4-Dimensional Frobenius manifolds and Painleve’ VI

A Frobenius manifold has tri-Hamiltonian structure if it is even-dimensional and its spectrum is maximally degenerate. We study the case of the lowest nontrivial dimension \(n=4\) and show that, under the assumption of semisimplicity, the corresponding isomonodromic Fuchsian system is described by the Painlevé \(\hbox {VI}\mu \) equation. Since the solutions of this equation are known to parametrize semisimple Frobenius manifolds of dimension \(n=3\) , this leads to an explicit procedure mapping 3-dimensional Frobenius structures of 4-dimensional ones, and giving all tri-Hamiltonian structures in four dimensions. We illustrate the construction by computing two examples in the framework of Frobenius structures on Hurwitz spaces.     

A tri-Hamiltonian formulation of a new soliton hierarchy associated with so(3,R)

A third Hamiltonian operator is presented for a new hierarchy of bi-Hamiltonian soliton equations, thereby showing that this hierarchy is tri-Hamiltonian. Additionally, an inverse hierarchy of common commuting symmetries is also presented.     

Separation of Variables in Multi-Hamiltonian Systems: An Application to the Lagrange Top

Starting from the tri-Hamiltonian formulation of the Lagrange top in a six-dimensional phase space, we discuss the reduction of the vector field and of the Poisson tensors. We show explicitly that after the reduction to each symplectic leaf, the vector field of the Lagrange top is separable in the Hamilton–Jacobi sense.     

Two New Classes of Hamiltonian Graphs: (Extended Abstract)

We prove that a triangular grid without local cuts is (almost) always Hamiltonian. This suggests an efficient scheme for rendering triangulated manifolds by graphics hardware. We also show that the Hamiltonian Cycle problem is NP-Complete for planar subcubic graphs of arbitrarily high girth. As a by-product we prove that there exist tri-Hamiltonian planar subcubic graphs of arbitrarily high girth.     

Subcomplexes in curved BGG-sequences

BGG-sequences offer a uniform construction for invariant differential operators for a large class of geometric structures called parabolic geometries. For locally flat geometries, the resulting sequences are complexes, but in general the compositions of the operators in such a sequence are nonzero. In this paper, we show that under appropriate torsion freeness and/or semi-flatness assumptions certain parts of all BGG sequences are complexes. Several examples of structures, including quaternionic structures, hypersurface type CR structures and quaternionic contact structures are discussed in detail. In the case of quaternionic structures we show that several families of complexes obtained in this way are elliptic.     

Invariant affinor metric structures on Lie groups

We introduce the class of special metric structures on Lie groups which are connected with the radical of a fixed 1-form on a Lie group. These structures are called affinor metric structures. We introduce and study some special classes of invariant affinor metric structures and generalize many results of the theory of contact metric structures on Lie groups.     

Some generalizations of the Riemann spaces of Einstein

We introduce a class of Riemann structures, called generalized Einstein structures of index 2e, of which Einstein spaces are a particular case. We show that these structures are stationary for functions introduced on a family of Riemann structures of the compact manifold of H. Weyl. This result solves a problem of M. Berger. As examples of structures which are generalized Einstein structures over all indices we cite homogeneous compact Riemann spaces with a nondecomposable isotropy group and products of such spaces.     

Computability Over Structures of Infinite Signature

Continuing the paper [7], in which the Blum-Shub-Smale approach to computability over the reals has been generalized to arbitrary algebraic structures, this paper deals with computability and recognizability over structures of infinite signature. It begins with discussing related properties of the linear and scalar real structures and of their discrete counterparts over the natural numbers. Then the existence of universal functions is shown to be equivalent to the effective encodability of the underlying structure. Such structures even have universal functions satisfying the s-m-n theorem and related features. The real and discrete examples are discussed with respect to effective encodability. Megiddo structures and computational extensions of effectively encodable structures are encodable, too. As further variants of universality, universal functions with enumerable sets of program codes and such ones with constructible codes are investigated. Finally, the existence of m-complete sets is shown to be independent of the effective encodability of structures, and the linear and scalar structures are discussed once more, under this aspect.     

Decomposable searching problems I. Static-to-dynamic transformation

Transformations that serve as tools in the design of new data structures are investigated. Specifically, general methods for converting static structures (in which all elements are known before any searches are performed) to dynamic structures (in which insertions of new elements can be mixed with searches) are studied. Three classes of such transformations are exhibited, each based on a different counting scheme for representing the integers, and a combinatorial model is used to show the optimality of many of the transformations. Issues such as online data structures and deletion of elements are also examined. To demonstrate the applicability of these tools, several new data structures that have been developed by applying the transformations are studied.     

Invariants of conformal transformations of almost contact metric structures

The so-called structure tensors of almost contact metric structures, which play a key role in the geometry of almost contact metric structures, are explicitly calculated. The transformations of these tensors under conformal transformations of almost contact metric structures are described. The results obtained are used to study the behavior of the most interesting classes of almost contact structures under conformal transformations.     

Almost Complex Poisson Manifolds

In this paper we consider complex Poisson manifolds and extendthe concept of complex Poisson structure, due to Lichnerowicz to themore general concept of almost complex Poisson structures. Examples ofsuch structures and the associated generalized foliation are given.Moreover, some properties of the complex symplectic structures as wellas of the holomorphic complex Poisson structures are studied.     

