<a name="docA001">　Index begin　Index this file
2009-06-08-19-10 start
This file http://freeman2.com/tute0008.htm
is Freeman's reading notes. Although Freeman
always keep correct view point. But Freeman's
capability is limited, plus no one proofread
this file. Then you can still find wrong view
point. When read, please put question mark as
often as possible. If you suspect any view 
point wrong, please ask a math expert near by.

<a name="docA002">,<a name="textbook">
This file is a note for reading inequality
book written by Professor J. Michael Steele
The Cauchy-Schwarz Master Class  　★★★★★
Below use 'textbook' as abbreviation.
Freeman also read web pages online, and will 
indicate the source URL at discussion point.

<a name="docA003">
Freeman study mechanical engineering. 
Engineering mathematics do not teach 
inequality. Above book is first inequality
book. First time read, it was very hard.
Although high school time learned
  a*a + b*b >= 2*a*b
But this little knowledge do not help.

<a name="docA004">
This file follow textbook chapter section
order, but not continuous, Freeman skip 
those uncertain sections/problems.

This file first function is to learn 
inequality. Second function is to learn
how to use html code to write math 
equations.

<a name="docA005">　Index begin　Index this file
On 2009-01-27-10-08 Freeman accessed
the next page
http://www.sftw.umac.mo/~fstitl/10mmo/inequality.html
save as sftw.umac.mo_text_math_eqn_good.htm
Above page is the main reference for html
math equation.

In order to let reader to build  html math
equation, previous file tute0007.htm page 
end has math symbol and internal code.

This file third function is to display how
to draw curves in web page. The main engine
is XYGraph v2.3 - Technical Figures

Thank you for read Freeman's inequality page.
2009-06-08-19-47 stop

<a name="docA006">
2009-08-23-15-00 start
□　Exercise 1.14 solution

Exercise 1.14 problem (a) is solved by hint
Exercise 1.14 problem (b) is out of LiuHH's
reach. What is "cardinality of a set B⊂Z3" ?
Without clear understand the definition, 
LiuHH is unable to solve problem (b)
LiuHH major in mechanical engineering.
LiuHH is mathematics admirer and outsider.

<a name="docA007">
To solve problems in
The Cauchy-Schwarz Master Class
LiuHH's goal is to solve 50% of the exercise
problems. In reality, solve 40% is good enough.
Please do not be surprise that LiuHH skip some
text section, skip some problems.
2009-08-23-15-12 stop

<a name="ch13a001">　Index begin　Index this file
2010-06-10-17-16 start
■■Chapter 13: Majorization and
　　Schur Convexity

Majorization is a new concept 
to most reader. Its definition
is simple. But its application
and related inequalities need
more time to understand.
Majorization definition A, B
<a name="ch13a002">
Textbook majorization definition
is recorded at tute0044.htm
The next is same thing but use
LiuHH's word write as a study
notes.

<a name="ch13a003">　Index begin　Index this file
■　Definition of Majorization
If two sequences α and β have
the following properties
(1) α_seq. and β_seq. both have
    non-negative elements. alert
(2) α_seq. and β_seq. both have
    same number of elements.
(3) α_sequence total sum and
    β_sequence total sum are
    equal. Blue alert
<a name="ch13a004">
(4) When arrange both α and β 
    to descending order,
    β partial sum is always
    greater than or equal to
    α partial sum.
<a name="ch13a005">
Assume both α_seq. and β_seq. 
have n elements. 
Condition (3) say total sum
  ∑[k=1,n]αk=∑[k=1,n]βk ---eqn.BN001
are equal.
<a name="ch13a006"> //numerical
Condition (4) say partial sum
         α1≦β1 ---eqn.BN002
      α1+α2≦β1+β2 ---eqn.BN003
   α1+α2+α3≦β1+β2+β3 ---eqn.BN004
.....
  ∑[k=1,n-1]αk≦∑[k=1,n-1]βk ---eqn.BN005
has β beat α inequalities.
<a name="ch13a007">
Above is in textbook page 191.
LiuHH dropped square bracket.
(Textbook use α[1]≦β[1])

<a name="ch13a008">　Index begin　Index this file
If α_seq. and β_seq satisfy 
above four conditions, we
say
  β_seq majorize α_seq
or
  α_seq is majorized by β_seq

<a name="ch13a009">
Remember above relations as
  greater partial sum sequence
  majorize
  smaller partial sum sequence
or
  more diverse sequence //β_seq
  majorize
  more average sequence //α_seq
Example [3,0,0] majorize [1,1,1]
<a name="ch13a010">
'majorize' is similar to '≧'
or '＞', but not be the same.
'≧' or '＞' have one comparison
'majorize'  has  n  successive
comparisons. (=) (n-1)
To differentiate this difference. 
Mathematician use "≺ ≻ ≼ ≽ ≾ ≿ ⊀ ⊁" 
for successive comparisons.
<a name="ch13a011">
Single comparison use '＞' which
is made of two straight lines.
Successive comparison use '≻'
which is made of two arcs.
<a name="ch13a012">
Liu,Hsinhan computer not instal
majorization fonts. In this page
use '(≦)' '(≧)' '(＜)' '(＞)' 
for the relation of successive 
comparison.

<a name="ch13a013">　Index begin　Index this file
Above notes has a requirement
which textbook NOT ask for.
[[
When arrange both α and β 
    to descending order
]]
<a name="ch13a014">
For example, if observed data
is 
  α=[3,6,2,8,7] ---eqn.BN006
Above definition need rearrange
α to
  α*=[8,7,6,3,2] ---eqn.BN007
<a name="ch13a015">
Textbook solve this ordering 
problem by assign
  α1=3,  α2=6,  α3=2,  α4=8,  α5=7 ---eqn.BN008
as original non-sorted sequence
and use square bracket
  α[1]=8,α[2]=7,α[3]=6,α[4]=3,α[5]=2 ---eqn.BN009
for sorted new order sequence.

<a name="ch13a016">
LiuHH use the condition
[[
When arrange both α and β 
    to descending order
]]
to avoid use square bracket,
not let equation become too
long.

<a name="ch13a017">
Descending order is important
to judge whether two sequences
have majorization relation.
2010-06-10-18-30 stop

<a name="ch13a018">　Index begin　Index this file
■　Definition of Schur's convexity
2010-06-10-20-32 start // A, B
Base on majorization definition,
Schur's convexity can be defined
as following. // symmetry alert

<a name="ch13a019">
If a d_dimensional real number 
space is written as Rd.
If A is a sub-space of Rd,
if function f map from A to real 
number (that is function f has 
d input variables and one real 
number output)
<a name="ch13a020">
   If for all α, β  ---eqn.13.2 begin
   in A, for which  //textbook print 
   they have α(＜)β //in one line
   if f(α)≦f(β)     ---eqn.13.2 end
Then we aptly call this problem
as Schur monotone. //●●discuss
(Textbook page 192,line 14)
Popular term is Schur convex.
// exp(x) and log(x) are monotone
// but they bend differently. ●●
<a name="ch13a021">
On the other hand, if we have 
α(＜)β and f(α)≧f(β) then
it is called Schur concavity.
(concavity example math
 convexity example math
 )

<a name="ch13a022">
 "α, β in A" mean that both α
and β have d elements (A is a
d_dimensional sub space)
Function f has d input variables.
(function f map from A to real)
Then f(α) and f(β) are defined.

