linear algebra
matrix
vector
linear algebra unsolved

The question is:
Suppose that $ V$ is finite-dimensional and $ T \in \mathcal{L}(V)$ ($ T$ is a linear operator on $ V$).  Prove that $ T$ is a scalar multiple of the identity if and only if $ ST \equal{} TS$ for all $ S \in \mathcal{L}(V)$.

Here is the solution I got:
The "only if" is obvious.  To prove the "if," suppose $ ST \equal{} TS$ for all $ S \in \mathcal{L}(V)$.
Let $ (v_1, \ldots, v_n)$ be an ordered basis for $ V$.  Let $ V_i \equal{} \text{span}(v_i), \quad i \equal{} 1,\ldots,n$.  Thus $ V \equal{} V_1 \oplus \cdots \oplus V_n$ is a decomposition of $ V$ into one-dimensional subspaces.
The first claim is that there exist scalars $ \lambda_1, \ldots, \lambda_n$ such that $ Tv_i \equal{} \lambda_iv_i, \quad i \equal{} 1,\ldots,n$.  To prove this, choose some $ i$ and define $ S$ so that for each $ v \equal{} a_1v_1 \plus{} \cdots \plus{} a_nv_n \in V$, we have $ Sv \equal{} a_iv_i$.  Since $ Tv_i \equal{} TSv_i \equal{} STv_i \in V_i$, we find that $ T|_{V_i}$ is a linear operator on $ V_i$.  Since $ V_i$ is one-dimensional, there exists $ \lambda_i$ such that $ Tv_i \equal{} \lambda_iv_i$.
Now it remains to show that $ \lambda_1 \equal{} \cdots \equal{} \lambda_n$.  To do this, choose arbitrary $ 1 \leq i,j \leq n$.  If $ w \equal{} a_1v_1 \plus{} \cdots \plus{} a_nv_n \in V$, let $ Sw \equal{} (a_i \plus{} a_j)v_i \plus{} a_jv_j$.  Then $ ST(v_i \plus{} v_j) \equal{} S(\lambda_iv_i \plus{} \lambda_jv_j) \equal{} (\lambda_i \plus{} \lambda_j)v_i \plus{} \lambda_jv_j$, while $ TS(v_i \plus{} v_j) \equal{} T(2v_i \plus{} v_j) \equal{} 2\lambda_iv_i \plus{} \lambda_jv_j$.  Since these values are equal, we must have $ \lambda_i \equal{} \lambda_j$.  Since $ i$ and $ j$ were arbitrary, we have $ \lambda_1 \equal{} \cdots \equal{} \lambda_n \equal{} \lambda$, and $ T \equal{} \lambda I$.

End of proof.

I feel like there is a simpler solution.  Can anyone think of one?
(Also, if you feel like checking my proof, I'd appreciate that too.)

I wrote out the two dimensional version [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?t=294985]here.[/url]

In general, that's the approach I'd take: write $ T$ as a matrix and test it for commuting with matrices with exactly one nonzero element.

For any $ v \in V$, complete $ v$ to a basis and let $ S$ be defined as the linear map taking $ v$ to $ v$ and every other basis element to zero.

Then $ Tv \equal{} T(Sv) \equal{} S(Tv)$; therefore $ Tv \equal{} \alpha_v v$. for some scalar $ \alpha_v$. Therefore every vector in $ V$ is an eigenvector of $ T$; so all must be in the same eigenspace and so $ T$ is a scalar multiple of the identity.

I proved not that every [i]basis[/i] vector is an eigenvector, but that every vector whatsoever is an eigenvector. There is a little work left to do to show that they must be in the same eigenspace, but that's not tremendously hard; look what happens when $ v$,$ w$, and $ v\plus{}w$ are all eigenvectors.

See also:
http://www.artofproblemsolving.com/Forum/viewtopic.php?t=287979
it's the same proof that Nukular has written.
Moreover the proof is valid in infinite dimension.

Nukular's proof is also valid in infinite-dimensional spaces, assuming the axiom of choice (without which the problem is impossible anyway, right?).  In the text I got this exercise from, the author does not assume the axiom of choice.

Yes benzi455.
Let $ L$ be a linearly independent subset of $ V$.
Here $ L\equal{}\{v\}$ for Nukular and $ L\equal{}\{u,Au\}$ for me.
According to Zorn's lemma the set of linearly independent subsets of $ V$ containing $ L$ has a maximal (for the inclusion) element $ B$ . $ B$ is a Hamel basis of $ V$.


modular arithmetic
AoPSwiki
number theory

wow, SplashD

u posted before i was done!

[quote="mathgeniuse^ln(x)"]How did yo guys do the question?

How do you do modular arithmetic with fractions and whole-numbers that are (mod n)?[/quote]

Will one of you two please explain how you did the problem?

[quote="mathgeniuse^ln(x)"]How did yo guys do the question?

How do you do modular arithmetic with fractions and whole-numbers that are (mod n)?[/quote]

AoPSWiki at:

[url=http://www.artofproblemsolving.com/Wiki/index.php/Mods]Modular Arithmetic[/url]

$\frac{1}{3}+\frac{1}{4}+\frac{1}{5}=\frac{47}{60}_{10}$
Now we individually change each of the denominator and numerator into base eleven. So using the remanider method, we get
$\frac{47}{60}_{10}=\frac{4\times 11+3}{5\times 11+5}_{10}=\frac{43}{55}_{11}$. :)


calculus
integration
function
real analysis
real analysis unsolved

Let $ f: [a,b]\to R$ be $ C^{1}$ function such that $ f(a) \equal{} f(b) \equal{} 0$. Prove that

(a). $ |\int_{a}^{b} f(x)d x|\le \frac {(b \minus{} a)^2}{4}\sup_{x\in [a,b]}\{ |f'(x)| \}.$
(b). When does the equality occur?

This is not very hard but very nice problem, I think.

It has been asked before, I believe. It looks quite familiar. I suppose after a month on this forum you know every problem known to man. :D

Since I can't edit anymore.. Are you sure the denominator is 4 and not 2? I'm getting 2 when I work it out.

[u]Solution.[/u] Given that $ f: [a,b]\rightarrow\mathbb{R}$ is $ C^1$, $ f'$ is continuous, and so the compactness of $ [a,b]$ implies that $ M = \sup_{x\in[a,b]}|f'(x)| = \max_{x\in[a,b]}|f'(x)|$.  The continuity of $ f$ and compactness of $ [a,b]$ imply that $ f$ is integrable.  The mean value theorem implies that $ |f(x) - f(u)|\leq|x - u|M$ for proper $ u\in[a,b]$.  Hence,
\begin{eqnarray*}\left|\int_a^bf(x)dx\right| & \leq & \left|\int_a^{\frac {a + b}{2}}f(x)dx\right| + \left|\int_{\frac {a + b}{2}}^bf(x)dx\right| \\
& = & \left|\int_a^{\frac {a + b}{2}}[f(x) - f(a)]dx\right| + \left|\int_{\frac {a + b}{2}}^b[f(x) - f(b)]dx\right| \\
& \leq & M\left(\int_a^{\frac {a + b}{2}}(x - a)dx + \int_{\frac {a + b}{2}}^b(b - x)dx\right) \\
& = & \frac {M(b - a)^2}{4},\end{eqnarray*}
as required. $ \Box$

Correct?


MATHCOUNTS
quadratics
LaTeX
calculus
derivative
continued fraction
algebra

In other words, $3+\frac{1}{3+\frac{1}{3+\frac{1}{...}}}$. This IS from the pre-TeX days!!
[hide]
We set it all equal to x, and substitute x into its own equation: $x=3+\frac{1}{x}$. Then we set a common denominator: $x^2=3x+1$ And prepare to use the Quadratic formula: $x^2-3x-1=0$ And the Quadratic formula gives $x=\frac{3\pm \sqrt{9-4(1)(-1)}}{2(1)}=\frac{3\pm \sqrt{13}}{2}$[/hide]

Now my example: What is the value of $3+\frac{3+\frac{...}{...}}{3+\frac{...}{...}}$? It is a BIT tougher :lol: But not a good example :blush:

All right, here is a more realistic one: $42+\frac{1}{42+\frac{1}{42+\frac{1}{...}}}$

Billy

[quote="MCrawford"]What is the value of  3 + 1/(3 + 1/(3 + 1/...))?

