Mathematics

The hyperbolic and circular trigonometric functions exhibit periodicity under repeated differentiation. The hyperbolic sine and cosine functions, whilst distinct from their first derivatives, are each equal to their own second derivative. Similarly, the circular sine and cosine functions are each equal to their own fourth derivative. Very little appears in the literature about functions of a real variable that are equal to their own third derivative. The purpose of this note is to define three such functions and to investigate their properties.

Il  testo Epistemologia matematica e Psicologia è la traduzione italiana (a cura di E. Gattico) del XIV volume degli Études d’Épistémologie Génétiqueediti  dal Centre di  studi  fondato da J. Piaget (1896-1980). È la testimonianza  e  il  risultato  di  un  percorso che  ha  legato  i  due  autori  fin  dal  1949, quando  lo  psicologo  svizzero  pubblica  un controverso Traité de logique: essai de logistique opératoiredi cui, tra i più convinti detrattori, vi è proprio il logico matematico olandese E.  W.  Beth  (1908-1964).  Le  sue  critiche serrate  costituiranno  l’inizio  di  una collaborazione  tra  i  due  che  culminerà nella  pubblicazione  a  quattro  mani  (una parte ciascuno, e una conclusione generale in  comune),  nel  1961,  di Épistémologie mathématique et Psychologie. Essi sur les relations entre la logique formelle et la pensée réelle. L’intento del testo è chiaro, ben sviluppato –soprattutto    nella    seconda    parte sull’epistemologia genetica–ed è il motivo per cui tradurlo, riproporlo e rileggerlo ancora oggi,  a  distanza  di  sessant’anni:  la matematica    (ed    equivalentemente,    per questo  discorso,  la  logica)  e  la  psicologia sono  indipendenti e autonome, ma hanno reciprocamente  un  positivo  e  fecondo influsso  (al  quale  le  due  riduzioni  storiche di  “psicologismo  logico”  e  “logicismo psicologico”  non  rendono  merito)  se studiate  dal  punto  di  vista coordinatodell’epistemologia.  Esse  collaborano  nel determinarsi    vicendevolmente    e    più precisamente  i  rispettivi  ambiti  di  validità normativa,    forza    deduttiva,    necessità oggettiva  (logica  matematica)  e  potenza esplicativa,  condotta  fattuale,  genesi  reale (psicologia)  e  si  chiarificano  i  rispettivi “fondamenti”. Sono dunque «complementari»  (p.  433).  È  soprattutto decisivo  il  nuovo  apporto  della  seconda sulla  prima,  ben  riassunto  dalla  metafora “scacchi  =  logica”:  «lo  psicologismo commette  lo  stesso  errore  simile  a  quello di  cui  sarebbe  vittima  un  giocatore,  nel caso    volesse    decidere    quali    siano    i problemi  suscettibili  di  soluzione  o  meno e    come    risolverli,    basandosi    sulle considerazioni  storiche  e  psicologiche  che spiegano  la  formazione  del  gioco  degli scacchi»  (p.  243).  Corretto.  Tuttavia  ciò non  toglie  che:  1)  il  giocatore  di  scacchi accetta il  codice  delle  regole  di  gioco  (e questo    «è    un    fatto    psicologico»);    di conseguenza:  2)  perché/come  le  accetta? Come  il  soggetto  si  appropria  di  regole logiche  che  hanno  validità  autonoma  e non dipendono da lui? Le risposte “usuali” (ha  imparato;  ne  ha  una  conoscenza innata;  le  scopre  per  intuizione...) sembrano insufficienti ancora e richiedono comunque una spiegazione del perché egli le  ritenga valide. Superati  questi  ostacoli  se ne presenta uno nuovo: 3) una volta che le regole    sono    riconosciute    valide    dal soggetto,  come  diventano normative per esso, modificandone il comportamento?L’ epistemologia  genetica  si  mette  al crocevia di tali questioni di validità (logica) e di  fatto (psicologia) cercando  «una  sorta di messa in corrispondenza» (p. 231) tra le strutture    deduttive    dell’attività  logico-matematica  e  le  strutture  causali-genetiche dei    processi    mentali...

