# Category Theory, Applications to Algebra, Logic, and by Kamps K.H., Pumplün D., Tholen W. By Kamps K.H., Pumplün D., Tholen W.

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Extra info for Category Theory, Applications to Algebra, Logic, and Topology: Proceedings, Gummersbach, FRG, 1981

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INVARIANT AND COMPLEMENTARY MEANS 49 if ∨ ≤ 3∧, while A(F 10) = 1 ∨ + 2 ∧ + ∧ (4 ∨ −3∨) 4 − ∨2 − 8 ∨ ∧ + 9 ∧2 +2 (∨ − 2∧) ∧ (4 ∨ −3∧) , if ∨ ≥ 3∧; G (F 10) = 1 2 √ √ √ √ ∧ 4 ∨ − ∧ − ∨∧ − ∨ − ∧ 4 ∨ −3∧ ; ∨ 1 2 ∨3 +9 ∨2 ∧ − 2 ∨ ∧2 − ∧3 − ∨2 − ∧2 ∧ (4 ∨ −3∧) ; 4 ∨ (∨ + ∧) ∨ 2 ∨3 + ∨2 ∧ + 6 ∨ ∧2 − ∧3 − ∨2 − ∧2 ∧ (4 ∨ −3∧) , = 2 (∨ + ∧) (∨2 + ∧2 ) H(F 10) = C (F 10) if ∨ ≤ t3 · ∧, while C (F 10) = 1 2 ∨2 + ∨ ∧ + 3 ∧2 + (∨ + ∧) 4 (∨ + ∧) ∧ (4 ∨ −3∧) − √ 2· √ 4 ∨4 −4 ∨3 ∧ − 3 ∨2 ∧2 + 2 ∨ ∧3 + 5 ∧4 + (2 ∨3 − ∨2 ∧ − 4 ∨ ∧2 − ∧3 ) 4 ∧ ∨ − 3∧2 if ∨ ≥ t3 · ∧; (F 10) F5 = 2∨−∧+ 1 2 ∨ + ∧ + ∧ (4 ∨ −3∧) · 4∧ (∨ − ∧)2 + 4∧2 − ∨ − ∧ ; if ∨ ≤ 2∧, while (F 10) F5 · ∨+ = 1 ∨+ 4 ∧ (4 ∨ −3∧) · ∧ (4 ∨ −3∧) + (∨ − ∧)2 + 4∧2 − √ 2 (∨ − ∧)2 + 4∧2 + ∨ − 4∧ − ∧ (3 ∨ −5∧) , if ∨ ≥ 2∧; (F 10) F6 = 1 ∧ ∧ (4 ∨ −3∧) + ∨2 + 2 ∨ ∧ · 2 4∨ (∨ − ∧)2 + 4∨2 + ∨ − ∧ 50 CHAPTER 2.

1) for some initial values a0 and b0 arbitrarily chosen. 1) are monotonously convergent to a common limit H G(a0 , b0 ). Proof. 2) by induction it follows that 0 < a0 < a1 < · · · < an < b n < · · · < b 1 < b 0 . The sequence (an )n≥0 is thus monotonic increasing and bounded above by b0 . So, it has a limit, say α. Similarly, the sequence (bn )n≥0 is monotonic decreasing and bounded below by a0 . It has so the limit β. Passing at limit in the relation bn+1 = G(an+1 , bn ) , we get β = G(α, β), thus α = β.

Mathieu, 1879; T. Nowicki, 1998]. Regarding the notation H ⊗ A we will see later the general √ definitions. We can illustrate Heron’s approximation process by computing 2. 00000. . 00000. . 33333. . 50000. . 41176. . 41666. . 41420. . 41421. . Heron’s method has been extended to roots of higher order in [A. N. Nikolaev, 1925; H. Ory, 1938; C. Georgakis, 2002]. Also an iterative method for approximating higher order roots by using square roots has been given in [D. Vythoulkas, 1949]. 3 Lagrange and the definition of the AGM A similar algorithm was developed in [J.