<a name="ch13a023">　Index begin　Index this file
  α(＜)β   indicate  n  comparisons
f(α)≦f(β) indicate one comparison.
(for "(＜)": =, n-1, more)
2010-06-10-21-05 here

<a name="ch13a024">
A simple example of Schur convex
function is the max(x) function
α1 is maximum value element in
   α sequence, 
β1 is maximum value element in
   β sequence, 
and given α(＜)β, then 
  max(α)＜max(β) ---eqn.BN010
is given.

<a name="ch13a025">
Another simple example of Schur 
convex function is textbook page 
192 and 193 example
  f(t1,t2,t3) ---eqn.BN011
  =1/t1 +1/t2 +1/t3
(convexity math)
2010-06-10-21-27 stop

<a name="ch13a026">
2010-06-12-10-28 start
Textbook page 191 line -2 give 
an example of sequences which 
have majorization relation. 
That is 
      (1,1,1,1) (＜) (2,1,1,0)
 (＜) (3,1,0,0) (＜) (4,0,0,0)
//above line ---eqn.13.1
<a name="ch13a027">
Majorization relation is transitive.
If α(＜)β(＜)γ, then α(＜)γ.
eqn.13.1 represent SIX relations.
other two missing relations are
(2,1,1,0) (＜) (4,0,0,0)
and
(1,1,1,1) (＜) (3,1,0,0)

<a name="ch13a028">　Index begin　Index this file
Please pay attention to their
total sum
  1+1+1+1 = 4
  2+1+1+0 = 4
  3+1+0+0 = 4
  4+0+0+0 = 4
Equal total sum is a necessary 
condition. 
<a name="ch13a029"> //symbolic
One element comparison give
  1＜2＜3＜4  //define sequence
Two elements comparison give
  1+1＜2+1＜3+1≦4+0
Three elements comparison give
  1+1+1＜2+1+1≦3+1+0≦4+0+0
<a name="ch13a030">
Four (all) elements comparison 
  1+1+1+1=2+1+1+0=3+1+0+0=4+0+0+0
The easy check confirmed that
eqn.13.1 is true.

<a name="ch13a031">
If four sequences were
(1,1,1,1) ---eqn.BN012 (four lines)
(0,1,1,2)
(1,3,0,0)
(0,0,4,0) 
we can not do comparison before
sort to descending order.
<a name="ch13a032">
Sort-first is a necessary 
procedure. With Sort-first
requirement in hand, we can
say that random ordered eqn.BN012
has eqn.13.1 majorization relation.
(Textbook page 192 line 3 )
<a name="ch13a033">　Index begin　Index this file
eqn.13.1 tell us one key point.
If first element has total sum
value [ (4,0,0,0) in eqn.13.1 ]
This sequence is greatest one
(most diverse sequence)
among same total sum sequences.
(4,0,0,0) majorize all other
4_sum and four_elements sequences.

<a name="ch13a034">
eqn.13.1 tell us second key point.
If all elements have average value
(total sum divide by total number 
 of elements)
[ (1,1,1,1) in eqn.13.1 ]
This sequence is smallest one
(most average sequence)
among same total sum sequences.
(1,1,1,1) is majorized by all 
other 4_sum and four_elements 
sequences.

<a name="ch13a035">
Define average value
  α_bar=(α1+α2+...+αn)/n ---eqn.BN013
We have

<a name="ch13a036">
eqn.BN014 is
 all element average value sequence
majorized by
 other same total sum sequences
majorized by
 first element has total sum sequence
2010-06-12-11-07 stop

<a name="ch13a037">　Index begin　Index this file
2010-06-12-11-55 start
■　Schur convex numerical example
Here present numerical example
for Schur convexity function.

<a name="ch13a038">
Let
   α=[α1,α2,...,αn] ---eqn.BN015
   β=[β1,β2,...,βn] ---eqn.BN016
be two sequences with the following
majorization relation
   α (＜) β ---eqn.BN017
One Schur convexity function 
is next

2010-06-12-12-22 here

<a name="ch13a041">
Please copy next code, paste to
[Box3, input JS command] of
calculator page, then click 
[Test box3 command, output to box4]
Box 4 has output result.

//<a name="ch13a042">
//2010-06-12-11-36
//[3,2,2,5,3] (<) [5,4,3,2,1]
//[[  program start
//calculator local how to use
p1=5,p2=4,p3=3,p4=2,p5=1; //if change, be alert
q1=3,q2=2,q3=2,q4=5,q5=3; //p_sum must= q_sum
var sump=1/p1+1/p2+1/p3+1/p4+1/p5;
var sumq=1/q1+1/q2+1/q3+1/q4+1/q5;
1/p1+1/p2+1/p3+1/p4+1/p5 //greater value
1/q1+1/q2+1/q3+1/q4+1/q5 //smaller value
var alert0='p,q same sum, OK';
if(abs(p1+p2+p3+p4+p5-q1-q2-q3-q4-q5)>1.e-8)alert0='p_sum not equal to q_sum'
alert0 //Schur convexity function
//calculator local how to use
//]] program end
//2010-06-12-11-38

2010-06-12-13-07 stop

<a name="ch13a043">　Index begin　Index this file
2010-06-12-17-00 start
■　What is in-hull coefficients?
　　was Unsolved problem
　　now computer program (local)
Textbook page 187, line 11 say
[[
<a name="ch13a044">
First, just to make the hypothesis
α∈H(β) //α is in hull of β.
concrete, we note that it is
equivalent to the assertion that
  (α1,α2,...,αn) ---eqn.BL081
 =∑[τ∈Sn]pτ(βτ(1),βτ(2),...,βτ(n))
(eqn.BL081 numerical example
 given α,β find matrix pτ local)
where pτ≧0 ---eqn.BL082 //α,β vs. pk
and   ∑[τ∈Sn]pτ=1 ---eqn.BL083
]]
<a name="ch13a045">
Above is chapter 12, page 187
Below is chapter 13, page 195
Problem 13.2 re-state eqn.BL081.
[[
<a name="ch13a046">
From Murihead's condition to a
special representation 

Here we should first recall that 
the notation α∈H(β) simply means 
that there are nonnegative weights
pτ which sum to 1 
<a name="ch13a047">
for which we have
  (α1,α2,...,αn) ---eqn.BL081
 =∑[τ∈Sn]pτ(βτ(1),βτ(2),...,βτ(n))
or, in other words, α is a 
weighted average of 
(βτ(1),βτ(2),...,βτ(n))
as τ run over the set Sn of
permutation of {1,2,...,n}
]]