Ključni pojmovi: matematička predznanja, očekivana znanja, retencija znanja, raskorak predznanja i očekivanog Pri kreiranju nastavnih programa za matematičke kolegije na visokoškolskim ustanovama polazi se od pretpostavke kako studenti imaju određena znanja i vještine nužne za uspješno usvajanje nastavnih tema sadržanih u tim kolegijima. Često ta pretpostavljena razina znanja i vještina ne odgovara stvarnom znanju studenata, iako se pretpostavke o razini predznanja temelje na nastavnim planovima i programima matematike u srednjoj školi. Je li uzrok tome činjenica da se pretpostavke o razini predznanja možda temelje na nastavnom planu i programu matematike za gimnazi-je ili činjenica da se razine znanja studenata razlikuju bez obzira na nastavni plan i program matematike u srednjoj školi koju su završili, ili oboje? Naime, budući da stu-denti dolaze iz različitih srednjih škola, dio njih nije ni bio u prilici steći zahtijevanu razinu predznanja, a kada i uče po istim nastavnim planovima i programima, praksa pokazuje da se razine njihovih predznanja razlikuju. Svjesni činjenice da često postoji raskorak između očekivanog predznanja stude-nata i njihovog stvarnog znanja, većina visokoškolskih nastavnika nastoji tu razliku samostalno premostiti. Njihova su nastojanja najčešće otežana činjenicom da se na-stavne teme koje bi im to omogućile uglavnom ne nalaze u okviru nastavnih progra-ma, a naknadno ih je nemoguće uvrstiti u zadani plan i program rada. Uz to, grupe studenata na predavanjima i vježbama iz matematičkih kolegija relativno su velike, što dodatno otežava procjenu njihovih predznanja kao i nastojanje da se njihovo zna-nje dovede na potrebnu razinu radi uspješnog nastavka učenja. Nekoliko je načina na koji visokoškolski nastavnici pokušavaju podići razinu predznanja studenata kako ne bi imali teškoća u svladavanju određenih matematičkih 1 Predavanje održano na 8. kongresu nastavnika matematike RH, 2018. godine u Zagreb 2 Nives Baranović, Filozofski fakultet, Sveučilište u Splitu 3 Ivo Baras, Sveučilišni odjel za stručne studije, Sveučilište u Splitu 4 Renata Kožul Blaževski, Sveučilišni odjel za stručne studije, Sveučilište u Splitu

The Royal Cubit is a unit of measurement obtained by the Egyptians, through their knowledge of our solar system, and applied to a geometric figure, called the "pyramid". The Egyptians not only knew the main values of our Earth and other values related to our Solar System, but they also knew how to set this knowledge up inside a monument that would enable us to recognize it by the measurements of the stones which are used to construct this building.

Organizadores: Maria Ivete Basniak, Sergio Rubio-Pizzorno

A obra apresenta e discute diferentes perspectivas teóricas utilizadas em pesquisas da América Latina em Educação Matemática e Tecnologias Digitais, mais especificamente em relação ao GeoGebra.

ISBN: 978-65-5939-047-2 (eBook)

DOI: 10.31560/pimentacultural/2020.472

The present paper tries to discuss and compare the discoveries made in past, about the new branch Geometrical Algebra" of mathematics by ancient mathematicians of different countries of the world. With its historical sketch and draw conclusion in the favor of ancient Hindu savants for its organization.

The book is available as a paperback via Amazon:

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This book is primarily intended for undergraduate students in Mathematics, pursuing courses that feature abstract algebra. All aspects of abstract algebra that you would expect to encounter in an undergraduate programme of study are covered, including ring theory, group theory and the beginnings of Galois theory. Unusually for an abstract algebra text, five chapters on linear algebra are also included, making the text a self-contained introduction to undergraduate algebra. The book will be of use throughout your undergraduate studies, and beyond. Very little is assumed by way of previous study.  It is assumed only that you are familiar with basic topics in algebra, such as complex numbers, matrices, and solving systems of linear equations. From these beginnings, you will be led step by step into the study of various algebraic structures, including groups, vector spaces and rings, and structure-preserving mappings between such objects such as homomorphisms of groups and linear transformations of vector spaces. Important theorems and results are presented and discussed at each stage, with proofs that will enable you to understand the logical development of the subject. Mathematical proof is an aspect of the subject which many students find challenging, and care has been taken to ensure that proofs are easy to follow with all steps clearly explained. In the last six chapters, you will be introduced to more novel topics that go beyond the typical undergraduate curriculum. These include exterior algebras, matrix groups and their associated Lie algebras, normed real algebras and Clifford algebras. Any of these topics could form the basis for a final year undergraduate dissertation, or provide a gentle introduction to graduate study in algebra. Throughout the text, the emphasis is upon presenting topics in an accessible manner, with worked examples which will help you to build a firm understanding of each new concept as it is introduced. Each chapter concludes with a comprehensive set of exercises designed to test and consolidate your understanding of the ideas presented. Solutions to the exercises are given at the end of the book. The book is based on lectures given by Michael Butler at the University of Bolton in the U.K. between 1994 and 2019. These were attended by students from an exceptionally wide range of backgrounds. The lecture notes that this book grew from were refined in the light of teaching and in response to feedback from students, over many deliveries of the material. Dr Butler has extensive experience of teaching abstract algebra to undergraduates, and his passion for teaching the subject has shaped this book.

