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At least 253 records · Page 14Linked to original sources
arXiv2009

Unshackling Linear Algebra from Linear Notation

This paper provides an introduction to trace diagrams at a level suitable for advanced undergraduates. Trace diagrams are a non-traditional notation for linear algebra. Vectors are represented by edges in a diagram, and matrices by markings along the edges of the diagram. The notation is rigorous and permits proofs more elegant than those written using traditional notation. We begin with the definition of trace diagrams, and move directly into two special cases that help orient the reader to the diagrammatic point-of-view. We then provide an explicit description of how they are calculated. Finally, we provide the diagrammatic perspective on some questions often posed by students seeing vectors and linear algebra for the first time. We also look at some questions inspired by the diagrammatic notation. We include several examples and exercises throughout, which are particularly important for adjusting to nonstandard notation.

math.HO↗
arXiv2011

Examples of Admissible Simplification of Mathematical Theories

"Mathematicians, like physicists, are pushed by a strong fascination. Research in mathematics is hard, it is intellectually painful even if it is rewarding, and you would not do it without some strong urge." [D. Ruelle]. We shall give some examples from our experience, when we were able to simplify some serious mathematical models to make them understandable by children, preserving both aesthetic and intellectual value. The latter is in particularly measured by whether a given simplification allows setting a sufficient list of problems feasible for school students.

math.HO↗
arXiv2009

Another look at e

This note describes a way of obtaining e that differs from the standard one. It could be used as an alternate way of showing how the value of e is obtained. No attempt is made to show the existence of the limit in the definition of e that appears in the final equation.

math.HO↗
arXiv2012

Approximate expressions for mathematical constants from PSLQ algorithm: a simple approach and a case study

In this note, I present a simple PSLQ code for finding null linear combinations, with the best rational coefficients, of mathematical constants, within some prescribed precision. As an example, I explore approximate expressions for the Apéry's constant $\,ζ{(3)} = \sum_{n\ge1}{\,1/n^3}$, an irrational number to which no exact, finite closed-form expression is known. % For this, I choose a suitable search basis composed by numbers which seem to be closely related to $\,ζ{(3)}$, namely $\,π$, $\,\ln{2}\,$, $\,\ln{(1+\sqrt{2}\,)}$, and $G$ (the Catalan's constant). On taking into account a suitable search basis, I have found a simple expression for $\,ζ{(3)}\,$ accurate to 21 decimal places, which is triply more accurate than the best previous one. As the short \emph{Maple}$^\mathrm{TM}$ code presented here can be easily adapted to study other constants, I decided to supply it to encourage the readers to conduct their own computational experiments, as well as to adopt it in projects of numerical analysis, number theory, or linear algebra.

math.NT↗
arXiv2009

Étienne Bézout : Analyse algébrique au siècle des Lumières

The topic of this paper is, on the one hand to introduce algebraic analysis results of Étienne Bézout (1730- 1783) not as we know them today but as he found them in his time, and on the other hand to emphasize his innovating viewpoints. We will be concerned with Bezout special way of reducing elimination for any degree systems to finding conditions for linear systems solutions, with his typical use of indeterminate coefficients that he doesn't compute but looks only for existence and number, with his idea to work on set of polynomials products sums, and with a very personal method to found two equations resultant.

math.HO↗
arXiv2009

Epistemography and algebra

We propose to address the problem of how to know students' knowledge in an entirely new approach called ?epistemography? which is, roughly, an attempt to describe the structure of this knowledge. We claim that what is to be known is made of five tightly interrelated organised systems: the mathematical universe, the system of semio-linguistic representations, the instruments, the rules of the mathematical game, and the identifiers.

math.HO↗
arXiv2009

Hamlet and Pfister forms (A tragedy in four acts)

In the mid-1960s A. Pfister discovered extraordinary, strongly multiplicative forms which are now called Pfister forms. From that time on, these forms played a dominant role in the theory of quadratic forms. One of the key properties of a Pfister form q is that q extended to a suitable transcendental extension, has the polynomial q as its similarity factor. Pfister's original proof used clever matrix calculations. Here we show that the desired isometry is induced by the multiplication of a suitable field element. We further consider the surprising possibility that Pfister's forms were already known by Hamlet, Rosencrantz and Guildenstern, and that they in fact led to a terrible tragedy which is yet filled with a haunting beauty and mystery that can still inspire us to this day.

math.HO↗
arXiv2010

Jacobi's bound and normal forms computations. A historical survey

Jacobi is one of the most famous mathematicians of his century. His name is attached to many results in various fields of mathematics and his complete works in seven volumes have been available since the end of the XIXth century and are very often quoted in many papers. It is then surprising that some of his results may have fallen into oblivion, at least in part. We will try to describe some of Jacobi's results on ordinary differential equations and the available, published or unpublished material he left. We will then expose the selective interests of his followers and their own contributions. There are in fact many interrelated results: a bound on the order of a differential system, a necessary and sufficient condition, given by a determinant, for the bound to be reached, an algorithm to compute the bound in polynomial time, and processes for computing normal forms using as few derivatives as possible. We give for all of them the form under which they could have been proved or rediscovered, sometimes independently of Jacobi's findings. In conclusion, we give the state of the art and suggest some possible applications of Jacobi's bound to improve some algorithms in differential algebra.

