The bi-periodic Horadam sequence and some perturbed tridiagonal 2-Toeplitz matrices: A unified approach.

In this note we consider the so-called bi-periodic Horadam sequences. Explicit formulas in terms of Chebyshev polynomials of the second kind and the determinant of some perturbed tridiagonal 2-Toeplitz matrices are established. Several illustrative examples are provided as well.

On the constant coefficients of a certain recurrence relation: A simple proof.

In this note, we provide a short proof for the explicit formulas of the coefficients of a particular 3-term recurrence relation derived from a k-periodic recurrence. Any of the recurrences can be naturally interpreted in terms of determinants of Hessenberg matrices families.

Comment on "Thermal transport in dimerized harmonic lattices: Exact solution, crossover behavior, and extended reservoirs".

In this Comment we recall the notion of a tridiagonal 2-Toeplitz matrix and its spectrum, confronting these results with others obtained recently on exact solutions of thermal transport in dimerized harmonic lattices.

The minimal-ABC trees with B1-branches.

The atom-bond connectivity index (or, for short, ABC index) is a molecular structure descriptor bridging chemistry to graph theory. It is probably the most studied topological index among all numerical parameters of a graph that characterize its topology. For a given graph G = (V, E), the ABC index of G is defined as [Formula: see text], where di denotes the degree of the vertex i, and ij is the edge incident to the vertices i and j. A combination of physicochemical and the ABC index properties are commonly used to foresee the bioactivity of different chemical composites. Additionally, the applicability of the ABC index in chemical thermodynamics and other areas of chemistry, such as in dendrimer nanostars, benzenoid systems, fluoranthene congeners, and phenylenes is well studied in the literature. While finding of the graphs with the greatest ABC-value is a straightforward assignment, the characterization of the tree(s) with minimal ABC index is a problem largely open and has recently given rise to numerous studies and conjectures. A B1-branch of a graph is a pendent path of order 2. In this paper, we provide an important step forward to the full characterization of these minimal trees. Namely, we show that a minimal-ABC tree contains neither 4 nor 3 B1-branches. The case when the number of B1-branches is 2 is also considered.

Greedy Hypervolume Subset Selection in Low Dimensions.

Given a nondominated point set [Formula: see text] of size [Formula: see text] and a suitable reference point [Formula: see text], the Hypervolume Subset Selection Problem (HSSP) consists of finding a subset of size [Formula: see text] that maximizes the hypervolume indicator. It arises in connection with multiobjective selection and archiving strategies, as well as Pareto-front approximation postprocessing for visualization and/or interaction with a decision maker. Efficient algorithms to solve the HSSP are available only for the 2-dimensional case, achieving a time complexity of [Formula: see text]. In contrast, the best upper bound available for [Formula: see text] is [Formula: see text]. Since the hypervolume indicator is a monotone submodular function, the HSSP can be approximated to a factor of [Formula: see text] using a greedy strategy. In this article, greedy [Formula: see text]-time algorithms for the HSSP in 2 and 3 dimensions are proposed, matching the complexity of current exact algorithms for the 2-dimensional case, and considerably improving upon recent complexity results for this approximation problem.

Class of cooperative stochastic models: Exact and approximate solutions, simulations, and experiments using ionic self-assembly of nanoparticles.

We present exact and approximate results for a class of cooperative sequential adsorption models using matrix theory, mean-field theory, and computer simulations. We validate our models with two customized experiments using ionically self-assembled nanoparticles on glass slides. We also address the limitations of our models and their range of applicability. The exact results obtained using matrix theory can be applied to a variety of two-state systems with cooperative effects.

Hypervolume Subset Selection in Two Dimensions: Formulations and Algorithms.

The hypervolume subset selection problem consists of finding a subset, with a given cardinality k, of a set of nondominated points that maximizes the hypervolume indicator. This problem arises in selection procedures of evolutionary algorithms for multiobjective optimization, for which practically efficient algorithms are required. In this article, two new formulations are provided for the two-dimensional variant of this problem. The first is a (linear) integer programming formulation that can be solved by solving its linear programming relaxation. The second formulation is a k-link shortest path formulation on a special digraph with the Monge property that can be solved by dynamic programming in [Formula: see text] time. This improves upon the result of [Formula: see text] in Bader ( 2009 ), and slightly improves upon the result of [Formula: see text] in Bringmann et al. ( 2014b ), which was developed independently from this work using different techniques. Numerical results are shown for several values of n and k.

Penalized likelihood and multi-objective spatial scans for the detection and inference of irregular clusters.

BACKGROUND: Irregularly shaped spatial clusters are difficult to delineate. A cluster found by an algorithm often spreads through large portions of the map, impacting its geographical meaning. Penalized likelihood methods for Kulldorff's spatial scan statistics have been used to control the excessive freedom of the shape of clusters. Penalty functions based on cluster geometry and non-connectivity have been proposed recently. Another approach involves the use of a multi-objective algorithm to maximize two objectives: the spatial scan statistics and the geometric penalty function. RESULTS & DISCUSSION: We present a novel scan statistic algorithm employing a function based on the graph topology to penalize the presence of under-populated disconnection nodes in candidate clusters, the disconnection nodes cohesion function. A disconnection node is defined as a region within a cluster, such that its removal disconnects the cluster. By applying this function, the most geographically meaningful clusters are sifted through the immense set of possible irregularly shaped candidate cluster solutions. To evaluate the statistical significance of solutions for multi-objective scans, a statistical approach based on the concept of attainment function is used. In this paper we compared different penalized likelihoods employing the geometric and non-connectivity regularity functions and the novel disconnection nodes cohesion function. We also build multi-objective scans using those three functions and compare them with the previous penalized likelihood scans. An application is presented using comprehensive state-wide data for Chagas' disease in puerperal women in Minas Gerais state, Brazil. CONCLUSIONS: We show that, compared to the other single-objective algorithms, multi-objective scans present better performance, regarding power, sensitivity and positive predicted value. The multi-objective non-connectivity scan is faster and better suited for the detection of moderately irregularly shaped clusters. The multi-objective cohesion scan is most effective for the detection of highly irregularly shaped clusters.