CONTENTS:
1 The Group Axioms and Examples
2 Subgroups and Group Homomorphisms
3 Vector Spaces
4 Subspaces and Linear Transformations
5 The Basis for a Vector Space
6 Eigenvalues and Eigenvectors
7 Inner Product Spaces
8 Cosets and Quotient Groups
9 Group Actions
10 Simple Groups
11 Soluble Groups
12 Fields and their Extensions
13 The Galois Group
14 The Ring Axioms and Examples
15 Subrings, Ideals and Ring Homomorphisms
16 Quotient Rings
17 Integral Domains and Fields
18 Finite Fields
19 Factorisation in an Integral Domain
20 Vector Spaces with Products
21 The Exterior Algebra of a Vector Space
22 Lie Algebras
23 Matrix Groups and their Tangent Spaces
24 Normed Real Algebras
25 Tensor Products and Clifford Algebras.

Evaluating the color of green coffee beans is an important method for determining their consistency and market price. This measurement is generally carried out by visual inspection or by using conventional instruments with some limitations. The purpose of this thesis was therefore to create a computer vision framework that would produce CIE (Commission International d'Eclairage) L*a*b * measurements of green coffee beans and identify them according to their color. Artificial Neural Networks (ANN) were used as a transformation model was used to classify coffee beans by the Bayes Classifier, in four distinct groups: white, cane green, green and blue-green.

In the ever-expanding field of technology and science, the extensive application of mathematics, statistics and numerical approach is also ever-increasing. The study of literature from past few decades shows that the use of mathematical modelling and numerical techniques provides systematic path to achieve more realistic results. The newly developed areas are employing mathematics and numeric to develop basic model while the existing areas are adopting these tools for optimising the various parameters and thereby improving the performance of the system. Now-a-days there is rare or no field where these study tools are not applied. This paper encompasses the different approaches developed and applied by research community is numerous fields and their achievements in improvement of results.

La pensée épistémologique de René Thom montre deux facettes qui correspondent grossièrement à deux périodes de sa recherche scientifique : (a) un platonisme mathématique qui recherche le corrélat des structures topologiques dans la réalité des phénomènes (la période de 1965 à 1977), et (b) la critique du paradigme galiléen en physique expérimentale et la pertinence d'une philosophie de la nature appliquée à la biologie et à la sémiotique (la période de 1978 à 1990). Les deux positions peuvent être analysées d'une part en partant du livre Stabilité structurelle et morphogenèse de 1972 et de son programme d'une analyse qualitative basée sur la théorie des catastrophes (ensemble avec Christopher Zeeman), d'autre part basé sur le livre de 1988 avec la vision d'une sémiophysique et un programme qui se met à la recherche des forces qui sélectionnent et canalisent la morphogenèse du sens (et qui permettent de constituer une sémantique en linguistique et une sémiotique dynamique). L'épistémologie de Thom peut être interprétée dans le contexte de la tradition philosophique de Leibniz à Kant et Husserl. Jean Petitot explique les rapports tout en considérant la sémiotique de Greimas et les recherches cognitives en linguistique et en neuropsychologie. Ces aspects historiques sont discutés brièvement. Cet article essaie de clarifier et d'évaluer la portée épistémologique des travaux de Thom pour la sémiotique et la linguistique. Il poursuit le but non seulement de faire comprendre cette contribution au débat épistémologique, mais aussi de circonscrire le potentiel épistémologique de la pensée morphodynamique de René Thom pour la sémiotique et la linguistique. Mots-clés : René Thom; théorie des catastrophes; platonisme; philosophie de la nature; prégnance.

According to Quantum Perspective Model, this article researches whether there is a link between the square root of five number and the genetic sequences. At first, the square root digits of the number five after the comma are added respectively. Secondly, the resulting sum corresponds to the nucleotide bases, the results obtained in this way are expressed as nucleotide bases. (A, T, C, G, and U). (A)Adenine, (T)Thymine, (C)Cytosine, (G)Guanine, (U)Uracil. From this point of view, when the first three hundred digits of the square root of the number five after the comma are calculated, the gene sequence is obtained as follows : [ATTTATTCAATACATAACCCCATTGA]. Thirdly, after researching this sequence at NCBI (National Biotechnology Information Center), the search result is similar to bony fishes, especially DANIO RERIO (Zebra fish). Fourthly, the genetic codes of Zebra fishes were found to be similar to human genetic codes. In summary, the connection between these results and the square root of the five in mathematical science and the genetic codes in biochemistry shed light on explaining irrational numbers.