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[[Category:Non-talk pages that are automatically signed]]<noinclude>{{Wikipedia:Reference desk/header|WP:RD/MA}}
[[Category:Wikipedia resources for researchers]]
[[Category:Wikipedia help forums]] 
</noinclude>

{{Wikipedia:Reference_desk/Archives/Mathematics/2010 August 16}}

{{Wikipedia:Reference_desk/Archives/Mathematics/2010 August 17}}

{{Wikipedia:Reference_desk/Archives/Mathematics/2010 August 18}}

= August 19 =

== Best Mathematicians of 2010 ==

Does anyone know who won the fields medal? I can't seem to find this information anywhere! ... If it hasn't been released, when will it be? I heard that it would be released on 19 August,2010 but I don't want to be waiting till midnight for the declaration of the world's best mathematicians! Please guys, put me out of my misery and tell me how many more hourse I need to wait? I've waited for 4 years already ... Thanks! Signed:THE DUDE. (BTW, I fixed my typsetting bug!) <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/110.20.26.228|110.20.26.228]] ([[User talk:110.20.26.228|talk]]) 08:37, 19 August 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot--> 

I think I know now ... Damn, why do applied mathematicians always get the awards? Where are the algebraists, topologists and geometers (and number theorists for that matter)? And why on Earth hasn't anyone received it for logic and set theory yet? (FOrget the the first question in above paragraph, I think I know who won it.) Thanks guys ... <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/110.20.26.228|110.20.26.228]] ([[User talk:110.20.26.228|talk]]) 08:37, 19 August 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->
:There is already a Wikipedia [[User:THE DUDE]], which apparently is not you. Both impersonating other users and pretending to be a user which doesn't really exist is forbidden. Your continued failure to [[Wikipedia:Signatures|sign your posts]], followed by vandalizing your automatically-supplied signatures, is disruptive. Please stop. You can [[Special:UserLogin|create an account]] if you wish. -- [[User:Meni Rosenfeld|Meni Rosenfeld]] ([[User Talk:Meni Rosenfeld|talk]]) 08:16, 19 August 2010 (UTC)
::Meni Rosenfeld, I just don't want to be known to the public. I want to be anonymous. You have a name, Meni Rosenfeld. If THE DUDE is taken, can I have another name. I never vandalized signatures. I just didn't want to be made public. Isn't that a simple rquest? Or are the people here at Wikipedia so serious. What's a change in signature between friends?  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/110.20.26.228|110.20.26.228]] ([[User talk:110.20.26.228|talk]]) 08:37, 19 August 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->
:::Dmcq has advised you to create an account if you want to remain anonymous. You ignored his advice, lied about your motivation, and now you are playing dumb. We have no reason to believe your intentions are in any way friendly. -- [[User:Meni Rosenfeld|Meni Rosenfeld]] ([[User Talk:Meni Rosenfeld|talk]]) 09:44, 19 August 2010 (UTC)
::::OK, I fixed up all signatures? Now can you please answer my question Meni Rosenfeld?  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/114.72.209.200|114.72.209.200]] ([[User talk:114.72.209.200|talk]]) 11:34, 19 August 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->
:::::I don't know the answer, but I'm sure that now those who do will be much happier to assist you. (Your next step is to sign your posts by typing 4 tildes, <nowiki>~~~~</nowiki> ). -- [[User:Meni Rosenfeld|Meni Rosenfeld]] ([[User Talk:Meni Rosenfeld|talk]]) 11:45, 19 August 2010 (UTC)

[[Special:Contributions/114.72.244.249|114.72.244.249]] ([[User talk:114.72.244.249|talk]]) 08:35, 20 August 2010 (UTC) Is this right? [[Special:Contributions/114.72.244.249|114.72.244.249]] ([[User talk:114.72.244.249|talk]]) 08:35, 20 August 2010 (UTC) How did you type 4 tildas without a signature coming up?

:Creating an account is the only way to hide you IP address. No matter what edit you do to this page its and easy matter just to look at the [http://en.wikipedia.org/w/index.php?title=Wikipedia:Reference_desk/Mathematics&curid=3271152&action=history history] to find your IP. You can create an account at [[Special:UserLogin/signup]] --[[User:Salix alba|Salix]] ([[User talk:Salix alba|talk]]): 08:46, 19 August 2010 (UTC)
:Using the [[Wikipedia:NOWIKI#Limiting_formatting.2Fescaping_wiki_markup|nowiki]] tag. -- [[User:Meni Rosenfeld|Meni Rosenfeld]] ([[User Talk:Meni Rosenfeld|talk]]) 13:24, 20 August 2010 (UTC)
'''Why isn't anyone answering my questions? Please answer guys ... Thanks guys ...''' [[Special:Contributions/114.72.244.249|114.72.244.249]] ([[User talk:114.72.244.249|talk]]) 08:37, 20 August 2010 (UTC)

:Can't speak for anyone else, but I haven't attaempted to answer your original questions because (a) they call for opinion, which is not the best use of Wikipedia's reference desks and (b) you come across as bad-tempered and argumentative, so even if I did venture an opinion, I think there is a good chance you would just use it as a springboard for an argument, so it would be a waste of my time. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 08:53, 20 August 2010 (UTC)

Please answer my question? I won't attack you. What makes you think I'm bad tempered? I just like to express my opinion 'is all. I'm as polite as I can be and yet I'm misunderstood.I won't argue. Come on?   <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/110.20.55.15|110.20.55.15]] ([[User talk:110.20.55.15|talk]]) 11:38, 20 August 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

