Maths Education

B i ε, B i = 2 x i 3 y i , y establecemos su unicidad. Definimos la base global M = mcm(B i) y demostramos la expansión polinómica única de N en potencias de M. Finalmente, formalizamos el método Puente ∆ basado en la reducción por M 2 y discutimos su rol en el contexto de factorización de módulos RSA.

Royaume du Maroc Académie régionale de l'enseignement Ministère de l'éducation nationale, et de la formation de Souss-Massa De l'enseignement supérieur, de la formation Délégation provinciale d'Agadir Niveaux : 1BAC S. EX, SM, STC Chargé de l'examen : Pr. Karam OUHAROU Devoir surveillé N°1, Notions de logique, et opérations fontionnelles Exercice 1 Se familiariser avec les notions de logique

A simple proof of Euler's formula which states that the sum of the reciprocals of all natural numbers squared equals ζ(2) = π^2/6 is presented based on the distribution theory introduced by Laurent Schwartz. Additional identities are obtained as a byproduct of the derivation.

This paper reports on the quantitative part of my doctoral study carried out in three state-funded secondary schools in England. Using Bourdieu’s trilogy of habitus, embodied cultural capital and field, I have explored the interaction of social class, gender, ethnicity and attitudes towards mathematics, and their impact on mathematical achievement. The findings suggest that although cultural capital correlates positively with mathematical achievement, social class and habitus, which has been operationalised as mathematics-related career aspirations, have a more significant impact on mathematical achievement.

In this paper we include topics which we consider are relevant building blocks to design a theory of mathematics education. In doing so, we introduce a pretheory consisting of a set of interdisciplinary ideas which lead to an understanding of what occurs in the "central nervous system" -our metaphor for the classroom and eventually in more global settings. In particular we highlight the crucial role of representations, symbols viewed from an evolutionary perspective and mathematics as symbolic technology in which representations are embedded and executable.

Mathematics support, the provision of additional learning opportunities to, primarily, nonmathematics specialist undergraduates has grown significantly since the early 1990s, particularly in the UK, Ireland and Australia. Alongside the growth in volume of provision, there has been a marked increase in the amount of research and scholarship relating to mathematics support that has been carried out and disseminated. This paper reviews this literature and in doing so identifies areas in which mathematics support has evolved. This evolution has taken place in response to a range of crucial changes in the external policy and general environment and, in particular, in response to the changing nature of the so-called 'Mathematics Problem'. Key themes that emerge from the literature review, which are explored in detail, are the characteristics of students who engage with mathematics support and reasons why others do not; the role of the mathematics support tutor, who undertakes the tutoring task and how they are trained; the positioning of mathematics support within higher education structures; and the evaluation of the effectiveness of mathematics support.

We are delighted to announce SpringerBriefs in Education, an innovative product type that combines elements of both journals and books. Briefs present concise summaries of cutting-edge research and practical applications in education. Featuring compact volumes of 50 to 125 pages, the SpringerBriefs in Education allow authors to present their ideas and readers to absorb them with a minimal time investment. Briefs are published as part of Springer's eBook Collection. In addition, Briefs are available for individual print and electronic purchase.

This paper reports on the quantitative part of my doctoral study carried out in three state-funded secondary schools in England. Using Bourdieu's trilogy of habitus, embodied cultural capital and field, I have explored the interaction of social class, gender, ethnicity and attitudes towards mathematics, and their impact on mathematical achievement. The findings suggest that although cultural capital correlates positively with mathematical achievement, social class and habitus, which has been operationalised as mathematics-related career aspirations, have a more significant impact on mathematical achievement.

Below is a self-contained summary of the principal equations found in the uploaded materials and our discussions, with step-by-step derivations and definitions of all symbols. Each equation is accompanied by references to the relevant sources.

