Compatible Lie Brackets and Integrable Equations of the Principal Chiral Model Type

We consider two classes of integrable nonlinear hyperbolic systems on Lie algebras. These systems generalize the principal chiral model. Each system is related to a pair of compatible Lie brackets and has a Lax representation, which is determined by the direct sum decomposition of the Lie algebra of Laurent series into the subalgebra of Taylor series and the complementary subalgebra corresponding to the pair. New examples of compatible Lie brackets are given.     

Lax Pairs for Equations Describing Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Reductions of the Lamé Equations

We solve the problem of describing compatible nonlocal Poisson brackets of hydrodynamic type. We prove that for nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type, there exist special local coordinates such that the metrics and the Weingarten operators of both brackets are diagonal. The nonlinear evolution equations describing all nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type are derived in these special coordinates, and the integrability of these equations is proved using the inverse scattering transform. The Lax pairs with a spectral parameter for these equations are found. We construct various classes of integrable reductions of the derived equations. These classes of reductions are of an independent differential-geometric and applied interest. In particular, if one of the compatible Poisson brackets is local, we obtain integrable reductions of the classical Lamé equations describing all orthogonal curvilinear coordinate systems in a flat space; if one of the compatible brackets is generated by a constant-curvature metric, the corresponding equations describe integrable reductions of the equations for orthogonal curvilinear coordinate systems in a space of constant curvature.     

Herz型Hardy空间上的一类交换子

证明一大类次线性算子与ＢＭＯ函数构成的交换子在Ｈａｒｚ型Ｈａｒｄｙ空间上的有界性。     

Rankin-Cohen Brackets and Invariant Theory

Using maps due to Ozeki and Broué-Enguehard between graded spaces of invariants for certain finite groups and the algebra of modular forms of even weight we equip these invariants spaces with a differential operator which gives them the structure of a Rankin-Cohen algebra. A direct interpretation of the Rankin-Cohen bracket in terms of transvectant for the group SL(2, C) is given.     

Seeking (and finding) Lagrangians

It is well known that for any second-order ordinary differential equation (ODE), a Lagrangian always exists, and the key to its construction is the Jacobi last multiplier. Is it possible to find Lagrangians for first-order systems of ODEs or for higher-order ODEs? We show that the Jacobi last multiplier can also play a major role in this case.     

一类Marcinkiewicz积分交换子的端点估计

证明了某种关于核的对数型Lipschitz条件下,带零次齐次核的Marcinkiewicz积分交换子是(H1,L1,∞),(Hb1,L1)和(Hb1,∞,L1,∞)型有界的.     

准径向Bochner-Riesz算子极大交换子的有界性

建立了准径向Bochner-Riesz算子与Lipschitz函数生成的极大交换子的(L~p,Λ_(β-n/p)和(L~(n/β),BMO)有界性.     

Commutator automorphisms of formal power series rings

For a big class of commutative rings , every continuous -automorphism of with the linear part the identity is in the commutator subgroup of . An explicit bound for the number of commutators involved and a -theoretic interpretation of this result are provided.

     

Involutive Solutions and Commutator Representations of the Dirac Hierarchy

ConsidertheDiracspectralproblemwherep,qaretwopotentials,Aisaspectralparameter.L*isaninjectivehomomorphism.ThefunctionalgradientVA=(2RR,ri-of)TofeigenvalueAwithrespecttop,qsatisfiesarecalledtheLenard'soperatorpairof(1).Theorem1LetG(1)(x),G(z)(x)betwoarbitarysmoothfunctions,G=(G(1),G(2))".ThenthefollowingoperatorequationwithrespecttoV=V(G),possessestheoperatorsolutionwhereL.'-jisthecommutator;L=L(p,q),K,Jaredefinedby(1)l(4)respectively.ProofSubstitute(6)into(5),directlycalculate.Defin…     

交换子在加权Herz型Hardy空间上的有界性

主要讨论由Lipschitz函数b与广义C-Z算子T生成的交换子[b,T]在加权Herz型Hardy空间上的有界性,证明了[6,T]从HKq1^α,p（w1,w2^q1）到HKq2^α,p（w1,w2^q2）的有界性．     

Two-dimensional systems that arise from the Noether classification of Lagrangians on the line

