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Στόχος της παρούσας εργασίας είναι η σύνδεση της μοντέρνας θεωρίας σολιτονίων με την κλασική διαφορική γεωμετρία. Ειδικότερα, αρχίζουμε με ένα εισαγωγικό μέρος, όπου παραθέτουμε τις βασικές έννοιες που αφορούν: α) Τις λύσεις μη-γραμμικών μερικών διαφορικών εξισώσεων (ΜΔΕ) που ονομάζονται σολιτόνια (solitons) και β) Την γεωμετρία των ομαλών καμπυλών και επιφανειών του Ευκλείδειου χώρου). Ακολουθεί, το δεύτερο και κύριο μέρος, στο οποίο μελετάμε την σχέση τριών χαρακτηριστικών μη-γραμμικών εξισώσεων εξέλιξης, της εξίσωσης sine-Gordon, της τροποποιημένης εξίσωσης Korteweg de Vries (mKdV) και της μη γραμμικής εξίσωσης Schrödinger (NLS), με την θεωρία καμπυλών και επιφανειών. Αναλυτικότερα, στο πρώτο μέρος και πιο συγκεκριμένα στο πρώτο κεφάλαιο παρουσιάζουμε μια ιστορική αναδρομή στην έννοια του σολιτονίου. Στην συνέχεια αναζητούμε κυματικές-σολιτονικές λύσεις για τις εξισώσεις KdV και NLS. Κλείνουμε παραθέτοντας τις προϋποθέσεις κάτω από τις οποίες μια μη γραμμική εξίσωση είναι ολοκληρώσιμη. Επιλέγουμε να αναλύσουμε δύο από αυτές τις προϋποθέσεις, χρησιμοποιώντας συγκεκριμένα παραδείγματα, ενώ, για τις άλλες δύο, περιοριζόμαστε σε μια συνοπτική περιγραφή . Στο δεύτερο κεφάλαιο του εισαγωγικού μέρους γίνεται μια εκτενής αναφορά σε θεμελιώδεις έννοιες της διαφορικής γεωμετρίας. Πιο συγκεκριμένα, οι έννοιες αυτές σχετίζονται με την θεωρία καμπυλών και επιφανειών και για ορισμένες από αυτές παρουσιάζουμε κάποια αντιπροσωπευτικά παραδείγματα. Ακολουθεί το κύριο μέρος και ειδικότερα το πρώτο κεφάλαιο, στο οποίο, μελετώντας υπερβολικές επιφάνειες, καταλήγουμε σε ένα κλασικό μη γραμμικό σύστημα εξισώσεων. Είναι αυτό που οφείλουμε στον Bianchi και το οποίο ενσωματώνει τις εξισώσεις Gauss-Mainardi-Codazzi. Στην συνέχεια, περιοριζόμαστε στις ψευδοσφαιρικές επιφάνειες και έτσι καταλήγουμε στην εξίσωση sine-Gordon. Ακολουθεί η ενότητα 1.2, στην οποία βρίσκουμε τον μετασχηματισμό auto-Bäcklund για την εξίσωση sine-Gordon και περιγράφουμε την γεωμετρική διαδικασία για την κατασκευή ψευδοσφαιρικών επιφανειών. Στην ενότητα 1.3, χρησιμοποιώντας τον παραπάνω μετασχηματισμό Bäcklund, καταλήγουμε στο Θεώρημα Αντιμεταθετικότητας του Bianchi. Συνεχίζουμε με την ενότητα 1.4, στην οποία παρουσιάζουμε ψευδοσφαιρικές επιφάνειες, οι οποίες αντιστοιχούν σε σολιτονικές λύσεις της εξίσωσης sine-Gordon. Πιο αναλυτικά, στην υποενότητα 1.4.1 κατασκευάζουμε την ψευδόσφαιρα του Beltrami, η οποία αντιστοιχεί στην στάσιμη μονο-σολιτονική λύση. Στην υποενότητα 1.4.2 μελετάμε το ελικοειδές που δημιουργείται από την έλκουσα καμπύλη, δηλαδή την επιφάνεια Dini, την οποία και κατασκευάζουμε. Ακολουθεί η υποενότητα 1.4.3, όπου, χρησιμοποιώντας το θεώρημα μεταθετικότητας, καταλήγουμε στην λύση δύο-σολιτονίων για την εξίσωση sine-Gordon και συνεχίζουμε με την υποενότητα 1.4.4, όπου κατασκευάζουμε περιοδικές λύσεις των δύο-σολιτονίων γνωστές ως breathers. Στο δεύτερο κεφάλαιο μελετάμε την κίνηση συγκεκριμένων καμπυλών και επιφανειών, οι οποίες οδηγούν σε σολιτονικές εξισώσεις. Ειδικότερα, στην ενότητα 2.1 καταλήγουμε στην εξίσωση sine-Gordon μέσω της κίνησης μιας μη-εκτατής καμπύλης σταθερής καμπυλότητας ή στρέψης. Ακολουθεί η ενότητα 2.2, όπου η εξίσωση sine- Gordon προκύπτει ως η συνθήκη συμβατότητας για το 2 2 γραμμικό σύστημα AKNS. Στην συνέχεια, στην ενότητα 2.3 ασχολούμαστε με την κίνηση ψευδοσφαιρικών επιφανειών. Πιο συγκεκριμένα, στην υποενότητα 2.3.1 συνδέουμε την κίνηση μιας ψευδοσφαιρικής επιφάνειας με ένα μη αρμονικό μοντέλο πλέγματος, το οποίο ενσωματώνει την εξίσωση mKdV. Επιπλέον, στην υποενότητα 2.3.2 δείχνουμε ότι η καθαρά κάθετη κίνηση μιας ψευδοσφαιρικής επιφάνειας, παράγει το κλασικό σύστημα Weingarten. Ολοκληρώνουμε την ενότητα 2.3 με την κατασκευή των μετασχηματισμών Bäcklund τόσο για το