:Fields Medal winners are listed at [[Fields Medal]].  [[User:Dragons flight|Dragons flight]] ([[User talk:Dragons flight|talk]]) 08:54, 20 August 2010 (UTC)

== Is the argument of a given unary function a singleton? ==

The argument of a given binary function is an (ordered) pair, so the argument of a given unary function is a [[singleton]], isn't it? [[User:Eliko|Eliko]] ([[User talk:Eliko|talk]]) 08:45, 19 August 2010 (UTC)
:Yes--[[User:Shahab|Shahab]] ([[User talk:Shahab|talk]]) 10:38, 19 August 2010 (UTC)
::Unfortunately, it's mentioned nowhere in Wikipedia, as far as I know. [[User:Eliko|Eliko]] ([[User talk:Eliko|talk]]) 10:47, 19 August 2010 (UTC)
:::Well, technically, it's an [[individual]] rather than a singleton.  But you could just as easily take it to be a singleton, and nothing important would change. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 10:47, 19 August 2010 (UTC)
::::I'm looking for the formal definition. "technically" (as you've put it), a binary function doesn't have two arguments each of which is an individual, but rather has an argument being an (ordered) pair of individuals (because the order of both individuals must be preserved), so I would expect that, "technically", a unary function won't have one argument which is an individual, but rather will have an argument being a singleton of an individual. [[User:Eliko|Eliko]] ([[User talk:Eliko|talk]]) 12:33, 19 August 2010 (UTC)
:::::There isn't necessarily a "the" formal definition.  You're talking about details at the "implementation" level, which may vary from formalization to formalization, but don't change the conceptual content for most purposes.
:::::I would say that a unary function takes an individual, and a binary function, at the most fundamental level, takes two individuals (distinguishing between them, as you note).  For many purposes it's convenient to package those two individuals into a single individual, that we then call an ordered pair.  But at the very fine level you seem to be examining this, that technically makes it a unary function that takes as its sole argument an ordered pair, rather than a binary function with two arguments. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 22:11, 19 August 2010 (UTC)
::::::You seem to distinguish between "the most fundamental level" and the "very fine level". So would you agree with the following? "At the most fundamental level", a binary ''real'' function takes two arguments each of which is a real number, while a unary ''real'' function takes one argument which is a real number. On the other hand, i.e. "at the very fine level", every ''real'' function is unary, the following way: a function that would be called "binary" at "the most fundamental level", is in fact (i.e. "at the very fine level") a unary function that "takes as its sole argument an ordered pair" of real numbers, while a function that would be called "unary" at "the most fundamental level", is in fact (i.e. "at the very fine level") a unary function that "takes as its sole argument a singleton" of a real number. [[User:Eliko|Eliko]] ([[User talk:Eliko|talk]]) 07:15, 20 August 2010 (UTC)
::::::: Dude, fair warning:  If you persist in trying to parse my rhetorical flourishes for some precise technical meaning, I'm going to get upset with you and leave the conversation.  I doubt I'm the only one who'll react that way.  Phrases like "at a very fine level" have a function, which you as a presumably competent social actor should be able to discern, but the indication of some exact mathematical or philosophical category is not it.
::::::: That said, here's how I see it:  A binary function, conceptually, is a way of (but not necessarily an expressible rule for) taking two things and getting a third thing, in such a way that the third thing depends only on the two inputs (with the slots for the two inputs being distinguishable).
::::::: How you code that concept into some specific framework, such as set theory, is another matter.  You have lots of choices, and most of the time it doesn't matter even slightly which choice you pick.  Occasionally it will matter (say, when you don't have a pairing function available), and for those instances it's good to practice mental flexibility and not get too hung up on the formalism.  --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 08:54, 20 August 2010 (UTC)

== [[Monolith (Space Odyssey)|Monoliths]] in [[2001:A Space Odyssey]] ==

In 2001: A Space Odyssey, the Monoliths are 1:4:9. That would be 1 by 4 by 9. But 1 ''what'' by 4 ''what'' by 9 ''what''? Centimetres? Metres? Yards? Feet? Inches? Can somebody help? --[[User:Editor510|<font color="red">Editor510</font>]] [[User talk:Editor510|<font color="#4CBB17"><small><sup>drop us a line, mate</sup></small></font>]] 08:58, 19 August 2010 (UTC)

PS: I know this is about an entertainment film but it is a mathematical question, so please don't ask me to move the question.
:The author carefully avoided that question by using the smaller side of the monolith as unit of measurement, and then noticing that the larger side was 9 units and the intermediate side was 4 units. [[User:Bo Jacoby|Bo Jacoby]] ([[User talk:Bo Jacoby|talk]]) 10:56, 19 August 2010 (UTC).

