A Hierarchy of Multidimensional Hénon-Heiles Systems

Abstract A hierarchy of multidimensional Hénon-Heiles (M-H-H) systems are constructed via the x- and t _n -higher-order-constrained flows of KdV hierarchy. The Lax representation for the M-H-H hierarchy is determined from the adjoint representation of the auxiliary linear problem for the KdV hierarchy. By using the Lax representation the classical Poisson structure and r-matrix for the hierarchy are found and the Jacobi inversion problem for the hierarchy is constructed. Supported by National Research Project “Nonlinear Sciences”     

Application of a semi-direct sum of Lie algebras to the TD hierarchy

We construct a Lie algebra G by using a semi-direct sum of Lie algebra G₁ with Lie algebra G₂. A direct application to the TD hierarchy leads to a novel hierarchy of integrable couplings of the TD hierarchy. Furthermore, the generalized variational identity is applied to Lie algebra G to obtain quasi-Hamiltonian structures of the associated integrable couplings.     

Bounded query classes and the difference hierarchy

LetA be any nonrecursive set. We define a hierarchy of sets (and a corresponding hierarchy of degrees) that are reducible toA based on bounding the number of queries toA that an oracle machine can make. WhenA is the halting problemK our hierarchy of sets interleaves with the difference hierarchy on the r.e. sets in a logarithmic way; this follows from a tradeoff between the number of parallel queries and the number of serial queries needed to compute a function with oracleK.Supported in part by NSF grant CCR-8808949. Part of this work was completed while this author was a student at Stanford University supported by fellowships from the National Science Foundation and from the Fannie and John Hertz FoundationSupported in part by NSF grant CCR-8803641Part of this work was completed while this author was on sabbatical leave at the University of California, Berkeley     

Semidefinite bounds for the stability number of a graph via sums of squares of polynomials

Lovász and Schrijver (SIAM J. Optim. 1:166–190, 1991) have constructed semidefinite relaxations for the stable set polytope of a graph G = (V,E) by a sequence of lift-and-project operations; their procedure finds the stable set polytope in at most α(G) steps, where α(G) is the stability number of G. Two other hierarchies of semidefinite bounds for the stability number have been proposed by Lasserre (SIAM J. Optim. 11:796–817, 2001; Lecture Notes in Computer Science, Springer, Berlin Heidelberg New York, pp 293–303, 2001) and by de Klerk and Pasechnik (SIAM J. Optim. 12:875–892), which are based on relaxing nonnegativity of a polynomial by requiring the existence of a sum of squares decomposition. The hierarchy of Lasserre is known to converge in α(G) steps as it refines the hierarchy of Lovász and Schrijver, and de Klerk and Pasechnik conjecture that their hierarchy also finds the stability number after α(G) steps. We prove this conjecture for graphs with stability number at most 8 and we show that the hierarchy of Lasserre refines the hierarchy of de Klerk and Pasechnik.      

From image processing to topological modelling with <Emphasis Type="Italic">p</Emphasis>-adic numbers

Encoding the hierarchical structure of images by p-adic numbers allows for image processing and computer vision methods motivated from arithmetic physics. The p-adic Polyakov action leads to the p-adic diffusion equation in low level vision. Hierarchical segmentation provides another way of p-adic encoding. Then a topology on that finite set of p-adic numbers yields a hierarchy of topological models underlying the image. In the case of chain complexes, the chain maps yield conditions for the existence of a hierarchy, and these can be expressed in terms of p-adic integrals. Such a chain complex hierarchy is a special case of a persistence complex from computational topology, where it is used for computing persistence barcodes for shapes. The approach is motivated by the observation that using p-adic numbers often leads to more efficient algorithms than their real or complex counterparts.     

Boolean Hierarchies of Partitions over a Reducible Base

The Boolean hierarchy of partitions was introduced and studied by Kosub and Wagner, primarily over the lattice of NP-sets. Here, this hierarchy is treated over lattices with the reduction property, showing that it has a much simpler structure in this instance. A complete characterization is given for the hierarchy over some important lattices, in particular, over the lattices of recursively enumerable sets and of open sets in the Baire space.     

A Lie algebra containing four parameters for the generalized Dirac hierarchy

A Lie algebra containing four parameters is obtained, whose commutation operation is concise, and the corresponding computing formula of constant γ in the variational identity is presented in this paper. As application, a new Liouville integrable hierarchy which can be reduced to Dirac hierarchy is derived by designing a special isospectral problem. We call it generalized Dirac hierarchy.     

