,
,
,
,
or even expressions of those terms combined as if that would make them defined. I have made this post to answer these questions once and for all, and I politely ask everyone to link this post to threads that are talking about this issue.
,
isn't even rigorously defined in this expression. What exactly do we mean by 
.
,
does not exist in the first place, although you will see why in a calculus course. So the point is that 
?
indeterminate? In analysis (the theory behind calculus and beyond), limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits. However, if the expression obtained after this substitution does not provide sufficient information to determine the original limit, then the expression is called an indeterminate form. For example, we could say that 
,
or ![\[0/0, \infty/\infty, \infty-\infty, \infty \times 0\]](http://latex.artofproblemsolving.com/2/6/0/260f97a1cf1ea8722ff97a29b83bac7bfe83e577.png)
etc etc. So what makes something undefined? In the broader scope, something being undefined refers to an expression which is not assigned an interpretation or a value. A function is said to be undefined for points outside its domain. For example, the function 
given by the mapping 
is undefined for 
.
is undefined because dividing by ![\[\overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty,+\infty\}.\]](http://latex.artofproblemsolving.com/9/9/3/993c78b62ec2c84b5d77daa8e9a8cd6f70fcf904.png)
So instead of treating infinity as an idea, we define infinity (positively and negatively, mind you) as actual numbers in the reals. The advantage of doing this is for two reasons. The first is because we can turn this thing into a totally ordered set. Specifically, we can let 
for each 
which means that via this order topology each subset has an infimum and supremum and 
is therefore compact. While this is nice from an analytic standpoint, extending the reals in this way can allow for interesting arithmetic! In ![\begin{align*}
a + \infty = \infty + a & = \infty, & a & \neq -\infty \\
a - \infty = -\infty + a & = -\infty, & a & \neq \infty \\
a \cdot (\pm\infty) = \pm\infty \cdot a & = \pm\infty, & a & \in (0, +\infty] \\
a \cdot (\pm\infty) = \pm\infty \cdot a & = \mp\infty, & a & \in [-\infty, 0) \\
\frac{a}{\pm\infty} & = 0, & a & \in \mathbb{R} \\
\frac{\pm\infty}{a} & = \pm\infty, & a & \in (0, +\infty) \\
\frac{\pm\infty}{a} & = \mp\infty, & a & \in (-\infty, 0).
\end{align*}](http://latex.artofproblemsolving.com/6/d/8/6d8fc1eca5c791d430a7168cd3b37a02104fa318.png)
So addition, multiplication, and division are all defined nicely. However, notice that we have some indeterminate forms here which are also undefined,![\[\infty-\infty,\frac{\pm\infty}{\pm\infty},\frac{\pm\infty}{0},0\cdot \pm\infty.\]](http://latex.artofproblemsolving.com/a/0/b/a0b67076efcc014c78b2023be1151c84b57c1503.png)
So while we define certain things, we also left others undefined/indeterminate in the process! However, in the context of measure theory it is common to define 
as greenturtle3141 noted below. I encourage to reread what he wrote, it's great stuff! As you may notice, though, dividing by ![\[\mathbb{C}^*=\mathbb{C}\cup\{\tilde{\infty}\}\]](http://latex.artofproblemsolving.com/8/4/e/84ed3d6ab237df970333b87beaef5311caa46185.png)
which we call the Riemann Sphere (it actually forms a sphere, pretty cool right?). As a note, 
means complex infinity, since we are in the complex plane now. Here's the catch: division by ![\[\frac{z}{0}=\tilde{\infty},\frac{z}{\tilde{\infty}}=0.\]](http://latex.artofproblemsolving.com/1/6/0/160cb939707708ff4e0930724df2928af11d7fd1.png)
where 
and 
Furthermore, we actually have some nice properties with multiplication that we didn't have before. In 
it holds that![\[\tilde{\infty}\times \tilde{\infty}=\tilde{\infty}\]](http://latex.artofproblemsolving.com/8/e/0/8e035b4841d20c5ad671c46355cc667ca01e533b.png)
but 
and 
are left as undefined (unless there is an explicit need to change that somehow). One could define the projectively extended reals as we did with ![\[{\widehat {\mathbb {R} }}=\mathbb {R} \cup \{\infty \}.\]](http://latex.artofproblemsolving.com/c/0/4/c044ff37849acad3b8b280b9753539225a1276ca.png)
They behave in a similar way to the Riemann Sphere, with division by 
be an acute triangle with circumcenter 
and orthocenter 
,
.
to the circumcicle of 
in 
.
be the midpoint of 
.
.
and 
Prove that
and 
be positive integers greater than 
grid lies a coin with its tail side up. A move consists of the following steps.
square in the grid;
for which it is possible that every coin shows head-side up after a finite number of moves.
and 
.![\[P=\frac{1}{5-2a}+\frac{1}{5-2b}+\frac{1}{5-2c}.\]](http://latex.artofproblemsolving.com/c/3/7/c3708768a84c2bbb1e4fae502a9bb1e92cb13d02.png)
feet and area 
,![\[ \textbf{(A)}\ 24 \qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 46 \qquad
\textbf{(D)}\ 66 \qquad
\textbf{(E)}\ 74
\]](http://latex.artofproblemsolving.com/8/2/f/82ffc15072ca99f6cd1abc312781eb4827817356.png)
degrees about one side. The space through which the hexagon travels during the rotation forms a solid. Find the volume of this solid.
and height 
:
Volume of cone frustum with base radii 
,
has parallel bases 
and 
,
is the midpoint of side 
and 
.
cm and 
cm, what is 
?
