What exactly can we understand by "structural mathematics"? Mathematical structures was a concept which dominated images of mathematics of the twentieth century. It was followed by an effort to transpose the work of Nicolas Bourbaki and his contemporaries into high school education. Progress and results of so-called "modernization of school mathematics" were often received with criticism and hesitation. Using didactical-mathematical concepts of constructivism, languages of mathematics and their various representations and instrumental realism we analyze the principal reasons of that failure. Concurrently we show how it is possible to revive its teaching by broader conception of "structuring" in mathematics. So from mathematical structures we gradually pass to structural mathematics.

Submitted by Franciele Moreira (francielemoreyra@gmail.com) on 2018-03-22T15:32:25Z No. of bitstreams: 2 Dissertação - Flávio Adalberto Gomes - 2018.pdf: 15036236 bytes, checksum: 6f6f7663415c300c3d9a27bd77d12300 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-03-23T14:55:24Z (GMT) No. of bitstreams: 2 Dissertação - Flávio Adalberto Gomes - 2018.pdf: 15036236 bytes, checksum: 6f6f7663415c300c3d9a27bd77d12300 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-03-23T14:55:24Z (GMT). No. of bitstreams: 2 Dissertação - Flávio Adalberto Gomes - 2018.pdf: 15036236 bytes, checksum: 6f6f7663415c300c3d9a27bd77d12300 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-02-21 / This work presents methodology for construction and optimization of mathematical structures that represent experimental data. A hybrid optimization process between an heuristic method and a direct search deterministic method is used. Input and output values of real systems are used and some mathematical expressions are formulated to be used as test and validation. The results are compared with traditional methods, analyzing the oscillations at the edges. The proposed method presents a feasible solution for experimental data analysis and extrapolation, with reduced mathematical expressions. / Este trabalho apresenta metodologia para construção e otimização de estruturas matemáticas que representem dados experimentais. É utilizado processo de otimização híbrido entre método heurístico e método determinístico de busca direta. São utilizados valores de entrada e saída de sistemas reais e são formuladas algumas expressões matemáticas para serem utilizadas como teste e validação. Os resultados são comparados com métodos tradicionais, analisando as oscilações nas bordas. O método proposto apresenta solução viável para análise de dados experimentais e extrapolação, com expressões matemáticas reduzidas.

The thesis covers methods for image segmentation. Fuzzy segmentation is based on the thresholding method. This is generalized to accept multiple criteria. The whole process is mathematically based on the free algebra theory. Free distributive lattice is created from poset of elements based on image properties and the lattice members are represented by terms used by the threshoding. Possible segmentation results compose the equivalence classes distribution. The thesis also contains description of resulting algorithms and methods for their optimization. Also the method of area subtracting is introduced.