A History of Mathematics
... El hombre para comprender y describir las formas que subyacen en la naturaleza, comenzó a dibujar figuras y diseños que permitían modelar las posibles formas y patrones que son parte de un orden o estructura que moldean el tejido físico de la realidad, y dentro de las figuras primordiales se encuentra el triángulo presente en las enseñanzas fundamentales de las civilizaciones antiguas hasta la época actual, representa un símbolo o ideograma que transmite un mensaje que está asociado al altruismo o control; los sumerios, egipcios, chinos, hindúes, judíos y cristianos diseñaron movimiento creacionistas que están arraigados a la historia y al crecimiento social del hombre, desde figuras como el Tetragrámaton, la Estrella de David, el Hombre de Vitruvio, la Tetractys Pitagórica, Teorema de Pitágoras, los Sólidos Platónicos, las grandes Pirámides en distintas partes de la geografía de la Tierra. y hasta en la imprenta de los billetes del Dólar [2] . Cuando el hombre comenzó a formular las características geométricas y algebraicas a través del método científico, esta enigmática figura como paso a dar respuestas para el crecimiento de la sociedad, como la escuadra, ángulos, teoremas, trigonometría, geometría aplicada, en construcción, navegación, ingeniería, arquitectura, astronomía, geodesia y química; ambas concepciones filosofías y matemáticas describieron el comportamiento externo y la interacción de sus cortes en los vértices, que dan soluciones y aportes a desafíos físicos. ...
... Los triángulos son uno de los símbolos más usados y de mayor importancia dentro de la geometría, dentro de las enseñanzas históricas en cualquier civilización y cultura, el triángulo es uno de los más conocidos, teniendo consideraciones especiales y versátiles, siendo el triángulo la primera figura que se puede dibujar usando solamente tres líneas rectas (Figura 3). Es uno de los primeros símbolos que el ser humano fue incorporando a su cultura hasta convertirse en atributo de dioses y reyes [2] . ...
... Triángulo equilatero [2] El triángulo esta asociado con el número 3, expresando divinidad y representando a todos los conceptos que están formados por tres componentes (ancho largo espesor -mineral vegetal animal -padre hijo espiritu santo). El triángulo esta en la base de la formación de cualquier pirámide. ...
El triángulo como figura geométrica compuesta por unión de tres rectas que forman un polígono de tres lados, con tres vértices y tres ángulos [1], ha sido muy estudiada desde la antigüedad por los sumerios, egipcios, chinos, hindúes, hasta llegar a movimientos filosóficos, creacionistas, escribas y matemáticos. Todos estos han dado grandes aportes a partir del estudio científico y metafísico esta enigmática figura que está arraiga a la historia y al crecimiento social del hombre, que se puede ver desde figuras como el Tetragrámaton, la Estrella de David, el Hombre de Vitruvio, la Tetractys Pitagórica, los Sólidos Platónicos, las grandes Pirámides en distintas partes de la geografía de la Tierra y hasta en la imprenta de los billetes del Dólar [2]. Fue en 1654 cuando el matemático francés Blaise Pascal introduce en la publicación (Tratado del triángulo Aritmético), aunque ya esta figura era conocida por los árabes como triangulo de Khayyám (1048-1131), en china como triangulo de Yang Hui (1238 - 1298) o triangulo de Tartaglia (1500-1577); hoy es conocido por Triangulo de Pascal, ya que formulo científicamente el estudio y desarrolló muchas de sus aplicaciones y fue el primero en organizar la información de manera conjunta, la cual se define como una secuencia triangular infinita de números enteros que comienza con uno (1) en el vértice superior y se expande hacia abajo con números calculados a partir de los números de la fila superior [3]; dentro de las aplicaciones se tiene: Palo de hockey, Fibonacci, Expresión Binomio, Combinatoria de subconjuntos, Pentágonos intercalados, Números de Mersen 2n-1, Potencia de 11, Triangulo de Sierpinski, Grilla en grafos Caminos, Números Triangulares, Números tetraédricos y más, Secuencia de Naranya, El Quincunx, Número de Pi (π) y Número de Euler (e) [3]. En esta presentación se estudia el comportamiento finito a través de la ampliación de raíz digital y la Teoría Tesla Zollner, la cual evidencia como dentro de la naturaleza esencial primordial subyacen patrones y figuras geométricas que se definen como Triangulo Fractal Primordial Pitágoras Pascal Tesla Zollner (Figura 1) y la Estrella Tetraédrica David Tetractys Pascal Tesla Zollner (Figura 2), por los modelos basados en las secuencias primordiales 147 – 258 – 369 y un patrón en particular de la secuencia de reproducción celular 124785 , que develan la conexión con el Tetractys Pitagórico y su conexión con la aritmética, geometría y armonía.
... The day-length was assumed to change at a constant rate of one muhurta per month. 2 Thus, the need for a calendar contributed to the development of two fundamental concepts in mathematics. One is periodicity, as in recurring cycles of seasons and day-lengths. ...
... Cubes of chert (a sedimentary rock) have been found in large numbers at Harappa and Mohenjo-Daro. If we use 13.6 g as a unit, their weights run as 1,2,4,10,20,40,100,200,400, 500 and 800 ...
... A case could also be made for the sixth weight, of 27.2 g, which in some places is the most common. If we take 27.2 g as unit, we have the values 1,2,5,10,20,50,100,200,250 and 400 whose integer values from 1 to 100 match the common choices for currency notes in our time. It is also interesting to see the variation from the smallest weight of 0.85 g to the highest weight of 10.9 kg. ...
... Al calcular esta proporción, siempre obtenemos el número de oro, que es una constante que forma parte de las figuras geométricas naturales y muchas creadas por el hombre, incluso en objetos cotidianos como tarjetas de crédito o identificaciones (Figura 1). (2)(3)(4)(5) ...