μοντέλο πλέγματος, όσο και για το σύστημα Weingarten. Το κεφάλαιο κλείνει με την ενότητα 2.4, όπου μέσω της κίνησης μιας μη εκτατής καμπύλης μηδενικής στρέψης, καταλήγουμε στην εξίσωση mKdV. Στην συνέχεια μελετάμε την κίνηση των επιφανειών Dini και τελικά κατασκευάζουμε επιφάνειες που αντιστοιχούν στο τριπλά ορθογώνιο σύστημα Weingarten. Στο τρίτο και τελευταίο κεφάλαιο επικεντρωνόμαστε στην εξίσωση NLS. Πιο συγκεκριμένα, στην ενότητα 3.1 καταλήγουμε στην εξίσωση NLS μ’ έναν καθαρά γεωμετρικό τρόπο. Επιπλέον, κατασκευάζουμε επιφάνειες, οι οποίες αντιστοιχούν στην μονο-σολιτονική λύση της εξίσωσης NLS και παρουσιάζουμε γι’ αυτές κάποιες γενικές γεωμετρικές ιδιότητες. Το κεφάλαιο 3 ολοκληρώνεται με την ενότητα 3.3 όπου αρχικά λαμβάνουμε ακόμη μια φορά την εξίσωση NLS, χρησιμοποιώντας την μελέτη στην κινηματική των Marris και Passman. Κλείνουμε και αυτό το κεφάλαιο με τον auto- Bäcklund μετασχηματισμό για την εξίσωση NLS και επιπλέον παρουσιάζουμε χωρικά περιοδικές λύσεις της, γνωστές ως smoke-ring (δαχτυλίδι-καπνού). / The aim of this diploma thesis is to find a connection between modern soliton theory and classical differential geometry. More particularly, we begin with an introductory section, where we present the basic concepts regarding soliton equations and the geometry of smooth curves ans surfaces. This is followed by the main body of the thesis, which focuses on three partial differential equations, namely, the sine-Gordon equation, the modified Korteweg de Vries equation (mKdV) and the nonlinear Scrödinger equation (NLS), and their connection to the theory of curves and surfaces. The first introductory chapter is a historical overview of the notion of solitons. We then seek travelling wave solutions for the KdV and NLS equations. Closing, we quote the conditions under which a nonlinear equation is integrable. We choose to analyze in detail two of these conditions while we settle for a brief description of the other two. The second chapter is an extensive report on fundamental concepts of differential geometry, namely, those associated with the theory of curves and surfaces in Euclidean three-dimensional space, and we present some representative examples. Chapter 1 of the main part, opens with the derivation of a classical nonlinear system which we owe to Bianchi and embodies the Gauss-Mainardi-Codazzi equations. We then specialise to pseudospherical surfaces and produce the sine-Gordon equation. Section 1.2 includes the derivation of the auto-Bäcklund transformation for the sine-Gordon equation along with the geometric procedure for the construction of pseudospherical surfaces. In section 1.3, we use the above transformation to conclude to Bianchi’s Permutability Theorem. We continue to section 1.4, where we present certain pseudospherical surfaces. These surfaces correspond to solitonic solutions of the sine- Gordon equation, i.e. in subsection 1.4.1 we construct the pseudosphere which corresponds to the stationary single soliton solution. Also, in subsection 1.4.2 we examine the helicoid that is created by the tractrix, namely, the Dini surface. In section 1.4.3, by use of Bianchi’s Permutability Theorem, we end up in the two-soliton solution for the sine-Gordon equation and continue in the next subsection, where we present periodic two-soliton solutions, known as breathers. In Chapter 2, we show how certain motions of curves and surfaces can lead to solitonic equations. More precisely, in section 2.1, we arrive at the sine-Gordon equation, through the motion of an inextensible curve of constant curvature or torsion. Then, section 2.2 displays how the sine-Gordon equation arises as the compatibility condition for the linear 2 2 AKNS system. In section 2.3 we study the movement of pseudospherical surfaces. In