This paper compiles and synthesizes a comprehensive set of both established and novel theoretical equations at the frontier of physics and information theory. Beginning with classical anchors—Einstein’s mass–energy equivalence
𝐸
=
𝑚
𝑐
2
E=mc
2
  and the Einstein field equations
𝐺
𝜇
𝜈
+
Λ
 
𝑔
𝜇
𝜈
=
𝜅
 
𝑇
𝜇
𝜈
G
μν
​
+Λg
μν
​
=κT
μν
​
—we extend the formalism by introducing an Information–Energy tensor
𝐼
𝜇
𝜈
I
μν
​
  into the stress–energy source. We explore Landauer’s principle to derive a “mass of a bit”
𝑚
b
i
t
=
𝑘
𝐵
𝑇
ln
⁡
2
 
/
 
𝑐
2
m
bit
​
=k
B
​
Tln2/c
2
, linking thermodynamic and informational measures. Central to our speculative framework is the modeling of the observer as a quaternion
𝑞
=
𝑤
+
𝑥
 
𝑖
+
𝑦
 
𝑗
+
𝑧
 
𝑘
q=w+xi+yj+zk, with unit‐quaternion rotations encoding measurement “collapse” events in a 24-dimensional Leech lattice. We formalize the Fold Operator
𝐹
^
F
  on a re-centered number line, define pre-actualized phase states
𝜓
𝑛
=
𝑒
𝑖
𝜃
𝑛
ψ
n
​
=e
iθ
n
​

  for digits 1–9, and introduce a nonstandard quaternionic convolution
⊗
⊗ to capture recursive harmonic collapse dynamics. Supporting constructs include a “folded Pythagorean” norm
ℎ
=
𝑟
2
+
𝑖
2
h=
r
2
+i
2

​
  and golden‐ratio scaling
𝜑
=
(
1
+
5
)
/
2
φ=(1+
5
​
)/2, integrated into a Recursive Arithmetic Engine that maps arithmetic to geometric phase convolution. Together, these equations weave a unified tapestry from quantum mechanics and relativity through information theory to a proposed quaternionic-geometric ontology of measurement.

My thanks to Christian, Erydir, Jean-Charles, Kirby and countless other for their support, insights and most importantly guidance for helping clarify these ideas. This work is a tribute to all who have also glimpsed the same underlying mathematical patterns of the universe.

These aren't meant as solid proofs, but as working hypothesis - open for further exploration. The more we push, the closer we must arrive at a fundamentally new way of looking at the universe; and one that multiple independent researchers have separately arrived at.
This work isn't to usurp from anyone's existing frameworks - but rather how the math beneath it must be correct and proves theirs.

We present a simple and accurate approximation for the square root of integers of the form x = n^2 + a, based on the structure of square numbers and the spacing between them. The estimate √ x ≈ n + a/(2n + 1) is easy to compute and yields small errors. We also derive an explicit expression for the approximation error and discuss its behavior as a function of n and a.

E-learning has been receiving increased attention since the advent of the Internet, following the hype of “online everything”: from administration, to communication to business to education. The promise of e-learning has, however, fallen short of statements like those of J. Chambers (CEO of CISCO Systems) that said in 1998 that “the biggest growth in the Internet, and the area that will prove to be one of the biggest agents of change, will be e-learning”[1]. This is not to say that e-learning should be abandoned as a concept. It ...

Cet article présente une méthode d’approximation pour résoudre des équations exponentielles dans lesquelles la variable apparaît à la fois dans la base et dans l’exposant, sous la forme
(
𝑎
+
𝑥
)
(
𝑏
+
𝑥
)
=
𝑐
(a+x)
(b+x)
=c. En développant l’équation logarithmique équivalente, nous montrons qu’il est possible d’approximer la solution
𝑥
x par une équation quadratique. Cette méthode est testée sur plusieurs cas, avec des taux de précision variables en fonction des valeurs de
𝑎
a,
𝑏
b et
𝑐
c. Une analyse de la position du minimum local de la fonction associée est également proposée. Le travail met en évidence les limites et les conditions d’efficacité de cette approximation.

