In the article,we provide a sharp lower bound for the weighted Lehmer mean of the complete p-elliptic integrals of the first and second kinds,which is the extension of the previous results for complete p-elliptic inte...In the article,we provide a sharp lower bound for the weighted Lehmer mean of the complete p-elliptic integrals of the first and second kinds,which is the extension of the previous results for complete p-elliptic integrals.展开更多
The N involutive integrals of motion with linearly independent gradients for the nonlin earized eigenvalue problem corresponding to the classical Boussinesq (CB) hierarchy are given. It is shown that when n=1,2,3, the...The N involutive integrals of motion with linearly independent gradients for the nonlin earized eigenvalue problem corresponding to the classical Boussinesq (CB) hierarchy are given. It is shown that when n=1,2,3, the nonlinearized time parts of Lax systems for the CB hierarchy are transformed into three finite-dimensional integrable Hamiltonian systems under the constraint of the nonlinearized spatial part.展开更多
The level 3 case for Ramanujan-type series has been considered as the most mysterious and the most challenging,out of all possible levels for Ramanujan-type series.This motivates the development of new techniques for ...The level 3 case for Ramanujan-type series has been considered as the most mysterious and the most challenging,out of all possible levels for Ramanujan-type series.This motivates the development of new techniques for constructing Ramanujan-type series of level 3.Chan and Liaw introduced an alternating analogue of the Borwein brothers’identity for Ramanujan-type series of level 3;subsequently,Chan,Liaw,and Tian formulated another proof of the Chan–Liaw identity,via the use of Ramanujan’s class invariant.Using the elliptic lambda function and the elliptic alpha function,we prove,via a limiting case of the Kummer–Goursat transformation,a new identity for evaluating the summands for alternating Ramanujan-type series of level 3,and we apply this new identity to prove three conjectured formulas for quadratic-irrational,Ramanujan-type series that had been discovered via numerical experiments with Maple in 2012 by Aldawoud.We also apply our identity to prove a new Ramanujan-type series of level 3 with a quartic convergence rate and quartic coefficients.展开更多
In this paper, a new completely integrable system related to the complex spectral problem -φ xx＋（i/4）wpx＋（i/4）（wp）x＋（1/4）vφ=iλφxand the constrained flows of the Boussinesq equations axe generated. Accor...In this paper, a new completely integrable system related to the complex spectral problem -φ xx＋（i/4）wpx＋（i/4）（wp）x＋（1/4）vφ=iλφxand the constrained flows of the Boussinesq equations axe generated. According to the viewpoint of Hamiltonian mechanics, the Euler-Lagrange equations and the Legendre transformations, a reasonable Jacobi-Ostrogradsky coordinate system is obtained. Moreover, by means of the constrained conditions between the potentiaJ u, v and the eigenfunction φ, the involutive representations of the solutions for the Boussinesq equation hieraxchy axe given.展开更多
Let K(r)be the complete elliptic integrals of the first kind for r∈(0,1)and f_(p)(x)=[(1−x)^(p)K(√x)].Using the recurrence method,we find the necessary and sufficient conditions for the functions−f′_(p),ln f_(p),−(...Let K(r)be the complete elliptic integrals of the first kind for r∈(0,1)and f_(p)(x)=[(1−x)^(p)K(√x)].Using the recurrence method,we find the necessary and sufficient conditions for the functions−f′_(p),ln f_(p),−(ln f_(p))^((i))(i=1,2,3)to be absolutely monotonic on(0,1).As applications,we establish some new bounds for the ratios and the product of two complete integrals of the first kind,including the double inequalities exp[r^(2)(1−r^(2))/^(64)]/(1+r)^(1/4)<K(r)/K(√r)<exp[−r(1−r)/4],π/2 exp[θ0(1−2r^(2))]<π/2 K(r′)/K(r)<π/2(r′/r)^(p)exp[θ_(p)(1−2r^(2))],K^(2)(1/√2)≤K(r)K(r′)≤1/√2rr′K^(2)(1/√2)for r∈2(0,1)and p≥13/32,where r′=√1−r^(2) and θ_(p)=2Γ(3/4)^(4)/π^(2)−p.展开更多