particular, we connect, in subsection 2.3.1, the motion of a pseudospherical surface to a continuum version of an unharmonic lattice model, which encorporates the mKdV equation. Moreover, in subsection 2.3.2, we show that a purely normal motion of a pseudospherical surface produces the classical Weingarten system. We conclude section 2.3 by constructing the Bäcklund transformation both for the lattice model and the Weingarten system. The chapter ends with section 2.4, where through the motion of an inextensible curve of zero torsion, we produce the mKdV equation. Furthermore, we investigate the motion of Dini surfaces and, finally, construct surfaces corresponding to the triply orthogonal Weingarten system. The third and final chapter focuses on the NLS equation. In section 3.1 we produce the NLS equation through a purely geometric manner. We then construct surfaces, that correspond to the single-soliton solution of this equation, and also present certain general geometric properties of them. We conclude the final chapter with the auto-Bäcklund transformation for the NLS equation and the presentation of spatially periodic solutions, known as smoke-ring.

Short interspersed nuclear elements (SINEs) are small non-autonomous and heterogeneous retrotransposons, widespread in animals and plants and usually differentially propagated in related species resulting in genome-specific copy numbers. Within the monocots, the Poaceae (sweet grasses) is the largest and economically most important plant family. The distribution of 24 Poaceae SINE (PoaS) families, five of which showing a subfamily structure, was analyzed in five important cereals (Oryza sativa, Triticum aestivum, Hordeum vulgare, Sorghum bicolor, Zea mays), the energy crop Panicum virgatum and the model grass Brachypodium distachyon. The comparative investigation of SINE abundance and sequence diversity within Poaceae species provides insights into their species‐specific diversification and amplification. The PoaS families and subfamilies fall into two length and structural categories: simple SINEs of up to 180 bp and dimeric SINEs larger than 240 bp. Of 24 PoaS families, 20 are structurally related across species, in particular either in their 5′ or 3′ regions. Hence, reshuffling between SINEs, likely caused by nested insertions of full-lengh and truncated copies, is an important evolutionary mechanism of SINE formation. Most striking, the recently evolved homodimeric SINE family PoaS‐XIV occurs exclusively in wheat (T. aestivum) and consists of two tandemly arranged PoaS‐X.1 copies. Exemplary for deciduous tree species, the evolutionary history of SINE populations was examined in six Salicaceae genomes (Populus deltoides, Populus euphratica, Populus tremula, Populus tremuloides, Populus trichocarpa, Salix purpurea). Four of eleven Salicaceae SINE (SaliS) families exhibit a subfamily organization. The SaliS families consist of two groups, differing in their phylogenetic distribution pattern, sequence similarity and 3’ end structure. These groups probably emerged at different evolutionary periods of time: during the ‘salicoid duplication’ (~ 65 million years ago) in the Salix-Populus progenitor, and during the separation of the genus Salix (~ 45 - 65 million years ago), respectively. Similar to the PoaS families, the majority of the 20 SaliS families and subfamilies share regions of sequence similarity, providing evidence for SINE emergence by reshuffling. Furthermore, they also contain an evolutionarily young dimeric SINE family (SaliS-V), amplified only in two poplar genomes. The special feature of the Salicaceae SINEs is the contrast of the conservation of 5’ start motifs across species and SINE families compared to the high variability of 3’ ends within the SINE