... Sin embargo, en muchos casos, la naturaleza presenta formas y espacios complejos que escapan a esta descripción. Para entender estos objetos intrincados, la geometría fractal es la herramienta adecuada.Esta ciencia permite analizar espacios y figuras irregulares y complejas, como los árboles, costas, nubes o montañas, que no pueden ser explicados con la geometría lineal de Euclides, basada en rectas, puntos, planos y esferas.(2,4,5,(8)(9)(10)(11)(12)(13) Los fractales se definen matemáticamente como objetos cuyas dimensiones son el resulta-do de una fórmula que usa números complejos, es decir, que tienen una parte real (cualquier número entre 0 y ∞, incluyendo números ra-cionales e irracionales) y una parte imaginaria (números con raíces cuadradas negativas). ...
... Por ejemplo, en la Figura 6 que muestra los cuerpos cavernosos del pene, se pueden medir curvas simples a gran escala (Figura 6A), pero una longitud diferente se obtiene al ver los detalles más pequeños con un microscopio de luz (Figura 6B) y aún una medida distinta con un microscopio electrónico (Figura 6C). Con cada escala más pequeña,la longitud aumenta debido a la infinitud de detalles.(2,6,9,14) ...
The penis is an organ with a unique and complex structure. It is formed by a fractal architecture, meaning its design is based on highly complex repetitive patterns. In addition, its dimensions follow what are known as "golden proportions", also known as "the divine proportion". These proportions are a mathematical relationship that is found in many natural structures and, in the case of the penis, provide stability and improve its performance. This design is not the result of chance, but rather the result of an underlying mathematical message. Fractal geometry, based on complex numbers, is a key tool in understanding the internal structure of the penis, which cannot be described using traditional Euclidean geometry. In addition to fractal geometry, the presence of golden proportions in the penis suggests that its design is influenced by mathematics. To fully understand this relationship, additional studies are required and the creation of innovative mathematical models that include tools such as artificial intelligence. In conclusion, the penis is an example of how mathematics and nature are closely related and how mathematical complexity can improve the functioning of natural structures.
... According to Fuson (1992), culture determines a basis for counting by using a succession of numbers, words, body positions, or gestures that can be linked to different labels to make it more meaningful. For example, in Papua New Guinea, different counting devices and techniques are used by different groups of people (Encyclopaedia of PNG, 1972), and according to Boyer and Merzbach (1991) prehistoric people made number records by cutting notches on a bone or a stick. ...
... Since the time of ancient Egyptians or earlier, there has been documented interest in trying to understand how counting began (Fauvel & Gray, 1987). It has been acknowledged that the principle of counting arose in connection with primitive religious rituals (Boyer & Merzbach, 1991;Everett, 2018;Groza, 1968). In religious ceremonies, it was essential to call the participants in a certain order and counting appears to have been invented to take care of this problem (Boyer & Merzbach, 1991). ...
... It has been acknowledged that the principle of counting arose in connection with primitive religious rituals (Boyer & Merzbach, 1991;Everett, 2018;Groza, 1968). In religious ceremonies, it was essential to call the participants in a certain order and counting appears to have been invented to take care of this problem (Boyer & Merzbach, 1991). Therefore, it is not surprising that the concept of whole numbers is one of the oldest and its origin lies in the prehistoric era. ...
Ethnomathematics is the study of mathematics that takes into consideration the culture in which mathematics arises. It is a subject that values and recognises the contributions of all cultures to the development of mathematics. The aim of this study was to explore the nature of indigenous thinking in the Maldives with respect to counting and measuring that are found in the Maldivian society and are related to traditional and cultural contexts, so that these ideas can be considered for inclusion in future primary mathematics curricula in the Maldives. The fieldwork and data collection was done in the Maldives. Data was collected through interviews with people who do practical work as part of their everyday life, and informal discussions held with historians, mathematicians, mathematics teachers, teacher educators and mathematics students. In total, 91 interviews and informal discussions were conducted. The study also involved the analysis of documents focussed on finding the sources of mathematics, and mathematics currently used in the Maldives. The data from interviews and document analysis show that counting and measuring are in the Maldivian culture even though people may not identify these as mathematics. Cultural contexts in the Maldivian society where counting and measuring are evident include fishing, boat building, building and construction, agriculture, astronomy and navigation, house work, mat weaving, rope making and toddy collecting. The evidence from informal interviews with historians and mathematicians, and document analysis show that initially Arabia and South Asia (mainly India) influenced Maldivian mathematics, and later the Britain. In conclusion, this study identified the Maldivian mathematical ideas related to counting and measuring thereby arguing that mathematics is not culture free. Mathematics exists in every culture even though the way ideas are expressed and emphasised vary from culture to culture.
... Since one cubit is 7 hand's breadths, the result is multiplied by 7 so that the result can be presented in hand's breadths. In contemporary terms, we would say that the scribe was using either similar triangles or an early version of trigonometric ratios (Boyer, 1968;Gillings, 1972). A similar process seems to have been used in two-dimensional draw-ings, except that a drop of six units was used instead of seven (Robins & Shute, 1985). ...
... Ratios were also used by the Hindus. In a book from the late fourth or early fifth century AD, Siddhãntas, a study of the relation between the half chord and the half angle subtended at the center was developed, thus producing our contemporary trigonometric functions (Boyer, 1968). Hindu theoretical astronomy measured radial distances using the same units as the length of the circumference. ...
... Van der Waerden (1983, p. 24) suggested that the Egyptians may have learnt about Pythagorean triples from the Babylonians. In support of this view, Boyer and Merzbach (1991) wrote: "It often is said that the ancient Egyptians were familiar with the Pythagorean theorem, but there is no hint of this in the papyri that have come down to us" (p. 17). ...
... Notice that gravitation's Newtonian description as a vectorial force was also a geometric concept, namely, a directed arrow in Euclidian space with a further notion of size (strength) attached.6 For an illuminating account of the experimental and now conventional theoretical foundations of quantum mechanics, see Chapters 1 and 2 of this reference.7 The inner product effectively turns in principle shapeless state vectors into arrows, since geometric magnitudes demand geometric objects. ...