,
,
Find
Y by jatin, az360, hctb00, dan23, dantx5, cobbler, dizzy, Einstein314, champion999, 62861, Kgxtixigct, richrow12, Adventure10, Mango247, and 18 other users
Since I was asked about it I'm posting it.
We begin with a definition:
1. Let
, 
, ..., 
, 
, 
, ..., 
be arbitrary real numbers satisfying 
and 
. Now, if we have all the following conditions fulfilled:
;
;
;
...
generally
for any natural k with 
;
and
(mind the equality sign in the last condition; it is not a
sign!), then we say that the number array 
majorizes the number array 
. We write this in the form 
.
Now, if instead of
;
;
;
...
generally for any natural k with ;
and,
we have the conditions
;
;
;
...
generally
for any natural k with ;
and
fulfilled (i. e., the same conditions with all's replaced by 
's), then we say that the number array minorizes the number array . We write this in the form 
. Of course, this is equivalent to 
.
2. Thus we have defined the notions "majorizes" and "minorizes" only for non-increasing arrays (i. e. for arrays satisfying and ). Now, assume that , , ..., , , , ..., are just arbitrary real numbers, without any conditions. Then, let 
be the non-increasing permutation of the array , i. e. the permutation of the array that satisfies 
. Similarly, let 
be the non-increasing permutation of the array , i. e. the permutation of the array that satisfies 
. The number arrays and are both non-increasing, and hence we have defined majorization for such arrays.
Then, we say that the number array majorizes the number array if and only if the number array majorizes the number array . In this case, we write .
Similarly, we say that the number array minorizes the number array if and only if the number array minorizes the number array . In this case, we write . Again this is equivalent to .
3. So we have defined the terms "majorize" and "minorize" for any two number arrays. It should be noted that majorization is a partial order on the set of number arrays, not a total order - i. e., not for every pair of two number arrays and one can say that either the first one majorizes the second one, or the second one majorizes the first one. It often happens that none of the arrays majorizes or minorizes the other one. But sometimes when you have some special arrays, you can prove that one of them majorizes the other one.
4. Now, the Karamata inequality, also called the Majorization Inequality or the Hardy-Littlewood inequality, states that if, , ..., , , , ..., are 
reals from an interval 
such that the number array majorizes the number array , and 
is any convex function, then
.
If is a concave function instead, then we instead have
.
If the number array minorizes the number array instead of majorizing it, then both inequalities are reversed.
5. The Jensen inequality for real numbers is a special case of the Karamata inequality. In fact, if
is the arithmetic mean of the numbers , , ..., , then it is easy to show that the number array majorizes the number array 
. Hence, the Karamata inequality yields:
If f(x) is any convex function, then
,
i. e.
.
If f(x) is a concave function instead, then we instead have
,
i. e.
.
Of course, this is exactly the Jensen inequality for
reals.
6. The definition of majorizing and minorizing number arrays given above is somewhat unsatisfying from an intuitive point of view, since it does not help one to imagine how an array majorizing another array looks like. Unfortunately, this is partly inherent to the notion of majorization, which indeed is quite unintuitive. For a - rather facile - visualization of the notion, you can imagine that a number array majorizes a number array if the two arrays have the same sum of numbers, but the numbers of the first array are set wider apart than those of the second array, while those of the second array lie closer together. From this intuitive viewpoint, it is clear why the number array majorizes the number array , where is the arithmetic mean of the numbers , , ..., : In fact, the two number arrays have the same sum of elements, but the elements of the second number array lie nearer to each other (they are all equal). Alas, this viewpoint does not help one to really understand what majorization is about.
Well, I know there is more to say. For instance, the Karamata inequality has a kind of converse, but I am not sure how it is formulated, so I leave this to the other MathLinkers more used to inequalities.
Darij
We begin with a definition:
1. Let
, 
, ..., 
, 
, 
, ..., 
be arbitrary real numbers satisfying 
and 
. Now, if we have all the following conditions fulfilled:
;
;
;...
generally
for any natural k with 
;and
(mind the equality sign in the last condition; it is not a
sign!), then we say that the number array 
majorizes the number array 
. We write this in the form 
.Now, if instead of
;;;...
generally
for any natural k with ;and
,we have the conditions
;
;
;...
