S.O.S. Mathematics CyberBoard

Your Resource for mathematics help on the web!
 It is currently Fri, 9 Dec 2016 06:49:22 UTC

 All times are UTC [ DST ]

 Page 1 of 1 [ 5 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: Repetitive Derivatives of PolynomialsPosted: Sat, 24 Apr 2010 00:11:20 UTC
 Member

Joined: Tue, 24 Mar 2009 04:48:47 UTC
Posts: 40
Let T map the space of 2nd degree polynomials into itself by the rule:

T(y)= a + (2*a+b)x + (2a+b+c)x^2 where y=ax^2+bx+c.

Informally, T maps y to the sum of itself, its first derivative, and its second derivative.
Find an explicit formula for T^n(y), i.e. the composition of T n times applied to y.

Spoiler:
Answer: T^n(y)=ax^2+(2na+b)x+((n+1)!a + nb + c)x^2

_________________
Euler is God!

Top

 Post subject: Posted: Sat, 24 Apr 2010 03:25:49 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Sun, 24 Jul 2005 20:12:39 UTC
Posts: 4933
Location: Ottawa Ontario
So?

_________________
I'm just an imagination of your figment...

Top

 Post subject: Re: Repetitive Derivatives of PolynomialsPosted: Sat, 24 Apr 2010 12:17:23 UTC
 S.O.S. Oldtimer

Joined: Sun, 4 Nov 2007 12:08:30 UTC
Posts: 246
Location: Bratislava, Slovakia
srjcstud wrote:
T(y)= a + (2*a+b)x + (2a+b+c)x^2 where y=ax^2+bx+c.

Informally, T maps y to the sum of itself, its first derivative, and its second derivative.

Did you mean T(y)=ax^2 + (2*a+b)x + (2a+b+c)?

Top

 Post subject: Re: Repetitive Derivatives of PolynomialsPosted: Sat, 24 Apr 2010 19:28:31 UTC
 S.O.S. Oldtimer

Joined: Sun, 4 Nov 2007 12:08:30 UTC
Posts: 246
Location: Bratislava, Slovakia
I guess you are talking about the map

since this is what you wrote informally.

This map can be described as linear map , or
in the matrix form
.

So your question is equivalent to computing . I tried this, however, my result is different from yours...?

Spoiler:
where
. We have

and .
From binomial theorem (which can be used, since every matrix commutes with ) we get

then .

Top

 Post subject: Posted: Sun, 25 Apr 2010 18:28:00 UTC
 Member

Joined: Tue, 24 Mar 2009 04:48:47 UTC
Posts: 40
I get a different matrix A for the transformation.

_________________
Euler is God!

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 5 posts ]

 All times are UTC [ DST ]

Who is online

Users browsing this forum: No registered users

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forum

Search for:
 Jump to:  Select a forum ------------------ High School and College Mathematics    Algebra    Geometry and Trigonometry    Calculus    Matrix and Linear Algebra    Differential Equations    Probability and Statistics    Proposed Problems Applications    Physics, Chemistry, Engineering, etc.    Computer Science    Math for Business and Economics Advanced Mathematics    Foundations    Algebra and Number Theory    Analysis and Topology    Applied Mathematics    Other Topics in Advanced Mathematics Other Topics    Administrator Announcements    Comments and Suggestions for S.O.S. Math    Posting Math Formulas with LaTeX    Miscellaneous