Spectral theory of approximate lattices in nilpotent Lie groups

We consider approximate lattices in nilpotent Lie groups. With every such approximate lattice one can associate a hull dynamical system and, to every invariant measure of this system, a corresponding unitary representation. Our results concern both the spectral theory of the representation and the topological dynamics of the system. On the spectral side we construct explicit eigenfunctions for a large collection of central characters using weighted periodization against a twisted fiber density function. We construct this density function by establishing a parametric version of the Bombieri–Taylor conjecture and apply our results to locate high-intensity Bragg peaks in the central diffraction of an approximate lattice. On the topological side we show that under some mild regularity conditions the hull of an approximate lattice admits a sequence of continuous horizontal factors, where the final horizontal factor is abelian and each intermediate factor corresponds to a central extension. We apply this to extend theorems of Meyer and Dani–Navada concerning number-theoretic properties of Meyer sets to the nilpotent setting.

Finding multi-objective supported efficient spanning trees

This article introduces a new algorithm for computing the set of supported non-dominated points in the objective space and all the corresponding efficient solutions in the decision space for the multi-objective spanning tree (MOST) problem. This algorithm is based on the connectedness property of the set of efficient supported solutions and uses a decomposition of the weight set in the weighting space defined for a parametric version of the MOST problem. This decomposition is performed through a space reduction approach until an indifference region for each supported non-dominated point is obtained. An adjacency relation defined in the decision space is used to compute all the supported efficient spanning trees associated to the same non-dominated supported point as well as to define the indifference region of the next points. An in-depth computational analysis of this approach for different types of networks with three objectives is also presented.

THE ISSUE OF LS-OPTIMIZATION SATELLITE GEODETIC NETWORKS IN THE TERRESTRIAL COORDINATE SYSTEM

The issue of processing GNSS measurements performed to add a new item into a "hard" network is discussed. Two variants of this add: «beam» and «network» are compared. A fragment of the satellite network, which includes five "hard" points of the main network and one defined, acted as the object on which the computational experiment was carried out, testing the theory of this question. The processing of observational materials is implemented in to Excel electronic computational table using an algorithm synthesized version of the parametric version of LS-optimization of spatial data. The «insert» coordinates obtained as a result of this processing and their average square errors were identical for both variants, which is what follows from the theory of LS-optimization of measurement results taking into account of the errors of the coordinates of «hard» points.

A Frobenius–Nirenberg theorem with parameter

The Newlander–Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the complex structure in the complex Euclidean space. We will show two results about the Newlander–Nirenberg theorem with parameter. The first extends the Newlander–Nirenberg theorem to a parametric version, and its proof yields a sharp regularity result as Webster’s proof for the Newlander–Nirenberg theorem. The second concerns a version of Nirenberg’s complex Frobenius theorem and its proof yields a result with a mild loss of regularity.

Parametric Jensen-Shannon Statistical Complexity and Its Applications on Full-Scale Compartment Fire Data

The order/disorder characteristics of a compartment fire are researched based on experimental data. From our analysis performed by new, pioneering methods, we claim that the parametric Jensen-Shannon complexity can be successfully used to detect unusual data, and that one can use it also as a means to perform relevant analysis of fire experiments. Thoroughly comparing the performance of different algorithms (known as permutation entropy and two-length permutation entropy) to extract the probability distribution is an essential step. We discuss some of the theoretical assumptions behind each step and stress that the role of the parameter is to fine-tune the results of the Jensen-Shannon statistical complexity. Note that the Jensen-Shannon statistical complexity is symmetric, while its parametric version displays a symmetric duality due to the a priori probabilities used.

Non-pharmacological motor-cognitive treatment to improve the mental health of elderly adults

SUMMARY OBJECTIVE: To propose a program of physical-cognitive dual task and to measure its impact in Chilean institutionalized elderly adults. METHOD: Experimental design study with pre and post-intervention evaluations, measuring the cognitive and depressive levels by means of the Pfeiffer test and the Yesavage scale, respectively. The program was applied for 12 weeks to adults between 68 and 90 years old. The statistical analysis was based on the nonparametric Wilcoxon test for paired samples and was contrasted with its parametric version. The statistical software R was used. RESULTS: Statistically significant differences were obtained in the cognitive level (p-value < 0.05) and highly significant (p-value < 0.001) in the level of depression with both tests (parametric and nonparametric). CONCLUSION: Due to the almost null evidence of scientific interventions of programs that integrate physical activity and cognitive tasks together in Chilean elderly adults, a program of physical-cognitive dual task was proposed as a non-pharmacological treatment, easy to apply and of low cost to benefit their integral health, which improves significantly the cognitive and depressive levels of institutionalized elderly adults.