Hierarchies Of Type Research Articles

Integrable hierarchies, Frölicher–Nijenhuis bicomplexes and Lauricella bi-flat F-manifolds

Given the Frölicher-Nijenhuis bicomplex associated with a -tensor field L with vanishing Nijenhuis torsion, we define a multi-parameter family of bi-flat structures . This result is obtained by combining the construction of integrable hierarchies of hydrodynamic type starting from Frölicher-Nijenhuis bicomplexes with the construction of flat F-manifold structures from integrable systems of hydrodynamic type. By construction L is the operator of multiplication by the Euler vector field E and the number of parameters coincides with the number of Jordan blocks appearing in its Jordan normal form. We call these structures Lauricella bi-flat structures since in the n-dimensional semisimple case flat coordinates of are Lauricella functions. The -tensor fields defining the corresponding integrable hierarchies have a similar block diagonal structure.

Travelling through the Natural Hierarchies of Type I Collagen with X-rays: From Tendons of Cattle, Horses, Sheep and Pigs.

Type I collagen physiological scaffold for tissue regeneration is considered one of the widely used biomaterials for tissue engineering and medical applications. It is hierarchically organized: five laterally staggered molecules are packed within fibrils, arranged into fascicles and bundles. The structural organization is correlated to the direction and intensity of the forces which can be loaded onto the tissue. For a tissue-specific regeneration, the required macro- and microstructure of a suitable biomaterial has been largely investigated. Conversely, the function of multiscale structural integrity has been much less explored but is crucial for scaffold design and application. In this work, collagen was extracted from different animal sources with protocols that alter its structure. Collagen of tendon shreds excised from cattle, horse, sheep and pig was structurally investigated by wide- and small-angle X-ray scattering techniques, at both molecular and supramolecular scales, and thermo-mechanically with thermal and load-bearing tests. Tendons were selected because of their resistance to chemical degradation and mechanical stresses. The multiscale structural integrity of tendons' collagen was studied in relation to the animal source, anatomic location and source for collagen extraction.

Soliton solutions to constrained nonlocal integrable nonlinear Schrödinger hierarchies of type (−λ,λ)

The paper aims to generate nonlocal integrable nonlinear Schrödinger hierarchies of type [Formula: see text] by imposing two nonlocal matrix restrictions of the AKNS matrix characteristic-value problems of arbitrary order. Based on the explored outspreading of characteristic-values and adjoint characteristic-values, exact soliton solutions are formulated by applying the associated reflectionless generalized Riemann–Hilbert problems, in which characteristic-values and adjoint characteristic-values could have a nonempty intersection. Illustrative models of the resultant mixed-type nonlocal integrable nonlinear Schrödinger equations are presented.

Flat F-Manifolds, F-CohFTs, and Integrable Hierarchies

We define the double ramification hierarchy associated to an F-cohomological field theory and use this construction to prove that the principal hierarchy of any semisimple (homogeneous) flat F-manifold possesses a (homogeneous) integrable dispersive deformation at all orders in the dispersion parameter. The proof is based on the reconstruction of an F-CohFT starting from a semisimple flat F-manifold and additional data in genus 1, obtained in our previous work. Our construction of these dispersive deformations is quite explicit and we compute several examples. In particular, we provide a complete classification of rank 1 hierarchies of DR type at the order 9 approximation in the dispersion parameter and of homogeneous DR hierarchies associated with all 2-dimensional homogeneous flat F-manifolds at genus 1 approximation.

Generalized Integrable Hierarchies of AKNS Type, Super Dirac Type and Super NLS–mKdV Type

Integrable hierarchies provide many important physical models. Firstly, with the help of symbolic computation software Maple, two generalized integrable hierarchies of Ablowitz–Kaup–Newell–Segur (AKNS) type are constructed from the matrix spectral problem associated with the Lie algebra s l ( 2 ) . Then, two generalized integrable hierarchies of super Dirac type and super nonlinear Schrodinger-modified Korteweg–de Vries (NLS–mKdV) type are obtained to illustrate the use of the super Lie algebra B ( 0 , 1 ) .

