A Finite-Dimensional Completely Integrable System Associated with Boussinesq Hierarchy

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.

Bohr–Sommerfeld Levels for Quantum Completely Integrable Systems

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.

A NEW POLYNOMIAL INVOLUTIVE SYSTEM AND A CLASSICAL COMPLETELY INTEGRABLE SYSTEM

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.

Sharp power-type Heronian and Lehmer means inequalities for the complete elliptic integrals

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.

Nuclearity and Finite-Representability in the System of Completely Integral Mapping Spaces

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.

An implicit symmetry constraint of the modified Korteweg-de Vries (mKdV) equation

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.

REFINEMENTS OF TRANSFORMATION INEQUALITIES FOR ZERO-BALANCED HYPERGEOMETRIC FUNCTIONS

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.

SHARP BOUNDS FOR TOADER-TYPE MEANS IN TERMS OF TWO-PARAMETER MEANS

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.

ON THE BOUNDS OF THE PERIMETER OF AN ELLIPSE

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.

ABSOLUTE MONOTONICITY INVOLVING THE COMPLETE ELLIPTIC INTEGRALS OF THE FIRST KIND WITH APPLICATIONS

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.

A COMPLETE BOUNDARY INTEGRAL FORMULATION FOR STEADY COMPRESSIBLE INVISCID FLOWS GOVERNED BY NONLINEAR EQUATIONS

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.

Geometric-Harmonic Mean and Characterizations of Some Mean-Values

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.

Binary Nonlinearization of the Nonlinear Schr?dinger Equation Under an Implicit Symmetry Constraint

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.

BACKLUND TRANSFORMATION OF THE SOLUTIONS OF DIFFERENT EQUATIONS

In [1], the author discussed the Bcklund transformation (in Darboux type) of nonlinear evolution equations by using the gauge transformation. This type can be considered as an explicit form of Bcklund transformation. In this note, we shall discuss the Bcklund transformation (in Darboux type) of the solutions of the two different equations.

Complete Moment and Integral Convergence for Sums of Negatively Associated Random Variables

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.

Complete Moment Convergence for Arrays of Rowwise Widely Orthant Dependent Random Variables

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.

ON THE COMPLETE INTEGRABILITY OF NONLINEARIZED LAX SYSTEMS FOR THE CLASSICAL BOUSSINESQ HIERARCHY

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.

Quasi-linearization and stability analysis of some self-dual,dark equations and a new dynamical system

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.

Overdetermined Systems of First Order Partial Differential Equations with Singular Solution

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.

Singular integrals in axisymmetric problems of elastostatics

Singular integral equations arisen in axisymmetric problems of elastostatics are under consideration in this paper.These equations are received after applying the integral transformation and Gauss-Ostrogradsky’s theorem to the Green tensor for equilibrium equations of the infinite isotropic medium.Initially,three-dimensional problems expressed in Cartesian coordinates are transformed to cylindrical ones and integrated with respect to the circumference coordinate.So,the three-dimensional axisymmetric problems are reduced to systems of one-dimensional singular integral equations requiring the evaluation of linear integrals only.The thorough analysis of both displacement and traction kernels is accomplished,and similarity in behavior of both kernels is established.The kernels are expressed in terms of complete elliptic integrals of first and second kinds.The second kind elliptic integrals are nonsingular,and standard Gaussian quadratures are applied for their numerical evaluation.Analysis of external integrals proved the existence of logarithmic and Cauchy’s singularities.The numerical treatment of these integrals takes into account the presence of this integrable singularity.The numerical examples are provided to testify accuracy and efficiency of the proposed method including integrals with logarithmic singularity,Catalan’s constant,the Gaussian surface integral.The comparison between analytical and numerical data has proved high precision and availability of the proposed method.