math.CA↗
arXiv2009

On a curious property of 3435

Folklore tells us that there are no uninteresting natural numbers. But some natural numbers are more interesting then others. In this article we will explain why 3435 is one of the more interesting natural numbers around. We will show that 3435 is a Munchausen number in base 10, and we will explain what we mean by that. We will further show that for every base there are finitely many Munchausen numbers in that base.

math.HO↗
arXiv2009

The Citation Impacts and Citation Environments of Chinese Journals in Mathematics

Based on the citation data of journals covered by the China Scientific and Technical Papers and Citations Database (CSTPCD), we obtained aggregated journal-journal citation environments by applying routines developed specifically for this purpose. Local citation impact of journals is defined as the share of the total citations in a local citation environment, which is expressed as a ratio and can be visualized by the size of the nodes. The vertical size of the nodes varies proportionally to a journal's total citation share, while the horizontal size of the nodes is used to provide citation information after correction for the within-journal (self-) citations. In this study, we analyze citation impacts of three Chinese journals in mathematics and compare local citation impacts with impact factors. Local citation impacts reflect a journal's status and function better than (global) impact factors. We also found that authors in Chinese journals prefer international instead of domestic ones as sources for their citations.

cs.DL↗
arXiv2011

On game psychology: an experiment on the chess board/screen, should you always "do your best", and why the programs with prescribed weaknesses cannot be our good friends?

It is noted that some unusual moves against a strong chess program greatly weaken its ability to see the serious targets of the game, and its whole level of play... It is suggested to create programs with different weaknesses in order to analyze similar human behavior. Finally, a new version of chess, "Chess Corrida" is suggested.

cs.AI↗
arXiv2011

Zariski decomposition: a new (old) chapter of linear algebra

In a 1962 paper, Zariski introduced the decomposition theory that now bears his name. Although it arose in the context of algebraic geometry and deals with the configuration of curves on an algebraic surface, we have recently observed that the essential concept is purely within the realm of linear algebra. In this paper, we formulate Zariski decomposition as a theorem in linear algebra and present a linear algebraic proof. We also sketch the geometric context in which Zariski first introduced his decomposition.

math.AG↗
arXiv2009

Best approximation with wavelets in weighted Orlicz spaces

Democracy functions of wavelet admissible bases are computed for weighted Orlicz Spaces in terms of its fundamental function. In particular, we prove that these bases are greedy if and only if the Orlicz space is a Lebesgue space. Also, sharp embeddings for the approximation spaces are given in terms of weighted discrete Lorentz spaces. For Lebesgue spaces the approximation spaces are identified with weighted Besov spaces.

math.HO↗
arXiv2009

A Eulogy for Jack Good

Irving John ("Jack") Good (9 December 1916 - 5 April 2009) was one of my greatest heroes and influencers. On Oct. 25, 2009, I gave a twenty-three minute talk with the present title, and this article is an extended transcript of that talk. As with all my papers, the "accompanying" Maple package is much more important (mathematically, of course, I also talk about the human side of Jack in this article, and this is even more important than any math). In particular, I taught Jack's brilliant ideas to my computer, and now it can do even better than Jack.

math.HO↗
arXiv2010

Intro To Logarithms After The Meltdown

This work is an attempt to 'reconstruct' logarithms in the hypothetical case that mankind has suffered a catastrophe through which all repositories of (mathematical as well as other) knowledge are lost, with the exception of simple arithmetic operations. It turns out that this is possible without re-inventing calculus!

math.HO↗
arXiv2009

Aspects Analytiques Dans la Mathematique De Sharaf Al-Dîn Al-TÛSÎ

The analytical aspects of the "Traité des équations" of Sharaf al-Dîn al-Tûsî (2nd half of the XIIth century) have been underlined by R. Rashed (1974, 1986). In the present paper, we consider again some of those aspects, when studying the "second part" of the "Traité". We find out that al-Tûsî introduces certain notions, develops reasonings and uses calculating procedures that may allow his work to be an important reference in the history of the mathematical analysis. As such, one is led to examine the foundations of a J.P. Hogendijk conjecture (1989), which situates the advanced mathematics of this "2nd Part" in the only domain of Euclidian geometry. Simultaneously, we shall try to shed some light on a passage of the "Traité", made unclear by al-Tûsî style and expression. Our study is essentially based on the texts to confirm the originality and the importance of these XIIth century mathematics. I would like to express my thanks to Prof. C. Houzel (Univ. Paris VII) and B. El-Mabsout (Univ. Paris VI), for reading the manuscript and providing their precious remarks.

math.HO↗