proof of rearrangement inequality for >2 sequences?
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From: hanrahan398 on 22 May 2010 09:14 Does someone know of an online proof of the rearrangement inequality for more than 2 sequences? Thanks! Michael
From: Gerry on 22 May 2010 20:28 On May 22, 11:14 pm, hanrahan...(a)yahoo.co.uk wrote: > Does someone know of an online proof of the rearrangement inequality > for more than 2 sequences? Never heard of it. -- GM
From: Ray Vickson on 22 May 2010 22:52 On May 22, 6:14 am, hanrahan...(a)yahoo.co.uk wrote: > Does someone know of an online proof of the rearrangement inequality > for more than 2 sequences? > > Thanks! > > Michael What do you mean? The usual rearrangement results refer to inner products of the form sum x_i * y_s(i) for two equi-length sequences {x_j} and {y_j} and for s(.) a permutation. Do you mean that you have "triple" products of the form sum x_i*y_s(i)*z_t(i)? Are you claiming that there are results about this, but you don't know their proofs? R.G. Vickson
From: hanrahan398 on 24 May 2010 06:49 On May 23, 3:52 am, Ray Vickson <RGVick...(a)shaw.ca> wrote: > On May 22, 6:14 am, hanrahan...(a)yahoo.co.uk wrote: > > Does someone know of an online proof of the rearrangement inequality > > for more than 2 sequences? > What do you mean? The usual rearrangement results refer to inner > products of the form sum x_i * y_s(i) for two equi-length sequences > {x_j} and {y_j} and for s(.) a permutation. Do you mean that you have > "triple" products of the form sum x_i*y_s(i)*z_t(i)? Are you claiming > that there are results about this, but you don't know their proofs? Yes. One result in particular: for any no. of ascending sequences of positive reals {a_1,...,a_n}, {b_1,...,b_n}, {c_1,...,c_n}, ..., {z_1,...,z_n}, and permutations {B_1,...,B_n}, {C_1,...,C_n}, ..., {Z_1,...,Z_n}, a theorem states that (a_1)(b_1)...(z_1) + ... + (a_n)(b_n)...(z_n) >= (a_1)(B_1)...(Z_1) + ... + (a_n)(B_n)...(Z_n) or in shorter notation: sum (a_i * b_i * ... *z_i) >= sum (a_i * b_s(i) * ... * z_s(i) ) This is used by Arthur Engel on p.170 of his book 'Problem Solving Strategies', but he doesn't give a proof. Michael
From: Ray Vickson on 24 May 2010 12:54
On May 24, 3:49 am, hanrahan...(a)yahoo.co.uk wrote: > On May 23, 3:52 am, Ray Vickson <RGVick...(a)shaw.ca> wrote: > > > On May 22, 6:14 am, hanrahan...(a)yahoo.co.uk wrote: > > > Does someone know of an online proof of the rearrangement inequality > > > for more than 2 sequences? > > What do you mean? The usual rearrangement results refer to inner > > products of the form sum x_i * y_s(i) for two equi-length sequences > > {x_j} and {y_j} and for s(.) a permutation. Do you mean that you have > > "triple" products of the form sum x_i*y_s(i)*z_t(i)? Are you claiming > > that there are results about this, but you don't know their proofs? > > Yes. One result in particular: > > for any no. of ascending sequences of positive reals {a_1,...,a_n}, > {b_1,...,b_n}, {c_1,...,c_n}, ..., {z_1,...,z_n}, and permutations > {B_1,...,B_n}, {C_1,...,C_n}, ..., {Z_1,...,Z_n}, a theorem states > that > > (a_1)(b_1)...(z_1) + ... + (a_n)(b_n)...(z_n) > > >= (a_1)(B_1)...(Z_1) + ... + (a_n)(B_n)...(Z_n) > > or in shorter notation: > > sum (a_i * b_i * ... *z_i) >= sum (a_i * b_s(i) * ... * z_s(i) ) If the same permutation s(.) is applied to all sequences {b},{c},..., {z}, isn't b_s(i) * ... * z_s(i) just a re-arrangement of b_i * ... z_i = P_i; in other words, can't we just apply the two-sequence result to the increasing sequences a_1, a_2, ..., a_n and P_1, P_2, ... , P_n? (The P_j are increasing because all b_j, c_j, ..., z_j are > 0 and they are increasing in j. R.G. Vickson > > This is used by Arthur Engel on p.170 of his book 'Problem Solving > Strategies', but he doesn't give a proof. > > Michael |