This article explores the overall geometric manner in which human beings make sense of the world around them by means of their physical theories; in particular, in what are nowadays called pregeometric pictures of Nature. In these, the pseudo-Riemannian manifold of general relativity is considered a flawed description of spacetime and it is attempted to replace it by theoretical constructs of a different character, ontologically prior to it. However, despite its claims to the contrary, pregeometry is found to surreptitiously and unavoidably fall prey to the very mode of description it endeavours to evade, as evidenced in its all-pervading geometric understanding of the world. The question remains as to the deeper reasons for this human, geometric predilection--present, as a matter of fact, in all of physics--and as to whether it might need to be superseded in order to achieve the goals that frontier theoretical physics sets itself at the dawn of a new century: a sounder comprehension of the physical meaning of empty spacetime.
... In 1824, the Norwegian mathematician Neils Abel (1802-1829) proved that it is impossible to solve the general equation of the fifth degree in terms of radicals, closing the door on further exploration in this direction. Around the same time, a French mathematician, Evarist Galois (1811-1832), extended this proof to all degrees greater than five [4]. ...
... The search for effective and elegant methods to find the roots of these equations has been an area of constant interest. According to [1], in the year 1545 the solution not only to cubic equations, but also to quartic equations, became widely known with the publication of "Ars Magna" by Girolamo Cardano (1501-1576). ...
Este artigo apresenta um método alternativo para resolver equações polinomiais de quarto grau. Embora tal resultado já tivesse sido almejado há algum tempo, ainda no século XVI, pelo matemático italiano Lodovico Ferrari, este trabalho ganha originalidade por estar relativamente fora de outros métodos discutidos anteriormente. Neste trabalho, apresentaremos dois teoremas originais e dois corolários. Começaremos introduzindo um modelo especial de polinômio de quarto grau, que permite visualizar todas as suas raízes de maneira clara. Em seguida, demonstraremos o resultado principal deste estudo: a capacidade de converter qualquer polinômio genérico de quarto grau em um formato especial, facilitando assim a identificação de suas raízes. Este método oferece uma perspectiva diferente na resolução de equações polinomiais complexas, proporcionando uma estrutura clara e sistemática para lidar com problemas que desafiaram métodos convencionais. Por fim, serão apresentados exemplos práticos que ilustram a aplicação deste método. Espera-se que este resultado possa servir de inspiração e base para trabalhos futuros que abordem este tema na matemática contemporânea.
... Once the flow calculation in the MLN is completed, the determination of the corresponding CF can be easily evaluated using Guldin's theorem (Merzbach, Boyer, 2011) for the distribution of P/P0 along the nozzle wall. ...
The aim of this work is to develop a numerical calculation program to determine the first critical buckling force of the axisymmetric supersonic MLN in the adaptation regime by the energetic method for the needs of the future aerospace construction. After designing the MLN by the HT MOC, the contour of the nozzle will be determined point by point meeting the requirements for eliminating thrust losses. The design essentially depends on the exit Mach number and the stagnation temperature of the combustion chamber. In this first step, the distribution of the wall pressure ratio at each internal point along the internal wall of the nozzle, as well as the thrust force are determined. Its resultant necessarily applied to the thrust center in relation to the throat of the nozzle, which must also be determined in order to assimilate the nozzle as being a beam of variable section embedded in the throat and free at the exit of the nozzle. The section of the beam is of the thin-walled circular shape having a constant thickness across the longitudinal axis and constructed of a same material. Using the energy method, one can numerically determine the value of the first critical buckling force of the nozzle corresponding to the data. The obtained integrals are calculated by the Simpson method. The cubic spine interpolation is used to interpolate the buckling deformation in eatch iteration. A new dimensionless coefficient of the critical force is introduced. We also consider a new other buckling coefficient λ as being the Logarithm of the ration between the critical buckling force under the thrust force, making it possible to directly indicate the existence or absence of buckling in the nozzle during the flow in the nozzle. Then if λ>0.0, the buckling phenomenon is absent in the flow, and it is present if λ≤0.00. Truncation of the nozzle is considered in order to see the possibility of increasing the critical buckling force. The application is made for Air. An example of the structural sizing of the nozzle to avoid the buckling phenomenon is considered in this study.
... Despite the Greek origins of the word "parameter", its history is much more recent. The concept (not the word) of a parameter arises first in the work of Jordanus Nemorarius (1225-1260), who started working with letters (as opposed to specific numbers) thereby being able to treat larger sets of cases to which the same principles can be applied (Boyer & Merzbach, 2011). The word "parameter" has probably been introduced by French mathematician Claude Mydorge (1585-1648), who used it to refer to the latus rectum of a parabola (Harris, 1708;Sugimoto, 2013). ...
Psychometrics and quantitative psychology rely strongly on statistical models to measure psychological processes. As a branch of mathematics, geometry is inherently connected to measurement and focuses on properties such as distance and volume. However, despite the common root of measurement, geometry is currently not used a lot in psychological measurement. In this paper, my aim is to illustrate how ideas from non-Euclidean geometry may be relevant for psychometrics.
... Additionally, examples from the history of mathematics help students better understand mathematical concepts and relate them to real-world applications. There are several important reasons why the history of mathematics is included in textbooks: (Boyer & Merzbach, 2011) • Understanding the historical context: Covering the history of mathematics provides an important perspective for understanding how mathematical concepts develop and evolve. By understanding the historical context of mathematical thought, students can see how mathematics has grown, changed, and shaped itself into its current form. ...