Sorry, it's hard to replicate continued fractions in text.

This question comes with a new rule: Make up your own example of a continued fraction and let somebody else solve that one.  Happy solving.[/quote]

[hide]
\[x = 3 + \cfrac{1}{3 + \cfrac{1}{3 + \cdots}}\]
\[x = 3 + \cfrac{1}{x}\]
\[x^2 = 3x + 1\]
\[x^2 - 3x - 1 = 0\]
\[x = \frac{3 \pm \sqrt{13}}{2}\]
[/hide]

solafidefarms, could you complete that LaTeX expression?  It gives an "Unparsable or potentially dangerous LaTeX formula..."

Anyway, here's a harder challenge that uses the same principle:

Find the value of
\[
1 + \cfrac{2}{2 + \cfrac{1}{1 + \cfrac{2}{2+ \cdots}}}
\]

let x = the fraction
so we get:
x = 1 + 2/(2+1/x)
   = 1 + 2/ (2x/x_
   = 1 + 1
    = 2
I'm not sure if it is right or not
but here is another one:
3+1/(4+3+1(4+...

[quote="shake9991"]let x = the fraction
so we get:
x = 1 + 2/(2+1/x)
   = 1 + 2/ (2x/x_
...
[/quote]

2+ 1/x =/=(not equal to) 2x/x

...unless x approaches infinity =o =o =o

[quote="solafidefarms"]In other words, $3+\frac{1}{3+\frac{1}{3+\frac{1}{...}}}$. This IS from the pre-TeX days!!
[hide]
$\ldots x=\frac{3\pm \sqrt{9-4(1)(-1)}}{2(1)} \ldots$[/hide]

Now my example: What is the value of ...? It is a BIT tougher ...
Billy[/quote]

Thanks for rewriting the original challenge in TeX, Solafide.  You have the right strategy for the solution, but check your arithmetic inside the radical.

Your own challenge is misformatted, and I can't figure out what you are trying to ask.

I got 3 :pm:  :sqrt: 13  /2.... how do you get the fractions into the forums (sry im new to these forums)?... for the person who got the answer with  :sqrt:5, you must have subtracted instead of adding in the derivative of your quadratic... but usually in mathcounts you can factor out the quadratics to make it easier to solve.... and cuts down on the stupid mistakes.. ;)

Yes, I subtracted. :( :blush: :?

For all that cool fractions and stuff, its called LaTeX.See the left side of the screen, where it says Latex? Go there for a tutorial on it. Then just put your latex in and it shows up nice and pretty!    :lol:

Billy

Thanks a lot... makes writing in these forums a LOT easier....

[quote="solafidefarms"]
Now my example: What is the value of $3+\frac{3+\frac{...}{...}}{3+\frac{...}{...}}$? It is a BIT tougher...[/quote]

[hide]
$x = 3 + \frac{x}{x} = 4$
[/hide]

[quote="DON FAVOR"][quote="hockeyadi23"]Here's a different problem but with a similar application:
${\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5+...}}}}}$
[hide="Hint"]Again substitution is the key here[/hide]

--Adi[/quote]

do it as that man did it on top of topik
iss more easear
o i cant do it then write my mail and i tell how to do it[/quote]

I don't exactly know what you want, but if you want the solution, see the spoiler link below:
[hide="spoiler"]x = ${\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5+...}}}}}$
$x = {\sqrt{5 + x} }$
You can take it from here...[/hide]


calculus
integration
LaTeX
logarithms
algebra
partial fractions
calculus computations

[quote="topper"]If anyone could help me with these, that'd be great.

1) integral from -3 to -1 of (5z-13)/(z^2-5z+6) dz

2) integral from 1 to 2 of (x-2x^3)lnx dx

(sorry, I don't know latex)

Thanks![/quote]

Done:

1) $\int_{-3}^{-1}\frac{5z-13}{z^2-5z+6}\,dz$

2) $\int_1^2(x-2x^3)\ln x\,dx$

1.  Notice how the denominator is easily factored and the numerator is a lower degree.  This usually implies [hide]partial fractions[/hide]

2.  Two very different parts, so integration by parts is nice.  [hide]$\ln{x}$ and $(x-2x^3)dx$[/hide]

I'll expand if you're not familiar with these techniques, but if so, give it a shot.


From a point P, two tangent, PT and PS are drawn to a circle with center O and radius 2r. If OP=25, Show that triangle TPS is equilateral.

pls giv me the answer ASP. :( [/img]

[quote="damselreena"]From a point P, two tangent[s], PT and PS are drawn to a circle with center O and radius 2r. If OP = 25, Show that triangle TPS is equilateral.

pls giv me the answer ASP. :( [/img][/quote]

This problem is not doable.  It needs to settle on a specific circle size,
so it has a specific radius.

TS is a chord of the circle.

Let the intersection point of PO and TS be N.

Let the measure of NO be x and let the measure of PN be (25 - x).
(The length of OP is given as 25.)

T = 2r, N = x, so TN = $ \sqrt {4r^2 \minus{} x^2}$ from the Pythagorean Theorem.

Using the Pythagorean Theorem, PT = PS = $ \sqrt {4r^2 \plus{} 625 \minus{} 50x}$ . . . (1)

TS = 2(TN) = $ 2\sqrt {4r^2 \minus{} x^2}$ . . . (2)

For triangle TPS to be equilateral, it is required for (1) to equal (2).

I get it to work down to:

$ 12r^2 \equal{} 4x^2 \minus{} 50x \plus{} 625$.

There are too many variables.


algebra
polynomial
Galois Theory
number theory
algebra open

Let $P \in \mathbb{Z}[x]$ a polynomial with integer coefficients.
Prove or disprove the following statement:
For each $n \in \mathbb{N}$ exists a polynomial with integer coefficients $Q \in \mathbb{Z}[x^{n}]$ such that:
- $Q$ is a multiple of $P$
- All powers of $x$ appear in $Q$ with an exponent divisible by n.

Example: take $P = x^{2}-x+1$, n=5. $Q = x^{30}-1$ is good.

I am very curious about this!!

$P(x_{1})\cdots P(x_{n})\in\mathbb{Z}[x_{1},\ldots,x_{n}]$ is a symmetric polynomial. Then
(*) $P(x_{1})\cdots P(x_{n})=Q(\sigma_{1},\ldots,\sigma_{n})$ for some $Q\in\mathbb{Z}[x_{1},\ldots,x_{n}]$, where
$\sigma_{k}(x_{1},\ldots,x_{n})=\sum_{1\leq j_{1}<j_{2}<\cdots<j_{k}\leq n}x_{j_{1}}\cdots x_{j_{k}}$ is the [url=http://en.wikipedia.org/wiki/Symmetric_polynomial]elementary symmetric polynomial[/url] of degree $k$ for $k=1,\ldots,n$.

Now if $\zeta=e^{2\pi i / n}$ is a primitive $n^\text{th}$ [url=http://en.wikipedia.org/wiki/Root_of_unity]root of unity[/url], then $x^{n}-1=\prod_{k=1}^{n}(x-\zeta^{k})$ and from [url=http://en.wikipedia.org/wiki/Vi%C3%A8te%27s_formulas]Viète's formulas[/url] follows
$\sigma_{k}(\zeta,\zeta^{2},\ldots,\zeta^{n})=0$ for $k=1,\ldots,n-1$ and $\sigma_{n}(\zeta,\zeta^{2},\ldots,\zeta^{n})=(-1)^{n-1}$.
So $\sigma_{k}(\zeta x,\zeta^{2}x,\ldots,\zeta^{n}x)=\sigma_{k}(\zeta ,\zeta^{2},\ldots,\zeta^{n})x^{k}=0$ for $k=1,\ldots,n-1$
and $\sigma_{n}(\zeta x,\zeta^{2}x,\ldots,\zeta^{n}x)=\sigma_{n}(\zeta ,\zeta^{2},\ldots,\zeta^{n})x^{n}=\pm x^{n}$
and $(x_{1},x_{2},\ldots,x_{n})=(\zeta x,\zeta^{2}x,\ldots,\zeta^{n}x)$ in (*) gives the result
$P(x)=P(\zeta^{n}x)|P(\zeta x)\cdots P(\zeta^{n}x)=Q(0,\ldots,0,\pm x^{n})$.