:(EC) [[Monolith (Space Odyssey)#Appearance and capabilities]] discusses this, the monoliths are not all the same size.[[Special:Contributions/81.98.38.48|81.98.38.48]] ([[User talk:81.98.38.48|talk]]) 10:58, 19 August 2010 (UTC)

::Ratios don't require units. [[User:HiLo48|HiLo48]] ([[User talk:HiLo48|talk]]) 11:09, 19 August 2010 (UTC)

== Good estimation of number of divisors d(n) ==

Hello!
Let d(n) be a number of positive divisors of ''n'', for example, d(6)=4.
In 1907 S. Wigert proved that for any <math>\epsilon > 0</math> holds two statements:<br>
1) for infinitely many ''n'' <math>d(n)>2^{(1-\epsilon)\frac{\ln n}{\ln\ln n}}</math>;<br>
2) for all sufficiently large ''n'' <math>d(n)<2^{(1+\epsilon)\frac{\ln n}{\ln\ln n}}</math>;

Does anybody know more precise estimations than these?<br>
Actually I am able to prove better estimation than 1), but I don't have any information about better results.<br>
Thank you!<br>
[[User:RaitisMath|RaitisMath]] ([[User talk:RaitisMath|talk]]) 12:52, 19 August 2010 (UTC)
:If you don't get an answer here, you might try [[MathSciNet#Online database|MathSciNet]].&mdash;[[User:msh210|msh210]]<span class="Unicode">&#x2120;</span> 17:32, 20 August 2010 (UTC)

:Our [[divisor function]] function article discusses estimates a little bit and has some references.  I'm not sure if there's enough to help you. [[Special:Contributions/67.122.209.167|67.122.209.167]] ([[User talk:67.122.209.167|talk]]) 09:28, 21 August 2010 (UTC)

== Nowhere dense subsets ==

Hi,
I have recently encountered this proposition that seems somewhat vague to me:
'''Proposition:''' Let X be a topological space without isolated points having countable <math> \pi </math>-weight and such that every nowhere dense subset in it is closed. Then it is a [[Pytkeev space]].

The thing which is not clear to me is this, if every nowhere dense subset is closed, doesn't that means that it has to be discrete? and doesn't discrete means that every point is isolated? So, is the condition given in this proposition is that, there arn't any nowhere dense subsets in X? Which means that every subset of X is dense somewhere? which doesn't make sense.. I mean, what about subsets of X that contain one point for instance?
Thanks!
[[User:Topologia clalit|Topologia clalit]] ([[User talk:Topologia clalit|talk]]) 21:55, 19 August 2010 (UTC)
:Consider an infinite set with the cofinite topology.  This space is not discrete, but every nowhere dense set is closed: if a set is infinite, its closure is the entire space, so the only nowhere dense sets are finite.  (This space also has no isolated points, and, if the underlying set is countable, has countable <math>\pi</math>-weight.)  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/130.195.5.7|130.195.5.7]] ([[User talk:130.195.5.7|talk]]) 00:09, 20 August 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->
:Alternatively, you can start with any topology and make a finer one by defining all the nowhere dense sets to be closed (and then taking appropriate unions).  This new space will have isolated points precisely if the old space did, will have no greater <math>\pi</math>-weight (maybe the same?), and all nowhere dense sets will be closed.  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/130.195.5.7|130.195.5.7]] ([[User talk:130.195.5.7|talk]]) 00:35, 20 August 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

I see.. Thanks for your example. But I am still confused. I mean, here is the proof of this proposition. If we take your example of the cofinite topology, under consideration, how can I explain the emphesized remark in brackets? I mean, the nowhere dense subsets in your example are closed and not discrete..
'''Proof:'''
Let <math> x \in  Cl(A) \setminus A</math>. Then <math> x \in Cl(Int(Cl(A))) </math>, because every nowhere dense set is closed '''(and hence discrete)'''. Let <math> P_n : n \in \omega </math> be a list of elements of a countable <math> \pi </math>-base in the space, which are contained in <math> Int(Cl(A)) </math>. Let <math> M_n \cap P_n </math>. Then <math> \{ M_n : n \in \omega \} </math> is a countable <math> \pi </math>-net, at x, and each <math> M_n </math> is infinite.
:The comment is saying that the nowhere dense subsets are discrete.  I gave you an example where the total space isn't discrete.  It's not hard to see that every finite set in my example is discrete, however.
:To see that nowhere dense sets are discrete under the hypothesis, suppose one wasn't.  Then it would have an accumulation point.  Remove this accumulation point, and the resulting set would still be nowhere dense, but would not be closed, contrary to assumption.

Also, something in your second remark bothers me. Suppose for example that I take <math> \mathbb R^2 </math> with the usual topology and try to define all the nowhere dense sets to be closed. Then, for example, the nowhere dense set <math> \{ {1 \over n }\}_{n \in \mathbb N} </math> will turn closed? But it cant be since it doesn't contain it's accumulation point 0..
what am I missing here?
Thanks!
[[User:Topologia clalit|Topologia clalit]] ([[User talk:Topologia clalit|talk]]) 08:10, 21 August 2010 (UTC)

:Could you please supply an explicit reference to the text? <span style="white-space:nowrap;">— [[User:Fly by Night|<span style="font-family:Segoe print">'''''Fly by Night''''' </span>]] <font color="#000000">([[User talk:Fly by Night|<span style="font-family:Segoe print">talk</span>]])</font></span> 16:11, 21 August 2010 (UTC)

Ya sure, it's "Weakly Frechest-Urysohn and Pytkeev spaces" by V.I. Malykhin and G. Tironi. From Topology and its Applications 104 (2000) 181-190
Let me know if you can't find it.
Thanks!
[[User:Topologia clalit|Topologia clalit]] ([[User talk:Topologia clalit|talk]]) 16:20, 21 August 2010 (UTC)

: No, I can't find it. If it's from a journal then it's highly unlikely that I'll be able to find it. I was thinking more along the lines of a book. <span style="white-space:nowrap;">— [[User:Fly by Night|<span style="font-family:Segoe print">'''''Fly by Night'''''</span>]] <font color="#000000">([[User talk:Fly by Night|<span style="font-family:Segoe print">talk</span>]])</font></span> 16:35, 21 August 2010 (UTC)

:When you define the nowhere dense sets to be closed, you're changing the topology.  Not everything which was an accumulation point still is.