families, differing in sequence and length, presumably resulting from mutations in the poly(A) tail as a possible route for SINE elongation. Periods of increased transpositional activity promote the dissemination of novel 3’ ends. Thereby, evolutionarily older motifs are displaced leading to various 3’ end subpopulations within the SaliS families. Opposed to the PoaS families with a largely equal ratio of poly(A) to poly(T) tail SINEs, the SaliS families are exclusively terminated by adenine stretches. Among retrotransposon-based markers, SINEs are highly suitable for the development of molecular markers due to their unidirectional insertion and random distribution mainly in euchromatic genome regions, together with an easy and fast detection of the heterogeneous SINE families. As a prerequisite for the development of SINE-derived inter-SINE amplified polymorphism (ISAP) markers, 13 novel Theaceae SINE families (TheaS-I - TheaS-VII, TheaS-VIII.1 and TheaS-VIII.2, TheaS-IX - TheaS-XIII) were identified in the angiosperm tree species Camellia japonica. Moreover, six Pinaceae SINE families (PinS-I.1 and PinS-I.2, PinS-II – PinS-VI) were detected in the gymnosperm species Larix decidua. Compared to the SaliS and PoaS families, structural relationships are less frequent within the TheaS families and absent in the PinS families. The ISAP analysis revealed the genetic identity of Europe’s oldest historical camellia (C. japonica) trees indicating their vegetative propagation from the same ancestor specimen, which was probably the first living camellia on European ground introduced to England within the 18th century. Historical sources locate the native origin of this ancestral camellia specimen either in the Chinese province Yunnan or at the Japanese Gotō Islands. Comparative ISAPs showed no accordance to the Gotō camellia sample pool and appropriate Chinese reference samples were not available. However, the initial experiments demonstrated the potential of ISAP to resolve variations among natural populations. The ISAP application on angiosperm trees also concerned fast growing Populus clones grown in short rotation coppice plantations for energy production. The species-specific P. tremula ISAP primers might also be applied for the discrimination of hybrid poplar clones involving P. tremuloides genome portions, since SINEs of these two species are highly related. However, due to lineage-specific SINE evolution during speciation, cross-species applications are generally only successful to limited extent. The analysis of poplar hybrids composed of P. maximowiczii with either P. trichocarpa or P. nigra based on P. tremula ISAP primers showed a strongly reduced resolution. In forestry, hybrid larch (e.g. Larix × eurolepis) genotypes have to be selected from the offspring of Japanese (Larix kaempferi) and European larch (Larix decidua) crosses, as they exhibit superior growth rates compared to the parental species. Initial ISAP-based examinations of European larch genotypes provided less polymorphic banding patterns, probably resulting from general high levels of synteny and collinearities reported for gymnosperm species. Hence, the ISAP was combined with the AFLP technique to the novel marker system inter-SINE-restriction site amplified polymorphism (ISRAP). The amplicons originating from genomic regions between SINEs and EcoRI cleavage sites were visualized with the sensitive capillary gel electrophoresis. The ISRAP assays, based on EcoRI adapter primers combined with two different SINE-derived primers, resulted in a sufficient number of polymorphic peaks to distinguish the L. decidua genotypes investigated. Compared to ISAPs, the ISRAP approach provides the required resolution to differentiate highly similar larch genotypes.

