What do these new symbols mean ?I can't start this without knowing

In summary, the conversation discusses the P_n^k polynomial and its properties, including a summation property and a proof involving induction. The participants also discuss the proof of the positivity of the polynomial, with one suggesting the use of induction and the other proposing a derivative test.
  • #1
flyingpig
2,579
1

Homework Statement

[PLAIN]http://img88.imageshack.us/img88/4418/unledekp.png [Broken]

The Attempt at a Solution



What is the [tex]P^n _{k}[/tex] part thing?

Someone should probably go over the i) for me too...
 
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  • #2
The [itex]P_n^k[/itex] is just a symbol. It is a name for a polynomial defined by

[tex]P_n^k(x)=\binom{n}{k}x^k(1-x)^{n-k}[/tex]

So it's just like we define [itex]P(x)=x^2[/itex]. But now our polynomial depends of n and k.
 
  • #3
For the i)

[tex]\sum^{n}_{k=0} \left( \alpha f\left(\frac{k}{n}\right) \binom{n}{k} x^k(1-x)^{n-k} + \beta g\left(\frac{k}{n}\right) \binom{n}{k} x^k(1-x)^{n-k}\right) = \alpha \sum^{n}_{k=0} f\left(\frac{k}{n}\right) \binom{n}{k} x^k(1-x)^{n-k} + \beta \sum^{n}_{k=0} g\left(\frac{k}{n}\right) \binom{n}{k} x^k(1-x)^{n-k}[/tex]

Summation property??
 
  • #4
Something is wrong...do I need induction? This was from proof class
 
  • #5
For (i), you need to start from

[tex]B_n(\alpha f+\beta g)=\sum_{k=0}^n{(\alpha f+\beta g)(\frac{k}{n})x^k(1-x)^{n-k}}[/tex]
 
  • #6
micromass said:
For (i), you need to start from

[tex]B_n(\alpha f+\beta g)=\sum_{k=0}^n{(\alpha f+\beta g)(\frac{k}{n})x^k(1-x)^{n-k}}[/tex]

What's wrong with what I did...? GO on grill me!
 
  • #7
flyingpig said:
What's wrong with what I did...? GO on grill me!

Nothing is wrong, it's all correct, but it's not finished yet.

You still need to show that the left-hand-side of your post equals

[tex]B_n(\alpha f+\beta g)[/tex]

and that the right-hand-side equals

[tex]\alpha B_n(f)+\beta B_n(g)[/tex]
 
  • #8
[tex]B_n(\alpha f+\beta g)=\sum_{k=0}^n{(\alpha f+\beta g)(\frac{k}{n})x^k(1-x)^{n-k}} [/tex]

[tex]= \sum_{k=0}^n{(\alpha f (\frac{k}{n})x^k(1-x)^{n-k}} + \beta g(\frac{k}{n})x^k(1-x)^{n-k} ) = \sum_{k=0}^n{\alpha f (\frac{k}{n})x^k(1-x)^{n-k}} + \sum_{k=0}^n \beta g(\frac{k}{n})x^k(1-x)^{n-k} ) = \alpha B_n(f)+\beta B_n(g)[/tex]

okay done, got lazy with the long tex
 
  • #9
Yeah, that looks good!
 
  • #10
How do I start ii) then?
 
  • #11
You need to prove

[tex]B_n(f)\leq B_n(g)[/tex]

Start by writing these things out according to the definition of the [itex]B_n[/itex].
 
  • #12
[tex] \sum^{n}_{k=0} f\left(\frac{k}{n}\right) \binom{n}{k} x^k(1-x)^{n-k} \leq \sum^{n}_{k=0} g\left(\frac{k}{n}\right) \binom{n}{k} x^k(1-x)^{n-k}/tex]

SOme kinda of cancellations...?
 
  • #13
Well, you know that

[tex]f(k/n)\leq g(k/n)[/tex]

Now try to introduce the terms needed to conclude that [itex]B_nf\leq B_ng[/itex].
 
  • #14
Can you multiply P to both sides...? I don't know if P is always positive
 
  • #15
flyingpig said:
Can you multiply P to both sides...?

That's the idea.

I don't know if P is always positive

Try to prove it then. Prove that [itex]P_k^n(x)[/itex] is positive if [itex]x\in [0,1][/itex]...
 
  • #16
So I have to prove two things...

OKay since [tex]x \in [0, 1] \leq 0[/tex], then it doesn't matter what i put in right? Now how do do that in proper English...?
 
  • #17
flyingpig said:
OKay since [tex]x \in [0, 1] \leq 0[/tex], then it doesn't matter what i put in right?

This makes no sense to me...
 
  • #18
I meant to say [tex]x \in [0, 1] \geq 0[/tex]

sorry
 
  • #19
flyingpig said:
I meant to say [tex]x \in [0, 1] \geq 0[/tex]

sorry

Yeah, that also makes no sense. How can [itex][0,1]\geq 0[/itex]?? [0,1] is a set.
 
  • #20
OKay I wanted to say that numbers in [0,1] are alll positive, so we never had to worry about odd powers messing up with negative numbers
 
  • #21
OK, you need to show that for all [itex]x\in [0,1][/itex] and all k that

[tex]x^k\geq 0[/tex]

and

[tex](1-x)^{n-k}\geq 0[/tex]

That shouldn't be too difficult??
 
  • #22
This is a long problem lol

I have to do two induction problems first??
 
  • #23
You can do it by induction if you want to. You can also say that it's obvious. Isn't it obvious that the power of a positive number is positive.