The Making Assessment Count (MAC) project started at the University of Westminster in 2008. It sought to align staff and student expectations of feedback and support greater use of feed-forward approaches. A baseline analysis of staff views in the School of Life Sciences suggested that students did not make strategic use of the feedback they received. A similar analysis of the student position revealed that as a group they felt that the feedback provided to them was often insufficiently helpful. To address this dichotomy, a MAC process was developed in the School of Life Sciences and trialled with a cohort of about 350 first year undergraduate students. The process was based on a student-centred, three-stage model of feedback: Subject specific, Operational, and Strategic (SOS model). The student uses the subject tutor's feedback on an assignment to complete an online self-review questionnaire delivered by a simple tool. The student answers are processed by a web application called e-Reflect to generate a further feedback report. Contained within this report are personalised graphical representations of performance, time management, satisfaction and other operational feedback designed to help the student reflect on their approach to preparation and completion of future work. The student then writes in an online learning journal, which is shared with their personal tutor to support the personal tutorial process and the student's own development plan (PDP). Since the initial development and implementation of the MAC process within Life Sciences at Westminster, a consortium of universities has worked together to maximise the benefits of the project outcomes and collaboratively explore how the SOS model and e-Reflect can be exploited in different institutional and subject contexts. This paper presents and discusses an evaluation of the use of the MAC process within Life Sciences at Westminster from both staff and student perspective. In addition, the paper will show how the consortium is working to develop a number of scenarios for utilisation of the process as a whole as well as the key individual process components, the SOS model and e-Reflect.

The clock Angle Problem has been a fascinating problem in mathematic for quite many years. It involves calculating the angle between the Hour and Minute hands. The formula to figure out this angle is, This formula is mostly used to figure out the inner angle between the hour and minute hand of a clock. And it works beautifully due to its beautiful structure. But there are some cases where this formula gives some unexpected results, more accurately, it gives the outer angle rather than the inner angle. While, it is ok for it to give it but the reason why it sometime gives inner and sometimes outer is quite fascinating and I have written this paper on this exact topic of why sometimes it gives us the outer angle. I have tried my best to make it as professional and as academic grade as possible but since it is my first paper, there will be some mistakes so, I hope the readers will be able to forgive me for my mistakes and that my paper helps the mathematical community to enhance its understanding and give a fresh new perspective on this.

🎓 Thinking of becoming a Maths teacher? I wrote a short self-assessment guide (based on my own doubts and struggles) to help people decide if teaching is really right for them.
I’ve included info about the 2025 IMA scholarship, plus checklists, alternatives, and affiliate tools to help with career planning.

La Conjetura de la Expresión Jerárquica de Rendimiento (HYE) propone una transformación radical en la teoría de números, reimaginando los números naturales como estructuras jerárquicas generadas a partir de potencias de 2 y 3, en lugar de meras magnitudes definidas por divisibilidad. Este artículo de opinión sostiene que la HYE ofrece un marco revolucionario para entender los números como genealogías algebraicas, desafiando el enfoque tradicional en la factorización prima. Aunque prometedora, la conjetura requiere una formalización rigurosa para cumplir su potencial. Exploramos sus implicaciones filosóficas y matemáticas, abogando por su estudio como una nueva base para la teoría de números.

Unified Mathematical Framework for Multidimensional Fixed Points in Quantum and Cognitive Systems

Table of Contents

1. Introduction 2. Theoretical Background and Conceptual Framework 3. Fixed Points in Quantum–Relativistic Dynamics 4. Scalar Complex Fixed Points in Cognitive Systems 5. Light–Matter Interactions and Topological Considerations 6. Observer Consciousness and Nonreductive Models 7. Computational and Methodological Approaches 8. Synthesis: A Unified Framework for Quantum and Cognitive Fixed Points 9. Conclusion

1. Introduction

The quest to construct a unified mathematical framework for multidimensional fixed points at the intersection of quantum dynamics and cognitive systems represents one of the boldest research challenges of our era. This article establishes an integrated perspective that bridges multidimensional scalar complex fixed points found in quantum–relativistic dynamics—with roots in topological and binary symmetries and geometric constructs like the Pythagorean hypotenuse—with neurophysical models of mind–brain interaction and observer consciousness.

Recent advances in the philosophy of mind and quantum physics have spurred renewed interest in understanding how invariant states, or fixed points, emerge in both the physical and cognitive domains. While the classical reductive methods of cognitive science have effectively explained the performance of functions in perceptual and behavioral tasks 1 , they fall short when addressing the subjective aspect of conscious experience; the “hard problem” of consciousness 1 . Similarly, quantum–relativistic dynamics explore the idea of fixed points as fundamental anchors in complex systems, present in phenomena ranging from superconductivity to topological insulators.

