Levi Hierarchy Research Articles

On (2+1)-dimensional expanding integrable model of the Davey-Stewartson hierarchy

This paper mainly investigates the reductions of an integrable coupling of the Levi hierarchy and an expanding model of the (2+1)-dimensional Davey-Stewartson hierarchy. It is shown that the integrable coupling system of the Levi hierarchy possesses a quasi-Hamiltonian structure under certain constraints. Based on the Lie algebras construct, The type abstraction hierarchy scheme is used to gener?ate the (2+1)-dimensional expanding integrable model of the Davey-Stewartson hierarchy.

A new approach for finding standard heat equation and a special Newell-whitehead equation

Under a frame of 2 ? 2 matrix Lie algebras, Tu and Meng [9] once established a united integrable model of the Ablowitz-Kaup-Newel-Segur (AKNS) hierarchy, the D-AKNS hierarchy, the Levi hierarchy and the TD hierarchy. Based on this idea, we introduce two block-matrix Lie algebras to present an isospectral problem, whose compatibility condition gives rise to a type of integrable hierarchy which can be reduced to the Levi hierarchy and the AKNS hierarchy, and so on. A united integrable model obtained by us in the paper is different from that given by Tu and Meng. Specially, the main result in the paper can be reduced to two new various integrable couplings of the Levi hierarchy, from which we again obtain the standard heat equation and a special Newell-Whitehead equation.

Definable orthogonality classes in accessible categories are small

We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopěnka's principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Levy hierarchy.

Nonlinear Integrable Couplings of Levi Hierarchy and WKI Hierarchy

With the help of the known Lie algebra, a type of new 8-dimensional matrix Lie algebra is constructed in the paper. By using the 8-dimensional matrix Lie algebra, the nonlinear integrable couplings of the Levi hierarchy and the Wadati-Konno-Ichikawa (WKI) hierarchy are worked out, which are different from the linear integrable couplings. Based on the variational identity, the Hamiltonian structures of the above hierarchies are derived.

Scales of minimal complexity in $${K(\mathbb{R})}$$

Using a Levy hierarchy and a fine structure theory for $${K(\mathbb{R})}$$ , we obtain scales of minimal complexity in this inner model. Each such scale is obtained assuming the determinacy of only those sets of reals whose complexity is strictly below that of the scale constructed.

A GENERALIZED LEVI HIERARCHY AND ITS INTEGRABLE COUPLINGS

In this paper, by making use of the generalized Tu scheme, we consider an isospectral problem, and then a new integrable hierarchy is constructed. It is shown that the generalized Levi hierarchy can be obtained as a reduction. Further, integrable couplings of the generalized Levi hierarchy is produced based on an enlarging isospectral problem.

SPECTRAL RADIUS ANALYSIS OF MATRICES AND THE ASSOCIATED WITH INTEGRABLE SYSTEMS

An isospectral problem is introduced, a spectral radius of the corresponding spectral matrix is obtained, which enlightens us to set up an isospectral problem whose compatibility condition gives rise to a zero curvature equation in formalism, from which a Lax integrable soliton equation hierarchy with constraints of potential functions is generated along with 5 parameters, whose reduced cases present three integrable systems, i.e., AKNS hierarchy, Levi hierarchy and D-AKNS hierarchy. Enlarging the above Lie algebra into two bigger ones, the two integrable couplings of the hierarchy are derived, one of them has Hamiltonian structure by employing the quadratic-form identity or variational identity. The corresponding integrable couplings of the reduced systems are obtained, respectively. Finally, as comparing study for generating expanding integrable systems, a Lie algebra of antisymmetric matrices and its corresponding loop algebra are constructed, from which a great number of enlarging integrable systems could be generated, especially their Hamiltonian structure could be computed by the trace identity.

Two types of multicomponent matrix loop algebras and its application

Two types of multicomponent matrix Lie algebras are constructed, which are devoted to obtain two types of new loop algebras ÃM−1. By making use of Tu scheme, integrable multicomponent Levi hierarchy and multicomponent Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy are generated, which contain an arbitrary positive integer M. Furthermore, we expanded the multicomponent matrix loop algebra into a large one and work out integrable coupling of Levi hierarchy and AKNS hierarchy. This method proposed in this paper can be used generally.

The quadratic-form identity for constructing Hamiltonian structures of the NLS–MKdV hierarchy and multi-component Levi hierarchy

The trace identity is extended to the quadratic-form identity. The Hamiltonian structures of the NLS–MKdV hierarchy, and integrable coupling of multi-component Levi hierarchy are obtained by the quadratic-form identity. The method can be used to produce the Hamiltonian structures of the other integrable couplings or multi-component hierarchies.

Coding by club-sequences

Given any subset A of ω 1 there is a proper partial order which forces that the predicate x ∈ A and the predicate x ∈ ω 1 ∖ A can be expressed by ZFC -provably incompatible Σ 3 formulas over the structure 〈 H ω 2 , ∈ , N S ω 1 〉 . Also, if there is an inaccessible cardinal, then there is a proper partial order which forces the existence of a well-order of H ω 2 definable over 〈 H ω 2 , ∈ , N S ω 1 〉 by a provably antisymmetric Σ 3 formula with two free variables. The proofs of these results involve a technique for manipulating the guessing properties of club-sequences defined on stationary subsets of ω 1 at will in such a way that the Σ 3 theory of 〈 H ω 2 , ∈ , N S ω 1 〉 with countable ordinals as parameters is forced to code a prescribed subset of ω 1 . On the other hand, using theorems due to Woodin it can be shown that, in the presence of sufficiently strong large cardinals, the above results are close to optimal from the point of view of the Levy hierarchy.