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal. / Chez l'humain, près de 30% de la masse génomique est constituée de séquences répétées dispersées qui se sont amplifiées par le mécanisme de rétroposition. Ce processus, présent dans tous les génomes eucaryotes, implique la transcription inverse de l'ARN d'un élément répété et l'intégration de l'ADNc qui en résulte dans une nouvelle localisation génomique. Les "Long Interspersed Elements" (LINE) codent pour les activités spécifiques de la rétroposition, telles que la transcriptase inverse et l'endonucléase. A l'inverse les "Short Interspersed Elements" (SINE) ne codent pour aucune activité enzymatique et sont considérés comme des "satellites" des éléments LINE. Nous avons caractérisé 5 nouvelles familles de rétroposon SINE chez les mammifères. Celles-ci font partie des SINE dérivés d'ARNt et ont, comme caractéristique commune, un domaine central nommé "core". Les régions 3 sont distinctes pour chacune des familles, mais fortement identiques aux extrémités 3' de différents LINE. D'autres séquences SINE possédant ces mêmes critères sont présentes dans les génomes d'oiseaux, de reptiles, de poissons et de céphalopodes. Nous avons ainsi identifié une nouvelle "superfamille" de rétroposon appelée CORE-SINE présente chez tous les vertébrés. L'étude du rôle de chaque segment des CORE-SINE ; région dérivée d'ARNt, "core" et région dérivée de LINE, nous a permis de donner de nouveaux éléments de réponse sur l'évolution des rétroposons dans les génomes eucaryotes. Enfin, nous avons décrit la présence d'un nouvel élément LINE dans les génomes de marsupiaux. Celui-ci est fortement identique au rétroposon Bov-B des génomes bovins et reptiles. Sa présence dans ces différents génomes soulève la possibilité d'un transfert horizontal de cet élément. / Almost 30% of the human genome consists of copies of interspersed repeats that amplified by retroposition, a process widely spread among eukaryotic taxa. Retroposition involves reverse transcription of the transcribed copies and reintegration of the resulting cDNAs into the host genome. Retroposition requires specific activities in addition to the enzymatic machinery commonly found in the host cells. The reverse transcriptase as well as the endonuclease involved in the cDNA synthesis and integration, are coded by the actively retroposing long elements such as LINEs. In contrast, short elements (SINEs) do not encode any protein facilitating their proliferation. However, these elements must have used both host-specific and retroposition-specific activities provided in trans to secure their efficient amplification. We have characterised 5 new SINE retroposon families from mammalian genomes. They belong to tRNA-derived SINEs and have also a common central domain called "core". The 3 end regions of all families are distinct but they display high identity with the 3'extremities of different LINEs. Several SINEs with the same characteristics have been found in bird, reptile, fish, and cephalopod genomes. These data point to the existence of a new "super-family" of SINE retroposons, named CORE-SINE, present in all vertebrate genomes. The study of each CORE-SINE segments, i.e. tRNA-derived region, "core" and LINE-derived region, gave new insight into the evolution of retroposon in eukaryotic genomes. Finally, we also described a new LINE element from marsupial genomes. It presents high identity with the Bov-B element from bovine and reptile genomes, which raises the possibility of a horizontal transfer of this element between genomes.

Chez les mammifères, l'infection par l'adénovirus conduit à l'activation de la transcription de certains éléments SINE Alu. Il s'agit d'un mécanisme conservé car il est observé avec d'autres familles de virus. Adénovirus code pour une protéine, E1A qui est capable d'interagir avec la protéine Rétinoblastome (RB). Cette interaction provoque l'inactivation de RB ce qui provoque probablement l'activation transcriptionelle des éléments Alu. Ainsi, chez les mammifères, certaines protéines virales agissent négativement sur RB ce qui a pour conséquence de déréguler le cycle cellulaire, la transcription pol III de manière générale et la transcription des éléments SINE en particulier. Chez les plantes, l'activation de la transcription des éléments SINE à la suite d'un stress comme l'infection virale reste à démontrer. Cependant, le virus FBNYV possède au sein de son génome une protéine, CLINK, qui est entre autre constituée d'un domaine de liaison à RB similaire à celui retrouvé chez E1A d'adénovirus. La première étape de ce travail de thèse a porté sur l'analyse de la protéine CLINK et plus particulièrement l'effet de cette protéine sur le cycle cellulaire, sur la transcription pol III en général et sur la transcription des SINE endogènes de plantes. La seconde partie de cette étude porte sur la fonction des éléments SINE de plantes au sein de la cellule. Chez les mammifères, l'élément SINE Alu est capable de jouer un rôle dans la physiologie de la cellule en réponse à certains stress. En effet, la transcription de ces éléments est activée à la suite d'une infection par certaines familles de virus. Les transcrits ainsi produits sont alors capables d'interagir avec la protéine PKR, une kinase d'eIF2α. Ainsi, les éléments SINE sont capables d'intervenir dans des processus clefs de la cellule comme le mécanisme de régulation de la traduction. La seule kinase d'eIF2α identifiée chez les plantes est la protéine GCN2. Ainsi, nous avons choisit de caractériser la fonction de cette protéine chez Arabidopsis. Nous avons déterminé les mécanismes de régulation de la protéine en mettant en évidence certains inducteurs spécifiques aux plantes. Ce travail a permis de montrer l'importance de la protéine pour la plante et de découvrir des fonctions potentielles de la protéine dans des voies de stress typiques des végétaux. Enfin, l'impact des SINE de plantes sur l'activité de GCN2 a été analysé.