Wow, really beautiful solution!  :o

Thank you.

Or you can simply consider $R(x)=P(x)P(\zeta x)\cdots P(\zeta^{n-1}x)\not\equiv 0$.
Then for $R(x)=\sum_{k\geq 0}\alpha_{k}x^{k}$ we have $R(\zeta x)=R(x)$ and $\alpha_{k}\zeta^{k}=\alpha_{k}$ for all $k\geq 0$.
So $\alpha_{k}=0$ for all $n\nmid k$ and $P(x)|R(x)=T(x^{n})$ for some $T\in\mathbb{C}[x]$.

Now basic [url=http://en.wikipedia.org/wiki/Galois_Theory]Galois theory[/url] and [url=http://en.wikipedia.org/wiki/Algebraic_number_theory]Algebraic number theory[/url] would give $R(x),T(x)\in\mathbb{Z}[x]$
(the coefficients are algebraic integers and fixed under the galois group),
or if that's too much you can take a $\mathbb{Q}$-basis $B$ of $\mathbb{C}$ and $U(x)={R(x)\over P(x)}=P(\zeta x)\cdots P(\zeta^{n-1}x)\in\mathbb{C}[x]$
and write $T(x)=T_{1}(x)b_{1}+\cdots+T_{r}(x)b_{r}$ and $U(x)=U_{1}(x)b_{1}+\cdots+U_{r}(x)b_{r}$
for some $r\in\mathbb{N}$ and $b_{k}\in B$ and $T_{k},U_{k}\in\mathbb{Q}[x]$ for $k=1,\ldots,r$.
Then $0=R(x)-P(x)U(x)=T(x^{n})-P(x)U(x)=\sum_{k=1}^{r}(T_{k}(x^{n})-P(x)U_{k}(x))b_{k}$
and $P(x)U_{k}(x)=T_{k}(x^{n})$ for all $k$.
Then $T_{k}\not\equiv 0$ for some $k$ and $mT_{k},mU_{k}\in\mathbb{Z}[x]$ for some $m\in\mathbb{N}$.
And then $P(x)$ divides $mT_{k}(x^{n})\not \equiv 0$ in $\mathbb{Z}[x]$.

(I like my first one better  :P )


Nitish, I don't understand your sum so well. Is it,
"How many ways can you select 3 subsets such that they have atleast one common element
?

TO make it clear I will reframe it as
In how many ways can the subsets be formed so that the intersection of any three of them is not null
  :)

Okay, now I find the no.of triples such that the intersection is always null

W/O LOG
Each element in the set can be distributed to either 2 of the subsets or 1 of them or none of them.

So each element has 3 ways

There are n elements so 3$ ^n$ ways.

Now subtract this from total no of triples to get answer.

Now....Now....
What u have done is u have fixed 3 subsets, but ur method doesnot guarantee that any 3 subsets' intersection would be null

But i have starting feeling that this sum is almost unsolvable.I think i have to satisfied with ur answer :idea:


hi i'm interested in learning graph theory but i dunno which book to buy and there aren't too many books about it at my local library, does anyone have any book recommendations?

Try to find
[b]
Claude BERGE, [i]" Graphs"[/i] , Third revision edition 1991,
North Holland,Elsevier Sci.Publishers,ISBN: 0444876030
[/b]

:!:  wow that better be one heck of a book, the lowest price i could find on it is 95 bucks

Free book on PDF:
http://www.ecp6.jussieu.fr/pageperso/bondy/books/gtwa/gtwa.html

Intro Graph Theory (good reviews):
http://www.amazon.com/Introductory-Graph-Theory-Gary-Chartrand/dp/0486247759

Hope this helps. Good luck.


Recently, [b]mathemonster[/b] posted a topic titled Nursery Rhyme. Now, I post a (children's) story!

This story is...

[hide]
Alice's Adventures in Wonderland
[/hide]

I read the book a few months ago and didn't realise this. Maybe that book in my school library was...er...simplified, or..was my eyes just moving too fast...

Well, here is the chapter [i]The Pool of Tears[/i]

[hide]
`Curiouser and curiouser!' cried Alice (she was so much surprised, that for the moment she quite forgot how to speak good English); `now I'm opening out like the largest telescope that ever was! Good-bye, feet!' (for when she looked down at her feet, they seemed to be almost out of sight, they were getting so far off). `Oh, my poor little feet, I wonder who will put on your shoes and stockings for you now, dears? I'm sure I shan't be able! I shall be a great deal too far off to trouble myself about you: you must manage the best way you can; --but I must be kind to them,' thought Alice, `or perhaps they won't walk the way I want to go! Let me see: I'll give them a new pair of boots every Christmas.'
 Alice stretched tall

And she went on planning to herself how she would manage it. `They must go by the carrier,' she thought; `and how funny it'll seem, sending presents to one's own feet! And how odd the directions will look!

            ALICE'S RIGHT FOOT, ESQ.
                HEARTHRUG,
                    NEAR THE FENDER,
                        (WITH ALICE'S LOVE).

Oh dear, what nonsense I'm talking!'
Just then her head struck against the roof of the hall: in fact she was now more than nine feet high, and she at once took up the little golden key and hurried off to the garden door.

Poor Alice! It was as much as she could do, lying down on one side, to look through into the garden with one eye; but to get through was more hopeless than ever: she sat down and began to cry again.

`You ought to be ashamed of yourself,' said Alice, `a great girl like you,' (she might well say this), `to go on crying in this way! Stop this moment, I tell you!' But she went on all the same, shedding gallons of tears, until there was a large pool all round her, about four inches deep and reaching half down the hall.

After a time she heard a little pattering of feet in the distance, and she hastily dried her eyes to see what was coming. It was the White Rabbit returning, splendidly dressed, with a pair of white kid gloves in one hand and a large fan in the other: he came trotting along in a great hurry, muttering to himself as he came, `Oh! the Duchess, the Duchess! Oh! won't she be savage if I've kept her waiting!' Alice felt so desperate that she was ready to ask help of any one; so, when the Rabbit came near her, she began, in a low, timid voice, `If you please, sir--' The Rabbit started violently, dropped the white kid gloves and the fan, and skurried away into the darkness as hard as he could go.

 Giant Alice watching Rabbit run away

Alice took up the fan and gloves, and, as the hall was very hot, she kept fanning herself all the time she went on talking: `Dear, dear! How queer everything is to-day! And yesterday things went on just as usual. I wonder if I've been changed in the night? Let me think: was I the same when I got up this morning? I almost think I can remember feeling a little different. But if I'm not the same, the next question is, Who in the world am I? Ah, that's the great puzzle!' And she began thinking over all the children she knew that were of the same age as herself, to see if she could have been changed for any of them.

`I'm sure I'm not Ada,' she said, `for her hair goes in such long ringlets, and mine doesn't go in ringlets at all; and I'm sure I can't be Mabel, for I know all sorts of things, and she, oh! she knows such a very little! Besides, she's she, and I'm I, and--oh dear, how puzzling it all is! I'll try if I know all the things I used to know. Let me see: four times five is twelve, and four times six is thirteen, and four times seven is--oh dear! I shall never get to twenty at that rate! However, the Multiplication Table doesn't signify: let's try Geography. London is the capital of Paris, and Paris is the capital of Rome, and Rome--no, that's all wrong, I'm certain! I must have been changed for Mabel! I'll try and say "How doth the little--"' and she crossed her hands on her lap as if she were saying lessons, and began to repeat it, but her voice sounded hoarse and strange, and the words did not come the same as they used to do:--

            `How doth the little crocodile
              Improve his shining tail,
            And pour the waters of the Nile
              On every golden scale!

            `How cheerfully he seems to grin,
              How neatly spread his claws,
            And welcome little fishes in
              With gently smiling jaws!'