<a name="ch13a048">　Index begin　Index this file
Textbook say that 
if α∈H(β) in-hull relation is 
true, then eqn.BL081 is true 
Textbook did not say how to 
find in-hull coefficients 
p1, p2, ..., pn. (textbook
 said in page 198 to 201)
<a name="ch13a049">
Study notes main work is to
fill in details to persuade 
myself that I understand the
meaning of textbook.
Some fill in detail work is
OK. But some fill in detail 
work is wrong. This is study 
notes, not thesis. It is OK
to have few errors :)
<a name="ch13a050">
Example of the OK study notes are 
■　Physics dimension consistency
■　one to one
Example of the wrong study notes:
A simple example

<a name="ch13a051">
The following is the question
What is in-hull coefficients?
Textbook did not say how to 
find in-hull coefficients pj. 
<a name="ch13a052">
LiuHH guessed that for α∈H(β) 
in-hull relation, use permuted 
β sequence as matrix entry.
use random-order α sequence as 
given vector. (β is bottle, α 
is parts) Solve a set of n linear 
equation for in-hull coefficients 
pj, j=1,2,...,n
<a name="ch13a053">　Index begin　Index this file
tute0044.htm set up a simple 
example
If α power is
  α1=[3,2,1] ---eqn.BL090
If β power is
  β1=[4,1,1] ---eqn.BL091
First check α1(＜)β1 is true.
Let us find interpolation
coefficients p1, p2, p3
<a name="ch13a054">
Permute β1=[4,1,1] to create
matrix, and use α1 as given
vector. (error vs. correct)
In matrix form it is next
  [3] [4  1  1] [p1] ---this is error
  [2]=[1  1  4]*[p2] ---eqn.BN108 OK
  [1] [1  4  1] [p3] ---eqn.BL095
<a name="ch13a055">
where p1,p2 and p3 are in-hull 
coefficients for
  3 = 4*p1 + 1*p2 + 1*p3 ---eqn.BN020
  2 = 1*p1 + 1*p2 + 4*p3 ---eqn.BN021
  1 = 1*p1 + 4*p2 + 1*p3 ---eqn.BN022
<a name="ch13a056">
Solve eqn.BL095 get
  p1=2/3 ---eqn.BL096
  p2=0   ---eqn.BL097
  p3=1/3 ---eqn.BL098
All three p's are in [0,1].
All three p's sum to one.

<a name="ch13a057">
Above is chapter 12 work.
LiuHH use above method get 
reasonable answer.
When come to chapter 13,
Majorization and Schur Convexity
LiuHH did greater scale
calculation.

<a name="ch13a058">　Index begin　Index this file
2010-06-09-14-17 found that
β sequence=[5,4,3,2,1]
α sequence=[3,2,2,5,3]
they have α∈H(β) in hull relation,
but interpolation coefficients are
p1=2/5; p2=1/5; p3=-2/5;
p4=3/5; p5=1/5;
Five p's sum to one. OK
however p3=-2/5 is out of [0,1]
<a name="ch13a059">
Program used is (local)
http://freeman2.com/jspico_e.htm#Arndt04
input is next
 5 4 3 2 1   3
 1 5 4 3 2   2
 2 1 5 4 3   2
 3 2 1 5 4   5
 4 3 2 1 5   3

Output is next
1  0  0  0  0  2/5  
0  1  0  0  0  1/5  
0  0  1  0  0  -2/5  ←negative, trouble
0  0  0  1  0  3/5  
0  0  0  0  1  1/5  

<a name="ch13a060">
LiuHH concluded that the method
used above is wrong.
Above method can not find correct
interpolation coefficients.

<a name="ch13a061">
How to find coefficients ?
Textbook did not say.
Online papers did not say.
LiuHH wait for further reading.
LiuHH wait for more thinking.
2010-06-12-18-01 stop
now computer program find 
coefficients (matrix) (local)

<a name="ch13a062">　Index begin　Index this file
2010-06-12-20-04 start
■　Problem 13.1
　(Schur's Criterion)
　　Textbook page 193
Given that the function 
f:(a,b)n→Real is continuously 
differentiable and symmetric.
<a name="ch13a063">
Show that it is Schur convex 
on (a,b)n if and only if for
all 1≦j＜k≦n and all x∈(a,b)n
one has

2010-06-12-20-23 here
<a name="ch13a065">
Next is a simple example for
Schur's Criterion.
Walker's inequality state that
given positive a,b,c and define
  x=b+c-a ---eqn.BN023
  y=c+a-b ---eqn.BN024
  z=a+b-c ---eqn.BN025
Require that x,y,z be strictly
positive.
(do not miss above two
 requirements)
<a name="ch13a066">　Index begin　Index this file
Two positivity requirements
give us following majorization
relation.
  (a,b,c) (＜) (x,y,z) ---eqn.BN026
Then one has the following
reciprocal bound

<a name="ch13a068">
2010-06-12-20-36 here
The requirement of a,b,c and
x,y,z all be positive in
eqn.13.3 let us pay attention
to triangle ABC with three 
sides a,b,c. Since 
<a name="ch13a069">
triangle has the condition
  b+c-a≧0 ---eqn.BN027
  c+a-b≧0 ---eqn.BN028
  a+b-c≧0 ---eqn.BN029

<a name="ch13a070">
eqn.13.3 suggest us consider
the following three variables
function
  f(t1,t2,t3) ---eqn.BN030
 = 1/t1 +1/t2 +1/t3

<a name="ch13a071">
Assume Schur's Criterion eqn.13.4
is true. (prove later)
Substitute eqn.BN030 into eqn.13.4
Calculate as following

<a name="ch13a074">
2010-06-12-21-11 here
Last line has three squares
[(tj－tk)*(tj－tk)]
and
[(tj*tj)
*(tk*tk)]
and one sum of two positives
(tj+tk)

<a name="ch13a075">
eqn.BN031 has positive value.
eqn.BN030 satisfy Schur's 
Criterion eqn.13.4. Then 
Walker's inequality eqn.13.3
is true.
2010-06-12-21-17 here

<a name="ch13a076">
How to verify eqn.BN026?
First check if two sum are equal
  x+y+z=(b+c-a)+(c+a-b)+(a+b-c)
  x+y+z=a+b+c ---eqn.BN032
Confirmed two sum are equal. Next
<a name="ch13a077">　Index begin　Index this file
we can assume that 
  a≦b≦c ---eqn.BN033
(a,b,c) sequence in descending 
order become (c,b,a)
(x,y,z) sequence in descending 
order become (x,y,z)
  x=b+c-a ---eqn.BN023
  x=a+b+c-2*a ---eqn.BN023 aux.
x = constant sum - smallest 'a'
then x is greatest in (x,y,z)
<a name="ch13a078">
Now compare two sequences
             (c,b,a) and (x,y,z)
[NOT compare (a,b,c) and (x,y,z)]

<a name="ch13a079">
One term comparison
  x-c=b+c-a-c=b-a≧0 ---eqn.BN034
Since we assumed a≦b≦c

<a name="ch13a080">
Two term comparison
  (x+y)-(c+b)
 =(b+c-a+c+a-b)-(c+b)
 =(+c+c)-(c+b)
 = c-b≧0 ---eqn.BN035
Same reason, assumed a≦b≦c