I don't know what your professor wants. If he wants you to prove every little thing, then you might want to do induction.
 
  • #24
Do I have to prove that x^k and (1-x)^{n-k} > 0 first?
 
  • #25
flyingpig said:
Do I have to prove that x^k and (1-x)^{n-k} > 0 first?

You don't need to show >0, you need to show [itex]\geq 0[/itex]. But yes, if that's how you want to start, try to prove that first...
 
  • #26
Is there another way
 
  • #27
flyingpig said:
Is there another way

Another way for what?
 
  • #28
To do thisproblem without doing two inductions?
 
  • #29
flyingpig said:
To do thisproblem without doing two inductions?

No, I don't think that's possible if you really want to prove everything rigorously.
 
  • #30
I do the first one first

[tex]S(k) : x^k \geq 0[/tex]

1)Base Case for [tex]x \in [0,1][/tex]

[tex]x^k \geq 0[/tex]
[tex]x^0 = 1 \geq 0[/tex]

Thus the base case is true

2) Inductive Step.

Inductive Hypothesis: Assume that [tex]x^k \geq 0[/tex] is true for all k, then S(k + 1) is

[tex]x^{k +1} \geq 0[/tex]

First

[tex]x^k \geq 0[/tex]

[tex]x^k x \geq 0[/tex]
[tex]x^{k+1} \geq 0[/tex]

Thus by Induction, [tex]x^k \geq 0[/tex] for all k and for [tex]x \in [0,1][/tex]
 
  • #31
flyingpig said:
I do the first one first

[tex]S(k) : x^k \geq 0[/tex]

1)Base Case for [tex]x \in [0,1][/tex]

[tex]x^k \geq 0[/tex]
[tex]x^0 = 1 \geq 0[/tex]

Thus the base case is true

2) Inductive Step.

Inductive Hypothesis: Assume that [tex]x^k \geq 0[/tex] is true for all k, then S(k + 1) is

[tex]x^{k +1} \geq 0[/tex]

First

[tex]x^k \geq 0[/tex]

[tex]x^k x \geq 0[/tex]
[tex]x^{k+1} \geq 0[/tex]

Thus by Induction, [tex]x^k \geq 0[/tex] for all k and for [tex]x \in [0,1][/tex]

That's ok! And the second case follows from this case.
 
  • #32
For the other one, Is it S(n + 1) or S(k + 1)...?
 
  • #33
flyingpig said:
To do thisproblem without doing two inductions?

micromass said:
No, I don't think that's possible if you really want to prove everything rigorously.

One could observe that xk(1-x)n-k is continuous, has zeroes only at 0 and 1, and is positive at x = 1/2.
 
  • #34
LCKurtz said:
One could observe that xk(1-x)n-k is continuous, has zeroes only at 0 and 1, and is positive at x = 1/2.

So should I include a derivative test saying it would be positive everywhere? I am not sure if it is everywhere yet because i haven't started on the proof, but you mentioned x = 1/2 being positive, so there is a negative point?
 
  • #35
LCKurtz said:
One could observe that xk(1-x)n-k is continuous, has zeroes only at 0 and 1, and is positive at x = 1/2.

flyingpig said:
So should I include a derivative test saying it would be positive everywhere? I am not sure if it is everywhere yet because i haven't started on the proof, but you mentioned x = 1/2 being positive, so there is a negative point?

What level of course are you taking? I'm getting the impression that you haven't given any actual thought to what I posted and you just post a silly question in response hoping to get more detailed steps.

You are trying to show that function is nonnegative on [0,1]. Don't you have any idea how my suggestion might be relevant to that? Again, please tell me what level course this is that you are taking.
 
<h2>1. What do these new symbols mean?</h2><p>The meaning of symbols can vary depending on the context and field of study. It is important to provide more information about the specific symbols in order to accurately determine their meaning.</p><h2>2. Can you explain the significance of these symbols?</h2><p>The significance of symbols can also vary depending on the context and field of study. It is important to provide more information about the specific symbols in order to accurately determine their significance.</p><h2>3. How do I interpret these new symbols?</h2><p>The interpretation of symbols can be complex and may require knowledge of the specific field of study. It is best to consult with experts or references in the field to accurately interpret the symbols.</p><h2>4. Why are these symbols important?</h2><p>The importance of symbols can vary depending on the context and field of study. They may represent important concepts, relationships, or data in a particular area of study. It is best to provide more information about the specific symbols to understand their importance.</p><h2>5. Where can I find more information about these symbols?</h2><p>There are many resources available to learn more about symbols, including textbooks, research articles, and experts in the field. It is also helpful to provide more specific information about the symbols in order to find relevant and accurate information.</p>

1. What do these new symbols mean?

The meaning of symbols can vary depending on the context and field of study. It is important to provide more information about the specific symbols in order to accurately determine their meaning.

2. Can you explain the significance of these symbols?

The significance of symbols can also vary depending on the context and field of study. It is important to provide more information about the specific symbols in order to accurately determine their significance.

3. How do I interpret these new symbols?

The interpretation of symbols can be complex and may require knowledge of the specific field of study. It is best to consult with experts or references in the field to accurately interpret the symbols.

4. Why are these symbols important?

The importance of symbols can vary depending on the context and field of study. They may represent important concepts, relationships, or data in a particular area of study. It is best to provide more information about the specific symbols to understand their importance.

5. Where can I find more information about these symbols?

There are many resources available to learn more about symbols, including textbooks, research articles, and experts in the field. It is also helpful to provide more specific information about the symbols in order to find relevant and accurate information.

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