`I'm sure those are not the right words,' said poor Alice, and her eyes filled with tears again as she went on, `I must be Mabel after all, and I shall have to go and live in that poky little house, and have next to no toys to play with, and oh! ever so many lessons to learn! No, I've made up my mind about it; if I'm Mabel, I'll stay down here! It'll be no use their putting their heads down and saying "Come up again, dear!" I shall only look up and say "Who am I then? Tell me that first, and then, if I like being that person, I'll come up: if not, I'll stay down here till I'm somebody else"--but, oh dear!' cried Alice, with a sudden burst of tears, `I do wish they would put their heads down! I am so very tired of being all alone here!'
As she said this she looked down at her hands, and was surprised to see that she had put on one of the Rabbit's little white kid gloves while she was talking. `How can I have done that?' she thought. `I must be growing small again.' She got up and went to the table to measure herself by it, and found that, as nearly as she could guess, she was now about two feet high, and was going on shrinking rapidly: she soon found out that the cause of this was the fan she was holding, and she dropped it hastily, just in time to avoid shrinking away altogether.

`That was a narrow escape!' said Alice, a good deal frightened at the sudden change, but very glad to find herself still in existence; `and now for the garden!' and she ran with all speed back to the little door: but, alas! the little door was shut again, and the little golden key was lying on the glass table as before, `and things are worse than ever,' thought the poor child, `for I never was so small as this before, never! And I declare it's too bad, that it is!'

As she said these words her foot slipped, and in another moment, splash! she was up to her chin in salt water. Her first idea was that she had somehow fallen into the sea, `and in that case I can go back by railway,' she said to herself. (Alice had been to the seaside once in her life, and had come to the general conclusion, that wherever you go to on the English coast you find a number of bathing machines in the sea, some children digging in the sand with wooden spades, then a row of lodging houses, and behind them a railway station.) However, she soon made out that she was in the pool of tears which she had wept when she was nine feet high.

 Alice in pool of tears

`I wish I hadn't cried so much!' said Alice, as she swam about, trying to find her way out. `I shall be punished for it now, I suppose, by being drowned in my own tears! That will be a queer thing, to be sure! However, everything is queer to-day.'

Just then she heard something splashing about in the pool a little way off, and she swam nearer to make out what it was: at first she thought it must be a walrus or hippopotamus, but then she remembered how small she was now, and she soon made out that it was only a mouse that had slipped in like herself.

 Alice with Mouse in pool of tears

`Would it be of any use, now,' thought Alice, `to speak to this mouse? Everything is so out-of-the-way down here, that I should think very likely it can talk: at any rate, there's no harm in trying.' So she began: `O Mouse, do you know the way out of this pool? I am very tired of swimming about here, O Mouse!' (Alice thought this must be the right way of speaking to a mouse: she had never done such a thing before, but she remembered having seen in her brother's Latin Grammar, `A mouse--of a mouse--to a mouse--a mouse--O mouse!' The Mouse looked at her rather inquisitively, and seemed to her to wink with one of its little eyes, but it said nothing.

`Perhaps it doesn't understand English,' thought Alice; `I daresay it's a French mouse, come over with William the Conqueror.' (For, with all her knowledge of history, Alice had no very clear notion how long ago anything had happened.) So she began again: `Ou est ma chatte?' which was the first sentence in her French lesson-book. The Mouse gave a sudden leap out of the water, and seemed to quiver all over with fright. `Oh, I beg your pardon!' cried Alice hastily, afraid that she had hurt the poor animal's feelings. `I quite forgot you didn't like cats.'

`Not like cats!' cried the Mouse, in a shrill, passionate voice. `Would you like cats if you were me?'

`Well, perhaps not,' said Alice in a soothing tone: `don't be angry about it. And yet I wish I could show you our cat Dinah: I think you'd take a fancy to cats if you could only see her. She is such a dear quiet thing,' Alice went on, half to herself, as she swam lazily about in the pool, `and she sits purring so nicely by the fire, licking her paws and washing her face--and she is such a nice soft thing to nurse--and she's such a capital one for catching mice--oh, I beg your pardon!' cried Alice again, for this time the Mouse was bristling all over, and she felt certain it must be really offended. `We won't talk about her any more if you'd rather not.'

`We indeed!' cried the Mouse, who was trembling down to the end of his tail. `As if I would talk on such a subject! Our family always hated cats: nasty, low, vulgar things! Don't let me hear the name again!'

`I won't indeed!' said Alice, in a great hurry to change the subject of conversation. `Are you--are you fond--of--of dogs?' The Mouse did not answer, so Alice went on eagerly: `There is such a nice little dog near our house I should like to show you! A little bright-eyed terrier, you know, with oh, such long curly brown hair! And it'll fetch things when you throw them, and it'll sit up and beg for its dinner, and all sorts of things--I can't remember half of them--and it belongs to a farmer, you know, and he says it's so useful, it's worth a hundred pounds! He says it kills all the rats and--oh dear!' cried Alice in a sorrowful tone, `I'm afraid I've offended it again!' For the Mouse was swimming away from her as hard as it could go, and making quite a commotion in the pool as it went.

So she called softly after it, `Mouse dear! Do come back again, and we won't talk about cats or dogs either, if you don't like them!' When the Mouse heard this, it turned round and swam slowly back to her: its face was quite pale (with passion, Alice thought), and it said in a low trembling voice, `Let us get to the shore, and then I'll tell you my history, and you'll understand why it is I hate cats and dogs.'

It was high time to go, for the pool was getting quite crowded with the birds and animals that had fallen into it: there were a Duck and a Dodo, a Lory and an Eaglet, and several other curious creatures. Alice led the way, and the whole party swam to the shore.

Source: [u]http://www-2.cs.cmu.edu/People/rgs/alice-table.html[/u]
[/hide]

[size=200]Did you notice some math inside? Explain.[/size]

Alice in Wonderland was authored by a mathematician. Lewis Carrol was the penname of "Charles Lutwidge Dodgson".

Well, the statements Alice makes are true; $ 4\cdot 5\equal{}12_{18}$ and $ 4\cdot 6\equal{}13_{21}$. However, let us examine the makeup of some $ n\in\mathbb{Z}$ in different bases.
For some base $ x$, the base representation $ n_x$ may be written in polynomial form; namely, $ n_x\equal{}a_0\plus{}a_1x\plus{}a_2x^2\plus{}\cdots a_nx^n$. In this case $ 4\mid{n}\Rightarrow n\equal{}4k$. Considering this in some base $ x$, we see that alice must find and ordered pair of integers $ (k,x)$ such that \[ 4k\equal{}2x\plus{}1\] which is [b]never[/b] satisfied by integers.

We read this in the special education (for like high iq ppl) class and i am like i read that 4 yrs ago why am i reading it?
Then all the math seemed really clear like base changing multiplication he even has something about recursive functions!  :ninja:


Do the Class problems stay, or get wiped out too, like the class transcripts?

[quote="minicon"]They get moved to an archive that should be there as long as AoPS is up.
Thats if they get posted to the forum; otherwise they will eventually get deleted.
:)[/quote]

This is correct; the message board posts will be moved to an Archive Forum that you will continue to have access to.  The transcripts will only be available for 4-5 weeks after the end of class.


if a,b,c,d,e are distinct integers such that (6-a)(6-b)6-c)(6-d)(6-e)=45.find a+b+c+d+e????

this is an AMC problem from a couple of years ago (don't remember which). do a search. plus, you could have at least typed the question correctly if you're in such a rush to get an answer.

sorry for that ,i typed it wrong in a hurry     :oops:


