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# jff's jff-bib [16 articles]

Recent papers added to jff's library classified by the tag jff-bib. You can also see everyone's jff-bib.

## Designing an Algorithmic Proof of the Two-Squares Theorem

[CiTO]
In Mathematics of Program Construction (2010)

### Abstract

We show a new and constructive proof of the two-squares theorem, based on a somewhat unusual but very effective, way of rewriting the so-called extended Euclid's algorithm. Rather than simply verifying the result---as it is usually done in the mathematical community---we use Euclid's algorithm as an interface to investigate which numbers can be written as sums of two squares. The precise formulation of the problem as an algorithmic problem is the key, since it allows us to use algorithmic techniques and ...

## On the Inexistence of a Unique Existential Binary Operator

[CiTO]
(August 2009)
posted to by jff on 2009-11-27 22:06:17

## ✔ Which Mathematics for the Information Society?

[CiTO]
Teaching Formal Methods In Teaching Formal Methods, Vol. 5846 (2009), pp. 39-56, doi:10.1007/978-3-642-04912-5_4
posted to by jff on 2009-11-27 21:53:37

### Abstract

MathIS is a new project that aims to reinvigorate secondary-school mathematics by exploiting insights of the dynamics of algorithmic problem solving. This paper describes the main ideas that underpin the project. In summary, we propose a central role for formal logic, the development of a calculational style of reasoning, the emphasis on the algorithmic nature of mathematics, and the promotion of self-discovery by the students. These ideas are discussed and the case is made, through a number of examples that show ...

## Students' Feedback on Teaching Mathematics Through The Calculational Method

[CiTO]
Frontiers in Education 2009 (2009)
posted to by jff on 2009-06-26 15:08:10

## On Euclid's Algorithm and Elementary Number Theory

[CiTO]
(2009)

### Abstract

Algorithms can be used to prove and to discover new theorems. This paper shows how algorithmic skills in general, and the notion of invariance in particular, can be used to derive many results from Euclid's algorithm. We illustrate how to use the algorithm as a verification interface (i.e., how to verify theorems) and as a construction interface (i.e., how to investigate and derive new theorems).The theorems that we verify are well-known and most of them are included in standard number-theory books. ...

### Note (first note only)

Submitted for publication. Available at http://joaoff.com/publications/2009/euclid-alg/

## ✔ Recounting the Rationals: Twice!

[CiTO]
Mathematics of Program Construction In Mathematics of Program Construction, Vol. 5133 (2008), pp. 79-91, doi:10.1007/978-3-540-70594-9_6
posted to by jff on 2008-07-28 11:57:32 along with 1 person

### Abstract

We derive an algorithm that enables the rationals to be efficiently enumerated in two different ways. One way is known and is credited to Moshe Newman; it corresponds to a deforestation of the so-called Calkin-Wilf tree of rationals. The second is new and corresponds to a deforestation of the Stern-Brocot tree of rationals. We show that both enumerations stem from the same simple algorithm. In this way, we construct a Stern-Brocot enumeration algorithm with the same time and space complexity as ...

## A contribution towards a Distributed Java Virtual Machine

[CiTO]
posted to by jff on 2008-04-13 20:09:47

## An exercise from "The Art of Computer Programming"

[CiTO]
(February 2008)
posted to jff-bib by jff  on 2008-02-29 18:28:43 along with 2 groups

## Resolução eficaz de problemas: quatro exemplos

[CiTO]
(2008)
posted to jff-bib by jff on 2008-02-29 18:27:58

## The Chinese Remainder Theorem

[CiTO]
(2007)
posted to jff-bib by jff  on 2008-02-29 18:26:29 along with 2 groups

## Exchanging the values of two variables

[CiTO]
(July 2007)
posted to jff-bib by jff  on 2008-02-29 18:24:32 along with 2 groups

## Distributivity and the greatest common divisor

[CiTO]
(2006)
posted to by jff  on 2008-02-29 18:22:41 along with 2 groups

## Elementary Number Theory

[CiTO]
(2007)
posted to by jff  on 2007-03-05 10:08:26 along with 4 groups

## JaSkel: A Java Skeleton-Based Framework for Structured Cluster and Grid Computing

[CiTO]
In CCGRID '06: Proceedings of the Sixth IEEE International Symposium on Cluster Computing and the Grid (CCGRID'06) (2006), pp. 301-304
posted to jff-bib by jff  on 2006-09-26 12:00:15 along with 4 groups

## ParC\#: Parallel Computing with C\# in .NET.

[CiTO]
In PaCT (2005), pp. 239-248
posted to jff-bib by jff  on 2006-09-26 12:00:15 along with 4 groups

## Camila Revival: VDM meets Haskell

[CiTO]
In First Overture Workshop (2005)
posted to jff-bib by jff  on 2006-09-26 12:00:15 along with 4 groups