Penalty functions over a cartesian product of lattices

In this work we present the concept of penalty function over a Cartesian product of lattices. To build these mappings, we make use of restricted dissimilarity functions and distances between fuzzy sets. We also present an algorithm that extends the weighted voting method for a fuzzy preference relation.

Defining Bonferroni means over lattices

In the face of mass amounts of information and the need for transparent and fair decision processes, aggregation functions are essential for summarizing data and providing overall evaluations. Although families such as weighted means and medians have been well studied, there are still applications for which no existing aggregation functions can capture the decision makers' preferences. Furthermore, extensions of aggregation functions to lattices are often needed to model operations on L-fuzzy sets, interval-valued and intuitionistic fuzzy sets. In such cases, the aggregation properties need to be considered in light of the lattice structure, as otherwise counterintuitive or unreliable behavior may result. The Bonferroni mean has recently received attention in the fuzzy sets and decision making community as it is able to model useful notions such as mandatory requirements. Here, we consider its associated penalty function to extend the generalized Bonferroni mean to lattices. We show that different notions of dissimilarity on lattices can lead to alternative expressions.

Negations generated by bounded lattices t-norms

From the birth of fuzzy sets theory, several extensions have been proposed changing the possible membership values. Since fuzzy connectives such as t-norms and negations have an important role in theoretical as well as applied fuzzy logics, these connectives have been adapted for these generalized frameworks. Perhaps, an extension of fuzzy logic which generalizes the remaining extensions, proposed by Joseph Goguen in 1967, is to consider arbitrary bounded lattices for the values of the membership degrees. In this paper we extend the usual way of constructing fuzzy negations from t-norms for the bounded lattice t-norms and prove some properties of this construction.

(S,N)-implications on bounded lattices

Since the birth of the fuzzy sets theory several extensions have been proposed. For these extensions, different sets of membership functions were considered. Since fuzzy connectives, such as conjunctions, negations and implications, play an important role in the theory and applications of fuzzy logics, these connectives have also been extended. An extension of fuzzy logic, which generalizes the ones considered up to the present, was proposed by Joseph Goguen in 1967. In this extension, the membership values are drawn from arbitrary bounded lattices. The simplest and best studied class of fuzzy implications is the class of (S,N)-implications, and in this chapter we provide an extension of (S,N)-implications in the context of bounded lattice valued fuzzy logic, and we show that several properties of this class are preserved in this more general framework.

On fuzzy ideals and fuzzy filters of fuzzy lattices

In the literature there are several proposals of fuzzi cation of lattices and ideals concepts. Chon in (Korean J. Math 17 (2009), No. 4, 361-374), using the notion of fuzzy order relation de ned by Zadeh, introduced a new notion of fuzzy lattice and studied the level sets of fuzzy lattices, but did not de ne a notion of fuzzy ideals for this type of fuzzy lattice. In this thesis, using the fuzzy lattices de ned by Chon, we de ne fuzzy homomorphism between fuzzy lattices, the operations of product, collapsed sum, lifting, opposite, interval and intuitionistic on bounded fuzzy lattices. They are conceived as extensions of their analogous operations on the classical theory by using this de nition of fuzzy lattices and introduce new results from these operators. In addition, we de ne ideals and lters of fuzzy lattices and concepts in the same way as in their characterization in terms of level and support sets. One of the results found here is the connection among ideals, supports and level sets. The reader will also nd the de nition of some kinds of ideals and lters as well as some results with respect to the intersection among their families. Moreover, we introduce a new notion of fuzzy ideals and fuzzy lters for fuzzy lattices de ned by Chon. We de ne types of fuzzy ideals and fuzzy lters that generalize usual types of ideals and lters of lattices, such as principal ideals, proper ideals, prime ideals and maximal ideals. The main idea is verifying that analogous properties in the classical theory on lattices are maintained in this new theory of fuzzy ideals. We also de ne, a fuzzy homomorphism h from fuzzy lattices L and M and prove some results involving fuzzy homomorphism and fuzzy ideals as if h is a fuzzy monomorphism and the fuzzy image of a fuzzy set ~h(I) is a fuzzy ideal, then I is a fuzzy ideal. Similarly, we prove for proper, prime and maximal fuzzy ideals. Finally, we prove that h is a fuzzy homomorphism from fuzzy lattices L into M if the inverse image of all principal fuzzy ideals of M is a fuzzy ideal of L. Lastly, we introduce the notion of -ideals and - lters of fuzzy lattices and characterize it by using its support and its level set. Moreover, we prove some similar properties in the classical theory of - ideals and - lters, such as, the class of -ideals and - lters are closed under intersection. We also de ne fuzzy -ideals of fuzzy lattices, some properties analogous to the classical theory are also proved and characterize a fuzzy -ideal on operation of product between bounded fuzzy lattices L and M and prove some results.