Bu araştırmanın temel amacı, Yunanistan ve Türkiye'deki 6. sınıf matematik ders kitaplarında matematik tarihine yer verilmesinin karşılaştırmalı olarak analiz edilmesidir. Bu amaçla araştırma nitel araştırma yöntemi esas alınarak yürütülmüş ve doküman analizinden yararlanılmıştır. Araştırmanın veri kaynakları 2020-2021 eğitim-öğretim yılında Yunanistan ve Türkiye'deki devlet okullarında okutulan 6. sınıf matematik ders kitaplarıdır. Araştırmanın verileri içerik analizi yöntemiyle analiz edilmiştir. Araştırmanın bulgularına göre, Yunanistan 6. sınıf matematik ders kitabındaki matematik tarihini içeren madde sayısı, Türkçe 6. sınıf matematik ders kitabından niceliksel olarak daha fazladır. Ayrıca Yunanistan 6. sınıf matematik ders kitabında "Geometrik" öğrenme alanında matematik tarihine odaklanırken, Türkçe 6. sınıf matematik ders kitabında "Sayılar ve İşlemler" konusuna odaklanmaktadır. Ancak her iki ülkede de matematik tarihi içeriklerinin benzer olduğu ve ders kitaplarında farklı bölümlerde matematik tarihine yer verildiği görülmüştür.
... Reality and personal existence are made up of the struggle for or balance between these opposites. The Greeks viewed an educated person as one who possessed the wisdom harmony brings (Boyer & Merzbach, 2011). One is in tune with the universe and its forces. ...
The purpose of this study is to investigate how learning theories and philosophical thought have changed from pre-modern to post-modern thought. A paradigm change is always essential to the development of a curriculum, education as well as nation. I used document analysis method to comprehensively analyze various philosophical and learning theories about mathematics, revealing the emergence of a new paradigm derived from mathematics education. It is critically investigated that the reform movement of the last century shared a common perspective on mathematics learning and knowledge. The postmodern paradigm has replaced the modern paradigm, and philosophy and theology will eventually evolve to reflect this shift. Social science as well as the humanities, management, literature, mathematics, and philosophy have all been influenced by postmodern ideas. The paradigm shift in mathematics education is a transition from passive memorization to critical thinking, problem-solving techniques, and active participation. In order to improve relevance and motivation, it entails embracing technology as a tool for inquiry and discovery, cultivating collaborative learning settings, and incorporating real-world applications. This change is a reflection of a wider understanding of the need to give followers the adaptable abilities they need to prosper in a world that is changing quickly due to complexity and innovation.
... In the 16th century, Girolamo Cardano provided a rule for solving systems of two linear equations, known as the Rule of Cramer for 2 × 2 [54] systems, essentially Cramer's rule. This method led to the definition of determinants. ...
Quaternions, discovered by Sir William Rowan Hamilton in the 19th century, are a significant extension of complex numbers and a profound tool for understanding three-dimensional rotations. This work explores the quaternion's history, algebraic structure, and educational implications. We begin with the historical context of quaternions, highlighting Hamilton's contributions and the development of quaternion theory. This sets the stage for a detailed examination of quaternion algebra, including their representations as complex numbers, matrices, and non-commutative nature. Our research presents some advancements compared to previous educational studies by thoroughly examining quaternion applications in rotations. We differentiate between left and right rotations through detailed numerical examples and propose a general approach to rotations via a theorem, clearly defining the associated morphism. This framework enhances the understanding of the algebraic structure of quaternions. A key innovation is presenting a three-dimensional example illustrating the rotation of a frame with strings, connecting quaternions to the quaternion group, half-integer spin phenomena, and Pauli matrices. This approach bridges theoretical concepts with practical applications, enriching the understanding of quaternions in scientific contexts. We emphasize the importance of incorporating the history and applications of quaternions into educational curricula to enhance student comprehension and interest. By integrating historical context and practical examples, we aim to make complex mathematical concepts more accessible and engaging for students at the undergraduate and graduate levels. Our study underscores the enduring relevance of quaternions in various scientific and technological fields and highlights the potential for future research and educational innovations.
... Additionally, examples from the history of mathematics help students better understand mathematical concepts and relate them to real-world applications. There are several important reasons why the history of mathematics is included in textbooks: (Boyer & Merzbach, 2011) • Understanding the historical context: Covering the history of mathematics provides an important perspective for understanding how mathematical concepts develop and evolve. By understanding the historical context of mathematical thought, students can see how mathematics has grown, changed, and shaped itself into its current form. ...
The main purpose of this research is to comparatively analyze the inclusion of history of mathematics in 6th grade mathematics textbooks in Greece and Türkiye. For this purpose, the research was conducted on the basis of qualitative research method and document analysis was used. The data sources for the research are 6th grade mathematics textbooks taught in public schools in Greece and Türkiye in the 2020-2021 academic years. The data of the research were analyzed by the content analysis method. According to the findings of the research, the number of items that include the history of mathematics in the Greek 6th grade mathematics textbook is quantitatively higher than the Turkish 6th grade mathematics textbook. In addition, while Greece focuses on the history of mathematics in the "Geometric" learning area in the 6th grade mathematics textbook, Turkish focuses on "Numbers and Operations" in the 6th grade
mathematics textbook. However, it has been observed that the contents of the history of mathematics in both countries are similar, and the history of mathematics is included in different sections of the textbooks
... (For a fascinationg account of the life and work of Whish see [12].) These achievements are now well understood and are widely known thanks to the efforts of scholars more than a century after the publication of Whish's paper(see, for example, [9], [10], [13] Ch. 7, [26] p.202, [28]). ...
This paper is intended to serve two purposes: one, to present an account of the life of Sangamagr\=ama M\=adhava, the founder of the Kerala school of astronomy and mathematics which flourished during the 15th - 18th centuries, based on modern historical scholarship and two, to present a critical study of the three enigmatic correction terms, attributed to M\=adhava, for obtaining more accurate values of while computing its value using the M\=adhava-Leibniz series. For the second purpose, we have collected together the original Sanskrit verses describing the correction terms, their English translations and their presentations in modern notations. The Kerala rationale for these correction terms are also critically examined. The general conclusion in this regard is that, even though the correction terms give high precision approximations to the value of , the rationale presented by Kerala authors is not strong enough to convince modern mathematical scholarship. The author has extended M\=adhava's results by presenting higher order correction terms which yield better approximations to than the correction terms attributed to M\=adhava. The various infinite series representations of obtained by M\=adhava and his disciples from the basic M\=adhava-Leibniz series using M\=adhava's correction terms are also discussed. A few more such series representations using the better correction terms developed by the author are also presented. The various conjectures regarding how M\=adhava might have originally arrived at the correction terms are also discussed in the paper.
