By Joseph Fels Ritt

A big job undertaken via J. F. Ritt and his collaborators within the 1930's was once to offer the classical concept of nonlinear differential equations, just like the speculation created through Emmy Noether and her college for algebraic equations and algebraic forms. the present ebook provides the result of twenty years of labor in this challenge. The booklet speedy turned a vintage, and to this point, it continues to be some of the most entire and useful debts of differential algebra and its purposes.

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Am ∈ Kn , m ∈ N, 1 m (Vn (A0 + · · · + Am )) n ≥ 1 (Vn (Ai )) n . 2, means of rotations do not decrease volumes of compact convex sets. 3. COROLLARY. Let T : C n → C n be a mean of rotations. For every A ∈ Kn , Vn (T (A)) ≥ Vn (A). 1. DEFINITION. For every A ∈ K0n , let T(A) := {T (A) | T is a mean of rotations}. 2. THEOREM. Let A ∈ K0n and 0 ∈ A. There exists a sequence in T(A) Hausdorff convergent to a ball B0 satisfying the following conditions: ¯ ¯ 0 ) = b(A); (i) b(B (ii) Vn (A) ≤ Vn (B0 ) ≤ (diamA)n κn .

THEOREM. Let X be a family of nonempty compact subsets of Rn that contains the set of singletons. If F = f ∗ : X → X for some f : Rn → Rn and F : X → X preserves the set of singletons, preserves inclusion, and F (A) ⊂ F(A) for every A ∈ X , then F = F. Proof. Let A ∈ X . For every x ∈ A the set F ({x}) is a singleton: F ({x}) = {y} for some y ∈ F (A). Since F ({x}) ⊂ F({x}) = { f (x)}, it follows that F ({x}) = { f (x)}. By the assumption, F (A) ⊂ f (A). It remains to prove that f (A) ⊂ F (A). 1) Let y ∈ f (A); then y = f (x) for some x ∈ A, whence F (A) ⊃ F ({x}) = { f (x)}.

Ii): A ⊂ β B n ⇒ λA ⊂ A + (λ − 1)A ⊂ A + (λ − 1)β B n . 14. THEOREM. For any hyperplane H in Rn , the function S H | K0n is continuous. Proof. 3 We have to show that S H (A) = lim S H (Ak ). 5, S H +v (X ) = S H (X ) + v for every X ∈ Kn and v ⊥ H , without loss of generality we may assume that 0 ∈ H . Obviously, 0 ∈ int(A + u) for some unit vector u. Let u = u 1 + u 2 , u 1 H, u 2 ⊥ H. 6, S H (A + u) = S H (A + u 1 ) = S H (A) + u 1 and analogously, for every k, S H (Ak + u) = S H (Ak ) + u 1 ; thus we may also assume that 0 ∈ intA.