I see.. OK thanks. I'll think about it. Prove to myself that this is a topology...
Here is a direct link to the article: http://www.f2h.co.il/307676563971
Proposition 2.1 there.. What do you think?
[[User:Topologia clalit|Topologia clalit]] ([[User talk:Topologia clalit|talk]]) 17:07, 21 August 2010 (UTC)

:I don't speak [[Hebrew language|Hebrew]], and I don't want to download the application because I don't understand the site. Sorry. <span style="white-space:nowrap;">— [[User:Fly by Night|<span style="font-family:Segoe print">'''''Fly by Night'''''</span>]] <font color="#000000">([[User talk:Fly by Night|<span style="font-family:Segoe print">talk</span>]])</font></span> 17:14, 21 August 2010 (UTC)
::Nothing to worry about, just your average infested-with-extremely-intrusive-ads-and-possibly-malware file-sharing site...
::Transferbigfiles below is more friendly. Anyway what the OP linked to is a pdf, not an application. -- [[User:Meni Rosenfeld|Meni Rosenfeld]] ([[User Talk:Meni Rosenfeld|talk]]) 18:34, 21 August 2010 (UTC)
Sorry, I haven't even noticed that this link is in Hebrew..
Here is another link in: https://www.transferbigfiles.com/53c3b8c5-f0f5-4bbc-93d2-a0030eaae50a?rid=gcIEkCL2J4VQPfWN7RipdA%3d%3d
It's in English..  <small><span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Topologia clalit|Topologia clalit]] ([[User talk:Topologia clalit|talk]] • [[Special:Contributions/Topologia clalit|contribs]]) 17:51, 21 August 2010 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->
[[User:Topologia clalit|Topologia clalit]] ([[User talk:Topologia clalit|talk]]) 18:07, 21 August 2010 (UTC)

= August 20 =

== How do you write this augmented matrix in reduced row echelon form? ==

<math>
\begin{bmatrix}
1 & 2 & 0 & 1 \\
0 & 1 & 1 & 2 \\
0 & 0 & 0 & 1
\end{bmatrix}.
</math>
<br />[[Special:Contributions/115.178.29.142|115.178.29.142]] ([[User talk:115.178.29.142|talk]]) 03:25, 20 August 2010 (UTC)

Subtract twice the second row from the first row

<math>
\begin{bmatrix}
1 & 0 &-2 &-3 \\
0 & 1 & 1 & 2 \\
0 & 0 & 0 & 1
\end{bmatrix}.
</math>

Subtract multiples of the third row from the other rows

<math>
\begin{bmatrix}
1 & 0 &-2 & 0 \\
0 & 1 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}.
</math>

This matrix is in [[reduced row echelon form]]. [[User:Bo Jacoby|Bo Jacoby]] ([[User talk:Bo Jacoby|talk]]) 06:42, 20 August 2010 (UTC).
:Do one more step: add twice the second row to the first row:
<math>
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}.
</math>
[[User:Staecker|Staecker]] ([[User talk:Staecker|talk]]) 11:47, 20 August 2010 (UTC)

That doesn't work since the (2,2)th entry x 2 + (1,2)th entry isn't 0; it's 2. [ [[WP:NPA|personal attack]] removed ]  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/110.20.55.15|110.20.55.15]] ([[User talk:110.20.55.15|talk]]) 12:00, 20 August 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->
:Your correction of Staecker's error (who does know the basic math but made a momentary mistake) is welcome. Your personal attacks are not welcome. Please stop this. -- [[User:Meni Rosenfeld|Meni Rosenfeld]] ([[User Talk:Meni Rosenfeld|talk]]) 13:08, 20 August 2010 (UTC)
::Heh- yeah my mistake. That's what I get for adding answers first thing when I wake up. Thanks Meni- [[User:Staecker|Staecker]] ([[User talk:Staecker|talk]]) 13:10, 20 August 2010 (UTC)

== Comprehensive Linear Algebra Book With Relations To Other Branches of Mathematics ==

I've always wanted to find "the" Linear algebra book. A book that would serve me for my entire math life. But I never could find it. Hence I ask the gurus at this reference desk.

My ideal linear algebra book would: (a) Cover all the fundamentals of linear algebra such as what's covered in Herstein's Topics in Algebra in a nice and elegant way (canonical forms, bilinear forms, hermitian matrices, you know advanced topics + the basics) (b) Be significantly more advanced than this and cover topics in linear algebra that pop up deeply in other branches of math (say tensor algebra) (c) Be a pure math textbook.