New approaches to integrable hierarchies of topological type

This survey is devoted to a large class of systems of partial differential equations which on the one hand appear in classical problems of mathematical physics and on the other hand provide an efficient tool for the description of enumerative invariants in algebraic geometry. Particular attention is paid to new approaches to these systems, in particular, to the approach proposed in a recent paper of the author. Bibliography: 57 titles.

New approaches to integrable hierarchies of topological type

Обзор посвящeн обширному классу систем уравнений в частных производных, которые, с одной стороны, возникают в классических задачах математической физики, а с другой стороны, являются эффективным инструментом для описания перечислительных инвариантов в алгебраической геометрии. Особое внимание уделено новым подходам к этим системам, в частности подходу, предложенному в недавней работе автора. Библиография: 58 названий.

Hierarchies of Manakov-Santini Type by Means of Rota-Baxter and Other Identities

The Lax-Sato approach to the hierarchies of Manakov-Santini type is formalized in order to extend it to a more general class of integrable systems. For this purpose some linear operators are introduced, which must satisfy some integrability conditions, one of them is the Rota-Baxter identity. The theory is illustrated by means of the algebra of Laurent series, the related hierarchies are classified and examples, also new, of Manakov-Santini type systems are constructed, including those that are related to the dispersionless modified Kadomtsev-Petviashvili equation and so called dispersionless r-th systems.

New Soliton Hierarchies Associated with the Lie Algebra so(3, [formula omitted]) and their BI-Hamiltonian Structures

We present three new matrix spectral problems, based on the real special orthogonal Lie algebra so (3, ℝ), and construct the corresponding asymmetric soliton hierarchies of AKNS type, KN type and WKI type with the aid of symbolic computation by Maple. Bi-Hamiltonian structures yielding Liouville integrability of the resulting soliton hierarchies are furnished by the trace identity, and so, all newly presented equations possess infinitely many commuting symmetries and conservation laws.

Novel soliton hierarchies of Levi type and their bi-Hamiltonian structures

We present two new matrix spectral problems, and construct the corresponding soliton hierarchies of Levi type with the aid of symbolic computation by Maple. Bi-Hamiltonian structures of the resulting soliton hierarchies are obtained by means of the trace identity. Thus it is shown that these soliton hierarchies are Liouville integrable.

Towards Lax Formulation of Integrable Hierarchies of Topological Type

To each partition function of cohomological field theory one can associate an Hamiltonian integrable hierarchy of topological type. The Givental group acts on such partition functions and consequently on the associated integrable hierarchies. We consider the Hirota and Lax formulations of the deformation of the hierarchy of N copies of KdV obtained by an infinitesimal action of the Givental group. By first deforming the Hirota quadratic equations and then applying a fundamental lemma to express it in terms of pseudo-differential operators, we show that such deformed hierarchy admits an explicit Lax formulation. We then compare the deformed Hamiltonians obtained from the Lax equations with the analogous formulas obtained in [1, 2], to find that they agree. We finally comment on the possibility of extending the Hirota and Lax formulation on the whole orbit of the Givental group action.

EVENT INFERENCE WITH RELATION AND META-RELATION TYPE HIERARCHIES IN CONCEPTUAL STRUCTURE THEORY

□ This article proposes a method to perform automated inference from event assertions through the use of an ontology formalized under Conceptual Structure Theory. An upper event ontology is first presented as an example of how the ontology is built and how real-life event assertions and their relations are modeled into its hierarchies of type. Three different types are distinguished: an event type as a basic classification of events of similar nature, an event relation type as a predicate on event types, and an event meta-relation type as a “predicate of predicates” on event types, with formal relationships between them and their properties elicited. The proposed formalization could be leveraged by search engines on the Semantic Web and query-answering systems in specific domains of discourse.