In the article,we prove that the inequalities H_(p)(K(r);E(r))>π/2;L_(q)(K(r);E(r))>π/2 hold for all r 2(0;1)if and only if p≥3=4 and q≥3=4,where Hp(a;b)and Lq(a;b)are respectively the p-th power-type Heroni...In the article,we prove that the inequalities H_(p)(K(r);E(r))>π/2;L_(q)(K(r);E(r))>π/2 hold for all r 2(0;1)if and only if p≥3=4 and q≥3=4,where Hp(a;b)and Lq(a;b)are respectively the p-th power-type Heronian mean and q-th Lehmer mean of a and b,and K(r)and E(r)are respectively the complete elliptic integrals of the first and second kinds.展开更多
A complete boundary integral formulation for steady compressible inviscid flows governed by nonlinear equations is established by using ρV as variable. Thus, the dimensionality of the problem to be solved is reduced ...A complete boundary integral formulation for steady compressible inviscid flows governed by nonlinear equations is established by using ρV as variable. Thus, the dimensionality of the problem to be solved is reduced by one and the computational mesh to be generated is needed only on the boundary of the domain.展开更多
We describe a class of self-dual dark nonlinear dynamical systems a priori allowing their quasilinearization,whose integrability can be effectively studied by means of a geometrically based gradient-holonomic approach...We describe a class of self-dual dark nonlinear dynamical systems a priori allowing their quasilinearization,whose integrability can be effectively studied by means of a geometrically based gradient-holonomic approach.A special case of the self-dual dynamical system,parametrically dependent on a functional variable is considered,and the related integrability condition is formulated.Using this integrability scheme,we study a new self-dual,dark nonlinear dynamical system on a smooth functional manifold,which models the interaction of atmospheric magnetosonic Alfvén plasma waves.We prove that this dynamical system possesses a Lax representation that allows its full direct linearization and compatible Poisson structures.Moreover,for this selfdual nonlinear dynamical system we construct an infinite hierarchy of mutually commuting conservation laws and prove its complete integrability.展开更多
In the article, we present some refinements of three classes of transformation inequalities for zero-balanced hypergeometric functions by use of the updated monotonicity criterion for the quotient of power series.
In the article,we prove that the double inequalities Gp[λ1a+(1-λ1)b,λ1 b+(1-λ1)a]A1-p(a,b)<T[A(a,b),G(a,b)]<Gp[μ1 a+(1-μ1)b,μ1b+(1-μ1)a]A1-p(a,b),Cs[λ^(2) a+(1-λ2)b,λ2 b+(1-λ2)a]A1-s(a,b)<T[A(a,b)...In the article,we prove that the double inequalities Gp[λ1a+(1-λ1)b,λ1 b+(1-λ1)a]A1-p(a,b)<T[A(a,b),G(a,b)]<Gp[μ1 a+(1-μ1)b,μ1b+(1-μ1)a]A1-p(a,b),Cs[λ^(2) a+(1-λ2)b,λ2 b+(1-λ2)a]A1-s(a,b)<T[A(a,b),Q(a,b)]<Cs[μ2 a+(1-μ2)b,μ2 b+(1-μ2)a]A1-p(a,b)hold for all a,b>0 with a≠b if and only ifλ1≤1/2-(1-(2/π)2/p)1/2/2,μ1≥1/2-(2p)1/2/(4 p),λ2≤1/2+(2(3/(2 s)(E(21/2/2)/π)1/s)-1)1/2/2 andμ2≥1/2+s1/2/(4 s)ifλ1,μ1∈(0,1/2),λ2,μ2∈(1/2,1),p≥1 and s≥1/2,where G(a,b)=(ab)1/2,A(a,b)=(a+b)/2,T(a,b)=∫0π/2(a2 cos2 t+b2 sin2)1/2 tdt/π,Q(a,b)=((a2+b2)/2)1/2,C(a,b)=(a2+b2)/(a+b)and E(r)=∫0π/2(1-r^(2) sin^(2))1/2 tdt.展开更多
In this paper,we present new bounds for the perimeter of an ellipse in terms of harmonic,geometric,arithmetic and quadratic means;these new bounds represent improvements upon some previously known results.