Stretched-exponential behavior and random walks on diluted hypercubic lattices

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

On a refinement of Craig's lattices

Let p be an odd prime. A family of (p - 1)-dimensional over-lattices yielding new record packings for several values of p in the interval [149... 3001] is presented. The result is obtained by modifying Craig's construction and considering conveniently chosen Z-submodules of Q(zeta), where zeta is a primitive pth root of unity. For p >= 59, it is shown that the center density of the (p - 1)-dimensional lattice in the new family is at least twice the center density of the (p - 1)-dimensional Craig lattice. (C) 2010 Elsevier B.V. All rights reserved.

An Extension of Craig's Family of Lattices

Let p be a prime, and let zeta(p) be a primitive p-th root of unity. The lattices in Craig's family are (p - 1)-dimensional and are geometrical representations of the integral Z[zeta(p)]-ideals < 1 - zeta(p)>(i), where i is a positive integer. This lattice construction technique is a powerful one. Indeed, in dimensions p - 1 where 149 <= p <= 3001, Craig's lattices are the densest packings known. Motivated by this, we construct (p - 1)(q - 1)-dimensional lattices from the integral Z[zeta(pq)]-ideals < 1 - zeta(p)>(i) < 1 - zeta(q)>(j), where p and q are distinct primes and i and fare positive integers. In terms of sphere-packing density, the new lattices and those in Craig's family have the same asymptotic behavior. In conclusion, Craig's family is greatly extended while preserving its sphere-packing properties.

Gap solitons in Bose-Einstein condensates in linear and nonlinear optical lattices

Properties of localized states on array of BEC confined to a potential, representing superposition of linear and nonlinear optical lattices are investigated. For a shallow lattice case the coupled mode system has been derived. We revealed new types of gap solitons and studied their stability. For the first time a moving soliton solution has been found. Analytical predictions are confirmed by numerical simulations of the Gross-Pitaevskii equation with jointly acting linear and nonlinear periodic potentials. (c) 2007 Elsevier B.V. All rights reserved.

Three-dimensional solitons in cross-combined linear and nonlinear optical lattices

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Low dimensional behavior in three-dimensional coupled map lattices

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Gap solitons in a model of a superfluid fermion gas in optical lattices

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Localized modes of binary mixtures of Bose-Einstein condensates in nonlinear optical lattices

The properties of the localized states of a two-component Bose-Einstein condensate confined in a nonlinear periodic potential (nonlinear optical lattice) are investigated. We discuss the existence of different types of solitons and study their stability by means of analytical and numerical approaches. The symmetry properties of the localized states with respect to nonlinear optical lattices are also investigated. We show that nonlinear optical lattices allow the existence of bright soliton modes with equal symmetry in both components and bright localized modes of mixed symmetry type, as well as dark-bright bound states and bright modes on periodic backgrounds. In spite of the quasi-one-dimensional nature of the problem, the fundamental symmetric localized modes undergo a delocalizing transition when the strength of the nonlinear optical lattice is varied. This transition is associated with the existence of an unstable solution, which exhibits a shrinking (decaying) behavior for slightly overcritical (undercritical) variations in the number of atoms.

Generalized neighbor-interaction models induced by nonlinear lattices

It is shown that the tight-binding approximation of the nonlinear Schrodinger equation with a periodic linear potential and periodic in space nonlinearity coefficient gives rise to a number of nonlinear lattices with complex, both linear and nonlinear, neighbor interactions. The obtained lattices present nonstandard possibilities, among which we mention a quasilinear regime, where the pulse dynamics obeys essentially the linear Schrodinger equation. We analyze the properties of such models both in connection to their modulational stability, as well as in regard to the existence and stability of their localized solitary wave solutions.