MathDB

5

Part of 1997 IMC

Problems(2)

p-norms on integer lattice

Source: IMC 1997 day 1 problem 5

10/1/2005
For postive integer nn consider the hyperplane R0n=x=(x1x2...xn)∈Rn:∑i=1nxi=0 R_0^n = {x=(x_1x_2...x_n)\in\mathbb{R}^n : \sum\limits^n_{i=1}x_i=0} and the lattice Z0n={y∈R0n: (∀i:yi∈N)} Z_0^n = \{y \in R^n_0 : \ (\forall i: y_i \in \mathbb{N})\} Define the quasi-norm in Rn\mathbb{R}^n by ∥x∥p=∑i=1n∣xi∣pp\|x\|_p= \sqrt[p]{\sum\limits^{n}_{i=1}|x_i|^p} if 0<p<∞0<p<\infty and ∥x∥∞=max⁡i∣xi∣\|x\|_{\infty} = \max\limits_i |x_i|. (a) If x∈R0nx\in R^n_0 so that max⁡xi−min⁡xi≤1\max x_i - \min x_i \le 1 then prove that ∀p∈[1,∞],∀y∈Z0n\forall p \in [1,\infty], \forall y \in Z^n_0 we have ∥x∥p≤∥x+y∥p\|x\|_p\le\|x+y\|_p (b) Prove that for every p∈]0,1[p\in ]0,1[, there exist n∈N,x∈R0n,y∈Z0nn \in \mathbb{N}, x\in R^n_0, y\in Z^n_0 with max⁡xi−min⁡xi≤1\max x_i - \min x_i \le 1 and ∥x∥p>∥x+y∥p\|x\|_p>\|x+y\|_p
real analysisreal analysis unsolved
cute from IMC

Source: IMC 1997/problem B5

6/13/2004
Let XX be an arbitrary set and ff a bijection from XX to XX. Show that there exist bijections g, g′:X→Xg,\ g':X\to X s.t. f=g∘g′, g∘g=g′∘g′=1Xf=g\circ g',\ g\circ g=g'\circ g'=1_X.
algebra proposedalgebra