This article develops a unified framework that connects the emergence of fixed points in these seemingly disparate fields. By integrating the principles of structural coherence and organizational invariance from the study of consciousness with mathematical models found in quantum theory, we propose that the invariant nature of multidimensional scalar complex fixed points is a crucial element for understanding both the physical and cognitive aspects of reality.

My Mathematics Classes and Saurabh was one of the Good Articles of mine. Initially, it was written in Hindi, But with in two three years it was translated by different organizations in three-four different languages. here, i am sharing its Gujrati translation, which was published by Sandarbh ( Gujrati )

me hizo comentarios acerca del bajo nivel de sus alumnos. Por estar de acuerdo con las presunciones de mi compañero, decidí realizar a mis alumnos unas pruebas básicas, que de alguna forma midieran y compararan los conocimientos de geometría de los niños de 14 años de ESO con mis alumnos de segundo curso de universidad. Por esta razón, en el examen final incluí un ejercicio que contenía tres problemas de una colección de 70 ejercicios elaborado para niños de 14 años por un colegio sevillano. Un ejercicio de aplicación directa del teorema de Pitágoras, otro del cálculo del área de un polígono, y un tercero con operaciones elementales con ángulos. Este ejercicio no influyo en la calificación del examen. El nivel de dificultad de estos problemas era o el básico o el medio de entre los problemas de la mencionada colección. Lo anterior puso de manifiesto las siguientes situaciones:

1. ΤΑΥΤΟΤΗΤΑ ΔΙΔΑΚΤΙΚΟΥ ΣΕΝΑΡΙΟΥ
Θεματική του διδακτικού σεναρίου: Ρητοί Αριθμοί
Δημιουργός : Γιάννης Πλατάρος
Βαθμίδα – Τάξη:  Γυμνάσιο , Α΄ Τάξη  Διδακτικές ώρες: Δύο (2)
Ενότητα του ΠΣ και Προσδοκώμενα Μαθησιακά Αποτελέσματα (ΠΜΑ):
Αρ.Ρ.7.3 : Να συγκρίνουν και να διατάσσουν ρητούς, να τους αναπαριστούν στην αριθμογραμμή  και να αναγνωρίζουν ότι ένας ρητός δεν έχει επόμενο.
Προαπαιτούμενες δυνατότητες μαθητών/τριών (γνωστικές και κοινωνικο-πολιτισμικές):
Αρ.Ρ.7.1 και Αρ.Ρ.7.2 του ΠΜΑ,  ως προς το γνωστικό.  Ως προς το κοινωνικοπολιτιστικό επίπεδο των μαθητών στα πλαίσια της διαφοροποιημένης και συμπεριληπτικής φιλοσοφίας του Π.Σ. , εμπεριέχεται στον σχεδιασμό για όλους τους μαθητές

2. ΔΙΔΑΚΤΙΚΗ ΠΛΑΙΣΙΩΣΗ ΚΑΙ ΔΙΑΧΕΙΡΙΣΗ
2.1. Περί μαθητή και μάθησης: Η έννοια της πυκνότητας των ρητών, είναι δύσκολη, ακόμα και για φοιτητές ακόμα και για αποφοίτους ΑΕΙ των θετικών επιστημών. Η (Αυγερή , 2020) σε σχετική μεταπτυχιακή διπλωματική της  εργασία, αναφέρει ότι
«…η έννοια της πυκνότητας ενός συνόλου φαίνεται να δυσκολεύει όχι μόνο τους μαθητές όλων των βαθμίδων, αλλά και φοιτητές των μαθηματικών τμημάτων, ακόμη και τους εκπαιδευτικούς»
Επικαλείται επί τούτω προγενέστερες έρευνες των  (Giannakoulias et. al., 2007· Hartnett & Gelman,1998· Malara, 2001) 
Αναφερόμενη μάλιστα στους ( Vamvakoussi  & Vosniadou, 2004,2010, 2017) αναφέρει ότι
«…οι μαθητές θεωρούν ότι υπάρχει πεπερασμένο πλήθος αριθμών ανάμεσα σε δύο ρητούς»
Στην ίδια εργασία αναφέρεται επίσης υπάρχει πρόβλημα  με την έννοια της  ύπαρξης του «επόμενου αριθμού» και με την  έννοια του απείρου.
...........................

