General relativity, differential geometry, and unitary theories in the work of Mira Fernandes
An analysis of the work of Mira Fernandes on unitary theories is presented.
physics.hist-ph↗
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Results for “math.HO”
Search indexed arXiv papers on artificial intelligence, large language models, computer vision and robotics. Read source abstracts and follow links to arXiv.
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Quaternions In Three Dimensions
We construct the quaternion algebra [10] "geometrically" by a three dimensional analogue of the classic two dimensional geometric description of the complex field. The algebraic description of the multiplication operation in three dimensions involves the addition of one term. The construction leads to novel methods for implementing and interpolating rotations and understanding their topology.
math.RA↗
Some fascinating series and their sums
In this paper we present some interesting results involving summation of series in particular trigonometric ones. We failed to locate these results in existing literature or in the web like MathWorld (http://mathworld.wolfram.com/) nor could we derive them using software for analytical computation like Maple. The identities are beautiful and involve finite series.
cs.DM↗
Solution for the problem of the game heads or tails
In this paper, we describe the solution for a problem dealing with definite properties of binary sequences. This problem, proposed by Xavier Grandsart in the form of a mathematical contest, has been solved also by Maher Younan, Ph.D. student of Theoretical Physics at the University of Geneva, and Pierre Deligne, professor at Princeton and Field Medals, using different approaches with respect to the one presented in this work.
math.HO↗
The `Real' Schwarz Lemma
The purpose of this note is to discuss the real analogue of the Schwarz lemma from complex analysis.
math.HO↗
Square Trisection
A square trisection is a problem of assembling three identical squares from a larger square, using a minimal number of pieces. This paper presents an historical overview of the square trisection problem starting with its origins in the third century. We detail the reasoning behind some of the main known solutions. Finally, we give a new solution and three ruler-and-compass constructions. We conclude with a conjecture of optimality of the proposed solution.
math.HO↗
If a prime divides a product
One of the greatest difficulties encountered by all in their first proof intensive class is subtly assuming an unproven fact in a proof. The purpose of this note is to describe a specific instance where this can occur, namely in results related to unique factorization and the concept of the greatest common divisor.
math.HO↗
Archimedes' calculations of square roots
We reconsider Archimedes' evaluations of several square roots in 'Measurement of a Circle'. We show that several methods proposed over the last century or so for his evaluations fail one or more criteria of plausibility. We also provide internal evidence that he probably used an interpolation technique. The conclusions are relevant to the precise calculations by which he obtained upper and lower bounds on pi.
math.HO↗
Mathematics and Economics of Leonid Kantorovich
This is a short overview of the contribution of Leonid Kantorovich into the formation of the modern outlook on the interaction between mathematics and economics.
math.HO↗
Galois Groups in the work of Mira Fernandes
In connection to a review of the writings on Galois Theory of the Portuguese mathematician Aureliano Mira Fernandes at turn of the XXth century a brief narrative is presented on how the group theory concepts discovered by Galois are connected to previous work that took centuries to evolve, and how fast they changed the landscape of theoretical and applied mathematics and physics since its publication. From the conference "Mira Fernandes and his age - An historical Conference in honour of Aureliano de Mira Fernandes (1884-1958)", IST Lisbon, June 2009
math.HO↗
A new approach to the real numbers
In this paper we provide a complete approach to the real numbers via decimal representations. Construction of the real numbers by Dedekind cuts, Cauchy sequences of rational numbers, and the algebraic characterization of the real number system by the concept of complete ordered field are also well explained in the new setting.
math.CA↗
Geometric Algebra: A natural representation of three-space
Historically, there have been many attempts to produce an appropriate mathematical formalism for modeling the nature of physical space, such as Euclid's geometry, Descartes' system of Cartesian coordinates, the Argand plane, Hamilton's quaternions and Gibbs' vector system using the dot and cross products. We illustrate however, that Clifford's geometric algebra (GA) provides the most elegant description of physical space. Supporting this conclusion, we firstly show how geometric algebra subsumes the key elements of the competing formalisms and secondly how it provides an intuitive representation of the basic concepts of points, lines, areas and volumes. We also provide two examples where GA has been found to provide an improved description of two key physical phenomena, electromagnetism and quantum theory, without using tensors or complex vector spaces. This paper also provides pedagogical tutorial-style coverage of the various basic applications of geometric algebra in physics.
physics.hist-ph↗
Centre for Mathematical Sciences India (CMS): Professor A.M. Mathai's 75th Birthday
A brief overview on the Centre for Mathematical Sciences India, established in 1977, and its teaching and research programme is given.
math.HO↗
Martin Gardner's Mistake
When Martin Gardner first presented the Two-Children problem, he made a mistake in its solution. Later he corrected the mistake in another publication, but unfortunately his incorrect solution is more widely known than his correction. In fact, a Tuesday-Child variation of this problem went viral in 2010, and the same flaw keeps reappearing in solutions for that problem as well. In this article, I would like to popularize Martin Gardner's correction and conduct a detailed discussion of the new problem.
math.PR↗
Is Zero a Natural Number?
It is argued that zero should be considered as a cardinal number but not an ordinal number. One should make a clear distinction between order types that are labels for well-ordered sets and ordinal numbers that are labels for the elements in these sets.
math.HO↗
A Brief Review of SIAM Review
SIAM Review is examined for referee delay, citations, and paper length after the reorganization of the journal in 1999. A single, very-highly cited article was responsible for all the increase to the impact factor during the past decade; the reorganization did not improve the journal overall. Some suggestions are made for additional changes.
math.HO↗
A simple proof of the Abel-Ruffini theorem
This paper is purely expositional. The statement of the Abel-Ruffini theorem on unsolvability of equations using radicals is simple and well-known. We sketch an elementary proof of this theorem. We do not use the terms 'field extension', 'Galois group' and even 'group'. However, our presentation is a good way to learn (or recall) starting idea of the Galois theory. Our exposition follows `Mathematical Omnibus' of S. Tabachnikov and D.B. Fuchs (in English, http://www.math.psu.edu/tabachni/Books/taba.pdf). The main difference is that we show how the proof could have been invented. The paper is accessible for students familiar with complex numbers, and could be an interesting easy reading for mature mathematicians. The material is presented as a sequence of problems, which is peculiar not only to Zen monasteries but also to advanced mathematical education; most problems are presented with hints or solutions.
math.HO↗
Upside Down Magic, Bimagic, Palindromic Squares and Pythagoras Theorem on a Palindromic Day - 11.02.2011
In this short paper we have produced different kinds of upside down magic squares based on a palindromic day 11.02.2011. In this day appear only the algorisms 0, 1 and 2. Some of the magic squares are bimagic and some are palindromic. Magic sums of the magic squares of order 3x3, 4x4 and 5x5 satisfies the Pythagoras theorem. Bimagic squares of order 9x9 are produced with 4, 6 and 8 digits. The bimagic square of order 9x9 with 8 digits is of palindromic numbers. We have given bimagic squares of order 16x16 and 25x25, where the magic sum S1 in both the cases is same. In order to make these magic squares upside down, i.e., 180 degrees rotation, we have used the numbers in the digital form. All these magic square are only with three digits, 0, 1 and 2 appearing in the day 11.02.2011.
math.HO↗