In this paper, an implicit symmetry constraint is calculated and its associated binary nonlinearization of the Lax pairs and the adjoint Lax pairs is carried out for the modified Korteweg-de Vries (mKdV) equation. Aft...In this paper, an implicit symmetry constraint is calculated and its associated binary nonlinearization of the Lax pairs and the adjoint Lax pairs is carried out for the modified Korteweg-de Vries (mKdV) equation. After introducing two new inde-pendent variables, we find that under the implicit symmetry constraint, the spatial part and the temporal part of the mKdV equation are decomposed into two finite-dimensional systems. Furthermore we prove that the obtained finite-dimensional systems are Hamiltonian systems and completely integrable in the Liouville sense.展开更多
In this paper,we investigate local properties in the system of completely integral mapping spaces.We introduce notions of injectivity,local reflexivity,exactness,nuclearity,finite-represent ability and WEP in the syst...In this paper,we investigate local properties in the system of completely integral mapping spaces.We introduce notions of injectivity,local reflexivity,exactness,nuclearity,finite-represent ability and WEP in the system of completely integral mapping spaces.First we obtain that any finite-dimensional operator space is injective in the system of completely integral mapping spaces.Furthermore we prove that C is the unique nuclear operator space and the unique exact operator space in this system.We also show that C is the unique operator space which is finitely representable in{T_(n)}n∈Nin this system.As corollaries,Kirchberg’s conjecture and QWEP conjecture in the system of completely integral mapping spaces are false.展开更多
For a sequence of identically distributed negatively associated random variables {Xn; n ≥ 1} with partial sums Sn = ∑i=1^n Xi, n ≥ 1, refinements are presented of the classical Baum-Katz and Lai complete convergenc...For a sequence of identically distributed negatively associated random variables {Xn; n ≥ 1} with partial sums Sn = ∑i=1^n Xi, n ≥ 1, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form ∑n≥n0 n^r-2-1/pq anE（max1≤k≤n｜Sk｜^1/q-∈bn^1/qp）^＋〈∞to hold where r 〉 1, q 〉 0 and either n0 = 1,0 〈 p 〈 2, an = 1,bn = n or n0 = 3,p = 2, an = 1 （log n） ^1/2q, bn=n log n. These results extend results of Chow and of Li and Spataru from the indepen- dent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence.展开更多
In this paper, complete moment convergence for widely orthant dependent random vari- ables is investigated under some mild conditions. For arrays of rowwise widely orthant dependent random variables, the main results ...In this paper, complete moment convergence for widely orthant dependent random vari- ables is investigated under some mild conditions. For arrays of rowwise widely orthant dependent random variables, the main results extend recent results on complete convergence to complete moment convergence. These results on complete moment convergence are shown to yield new results on complete integral convergence.展开更多
In this paper we will show how the Bohr–Sommerfeld levels of a quantum completely integrable system can be computed modulo O(h^(∞))by an inductive procedure starting at stage zero with the Bohr–Sommerfeld levels of...In this paper we will show how the Bohr–Sommerfeld levels of a quantum completely integrable system can be computed modulo O(h^(∞))by an inductive procedure starting at stage zero with the Bohr–Sommerfeld levels of the corresponding classical completely integrable system.展开更多
Under the constrained condition induced by the eigenfunctions and the potentials, the Laxsystems of nonlinear evolution equations in relation to a matris eigenvalue problem are nonlin-earized to be a completely integr...Under the constrained condition induced by the eigenfunctions and the potentials, the Laxsystems of nonlinear evolution equations in relation to a matris eigenvalue problem are nonlin-earized to be a completely integrable system (R^(zN),dp∧dq, H), while the time part of it isnonlinearized to be its N-involutive system {F_m}. The involutive solution of the compatiblesystem (F_0), (F_m) is transformed into the solution of the m-th nonlinear evolution equation.展开更多