The contribution illustrates the outcomes of an interdisciplinary teaching project that involved students in a second class of a scientific high school in Palermo in s.y. 2018-2019. The activity was coordinated by the teachers of Latin (Prof. Li Causi) and Mathematics (Prof. Modica), and ended with the publication on a web page of the cooperative translation of a group of problems taken from the Propositiones ad acuendos iuvenes of Alcuin of York (735-804 AD), a philosopher and theologian active at the court of Charlemagne. The main objective was to approach the study of the texts chosen by the teachers, analyzing not only the transition from an ancient language (Latin) to Italian, but also the transition from the natural to the symbolic language of modern mathematics.

On November 10, 2022, after numerous attempts, I discovered a particular expression that allowed me—upon further exploration—to accurately recover the golden ratio from a definite integral. This personal discovery is not merely a numerical coincidence, but rather a gateway to an elegant interpretation of phi through analysis. The aim of this research is to document this original approach and to extend it further, ultimately deriving an infinite series converging to the golden ratio—a surprising link between continuous analysis and discrete sequences.

têm tido importante papel ao relacionar as disciplinas pedagógicas e as de conteúdo específico da matemática. Além disso, estas disciplinas permitem que os alunos reflitam sobre a teoria e tenha a oportunidade de fazer uma prática simulada, através

Το παρόν άρθρο έχοντας ως βάση τη θεωρητική τεκμηρίωση και τις προτάσεις του συγγράμματος: «Απο-φυλακοποίηση: Βασικές αρχές για μία δημόσια πολιτική μείωσης της φυλακής υπό την οπτική του ριζοσπαστικού Εγγυητισμού» του καθηγητή Ποινικού Δικαίου του Πανεπιστημίου της Βαρκελώνης Iñaki Rivera Beiras επιδιώκει να προβάλλει και να διανθίσει το πρόταγμα μίας ριζοσπαστικής αλλαγής στην οπτική και πρακτική που αφορά τον θεσμό της φυλακής, την σωφρονιστική μεταχείριση και την επανένταξη των κρατούμενων ατόμων: αυτή της ελαχιστοποίησης της φυλακής ως προϋπόθεση για την μείωση της κοινωνικής βλάβης και την κοινωνική επανένταξη. Σε αυτό το πλαίσιο, οι αρχές, πρακτικές και δράσεις που παρουσιάζονται βασίζονται στις αρχές του ποινικού εγγυητισμού-προτύπου ελάχιστης ποινικής παρέμβασης, του καταργητισμού, στις υπάρχουσες διεθνείς νομικές κατακτήσεις στο πεδίο των θεμελιωδών δικαιωμάτων, στην εμπειρία και παρακαταθήκη κοινωνικών εγχειρημάτων και κινημάτων καθώς και των αιτημάτων των ίδιων των κρατουμένων.

The present article, largely building on the theoretical perspectives and suggestions set forward in the work by Iñaki Rivera Beiras (2017) “Descarcelation. Principles for a public reductionist policy for prisons (from the radical garantism point of view)” aspires to showcase and contribute to the discussions of the imperative for a radical change in the narrative and practice of (i) the institution of prison, (ii) correctional treatment, and (iii) the rehabilitation of inmates: that of the minimization of prison as a prerequisite for the substantial reduction of social harm and for social re-integration. Within this framework, the principles, practices and actions presented in this article are based on the theory of penal garantism-minimum intervention and abolitionism, while accounting established international legal accomplishments in the field of fundamental human rights, the experiences and legacies of social movements, as well as the demands of inmates themselves.