This thesis is concerned with solitons (solutions of certain nonlinear partial differential equations) in certain cases when the underlying space is either a lattice or curved. Chapter 2 of the thesis is concerned with the outcome of collisions between a kink (a 1-dimensional soliton) and an antikink for certain topological discrete (TD) systems. The systems considered are the TD sine-Gordon and the TD ø(^4) For the TD sine-Gordon system it is found that the kink can support an internal shape mode which plays an important role during the collisions. In particular, this mode can be excited during collisions and this leads to spectacular resonance effects. The outcome of any particular collision has sensitive dependence on the initial conditions and could be either a trapped kink-antikink state, a "reflection" or a "transmission”. Such resonance effects are already known to exist for the conventional discrete ø(^4) system, and the TD ø(^4) system is no different, though the results for the two are not entirely similar. Chapter 3 considers the question of the existence of explicit travelling kink solutions for lattice systems. In particular, an expression for such a solution for the integrable lattice sine-Gordon system is derived. In Chapter 4, by reducing the Yang-Mills equations on the (2 + 2)-dimensional ultrahyperbolic space-time, an integrable Yang-Mills-Higgs system on (2 + 1) dimensional de Sitter space-time is derived. It represents the curved space-time version of the Bogomolny equations for monopoles on R(^3) . Using twister methods, various explicit solutions with gauge groups U(l) and SU(2) are constructed. A multi-solution SU(2) solution is also presented.

Accelerated durability testing of automotive components has become a major interest as it may predict the life characteristics of the vehicle by testing fatigue failure at higher stress level within a shorter period of time. In this work, a specially designed sub-scaled experimental testing bed with the rigid and elastic supports of a simply supported beam was designed and built to compare the effects of the elastic foundation on the change of modal parameters of the tested structure which was later used to tune the FE model. Afterwards, the accelerated loading profiles of both sine sweep and random vibration were applied on the FE model to compare the deviation of the cumulative fatigue damage between the elastic and rigid supports. This work reveals a significant amount of inaccuracy in the current laboratory testing system where the dynamic properties of the tested structure are not maintained close to the real situation. / October 2016

Made available in DSpace on 2016-04-27T16:57:34Z (GMT). No. of bitstreams: 1 Ricardo Ferreira dos Santos.pdf: 3136122 bytes, checksum: e26236b0fe2ca0324d70372150107eea (MD5) Previous issue date: 2014-11-13 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This research forms part of the studies for the use of Mathematical Modeling as a teaching strategy. In her a modeling activity for teaching the sine function is presented. The research had two main objectives: to analyze the effects of mathematical modeling in high school in order to achieve meaningful learning; and evaluate a proposed approach for modeling, through stages and phases. In this second case it was intended to verify that the protoganismo teacher in presenting the phenomenon preserves the desired features of modeling, increase students' interest in mathematics and motivates you to construct new knowledge. The subjects were fifteen students of the second year of high school to a public school in Sao Paulo, with voluntary participation. This is a qualitative research, developed through participant observation. The model used is shown in Curriculum Proposal of the State of São Paulo. The research is guided by conceptions of modeling Beltrão (2009), Bassanezi (2006) and Ausubel learning theory. The activities were developed using as anchor metric relations in right triangle, the coordinates of points on the Cartesian plane and the study of angles in the trigonometric circle. As a result it can be concluded that modeling can be used in basic education such as teaching methodology, brings results for student participation in the construction of their knowledge, but it is not an easy task because it requires changes in teacher pedagogical practice and student to have to take a participatory attitude. The approach