... To face these plagues, scientists have developed various approaches and techniques, such as meteorology [194], seismology [170], medicine [33], nanotechnology [82], cybernetics [7], computer science [30], and the development of powerful telescopes like Hubble and James Webb [72]. All this required the development of an appropriate theoretical arsenal and the adaptation of several disciplines, such as biophysics, biomathematics [145], bioinformatics [68], astrophysics [32], epidemiology [160], geology [155,9], mathematics [22], electronics [40], chemistry [17], mechanics [99], and quantum mechanics [123]. As a result, the 21st century has seen an unprecedented increase in our ability to understand and predict natural phenomena [174], which has allowed, among other things, to face the challenges of global climate change, pandemics, ... and thus to reduce the vulnerability of our species and our environment [139,105]. ...
This dissertation is a comprehensive interdisciplinary investigation that leverages optimal control theory to address complex issues across multiple fields, including air navigation, epidemiology, dynamic systems, and forest fire management.
The first chapter, titled "Mathematical Preliminaries," lays the groundwork for the thesis by providing a thorough review of mathematical concepts and tools essential for understanding the subsequent chapters. This chapter establishes the theoretical foundation necessary for approaching the practical applications that follow.
The second chapter is dedicated to air navigation, featuring two main articles. The first explores risk mitigation strategies for bird-aircraft collision avoidance, investigating control strategies both on the ground and in the air. The second article examines the management of panic in flight using a discrete model inspired by the SIRS epidemiological model.
The third chapter addresses epidemiological issues using a continuous SIR model. This model is utilized to develop control strategies to minimize the human and financial impact of epidemics. An additional article in this chapter extends the analysis by introducing awareness programs and health interventions.
The fourth chapter focuses on sensitivity analysis in linear discrete-time fractional-order dynamic systems. The aim is to characterize the gain matrices that make the system robust against uncertainties in initial conditions.
Finally, the fifth and last chapter proposes an innovative method for combating forest fires by controlling the spread of dry grass in a forest, considering factors such as wind intensity and direction to develop effective control strategies.
In summary, this thesis is not merely an academic exercise but aims to solve real and diverse problems using advanced mathematical techniques. It offers pragmatic solutions to concrete challenges and lays the groundwork for future research in each of the studied fields.
... Se considera que la filosofía de Ockham sentó las bases para la ciencia experimental (Hooykaas, 2000;Harrison, 2002;Riddle, 2008;Merzbach, Boyer, 2011). Entre los seguidores de Ockham están Jean Buridán (el autor del concepto del ímpetus en la mecánica, es decir, la conservación por un objeto del movimiento que le es impartido, es decir-el re-enfoque del conocimiento hacia las causas activas en lugar de las causas finales, aplicado a cuestiones tales como si son los polluelos la causa de que los pájaros construyan nidos, o de que si una planta es la causa de los procesos que ocurren en la semilla), Thomas Bradwardine (con obras sobre geometría), William Heytesbury y Richard Swineshead (con el teorema de la velocidad media), Nikolás Oresme (con el descubrimiento de la velocidad instantánea), Alberto de Sajonia (con la teoría del ímpetus) y Nicolás de Autrecourt (con el atomismo). ...
[...] durante la Edad Media no fue creada una concepción de algún sistema taxonómico (tal como lo entendemos ahora), pero se reprodujo de manera estable —para describir diferentes objetos— una estructura jerárquica arborescente de conceptos dispuestos de forma dicotómica. Los rangos en esta jerarquía eran relativos y el lugar de una misma cosa podía cambiar dependiendo de cómo o con qué definiciones se explicara la cosa, y con qué otras cosas se comparaba. El término “jerarquía” apareció de manera explícita en los siglos V-VI en las obras de Dionisio Areopagita. En el sistema de Areopagita se propuso una jerarquía con rangos absolutos y estos rangos absolutos desde entonces a veces se utilizaron en la presentación de algunos sistemas jerárquicos, especialmente sistemas que describen la jerarquía eclesiástica. La idea de los rangos, de los niveles de descripción de la realidad, surgieron en la escolástica en diferentes áreas, en particular—al desarrollar el metalenguaje de la cognición. Por primera vez surgió y se volvió cada vez más común la cuestión de la lista completa de cualquier tipo de objetos naturales, de los organismos vivos. Una de las primeras menciones de la lista completa (un catálogo completo) en relación con la enumeración de los seres vivos puede ser hallada en la obra de Averroes. Al final de la Edad Media, la disputa entre realistas y nominalistas terminó con la victoria de los nominalistas, y esta rama de la escolástica (los franciscanos) formó la base que creó la atmósfera espiritual y una serie de conceptos funcionales que fueron utilizados para describir la realidad de la ciencia emergente.
We define a system in two ways: using a general definition as a set of parts and their relationships and a more specific definition as a set of interacting agents. The latter definition is helpful in self-organizing systems, called complex adaptive systems. We construct a roadmap and vision, usually for the next ten years. It is then easier to define the research problems and the actual research. The emergence phenomenon in complex systems makes reduction and deduction impossible, and the systems are analytically intractable. There are some limitations for human knowledge and fundamental limits of nature. We present a hierarchy of core sciences and information technology.
Knowledge of history belongs to the general knowledge of all researchers and is, therefore, essential for understanding the present and envisioning the future. We present the history of information retrieval and the writing of scientific papers. Many modern concepts were invented before the scientific revolution of the 1600s, but their significance was not widely understood. Galileo, Francis Bacon, Rene Descartes, and Isaac Newton developed the modern analytical thinking combining rationalism and empiricism. The newest idea is the system concept of the last century by Ludwig von Bertalanffy, Norbert Wiener, Kenneth E. Boulding, and many others since about 1950. Analytical thinking is about 400 years old, but systems thinking is about 75 years old.