So is there such a book? I like the style of Herstein's topics in algebra so a book with this style would be desirable. My problem with Herstein's topics in algebra is: it's great and solid but I want a more advanced book. I like  my book to give some calculations but also lots of nice theory in a elegant way. + the book should cover as many advanced topics as possible. This is important. It should be comprehensive. If a series of texts is required, that's fine. I don't mind if my book requires math background in general algebra, in fact I want a book like that. Thanks guys ... Oh and I forget to add: this book should be primarily on finite dimensional vector spaces. It's alright if other vector spaces are covered, but I want the emphasis on finite. Thanks guys again. ..  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/114.72.248.27|114.72.248.27]] ([[User talk:114.72.248.27|talk]]) 07:29, 20 August 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

''Please answer my question? I request you as politely as I can. Why do people avoid me? What have I done? Please answer my question. I've made my question sound as polite as possible Thanks guys ...'' [[Special:Contributions/114.72.244.249|114.72.244.249]] ([[User talk:114.72.244.249|talk]]) 08:37, 20 August 2010 (UTC)
:Presumably no one can think of any such book ATM. This is not a mathematical question, hence it's not possible to ponder it and figure out a solution; if no one here happens to have encountered a book meeting your rather high demands, you may well get no answer.—[[User:EmilJ|Emil]]&nbsp;[[User talk:EmilJ|J.]] 13:12, 20 August 2010 (UTC)
::I can't think of any myself but you might find something at [[List of important publications in mathematics]].--[[User:Salix alba|Salix]] ([[User talk:Salix alba|talk]]): 00:50, 21 August 2010 (UTC)

== simultaneous equation ==

Find the values of <math>k</math> for which the simultaneous equations:
<br />
<math>(k+1)x-3y=0</math><br /><math>
2x-ky=0</math><br /> have solutions other than x=0, y=0<br />
--[[Special:Contributions/220.253.222.146|220.253.222.146]] ([[User talk:220.253.222.146|talk]]) 09:05, 20 August 2010 (UTC)

:Geometrically, what is going on here is that each equation represents a line through the origin (0,0). A solution to the simultaneous equations is the co-ordinates of a point at which the lines cross. The lines will not cross at any other point apart from (0,0) <u>unless</u> the two equations actually represent the same line, in which case any pair of co-ordinates that satisfies one equation will also satisfy the other. From the condition that the two equations represent the same line, you can get a quadratic equation in ''k'' which has two (real) roots. The same quadratic equation falls out if you take an algebraic approach and think in terms of matrices and determinants instead. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 09:41, 20 August 2010 (UTC)
::Or simply solve for one variable in the one and then substitute into the other. <math>k = -3, 2</math>.  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/203.97.79.114|203.97.79.114]] ([[User talk:203.97.79.114|talk]]) 10:32, 20 August 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

:::This looks like a homework question.  Wikipedia Reference Desks do not do people's homework for them.  We are happy to point people in the direction of suitable information on Wikipedia so they can read and learn, but there is no benefit to you if we do your homework.  [[User:Dolphin51|<font color="green">''Dolphin''</font>]] ''([[User talk:Dolphin51|<font color="blue">t</font>]])'' 11:33, 20 August 2010 (UTC)

[ [[WP:CIVIL|vulgarisms]] removed ]

For the record, the answer is to use matrixes. It's as imple as 1,2,3 ... [ [[WP:NPA|personal attack]] removed ] Your system of equations is

k+1  -3 

2    -k

and your goal is to determine when this matrix above kills something to 0. (like maps (x,y) to 0) But it does so when it's determinent is 0 and this is like when -k (k+1) = -6 or -k^2 - k + 6 = 0.
But the quadratic has a solution = 1 +,- sqrt(1 + 24)/-2. So k = 3 or k = -2. I repeat

<<<<<k = -3 or k = 2>>>>>

Write that on your homework assignment with the advanced matrix stuff and 'all. Your teacher will be very impressed and you'll get an A+. If she's a hot lady, maybe she'll go a bit further than that for your own pleasure ...<span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/110.20.55.15|110.20.55.15]] ([[User talk:110.20.55.15|talk]]) 12:01, 20 August 2010</span><!-- Template:UnsignedIP -->
:Please stop disrupting this reference desk, or you will be reported and subsequently [[Wikipedia:Blocking policy|blocked]].  -- [[User:Meni Rosenfeld|Meni Rosenfeld]] ([[User Talk:Meni Rosenfeld|talk]]) 13:13, 20 August 2010 (UTC)

I'm just speaking my voice. Maybe I was vulgar in the 80's but nowadays the words I used (I won't state them cause clearly people have problems with it) are common. If you get out more Meni Rosenfeld, you'll see that. And yes, the guy who asked the question is a ******* if he wants his homework done for him. But ******** are ******** and so we might as well give the ******* the answers. I CAN'T see what I've done wrong. I CAN see why other people *THINK* I've done wrong. For some odd reason society frowns down upon telling the truth ... But why shouldn't I voice my opinions? BTW, now that you've told me I'll stop, but all I'm saying is that you guys are all against me. First some dude called Tango removes my question for no particular reason. Then some annonymous guy tries to block me, and then you tell me to "sign my post". Well, I complied. But now people don't answer my questions anymore. I mean what do I need to do to get some service around here? Don't you guys get paid for answering my questions? I'll report you guys to Wikipedia for not doing your job right ... I mean I don't want to, but I just lost my temper. I'm ignored. And some guy called Gandalf insults me above. So everyone hates me. I think I should just leave and save you all the miseries. Goodbye. Block me, I don't really care. But I'm not posting here anymore until you guys start treating me with some respect.  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/110.20.55.15|110.20.55.15]] ([[User talk:110.20.55.15|talk]]) 13:28, 20 August 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->
:You seem to misunderstand a lot of aspects of how the reference desk operates. One of these is that no one here gets paid for anything. Another is that we don't do homeworks, which you'd know if you read the prominent box on top of this page.—[[User:EmilJ|Emil]]&nbsp;[[User talk:EmilJ|J.]] 13:50, 20 August 2010 (UTC)
:: right: homework exists to help keep class distinctions in a country (United States) that has all too many social upstarts and "mobility".  The only thing keeping any uppity declasse' from rising through the ranks of education is the assignment of homework that other kids with better-educated parents will have help in completing but which that kid will not.  I think it is the last barrier to total communist anarchy dictatorship of the proletariat, and any cheating, or attempts by students to coopt other classes' parents (such as the parents who would contribute to the refernece desk) must be smitten strongly, quickly, and decisively.  If someone can't ask their own parent this question, then maybe that person should accept his lot in life instead of trying to pull down the average in a higher class to which he, of rights, would never accede. [[Special:Contributions/92.230.70.110|92.230.70.110]] ([[User talk:92.230.70.110|talk]]) 21:23, 20 August 2010 (UTC)