Asymptote
LaTeX
articles
geometry
3D geometry
sphere

Hi everybody!
First of all, i would like to say for you, that i haven't got a really good english. Sorry about this...
This is a fantastic program (Asymptote), and I'm really happy I found it. This forum gave me a lot of knowledge, thank you!
But now I have a big problem with the printing of labels.
MikTex 2.7 is updated and installed on drive F:(like all of my programs), Asymptote on drive C (try default):
When I want to view a very simple asy...:
[code]draw((0,0)--(100,100));
draw((0,100)--(100,0));
dot((50,50));
label("$P$",(50,50),S);[/code]
... I got an error message... (XXX.asy is in the C:\...\Asymptote directory):
[code]c:\Program Files\Asymptote>asy XXX
latex.exe: Bad file descriptor
This is pdfTeX, Version 3.1415926-1.40.9 (MiKTeX 2.7)
entering extended mode
LaTeX2e <2005/12/01>
Babel <v3.8l> and hyphenation patterns for english, dumylang, nohyphenation, ge
rman, ngerman, german-x-2008-06-18, ngerman-x-2008-06-18, french, hungarian, lo
aded.
(XXX_.tex ("F:\MiKTeX 2.7\tex\latex\base\article.cls"
Document Class: article 2005/09/16 v1.4f Standard LaTeX document class
("F:\MiKTeX 2.7\tex\latex\base\size12.clo"))
("F:\MiKTeX 2.7\tex\latex\graphics\graphicx.sty"
("F:\MiKTeX 2.7\tex\latex\graphics\keyval.sty")
("F:\MiKTeX 2.7\tex\latex\graphics\graphics.sty"
("F:\MiKTeX 2.7\tex\latex\graphics\trig.sty")
("F:\MiKTeX 2.7\tex\latex\00miktex\graphics.cfg")
("F:\MiKTeX 2.7\tex\latex\graphics\dvips.def")))
("F:\MiKTeX 2.7\tex\latex\graphics\color.sty"
("F:\MiKTeX 2.7\tex\latex\00miktex\color.cfg")
("F:\MiKTeX 2.7\tex\latex\graphics\dvipsnam.def"))
No file XXX_.aux.
<XXX_0.eps> [1] (XXX_.aux) )
Output written on XXX_.dvi (1 page, 424 bytes).
Transcript written on XXX_.log.
plain_xasy.asy: 93.7: runtime: shipout failed

c:\Program Files\Asymptote>[/code]
:what?:
I have a lot of file with different extensions (XXX_.dvi, .aux, .log), but I haven't got any .pdf or .ps file. When I open the .dvi, I see, that it contains a lonely "math style" P on the page...
(If I install Asymptote on drive F, the situation is the same... With a simple "P" without $ \$$ = same situation, i see the label, but don't see the lines and the dot in .dvi... No .ps, no pdf... )
However if I don't type in the latest command line (label...etc.), everything works fine, I have an X with a dot in GSview...
How can I solve this problem? Please  :help: ! Without this feature Asymptote is not the same, and I would like to make figures with labels.

Best wishes!

knocknick

[quote="stevem"]See [[Asymptote:_Help]] Problem 3[/quote]

Thanx [i]stevem[/i], i have try it, but don't works...  :(
I found a "solution", which works, but only in 2D.  :mad:
[url]http://sourceforge.net/forum/message.php?msg_id=3794491[/url]
I can make the figure with labels in eps with the [code]asy -vvv filename.asy[/code] command, but when I try to draw a sphere or something else in 3D, I got only the sphere, no labels...  :wallbash_red:  :wallbash_red:
I can't believe, that I cannot use this powerful feature of Asymptote. What is the different between 2D labels and 3D labels?
I know, that it possible, but still don't know how...  :wacko:
Any idea why get I always this  error message? (see post #1)

Best wishes!

knocknick


articles inequalities parameterization Support geometry

Hello,

living in a capitalist country, and never having been to any communist country, except a brief trip into east germany in 2000 (>1991 I know), communism is something I have to read about or watch documentaries about.

Many people (like the political party PVDA in Belgium) seem to praise communism, although many of them have never been in a communist nation.  On the other hand, it is obvious that countries like USA don't prefer a positive image of communist nations.  Therefore a few questions :

1) Pure factual, rather than ideological : has communism ever (now or in past) been democratically established?  I'm sure many well informed people( Tokenadult for one, and many others) can point out how the different regimes came to power.  And has there ever been a nation that kept re-electing a communist regime (regardless of whether or not it came to power democratically) even though they could have elected noncommunist politicians.

2) Do you think that in the end, communism is a temporary status for any country?  When you look at the world map, it seems that if you consider only Laos Vietnam China, North Korea and Cuba as communist , no communist nations of the 'first generation' still exist.  All of those five became communist after the world war , vietnam and laos really late actually.  Could it be that right now communism is declining already, and that it's only a matter of time when those communist regimes disappear as well?

However, although my first question is meant as factual, democratic and communist are not strictly defined.  Also, regardless of what one thinks of communism, dumping a communist system does not necessarily imply that everything changes (like in russia)

I think that communism will dissapear in less than 50 years... just matter of time!!!!

The rise of communism in Vietnam could have been democratic.  That is, there was very much time during which the people of Vietnam, had they voted, most likely would have elected the North Vietnamese.  Many South American leftist regiemes have been freely elected as well.  More important than being freely elected, however, is to allow opportunities for others to be elected, something that hasn't happened in any of the states that you names (although it has happened with various South American leftists -- Hugo Chavez, for instance, is vilified by the American government, but is wildly popular in Venezuela.  In an earlier age, he would probably be receiving aid from the Soviet Union.)

Do you consider Chavez,Allende and Morales communist?  I don't think they are generally accepted as communist.  It is a bit vague, and I knew that was gonna be a difficulty about that question.
About the vietnamese, weren't the Vietcong south vietnamese supporting a communist regime?  I know the south vietnamese regime was far from perfect, but didn't the south vietnamese suffer from the vietcong too?  Nguyen Van Lem's infamous execution on Tv, wasn't that a consequence of the killing of wives and children of south vietnamese police officers by communists?

I personally wouldn't consider any of those three to be communists.  Chavez is obviously living in a post-Soviet world in which the pressures against it are much stronger than the pressures for it; this also goes for Morales (who hasn't had very much time to make policy, obviously, so I can't comment on that) but who seems less in that mold.  I don't know very much about Allende, but I think of him as a democrat.  I think, however, there are many people who viewed or view Allende as a communist -- he was, after all, a member of the Socialist party.  That was a fairly small sampling you chose, though, and 2 out of 3 were from the past 5 years.  There were plenty of other leftist movements or leaders in South America -- the Sandanistas, for instance.

[quote]Could it be that right now communism is declining already, and that it's only a matter of time when those communist regimes disappear as well?
[/quote]

I think communism has been declining for a long time now...For example, isn't the collapse of the Soviet Union considered to have been a major blow for communism?  That event, as I understand it, made it much more difficult for intellectuals to hope that communism was somehow a workable system.

My impression is that whenever anything like communism has been tried, it has led to famine and mass murder.  I've read from many different sources that something like 100 million people were murdered by their own governments in the 20th century.  A truly staggering figure.  I don't think communism can last...a country will eventually have to embrace some free market principles, or else face total economic collapse.  If communism does last, it will be only in a state of extreme squalor.

I don't know if any communist regime has ever been democratically elected...but I have read that Hitler was democratically elected.  So I think it could be possible.  I don't believe in unlimited majority rule, like what they had in Ancient Greece when they voted to kill Socrates.

My basic idea about politics is that my life belongs to myself and the government should leave me alone and let me live my own life and pursue my own happiness however I choose.  (So long as I interact with others on a strictly voluntary basis.)

I'm still learning about politics and history though.  I don't know much about them yet.

Germany was communist now?

fascists were clear opposers of communism

I didn't mean that Germany was communist...If it sounded that way, I should've been more clear.

I think communism will be around for a long time, because it has enough good points to convince people that it might work out.

Maybe so but one of my questions is do they tend to be chosen by the people too
if not it does not really matter

I'm a little confused by your question.  Are you asking if a communist state has ever been established through a democratic process?  Most democratic countries have laws or constitutions that prevent many of the basic tenets of communism (take ours and the limits on internal state trade, for example, though this is often circumvented).  In fact, it seems like such a democratic process would be contrary to the ideals of communism.  I guess you could say that a national "mood" or climate could illustrate the prevailing popularity of communism, but I don't think that an political or legal justification for such a complete revolution could exist.

I'm not sure but that was basically my question (though worded more politely) : has a nation ever thanked its leaders for communism?

I'm not justifying revolution against it.

Actually that brings me to another question : I already asked it is wanted by the people and whether or not you think it will vanish, but now I also ask : has a communist nation ever prospered (or improved its status) after removal of communism?

miraculouspostmaster, you refer to your own situation, but to my shame I must admit I had never heard about Saint Kitts and Nevis (though I try to know at least some basic facts about every country on earth).  Wikipedia however points out secessionism, which I can relate to very much being a Belgian.
Please elaborate.