Este trabajo es la culminación de un gran esfuerzo por parte de todos y cada uno de mis familiares, en concreto quisiera agradecer primeramente a mis padres que siempre me han apoyado y estado de cerca alentandome en cada situación de la vida , a mi hermana, tías Nena, Rosa, Blanca, Yuya quienes nunca han dudado en darme lo mejor para ser la persona que soy,a mis abuelas Sofi y Conchita, a mis profesores en especial al Dr. Valero que con paciencia ha estado guiando este trabajo y compartiendo sus conocimientos desde prontos semestres en la facultad, a Fer G. quién me apoyó desde antes de comenzar la licenciatura en esta hermosa ciencia que es la matemática y por último pero no menos importantes a mis amigos de toda la vida Chuy, Tony ,Joshy, Willy ,Adriana ,David que han estado presentes desde el comienzo de esta empresa.

This study investigates the effectiveness of a tessellation-based instructional program supported by digital tools in enhancing geometric learning in primary school students. While tessellation activities have been shown to support geometry learning at both primary and secondary levels, their use in digital environments has only been explored in secondary education. Digital tools like GeoGebra and 3D printing can improve engagement and understanding, yet their impact on younger students remains under-researched. A quasi-experimental design was employed with fourth-grade students, comparing an experimental group using GeoGebra and 3D printing with a control group. Pre-tests and post-tests measured learning outcomes. The results showed that the experimental group achieved significantly higher post-test scores, while the control group showed no significant improvement. Additionally, students reported high engagement, particularly with digital tools, reinforcing their role in enhancing learning. This study extends research on digital tessellation to primary education, demonstrating the potential of GeoGebra and 3D printing in fostering geometric understanding. Findings suggest that integrating these technologies into primary curricula can enhance engagement and learning, calling for further research and teacher training in digital geometry instruction.

In this article we share the story of a proposal for permanent training for mathematics teachers who are in charge of the training of students between 12 and 18 years old. We present a task and the fundamentals that lead us to develop it in this training scenario, for the collective construction of specialized knowledge of those who teach topics related to metric geometry and its relationship with optical illusions. We comment on the planning of the scenario and the results achieved from the perspective of the Mathematics Teacher's Specialized Knowledge Model (MTSK Model), which we relate to studies on the use of technologies in mathematics classrooms. We conclude with an analysis of the strengths and weaknesses, opportunities and threats that may arise in scenarios such as the one described for the permanent training of teachers considered teaching professionals.

En este trabajo, se estudiaron los conjuntos flexibles aplicados en el contexto
de la teoría de anillos. Primero definimos las nociones clásicas de los conjuntos
flexibles, siguiendo las ideas b´asicas de Molodtsov, a continuaci´on se
introducen los conceptos de funciones flexibles y composici´on de funciones
flexibles, y se dedujeron sus propiedades, recordamos los conceptos cl´asicos de
la teor´ıa de anillos, anillos cocientes y los teoremas de isomorfismos. Enunciamos
el concepto de anillos cocientes flexibles, siguiendo las ideas de Xiaolong
y damos las primeras nociones de homomorfismo natural cuasi-flexible y flexible.
En particular, demostramos el teorema fundamental del homomorfismo
natural flexible junto con los dos teoremas de isomorfismo para anillos flexibles.
Por ´ultimo, establecemos una correspondencia uno a uno entre los
homomorfismos de anillos y los homomorfismos de anillos flexibles.

Confrontés à un problème de recherche visant à faire une contribution à l'amélioration de l'enseignement/apprentissage de la géométrie dans les séries C et D de l'enseignement secondaire nigérien, nous avons réalisé une exploration suivant Quivy et Van Campenhoudt (1997), pour recenser les difficultés liés à l'enseignement/apprentissage de la géométrie dans les lycées de la région de Niamey et d'identifier leurs approches solutions. Notre cadre théorique s'appuie sur les trois registres de la didactique proposés par Martinant (1987) dans Astolfi et al. (2008), où un fait didactique est soit épistémologique, psychologique, ou pédagogique. Ce cadre a été utilisé pour classer les difficultés, suivant qu'elles soient du côté du savoir, de l'apprenant ou de l'enseignant. Les 25 participants, ont été soumis en entrevues semistructurées ou en focus group, tous intervenant d'une manière ou d'une autre à l'amélioration de l'enseignement des mathématiques dans les lycées. Les résultats montrent que les difficultés sont variées et elles émanent de la situation épistémologique du savoir géométrique, de la préparation mathématique didactique et pédagogique de l'enseignant et de la psychologie de l'apprenant. À cela s'ajoute un problème d'infrastructures scolaires. Ceux-ci ont un impact sur le sentiment d'efficacité personnelle des enseignants, l'engagement des apprenants, et engendrent une difficile création des conditions d'apprentissage. Pour une solution de préparation des enseignants, il est nécessaire de faire un profilage pour mieux diagnostiquer les lacunes en préparations mathématique, didactique et pédagogique, afin de développer un modèle de formation continue leur permettant de se sentir efficace dans leurs pratiques, face aux savoirs géométriques et de construire des séquences d'enseignement motivant pour les apprenants.

