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"ECKART SCHMIDT"
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n-Gons
This book, a translation of the German volumen-Ecke, presents an elegant geometric theory which, starting from quite elementary geometrical observations, exhibits an interesting connection between geometry and fundamental ideas of modern algebra.
eBook
Jesu, meine Freude“. Zu einer theologischen und pragmatischen Hermeneutik von Text und Musik in J.S. Bachs Motette BWV 227
J.S. Bach's motet \"Jesu, meine Freude\" (BWV 227) combines three textual layers: two language texts (verses from Paul's letter to the Romans, and Franck's sacred hymn from the 17th century) and Bach's musical \"text\". This essay investigates the relation between these three \"texts\". An introductory chapter (I.) considers the historical background of the composition of the motet. Then (II.), both literary layers are analysed through a comparison of the texts. The (significant) differences between them are highlighted and it is shown how they relate to each other in combination, sometimes even resulting in a shiftof their original meaning. In the next chapter (III.) the layer of Bach's music is added to the hermeneutical analysis by exploring its structural function and how the music can support, or even reshape, the meaning of the words. The last chapter (IV.) begins by investigating possibilities of a theological hermeneutics of \"mere\" music independently from words. Unlike hermeneutical approaches that too quickly identify musical experiences with \"religious\" experiences, the thesis suggested here is that musical experience may indeed be understood as analogous to legitimate religious experience, but is called to accountability before the \"Logos\" to remain theologically responsible. Just as Franck's poem individualizes and emotionalizes Paul's dogmatics, so does Bach's music individualize and emotionalize both texts, instigating or even necessitating a combination of an aesthetics of work and reception. [PUBLICATION ABSTRACT]
Journal Article
Rational components of an n-gon
Ann-gonAis said to beQ-regularif, for each divisord≠ 1 ofn, all omitting sub-d-gons ofAare isobaric.¹ Let us denote the set ofQ-regularn-gons by n.
For the divisord=nthis definition merely requiresAto be isobaric to itself. Since this is always the case, we can limit ourselves to non-trivial divisorsdofn(d ≠ 1,n). Every monogon, and everyp-gon (primep), isQ-regular: 1= p. Thus the concept ofQ-regularity says nothing for monogons andp-gons. In order to see the strength or weakness
Book Chapter
The main theorem about cyclic classes
First letRbe a commutative ring with 1. The units (invertible elements) ofRform an abelian groupUwith respect to multiplication. Elementsa,b
ԑ
Rare calledassociated(in symbolsa~b) if there existsu
ԑ
Usuch thatua=b. This association is an equivalence relation onR. The class of associates in whichalies isUa. (Note thatacan be a zero divisor, and then the equationxa=bforb=Ocan have several solutions for the oneb. Thus, ifaandbare associated, there may still be additional equations
Book Chapter
Boolean algebras of the n-gonal theory I
Assuming the data of §1.1, let us form as in §6.2 the polynomial ringK[x] over the given fieldK, which is also an algebra overK. As before, letkbe the number of prime factors of the polynomialxn
–1 inK[x]. By chapter 6, theorem 3,xn
–1 is square-free. ThereforeL(xn
–1), the lattice of divisors ofxn
–1, is a Boolean algebra with 2kelements.
Substitution of the cyclic mapping ζ in the polynomialsf(x)ԑ K[x],
\\[x//\\zeta :f(x)\\to f(\\zeta ),\\] (1)
is a homomorphism ofK[x] ontoK[ζ], the algebra of cyclic mappings. The kernel of this homomorphism
Book Chapter
Cyclic classes of n-gons
Letnbe a natural number andKa (commutative) field whose characteristic does not divide the numbern. The elements ofKare denoted bya,b, … , the zero element by 0, and the unit element by 1. By the condition imposed upon the characteristic, the element l/n exists inK. (This condition is fulfilled for everynin fields of characteristic 0, in particular the fieldQof rational numbers.)
LetVbe a vector space overK. The elementsa,b, … ofVare also calledpoints, and in particular the zero vectoro
Book Chapter
Boolean algebras of the n-gonal theory II
First let us recall a well-known result:
Lemma Let M, N be sets and φ a mapping of M into N, ψ a mapping of N into M, such that
(a) \\[\\psi \\varphi x=x\\quad for\\ all\\ x\\in M,\\]
(b) \\[\\varphi \\psi y=y\\quad for\\ all\\ y\\in N.\\]
Then φ is a one-one mapping of M onto N and ψ is its inverse.
For the proof note thatφis one-one:forφx
1=φx
2impliesψφpx
l=ψφpx
2, and thusx
1=x
2from (a). Also everyy ԑ Nis, by (b), theφ-image ofψy ԑ M.
Now letRbe a ring with 1 and 𝕬 anR-module (cf. §7.1). We consider
Book Chapter