Stop talking bullcrap 92.230.70.110. If you want to see bullshit in it's finest form look above LOL.  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/110.20.11.94|110.20.11.94]] ([[User talk:110.20.11.94|talk]]) 00:57, 21 August 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

== Yoplait Yogurt commercial ==

In a Yoplait Yogurt commercial, the development of a woman's bone strength is represented as a curve that starts at (0,0) and rises quickly before dropping at a slightly slower rate which decreases, and then approaching the X-axis. Whether this is accurate or not I don't know, particularly since the curve breaks like a bone when it reaches the point where a woman's bones would break easily. But I recall seeing a function like this, which for values of x less than 0, F(-x) = -F(x). What might this function be?[[User:Vchimpanzee|<font color="Green">Vchimpanzee</font>]]&nbsp;'''·''' [[User talk:Vchimpanzee|<span style="color: orange"> talk</span>]]&nbsp;'''·''' [[Special:Contributions/Vchimpanzee|<span style="color: purple">contributions</span>]]&nbsp;'''·''' 21:36, 20 August 2010 (UTC)
:<math>x e^{-x^2}</math> ? --[[User:Qwfp|Qwfp]] ([[User talk:Qwfp|talk]]) 21:56, 20 August 2010 (UTC)
::Thanks. That might be it. Where could I find a graph of that, because Google has failed me.[[User:Vchimpanzee|<font color="Green">Vchimpanzee</font>]]&nbsp;'''·''' [[User talk:Vchimpanzee|<span style="color: orange"> talk</span>]]&nbsp;'''·''' [[Special:Contributions/Vchimpanzee|<span style="color: purple">contributions</span>]]&nbsp;'''·''' 17:59, 21 August 2010 (UTC)
:::http://www.wolframalpha.com/input/?i=x*e^%28-x^2%29. Note that this is just one example, there are infinitely many functions matching your description. -- [[User:Meni Rosenfeld|Meni Rosenfeld]] ([[User Talk:Meni Rosenfeld|talk]]) 18:18, 21 August 2010 (UTC)

== Irreducible polynomials and the tower law ==

Hi all!

I'm trying to solve the following:

Let K be a field and c ∈ K. If m, n ∈ Z>0 are coprime, show that X<sup>mn</sup>−c is irreducible if and only if both X<sup>m</sup> − c and X<sup>n</sup> − c are irreducible. (Hint: Use the Tower Law.)

Problem is, I don't really understand how you can use the tower law when you don't know something is irreducible: surely you require irreducibility of f(x) for K[x]/(f(x)) to actually be a field? In which case, you can't really say anything about the degree of the relevant field extensions before you know they're irreducible? And obviously you can't say they're irreducible until you've proved the result, at which point it's moot. 

For example, if K<sub>mn</sub>, K<sub>m</sub> and K<sub>n</sub> represent K[x]/(x<sup>mn</sup>-c), and so on, respectively, and we assume K<sub>m</sub>, K<sub>n</sub> are irreducible, then [K<sub>n</sub>:K][K<sub>m</sub>:K]=[K<sub>mn</sub>:K<sub>m</sub>][K<sub>m</sub>:K]=mn I believe, but we can't then say '[K<sub>mn</sub>:K]=mn' because we don't know that's a field necessarily, do we? I'd appreciate a nudge in the right direction, as I've been stuck on this for a while and I think it's probably a very short argument, just one I'm not getting - I believe I've proved X<sup>mn</sup>−c is irreducible implies both X<sup>m</sup> − c and X<sup>n</sup> − c are irreducible via the contrapositive, but I didn't use the tower law, or the fact that m, n are coprime: simply write Y=X<sup>m</sup> (assuming X<sup>n</sup>-c reducible), then X<sup>mn</sup> - c = Y<sup>n</sup> - c is reducible. Am I doing something wrong here? Or just missing the obvious?

