By Arieh Iserles

ISBN-10: 0521443563

ISBN-13: 9780521443562

This is often the second one quantity within the annual sequence, Acta Numerica, a suite of invited survey papers encompassing all elements of numerical research. Following the luck of the 1st quantity, this year's papers disguise a few of the center subject matters in numerical tools this day. Papers on annealing difficulties, parallel tools, area decomposition, and multigrid equipment will let a person with an curiosity in computational and mathematical numerics to fast grab the most recent advancements and rising tendencies during this varied topic.

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2 II. 15) The task is to inscribe a cone of maximal volume into a sphere of radius r (Fig. 3). r be the radius of the sphere, x the height of the inscribed cone. After setting V( x1) = V( x2) which implies V( x1) − V( x2) = 0 the latter difference is factored into a product ( x1 − x2 ) ⋅ g ( x1 , x2 ) = 0 Then setting x1 = x2 = x0 the equation g ( x0 , x0 ) = 0 has to be solved to determine the extremal point. The following steps demonstrate an example: 1 1 x1 (2r − x1 ) x1 = x2 (2r − x2 ) x2 3 3 2rx12 − x13 = 2rx2 2 − x23 ( x13 − x23 ) − 2r ( x12 − x2 2 ) = 0 x12 + 2 x1 x2 + x2 2 − 2r ( x1 + x2 ) = 0 x1 = x2 = x0 3 x0 2 − 4rx0 = 0 22 H.

The fact that these two sides of mathematics are reflected at school and at university in quite a different way has serious consequences for the role of important characteristics of mathematics like proving, rigor, or formalism. For example, most of the mathematical concepts in school are introduced and used informally (Engelbrecht 2010). Accordingly, students mainly work with a “concept image” of a concept (in the sense of Tall and Vinner 1981), whereas the “concept definition” of most of the concepts does not play a prominent role.

In this contribution, we describe an approach to examine and describe the two discontinuities and the related requirements for the corresponding transformations through textbook comparisons. Mathematics textbooks play a decisive role for students’ learning processes at school (Rezat 2009) as well as at university (Alsina 2001). Accordingly, we first assume that the way mathematics is presented in the textbooks can be considered as an indicator of the character of school mathematics and academic mathematics.

### Acta Numerica 1993: Volume 2 by Arieh Iserles

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