generally
for any natural k with ;and
fulfilled (i. e., the same conditions with all
's replaced by 
's), then we say that the number array minorizes the number array . We write this in the form 
. Of course, this is equivalent to 
.2. Thus we have defined the notions "majorizes" and "minorizes" only for non-increasing arrays (i. e. for arrays satisfying
and ). Now, assume that , , ..., , , , ..., are just arbitrary real numbers, without any conditions. Then, let 
be the non-increasing permutation of the array , i. e. the permutation of the array that satisfies 
. Similarly, let 
be the non-increasing permutation of the array , i. e. the permutation of the array that satisfies 
. The number arrays and are both non-increasing, and hence we have defined majorization for such arrays.Then, we say that the number array
majorizes the number array if and only if the number array majorizes the number array . In this case, we write .Similarly, we say that the number array
minorizes the number array if and only if the number array minorizes the number array . In this case, we write . Again this is equivalent to .3. So we have defined the terms "majorize" and "minorize" for any two number arrays. It should be noted that majorization is a partial order on the set of number arrays, not a total order - i. e., not for every pair of two number arrays
and one can say that either the first one majorizes the second one, or the second one majorizes the first one. It often happens that none of the arrays majorizes or minorizes the other one. But sometimes when you have some special arrays, you can prove that one of them majorizes the other one.4. Now, the Karamata inequality, also called the Majorization Inequality or the Hardy-Littlewood inequality, states that if
, , ..., , , , ..., are 
reals from an interval 
such that the number array majorizes the number array , and 
is any convex function, then
.If
is a concave function instead, then we instead have
.If the number array
minorizes the number array instead of majorizing it, then both inequalities are reversed.5. The Jensen inequality for real numbers is a special case of the Karamata inequality. In fact, if
is the arithmetic mean of the numbers , , ..., , then it is easy to show that the number array majorizes the number array 
. Hence, the Karamata inequality yields:If f(x) is any convex function, then
,i. e.
.If f(x) is a concave function instead, then we instead have
,i. e.
.Of course, this is exactly the Jensen inequality for
reals.6. The definition of majorizing and minorizing number arrays given above is somewhat unsatisfying from an intuitive point of view, since it does not help one to imagine how an array majorizing another array looks like. Unfortunately, this is partly inherent to the notion of majorization, which indeed is quite unintuitive. For a - rather facile - visualization of the notion, you can imagine that a number array
majorizes a number array if the two arrays have the same sum of numbers, but the numbers of the first array are set wider apart than those of the second array, while those of the second array lie closer together. From this intuitive viewpoint, it is clear why the number array majorizes the number array , where is the arithmetic mean of the numbers , , ..., : In fact, the two number arrays have the same sum of elements, but the elements of the second number array lie nearer to each other (they are all equal). Alas, this viewpoint does not help one to really understand what majorization is about.Well, I know there is more to say. For instance, the Karamata inequality has a kind of converse, but I am not sure how it is formulated, so I leave this to the other MathLinkers more used to inequalities.
Darij
This post has been edited 5 times. Last edited by darij grinberg, Mar 29, 2009, 12:10 AM
a 
matrix having properties:
,
for 
is given in 
and 
( where 
denotes the transpose of 
)
?
.
and 
such that 
with the properties 1) and 2) such that 
.
![\[a_1f(x_1)+a_2f(x_2)+\dots+a_nf(x_n)\geq f(a_1x_1+\dots+a_nx_n)\]](http://latex.artofproblemsolving.com/5/7/5/5754cb3b57716e75963d3135c67dd2c546c493b0.png)
,
is convex function, ![$a_i\in[0,1]$](http://latex.artofproblemsolving.com/c/f/5/cf5c874062885702b41ac30bb2066012d49bb8c5.png)
and 
.
is a particular case only.
![$f:[a,b]\to \mathbb{R}$](http://latex.artofproblemsolving.com/b/7/3/b7344c80f2c235aaadafedfbca0e2734001b156d.png)
.![\[k\cdot f(x)+(1-k)\cdot f(y)\geq f(kx+(1-k)y)\quad \forall x,y\in [a,b],\ \forall k\in[0,1].\]](http://latex.artofproblemsolving.com/a/0/a/a0a7b7d852cb8965215cb2261f310aef9010310f.png)
for all ![$x\in[a,b]$](http://latex.artofproblemsolving.com/4/9/4/49432e6cff349a39878727bec9bfcb0efc9fbb7c.png)
.
.
for every convex function 
.
and 
have equal number of components),
)
(conclusion: the majorization conditions).
holds for every function f of the form f(x) = |x - u| (with u constant), then 
.
is the arithmetic mean of the numbers 
,
,
,
majorizes the number array 
.
,
.
,
.
T
and 
,
would need to be convex, to convert it back to the additive form.
and 
be two functions defined on interval 
satisfying:
for any 
;
.![\[\int_{a}^{b}f\left(\varphi(x)\right)\geq \int_{a}^{b}f\left(\psi(x)\right).\]](http://latex.artofproblemsolving.com/e/0/e/e0e96f79a8227a0de04ccaaf236619b96cdb4c65.png)
I hope someone can prove it or give a counterexample.
be a smooth nonnegative function which is convex on ![$[a,b]$](http://latex.artofproblemsolving.com/8/e/c/8ecbd1ba3da8f2adef66a63f2ab32c47e63fa734.png)
and let ![$[x,y]\subset[a,b]$](http://latex.artofproblemsolving.com/c/e/7/ce70161202a64abd224f277a3bebb8d24f6673f5.png)
such that 
.
?
