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Algebraic K-Theory: Conference on Algebraic K-Theory : by Grzegorz Banaszak, Wojciech Gajda, Piotr Krason

By Grzegorz Banaszak, Wojciech Gajda, Piotr Krason

This ebook comprises court cases of the study convention on algebraic $K$-theory that came about in Poznan, Poland, in September 1995. The convention concluded the task of the algebraic $K$-theory seminar held on the Adam Mickiewicz collage within the educational 12 months 1994-1995. Talks on the convention coated quite a lot of present examine actions in algebraic $K$-theory. specifically, the next subject matters have been lined: $K$-theory of fields and earrings of integers; $K$-theory of elliptic and modular curves; concept of reasons, motivic cohomology, Beilinson conjectures; and, algebraic $K$-theory of topological areas, topological Hochschild homology and cyclic homology. With contributions through a few best specialists within the box, this e-book offers a glance on the country of present learn in algebraic $K$-theory

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Additional resources for Algebraic K-Theory: Conference on Algebraic K-Theory : September 4-8, 1995 the Adam Mickiewicz University, Poznan, Poland

Example text

2 Prodotto righe per colonne 29 Osserviamo che il numero delle colonne di A coincide con il numero delle righe di B ed `e 3. Quindi si pu` o procedere e si ottiene    2 −1  3·0+2·1+0·1 3 · (−1) + 2 · 1 + 0 · 1 3 2 1 · (−1) + 2 · 1 + 1 · 1    1·0+2·1+1·1     −1 −1   0 · 0 + 0 · 1 + (−1) · 1 0 · (−1) + 0 · 1 + (−1) · 1     A·B = = 2 6 3 · (−1) + 2 · 1 + 7 · 1    3·0+2·1+7·0     1 1  1·0+1·1+1·0 1 · (−1) + 1 · 1 + 1 · 1 2 0 2·0+2·1+0·0 2 · (−1) + 2 · 1 + 0 · 1 Si osservi che, come previsto dal discorso generale, il numero di righe della matrice prodotto `e 6 (come A) e il numero delle sue colonne `e 2 (come B).

Questa idea ci porter` a allo studio delle matrici elementari, metter`a ancora in risalto l’importanza del prodotto righe per colonne e permetter`a di elaborare un algoritmo, detto metodo di Gauss, basato sulla scelta di speciali elementi chiamati pivot. Saremo in grado di calcolare l’inversa di una matrice, nel caso che esista, faremo una digressione sul costo computazionale del metodo di Gauss, impareremo quando e come decomporre una matrice quadrata in forma LU , ossia prodotto di due speciali matrici triangolari.

Nell’ultimo capitolo vedremo un metodo pi` u sofisticato, che usa gli autovalori, per risolvere gli Esercizi 15 e 16. @ @ Esercizio 15. Calcolare A100 nei seguenti casi (a) A= (b) A= „ 1 0 1 2 « „ 3 2 0 −1 « Esercizio 16. Si considerino le seguenti matrici 0 0 A = @0 3 1 1 − 12 0 12 A 1 8 5 0 1 3 B=@2 0 e si provino le seguenti uguaglianze 0 281457596383971 8243291212479289 B A13 = @ 0 B B 13 = @ 1 1 1 1 −21 A 3 1 4 1000000 13791079790208861 125000 394993103775412801 5000000 257961125226942479 2000000 431570585554290003 250000 154508738617589077 125000 1 6250 1883521814429871 3125 431570585554290003 1000000 2075574373808189 3265173504 −22589583602079623 1088391168 46412434031431 120932352 −2771483961974593 272097792 −7482652061373805 725594112 −2468698236647575 322486272 −34285516978000235 2176782336 155899288381048673 725594112 −872661281513917 80621568 1 C A C A i calcolatori non sono intelligenti, ma pensano di esserlo 3 Soluzioni dei sistemi lineari in teoria non c’`e alcuna differenza tra teoria e pratica, in pratica invece ce n’`e In questo capitolo affronteremo la questione di come risolvere in pratica i sistemi lineari.

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