<a name="ch13a081">
eqn.BN032,eqn.BN034 and eqn.BN035
confirmed that
  (a,b,c) (＜) (x,y,z) ---eqn.BN026
is true.
Sequence (x,y,z) majorize
sequence (a,b,c).
2010-06-12-21-56 stop

<a name="ch13a082">　Index begin　Index this file
2010-06-13-08-11 start
■　Gradual sequence AM, GM, SUM
Given a sequence  
  w=(w1,w2,...,wn) ---eqn.BN036
we can build a new sequence 
from the given sequence. For
example, we can build gradually
AMed sequence from w

<a name="ch13a089">　Index begin　Index this file
2010-06-13-13-35 start
■　Proof of Schur's Criterion
　　Interpretation of a derivative
　　condition
eqn.13.4 is Schur's Criterion for
Schur convexity. The following 
try to explain the meaning of
eqn.13.4. 
<a name="ch13a090">
Assume we have a sequence  
  w=(w1,w2,...,wn) ---eqn.BN036
It could be random ordered, 
not in descending order. But 
for majorization purpose, let
w be in descending order. 
<a name="ch13a091">
Apply gradually SUMed tilde 
sequence eqn.BN040 to eqn.BN045
The definition of majorization
become eqn.BN048 to eqn.BN051.
This tilde transformation let
us write partial summed equation
like ordinary equation. The sign
of summation ∑ become hidden.

<a name="ch13a092">
Schur Criterion require function
f:(a,b)n→Real is continuously 
differentiable and symmetric.
Please pay attention to the
condition symmetric. For a
non-symmetric function, do not
bother to test Schur Criterion.

<a name="ch13a093">
Schur Criterion require function
f(x) has n variables. If n=1, 
single variable calculus? Do not
bother to test Schur Criterion.
In other words, it is not easy to
draw curves for Schur Criterion.
If do so, must set n-1 variables 
to constants, allow only one vary.

<a name="ch13a094">　Index begin　Index this file
Define a order condition for a
sequence (textbook p.193 line-6)
  D={(x1,x2,...,xn):
    x1≧x2≧...≧xn} ---eqn.BN052
Define a set with constraint 
  B=(a,b)n∩D ---eqn.BN053
which says that B is a set, its
element has n components. 
<a name="ch13a095">
Each component range in (a,b). 
These n components satisfy ('∩')
the condition ('D') of
  x1≧x2≧...≧xn ---eqn.BN054

<a name="ch13a096">
Based on set B, define another
set B~ as following
  B~={x~:x∈B} ---eqn.BN055
It says that set B~ contains 
sequence x~. The original form
of x~ is x. x is in set B.

<a name="ch13a097">
Based on set B~, define new
function
  f~:B~ to Real ---eqn.BN056
Set (textbook page 193 line -4)
  f~(x~)=f(x) ---eqn.BN057
  for all x~ in B~
<a name="ch13a098">
If we work with f(x) for 
majorization purpose, we 
write a lot of summation 
sign ∑[j=1,k＜n].
Above tilde definition f~(x~)
allow us do same thing without
write summation sign. because
tilde '~' absorbed summation 
sign.

<a name="ch13a099">　Index begin　Index this file
Majorization is a relation 
between TWO arrays x and y.
Both x and y must have same
number of elements.
Both x and y must have same
total sum values.
Above two points are the base
of comparison. Without above 
two points, comparison has 
no meaning.

<a name="ch13a100">
If both no-tilde sequences x 
and y are in set B eqn.BN053.
If x and y have majorization
relation
  x (＜) y ---eqn.BN058
then no-tilde relation
  f(x) ≦ f(y) ---eqn.BN059
is true, if and only if the 
following has-tilde relations
are true. They are

<a name="ch13a101">
For all x~ and y~ be in set B~ 
eqn.BN055. Such that eqn.BN048 
to eqn.BN051 tilde relations
are true for x~ and y~. Follow 
textbook page 194, line 2
<a name="ch13a102">
  x~n = y~n ---eqn.BN060
and
  x~j ≦ y~j ---eqn.BN061
  for all 1≦j＜n
then
  f~(x~)≦f~(y~) ---eqn.BN062

Above is mathematics equation
for "if and only if" condition.
<a name="ch13a103">
Simpler verbal description is

Function f is Schur convex on B
if and only if
function f~ on B~ is a
non-decreasing function for its
first n-1 coordinates.

2010-06-13-15-10 stop

<a name="ch13a104">　Index begin　Index this file
2010-06-13-17-16 start
The original problem is: for 
TWO sequences x and y if their
gradual sum have majorization
relation
  x (＜) y ---eqn.BN058
<a name="ch13a105">
then no-tilde relation
  f(x) ≦ f(y) ---eqn.BN059
is true.
eqn.BN058 is multiple comparison
with summation involved.
eqn.BN059 is single comparison.
Two equations are on different
base points.

<a name="ch13a106">
The modified problem is 
For all x~ and y~ be in set B~ 
eqn.BN055. Such that
  x~n = y~n ---eqn.BN060
and
  x~j ≦ y~j ---eqn.BN061
  for all 1≦j＜n
then
  f~(x~)≦f~(y~) ---eqn.BN062

<a name="ch13a107">
The modified problem equations 
has multiple comparison without
summation involved.
2010-06-13-17-40 here

Textbook page 194 line 8 has
equation for ONE sequence.

<a name="ch13a109">　Index begin　Index this file
Eventually we need link tilde
variable x~ with original
variable x 

<a name="ch13a110">
Re-write
  f~(x~)=f(x) ---eqn.BN057
as
  f~(x~)=f(x1,x2,x3,...,xn) ---eqn.BN064
Recall eqn.BN047 for next change
  f~(x~) ---eqn.BN065
  =f(x~1,x~2－x~1,x~3－x~2,..., x~n－x~n-1)
Next step, substitute eqn.BN065
into eqn.BN063 get

<a name="ch13a115">
2010-06-13-19-33 here
Sequence 1, j, 3 is shorter
than j-1, j, j+1. Therefore
above derivation assumed j=2
for convenience.

<a name="ch13a116">
Compare eqn.13.5 with eqn.13.4
(eqn.13.4 is Schur's Criterion)
eqn.13.4 require xj and xk
eqn.13.5 give us xj and xj+1
What to do ?
We sum eqn.13.5 over the indices
j, j+1, ..., k-1. and
equire that  1≦j＜k≦n

<a name="ch13a117">　Index begin　Index this file
Assume j=2 and k=6. The sum 
of eqn.13.5 is
 +∂[f(x)]/∂[x2] - ∂[f(x)]/∂[x3]
 +∂[f(x)]/∂[x3] - ∂[f(x)]/∂[x4]
 +∂[f(x)]/∂[x4] - ∂[f(x)]/∂[x5]
 +∂[f(x)]/∂[x5] - ∂[f(x)]/∂[x6] ---eqn.BN069
red cancel red, blue cancel blue
purple cancel purple. Just like
telescoping. 
<a name="ch13a118">
The net result is
 0 ≦ +∂[f(x)]/∂[x2]
     -∂[f(x)]/∂[x6] ---eqn.BN070
or
 0 ≦ +∂[f(x)]/∂[xj]
     -∂[f(x)]/∂[xk] ---eqn.BN071
for all x in B.