Refining the polynomial hierarchy

By a result of Kadin, the difference hierarchy over NP does not collapse if the polynomial hierarchy does not. We extend this result to a natural refinement of the polynomial hierarchy. The refinement is obtained from levels of the polynomial hierarchy by validly applying addition modulo 2, and so here we call it the plus-hierarchy. Also, we consider two refinements of the plus-hierarchy, which may be relevant to a classification of some languages. Supported by the A. von Humboldt Foundation and by RFFR grant No. 96-01-00257. Translated fromAlgebra i Logika, Vol. 38, No. 4, pp. 456–475, July–August, 1999.     

On complete integrability of a hierarchy of finite-dimensional Hamiltonian systems

A hierarchy of Hamiltonian systems obtained from the Lax pair of KdV hierarchy under the constraint condition on potentialu = q, q is presented. The independent integrals for these Hamiltonian systems are constructed by using recursion operator and shown to be in involution. Thus this hierarchy of Hamiltonian systems is completely integrable in the sense of Liouville, and they commute with each other.This project is supported by the National Natural Science Foundation of China.     

A triangular norm hierarchy

Triangular norms and their hierarchy are investigated. A localisation of classical T_s norms and various other norms is obtained. The class of t-norms proposed by Zimmermann, Fuzzy Set Theory and its Applications, Kluwer, Dordrecht, 1991 is placed in the above hierarchy and a proof that it is weaker (see Butnariu, Fuzzy Sets and Systems 69 (1995) 241–255) then T₁ concludes the paper.     

The Hausdorff-Ershov Hierarchy in Euclidean Spaces

The topological arithmetical hierarchy is the effective version of the Borel hierarchy. Its class Δ^ta₂ is just large enough to include several types of pointsets in Euclidean spaces ℝ^k which are fundamental in computable analysis. As a crossbreed of Hausdorff's difference hierarchy in the Borel class Δ^B₂ and Ershov's hierarchy in the class Δ⁰₂ of the arithmetical hierarchy, the Hausdorff-Ershov hierarchy introduced in this paper gives a powerful classification within Δ^ta₂. This is based on suitable characterizations of the sets from Δ^ta₂ which are obtained in a close analogy to those of the Δ^B₂ sets as well as those of the Δ⁰₂ sets. A helpful tool in dealing with resolvable sets is contributed by the technique of depth analysis. On this basis, the hierarchy properties, in particular the strict inclusions between classes of different levels, can be shown by direct constructions of witness sets. The Hausdorff-Ershov hierarchy runs properly over all constructive ordinals, in contrast to Ershov's hierarchy whose denotation-independent version collapses at level ω². Also, some new characterizations of concepts of decidability for pointsets in Euclidean spaces are presented.     

Quotients of partial abelian monoids

A hierarchy of partial abelian structures is considered. In an order of decreasing generality, these structures include partial abelian monoids (PAM), cancellative PAMs (CPAM), effect algebras (or D-posets), orthoalgebras, orthomodular posets (OMP) and orthomodular lattices (OML). If P is a PAM, the concepts of a congruence on P and a quotient P are defined. Similar definitions are given for quotients of higher level categories in the hierarchy. The notion of a Riesz ideal I on a CPAM P is defined and it is shown that I generates a congruence on P. The corresponding quotients P/I for categories in the hierarchy are studied. It is shown that a subset I of an OML is a Riesz ideal if and only if I is a p-ideal. Moreover, for effect algebras, we show that congruences generated by Riesz ideals are precisely those that are given by a perspectivity. The paper includes a large number of counterexamples and examples that illustrate various concepts. Received April 14, 1997; accepted in final form January 19, 1998.     

Symmetries for the Ablowitz–Ladik Hierarchy: Part II. Integrable Discrete Nonlinear Schrödinger Equations and Discrete AKNS Hierarchy

In the paper, we continue to consider symmetries related to the Ablowitz–Ladik hierarchy. We derive symmetries for the integrable discrete nonlinear Schrödinger hierarchy and discrete AKNS hierarchy. The integrable discrete nonlinear Schrödinger hierarchy is in scalar form and its two sets of symmetries are shown to form a Lie algebra. We also present discrete AKNS isospectral flows, non‐isospectral flows and their recursion operator. In continuous limit these flows go to the continuous AKNS flows and the recursion operator goes to the square of the AKNS recursion operator. These discrete AKNS flows form a Lie algebra that plays a key role in constructing symmetries and their algebraic structures for both the integrable discrete nonlinear Schrödinger hierarchy and discrete AKNS hierarchy. Structures of the obtained algebras are different structures from those in continuous cases which usually are centerless Kac–Moody–Virasoro type. These algebra deformations are explained through continuous limit and degree in terms of lattice spacing parameter h.     

