Finite-dimensional algebras with smallest resolutions of simple modules

Let $X$ be a finitely generated left module over a left artinian ring $R$, and let $p(X)=\{l_i\}$ be the infinite sequence of nonnegative integers where $l_i$ is the length of the $i$-th term of the minimal projective resolution of $X$. We introduce a preorder relation $\le$ on the set $\{p(X)\}$ and characterize the elementary finite-dimensional algebras $\Lambda$ with the following property. Let $S$ be a simple $\Lambda$-module, and let $T$ be a finitely generated module over an arbitrary left artinian ring $R$. If the projective dimension of $S$ does not exceed the projective dimension of $T$, then $p(S)\le p(T)$. We characterize the indicated algebras by quivers with relations.Comment: Minor revisions, to appear in Journal of Algebr

From regular pentagons to the icosahedron via the golden ratio - part I

This series of articles will explore an amazing connection between three different objects in mathematics: the regular pentagon, the Golden Ratio and the icosahedron. Obviously, if the Golden Ratio is involved then the Fibonacci sequence can’t be far behind and, if the icosahedron is, so is its dual the dodecahedron! In the first of these articles we will begin with the question of how to construct regular polygons, and restrict our attention to the regular pentagon and some of its properties. The regular pentagon serves as a doorway to a veritable treasure house of interconnected mathematical ideas and it never fails to astonish me. At the outset I would like to acknowledge that almost all the ideas discussed here can be found in [3]. In this series of articles we expand on some of the ideas and take a few digressions along the way

Review of "math! encounters with high school students" by serge lang.

The notion of dialogue and mathematics may at first seem a strange combination, but if one thinks about it, often in a lively interactive classroom this is exactly what is transpiring. According to the late physicist David Bohm, the root of the word dialogue comes from the Greek dialogus. The word logos in turn can be interpreted as ‘meaning of the word’ and dia means ‘through’. So dialogue can then be seen as a process where there is a flow of meaning. All teachers would agree that this is what they would like in their classroom. The book under review, Math! Encounters with High School Students by Serge Lang, is an old one, published in 1985, but well worth bringing to the notice of students and teachers of mathematics. It is a series of seven dialogues on mathematics with school students and a postscript discussing mathematics teaching

The story of maths : a brief review

If you want a whirlwind tour of the history of mathematics in four hours, where you will be taken to marvellous sites all around the globe, meet historians of mathematics, mathematicians, curators of museums, descendants of famous mathematicians and also learn some mathematics, then you must see the four-part series called The Story of Maths

A review of the cartoon guide to algebra and the cartoon guide to calculus

I don’t know what your experience of learning mathematics was—but for me, till I went to graduate school and except for a few courses as an undergraduate, it was a very heavy affair. First of all, there was an overwhelming sense of being weighed down that was associated with ‘knowledge’. There was so much to learn, so much to remember and so much to be tested on! There was no sense of lightness associated with learning, no sense of play or joy in discovering and understanding. I remember once, wandering around a huge library with this feeling, and coming upon a lovely poem by Justin Richardson (Punch, 1952), which gave me immense relief. It goes like this

From regular pentagons to the icosahedron and dodecahedron via the golden ratio – ii

In the previous article (At Right Angles, Issue 4, July 2019, pages 5-9 ) we saw how we could construct a regular pentagon using a ruler and compass, and discovered a nested sequence of pentagons that can be built up by extending the sides of a given regular pentago

A culture of enjoying mathematics

It appears that whether we like or not, Mathematics pervades all aspect of our life. Whether you are a farmer or a techie, a comfortable relationship with mathematics, and competency at the level at which one uses it,is a requisite in a equitable society

Introducing numberphile

Ever since we decided to reintroduce our readers to Numberphile(https://www.numberphile.com/), I have been obsessively watching videos hosted via YouTube on the site! I have also listened to some podcasts. The content is both mathematics and interviews with mathematicians about a range of topics from “Why do people hate mathematics?” to “Fame and Admiration.

Introducing Steven Strogatz

He is the author of several best-selling books like The Joy of X: A Guided Tour of Math, From One to Infinity. He writes frequently for the New York Times, and appears regularly on National Public Radio in the US. The purpose of this short review is to alert readers to Steven Strogatz’s work (http://www.stevenstrogatz.com) in general, and to draw special attention to 15 pieces, under the broad heading of ‘Elements of Maths,' that he wrote for the New York Times from January 2010 to May 2010(http://www.stevenstrogatz.com/essays/tag=Elements+of+Math)

Towards mathematical disposition notes from a small school supportive learning spaces

f there is one thing mathematicians or math educators are agreed upon, it is that the state of math education the world over is very unsatisfactory. Many schools have poor infrastructure and often an absent teacher. Even in schools with good infrastructure and teachers present, the curriculum is often dry and unimaginative and the textbooks more so