<a name="ch13a119">
Majorization sort method is
  x1≧x2≧x3≧...≧xn ---eqn.BN072
with 1≦j＜k≦n requirement, we
know that for j＜k we have
  xj≧xk ---eqn.BN073
or 
 0 ≦ xj - xk ---eqn.BN074
<a name="ch13a120">
Put eqn.BN074 and eqn.BN071 
together, we get 
Schur's Criterion eqn.13.4 

Problem 13.1 is done.
2010-06-13-20-10 stop



<a name="ch13a121">　Index begin　Index this file
2010-06-14-12-32 start
Definition of Majorization 
condition (1) is
(1) α_seq. and β_seq. both have
    non-negative elements.
Please pay attention to the 
following difference.
<a name="ch13a122">
In Muirhead inequality, we allow
α_seq. and β_seq. both have
negative/zero/positive elements.

In Schur Convexity analysis,
we insist
α_seq. and β_seq. both have
    non-negative elements
<a name="ch13a123">
Muirhead case is recorded in
textbook page 186, line -2.
[[
there is no constraint on the
sign of the coordinates of α
and β in Muirhead's inequality.
]]
Top of textbook page 187 has an
example illustrate negative β
sequence element still work OK.

<a name="ch13a124">
Schur Convexity analysis case,
textbook page 191 line 11, 
initially use "real α and β". 
In errata page,  change
from [for any pair of real]
to [for any pair of nonnegative
    real]

<a name="ch13a125">　Index begin　Index this file
Why Muirhead different from Schur?
Muirhead inequality use α and 
β as power of data sequence.
Schur Convexity use α 
and β as data sequence.
(NO "power of")
<a name="ch13a126">
A negative power bring positive
data from numerator to denominator.
A negative power do not generate 
complex number. On the other hand,
<a name="ch13a127">
data can be raised to non-integer 
power. If data is negative, then
(negative data)^(0.25) is complex.
We do not want to see complex
answer. Therefore limit data
α and β be non-negative.

<a name="ch13a128">
Do not just read and believe
LiuHH's words !! Read and turn
around in your brain for a while,
judge whether LiuHH opinion is
correct or not. 
If I find out other reason for
why α and β must be non-negative
I will record at later time.
2010-06-14-12-54 stop

<a name="ch13a129">　Index begin　Index this file
2010-06-14-13-59 start 
In Muirhead inequality, power
sequence α_seq. and β_seq. both 
allow negative/zero/positive 
elements.
In Muirhead inequality, data
sequence x_seq. all be POSITIVE.
See Problem 12.3

<a name="ch13a130">
In Schur Convexity analysis,
both α_seq. and β_seq. are data
sequence.
both α_seq. and β_seq. are non-
negative sequence. (see errata)
<a name="ch13a131">
Can you guess
How Schur regulate his power
sequence?
2010-06-14-14-05 stop

2010-06-14-15-13 done first  proofread
2010-06-14-18-16 done second proofread
2010-06-14-18-32 done spelling check



<a name="ch13a132">　Index begin　Index this file
2010-06-15-10-02 start
■　From Schur concavity to AM-GM
　　(textbook page 194 lower half)

We can test Schur's Criterion 
eqn.13.4 with the most basic 
inequality, AM-GM inequality.

<a name="ch13a133">
Given a sequence
  x=[x1,x2,...,xn] ---eqn.BN075
Different calculation method,
give us different answer, The
Arithmetic Mean (AM) method is
  am=(x1+x2+...+xn)/n ---eqn.BN076
Geometric Mean (GM) method is
  gm=(x1*x2*...*xn)1/n ---eqn.BN077
(both equation are equal weight,
 non-equal weight is possible)

<a name="ch13a134">
If x is observed length data,
eqn.BN076 am give us averaged
length.
eqn.BN077 gm give us another
averaged length. Calculation 
of GM case, raised to length 
n-th dimension and use root
power (1/n) bring n-th dim.
back to length first power.

<a name="ch13a135">
Key point is that both AM and
GM are length, let us be able
to compare length with length.

<a name="ch13a136">
Second thing worth to note is
that both eqn.BN076, eqn.BN077
treat x elements symmetrically.
If it were non-symmetric 
function, we do not need to
do Schur's Criterion problem.

<a name="ch13a137">　Index begin　Index this file
We use eqn.BN077 raise to n-th
power as Schur's Criterion test
function. That is
  f(x)=x1*x2*...*xn ---eqn.BN078
Here we require that all xj be
positive for 1≦j≦n. 
<a name="ch13a138">
If any one xj be negative,
(negative number)^(1/n)=complex
We do not want to see complex
answer.
If any one xj be zero, GM value 
is zero, a dull case. Exclude 
both cases, what left is all 
elements be positive.

<a name="ch13a139">
Now substitute eqn.BN078 into
Schur's Criterion eqn.13.4 
BUT ignore the '0 ≦' sign. 
Because eqn.13.4 is for Schur
convex, in that case '0 ≦' is
proper. The following will find
Schur concave. With this change
in mind, proceed as following.

<a name="ch13a144">
2010-06-15-11-23 here
eqn.BN083 tell us that 
Schur's Criterion eqn.13.4 has
non-positive value. therefore
  f(x)=x1*x2*...*xn ---eqn.BN078
is Schur concave.

<a name="ch13a145">
Above determined f(x)'s Schur 
concavity. 
Schur's Criterion work with two 
majorize-related sequences only.
Next examine the majorization 
property between AM sequence 
and GM sequence.
Define x_bar as following
  x_bar=(x1+x2+...+xn)/n ---eqn.BN084
<a name="ch13a146">
Now AM sequence is
  x_bar=[x_bar,x_bar,...,x_bar] ---eqn.BN085
GM sequence is
  x=[x1,x2,...,xn] ---eqn.BN086
<a name="ch13a147">　Index begin　Index this file
Average [x_bar,x_bar,...,x_bar]
is majorized by every other
same element number, same total
sum sequence. Then
  x_bar(＜)x ---eqn.BN087
That is
  AM_seq. majorized by GM_seq.

<a name="ch13a148">
Schur convexity say that if
  AA_seq. (＜) BB_seq. ---eqn.BN387
then //proofread inserted eqn. use 300+
  f(AA_seq) ≦ f(BB_seq) ---eqn.BN388
Schur concavity say that if
  AM_seq. (＜) GM_seq. ---eqn.BN087
then
  f(AM_seq) ≧ f(GM_seq) ---eqn.BN088
This function
  f(x)=x1*x2*...*xn ---eqn.BN078
is Schur concave. So we have
  x1*x2*...*xn ≦  ---eqn.BN089
  x_bar*x_bar*...*x_bar
Is this AM-GM inequality ?!
<a name="ch13a149">
Take n-th power root, we get
  (x1*x2*...*xn)1/n≦x_bar ---eqn.BN090
Recover x_bar from eqn.BN084
eqn.BN090 is GM≦AM exactly.
2010-06-15-11-57 stop

<a name="ch13a150">
2010-06-15-15-11 start
  am=(x1+x2+...+xn)/n ---eqn.BN076
and
  gm=(x1*x2*...*xn)1/n ---eqn.BN077
are two symmetric functions.
Above use gm eqn.BN077 to test
Schur's Criterion eqn.13.4
What if we choose eqn.BN076 as
start point? What we are going 
to get?

