Mathematics

El propósito de este documento consiste en implementar una metodología para encontrar el lado y la apotema de un pentágono regular inscrito en una circunferencia de radio conocido. Se describe un análisis geométrico basado en el estudio de la proporción áurea con la utilidad del segundo teorema de Tales, el cual se encuentra inherente a la geometría del pentágono regular inscrito,  facilitando su comprensión y desarrollo en su construcción y generación de novedosas fórmulas que involucran el número áureo. Se pretende que esta metodología pueda implementarse también  en el análisis de cualquier polígono regular de n lados inscrito en un círculo de radio conocido, para determinar el lado y la apotema del mismo. Estos conceptos pueden incorporarse en un currículo de estudio relacionado con las ciencias afines a la medición, el diseño y la construcción; además, debido a la precisión y sencillez en su proceso, puede implementarse en el diseño de modelos matemáticos y algorítmicos para la geometría computacional, informática gráfica y hardware de programas CAD.

This paper studies connections between two alternatives to the standard probability calculus for representing and reasoning about uncertainty: imprecise probability and comparative probability. The goal is to identify complete logics for reasoning about uncertainty in a comparative probabilistic language whose semantics is given in terms of imprecise probability. Comparative probability operators are interpreted as quantifying over a set of probability measures. Modal and dynamic operators are added for reasoning about epistemic possibility and updating sets of probability measures.

Publication Date: February 8 , 2021                        Number of Pages : 5
In this short paper , we solve a purely numerical; a purely computational
problem that was featured in the 1971 Mathematical Olympiad of
    Sweden. We state it below as PROBLEM 1:
                                                  PROBLEM 1
                      11th Swedish Mathematical  Olympiad 1971
        Problem#4: Find ,
[(65533)^3 + (65534)^3 + (65535)^3 + ...+ (65539)^3]/
[(32765)(32766)+(32767)(32768)+(32768)(32769)+(32770)(32771)] .
Note the numerator of the given fraction is the sum of the cubes of
  seven consecutive integers: 65533 , 65534 , ...... , 65539.
The denominator is the sum of four products:
  (32765)(32766)  , (32767)(32768) , (32768)(32769) ,  and
    (32770)(32771) .
  There are two results in this paper. Result1 , is simply the familiar
    formula for the sum of cubes of the first n positive integers:
        1^3 + 2^3 + ... + n^3 = [n(n+1)]^2/4  .
  Using Result1 , we prove Result 2 in Section 2 , pages 3-5.
    Result 2 provides the answer to the above problem , PROBLEM 1:
        The given rational number is equal to ,
                                    (14)(32768) = 458752 .

The present study tries to find out the effectiveness of inverted classroom strategies in learning mathematics among secondary teacher education students. The study was conducted to develop a video lesson for the methods of teaching mathematics and experimenting the same with a set of students studying for B.Ed. and finding out its effectiveness over the conventional method of teaching. Two equivalent group experimental designs are employed for this study. The investigator has chosen 46 B.Ed students for the study. The study revealed that there was a significant difference between control and experimental group students in their gain scores and for the attainment of knowledge, understanding, and applicaiton objectives.

Mental health occupies a prominent place in the success of human being. The study explored the
relationship between mental health and academic achievement among college students. The
population of the study is under graduate students of Malappuram District, Kerala. The study
conducted on a sample of 108 under graduate students which have been selected from arts and
science subjects through a stratified random sampling. The tool used for assessing mental health
of prospective teachers is Mental Health Questionnaire for College Students developed by the
researcher. The tool is a four point scale having 20 questions from dimensions of Stress,
Depression, Anxiety and Worries. Academic achievement of graduate students was assessed by
their CGPA score. The statistical techniques used for analysing the data are mean, standard
deviation, t-test and correlation. The study showed that there is positive and significant correlation
between mental health and academic achievement among college students. It is also found that
there is no significant difference in mental health and academic achievement among college
students based on the stream of subjects. It is also found that there is significant difference in
academic achievement and mental health based on gender.