We describe the whole research process using the conventional analytical thinking. Research is an advanced learning process that usually proceeds inductively bottom-up and from inside out. However, we present the results deductively top-down and from outside in. Research is different from development and innovation. Researchers must have specific properties, including independence and creativity, and follow high ethical standards. We divide research into the formation of concepts and theories. Conceptual analysis starts any research: we must understand the concepts and their definitions and relationships. The formation of theories consists of experimental-abductive and hypothetico-deductive methods, corresponding discovery, and verification or falsification.
The purpose of this paper is to develop a new numerical calculation program allowing determining the position of the gravity ( x GC ) and thrust ( x TC ) centers of an axisymmetric supersonic nozzle, aiming to relate them, and with the nozzle pressure ratio ( NPR ). However, the x TC changes position with the change in NPR and x GC does not change with the flow. First, one designs the nozzle to determine numerically point by point its contour, and P / P 0 distribution along the wall, giving a uniform flow and parallel to the exit section. One made the design by HT MOC . One made the distribution of P / P 0 in the case of adaptation. The position of x TC and x GC are determined with the variation of exit Mach number ( M E ), stagnation temperature ( T 0 ), nozzle throat ratio ( λ ), the truncate of the nozzle, and the effect of NPR . The x GC changes also with M E , T 0 , λ , truncate of the nozzle, and do not change with NPR . One made application on MLN given its current aerospace application, and with BPN and DEN , aiming to make a comparison and improvement for our latest DEN . One has done the validation of the results by calculating x GC of BPN central body, where it is on the middle of the nozzle according to the exact solution. For the x TC , one made the validation for a nozzle having a constant section and P / P 0 distribution, where it is on the middle of the nozzle, in accordance with the exact solution. One makes the application for air.
This chapter elaborates on the integral role mathematics plays in shaping sustainable solutions for our evolving world. The mathematical significance and historical development of optimization and sustainability have been discussed. A classification of basic optimization problems has been provided. Emphasizing the symbiotic relationship between optimization and sustainability, it has been discussed how mathematical frameworks, from linear programming to stochastic modeling, provide tools for informed decision-making. Such models not only aim for optimal outcomes but also embed long-term viability. As global challenges amplify, including unpredictable environmental shifts, emerging methodologies, particularly at the intersection of machine learning and optimization, become crucial. Through exploring various mathematical concepts, their applications in sustainability, and future research directions, this article underscores the profound significance of mathematics in charting a sustainable path forward.
Artificial and natural computations and their scope are highlighted. Stressed are the finite- and the infinite-precision computations and their precise differences vis-a-vis the eternity of Nature and her laws along with that of computations by living and non-living entities. Further, the importance of the present day computational scenario is discussed in comparison with that of air and water for the existence of any form of life anywhere and its evolution.
Extensive cartometric analyses were performed on the portolan atlases of Pietro Vesconte (1313), Andrea Bianco (1436), and Battista Agnese (1538), along with their manually assembled composites and contemporaneously made portolan charts. The results indicate that none of the small-scale atlas sheets were fully aligned with the contemporary magnetic north, that they typically exhibit coastline contours mapped to different scales, and that the spatiums (50 miglia intervals) in Vesconte's and Bianco's atlases differ in length subsequent to the assembly of their composites. Consequently, the atlas sheets could not provide sailors with consistent navigational accuracy across the whole displayed area. Moreover, notable geometric differences can be observed between the composite of Vesconte's atlas and his 1311 portolan chart, as well as between the composite of Bianco's atlas and his smaller-scale chart bound in the same atlas, indicating that their creators had a limited understanding of their geometry. The discovery of cartometrically determined underlying mosaics of subsections encapsulated within their composites with accuracy twice as great—whose spatial extents have remained almost unchanged during more than two centuries of their production and geometrically correspond to portolan charts created before them—suggests that portolan atlases are medieval and early modern copies of earlier-made cartographic sources.
This research examines a 58-year, 202 sequential issues, longitudinal published record of the Ontario Mathematics Gazette. The publication is oriented to K–12 teachers of mathematics in Ontario. A total of 3609 units of analysis were identified that led to 447 unique index categories. A chi-squared goodness of fit test found, within the limits of the test, that these categories arise continuously, independently, and at a steady rate. The frequency of occurrence in both absolute and relative terms showed a few trends, such as the shift of curriculum development away from teachers and toward research findings. A key finding was the absence of significant biases or short-term trends within the publication’s history. This speaks to the association being an independent voice, but the breadth of topics indicates it is an important support for teachers.
In 1928 Henry Scudder described how to use a carpenter's square to trisect an angle. We use the ideas behind Scudder's technique to define a trisectrix---a curve that can be used to trisect an angle. We also describe a compass that could be used to draw the curve.
The aim of this work is to develop a numerical calculation program to study the effect of gas on the supersonic flow around a sharp airfoil in the context of a gas at high temperature (HT) below the threshold of dissociation of molecules by use of successive shocks and expansions on the airfoil surface. The airfoil is discretized into several consecutive segments on the extrados and intrados of the airfoil. The relations of an oblique shock wave (OSW) and the Prandtl Meyer expansion (PME) are used to determine the thermo-physical parameters on all segments including the Mach number, the pressure ratio, the temperature ratio, and the total pressure ratio in each segment considered to determine the aerodynamic coefficients of drag, lift, and pitching moment of the airfoil. The application is made for air and with 6 other gases which are H2, O2, N2, CO, CO2, and H2O. Three new comparison coefficients λD, λL and λM have been introduced to compare respectively the drag, lift and pitching moment coefficients of any gas compared to air for any sharp airfoils. So when a coefficient is less than unity, the gas considered has better performance compared to air for the corresponding gas dynamics coefficient. The application is made again for symmetrical curved airfoil given its practical interest in the international community. The validation is made by considering the calculation of the flow using the HT model at low temperature T0=500 K approximately, which represents the PG model threshold, the results of which for this last model can be found in the literature. The comparison of the results between the gases mentioned above and air shows the existence of a difference of 18.56%, 8.87%, 5.00% 2.39% 73.62% and 56.13% between the gases and air when M∞=5.00, T0=3500 K, α=2° and t/C=4% demonstrating the existence of gases more efficient than air in the field of aerospace construction. The application can be expanded for the design of future atmospheric and space vehicles.