The purpose of this paper is to provide a direct proof on the fact that the geometric-harmonic mean of any two positive numbers can be calculated by a first complete elliptical integral, and then to give new character...The purpose of this paper is to provide a direct proof on the fact that the geometric-harmonic mean of any two positive numbers can be calculated by a first complete elliptical integral, and then to give new characterizations of some mean-values.展开更多
We study overdetermined systems of first order partial differential equations with singular solutions.The main result gives a characterization of such systems and asserts that the singular solution is equal to the con...We study overdetermined systems of first order partial differential equations with singular solutions.The main result gives a characterization of such systems and asserts that the singular solution is equal to the contact singular set.展开更多
By modifying the procedure of binary nonlinearization for the AKNS spectral problem and its adjoint spectral problem under an implicit symmetry constraint, we obtain a finite dimensional system from the Lax pair of th...By modifying the procedure of binary nonlinearization for the AKNS spectral problem and its adjoint spectral problem under an implicit symmetry constraint, we obtain a finite dimensional system from the Lax pair of the nonlinear Schrodinger equation. We show that this system is a completely integrable Hamiltonian system.展开更多
基金Supported by the National Natural Science Foundation of China(11971142)the Natural Science Foundation of Zhejiang Province(LY19A010012)the key Scientific Research Projects of Hunan Provincial Department of Education in 2021(21A0526)。
文摘In the article,we provide a sharp lower bound for the weighted Lehmer mean of the complete p-elliptic integrals of the first and second kinds,which is the extension of the previous results for complete p-elliptic integrals.
文摘The N involutive integrals of motion with linearly independent gradients for the nonlin earized eigenvalue problem corresponding to the classical Boussinesq (CB) hierarchy are given. It is shown that when n=1,2,3, the nonlinearized time parts of Lax systems for the CB hierarchy are transformed into three finite-dimensional integrable Hamiltonian systems under the constraint of the nonlinearized spatial part.
基金supported by a Killam Postdoctoral Fellowship from the Killam Trusts.
文摘The level 3 case for Ramanujan-type series has been considered as the most mysterious and the most challenging,out of all possible levels for Ramanujan-type series.This motivates the development of new techniques for constructing Ramanujan-type series of level 3.Chan and Liaw introduced an alternating analogue of the Borwein brothers’identity for Ramanujan-type series of level 3;subsequently,Chan,Liaw,and Tian formulated another proof of the Chan–Liaw identity,via the use of Ramanujan’s class invariant.Using the elliptic lambda function and the elliptic alpha function,we prove,via a limiting case of the Kummer–Goursat transformation,a new identity for evaluating the summands for alternating Ramanujan-type series of level 3,and we apply this new identity to prove three conjectured formulas for quadratic-irrational,Ramanujan-type series that had been discovered via numerical experiments with Maple in 2012 by Aldawoud.We also apply our identity to prove a new Ramanujan-type series of level 3 with a quartic convergence rate and quartic coefficients.
文摘In this paper, a new completely integrable system related to the complex spectral problem -φ xx＋（i/4）wpx＋（i/4）（wp）x＋（1/4）vφ=iλφxand the constrained flows of the Boussinesq equations axe generated. According to the viewpoint of Hamiltonian mechanics, the Euler-Lagrange equations and the Legendre transformations, a reasonable Jacobi-Ostrogradsky coordinate system is obtained. Moreover, by means of the constrained conditions between the potentiaJ u, v and the eigenfunction φ, the involutive representations of the solutions for the Boussinesq equation hieraxchy axe given.
文摘Let K(r)be the complete elliptic integrals of the first kind for r∈(0,1)and f_(p)(x)=[(1−x)^(p)K(√x)].Using the recurrence method,we find the necessary and sufficient conditions for the functions−f′_(p),ln f_(p),−(ln f_(p))^((i))(i=1,2,3)to be absolutely monotonic on(0,1).As applications,we establish some new bounds for the ratios and the product of two complete integrals of the first kind,including the double inequalities exp[r^(2)(1−r^(2))/^(64)]/(1+r)^(1/4)<K(r)/K(√r)<exp[−r(1−r)/4],π/2 exp[θ0(1−2r^(2))]<π/2 K(r′)/K(r)<π/2(r′/r)^(p)exp[θ_(p)(1−2r^(2))],K^(2)(1/√2)≤K(r)K(r′)≤1/√2rr′K^(2)(1/√2)for r∈2(0,1)and p≥13/32,where r′=√1−r^(2) and θ_(p)=2Γ(3/4)^(4)/π^(2)−p.
