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195
CauchyBinet for pseudo determinants
, 2013
"... Abstract. The pseudodeterminant Det(A) of a square matrix A is defined as the product of the nonzero eigenvalues of A. It is a basisindependent number which is up to a sign the first nonzero entry of the characteristic polynomial of A. We extend here the CauchyBinet formula to pseudodeterminants ..."
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Abstract. The pseudodeterminant Det(A) of a square matrix A is defined as the product of the nonzero eigenvalues of A. It is a basisindependent number which is up to a sign the first nonzero entry of the characteristic polynomial of A. We extend here the CauchyBinet formula to pseudodeterminants. More specifically, after proving some properties for pseudodeterminants, we show that for any two n × m matrices F, G, the formula Det(F T G) = P det(FP)det(GP) holds, where det(FP) runs over all k×k minors of A with k = min(rank(F T G), rank(GF T)). A consequence is the following Pythagoras theorem: for any selfadjoint matrix A of rank k one has Det 2 (A) = ∑ P det2 (AP), where det(AP) runs over all k × k minors of A. 1.
Compass and Straightedge in the Poincaré Disk
, 2001
"... The spirit of this article belongs to another age. Today, “geometry” is most often analytic; this is especially suited to the making of nice pictures by computer. But here we give a synthetic approach to the development of hyperbolic geometry; our constructions use only a Euclidean compass and Eucli ..."
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The spirit of this article belongs to another age. Today, “geometry” is most often analytic; this is especially suited to the making of nice pictures by computer. But here we give a synthetic approach to the development of hyperbolic geometry; our constructions use only a Euclidean compass and Euclidean straightedge, and can be carried out by hand. Indeed, M.C. Escher used something like the methods we give here to produce his well known Circle Limit I, II, III, and IV prints [9]. In [6], H.S.M. Coxeter describes a remarkable correspondence with Escher. Having met at the 1954 International Congress of Mathematics in Amsterdam, Coxeter apparently sent Escher a paper in which a drawing of part of a tiling of the Poincaré disk appeared. Coxeter must have been quite pleased and surprised to find a print of Circle Limit I in his mail in December 1958. It is quite remarkable that the drawing in the paper Coxeter had sent Escher was not even as detailed as our Figure 1, and did not show the “scaffolding ” Coxeter had used in its construction. Nonetheless Escher deduced and generalized the technique of its construction, producing incredibly fine tesselations of the Poincarédisk.
Gröbner Bases and Invariant Theory
"... This paper was Hilbert's quick answer to those of his fellow mathematicians who harshly criticized the nonconstructiveness of his first proof. The second paper contains the Nullstellensatz, the HilbertMumford criterion ..."
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This paper was Hilbert's quick answer to those of his fellow mathematicians who harshly criticized the nonconstructiveness of his first proof. The second paper contains the Nullstellensatz, the HilbertMumford criterion
Technology as a Transformative Force in Education: What Else Is Needed to Make It Work? 12
"... Looking Beyond the RearView Mirror Computational technology is a profoundly transformative force across our society. In importance, it is analogous to technological inventions such as the printing press or the internal combustion engine, and such representational inventions such as alphabetic writi ..."
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Looking Beyond the RearView Mirror Computational technology is a profoundly transformative force across our society. In importance, it is analogous to technological inventions such as the printing press or the internal combustion engine, and such representational inventions such as alphabetic writing. Impacts of such enormous
Infinite series from history to mathematics education, International Journal for Mathematics Teaching and Learning, ISSN 1473 – 0111, http://www.ex.ac.uk/cimt/ijmtl/bagni.pdf. Bagni, G.T. (forthcominga), Historical roots of limit notion. Development of i
 Educational Studies in Mathematics
, 2005
"... Abstract: In this paper an example from the history of mathematics is presented and its educational utility is investigated, with reference to pupils aged 1618 years. Students ’ behaviour is examined: we conclude that historical examples are useful in order to improve teaching of infinite series; h ..."
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Abstract: In this paper an example from the history of mathematics is presented and its educational utility is investigated, with reference to pupils aged 1618 years. Students ’ behaviour is examined: we conclude that historical examples are useful in order to improve teaching of infinite series; however their effectiveness must be verified by the teacher using experimental methods, and the primary importance of the cultural context must be taken into account.
The calculus as algebraic analysis: Some observations on mathematical analysis in the 18th century. Archive for History of Exact Sciences
, 1989
"... (a) Equality of mixed partial differentials.................. 319 ..."
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(a) Equality of mixed partial differentials.................. 319
The Critical Analysis of the Pythagorean Theorem and of the Problem of Irrational Numbers
"... Abstract. The critical analysis of the Pythagorean theorem and of the problem of irrational numbers is proposed. Methodological basis for the analysis is the unity of formal logic and of rational dialectics. It is shown that: 1) the Pythagorean theorem represents a conventional (conditional) theoret ..."
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Abstract. The critical analysis of the Pythagorean theorem and of the problem of irrational numbers is proposed. Methodological basis for the analysis is the unity of formal logic and of rational dialectics. It is shown that: 1) the Pythagorean theorem represents a conventional (conditional) theoretical proposition because, in some cases, the theorem contradicts the formallogical laws and leads to the appearance of irrational numbers; 2) the standard theoretical proposition on the existence of incommensurable segments is a mathematical fiction, a consequence of violation of the two formallogical laws: the law of identity of geometrical forms and the law of lack of contradiction of geometrical forms; 3) the concept of irrational numbers is the result of violation of the dialectical unity of the qualitative aspect (i.e. form) and quantitative aspect (i.e. content: length, area) of geometric objects. Irrational numbers represent a calculation process and, therefore, do not exist on the number scale. There are only rational numbers.