Narrow Results

Based on the study of the dynamics of a dissipation-modified Toda anharmonic (one-dimensional, circular) lattice ring we predict here a new form of electric conduction mediated by dissipative solitons. The electron-ion-like interaction permits the trapping of the electron by soliton excitations in the lattice, thus leading to a soliton-driven current much higher than the Drude-like (linear, Ohmic) current. Besides, as we lower the values of the externally imposed field this new form of current survives, with a field-independent value.

Assuming the quantum mechanical "tight binding" of an electron to a nonlinear lattice with Morse potential interactions we show how electric conduction can be mediated by solitons. For relatively high values of an applied electric field the current follows Ohm's law. As the field strength is lowered the current takes a finite, constant, field-independent value.

We first summarize features of free, forced and stochastic harmonic oscillations and, following an idea first proposed by Lord Rayleigh in 1883, we discuss the possibility of maintaining them in the presence of dissipation. We describe how phonons appear in a harmonic (linear) lattice and then use the Toda exponential interaction to illustrate solitonic excitations (cnoidal waves) in a one-dimensional nonlinear lattice. We discuss properties such as specific heat (at constant length/volume) and the dynamic structure factor, both over a broad range of temperature values. By considering the interacting Toda particles to be Brownian units capable of pumping energy from a surrounding heat bath taken as a reservoir we show that solitons can be excited and sustained in the presence of dissipation. Thus the original Toda lattice is converted into an active lattice using Lord Rayleigh's method. Finally, by endowing the Toda–Brownian particles with electric charge (i.e. making them positive ions) and adding free electrons to the system we study the electric currents that arise. We show that, following instability of the base linear Ohm(Drude) conduction state, the active electric Toda lattice is able to maintain a form of high-T supercurrent, whose characteristics we then discuss.

This work provides a description of the main computational tools for the study of discrete breathers. It starts with the observation of breathers through simple numerical runs, the study uses targeted initial conditions, and discrete breather impact on transient processes and thermal equilibrium. We briefly describe a set of numerical methods to obtain breathers up to machine precision. In the final part of this work we apply the discussed methods to study the competing length scales for breathers with purely anharmonic interactions — favoring superexponential localization — and long range interactions, which favor algebraic decay in space. As a result, we observe and explain the presence of three different spatial tail characteristics of the considered localized excitations.

An interesting sequence of arrays on hierarchical hexagonal grids was employed by PYXIS Innovation Inc. for an efficient digital earth model. These arrays constitute a Cauchy sequence in terms of Hausdorff distance. In this paper, we show that the box-counting dimension for the boundary of the limit of this sequence is .

This paper presents three main ideas. They are the Metatheorem, the lattice embedding for sets, and the lattice embedding for algebras.

The Metatheorem allows you to convert existing theorems about classical subsets into corresponding theorem about fuzzy subsets. The concept of a fuzzyfiable operation on a powerset is defined. The main result states that any implication or identity which can be stated using fuzzyfiable operations is true about fuzzy subsets if and only if it is true about classical subsets.

The lattice embedding theorem for sets shows that for any set X, there is a set Y such that the lattice of fuzzy subsets of X is isomorphic to a sublattice of the classical subsets of Y. In fact it is further proved that if X is infinite, then we can choose Y = X and get the surprising result that the lattice of fuzzy subsets of X is isomorphic to a sublattice of the classical subsets of X itself. The idea is illustrated with an example explicitly showing how the lattice of fuzzy subsets of the closed unit interval 𝕀 = [0,1] embeds into the lattice of classical subsets of 𝕀.

The lattice embedding theorem for algebras shows that under certain circumstances the lattice of fuzzy subalgebras of an algebra A embeds into the lattice of classical subalgebras of a closely related algebra A′. The following sample use of this embeding theorem is given. It is a well known fact that the lattice of normal subgroups of a group is a modular lattice. The embeding theorem is used here to conclude that lattice of fuzzy normal subgroups of a group is a modular lattice too.