Although communism has never been implemented at a national level in a democratic fashion, there is nothing inherently undemocratic about communism.  (By the way, Fredbel, his reference was almost certainly to the constitution of the United States, although I'm not exactly sure what he's referencing or why he thinks it's inherently inimicable to communism.)

As for your most recent question, there are a number of countries which have improved their economic status since the end of communism, most of them in Eastern Europe: Slovenia, the Czech Republic, and those other former Soviet bloc nations which are now part of the EU are all better off now then they were before 1990.  Of these, the two that I named are both fairly prosperous -- Slovenia, for example, was the first country to switch from a borrower to a lending partner at the World Bank and has a GDP per capita of over $\$20000$, six times that of India (3400 dollars), more than triple that of China (6200 dollars) and more than double that of Brazil (8,500 dollars) (according to the [url=http://www.cia.gov/cia/publications/factbook/geos/si.html]CIA World Factbook[/url]).  The Czech Republic is comparably well-off.  (Slovakia, Hungary, Estonia, Latvia, Lithuania and Poland are all somewhat less-well off (12000 to 15000 dollars GDP per capita), but still doing better than, say, Croatia (11600 dollars) or Romania and Bulgaria, which are in the same neighborhood as Brazil.)  Also, Slovene and Czech society are less stratified then, say, Russia, so the GDP per capita number isn't hiding anything terrible, at least as far as I know.

I should also note that one can find in the formerly communist Eastern and Central European states (Hungary and Slovenia for sure) graffiti lamenting the new order.  While I don't think those views represent majority opinion, there is growing political dissatisfaction in many of those states which bungled their transitions, both in those states which successfully implemented democracy (Hungary) and those in which it was not-so-successfully implemented (think about the recent protests and mini-revolution in Ukraine, for instance).

[quote="JBL"]Of these, the two that I named are both fairly prosperous -- Slovenia, for example, was the first country to switch from a borrower to a lending partner at the World Bank and has a GDP per capita of over , six times that of India (3400 dollars), [/quote]
Kidding? Six times of India's, the biggest democratic country in this world?
So, the conclusion is, in order to become "fairly prosperous", the best way is first choose a communistic government, then switch to a democratic one, repeat if this does not work. Is this what you meant?

Six times India's, [i]per capita[/i].  Per capita is Latin -- it means per person, so you divide the total GDP by the population to get the GDP per capita.  The population of India is roughly 500 times as large as the population of Slovenia, so India's total economy is (unsurprisingly) vastly larger than Slovenia's.  (In fact, Slovenia is the 49th richest country/region in the world, measured by per capita GDP.  The Czech Republic is 58th.  Brazil, China and India are respectively 93rd, 118th and tied for 155th (with Iraq, interestingly enough).)

I think the better lesson to learn from Czech and Slovene society is this:  if you are a reasonably homogenous society with a well-established middle class and ties (historic and geographic) to the West, even fifty years of Communist domination won't totally ruin you, and it helps to have been run relatively well compared to, say, Romania.

excuse my imperfect english GDP per capita, that's what you use as a number to evaluate prosperity right?

Could you please explain the abbreviation and explain in very simple terms what exactly you calculate then?

GDP per capita = Gross Domestic Product per capita = GDP/total population.

GDP is a simple way of measuring the size of an economy.  Wikipedia has a good article about it, but the basic gist is that it measures the total amount of production of goods, services, etc., inside the country.

Obviously it's not a perfect measure of national wealth -- for example, it would utterly fail to capture the levels of poverty or inequality in a country -- it tells you how much wealth is being created, but not who it belongs to.  In the cases I've mentioned, though, this pitfall is not a problem.  Also, it misses out on transactions that occur illegally (the black market) and transactions in which no money changes hands (barter).  Most importantly, people have questioned it as a meaningful measure of the standard of living.  However, it's still probably the most commonly applied measure of national wealth, and I think that in general it paints an accurate picture.

I DID notice that it's GDP per capita, and it's the right parameter to make a rough estimation  of  living-standard, not the total GDP.
How many years has India "practiced" democracy? Any how many years has Slovenia done? And now Slovenian live much better than the Indian, what this implies? Communism is so evil, once we removed it and replace by democracy,
an economic upsurge will arrive immediately, it can even catch up with those countries which are always practicing democracy. That's why I said "in order to become "fairly prosperous", the best way is switching regime every now and then.

As far as I know, communist party is illegal in the US, let alone being elected.

Last in, first out: from time to time I walk past the headquarters of the Socialist Party USA.  They have a ballot line in New York City.

Sorry, I did not fully understand you.

Your argument doesn't make any sense -- you're extrapolating from the two data points of Slovenia and India, each of which has a significant history which have affected their current economic status.  The "frequent changes of regieme" has not been an obviously positive part of this: just compare Slovenia to the other former Yugoslav states, with similarly complex histories.  Slovenia and Croatia are each significantly better off than Serbia, Bosnia and Macedonia.  Why the difference, if they all have had so many alterations in government?  Who the governments were is much more significant then how many of them.

Interesting discussion.

There has never been a comunist government that has come to power except through violent overthrow of an existing government (sort of how the United States started its democracy).   While current laws and consitiutions in the United States would conflict with communist ideas (everything belongs to the state) they can be changed by supermajority votes over a period of time.

Karl Marx and the theory behind communisim have never worked right.  (Keep in mind that when Karl wrote, the largest company was owned by one of his personal friends in England and employed less than 6,000 people.)  The theory in short is that the "natural" evolution goes king - democracy - communism - socialism.  There is no support, except those who like the idea of a leading a strong government that ownes everything, can do anything to stay in power because they are working for the long term better good, like a surgeon cutting out a cancer to save a person's life, for a long established communist government.

One thing you are confusing is that GDP is largely a result of the position of a country with respect to its neighbors and international investment.  The government history is of small impact in setting the EU nation's incomes.  These governemnts adopted rule of law and property rights that supported massive investment to take advantage of low worker wages and high skills, which quickly transformed the communist destroyed economies.  Their neighbors were happy to welcome the nations into the "friendly" camp, even if it meant spending more money to help their neighbors that was justified on a purely economic basis.  India, China and other countries have real problems because they do not have rule of law and property right respected to an extent that investors are willing to invest the same way they were in the former communist countries.  Investment is going well, but these countries are severly restrained by the legal structures and history they have.

Note that Hitler was elected to office, had the laws changed to make him a dictator.  Many people have said many words about why and the like, but this has never happened to communism.  In fact the Russian revolution was not even led by the communists, they just seized power in the confusion that took place after the revolution.

Vietnam can't be judged without taking into account the effect of nearby communist China.

[quote="wizard2378"]There has never been a comunist government that has come to power except through violent overthrow of an existing government (sort of how the United States started its democracy).[/quote]

I think this comment is rather simplistic.  Tito in Yugoslavia came to power by successfully resisting the Nazis.  The North Vietnamese came to power by successfully resisting the French, then the Americans.  So, in those cases we are starting with an existing government of seriously questionable legitimacy.  In many of the Soviet satellite states, however, the communists gained power in large part through (abuses of) the democratic process in the fallout from WWII, pressing the advantage gained by having the Red Army sitting around, but not literally having a military revolution.  In Hungary, for example, there was no revolution, just the systematic dismantling of opposing political parties following elections, and then the cementing of power.

[quote]Communism is so evil, once we removed it and replace by democracy,
an economic upsurge will arrive immediately, it can even catch up with those countries which are always practicing democracy. [/quote]

I disagree with your statement. I don't think communism can be [i]evil[/i], at best, it can not work, and even act as a platform toward evil, but in and of itself it can't be. But I disagree that replacing it by democracy creates an immediate upsurge. Ask the many Romanians on this forum - economic strenght doesn't come quickly. (What does, however, after regime changes, is inflation =p).

Maybe you're trying to argue that only a human can be evil, or something, but I would say that communism is an evil system.  A non-evil system would leave me alone, let me create any kind of wealth that I like, let me keep at least most of it once I've created it (after all, it didn't even exist until I created it..), and let me trade it in whatever manner I like, according to whatever agreement I make with my trading partners.  A communist government systematically uses force against innocent people in order to prevent them from engaging in this kind of voluntary behavior.