<a name="ch13a151">
Above use all product eqn.BN077
am disguise as product of x_bar.
Now eqn.BN076 is a all sum 
equation. How to disguise gm as
all sum? GM≦AM is
<a name="ch13a152">　Index begin　Index this file
  (x1*x2*...*xn)1/n
 ≦(x1+x2+...+xn)/n  ---eqn.BN091
If whole equation take log(),
gm side change to sum of log()
am side is log of sum. 
Not compatible.

<a name="ch13a153">
Second trouble point, substitute 
am function eqn.BN076 into 
Schur's Criterion eqn.13.4
we find

<a name="ch13a156">
Above calculation tell us that
from Schur concavity to AM-GM
The choice of
  gm=(x1*x2*...*xn)1/n ---eqn.BN077
is a through path.
The choice of
  am=(x1+x2+...+xn)/n ---eqn.BN076
achieve nothing.
2010-06-15-15-41 stop

<a name="ch13a157">　Index begin　Index this file
2010-06-15-16-22 start
■　Problem 13.2 (Muirhead 
　　Implies Majorization)
Numerical example for Problem 13.2

Show that Muirhead's condition
implies that α is majorized by
β; that is, 
<a name="ch13a158">
show that one has 
the implication
  α∈H(β) → α(＜)β ---eqn.13.6
//given α∈H(β) mathematics equation
//target α(＜)β mathematics equation

Next six lines from textbook
is recorded here
2010-06-15-16-33 stop

<a name="ch13a159">
2010-06-16-11-04 start
Problem 13.2 (Muirhead 
Implies Majorization) will be
delayed. or it is NOT DONE
at present time 2010-06-16.
2010-06-25 improved.
<a name="ch13a160">
 To understand a problem,
Liu,Hsinhan need numerical
example to help. Five days 
intensive online search,
saved/read 52 pdf files,
not find
[[
<a name="ch13a161">　Index begin　Index this file
numerical example of
given two sequences
α1,α2,...,αn
β1,β2,...,βn
find doubly stochastic matrix D
such that
  α=Dβ ---eqn.13.10
]]
α=Dβ is textbook page 196, 
equation (13.10)

<a name="ch13a162">
Textbook page 195 line -8 has
  (α1,α2,...,αn)= ---eqn.BN094
  ∑[τ∈Sn]pτ(βτ(1),βτ(2),...,βτ(n))
LiuHH believe eqn.BN094 is true,
need numerical verification. But
on 2010-06-16 LiuHH is unable to 
do so. 

<a name="ch13a163">
Textbook page 195 line -2 has
  djk=∑[τ:τ(j)=k]pτ ---eqn.BN095
without numerical check, it is
hard to see in a two dimensional 
matrix, one element djk need 
further third index τ summation.
There must be good reason, but
how?
<a name="ch13a164">
Spend too much time for one
problem. Liu,Hsinhan wait for
future reading. On 2010-06-16 
Problem 13.2 is NOT DONE
2010-06-16-11-30 stop

<a name="ch13a165">　Index begin　Index this file
2010-06-16-11-33 start
■　Final consideration of the
　　Walker example
Textbook page 197.

given positive a,b,c and define
  x=b+c-a ---eqn.BN023
  y=c+a-b ---eqn.BN024
  z=a+b-c ---eqn.BN025
Require that x,y,z be strictly
positive.
<a name="ch13a166">
If we add y with z get
  y+z=(c+a-b)+(a+b-c)=2*a ---eqn.BN096
similarly
  z+x=(a+b-c)+(b+c-a)=2*b ---eqn.BN097
  x+y=(b+c-a)+(c+a-b)=2*c ---eqn.BN098
<a name="ch13a167">
We can rewrite above three
equations as
  a=(y+z)/2 ---eqn.BN099
  b=(z+x)/2 ---eqn.BN100
  c=(x+y)/2 ---eqn.BN101
<a name="ch13a168">
Put it in column vector form as
  [a]   1 [y]   1 [z] 
  [b] = -*[z] + -*[x] ---eqn.BN102
  [c]   2 [x]   2 [y] 
 left side vector  is [a,b,c]
right side vector are [y,z,x]
and [z,x,y]. 

<a name="ch13a169">
We like to see an equation in 
same order, 
use [a,b,c] and [x,y,z]. This
can be done by using 
permutation matrix. 
<a name="ch13a170">　Index begin　Index this file
Rewrite eqn.BN102 as
  [a]   1 [0  1  0] [x]
  [b] = -*[0  0  1]*[y]
  [c]   2 [1  0  0] [z]

        1 [0  0  1] [x]
      + -*[1  0  0]*[y] ---eqn.BN103
        2 [0  1  0] [z] 
<a name="ch13a171">
Please verify 
red  BN103 equal red  BN102 and 
blue BN103 equal blue BN102 .
Please see
Matrix multiply with vector
eqn.AB52, eqn.AB53 for help.

<a name="ch13a172">
In eqn.BN103, add two permutation
matrix with coefficient 1/2 get
  [a]   1 [0  1  1] [x]
  [b] = -*[1  0  1]*[y] ---eqn.BN104
  [c]   2 [1  1  0] [z]
eqn.BN104 is in textbook 
page 197, line -9.
textbook put 1/2 inside matrix.
eqn.BN104 put 1/2 outside matrix.

<a name="ch13a173">
All of above calculation, what
do we get?
We get the D matrix in
  α=Dβ ---eqn.13.10
Here 
α sequence is [a,b,c]
β sequence is [x,y,z]
<a name="ch13a174">
If put 1/2 into matrix, 
3x3 D matrix in eqn.BN104 has
the property that
each column sum to one
each   row  sum to one
<a name="ch13a175">　Index begin　Index this file
But eqn.BN104 is trivial
It start from given relation
eqn.BN023 to eqn.BN025
In general, α sequence and 
β sequence element-by-element
relation is not given.

<a name="ch13a176">
Another point worth attention 
is that for
  α (＜) β ---eqn.BN105
majorization relation, 
write in
  α=Dβ ---eqn.13.10
is correct.
<a name="ch13a177">
write in
  β=Eα ---eqn.BN106
is wrong.
We can build wrong one from
eqn.BN023 to eqn.BN025. It is
  [x]   1 [-1  1  1] [a]
  [y] = -*[ 1 -1  1]*[b] ---eqn.BN107
  [z]   1 [ 1  1 -1] [c]
<a name="ch13a178">
eqn.BN107 matrix is wrong, 
because 
Matrix has negative element -1.

<a name="ch13a179">
Relation 
between matrix D in eqn.13.10
 and    matrix E in eqn.BN106
is that D*E=Identity or
D = inverse of E
E = inverse of D

<a name="ch13a180">
If write
  α (＜) β ---eqn.BN105
in verbal words as
  more average seq. ＜ more diverse seq.
then the correct one between
  α=Dβ ---eqn.13.10
  β=Eα ---eqn.BN106
is
<a name="ch13a181">　Index begin　Index this file
  average seq. = D * diverse seq.
the other way is wrong.
Matrix D has 
all elements be non-negative 
and 
all column sum to one
all   row  sum to one
Doubly stochastic matrix satisfy
above three properties.