Thankyou very much for any responses, much appreciated indeed - [[Special:Contributions/84.45.219.231|84.45.219.231]] ([[User talk:84.45.219.231|talk]]) 22:52, 20 August 2010 (UTC)
:Just let α be any root of X<sup>mn</sup>-c and consider K[α]. This is definitely a field; it's isomorphic to K[x]/(f), where f is the minimal polynomial of α (which could in general be any irreducible factor of X<sup>mn</sup>-c). [[User talk:Algebraist|Algebraist]] 23:02, 20 August 2010 (UTC)
::Right, well I got the half of the proof which I was missing, so thankyou! (Something along the lines of 'm divides this, n divides this, they're coprime so mn divides this and it's less than or equal to mn, so equals mn'...) - Out of interest, does this argument also enable us to prove the direction I succeeded in proving without the tower law etc? I can't see how you could use it to prove '<=>', rather than just '<=', but I could be overlooking something. Either way, thanks for the help, I suppose I should probably press on with this fiasco which is teaching myself Galois theory, though I'm quite convinced I'm totally useless at it! Cheers :) [[Special:Contributions/84.45.219.231|84.45.219.231]] ([[User talk:84.45.219.231|talk]]) 00:09, 21 August 2010 (UTC)

If you can't do Galois theory what hope do you have to suceed in math? So to help you, take the following advice: Compute as many Galois groups as possible. That should aid your disorder. No need to thank me.  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/110.20.11.94|110.20.11.94]] ([[User talk:110.20.11.94|talk]]) 01:01, 21 August 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->
:I'm not sure to what extent that's sarcastic, but if I intended to remain useless at Galois theory, I probably wouldn't be here trying to improve :) [[Special:Contributions/84.45.219.231|84.45.219.231]] ([[User talk:84.45.219.231|talk]]) 01:40, 21 August 2010 (UTC)
::Please ignore the comments of 110.20.11.94, who unfortunately is now [[meta:What is a troll?|trolling]] this reference desk. I am sorry that you had to see this. -- [[User:Meni Rosenfeld|Meni Rosenfeld]] ([[User Talk:Meni Rosenfeld|talk]]) 17:26, 21 August 2010 (UTC)
:::Much appreciated Meni, I shall do! Always glad there are dedicated 'refdesk-ees' such as yourself and Algebraist about to help. [[Special:Contributions/84.45.219.231|84.45.219.231]] ([[User talk:84.45.219.231|talk]]) 20:18, 21 August 2010 (UTC)

= August 21 =

== A function ==

What functions f(x) are there such that f[f(x)]=x? Obviously excluding f(x)=''constant'' of course. [[Special:Contributions/76.230.213.76|76.230.213.76]] ([[User talk:76.230.213.76|talk]]) 00:13, 21 August 2010 (UTC)

Assuming you can graph these, have a look at functions which are symmetric about the line y=x: for example, f(x)=1/x, f(x)=1-x etc. Your example is incorrect - for example, f(x)=0 gives f(f(1))=f(0)=0 which does not equal 1. Maybe you were thinking of f(x)=x, which is trivially symmetric about y=x (since it 'is' y=x). [[Special:Contributions/84.45.219.231|84.45.219.231]] ([[User talk:84.45.219.231|talk]]) 00:31, 21 August 2010 (UTC)
:Such a function is called an [[involution (mathematics)|involution]]. [[User talk:Algebraist|Algebraist]] 00:36, 21 August 2010 (UTC)
::(also called a ''self-inverse'' function)  [[User:Dbfirs|''<font face="verdana"><font color="blue">D</font><font color="#00ccff">b</font><font color="#44ffcc">f</font><font color="66ff66">i</font><font color="44ee44">r</font><font color="44aa44">s</font></font>'']] 11:51, 21 August 2010 (UTC)

:Another interesting concept to think about is the [[functional square root]]. That is, given a function ''f'', can we find or create a function ''g'' such that ''g''(''g''(x))=''f''(x)? After that you might consider the idea [[fractional derivative]]s. Not all these avenues of thought have proved fruitful but they are interesting to think about nonetheless. [[User Talk:Zunaid|Zunaid]] 07:01, 21 August 2010 (UTC)
::This is similar to [[Fixed point (mathematics)]].[[User:Smallman12q|Smallman12q]] ([[User talk:Smallman12q|talk]]) 02:13, 22 August 2010 (UTC)

== Graphically plotting a flight course ==

What would be the best way to graphically display a flight course given data (per second) of elements such as airspeed, altitude, vertical acceleration, heading, pitch, etc?

Thank You  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/74.113.218.132|74.113.218.132]] ([[User talk:74.113.218.132|talk]]) 00:16, 21 August 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->
:You mean on a [[graph]] or [[chart]]?[[User:Smallman12q|Smallman12q]] ([[User talk:Smallman12q|talk]]) 02:15, 22 August 2010 (UTC)
::If your data consists only of speeds and accelerations then you need to integrate to be able to plot the course.  Someone might point you towards some special software, but you could easily set up the approximate calculations on a spreadsheet, and then graph the resulting displacements to display a flight course.  Errors would accumulate over a long course.  [[User:Dbfirs|''<font face="verdana"><font color="blue">D</font><font color="#00ccff">b</font><font color="#44ffcc">f</font><font color="66ff66">i</font><font color="44ee44">r</font><font color="44aa44">s</font></font>'']] 06:41, 22 August 2010 (UTC)

== Hiya! ==

Does anyone know a proof that if A is a square matrix then there's a matrix B s.t. AB = BA = det(A) I. I know B is called the [[adjugate]] of A, but how do you find such a B or proof such a B exists/ Thanks.  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/110.20.59.41|110.20.59.41]] ([[User talk:110.20.59.41|talk]]) 12:29, 21 August 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

:You might want to take a look at the [[adjugate matrix]] article. -- [[User:The Anome|The Anome]] ([[User talk:The Anome|talk]]) 12:49, 21 August 2010 (UTC)

:<small>([[wp:edit conflict|edit conflict]])</small> Try the [[adjugate matrix]] article. It's mentioned in [[Adjugate_matrix#Applications|this subsection]]. It's a consequence of the [[Laplace expansion]]. <span style="white-space:nowrap;">— [[User:Fly by Night|<span style="font-family:Segoe print">'''''Fly by Night'''''</span>]] <font color="#000000">([[User talk:Fly by Night|<span style="font-family:Segoe print">talk</span>]])</font></span> 12:51, 21 August 2010 (UTC)