Formulação numérica de estruturas compósitas amortecidas utilizando as teorias da Deformação Cisalhante de Primeira Ordem e de Alta Ordem

Composite materials have been used in the design of the aircrafts structures because their low weight and high mechanical strength. However, structures made in composite material are exposed to dynamical and/or static loading environments. Therefore, a major research effort is undertaken in the development of tools numerical for analysis and design of composite structures. This paper presents a numerical formulation of the composite structures using the Finite Element Method (FEM). The damped composite structures, using inserted viscoelastic devices, and undamped composite structures are formulated by FEM. Viscoelastic materials are applied as continuous layers inserted on composite structures. The intrinsic damping of the composite material is included in the studies, too. The First‐order (FSDT) and Higher‐order Shear Deformation (HSDT) theories are formulated. They are distinguished by order of the approximation functions used in the mechanical displacements field. Both theories are computationally implemented using the Serendipity finite element. This is a rectangular finite element with 8 nodes, 5 or 11 degrees of freedom per node. The results are compared with papers predictions. The advantages and disadvantages of using each theory in the modeling of composite (thin or thick) and thick sandwiches structures, including the intrinsic and the viscoelastic damping, are discusses.     

Measurement and comparison of soil structures.

A model is established to describe the structures of tilled soils using Markov chain theory. The effectiveness of the model in describing soil structures, and its accuracy when the model parameters are determined from limited field data is investigated by a consideration of variances of the transition probabilities and Markov chain state occurances in finite length chains. Criteria for correlation of soil structures at small horizontal and vertical displacements are derived, in order to establish distances at which soil structures become effectively independent. In this, a mathematical analysis is made of limiting covariances, generally applicable to the type of Markov chain used in describing these structures, in order to drastically reduce computing time in processing field data. Similarity coefficients are defined from the theory to measure similarity in different soil structures, and are compared in practice.     

Linearization of Nambu Structures

Nambu structures are a generalization of Poisson structures in Hamiltonian dynamics, and it has been shown recently by several authors that, outside singular points, these structures are locally an exterior product of commuting vector fields. Nambu structures also give rise to co-Nambu differential forms, which are a natural generalization of integrable 1-forms to higher orders. This work is devoted to the study of Nambu tensors and co-Nambu forms near singular points. In particular, we give a classification of linear Nambu structures (integral finite-dimensional Nambu-Lie algebras), and a linearization of Nambu tensors and co-Nambu forms, under the nondegeneracy condition.     

Metric spaces are universal for bi-interpretation with metric structures

In the context of metric structures introduced by Ben Yaacov, Berenstein, Henson, and Usvyatsov [3], we exhibit an explicit encoding of metric structures in countable signatures as pure metric spaces in the empty signature, showing that such structures are universal for bi-interpretation among metric structures with positive diameter. This is analogous to the classical encoding of arbitrary discrete structures in finite signatures as graphs, but is stronger in certain ways and weaker in others. There are also certain fine grained topological concerns with no analog in the discrete setting.     

An algebraic and combinatorial approach to the analysis of line drawings of polyhedra

Mathematical structures of line drawings of polyhedral scenes are studied from the viewpoint of scene analysis. First, algebraic structures of line drawings are elucidated, and a necessary and sufficient condition is obtained for a line drawing to represent a polyhedron. Next, combinatorial structures are investigated and the class of pictures that represent nontrivial three-dimensional configurations when vertices are drawn in general position is characterized by incidence structures of polyhedra. The results are furthermore applied to correction of vertex-position errors, discrimination between correct and incorrect line drawings, recognition of unique solvability of some figure-construction problems, classification of line drawings, and other related problems.     

Frobenius–Ehresmann structures and Cartan geometries in positive characteristic

The aim of the present paper is to lay the foundation for a theory of Ehresmann structures in positive characteristic, generalizing the Frobenius-projective and Frobenius-affine structures defined in the previous work. This theory deals with atlases of étale coordinate charts on varieties modeled on homogeneous spaces of algebraic groups, which we call Frobenius–Ehresmann structures. These structures are compared with Cartan geometries in positive characteristic, as well as with higher-dimensional generalizations of dormant indigenous bundles. In particular, we investigate the conditions under which these geometric structures are equivalent to each other. Also, we consider the classification problem of Frobenius–Ehresmann structures on algebraic curves. The latter half of the present paper discusses the deformation theory of indigenous bundles in the algebraic setting. The tangent and obstruction spaces of various deformation functors are computed in terms of the hypercohomology groups of certain complexes. As a consequence, we formulate and prove the Ehresmann–Weil–Thurston principle for Frobenius–Ehresmann structures. This fact asserts that deformations of a variety equipped with a Frobenius–Ehresmann structure are completely determined by their monodromy crystals.     

Symmetries of flat rank two distributions and sub-Riemannian structures

Flat sub-Riemannian structures are local approximations -- nilpotentizations -- of sub-Riemannian structures at regular points. Lie algebras of symmetries of flat maximal growth distributions and sub-Riemannian structures of rank two are computed in dimensions 3, 4, and 5.

     

Subcomplex and sub-Kähler structures

We introduce the notion of subcomplex structure on a manifold of arbitrary real dimension and consider some important particular cases of pseudocomplex structures: pseudotwistor, affinor, and sub-Kähler structures. It is shown how subtwistor and affinor structures can give sub-Riemannian and sub-Kähler structures. We also prove that all classical structures (twistor, Kähler, and almost contact metric structures) are particular cases of subcomplex structures. The theory is based on the use of a degenerate 1-form or a 2-form with radical of arbitrary dimension.     

On inner tensor structures in Cartan spaces

Cartan spaces equipped with almost complex and almost antiquaternion structures are considered. The Hermitian metrics of almost complex Cartan spaces are found. It is proved that in the Cartan spaces there always exist Kählerian metrics. The conditions of the complete integrability of these structures are established, and the method of constructing affine connections consistent with these structures is given.