Let f be a polynomial in Q[x]. We say that f is dynamically irreducible or
stable over Q if all its iterates f^n
:= f ◦ f ◦ ... ◦ f are irreducible over Q. Generally,
a polynomial is called eventually stable if the number of irreducible factors of any
iterate f^n
is bounded by some c ∈ Z^+, in particular, if c = 1, then f is dynamically
irreducible. A polynomial defined over Q is said to be pure with respect to a prime
p if its Newton polygon consists of exactly one line, e.g., p^r
-Eisenstein polynomials for some r ≥ 1. In 1985, Odoni showed that Eisenstein polynomials are dynamically irreducible over
Q. Ali extended this result to include p^r-Eisenstein polynomials for any r ≥ 1. In
this thesis, we present families of pure polynomials that are dynamically irreducible
in Q[x]. Under some conditions, we characterize certain families and develop some
criteria of dynamically irreducible polynomials that possess a pure iterate. In addition, we describe some iterative techniques to produce irreducible polynomials in
Q[x] from pure polynomials by composition.
Recently, Demark et al. investigated the eventual stability of a quadratic binomial
of the form x^2 −1/c ∈ Q[x] for some c ∈ Z\{0,−1}. In this work, we prove that pure
polynomials are eventually stable in Q[x]. Also, we display a family of eventually
stable polynomials that possess a pure iterate.
This chapter revises coordinate systems which include Cartesian coordinates, polar, spherical polar, cylindrical and barycentric coordinate systems. It also includes the distance between two points in space, and the area of simple 2D shapes. It concludes with a collection of worked examples.
Within the context of Indian religions, Jainism has long been recognized for its extensive use of permutations and combinations. However, the application of these principles within Buddhist scriptures has received relatively little scholarly attention. This paper introduces a new example of the specific application of permutations and combinations in Buddhist scriptures. In this paper, we focus on the first saṅghādisesa rule in the Theravāda-vinaya, which lists a series of element sets and arranges these elements according to a certain pattern known as “ten-roots” (mūla), and we discover that these arrangements form a regular numerical sequence, called “oblong numbers”. Moreover, similar patterns with different quantities are also found in the fourth Pārājika and the fifth saṅghādisesa rules. This indicates that the compilers of the Theravāda-vinaya did not use this mathematical knowledge without basis. Interestingly, we also found the use of this sequence in the Bakhshālī manuscript. Therefore, in this article, after summarizing and verifying the arrangement rules of the Theravāda-vinaya, we discuss whether the oblong numbers were influenced by Greek mathematics.
Introducción En este ensayo se explora la evolución de las funciones matemáticas a lo largo de la historia, desde sus orígenes en las antiguas civilizaciones hasta su papel fundamental en la ciencia y la tecnología contemporáneas. Para iniciar, es esencial definir qué se entiende por función matemática. Según Daniel(2022), una función matemática es una relación entre dos conjuntos, en la que a cada elemento del primer conjunto (dominio) le corresponde un único elemento del segundo conjunto (codominio o rango). Este concepto ha sido clave en el desarrollo de las matemáticas y sus aplicaciones en diversas áreas del conocimiento. Con el objetivo de entender mejor la evolución histórica de las funciones matemáticas, se ha adoptado un enfoque historiográfico inspirado en la propuesta de (Kolmogorov, 1936, como se citó en Otero, 2006), quien divide la historia de las matemáticas en cuatro grandes períodos: 1. El origen de las matemáticas. 2. El período de las matemáticas elementales (hasta el siglo XVI). 3. La historia de las matemáticas en el siglo XIX y principios del XX. 4. Las matemáticas contemporáneas.
In this chapter we introduce (PAR), the Principle of Arbitrary Reference. According to PAR any object of the universe of discourse is capable of been picked out by an act of arbitrary reference. We argue that PAR is essential for both formal and informal logical deduction, as well as for the semantics of quantifiers. We propose to understand arbitrary reference as direct reference via an ideal act of choice, setting the stage for further developments in later chapters.
The aim of this work is to develop a new numerical calculation program making it possible to correct the thrust coefficient CF and the exit Mach number ME of an existing MLN by studying the effect of the stagnation temperature T0 of the combustion chamber below the dissociation threshold of the molecules, based on the use of the HT model to, firstly, correct the flow in the nozzle, and then deduce the new corresponding values of (CF)C and (ME)C. The nozzle equipped with missiles and supersonic aircraft is determined using the PG model. This model is developed without dependence of T0. In reality, and depending on the used propellants, the combustion chamber T0 value can reach high values exceeding 500 K. Since T0 can be high, the (CF)C and (ME)C cannot be those obtained by the PG model. They must respond to the behavior of T0 effect, since the gas actually behaves like a gas at HT. Since the MLN has an unchanged contour, that is to say the mass of the nozzle is unchanged, it is necessary to correct the (CF)PG as well as, (ME)PG in order to further correct the other performances of the aerospace machines using the MLN, like the range, flight time, and maximum altitude of the supersonic missiles. An introduction of two new dimensionless coefficients λ and ψ making it possible to respectively determine the correction rate of (CF)C and (ME)PG compared to those determined by the manufacturers during the design of the MLN by the PG model. The calculation of the error between the obtained correction value and that given by the manufacturer is done for all parameters. The application is made for air and for three other gases H2O (gas), CO and N2. For example when (ME)PG = 3.00, T0 = 3500 K, r* = 1.00 and air, we will have a correction until 21.16% for (CF)C and 9.50% for (ME)C.