Integrable Hierarchy, 3×3 Constrained Systems, and Parametric Solutions

This paper provides a new integrable hierarchy. The DP equation: m _t +um _x +3mu _x =0, m=u–u _xx , proposed recently by Degasperis and Procesi, is the first member in the negative order hierarchy while the first equation in the positive order hierarchy is: m _t =4(m ^–2/3) _x –5(m ^–2/3) _xxx +(m ^–2/3) _xxxxx . The whole hierarchy is shown Lax-integrable through solving a key matrix equation. To obtain the parametric solutions for the whole hierarchy, we separately discuss the negative order and the positive order hierarchies. For the negative order hierarchy, its 3×3 Lax pairs and corresponding adjoint representations are cast in Liouville-integrable Hamiltonian canonical systems under the Dirac–Poisson bracket defined on a symplectic submanifold of R ^6N . Based on the integrability of those finite-dimensional canonical Hamiltonian systems we give the parametric solutions of all equations in the negative order hierarchy. In particular, we obtain the parametric solution of the DP equation. Moreover, for the positive order hierarchy, we consider a different constraint and process a procedure similar to the negative case to obtain the parametric solutions of the positive order hierarchy. In a special case, we give the parametric solution of the 5th-order PDE m _t =4(m ^–2/3) _x –5(m ^–2/3) _xxx +(m ^–2/3) _xxxxx . Finally, we discuss the stationary solutions of the 5th-order PDE, which may be included in the parametric solution.     

A correspondence between the AKNS hierarchy and the JM hierarchy

Reference [1] presented a gauge transformation between thex parts of the AKNS eigenvalue problem and those of the JM (Jaulent-Miodek) eigenvalue problem. In this paper we discuss the correspondence between thet parts of the AKNS eigenvalue problem and thet parts of the JM eigenvalue prohlem under the gauge transformation, and give a correspondence between the AKNS hierarchy and the JM hierarchy and also three types of Darboux transformation for the JM hierarchy.Project supported by the Science Fund of the Ministry of Education.     

Strengthened semidefinite programming bounds for codes

We give a hierarchy of semidefinite upper bounds for the maximum size A(n,d) of a binary code of word length n and minimum distance at least d. At any fixed stage in the hierarchy, the bound can be computed (to an arbitrary precision) in time polynomial in n; this is based on a result of de Klerk et al. (Math Program, 2006) about the regular ∗-representation for matrix ∗-algebras. The Delsarte bound for A(n,d) is the first bound in the hierarchy, and the new bound of Schrijver (IEEE Trans. Inform. Theory 51:2859–2866, 2005) is located between the first and second bounds in the hierarchy. While computing the second bound involves a semidefinite program with O(n ⁷) variables and thus seems out of reach for interesting values of n, Schrijver’s bound can be computed via a semidefinite program of size O(n ³), a result which uses the explicit block-diagonalization of the Terwilliger algebra. We propose two strengthenings of Schrijver’s bound with the same computational complexity. Supported by the Netherlands Organisation for Scientific Research grant NWO 639.032.203.     

On a new generalization of dispersionless Kadomtsev–Petviashvili hierarchy and its reductions

The dispersionless Kadomtsev–Petviashvili hierarchy is generalized by introducing two new time series γ_n and σ_k with two parameters η_n and λ_k. By this hierarchy, we obtain the first type, the second type as well as mixed type of dispersionless Kadomtsev–Petviashvili equation with self‐consistent sources and their related conservation equations. In addition, the reduction and constrained flow of this new hierarchy are studied. The first type, the second type and the mixed type of dispersionless Korteweg–de Vries equation with self‐consistent sources and of dispersionless Boussinesq equation with self‐consistent sources are obtained. Copyright © 2013 John Wiley & Sons, Ltd.     

On a hierarchy of models for electrical conduction in biological tissues

In this paper we derive a hierarchy of models for electrical conduction in a biological tissue, which is represented by a periodic array of period ε of conducting phases surrounded by dielectric shells of thickness εη included in a conductive matrix. Such a hierarchy will be obtained from the Maxwell equations by means of a concentration process η → 0, followed by a homogenization limit with respect to ε. These models are then compared with regard to their physical meaning and mathematical issues. Copyright © 2005 John Wiley & Sons, Ltd.     

On the Fourth Painlevé Hierarchy

We use the inverse monodromy transform to find the fourth Painlevé hierarchy. The second and third members of this hierarchy are given. Special and rational solutions of the second and third members for the P ₄ hierarchy are discussed. We apply the Painlevé test to the second member of the fourth Painlevé hierarchy.