using this inequality and the Hajeck Renyi inequality we have obtained recently.Some known results are extended and improved.
基金Project supported by the National Natural Science Foundation of China.
文摘A stochastic averaging method is proposed for quasi-Hamiltonian systems (Hamiltonian systems with light dampings subject to weakly stochastic excitations). Various versions of the method, depending on whether the associated Hamiltonian systems are integrable or nonintegrable, resonant or nonresonant, are discussed. It is pointed out that the standard stochastic averaging method and the stochastic averaging method of energy envelope are special cases of the stochastic averaging method of quasi-Hamiltonian systems and that the results obtained by this method for several examples prove its effectiveness.
基金Project financed by the National Natural Science Foundation of China.
文摘The Liouville's integrability of the second order autonomous system is studied. It isproved that a second order polynomial system is Liouville integrable if and only if thereis an integral factor μ(x, y), such that or a rational function in x and y.
基金sponsored by the National Natural Science Foundations of China(Nos.12235007,11975131,11435005,12275144,11975204)KC Wong Magna Fund in Ningbo UniversityNatural Science Foundation of Zhejiang Province No.LQ20A010009。
文摘The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability.The well-known modified Kd V equation is a prototypical example of an integrable evolution equation in one spatial dimension.Do there exist integrable analogs of the modified Kd V equation in higher spatial dimensions?In what follows,we present a positive answer to this question.In particular,rewriting the(1+1)-dimensional integrable modified Kd V equation in conservation forms and adding deformation mappings during the process allows one to construct higher-dimensional integrable equations.Further,we illustrate this idea with examples from the modified Kd V hierarchy and also present the Lax pairs of these higher-dimensional integrable evolution equations.
文摘We present an introduction to the Darboux integrability theory of planar complex and real polynomial differential systems containing some improvements to the classical theory.
基金supported by the National Natural Science Foundation of China (11071190)
文摘Let x （xn）≥1 be a martingale on a noncommutative probability space n （M, r） and （wn）n≥1 a sequence of positive numbers such that Wn = ∑ k=1^n wk →∞ as n →∞ We prove that x = （x.）n≥1 converges in E（M） if and only if （σn（x）n≥1 converges in E（.hd）, where E（A//） is a noncommutative rearrangement invariant Banach function space with the Fatou property and σn（x） is given by σn（x） = 1/Wn ∑k=1^n wkxk, n=1, 2, .If in addition, E（Ad） has absolutely continuous norm, then, （an（x））≥1 converges in E（.M） if and only if x = （Xn）n≥1 is uniformly integrable and its limit in measure topology x∞∈ E（M）.
文摘In this article,we explore the famous Selkov–Schnakenberg(SS)system of coupled nonlinear partial differential equations(PDEs)for Lie symmetry analysis,self-adjointness,and conservation laws.Moreover,miscellaneous soliton solutions like dark,bright,periodic,rational,Jacobian elliptic function,Weierstrass elliptic function,and hyperbolic solutions of the SS system will be achieved by a well-known technique called sub-ordinary differential equations.All these results are displayed graphically by 3D,2D,and contour plots.
文摘In the present paper,we mostly focus on P_(p)^(2)-statistical convergence.We will look into the uniform integrability via the power series method and its characterizations for double sequences.Also,the notions of P_(p)^(2)-statistically Cauchy sequence,P_(p)^(2)-statistical boundedness and core for double sequences will be described in addition to these findings.
文摘This paper studies the properties of Nambu-Poisson geometry from the(n-l,k)-Dirac structure on a smooth manifold M.Firstly,we examine the automorphism group and infinitesimal on higher order Courant algebroid,to prove the integrability of infinitesimal Courant automorphism.Under the transversal smooth morphismΦ:N-→M and anchor mapping of M on(n-1,k)-Dirac structure,it's holds that the pullback(n-1,k)-Dirac structure on M turns out an(n-1,k)-Dirac structure on N.Then,given that the graph of Nambu-Poisson structure takes the form of(n-1,n-2)-Dirac structure,it follows that the single parameter variety of Nambu-Poisson structure is related to one variety closed n-symplectic form under gauge transformation.WhenΦ:N-→M is taken as the immersion mapping of(n-1)-cosymplectic submanifold,the pullback Nambu-Poisson structure on M turns out the Nambu-Poisson structure on N.Finally,we discuss the(n-1,O)-Dirac structure on M can be integrated into a problem of(n-1)-presymplectic groupoid.Under the mapping II:M-→M/H,the corresponding(n-1,O)-Dirac structure is F and E respectively.If E can be integrated into(n-1)-presymplectic groupoid(g,2),then there exists the only,such that the corresponding integral of F is(n-1)-presymplectic groupoid(g,).
基金Supported by National Natural Science Foundation of China(Grant Nos.12071003,12201294)Natural Science Foundation of Jiangsu Province,China(Grant No.BK20220865)。
文摘In this paper,we consider the derivatives of intersection local time for two independent d-dimensional symmetricα-stable processes X^(α) and X^(α)with respective indices α and α.We first study the sufficient condition for the existence of the derivatives,which makes us obtain the exponential integrability and H?lder continuity.Then we show that this condition is also necessary for the existence of derivatives of intersection local time at the origin.Moreover,we also study the power variation of the derivatives.
基金The project supported by the Scientific Research Award Foundation for Outstanding Young and Middle-Aged Scientists of Shandong Province of China
文摘A discrete matrix spectral problem and the associated hierarchy of Lax integrable lattice equations are presented, and it is shown that the resulting Lax integrable lattice equations are all Liouville integrable discrete Hamiltonian systems. A new integrable symplectic map is given by binary Bargmann constraint of the resulting hierarchy. Finally, an infinite set of conservation laws is given for the resulting hierarchy.