It is often said that communism is a good ideal that just doesn't happen to work in practice.  I disagree with this.  There is a large amount of historical evidence and theory that seems to show communism is a terrible failure in practice, but I would say that even as an idea communism is immoral, because it systematically uses force in order to control the way people live their lives (specifically, in order to control the parts of peoples' lives related to wealth creation and trade.)

Two questions: if taxation isn't evil, but state appropriation inherently is, where's the dividing line?

What about when restraints on individual action are necessary for the common good?  (This is posed hypothetically, but it need not be -- consider the one-child policy in China, for instance, or the various methods Tito used which prevented Yugoslavia from imploding  for a good long while.)

[quote]Two questions: if taxation isn't evil, but state appropriation inherently is, where's the dividing line?[/quote]

Well...The way I look at it is, taxation at best is a necessary evil.  Ideally there would be some method of voluntarily funding the government.  But it's not clear to me that this is possible, so I would like for taxation to be as low as possible while still allowing for the government to provide a police force, a justice system, and a military, which exist only in order to protect individual rights.

[quote]What about when restraints on individual action are necessary for the common good? (This is posed hypothetically, but it need not be -- consider the one-child policy in China, for instance, or the various methods Tito used which prevented Yugoslavia from imploding for a good long while.)[/quote]

I'm usually very skeptical when someone suggests that a restraint on individual action is necessary for the "common good."  Perhaps there are some extreme cases where it may be a good idea, but I really don't like the idea of the government using violence (or the threat of violence) in order to control the behavior of a peaceful individual who interacts with others on a voluntary basis.  In my mind, it would take an extreme situation indeed in order to justify a use of violence like that.  It's not easy to justify such a thing on moral grounds.

I know nothing about Tito in Yugoslavia, and I don't know much about the one-child policy in China.  (And maybe I should say again, I really don't know much about politics.)  But one thought about the one-child policy in China is this...In prosperous nations, with a lot of individual freedom, I have the impression that the population growth seems to control itself automatically, with no need for government force.  Maybe this is because in a prosperous nation, you have so many choices for what to do with your life, it may turn out there are some things you would rather do than have kids.  Or maybe it's the wide availability of birth control, I really don't know.  But I wonder if China's population problem wasn't a symptom of the more fundamental problem that its economy had been destroyed by years of communism.  Given the situation that China was in, after years of communism, perhaps the population problem was extreme enough to justify the one-child law, I don't know....but I wish that China had had capitalism all along, so that it would've become so prosperous that the population growth would've automatically controlled itself, which seems to be the trend.  (And I hope that China will continue to embrace free markets, and that the one-child law will become unnecessary...)

Really my whole spirit revolts at the idea of a one-child law.  Talk about using force in order to prevent peaceful people from pursuing their own happiness the way they choose.  I would be extremely hesitant to accept such a thing.

I have so much to say about this being communist in spirit.

first communism is not illegal in the US nor are communist parties of which there are several (many were puppets of ths Soviet Union and dissapeared with it but there are still communist parties they are just unimportant in a two party system).  In some countries communism is illegal (Poland) but communism doesn't naturally conflict with democracy.  Communism is an economic system, democracy is a political system, they are connected but not inseperable.

Second most countries wind up socialist either with the governement or the pseudo governement.  The US is socialist we just dont admit it and the conservative republicans try to change it.  (examples welfare, medicare, social security)  In fact America has really been socialist since shortly after the emergence of the city (prior to this time america wasnt socialist due to the lack of need, most people had jobs or moved west, and there wasnt real poverty but due to population booms and the industrial revolution jobs dissapeared and wages fell so the working class began to suffer and needed support).  The socialism at first was not a product of the government which did nothing after the civil war until Teddy Roosevelt but a result of the party system which provided the services people needed for their votes.  In fact socialism was introduced to increase democracy by attacking the parties.  (this information is from my ap history class)

third one reason communism was never democratically implemented was that the Soviet Union was the defender of communism and simultaneously totalitarian dictators while the United states defended "democracy" and capitalism.  Many countries who would have elected communist regimes did not because elections weren't held because the US knew communists would win.  (Allende was democratically elected but then overthrown and while he didnt run on a communist ticket he was a marxist)

finally my personal opinions (i realize tbe above are biased but here i will not attempt to distingusih my opinions and fact)  I believe that communism works in small cases but when a region becomes too large (almost any country) the system collapses due to corruption (this is why the kibbutzim were successful but communism fails when a country becomes communist).  Also i favor communism/strong socialism with democracy because in communism the government controls industry but in capitalism industry controls the govenrment.  In response to Aunt Sally while i do not want the government spying on me i want the government regulating the power of individuals and helping hte poor.

[quote="jehr314pi"]

third one reason communism was never democratically implemented was that the Soviet Union was the defender of communism and simultaneously totalitarian dictators while the United states defended "democracy" and capitalism.  Many countries who would have elected communist regimes did not because elections weren't held because the US knew communists would win.  (Allende was democratically elected but then overthrown and while he didnt run on a communist ticket he was a marxist)

.[/quote]

I don't understand this way of thinking.  Please explain.

If USA is socialist, what countries are not socialist?  They teach you that in history?

About Tito, it would be interesting if he would be discussed a little more, I am getting very interested in the nonclassical communist nations: Yugoslavia and Albania (weren't allies with each other, but in the end both were neither allied with China or USSR)

[quote="fredbel6"][quote="jehr314pi"]

third one reason communism was never democratically implemented was that the Soviet Union was the defender of communism and simultaneously totalitarian dictators while the United states defended "democracy" and capitalism.  Many countries who would have elected communist regimes did not because elections weren't held because the US knew communists would win.  (Allende was democratically elected but then overthrown and while he didnt run on a communist ticket he was a marxist)

.[/quote]

I don't understand this way of thinking.  Please explain.

If USA is socialist, what countries are not socialist?  They teach you that in history?

[/quote]

My point is that there is no example of communism or capitalism in modern society (ie since the industrial revolution) in a non primitive society, non volitile (ie people who engage in international trade with a well established government and true economy, so I'm not counting tribes/decentralized areas or places that are about to have a revolution)  In any country once they are industrialized and job differntiation occurs ultimately society reaches the point where there exist a substantial population who needs support.  Once htis happens someone needs to take care of those people or else there will be a revolution, in communism and procalaimed socialism the governement will do it, while in pure capitalism someone does it (in the US the political machines) and gains the support of the people becoming in effect the government.  In the modern day US the govenrment provides a lot of services that make it socialist.  Socialism is a scale and some countries are more socialist than others but if there is competition and free market while the govenrment regulates and controls important industires there is socialism.  (The US is socialist because of the postal system as a major essential industry)  the scale goes
capitalism s......o......c......i......a......l.....i......s.......m communism where unless there is no government intervention or complete government control there is socialism.

As a result much of the debate is how socialist to be:  In hte US conservative republicans want virtually free market with government control of post office (i dont think anyone objects to this) some regulation of business (very little regulation) and few services while compassionate conservatives (Bush) and democrats want moderate to heavy regulation and many services.

with that said people dont like to consider the US socialist and so they say that we're capitalist, this reminds me of the trust claiming they supported competition at the end of the 19th century when in reality they tried to stop it.

and no they dont teach us this in school they dont teach us anything in history, but i read the book.

A lot of what you've said is very strange.  In particular, you seem to abuse terminology, especially with respect to the question of socialism versus capitalism.  The United States is by no stretch of the imagination a socialist state.  The United States does provide some social services to its citizens, which is a common feature of all developed nations at this point, but social services $\neq$ socialism under the common understanding of the term.  Even the Scandinavian nations aren't truely socialistic, they just have far larger social welfare systems than this country.  While there is a wide range of options between "pure capitalism" and a command economy, it is simply not accurate to describe all of them as socialism, at least not if you want other people to understand you.  It also dilutes the meaning of the term socialism.

Also, your basic descriptions of the political positions of contemporary political groups are very, very simplistic.

