Tiago Mendonça da Costa

Calculating the possible conformations arising from uncertainty in the molecular distance geometry problem using constraint interval analysis

Abstract The calculation of the 3D structure of a protein molecule is important because it is associated to its biological function. Nuclear Magnetic Resonance (NMR) experiments can provide distance information between atoms that are close enough in a given protein and the problem is how to use these distances to determine the protein structure. Using the chemistry of proteins and supposing all the distances are precise values, it is possible to define an atomic order v 1 , ⋅⋅⋅, v n , such that the distances related to the pairs { v i − 3 , v i } , { v i − 2 , v i } , { v i − 1 , v i } are available, and solve the problem iteratively using a combinatorial method, called Branch-and-Prune (BP). However, due to uncertainty in NMR data, the distances associated with pairs { v i − 3 , v i } may not be precise, which implies that there are many difficulties in applying the BP algorithm to this scenario. The use of standard interval arithmetic can be directly applied to the algorithm, but it is known that it generates overestimations. This paper proposes a new methodology to compute possible conformations on the presence of uncertainties arising from NMR distance measurements using a constraint interval analysis approach. Some numerical examples are presented.

Generalized interval vector spaces and interval optimization

This paper presents a method of endowing the generalized interval space with some different structures, such as vector space, topological space, order relations and algebraic calculus.We formulate Interval optimization problems and relate them to classic multi-objective optimization problems.We also present a version of the Mini-maxs Theorem in the interval space context. This paper presents a method for endowing the generalized interval space with some different structures, such as vector spaces, order relations and an algebraic calculus. With these concepts we formulate interval optimization problems and relate them to classic multi-objective optimization problems. We also present a version of the Von Neumanns Mini-max Theorem in the interval context.

Jensen's inequality type integral for fuzzy-interval-valued functions

Abstract This study presents fuzzy versions of Jensen inequalities type integral for convex and concave fuzzy-interval-valued functions. To this end, the concepts of fuzzy inclusion order relation, convexity, and concavity for fuzzy-interval-valued functions are used. Some examples showing the applicability of the theory developed in this study are presented. Since the fuzzy results are obtained through level sets of fuzzy-interval elements, the versions of these results in the interval context are presented here for the first time.

A new approach to linear interval differential equations as a first step toward solving fuzzy differential

Abstract This study uses a concept of interval differentiability, which was recently introduced, to formulate interval initial value problems involving linear interval differential equations. Differently from the approaches that use the gH-differentiability, this study does not make use of a criterion of choice for switching points in order to obtain solutions for such problems. The method herein presented provides solutions in a simple, straightforward, and computationally tractable way. Moreover, these solutions are intuitive because they coincide with the solutions given by a differential inclusion method. The efficiency and practicality of our approach are illustrated through some examples that have appeared in other articles.

A new approach for differentiability of interval-valued functions as first step toward fuzzy differentiability

This manuscript presents a new approach to work with concepts of interval calculus, such as interval differentiability. Moreover, we present the relationship that there exists between our approach to interval differentiability and other interval differentiability concepts known in the literature. We consider that this is a first step toward a new way of formulating fuzzy differentiability.

The Discretizable Molecular Distance Geometry Problem (DMDGP)

We know that to ensure the finiteness of the solution set of the DGP, we can impose an order on the vertices of the associated graph. If such an order exists, it is not hard to find it in the DGP graph.

The Discretizable Distance Geometry Problem (DDGP 3 )

We begin this chapter by describing an important class of the DGP in \(\mathbb{R}^{3}\) having a vertex order as described in Sect. 3.3, called the Discretizable DGP3 (DDGP3). Even though this definition can be extended to \(\mathbb{R}^{K}\) [65], we will consider just the case K = 3.

The Distance Geometry Problem (DGP)

The fundamental problem of DG, as we have previously stated, is to determine all the coordinates of a set points, in a given geometric space, for which some of the distances are known. Depending on the application, these points can represent stars, reachable points for a robot arm, atoms, or people. Each one of these objects can be represented by a vertex of a graph, and if the distance between them is known, we have an edge connecting the correspondent vertices. Formally, we have the following definition of the Distance Geometry Problem (DGP) [57].

From Continuous to Discrete

One approach that has been used to solve the DGP is to represent it as a continuous optimization problem [59]. To understand it, we consider a DGP with K = 2, V = {u, v, s}, E = {{ u, v}, {v, s}}, where the associated quadratic system is