En este articulo presentamos dos algoritmos, el primero permite escribir un polinomio simetrico f en k[x1, · · · , xn], con k un cuerpo, en terminos de las funciones simetricas elementales; el segundo, determina si un polinomio f en k[x1, · · · , xn], con k un cuerpo, es simetrico, y si este es el caso, como escribirlo en terminos de las funciones simetricas elementales. Ademas, probamos de manera detallada como se obtiene una base de Grobner G en el caso particular cuando se considera el orden lex sobre los terminos, herramienta necesaria para presentar el segundo algortimo. Adicionalmente, mostramos una pequena aplicacion de los polinomios simetricos en el calculo del anillo de invariantes de un grupo finito de matrices dado. Ilustramos los resultados con variados ejemplos.

The cast-resin dry type transformer which had its development in Europe began to be widely used in the United States in the 1980's. This type unit was incorporated into the first IEEE standard in 1989. One of the problems facing the IEEE working groups developing product standards and loading guides for the cast-resin transformer was the correct value of the hottest spot temperature allowance used to determine permissible average winding temperature rises so that insulation system temperatures would not be exceeded. Different values of the hottest spot temperature allowance are used in IEEE and IEC standards. Published literature on this subject is minimal. A comprehensive thermal test program was conducted on a 2000 kVA cast-resin dry type transformer with 300 embedded thermocouples to determine the hottest spot temperature allowance and to obtain data to develop mathematical models. Nineteen thermal test runs were performed at different loads, tap positions, and core excitations under natural and forced air operating conditions. The test data indicated that the hottest spot temperature allowance used in the IEC standard was too low. In the IEEE standard the hottest spot temperature allowance appears conservative up to about 115 OC average winding temperature rise. The locations of the hottest spot for the primary and secondary winding were determined. Conclusions about thermal testing and loading are presented.

• To equip students with the use of litmus paper to determine the acidity/basicity of solutions. • To determine the solubility of samples in different solutions. • To know the processes involved in identifying organic functional groups in compounds. • To be able to study the chemical properties of commonly encountered functional groups.

ABSTRAKMinyak kelapa merupakan sumber medium chain triglycerides(MCT) utama. Melalui proses fraksinasi dapat dihasilkan fraksi minyakdengan kandungan MCT tinggi. Penelitian ini bertujuan untuk mempelajaripengaruh berbagai faktor perlakuan dingin terhadap kristalisasi danfraksinasi minyak kelapa, serta untuk menetapkan prosedur pendinginanyang efektif dalam menghasilkan fraksi minyak dengan kandungan MCTtinggi. Penelitian dilaksanakan di Laboratorium SEAFAST CENTER IPBdari bulan Maret 2012 sampai bulan Februari 2013. Fraksinasi dilakukandengan memanaskan minyak pada suhu 70°C lalu didinginkan padaberbagai laju pendinginan untuk mencapai beberapa variasi suhukristalisasi, diaduk dengan kecepatan 15 rpm, dibiarkan mengkristal padalama waktu yang berbeda (hingga 900 menit), serta difraksinasi denganpenyaringan vakum menggunakan kertas Whatman 40. Tiga tahappendinginan yang merupakan faktor kunci keberhasilan proses kristalisasiminyak kelapa yaitu tahap pendinginan awal dari suhu 70 hing...