Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor

The integrability of an m-component system of hydrodynamic type, u _t = V(u)u _x , by the generalized hodograph method requires the diagonalizability of the m ×  m matrix V(u). This condition is known to be equivalent to the vanishing of the corresponding Haantjes tensor. We generalize this approach to hydrodynamic chains—infinite-component systems of hydrodynamic type for which the ∞ ×  ∞ matrix V(u) is ‘sufficiently sparse’. For such systems the Haantjes tensor is well-defined, and the calculation of its components involves finite summations only. We illustrate our approach by classifying broad classes of conservative and Hamiltonian hydrodynamic chains with the zero Haantjes tensor. We prove that the vanishing of the Haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi-Hamiltonian hydrodynamic reductions, thus providing an easy-to-verify necessary condition for the integrability.     

The Liouville integrability of integrable couplings of Volterra lattice equation

A hierarchy of integrable couplings of Volterra lattice equations with three potentials is proposed, which is derived from a new discrete six-by-six matrix spectral problem. Moreover, by means of the discrete variational identity on semi-direct sums of Lie algebra, the two Hamiltonian forms are deduced for each lattice equation in the resulting hierarchy. A strong symmetry operator of the resulting hierarchy is given. Finally, we prove that the hierarchy of the resulting Hamiltonian equations are all Liouville integrable discrete Hamiltonian systems.     

A lattice approach to narrow operators

It is known that if a rearrangement invariant function space E on [0,1] has an unconditional basis then each linear continuous operator on E is a sum of two narrow operators. On the other hand, the sum of two narrow operators in L₁ is narrow. To find a general approach to these results, we extend the notion of a narrow operator to the case when the domain space is a vector lattice. Our main result asserts that the set N_r(E, F) of all narrow regular operators is a band in the vector lattice L_r(E, F) of all regular operators from a non-atomic order continuous Banach lattice E to an order continuous Banach lattice F. The band generated by the disjointness preserving operators is the orthogonal complement to N_r(E, F) in L_r(E, F). As a consequence we obtain the following generalization of the Kalton-Rosenthal theorem: every regular operator T : E → F from a non-atomic Banach lattice E to an order continuous Banach lattice F has a unique representation as T = T_D + T_N where T_D is a sum of an order absolutely summable family of disjointness preserving operators and T_N is narrow. Supported by Ukr. Derzh. Tema N 0103Y001103.     

A unified approach to projective lattice geometries

The interest in pursuing projective geometry on modules has led to several lattice theoretic generalizations of the classical synthetic concept of projective geometry on vector spaces.Introduced in this paper is an approach that is capable of unifying various attempts within a new conceptual frame. This approach reflects algebraic properties from a lattice-geometric point of view. Together with new results we are presenting results from previous publications which have been improved in the frame of this work.     

Generalized curve approach to elementary particles

The theory developed in Ref. 1 is applied in a simplified form to a relativistic one-particle universe, and boundary symmetry conditions are imposed. The existence, for the particle, of a series of resonance states and a series of gauge states is derived. For the proton and the electron, a selection of such states accounts for the elementary particles observed to date.Dedicated to L. CesariThis research was sponsored by the United States Army under Contracts Nos. DAAG29-75-C-0024 and DAAG29-80-C-0041.This work was carried out at IMPA while the first author was visiting there in November 1981.     

A constructive approach to the finite congruence lattice representation problem

A finite lattice is representable if it is isomorphic to the congruence lattice of a finite algebra. In this paper, we develop methods by which we can construct new representable lattices from known ones. The techniques we employ are sufficient to show that every finite lattice which contains no three element antichains is representable. We then show that if an order polynomially complete lattice is representable then so is every one of its diagonal subdirect powers. Received August 30, 1999; accepted in final form November 29, 1999.     