From additional symmetries to linearization of Virasoro symmetries

We construct the additional symmetries and derive the Adler–Shiota–van Moerbeke formula for the two-component BKP hierarchy. Considered as certain reductions of the two-component BKP hierarchy, the Drinfeld–Sokolov hierarchies of type D are proved to possess symmetries written as the linear action of a series of Virasoro operators on the tau function. It results in that the Drinfeld–Sokolov hierarchies of type D coincide with Dubrovin and Zhang’s hierarchies associated to the Frobenius manifolds for Coxeter groups of type D, and that every solution of such a hierarchy together with the string equation is annihilated by certain combinations of the Virasoro operators and the time derivations of the hierarchy.

Drinfeld-Sokolov Hierarchies of Type A and Fourth Order Painleve Systems

We study the Drinfeld-Sokolov hierarchies of type An(1) associated with the regular conjugacy classes of W(An). A class of fourth order Painlevé systems is derived from them by similarity reductions.

On the moduli space of deformations of bihamiltonian hierarchies of hydrodynamic type

We investigate the deformation theory of the simplest bihamiltonian structure of hydrodynamic type, that of the dispersionless KdV hierarchy. We prove that all of its deformations are quasi-trivial in the sense of B. Dubrovin and Y. Zhang, that is, trivial after allowing transformations where the first partial derivative ∂ u of the field is inverted. We reformulate the question about deformations as a question about the cohomology of a certain double complex, and calculate the appropriate cohomology group.

Nonlocal Hamiltonian operators of hydrodynamic type with flat metrics, integrable hierarchies, and the associativity equations

We solve the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics. This problem is equivalent to describing all flat submanifolds with flat normal bundle in a pseudo-Euclidean space. We prove that every such Hamiltonian operator (or the corresponding submanifold) specifies a pencil of compatible Poisson brackets, generates bihamiltonian integrable hierarchies of hydrodynamic type, and also defines a family of integrals in involution. We prove that there is a natural special class of such Hamiltonian operators (submanifolds) exactly described by the associativity equations of two-dimensional topological quantum field theory (the Witten-Dijkgraaf-Verlinde-Verlinde and Dubrovin equations). We show that each N-dimensional Frobenius manifold can locally be represented by a special flat N-dimensional submanifold with flat normal bundle in a 2N-dimensional pseudo-Euclidean space. This submanifold is uniquely determined up to motions.

The constrained dispersionless mKP hierarchy and the dispersionless mKP hierarchy with self-consistent sources

We first show that the quasiclassical limit of the squared eigenfunction symmetry constraint of the Sato operator for the mKP hierarchy leads to a reduction of the Sato function for the dispersionless mKP hierarchy. The constrained dispersionless mKP hierarchy (cdmKPH) is obtained and it is shown that the ( 2 + 1 ) -dimensional dispersionless mKP hierarchy is decomposed to two ( 1 + 1 ) -dimensional hierarchies of hydrodynamical type. The dispersionless mKP hierarchy with self-consistent sources (dmKPHSCS) together with its associated conservation equations are also constructed. Some solutions of dmKPESCS are obtained by hodograph reduction method.

A cohomological construction of integrable hierarchies of hydrodynamic type

We explain how to use the theory of bidifferential ideals to construct integrable hierarchies of hydrodynamic type

Solutions of the WDVV Equations and Integrable Hierarchies of KP Type

We show that reductions of KP hierarchies related to the loop algebra of SL n with homogeneous gradation give solutions of the Darboux-Egoroff system of PDE's. Using explicit dressing matrices of the Riemann-Hilbert problem generalized to include a set of commuting additional symmetries, we construct solutions of the Witten– Dijkgraaf–E. Verlinde–H. Verlinde equations.

Transformations of solutions for equations and hierarchies of pseudo-spherical type

It is known that if an equation describes non-trivial one-parameter families of pseudo-spherical surfaces, its conservation laws, (generalized, nonlocal) symmetries and Backlund transformations can be studied by geometrical means [4, 10]. In this letter it is pointed out that there exist correspondences, or 'generalized Backlund transformations', between arbitrary solutions (satisfying some genericity conditions) of any two single equations describing pseudo-spherical surfaces. Then, the notion of a hierarchy of equations of pseudo-spherical type is introduced, and a theorem stating that there also exist correspondences between arbitrary solutions of any two such hierarchies is presented. A full account of these results appears elsewhere [12, 13].