In this paper, we investigate the algebraic counterpart of the Fundamental Theorem of Algebra. We explore the concept of real-closed fields and quadratic forms. We show, by means of Galois theory, that F (√ −1) is algebraically closed if F is real-closed. Lastly, we explain the algebraic closure of R(√ −1) = C by demonstrating the real-closeness of R.
Bu araştırmanın amacı matematiksel ilişkilendirmeye yönelik verilen eğitimin öğretmen adaylarının matematiksel ilişkilendirmeye yönelik öz yeterliklerine katkısını incelemektir. Araştırmanın katılımcılarını 2021-2022 eğitim-öğretim yılının bahar döneminde bir devlet üniversitesinde öğrenim görmekte olan 50 ortaokul matematik öğretmeni adayı oluşturmaktadır. Katılımcıların belirlenmesinde uygun örnekleme yönteminden faydalanılmıştır. Bu araştırmada ön test ve son test uygulanması sebebiyle yarı deneysel desen kullanılmıştır. Matematik öğretmen adaylarının öz yeterlik düzeyleri matematiksel ilişkilendirme dersini almadan önce ölçülmüş daha sonra araştırmacılar tarafından 6 haftalık matematiksel ilişkilendirme dersi eğitimi planlanmıştır. Verilen eğitimin sonunda matematik öğretmen adaylarının matematiksel ilişkilendirme öz yeterlik düzeyleri tekrar ölçülmüştür. Veriler “Matematiksel İlişkilendirme Öz Yeterlik Ölçeği” kullanılarak toplanmıştır. Araştırmadan elde edilen verilerin analizinde yüzde, frekans, aritmetik ortalama, standart sapma, minimum ve maksimum değerler ve bağımlı örneklem t testi kullanılmıştır. Matematiksel ilişkilendirmeye yönelik öz yeterlik ölçeğinin ön test uygulamasının analiz edilmesi ile elde edilen sonuçta öğretmen adaylarının öz yeterlikleri orta düzey olarak bulunurken, son test uygulamasının analizinde ise yüksek düzey bulunmuştur. Araştırma sonucunda öğretmen adaylarının öz yeterlik düzeyleri arasında son test lehine anlamlı bir fark bulunmuştur.
Capítulo 3 do texto didático "Lições de Cálculo Diferencial e Integral - Volume 1" elaborado para a disciplina de IC241 - Cálculo 1 da UFRRJ. Conteúdo: Definição de Continuidade; Propriedades Operatórias; Composição de Funções Continuas; Teorema do Valor Intermediário e Aplicações
This work is a segment of an ongoing doctoral research in Brazil. The Leonardo numbers and the Leonardo sequence have gained attention from mathematicians and the academic community. Despite being a relatively new sequence within mathematical literature, its discussion has intensified over the past five years, giving rise to other branches, with contributions and associations to other topics in mathematics. Thus, the aim of this study was to construct and present the state of the art of the Leonardo sequence, considering its historical aspects and highlighting works on its evolutionary process in the epistemic-mathematical field, regarding its generalization, complexification, hyper complexification, and combinatorial model during the last five years (2019-2023). The methodology used was a bibliographic study, where the state of the art was carried out through the mapping of publications on the subject. Twenty-four research works related to the key descriptors “Leonardo sequence”, “Leonardo numbers”, “complexification”, “generalization”, “hybrids”, and “combinatorial model” were found, cataloged, and discussed. From the analysis of these studies, it is noted that its development in pure mathematics has advanced to other branches and discoveries, and that, albeit timidly, research on the subject has emerged directed towards the field of education, especially in the initial teacher training and, particularly, in Brazil.
Capítulo 2 do texto didático "Lições de Calculo Diferencial e Integral - Volume 1" elaborado para a disciplina de IC241 - Cálculo 1 da UFRRJ. Conteúdo: Limite de funções de uma variável; Teorema do Confronto; Limite Fundamental; Definição Formal
In this essay we describe Torricelli’s life and works in their scientific context, including the Archimedean heritage in Torricelli’s works. We analyse the changes of Torricelli’s works and explain the novelties of the edition we are offering. Then, we provide a picture of the most significant results obtained by Torricelli (1608–1647), particularly in mechanics and geometry. Furthermore, we also focus on the Torricelli’s methodology, specifying how he proved two of his achievements, given their novelty and mathematical meaning: (a) the volume of the “solido acutissimo”; (b) the geometry of the logarithmic spiral. An ad hoc Content of the Opera geometrica is proposed as well. It is based on the original chronological sequences of the contents wrote by Torricelli. This is important to evaluate the birth of the scientific matters presented by Torricelli and the editorial process of the publication.
Capítulo 1 do texto didático "Lições de Calculo Diferencial e Integral - Volume 1" elaborado para a disciplina de IC241 - Cálculo 1 da UFRRJ. Conteúdo: Conjuntos Numéricos, Funções de uma Variável real, Funções Algébricas, Funções Transcendentes e Exercícios
The aim of this paper is to gain a better understanding of the nature of imaginary numbers, considering them as mathematical objects characterized by a lower dimensionality than that of real numbers. For this purpose, in addition to an article written a few years ago on the dimensional aspect of mathematical objects, reference will be made to the rules of arithmetic presented by Brahmagupta. In particular, it will be emphasized that even the nature of the arithmetic operation of multiplication has been misunderstood with regard to the roots of negative numbers. The negative numbers from which the square root is extracted are not to be seen as the result of a multiplication performed on the real straight line, i.e. without changing the nature of the multiplicand, but as the product between quantities of the same nature, a product that generates mathematical objects characterized by a nature different from that of the multiplicand. Sn alternative graphic approach to the usual one will finally better clarify the nature of imaginary numbers, highlighting their concreteness.
Capítulo 8 do texto didático "Um segundo curso de Cálculo" elaborado para a disciplina de IC869 -Cálculo 2a da UFRRJ. Conteúdo: Máximo e Mínimos; Método dos Multiplicadores de Lagrange.
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