Noether-like operators play an essential role in writing down the first integrals for Euler-Lagrange systems of ordinary differential equations (ODEs). The classification of such operators is carried out with the help of analytic continuation of Lagrangians on the line. We obtain the classification of 5, 6 and 9 Noether-like operators for two-dimensional Lagrangian systems that arise from the submaximal and maximal dimensional Noether point symmetry classification of Lagrangians on the line. Cases in which the Noether-like operators are also Noether point symmetries for the systems of two ODEs are mentioned. In particular, the 8-dimensional maximal Noether algebra is remarkably obtained for the simplest system of the free particle equations in two dimensions from the 5-dimensional complex Noether algebra of the standard Lagrangian of the scalar free particle equation. We present the effectiveness of Noether-like operators for the determination of first integrals of systems of two nonlinear differential equations which arise from scalar complex Euler-Lagrange ODEs that admit Noether symmetry.     

Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies Related to Them

We construct integrable bi-Hamiltonian hierarchies related to compatible nonlocal Poisson brackets of hydrodynamic type and solve the problem of the canonical form for a pair of compatible nonlocal Poisson brackets of hydrodynamic type. A system of equations describing compatible nonlocal Poisson brackets of hydrodynamic type is derived. This system can be integrated by the inverse scattering problem method. Any solution of this integrable system generates integrable bi-Hamiltonian systems of hydrodynamic type according to explicit formulas. We construct a theory of Poisson brackets of the special Liouville type. This theory plays an important role in the construction of integrable hierarchies.     

分数次积分算子的交换子在齐型空间上的弱型估计

本文研究分数次积分交换子,其中Kα(x,y)=d(x,y)α-1,m∈N且b(x)∈BMO(X,μ),证明了Iα,bm是从Orlicz空间L(log L)m(X)到弱Lq(X)空间的映照．同时还证明了分数次极大算子交换子Mα,bm也有类似性质．     

Littlewood-Paley g-函数交换子的加权估计

设g_(φ,b)是Littlewood-Paley g-函数与b生成的交换子,ω∈A_1.证明了若b属于加权BMO空间BMO(ω),则g_(φ,b)是L~p(ω)到L~p(ω~(1-p))(1p∞)有界的;若b属于加权Lipschitz空间Lip_β(ω)(0β1),则g_(φ,b)是L~p(ω)到L~q(ω~(1-q))的有界算子,其中1pq∞,1/q=1/p-β/n.     

Commutator of hypersingular integral with rough kernels on Sobolev spaces

In this paper, the authors give the boundedness of the commutator of hypersingular integral T γ from the homogeneous Sobolev space Lpγ (Rn) to the Lebesgue space Lp(Rn) for 1p∞ and 0 γ min{ n/2 , n/p }.     

A novel result on regularity of time-periodic Lagrangians

The regularity of the Peierls barrier $h_c^{\infty }$ for time-periodic Lagrangian systems is equivalent to the convergence of the associated Lax-Oleinik semigroup. In this paper, we established a novel result on the regularity, which has been applied to the convergence and nonconvergence of the Lax-Oleinik semigroup for time-periodic Lagrangians.     

奇异积分交换子的加权Hardy型估计

T_b表示由加权Lipschitz函数b∈Lip_β(μ)(0β1)与Calderon-Zygmund奇异积分算子T生成的交换子.当μ∈A_1,n/(n+β)p≤1时研究了T_b在经典加权Hardy空间H~p(μ))上的有界性质,在端点p=n/(n+β)处研究了T_b在加权Hardy空间上的弱型估计.     

广义Calderon-Zygmund算子交换子的加权有界性

讨论了广义Calderon-Zygmund算子与加权Lipschitz函数生成的交换子的有界性。     

Integrating factors, adjoint equations and Lagrangians

Integrating factors and adjoint equations are determined for linear and non-linear differential equations of an arbitrary order. The new concept of an adjoint equation is used for construction of a Lagrangian for an arbitrary differential equation and for any system of differential equations where the number of equations is equal to the number of dependent variables. The method is illustrated by considering several equations traditionally regarded as equations without Lagrangians. Noether's theorem is applied to the Maxwell equations.     

Invariants of Lagrangians on Weil Bundles and Their Classifications

Let Q:BA be an algebra epimorphism, where A, B are Weil algebras. Let Q M:T B M T AM denote the canonical extension of Q, from the bundle of B-velocities onto the bundle of A-velocities, over a manifold M. Classifications of all natural operators transforming real-valued functions on T AM into real-valued functions on T BM or into 1-forms on T BM of finite order with respect to Q M are given, provided the dimension of M is sufficiently large.