基金Supported by the National Natural Science Foundation of China(11971142)the Natural Science Foundation of Zhejiang Province(LY19A010012)。
文摘In the article,we prove that the inequalities H_(p)(K(r);E(r))>π/2;L_(q)(K(r);E(r))>π/2 hold for all r 2(0;1)if and only if p≥3=4 and q≥3=4,where Hp(a;b)and Lq(a;b)are respectively the p-th power-type Heronian mean and q-th Lehmer mean of a and b,and K(r)and E(r)are respectively the complete elliptic integrals of the first and second kinds.
文摘A complete boundary integral formulation for steady compressible inviscid flows governed by nonlinear equations is established by using ρV as variable. Thus, the dimensionality of the problem to be solved is reduced by one and the computational mesh to be generated is needed only on the boundary of the domain.
文摘We describe a class of self-dual dark nonlinear dynamical systems a priori allowing their quasilinearization,whose integrability can be effectively studied by means of a geometrically based gradient-holonomic approach.A special case of the self-dual dynamical system,parametrically dependent on a functional variable is considered,and the related integrability condition is formulated.Using this integrability scheme,we study a new self-dual,dark nonlinear dynamical system on a smooth functional manifold,which models the interaction of atmospheric magnetosonic Alfvén plasma waves.We prove that this dynamical system possesses a Lax representation that allows its full direct linearization and compatible Poisson structures.Moreover,for this selfdual nonlinear dynamical system we construct an infinite hierarchy of mutually commuting conservation laws and prove its complete integrability.
基金supported by the Natural Science Foundation of China(61673169,11401191,11371125)the Tianyuan Special Funds of the Natural Science Foundation of China(11626101)the Natural Science Foundation of the Department of Education of Zhejiang Province(201635325)
文摘In the article, we present some refinements of three classes of transformation inequalities for zero-balanced hypergeometric functions by use of the updated monotonicity criterion for the quotient of power series.
基金supported by the Natural Science Foundation of China(61673169,11301127,11701176,11626101,11601485)。
文摘In the article,we prove that the double inequalities Gp[λ1a+(1-λ1)b,λ1 b+(1-λ1)a]A1-p(a,b)<T[A(a,b),G(a,b)]<Gp[μ1 a+(1-μ1)b,μ1b+(1-μ1)a]A1-p(a,b),Cs[λ^(2) a+(1-λ2)b,λ2 b+(1-λ2)a]A1-s(a,b)<T[A(a,b),Q(a,b)]<Cs[μ2 a+(1-μ2)b,μ2 b+(1-μ2)a]A1-p(a,b)hold for all a,b>0 with a≠b if and only ifλ1≤1/2-(1-(2/π)2/p)1/2/2,μ1≥1/2-(2p)1/2/(4 p),λ2≤1/2+(2(3/(2 s)(E(21/2/2)/π)1/s)-1)1/2/2 andμ2≥1/2+s1/2/(4 s)ifλ1,μ1∈(0,1/2),λ2,μ2∈(1/2,1),p≥1 and s≥1/2,where G(a,b)=(ab)1/2,A(a,b)=(a+b)/2,T(a,b)=∫0π/2(a2 cos2 t+b2 sin2)1/2 tdt/π,Q(a,b)=((a2+b2)/2)1/2,C(a,b)=(a2+b2)/(a+b)and E(r)=∫0π/2(1-r^(2) sin^(2))1/2 tdt.
基金supported by the Natural Science Foundation of China(11971142)the Natural Science Foundation of Zhejiang Province(LY19A010012)。
文摘In this paper,we present new bounds for the perimeter of an ellipse in terms of harmonic,geometric,arithmetic and quadratic means;these new bounds represent improvements upon some previously known results.
