By Lev D. Elsgolc

ISBN-10: 0486457990

ISBN-13: 9780486457994

Topics comprise the tactic of edition in issues of mounted obstacles; variational issues of movable barriers and different difficulties; sufficiency stipulations for an extremum; variational difficulties of restricted extrema; and direct equipment of fixing variational difficulties. each one bankruptcy beneficial properties quite a few illustrative difficulties, and suggestions look on the end.

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**Example text**

8. Two multiplicative norms | · |, | · | on a field F are topologically equivalent if and only if there exists c > 0 such that |x| = |x|c for all x ∈ F. Proof. 2]. 9. Let G be an abelian group equipped with a seminorm | · |G , and let G be a subgroup of G. The quotient seminorm on the quotient G/G is defined by the formula |g + G |G/G = inf {|g + g |G }. 1) If | · |G is a norm, then | · |G/G is a norm if and only if G is closed in G. 2 Valuations and nonarchimedean norms We now restrict our attention to nonarchimedean absolute values, which can be described additively (using valuations) as well as multiplicatively (using norms).

For (u, v) ∈ Bμ we have |a||v| ≤ |(u, v)| ≤ μ|c| = μ|a||b|, so |v| ≤ μ|b|. Similarly |u| ≤ μ|a|. As a result, | f −1 (c − ab − uv)| ≤ λ−1 |c − ab − uv| ≤ λ−1 max{|c − ab|, |uv|} ≤ λ−1 max{λμ|c|, μ2 |a||b|} = μ|c|. Consequently, the map g(u, v) = f −1 (c − ab − uv) carries Bμ into itself. We next show that g is contractive. For (u, v), (t, s) ∈ Bμ , |g(u, v) − g(t, s)| ≤ | f −1 (ts − uv)| ≤ λ−1 |ts − uv| = λ−1 |t (s − v) + (t − u)v| ≤ λ−1 max{μ|a||s − v|, μ|t − u||b|} = λ−1 μ|(u − t, v − s)| = λ−1 μ|(u, v) − (t, s)|, which has the desired effect because λ−1 μ < 1.

Conversely, for any nonarchimedean (semi)norm | · |, v(·) = − log | · | is a real (semi)valuation. We will apply various definitions made for seminorms to semivaluations in this manner; for instance, if R is a ring and v is a real (semi)valuation on its additive group, we say that v is (sub)multiplicative if the corresponding nonarchimedean (semi)norm is. 2. We say that a group is nonarchimedean if it is equipped with a nonarchimedean norm; we say that a ring or field is nonarchimedean if it is equipped with a multiplicative nonarchimedean norm.

### Calculus of Variations by Lev D. Elsgolc

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