Mathematics

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

All scientists are very much known with statistics and that is a great thing. But statistics can also work against science as also is known when statistics is used in a wrong way. There is however also a way in which statistics can work against reality in a less known way. To that, this paper is conveying the message that a mathematical correct probability does not necessarily equals measurable reality. What is the probability that 1.000.000.000 coins all land heads up ? That is fairly easy to calculate if we have a perfect 50/50 coin. That chance is (1/2)^1.000.000.000 which is a  slim chance. The mathematical probability is real ! No argue about that. But how real/realistic is the actual happening that 1 billion coins will observably all land heads up ?. Is physical reality the same as mathematical probability ? Does math always line up with reality ? and do we scientists dare to be honest about that in all cases ?

keywords: probability, statistics, reality, mathematics, measurements, quantum mechanics, 5S, 6S, outliers, natures lumpyness, π, e, √2, mathematical, coins, scientific truth/honesty, amino acids, proteins, life.

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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 Goldbach Conjecture remains one of the several unsolved mathematical problems today, along with the Twin Prime Conjecture. It is said to be one of the simplest mathematical problems to state yet the most difficult to prove.  In this paper, the author explores a few even numbers about the problem, and, using them as his sample size, juxtaposes his ideas with those of others through literature review. The author explores a solution to the Goldbach conjecture which was put forward about 275 years ago, by a German mathematician, Christian Goldbach, who was a contemporary of the genius German mathematician, Leonhard Euler. Euler in a letter to Goldbach, had confessed at the time that unfortunately, he (Euler) could not prove Goldbach’s mathematical poser. Through fundamental analysis and critical thinking, this author attempts to share his thoughts on how to resolve this long-standing mathematical debacle. The methodology used is basic, experimental, and exploratory, with the use of inferences through recognition of number patterns. We hope that this paper will make a small contribution to the discourse on Goldbach’s conjecture, whose solution has eluded mathematicians for about 275 years.

Our project aims to enlighten the people running an election campaign about how popular they are and how their campaign is, based on reviews fetched from various social media platforms such as Twitter, Facebook, Instagram and so on. The idea behind this project is to give the election contestants a fair idea about how well they are faring and also get an understanding of what they need to work on based on the comments from the social media sites. The project is very user friendly and simple to use as it requires only keywords that must be input into the machine and the algorithm will do the rest. We provide a visual, graphic representation of the output so that it is simpler to understand.

The computerized world we are living in has a ton of information and data that is utilized by an assortment of clients, for example, videos, books and articles. Different users like different content. Getting what each user likes can be irritating. Each online services provider always aims in having many clients. Recommender systems importance’s arises in such situations. The recommender framework proposes to a client some substance for example e.g. movies and books depending on what the user likes. In this research, a new movie recommender system is proposed, that will be able to improve the existing recommender systems. With this new recommender framework, the client will receive an improved forecast contrasted with different frameworks that as of now exist for example content-based filtering and collaborative-filtering. In order to overcome the disadvantages of the both methods (collaborative filtering algorithm (unsupervised learning) and content-based filtering algorithm (supervised learning)), the new system combines both methods. This will bring up a more stable system compared to the existing ones.

Les racines culturelles du message de Jésus proviennent de l’antique Calabre, où le roi Italo fonda l’Italie avec le sissize, le banquet commun et égalitaire. Autour du sissize s’est développé les valeurs de l'amitié, de la liberté, de la justice sociale, du refus du sacrifice de sang, de l'alimentation végétarienne, de la communauté de vie et de biens, puis adoptées par Pythagore et résumées dans le "Sissize Pythagoricien", le dîner rituel avec pain et vin. La doctrine de l'École Pythagoricienne de Crotone s'est répandue en Méditerranée, arrivant également aux Esséniens, qui célébraient le même sissize, qui est ensuite devenue le repas de Jésus “la Cène”. La Calabre fut donc le berceau de l'Eucharistie. De là vient aussi l'hostie ronde et blanche de la messe et la nappe en lin blanc sur les autels. Ce livre explique les cultures à travers lesquelles la doctrine de Jésus s'est formée et donne un nouvel éclairage sur le destin de l'humanité : tant que les évangiles seront lus, ce livre sera également lu.
Le message de Jésus trouve ses racines dans l'Italie antique, née en Calabre, et dans l'École Pythagoricienne de Crotone : amitié, liberté, justice sociale, refus du sacrifice de sang, alimentation végétarienne, communauté de vie et de biens, dignité de la femme. Ce livre, avec la découverte de ces racines culturelles, révèle un Nouveau Christ et annonce sa Seconde Venue.

The safety and short-term therapeutic benefit of multilesion percutaneous transluminal coronary angioplasty was assessed in 135 patients, 66 of whom had a minimum of 6 months of follow-up study. Primary success, defined as successful dilation of the most critical lesion or all lesions attempted without major in-hospital complications was obtained in 117 (87%) of the 135 patients. Cardiac complications associated with the procedure were uncommon; prolonged angina occurred in 5% and myocardial infarction in 3%; emergency coronary bypass surgery was performed in 4% of the patients. There were no deaths. Complete revascularization was achieved in 46% of the 117 patients with a primary success. Of the 66 patients eligible for 6 month follow-up, 80% had an uncomplicated course and required no further procedures. Clinical improvement by at least one angina functional class was observed in 90% of the patients. Cardiac events such as the need for a second revascularization procedure were significantly more common in patients who had incomplete versus complete revascularization (35 versus 9%; p = 0.018). Repeat coronary angiography performed an average of 5 months after angioplasty revealed restenosis in 18 of 22 symptomatic patients and 3 of 9 asymptomatic patients. Restenosis occurred at the site of a single dilation in 12 patients, at two sites in 8 patients and at three sites in 1 patient. Thus, multilesion coronary angioplasty is an important therapeutic option for selected patients with multivessel disease and can be performed with relatively low risk. Improvement in angina status can be expected even in patients who have incomplete revascularization.(ABSTRACT TRUNCATED AT 250 WORDS)