I know but how do you PROVE that the adjugate exists? Sorry I wasn't clear and thanks for youre answer.  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/110.20.9.174|110.20.9.174]] ([[User talk:110.20.9.174|talk]]) 12:51, 21 August 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

:It has an ''explicit definition'', and so must exist. The definition involves taking [[determinant]]s of [[cofactor matrix|sub-matrices]]. That's just elementary arithmetic and algebra. Try [[Adjugate_matrix#Definition|this subsection]] of the [[adjugate matrix]] article. The only thing that needs to be proved is that a square matrix ''M'', multiplied by its adjugate matrix adj(''M''), is the determinant of ''M'' multiplied by the identity matrix: i.e. {{nowrap|1=''M''⋅adj(''M'') = det(''M'')⋅''I''}}. <span style="white-space:nowrap;">— [[User:Fly by Night|<span style="font-family:Segoe print">'''''Fly by Night'''''</span>]] <font color="#000000">([[User talk:Fly by Night|<span style="font-family:Segoe print">talk</span>]])</font></span> 12:52, 21 August 2010 (UTC)


:To be precise, note that the existence of such a <math>B</math> is fairly obvious, and the relation does ''not'' characterize the adjugate. Precisely, if <math>A</math> is not invertible, <math>B:=0</math> obviously verifies the relation, although in general it's not the adjugate of <math>A.</math> On the other hand, if <math>A</math> is invertible, <math>B:=\det(A)A^{-1}</math> obviously verifies the relation too (and in this case, it is the adjugate). Just to say that the existence of such a <math>B</math> is fairly obvious, even though it does not characterize the adjugate. --[[User:PMajer|pm]][[User talk:PMajer|<span style="color:blue;">a</span>]] 21:53, 21 August 2010 (UTC)

Thanks pm. Bt can someone give the explciit calculation? I can't find this in linear algebra books. In one book, it says the existence of teh adjugate is another form of the Cayley-Hamilton theorem. Is this true? If so how? Thanks for you answer.  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/114.72.245.87|114.72.245.87]] ([[User talk:114.72.245.87|talk]]) 00:11, 22 August 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

= August 22 =

== arithmetic progressions and primes ==

I want to show that any arithmetic progression if continued long enough wouldn't contain primes. How should I proceed. Thanks-[[User:Shahab|Shahab]] ([[User talk:Shahab|talk]]) 07:46, 22 August 2010 (UTC)

:[personal attack removed] If it contained primes over and over you'd clearly get a contradiction. Think about it and maybe you'll see it  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/114.72.236.126|114.72.236.126]] ([[User talk:114.72.236.126|talk]]) 08:10, 22 August 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->
:Do you mean that any arithmetic progression eventually contains a number which is not prime? This follows from the [[prime number theorem]]. -- [[User:Meni Rosenfeld|Meni Rosenfeld]] ([[User Talk:Meni Rosenfeld|talk]]) 08:53, 22 August 2010 (UTC)

Ya don't need any [profanity removed] prime number theory to see it. Besides, it doesn't follow from th eprime number theorem since ya could get primes appearing in the sequence over and over but less frequently as n goes to INFINITY WITHOUT contradicting any bullcrap prime number theory. So no, think harder Meni Rosenfeld and Shahb.  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/114.72.192.74|114.72.192.74]] ([[User talk:114.72.192.74|talk]]) 09:02, 22 August 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

::Yes I did mean so Meni, like the AP 5,11,17,23,29 has primes only but then the next term is non-prime. Uptil now the biggest such prime AP that has been found has around 20 terms. Not sure how the prime number theorem applies. -[[User:Shahab|Shahab]] ([[User talk:Shahab|talk]]) 09:05, 22 August 2010 (UTC)
:::Prove by contradiction. If there was a sequence with only primes, what would a lower bound for <math>\lim_{n\to\infty}\frac{\pi(n)}{n}</math> be? What is the limit in light of the PNT? -- [[User:Meni Rosenfeld|Meni Rosenfeld]] ([[User Talk:Meni Rosenfeld|talk]]) 09:19, 22 August 2010 (UTC)

[irrelevant vulgarities and personal attack removed]  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/110.20.58.220|110.20.58.220]] ([[User talk:110.20.58.220|talk]]) 10:25, 22 August 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

[personal attacks removed]  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/114.72.192.74|114.72.192.74]] ([[User talk:114.72.192.74|talk]]) 09:09, 22 August 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

:::Because if an arithmetic progressions with difference ''d'' contained only primes then the asymptotic density of primes would be at least 1/''d'', which contradicts the prime number theorem. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 09:19, 22 August 2010 (UTC)

:For a constructive proof, recall that an arithmetic progression looks like <math>a + bn</math> for some choice of <math>a</math> and <math>b</math>.  You want a choice of <math>n</math> that will make this be obviously non-prime.  Since all you have to work with are <math>a</math> and <math>b</math>, try choosing one of those for <math>n</math> and seeing what you can get.  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/203.97.79.114|203.97.79.114]] ([[User talk:203.97.79.114|talk]]) 09:20, 22 August 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

::... but make sure your proof covers the case ''a'' = 1. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 09:30, 22 August 2010 (UTC)

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