<a name="ch13a182">
Majorization requirement
is two sorted α,β sequences
α sequence successive sum
is always less or equal to
β sequence successive sum.
Compare α with β ?
<a name="ch13a183">
eqn.13.10 help us convert
from
Compare α with β 
to
Compare coef.*β with β 
Textbook page 196 bottom
half page and page 197
first few lines use above
conversion method and from
the order of β sequence
draw conclusion.
2010-06-16-12-48 here



2010-06-16-12-52 start
<a name="ch13a184">
After read 
Problem 13.2 
Muirhead Implies Majorization
and
Final consideration of the
Walker example.
<a name="ch13a185">
Now know why
A simple example is wrong
LiuHH should write
  [α1]   [p11  p12  p13] [β1]
  [α2] = [p21  p22  p23]*[β2] ---eqn.BN108
  [α3]   [p31  p32  p33] [β3]
should NOT write
  [α1]   [β1  β2  β3] [p1]
  [α2] = [β3  β1  β2]*[p2] ---eqn.BN109
  [α3]   [β2  β3  β1] [p3]
<a name="ch13a186">
here α and β are given and
p's are unknown.
How to build pjk matrix?
still wait for further reading 
and thinking. //solved
2010-06-16-13-06 stop

<a name="ch13a186a">
2010-06-26-17-54 proofread note start
From 2010-06-16-19-07 create matMVecf.htm
(matMVecf=Matrix Multiply Vector Function)
2010-06-17-21-43 save matMVecf.htm as jsmajor2.htm 
 to  2010-06-23-23-24 get all non-negative
 matrix. (correct output) Now still write
jsmajor2.htm document. There are nine days
no work record in tute0046.htm. Next few
days will continue jsmajor2.htm document.
2010-06-26-18-00 proofread note stop

<a name="ch13a187">　Index begin　Index this file
2010-06-25-09-52 start
■　Numerical example for 
　　Problem 13.2
key, Case k=1,k=2,k=3,k=4, go k=3
Build c, choke, add zero

Majorization definition has two
version. Prepare an example first.
(following use 'alfa' which is
 equal length with 'beta')
Assume
α_sequence = 18 14 12  6 ---eqn.BN110
β_sequence = 20 15 10  5 ---eqn.BN111

<a name="ch13a188">
Define
  alfa_1=18, alfa_2=14,  ---eqn.BN112
  alfa_3=12, alfa_4= 6.  ---eqn.BN113
and
  beta_1=20, beta_2=15,  ---eqn.BN114
  beta_3=10, beta_4= 5.  ---eqn.BN115

<a name="ch13a189">
Define partial sum as following
  alfa1S=alfa_1          ---eqn.BN116
  alfa2S=alfa_1+alfa_2     four lines
  alfa3S=alfa_1+alfa_2+alfa_3
  alfa4S=alfa_1+alfa_2+alfa_3+alfa_4
<a name="ch13a190">
and
  beta1S=beta_1          ---eqn.BN117
  beta2S=beta_1+beta_2     four lines
  beta3S=beta_1+beta_2+beta_3
  beta4S=beta_1+beta_2+beta_3+beta_4
<a name="ch13a191">
So we have
  alfa1S=18              ---eqn.BN118
  alfa2S=32=18+14          four lines
  alfa3S=44=18+14+12
  alfa4S=50=18+14+12+6
<a name="ch13a192">　Index begin　Index this file
and
  beta1S=20              ---eqn.BN119
  beta2S=35=20+15          four lines
  beta3S=45=20+15+10
  beta4S=50=20+15+10+5

<a name="ch13a193">
Majorization definition first
version is called //2nd version
"Muirhead's condition"
which is
  α∈H(β)               ---eqn.BN120
or
  α=D*β                 ---eqn.BN121
Numerical example: A B
<a name="ch13a194">
For this specific example
α_sequence = 18 14 12  6 ---eqn.BN110
β_sequence = 20 15 10  5 ---eqn.BN111
Muirhead's condition is a matrix
equation  // next equation ---eqn.BN122
[18] [52/63  13/630  52/630  45/630 ] [20]
[14]=[ 0      4/5     1/5      0    ]*[15]
[12] [ 1/9   16/90   64/90     0    ] [10]
[ 6] [ 4/63   1/630   4/630  585/630] [ 5]
<a name="ch13a195">
Please verify that above matrix
(1) all elements are nonnegative
(2) each  row   sum to one
(3) each column sum to one
It is doubly stochastic matrix D.

<a name="ch13a196">
Majorization definition second
version is called //1st version
"partial sum inequality" //BN002
which is
  alfa1S≦beta1S ---eqn.BN123
  alfa2S≦beta2S ---eqn.BN124
  alfa3S≦beta3S ---eqn.BN125
  alfa4S＝beta4S ---eqn.BN126
<a name="ch13a197">　Index begin　Index this file
in numerical value, it is
   18 ≦ 20 ---eqn.BN127
   32 ≦ 35 ---eqn.BN128
   44 ≦ 45 ---eqn.BN129
   50 ＝ 50 ---eqn.BN130
Total sum must be equal.

<a name="ch13a198">
Problem 13.2 say :
given matrix equation eqn.BN122
prove that partial sum inequality 
eqn.BN123 to eqn.BN126 are a 
necessary result.

<a name="ch13a199">
Textbook page 196 and 197 has 
proof. Following is a numerical
illustration, aid to textbook 
proof. It is a copy of LiuHH's
draft record. 
<a name="ch13a200">
On 2010-06-17-21-43 LiuHH created
a file jsmajor2.htm, page title
"Given two sequences, find
 doubly stochastic matrix"
On 2010-06-23-23-24 get all 
non-negative matrix. (correct
output) Today (2010-06-25) 
still write document.
LiuHH will upload to (local)
http://freeman2.com/jsmajor2.htm
few days later. 
<a name="ch13a201">
The matrix at eqn.BN122 is from 
jsmajor2.htm output.
2010-06-25-10-55 here
This numerical example use four
elements in each sequence. Then
we have n=4. Index k vary from
k=1 to k=4.

<a name="ch13a202">　Index begin　Index this file
Box 1 α_seq.= 18 14 12  6 ---eqn.BN110
Box 2 β_seq.= 20 15 10  5 ---eqn.BN111

<a name="ch13a203">
Above is jsmajor2.htm output. 
Because α_seq. and β_seq. are 
integers. Stochastic matrix
calculation use only +-*/ .
It is easy to find out the 
shorter quotient matrix below.

