Alice Kozakevicius

A Geometria Fractal (GF), uma área matemática emergente, tem se destacado na Educação Básica, conforme indicado por documentos oficiais. Este estudo qualitativo investiga a percepção de futuros professores de Matemática sobre a aplicabilidade da GF no ensino básico. Para isso, foi realizada uma oficina de construção de fractais utilizando origami, integrando conceitos teóricos da GF e do origami. O trabalho também incluiu uma análise dos currículos dos cursos de Licenciatura em Matemática das universidades federais do Rio Grande do Sul. Os resultados indicam a necessidade de incluir a GF nos currículos de formação de professores, destacando a viabilidade de se trabalhar esses conceitos durante a graduação para aprimorar as habilidades de visualização espacial dos educadores. Essa inserção pode enriquecer o ensino de Matemática, oferecendo uma nova perspectiva sobre sua relação com o mundo real. O estudo serve como ponto de partida, ressaltando a necessidade de dar continuidade à investigação sobre a presença da GF na Educação Básica e na formação de docentes, visando a uma compreensão mais integrada e robusta do tema no contexto educacional. A continuidade dessas investigações é crucial para a implementação eficaz da GF na prática pedagógica.

This paper introduces an adaptive filtering process based on shrinking wavelet coefficients from the corresponding signal wavelet representation. The filtering procedure considers a threshold method determined by an iterative algorithm inspired by the control charts application, which is a tool of the statistical process control (SPC). The proposed method, called SpcShrink, is able to discriminate wavelet coefficients that significantly represent the signal of interest. The SpcShrink is algorithmically presented and numerically evaluated according to Monte Carlo simulations. Two empirical applications to real biomedical data filtering are also included and discussed. The SpcShrink shows superior performance when compared with competing algorithms.

In the current work a WENO multiresolution scheme is considered for solving a seven-equation model of two- dimensional two-phase flow problems. A third order interpolating wavelet transform associated to a threshold operator is the main tool for obtaining the sparse representation of the solution at each time step. The spatial operator is solved on the sparse grid by the Lax-Friedrich flux splitting scheme in which the flux derivatives are approximated by the standard WENO scheme. A parallel formulation of the code based on OpenMP allows long term simulations to be obtained with shorter computational times. This type of adaptive parallel scheme can accurately capture the formation of shocks and the evolution of rarefaction waves. As a consequence, more detailed structures of the solution are identified, enriching the understanding of the analyzed two-phase flow phenomena.

Purpose The current work aims to present a parallel code using the open multi-processing (OpenMP) programming model for an adaptive multi-resolution high-order finite difference scheme for solving 2D conservation laws, comparing efficiencies obtained with a previous message passing interface formulation for the same serial scheme and considering the same type of 2D formulations laws. Design/methodology/approach The serial version of the code is naturally suitable for parallelization because the spatial operator formulation is based on a splitting scheme per direction for which the flux components are numerically computed by a Lax–Friedrichs factorization independently for each row or column. High-order approximations for numerical fluxes are computed by the third-order essentially non-oscillatory (ENO) and fifth-order weighted essentially non-oscillatory (WENO) interpolation schemes, assuming sparse grids in each direction. The grid adaptivity is obtained by a cubic interpolating wavelet transform applied in each space dimension, associated to a threshold operator. Time is evolved by a third order TVD Runge–Kutta method. Findings The parallel formulation is implemented automatically at compiling time by the OpenMP library routines, being virtually transparent to the programmer. This over simplifies any concerns about managing and/or updating the adaptive grid when compared to what is necessary to be done when other parallel approaches are considered. Numerical simulations results and the large speedups obtained for the Euler equations in gas dynamics highlight the efficiency of the OpenMP approach. Research limitations/implications The resulting speedups reflect the effectiveness of the OpenMP approach but are, to a large extension, limited by the hardware used (2 E5-2620 Intel Xeon processors, 6 cores, 2 threads/core, hyper-threading enabled). As the demand for OpenMP threads increases, the code starts to make explicit use of the second logical thread available in each E5-2620 processor core and efficiency drops. The speedup peak is reached near the possible maximum (24) at about 22, 23 threads. This peak reflects the hardware configuration and the true software limit should be located way beyond this value. Practical implications So far no attempts have been made to parallelize other possible code segments (for instance, the ENO|-WENO-TVD code lines that process the different data components which could potentially push the speed up limit to higher values even further. The fact that the speedup peak is located close to the present hardware limit reflects the scalability properties of the OpenMP programming and of the splitting scheme as well. Consequently, it is likely that the speedup peak with the OpenMP approach for this kind of problem formulation will be close to the physical (and/or logical) limit of the hardware used. Social implications This work is the result of a successful collaboration among researchers from two different institutions, one internationally well-known and with a long-term experience in applied mathematics for industrial applications and the other in a starting process of international academic insertion. In this way, this scientific partnership has the potential of promoting further knowledge exchange, involving students and other collaborators. Originality/value The proposed methodology (use of OpenMP programming model for the wavelet adaptive splitting scheme) is original and contributes to a very active research area in the past years, namely, adaptive methods for conservation laws and their parallel formulations, which is of great interest for the entire scientific community.

A low-complexity orthogonal multiplierless approximation for the 16-point discrete cosine transform (DCT) was introduced. The proposed method was designed to possess a very low computational cost. A fast algorithm based on matrix factorization was proposed requiring only 60~additions. The proposed architecture outperforms classical and state-of-the-art algorithms when assessed as a tool for image and video compression. Digital VLSI hardware implementations were also proposed being physically realized in FPGA technology and implemented in 45 nm up to synthesis and place-route levels. Additionally, the proposed method was embedded into a high efficiency video coding (HEVC) reference software for actual proof-of-concept. Obtained results show negligible video degradation when compared to Chen DCT algorithm in HEVC.

The main objective of this study was to enhance the performance of sleep stage classification using single-channel electroencephalograms (EEGs), which are highly desirable for many emerging technologies, such as telemedicine and home care. The proposed method consists of decomposing EEGs by a discrete wavelet transform and computing the kurtosis, skewness and variance of its coefficients at selected levels. A random forest predictor is trained to classify each epoch into one of the Rechtschaffen and Kales’ stages. By performing a comprehensive set of tests on 106,376 epochs available from the Physionet public database, it is demonstrated that the use of these three statistical moments has enhanced performance when compared to their application in the time domain. Furthermore, the chosen set of features has the advantage of exhibiting a stable classification performance for all scoring systems, i.e., from 2- to 6-state sleep stages. The stability of the feature set is confirmed with ReliefF tests which show a performance reduction when any individual feature is removed, suggesting that this group of feature cannot be further reduced. The accuracies and kappa coefficients yield higher than 90 % and 0.8, respectively, for all of the 2- to 6-state sleep stage classification cases.

The current work presents a parallel formulation using the MPI protocol for an adaptive high order finite difference scheme to solve 2D conservation laws. Adaptivity is achieved at each time iteration by the application of an interpolating wavelet transform in each space dimension. High order approximations for the numerical fluxes are computed by ENO and WENO schemes. Since time evolution is made by a TVD Runge-Kutta space splitting scheme, the problem is naturally suitable for parallelization. Numerical simulations and speedup results are presented for Euler equations in gas dynamics problems