Extending lattice: using generics and methods to implement new visualization methods within the Trellis framework

The lattice add-on package implements Trellis graphics in R. One of the major recent changes to the package API has been to make all high level functions generic, with the traditional implementations available as the “formula” method. This allows for cleaner and more flexible implementations for certain uses that were permitted in the original S-PLUS version on a one-off basis. For example, dotplot could be used to display one-way tables, but the new approach naturally extends to multi-way tables as well. More importantly, it opens up the possibility of new Trellis displays specifically designed for previously unsupported classes. We present some examples of such extensions and describe a few issues generally relevant to the development of new Trellis-style visualizations using the lattice infrastructure.     

Differential Galoisian approach to Jacobi integrability of general analytic dynamical systems and its application

The Morales-Ramis theory provides an effective and powerful non-integrability criterion for complex analytic Hamiltonian systems via the differential Galoisian obstruction. In this paper, we give a new Morales-Ramis type theorem on the meromorphic Jacobi non-integrability of general analytic dynamical systems. The key point is to show that the existence of Jacobian multipliers of a nonlinear system implies the existence of common Jacobian multipliers of Lie algebra associated with the identity c...     

A new approach to free vibration analysis using boundary elements

A boundary element method for the analysis of free vibrations in solid mechanics is developed using a non-standard boundary integral approach. It is shown that, utilizing the statical fundamental solution and employing a special class of coordinate functions, the algebraic eigenvalue problem results. The method has been realized numerically for the two-dimensional elastodynamics, and a number of examples demonstrating its accuracy have been included.     

A new approach to the characterization of the nano-surface structure by using factor analysis

The study provides a new approach to explore the existence of a nanostructure on the benzoic acid-modified glassy carbon electrode (BA-GC) system by applying factor analysis with varimax rotation of factors. The infrared spectra (IR) spectra of the bare glassy carbon electrodes (GC) and the BA-GC electrode systems with benzoic acid-nanofilm (BA-FILM) were obtained by using 10 different surface modification experiments. The IR data of the bare GC, BA-GC and BA-FILM were categorized into three series corresponding to the GC, BA-GC and BA-FILM. The factor analysis method was separately applied to each group to extract the factor loadings. By making use of these loadings, the rotated factor loadings corresponding to the first two factors were computed to estimate the relationship among and within groups. The results provided by the factor analysis method indicate the existence of the BA-FILM on the BA modified glassy carbon electrode.     

Friction dominated dynamics of interacting particles locally close to a crystallographic lattice

A system of particles, in general d‐dimensional space, that interact by means of pair potentials and adjust their positions according to the gradient flow dynamics induced by the total energy of the system is studied. The case when the range of the interaction is of the same order as the mean interparticle distance is considered. It is also assumed that particles, locally, are located close to some crystallographic lattice. An appropriate system of equations that describes the evolution of macroscopic deformation of the crystallographic lattice, as well as the system that describes the evolution of the main crystallographic directions is derived. Well‐posedness of the derived system is studied as well as the stability of the particle system. Same examples of potentials that yield stable and unstable systems are given. Copyright © 2012 John Wiley & Sons, Ltd.     

A new approach to solving the multiple traveling salesperson problem using genetic algorithms

The multiple traveling salesperson problem (MTSP) involves scheduling m > 1 salespersons to visit a set of n > m locations so that each location is visited exactly once while minimizing the total (or maximum) distance traveled by the salespersons. The MTSP is similar to the notoriously difficult traveling salesperson problem (TSP) with the added complication that each location may be visited by any one of the salespersons. Previous studies investigated solving the MTSP with genetic algorithms (GAs) using standard TSP chromosomes and operators. This paper proposes a new GA chromosome and related operators for the MTSP and compares the theoretical properties and computational performance of the proposed technique to previous work. Computational testing shows the new approach results in a smaller search space and, in many cases, produces better solutions than previous techniques.     

Galoisian obstruction to the integrability of general dynamical systems

The Galoisian approach to study the integrability of classical Hamiltonian systems, the so-called Morales–Ramis theory, has been proved to be useful and powerful by many applications. Here, two analogous forms of the Morales–Ramis theory for general dynamical systems both in vector field and mapping forms are given. Galois groups of the corresponding variational equations are studied, and some necessary conditions of the system to possess a certain number of integrals are presented. Several applications are given at last to illustrate our results.