Box 1 α_seq.= 18 14 12  6
Box 2 β_seq.= 20 15 10  5
<a name="ch13a204">
doubly stochastic matrix=
[[ // next quotient matrix ---eqn.BN132
52/63   13/630  52/630   45/630
 0       4/5     1/5       0 
 1/9    16/90   64/90      0
 4/63    1/630   4/630  585/630
]] //above is same as eqn.BN131

<a name="ch13a205">
  α=D*β                 ---eqn.BN121
eqn.BN121 expanded form is next

<a name="ch13a206">
k=1 case // next ten lines ---eqn.BN134
alfa1S=SUM[j=1,1]=18
     = 20*(52/63  )
      +15*(13/630 )
      +10*(52/630 )
      + 5*(45/630 )
//α sum from 1 to 1, β sum from 1 to 4
<a name="ch13a207">　Index begin　Index this file
              //define c's for k=1
     = 20*c1  // c1=(52/63  )
      +15*c2  // c2=(13/630 )
      +10*c3  // c3=(52/630 )
      + 5*c4  // c4=(45/630 )

<a name="ch13a208">
alfa1S=(beta_1*c1+beta_2*c2
       +beta_3*c3+beta_4*c4) ---eqn.BN135
where
c1+c2+c3+c4=k=1 ---eqn.BN136

<a name="ch13a209">
k=2 case // next ten lines ---eqn.BN137
alfa2S=SUM[j=1,2]=18+14
     = 20*(52/63 + 0  )
      +15*(13/630+4/5 )
      +10*(52/630+1/5 )
      + 5*(45/630+ 0  )
//α sum from 1 to 2, β sum from 1 to 4
<a name="ch13a210">
              //define c's for k=2
     = 20*c1  // c1=(52/63 + 0  )
      +15*c2  // c2=(13/630+4/5 )
      +10*c3  // c3=(52/630+1/5 )
      + 5*c4  // c4=(45/630+ 0  )

<a name="ch13a211">
alfa2S=(beta_1*c1+beta_2*c2
       +beta_3*c3+beta_4*c4) ---eqn.BN138
where
c1+c2+c3+c4=k=2 ---eqn.BN139

<a name="ch13a212">　Index begin　Index this file
k=3 case // next ten lines ---eqn.BN140
alfa3S=SUM[j=1,3]=18+14+12
     = 20*(52/63 + 0  + 1/9 )
      +15*(13/630+4/5 +16/90)
      +10*(52/630+1/5 +64/90)
      + 5*(45/630+ 0  +  0  )
//α sum from 1 to 3, β sum from 1 to 4
<a name="ch13a213">
              //define c's for k=3
     = 20*c1  // c1=(52/63 + 0  + 1/9 )
      +15*c2  // c2=(13/630+4/5 +16/90)
      +10*c3  // c3=(52/630+1/5 +64/90)
      + 5*c4  // c4=(45/630+ 0  +  0  )
<a name="ch13a213a">
Attention: c1 to c4 each sum part
of one column. Column sum is one.
c1 to c4 all be ≦1 so, 1-c1≧0,
 1-c2≧0,  1-c3≧0,  1-c4≧0 ---eqn.BN340
<a name="ch13a214">
alfa3S=(beta_1*c1+beta_2*c2
       +beta_3*c3+beta_4*c4) ---eqn.BN141
where
c1+c2+c3+c4=k=3 ---eqn.BN142

<a name="ch13a215">
k=4 case // next five lines ---eqn.BN143
alfa4S=SUM[j=1,4]=18+14+12+6
     = 20*(1)
      +15*(1)
      +10*(1)
      + 5*(1)

<a name="ch13a216">
Textbook page 196 line -5, -3
and page 197 line 2, 3, 4.
fix k value. The following
choose k=3 (among k=1,2,3)
and parallel to textbook.

<a name="ch13a217">　Index begin　Index this file
for k=3  //define delta_k
delta_k = delta_3 ---eqn.BN144
        = alfa3S - beta3S 
 =       //expand alfa3S, beta3S
 (alfa_1 + alfa_2 + alfa_3) ---eqn.BN145
-(beta_1 + beta_2 + beta_3)
Above is textbook page 196 
line -3 equation for k=3 case.

<a name="ch13a220">
for k=3
Next equation is most important 
step. alfa_j change to beta_j*cj
delta_k = delta_3 //used eqn.BN141
= (beta_1*c1+beta_2*c2+beta_3*c3+beta_4*c4)
 - beta3S  ---eqn.BN146
= (beta_1*c1+beta_2*c2+beta_3*c3+beta_4*c4)
 -(beta_1   +beta_2   +beta_3) ---eqn.BN147
Above is still textbook page 196 
line -3 equation for k=3 case.

// next equation ---eqn.BN122
[18] [52/63  13/630  52/630  45/630 ] [20]
[14]=[ 0      4/5     1/5      0    ]*[15]
[12] [ 1/9   16/90   64/90     0    ] [10]
[ 6] [ 4/63   1/630   4/630  585/630] [ 5]

<a name="ch13a234">
Following is re-group.
delta_k = delta_3
=+(beta_3-beta_1)*(1-c1)
 +(beta_3-beta_2)*(1-c2)
 +(beta_3-beta_3)*(1-c3)
 +(beta_4-beta_3)*c4  ---eqn.BN159
eqn.BN340: 1-c1≧0, 1-c2≧0, 1-c3≧0
Matrix be nonnegative, c4≧0
<a name="ch13a235">
recall, n=4, k=3
  β_seq.= 20 15 10  5    ---eqn.BN111
  beta_1=20, beta_2=15,  ---eqn.BN114
  beta_3=10, beta_4= 5.  ---eqn.BN115
beta_3 is pivoting point
<a name="ch13a236">
for 1≦j≦k=3  beta_j ＞ beta_k
that is  (beta_3-beta_1)≦0 ---eqn.BN160
         (beta_3-beta_2)≦0 ---eqn.BN161
         (beta_3-beta_3) =0 ---eqn.BN162
for 3=k＜j≦n=4  beta_j ≦ beta_k
that is  (beta_4-beta_3)≦0 ---eqn.BN163
then eqn.BN159 tell us
  delta_3 ≦ 0  ---eqn.BN164
is true.

2010-06-25-13-25 document here
<a name="ch13a237">　Index begin　Index this file
Similarly
 delta_1 ≦ 0 
 delta_2 ≦ 0 

Above showed that
given Muirhead's condition
then  partial sum inequality
is true.

<a name="ch13a238">
For a proof of general case,
(not special case n=4,k=3)
Please read textbook page 196, 197.
Above special case analysis may
help you to understand textbook
proof.

Problem 13.2 is done.
2010-06-25-13-58 document stop

2010-06-25-21-55 done first  proofread
2010-06-26-18-58 done second proofread
2010-06-26-19-18 done spelling check




<a name="Copyright">　Index begin　Index this file
2009-06-19-10-48
If you are interested in inequality,
suggest you buy the book
The Cauchy-Schwarz Master Class
written by Prof. J. Michael Steele
The Cauchy-Schwarz Master Class is
this web page's textbook.

To buy textbook, that is to show thanks
to Prof. J.M. Steele's great work.
and it is also respect copyright law.
Thank you. Freeman 2009-06-19

The Cauchy-Schwarz Master Class
J. Michael Steele ★★★★★
ISBN 978-0-521-54677-5
2009-06-19-10-56