Multi-component Levi Hierarchy and Its Multi-component IntegrableCoupling System**The project supported by National Natural Science Foundationof China under Grant No. 10371070, the Liuhui Center for AppliedMathematics, Nankai University, and Tianjin University and the EducationalCommittee of Liaoning Province of China under Gant No. 2004C057

A simple 3M-dimensional loop algebra X̃ is produced,whose commutation operation defined by us is as simple andstraightforward as that in the loop algebra Ã1. Itfollows that a general scheme for generating multi-componentintegrable hierarchy is proposed. By taking advantage ofX̃, a new isospectral problem is established, and then bymaking use of the Tu scheme the well-known multi-component Levihierarchy is obtained. Finally, an expanding loop algebraF̃M of the loop algebra X̃ is presented, basedon the F̃M, the multi-component integrable couplingsystem of the multi-component Levi hierarchy is worked out. Themethod in this paper can be applied to other nonlinear evolutionequation hierarchies.

Levy and set theory

Azriel Levy (1934–) did fundamental work in set theory when it was transmuting into a modern, sophisticated field of mathematics, a formative period of over a decade straddling Cohen’s 1963 founding of forcing. The terms “Levy collapse”, “Levy hierarchy”, and “Levy absoluteness” will live on in set theory, and his technique of relative constructibility and connections established between forcing and definability will continue to be basic to the subject. What follows is a detailed account and analysis of Levy’s work and contributions to set theory.

Integrable couplings of the hierarchies of evolution equations

An efficient, straightforward method for obtaining integrable couplings of a hierarchy of evolution equations is proposed in this paper. First a transformation of a kind of Lax pairs is presented, a new loop algebra is constructed, which can be used to obtain the integrable couplings of the Levi hierarchy. Using this method can also obtain the integrable couplings of the well-known generalized AKNS hierarchy. As special cases, the integrable couplings of KdV equation and mKdV equation are obtained, respectively. This method can be used generally.

A Bargmann system and the involutive representation of solutions of the Levi hierarchy

The author considers a linear spectral problem associated with the Levi hierarchy of equations. A Bargmann system under the so-called Bargmann constraint between the eigenfunctions and potentials is proposed, and is proved to be a completely integrable Hamiltonian system in the Liouville sense. Moreover, by the involutive solution of the compatible systems under the Bargmann constraint he presents the involutive representation of solutions of the Levi hierarchy. Especially, he obtains the involutive representation of the solution of the well known Burgers equation ut=-uxx+2uux.

A nonconfocal generator of involutive systems and Levy hierarchy

A new nonconfocal generator of involutive systems is obtained, and the relation between it and Levy equation hierarchy is discussed. Furthermore, the representation of the solution for the Levy hierarchy is given.

Partially Conservative Extensions of Arithmetic

Let T be a consistent r.e. extension of Peano arithmetic; $\Sigma _n^0$, $\Pi _n^0$ the usual quantifier-block classification of formulas of the language of arithmetic (bounded quantifiers counting “for free"); and $\Gamma$, $\Gamma ’$ variables through the set of all classes $\Sigma _n^0$, $\Pi _n^0$. The principal concern of this paper is the question: When can we find an independent sentence $\phi \in \Gamma$ which is $\Gamma ’$-conservative in the following sense: Any sentence $\chi$ in $\Gamma ’$ which is provable from $T + \phi$ is already provable from T? (Additional embellishments: Ensure that $\phi$ is not provably equivalent to a sentence in any class “simpler” than $\Gamma$; that $\phi$ is not conservative for classes “more complicated” than $\Gamma ’$.) The answer, roughly, is that one can find such a $\phi$, embellishments and all, unless $\Gamma$ and $\Gamma ’$ are so related that such a $\phi$ obviously cannot exist. This theorem has applications to the theory of interpretations, since “$\phi$ is $\Gamma$-conservative” is closely related to the property “$T + \phi$ is interpretable in T"-or to variants of it, depending on $\Gamma$. Finally, we provide simple model theoretic characterizations of $\Gamma$-conservativeness. Most results extend straightforwardly if extra symbols are added to the language of arithmetic, and most have analogs in the Levy hierarchy of set theoretic formulas (T then being an extension of ZF).

Partially conservative extensions of arithmetic

Let T be a consistent r.e. extension of Peano arithmetic; Σ n 0 \Sigma _n^0 , Π n 0 \Pi _n^0 the usual quantifier-block classification of formulas of the language of arithmetic (bounded quantifiers counting “for free"); and Γ \Gamma , Γ ′ \Gamma ’ variables through the set of all classes Σ n 0 \Sigma _n^0 , Π n 0 \Pi _n^0 . The principal concern of this paper is the question: When can we find an independent sentence ϕ ∈ Γ \phi \, \in \,\Gamma which is Γ ′ \Gamma ’ -conservative in the following sense: Any sentence χ \chi in Γ ′ \Gamma ’ which is provable from T + ϕ T + \phi is already provable from T? (Additional embellishments: Ensure that ϕ \phi is not provably equivalent to a sentence in any class “simpler” than Γ \Gamma ; that ϕ \phi is not conservative for classes “more complicated” than Γ ′ \Gamma ’ .) The answer, roughly, is that one can find such a ϕ \phi , embellishments and all, unless Γ \Gamma and Γ ′ \Gamma ’ are so related that such a ϕ \phi obviously cannot exist. This theorem has applications to the theory of interpretations, since “ ϕ \phi is Γ \Gamma -conservative” is closely related to the property “ T + ϕ T + \phi is interpretable in T"-or to variants of it, depending on Γ \Gamma . Finally, we provide simple model theoretic characterizations of Γ \Gamma -conservativeness. Most results extend straightforwardly if extra symbols are added to the language of arithmetic, and most have analogs in the Levy hierarchy of set theoretic formulas (T then being an extension of ZF).