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#1 2009-04-17 23:33:01

Kurre
Member
Registered: 2006-07-18
Posts: 280

Kurre's Exercises

#1 Solve:
a)


b)

#2 Find all functions satisfying:

#3 Find all functions satisfying:

#4 Find all functions satisfying:

#5 Let a be a given real number. Find all functions satisfying:

#6 Find all injective functions satisfying:

Last edited by Kurre (2009-05-05 05:21:26)

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#2 2009-05-04 06:45:12

Kurre
Member
Registered: 2006-07-18
Posts: 280

Re: Kurre's Exercises

Well I would like to change the topic to "kurres exercises" or something, but I cant, so here comes a few more that are not related to functional equations.

#7 Let


Find
expressed in A

#8 Let H be a subgroup of G. prove that the following statements are equivalent:
a) H is a normal subgroup
b) for all a,b in G,

if and only if

#9 Solve the following equation in primes p,q,r:

Last edited by Kurre (2009-05-06 02:07:52)

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#3 2009-05-04 08:50:47

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Kurre's Exercises

Changed! You can edit a topic's title by editing its first post.


Why did the vector cross the road?
It wanted to be normal.

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#4 2009-05-04 10:30:08

Kurre
Member
Registered: 2006-07-18
Posts: 280

Re: Kurre's Exercises

Thanks! But I still cant find a field where to edit the title? i know what u mean, i have seen it before. I thought it had been too long since i created the topic..

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#5 2009-05-04 11:03:26

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Kurre's Exercises

Hmm. Maybe the software has changed then. I'm certain it used to be possible that way.
(It's how I changed the title, but I probably follow different rules)

Good point about my answer, I forgot about that detail. I can tweak it though!


Why did the vector cross the road?
It wanted to be normal.

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#6 2009-05-04 20:50:17

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Kurre's Exercises

mathsyperson wrote:

Changed! You can edit a topic's title by editing its first post.

You can edit a thread title for something like the first five minutes after you start the thread; after that period, you can’t edit the title any more (except for moderators).

I know, it’s hard for moderators to know all the problems that non-moderators go through. neutral


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

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#7 2009-05-04 23:29:08

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Kurre's Exercises

Thanks for clarifying!


Why did the vector cross the road?
It wanted to be normal.

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#8 2009-05-06 02:18:20

Kurre
Member
Registered: 2006-07-18
Posts: 280

Re: Kurre's Exercises

correct mathsy up

hints



Last edited by Kurre (2009-05-06 02:20:42)

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#9 2009-05-06 06:16:44

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Kurre's Exercises


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

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#10 2009-05-06 06:54:14

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Kurre's Exercises


Why did the vector cross the road?
It wanted to be normal.

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#11 2009-05-06 08:19:49

Kurre
Member
Registered: 2006-07-18
Posts: 280

Re: Kurre's Exercises

nice smile


Last edited by Kurre (2009-05-06 08:27:15)

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#12 2009-05-06 08:27:55

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Kurre's Exercises


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

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#13 2009-05-06 08:52:51

Kurre
Member
Registered: 2006-07-18
Posts: 280

Re: Kurre's Exercises

#10 let f be a function from the natural numbers to the natural numbers satisfying

if n-1>m>0. Find the least possible value of f(2009) and f(2011)

Last edited by Kurre (2009-05-06 08:55:34)

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#14 2009-05-06 09:07:40

Kurre
Member
Registered: 2006-07-18
Posts: 280

Re: Kurre's Exercises

uhm okey that was much nicer than what I did neutral smile

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#15 2009-05-07 05:18:05

Kurre
Member
Registered: 2006-07-18
Posts: 280

Re: Kurre's Exercises

#11 Find all functions from the positive rational numbers to the positive rational numbers satisfying:


Last edited by Kurre (2009-05-07 05:28:36)

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#16 2009-05-08 00:15:18

Kurre
Member
Registered: 2006-07-18
Posts: 280

Re: Kurre's Exercises

#12
Let n be an integer with m prime factors (not necessarily distinct). suppose d|n and define

where
and
are the multiplicities for p in the prime factorization for n and d respectively. Show that:

and determine when equality occurs.

Last edited by Kurre (2009-05-08 00:15:34)

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#17 2009-05-12 03:33:32

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,455

Re: Kurre's Exercises

Hi Kurre;

  I have been working very hard on your #7. I will continue to work on it, but in case I don't get there after a couple of weeks more. I would very much like to see your method.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#18 2009-05-15 08:12:30

Kurre
Member
Registered: 2006-07-18
Posts: 280

Re: Kurre's Exercises

bobbym wrote:

Hi Kurre;

  I have been working very hard on your #7. I will continue to work on it, but in case I don't get there after a couple of weeks more. I would very much like to see your method.

Im really glad to hear you are working on it smile

If you want, I can post my solution.

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#19 2009-05-15 08:26:45

Kurre
Member
Registered: 2006-07-18
Posts: 280

Re: Kurre's Exercises

#13 Let M be a (p-1)x(p-1) matrix where p is an odd prime number and each row has each element from {1,2,3,...,p-2,p-1} exactly once. Prove that p|det(M)

Last edited by Kurre (2009-05-15 09:13:12)

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#20 2009-05-15 09:07:22

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Kurre's Exercises

#13 is false for p=2.


Why did the vector cross the road?
It wanted to be normal.

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#21 2009-05-15 09:15:43

Kurre
Member
Registered: 2006-07-18
Posts: 280

Re: Kurre's Exercises

mathsyperson wrote:

#13 is false for p=2.

true, forgot about that. Fixed! smile

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#22 2009-05-15 19:46:48

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Kurre's Exercises

can be generalized. For each integer
,
divides the determinant of a
matrix in which each positive integer less than and coprime with
appears exactly once in each row and in each column.


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

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#23 2009-05-15 23:15:07

Kurre
Member
Registered: 2006-07-18
Posts: 280

Re: Kurre's Exercises

JaneFairfax wrote:

can be generalized. For each integer
,
divides the determinant of a
matrix in which each positive integer less than and coprime with
appears exactly once in each row and in each column.

Yea I realized that when trying to sleep yesterday tongue
edit: you dont need exactly once in the columns also

Last edited by Kurre (2009-05-15 23:18:45)

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#24 2009-05-16 02:48:54

Kurre
Member
Registered: 2006-07-18
Posts: 280

Re: Kurre's Exercises

#14
let

be the n roots of unity. For which n and m does it hold that:
?

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#25 2009-05-16 11:05:43

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,455

Re: Kurre's Exercises

Incomplete answer so everyone else please continue to look

Last edited by bobbym (2009-05-16 11:11:55)


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

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