Formal Stiences are mind languages, mirrors of the Universe of ∆ST, as such among those huminds use, all of them based in the same Universal Grammar S<=>T and Pentalogic, there are ∆-scalar languages (Algebra of Scalar Numbers, Music of social frequency sounds), Temporal languages (verbal words and Space=Name<=>Time=Verb sentences; and equations whose operands represent the dimotions of existence), and spatial languages (art of human I=eyes, from 2D painting and sculpture, which only shows the 'external membrane of a 1D topologic sphere, through 3D architecture and Film, which has 1 Dimotion of 'hinformation'). A special place among space languages is occupied by Geometry; from the reversed Projective Geometry of the Mind, through the hyperbolic Geometry of the 5th dimension into the Elliptic Geometry of the Larger cosmological Universe. We shall thus study all the languages of the humind and introduce AI, Algorithms of Information, the Languages of the Digital Mind of the chip homoctonos...

Toeplitz Hermitian Positive Definite (THPD) matrices play an important role in signal processing and computer graphics and circular models, related to angular / periodic data, have vide applications in various walks of life. Visualizing a circular model through THPD matrix the required computations on THPD matrices using single bordering and double bordering are discussed. It can be seen that every tridiagonal THPD leads to Cardioid model.

Publication Date: February 1 , 2021                    Number of Pages: 31
In this work , we study and solve eleven problems that were part of the
Swedish Mathematical Olympiads in the years 1962- 1966 , 1969 , 1972 ,
1974 , and 1976 - 1978. The eleven problems are stated in the
introduction of this paper , on pages 2 - 4 . Eight of the eleven problems
are number theory problems; the other three algebra problems ,
  including a summation problem . In Section 2 , pages 5 and 6 , an
    Answer Key is offered. We present solutions to the eleven problems
in Sections 3 - 13 ; pages 7 through 31.
The eleven problems are stated as PROBLEMS 1 - 11.  Below , in this
abstract , we state six among the eleven problems; PROBLEMS 9 , 10 ,
  11 , 5 , 6 and 8 .
                                          PROBLEM 9
                          16thSwedish  Mathematical  Olympiad  1976
        Problem#2:  For which real  a  are there distinct real x , y such
                          that  x= a - ( y^2)    and  y= a - (x^2)  .
      PLEASE NOTE: With regard to PROBLEM 9 , which is stated on
              page 4 of this paper , there is some text missing: The missing
                                  part is ,  " and  y= a - (x^2)" .
                                            PROBLEM 10
                      17th  Swedish Mathematical Olympiad  1977
            Problem#1: p is a prime. Find the largest integer d  such that
                                          p^d  divides      (p^4) !    .
                                                  PROBLEM 11
                            18th Swedish  Mathematical Olympiad  1978
          Problem#2:  Let  s(m)    be the number  66......6  , with  m  6s.
                                      Find  s(1) + s(2) + .....+ s(n)  .
                                                    PROBLEM 5
                          6th Swedish  Mathematical  Olympiad  1966
              Problem#3:  Show that an integer congruent to  7(mod8) ,
                                cannot be a sum of three integer squares.
                                                        PROBLEM 6
                            9th Swedish  Mathematical Olympiad  1969
              Problem#1: Find all integers  m , n such that  m^3 = n^3 + n .
                                                        PROBLEM 8
                        14th Swedish  Mathematical Olympiad  1974
        Problem#3:  Let  a(1)= 1 ,  a(2) = 2^[a(1)]  , a(3)= 3^[a(2)] ,
                                a(4)= 4^[a(3)] , .......,  a(9)= 9^[a(8)]  .
                                  Find  the last digits of  a(9) .
      To conclude this abstract , it is worth mentioning that  Swedish
                  Mathematical Olympiad problems and relevant information
                        can be found in the website ,
                                            webee.technion.ac.il

El método homotópico (o del camino homotópico) puede emplearse para el cálculo de las raíces de una función. El trabajo a continuación muestra cuáles son las condiciones que mínimamente se requiere cumplir para garantizar la obtención satisfactoria de las raíces.

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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.

The present study, conducted during 2005-06, in two districts, viz. Amritsar and Faridkot of Punjab, (former having pre-dominantly tubewell-irrigated area and the latter having canal + tubewell irrigation facilities) has assessed the extent of water depletion and has measured ...

International Journal on Computational Science & Applications (IJCSA) is a Bi-monthly peer-reviewed and refereed open access journal that publishes articles which contribute new results in all areas of the Computational Science & Applications. Computational science is interdisciplinary fields in which mathematical models are combined with scientific computing methods to study of a wide range of problems in science and engineering. Modeling and simulation tools find increasing applications not only in fundamental research, but also in real-world design and industry applications.