文摘In this paper, an implicit symmetry constraint is calculated and its associated binary nonlinearization of the Lax pairs and the adjoint Lax pairs is carried out for the modified Korteweg-de Vries (mKdV) equation. After introducing two new inde-pendent variables, we find that under the implicit symmetry constraint, the spatial part and the temporal part of the mKdV equation are decomposed into two finite-dimensional systems. Furthermore we prove that the obtained finite-dimensional systems are Hamiltonian systems and completely integrable in the Liouville sense.
基金Supported by the National Natural Science Foundation of China(Grant No.11871423)Zhejiang Provincial Natural Science Foundation of China(Grant No.LQ21A010015)。
文摘In this paper,we investigate local properties in the system of completely integral mapping spaces.We introduce notions of injectivity,local reflexivity,exactness,nuclearity,finite-represent ability and WEP in the system of completely integral mapping spaces.First we obtain that any finite-dimensional operator space is injective in the system of completely integral mapping spaces.Furthermore we prove that C is the unique nuclear operator space and the unique exact operator space in this system.We also show that C is the unique operator space which is finitely representable in{T_(n)}n∈Nin this system.As corollaries,Kirchberg’s conjecture and QWEP conjecture in the system of completely integral mapping spaces are false.
基金supported by National Natural Science Foundation of China (Grant No. 10871146)supported by Natural Sciences and Engineering Research Council of Canada
文摘For a sequence of identically distributed negatively associated random variables {Xn; n ≥ 1} with partial sums Sn = ∑i=1^n Xi, n ≥ 1, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form ∑n≥n0 n^r-2-1/pq anE（max1≤k≤n｜Sk｜^1/q-∈bn^1/qp）^＋〈∞to hold where r 〉 1, q 〉 0 and either n0 = 1,0 〈 p 〈 2, an = 1,bn = n or n0 = 3,p = 2, an = 1 （log n） ^1/2q, bn=n log n. These results extend results of Chow and of Li and Spataru from the indepen- dent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11671012,11501004 and 11501005)the Natural Science Foundation of Anhui Province(Grant No.1508085J06)+1 种基金the Key Projects for Academic Talent of Anhui Province(Grant No.gxbjZD2016005)the Research Teaching Model Curriculum of Anhui University(Grant No.xjyjkc1407)
文摘In this paper, complete moment convergence for widely orthant dependent random vari- ables is investigated under some mild conditions. For arrays of rowwise widely orthant dependent random variables, the main results extend recent results on complete convergence to complete moment convergence. These results on complete moment convergence are shown to yield new results on complete integral convergence.
基金Supported by the National Key R and D Program of China(Grant No.2020YFA0713100)NSFC(Grant Nos.12026409,11721101,11926313)。
文摘In this paper we will show how the Bohr–Sommerfeld levels of a quantum completely integrable system can be computed modulo O(h^(∞))by an inductive procedure starting at stage zero with the Bohr–Sommerfeld levels of the corresponding classical completely integrable system.
文摘Under the constrained condition induced by the eigenfunctions and the potentials, the Laxsystems of nonlinear evolution equations in relation to a matris eigenvalue problem are nonlin-earized to be a completely integrable system (R^(zN),dp∧dq, H), while the time part of it isnonlinearized to be its N-involutive system {F_m}. The involutive solution of the compatiblesystem (F_0), (F_m) is transformed into the solution of the m-th nonlinear evolution equation.
文摘The purpose of this paper is to provide a direct proof on the fact that the geometric-harmonic mean of any two positive numbers can be calculated by a first complete elliptical integral, and then to give new characterizations of some mean-values.
文摘We study overdetermined systems of first order partial differential equations with singular solutions.The main result gives a characterization of such systems and asserts that the singular solution is equal to the contact singular set.
基金Supported by the National Natural Science Foundation of China(No.11001069,61273077,11271210 and 10971109)Program for NCET under Grant No.NCET-08-0515Zhejiang Provincial Natural Science Foun-dation of China under Grant No.LQ12A01002 and LQ12A01003
文摘By modifying the procedure of binary nonlinearization for the AKNS spectral problem and its adjoint spectral problem under an implicit symmetry constraint, we obtain a finite dimensional system from the Lax pair of the nonlinear Schrodinger equation. We show that this system is a completely integrable Hamiltonian system.