of steps and phases was facilitator / Esta pesquisa se insere nos estudos de utilização da Modelagem Matemática como estratégia de ensino. Nela é apresentada uma atividade de modelagem para o ensino da função seno. A pesquisa teve dois objetivos principais: analisar os efeitos de uma modelagem matemática no Ensino Médio com vistas à alcançar uma aprendizagem significativa; e avaliar uma proposta de abordagem para a modelagem, por meio de etapas e fases. Nesse segundo caso pretendeu-se verificar se o protoganismo do professor na apresentação do fenômeno preserva as desejadas características da modelagem, ampliar o interesse dos alunos pela Matemática e motivá-los para a construção de um conhecimento novo. Os sujeitos da pesquisa foram quinze alunos do segundo ano do Ensino Médio de uma escola pública de São Paulo, com participação voluntária. A pesquisa é de natureza qualitativa, desenvolvida por meio da observação participante. O modelo utilizado é o apresentado na Proposta Curricular do Estado de São Paulo. A pesquisa norteia-se nas concepções de modelagem de Beltrão (2009), Bassanezi (2006) e na teoria de aprendizagem de Ausubel. As atividades foram desenvolvidas utilizando como âncora as relações métricas no triângulo retângulo, as coordenadas de pontos no plano cartesiano e o estudo de ângulos na circunferência trigonométrica. Como resultado pode-se concluir que a modelagem pode ser utilizada na Educação Básica como metodologia de ensino, traz resultados para a participação dos alunos na construção de seus conhecimentos, porém não é tarefa fácil, pois exige do professor mudanças em sua prática pedagógica e do aluno por ter que assumir uma atitude participativa. A abordagem de etapas e fases foi facilitadora

The phase and amplitude gradient estimator (PAGE) method has proven successful in improving the accuracy of measured energy quantities over the p-p method, which has traditionally been used, in several applications. One advantage of the PAGE method is the use of phase unwrapping, which allows for increased measurement bandwidth above the spatial Nyquist frequency. However, phase unwrapping works best for broadband sources in free-field environments with high coherence. Narrowband sources often do not have coherent phase information over a sufficient bandwidth for a phase unwrapping algorithm to unwrap properly. In fact, phase unwrapping processing can cause significant error when there is no coherent signal near and above the spatial Nyquist frequency. However, for signals at any frequencies up to the spatial Nyquist frequency, the PAGE method provides correct intensity measurements regardless of the bandwidth of the signal. This is an improved bandwidth over the traditional method. For narrowband sources above the spatial Nyquist frequency, additional information is necessary for the PAGE method to provide accurate acoustic intensity. With sufficient bandwidth and a coherence of at least 0.1 at the spatial Nyquist frequency, a relatively narrowband source above the spatial Nyquist frequency can be unwrapped accurately. One way of using extra information, called the extrapolated PAGE method, uses the phase of a tone below the spatial Nyquist frequency and an assumption of a propagating field, and therefore linear phase, to extrapolate the phase above the spatial Nyquist frequency. Also, within certain angular and amplitude constraints, low-level broadband noise can be added to the field near a source emitting a narrowband signal above the spatial Nyquist frequency. The low-level additive broadband noise can then provide enough phase information for the phase to be correct at the frequencies of the narrowband signal. All of these methods have been shown to work in a free-field environment.

A 4 bit, Rom-Less Direct Digital Frequency Synthesizer (DDFS) is designed in 65nm CMOS technology. Interleaving with Return-to-Zero (RTZ) technique is used to increase the output bandwidth and synthesized frequencies. The performance of the designed synthesizer is evaluated using Cadence Virtuoso design tool. With 3.2 GHz sampling frequency, the DDFS achieves the spurious-free dynamic range (SFDR) of 60 dB to 58 dB for synthesized frequencies between 200 MHz to 1.6 GHz. With 6.4 GHz sampling frequency, the synthesizer achieves the SFDR of 46 dB to 40 dB for synthesized frequencies between 400 MHz to 3.2 GHz. The power consumption is 80 mW for the designed mixed-signal blocks.