Since I am now trying hard to work Tito, Yugoslavia or Slovenia into every post I make, I should point out that communist Yugoslavia was as (relatively) well-to-do as it was in part because of limited market reforms -- after a certain point in time, private businesses were allowed as long as they employed a very small (I don't know if it was 3 or 5 or 10) number of people.  Also, by not joining the Soviet bloc, Yugoslavia was open to aid from the West, and because of open borders Yugoslavs could leave the country and earn hard currency in Austria, Italy or other European states to bring back home.

[quote="Aunt Sally"]I really don't like the idea of the government using violence (or the threat of violence) in order to control the behavior of a peaceful individual who interacts with others on a voluntary basis.[/quote] Fair enough, neither do I, but I think your basic skepticism is severely misplaced.  One simple example is over-exploitation of resources, the "tradgedy of the commons."  Given any finite resource, it's in society's best interest to make sure that we use it sustainably.  However, anyone who mines (or fishes, or farms, etc.) the resource can correctly reason that if, in light of this common good, she were to not mine it as fully as possible, someone else would come along and do so.  Thus, in a fishery, any fisherman has no incentive to reduce his own haul (since anything he doesn't catch, someone else will) even if it is in the interest of all fishermen (and all consumers) that the total haul be maintained at a sustainable level.  I don't know of any large-scale solutions to this problem that don't rely upon government involvement, and it's a problem that is related to the vast majority of hypothetically sustainable resources.

For the specific example of China which you addressed: I also find draconian measures impinging on the exceedingly personal decision of whether or how many children I should have to be a serious affront.  However, increasing the standard of living of the Chinese is not a real solution.  I don't say this because I don't think it would limit population growth (indeed, it might), but because it doesn't address the real problem at all.  Many people is not inherently bad, it's bad because the world doesn't have enough resources to support so many.  To borrow from Jared Diamond, if everyone were in cryogenic facilities consuming minimal amounts of food, oxygen, space and energy, the world could easily support many billions of us.  But if China reaches a first world standard of living (with zero population growth), it will approximately double the total world consumption of everything.  It's not clear that current world levels of consumption are sustainable -- twice those levels are almost certainly not.  This is also a tradgedy of the commons problem: I, as an enlightened first-world occupant, know that the world-wide standard of living will inevitably go down as we hit and pass consumptive landmarks (peak oil being the one most frequently mentioned nowadays).  However, reducing my own consumption isn't going to help: someone else will use the extra calories and kilowatts that I choose to forego.  How do you propose a solution to this dilemma without impinging on individual rights?

(The Tito reference I made was this: Tito threw nationalists of various stripes in prison throughout his rule (along with democrats, liberals, etc.).  This is clearly an infringement on their persons.  However, within about ten years of his death, the system that he had designed to replace him collapsed, with the result being several years of vicious secretarian warfare led (for the most part) by nationalist Serbs and Croats.  (I personally lay blame primarily on the Serbs, although obviously many Serbs (and perhaps others) would tend to disagree.  Just laying the personal biases out there.)  So, the question is: would continued restraint of individuals have been acceptable if they would have avoided or limited the scope of the Balkan wars?

Heh, that makes two Tito paragraphs in one post!  Also, I am very fond of your screen name because I happen to have an Aunt Sally.  Go figure.

[quote="JBL"]Tito threw nationalists of various stripes in prison throughout his rule (along with democrats, liberals, etc.). This is clearly an infringement on their persons. However, within about ten years of his death, the system that he had designed to replace him collapsed, with the result being several years of vicious secretarian warfare led (for the most part) by nationalist Serbs and Croats.[/quote]
The interesting thing is that during Tito's reign, Milosevic was in no danger of being thrown in prison. Milosevic was an insider - a Communist Party member, a bureaucrat, a part of the system. As the ground shifted after Tito's death, and as the other Eastern European nations were breaking away from Soviet influence and from Communism, Milosevic discovered that ugly appeals to Serb nationalism gained him a certain kind of popularity and power. How much of that was his personal beliefs and how much was political expediency? I don't really know. What was Tudjman's background? Was he also once a party insider, or did he come to power in Croatia from the outside of what had been the Communist Party?

Milosevic's appeals to the worst aspects of nationalism unleashed the likes of Karadzic, Mladic, and others to do their worst. Evil is usually not spontaneous; it requires leadership.

I know very little about Tudjman, but wikipedia seems to -- he was one of Tito's generals during WWII but was later expelled from the party and imprisoned for his Croatian nationalism.  Reading through the article, it sounds like he was about two parts anti-Serb to one part pro-Croatian -- he seems to have felt that the quashing of nationalist sentiments was in fact encouraging Serbian domination.  This being wikipedia, I am inherently untrustworthy, and parts of the entry sound like they were written by someone with a Croatian-nationalist bent, but who knows.  Anyhow, it's a good point that you raise.

sorry if i've been using socialism wrong but i thought these were the definitions

capitalism=no government regulation
socialism=free market and competition but government regulation and government controls "essential" industries (post office, healthcare, water, power, depends on perspective of whats essential)
communism=government controls entire economy allows no competition, sets wages and prices

if these are not the correct definitions please tell me

Thanks for the insightful reply JBL.

[quote][quote]Aunt Sally wrote:
I really don't like the idea of the government using violence (or the threat of violence) in order to control the behavior of a peaceful individual who interacts with others on a voluntary basis.[/quote]

Fair enough, neither do I, but I think your basic skepticism is severely misplaced.[/quote]

You might be able to find some examples where government coercion against peaceful individuals seems to be necessary, but I believe there are millions of examples where people to try to invoke the "common good" in order to justify forceful violations of individual rights, even in situations where the free market would be more effective and efficient anyway (not to mention less violent).  So I do think it's a good idea to be skeptical whenever someone claims the systematic use of violence by the government is justified in order to promote the "common good."

[quote]However, reducing my own consumption isn't going to help: someone else will use the extra calories and kilowatts that I choose to forego. How do you propose a solution to this dilemma without impinging on individual rights?[/quote]

That's a good question.  I've thought about that before, but I think I'll need to learn more before I have a good answer.  I'm sure that libertarians have a lot of answers to this question, so if anyone is curious they could see if libertarians have a good answer or not.  For now all I can say is that I really dislike the use of violence against innocent people by the government, and I'm hopeful that a system could be devised which resorts to the use of violence very little.  And in my experience, people are often too quick to assume that the free market will fail, and too quick to resort to government force.

In a free market, as limited resources become more scarce, their prices go up, which leads people to conserve, and also encourages innovative scientists and businessmen to come up with more efficient methods of energy production and alternative sources of energy.  In a free market, perhaps this combination of conservation and innovation due to price constraints would manage to handle the whole energy problem pretty well.  Is that possible?

A country's economy is a hugely complex system, and how to allocate limited resources in the best way is a hugely complex problem.  It may be impossible for a human advisory board or committee to figure out how best to allocate limited resources for a whole economy without leading to shortages in crucial areas at crucial moments, but the price system handles this complex problem automatically.  Given the historical lessons we've learned about the difficulties of planned economies, it scares me to abandon the price system (or partially abandon the price system) and let the government decide how resources are allocated.

The government seems to have a terrible record at doing things.  In practice, the government messes everything up.  So I'm not sure the government would do better than the free market at managing the limited natural resources problem.  Often people are too quick to give up on the free market, and too quick to assume the government can handle the problem, despite the government's terrible record of handling and solving practical problems.

One possible thing to consider is this quote from Amory Lovins in the Feb. 2006 issue of Discover magazine, in an article called "The Energizer":

"If I could do just one thing to solve our energy problems, I would allow energy to compete fairly at honest prices regardless of which kind it is, what technology it uses, how big it is, or who owns it.  If we did that, we wouldn't have an oil problem, a climate problem, or a nuclear proliferation problem.  Those are all artifacts of public policies that have distorted the market into buying things it wouldn't otherwise have bought because they were turkeys."

That was an interesting article.

Okay, I should really stop debating about this because I really don't know what I'm talking about.  I'd like to encourage people to consider the moral stance expressed by the slogans "you should never initiate the use of force" and "all human interaction should be strictly voluntary", and to question more often whether government force is morally justified (when there is usually a laissez-faire alternative that would probably work better in practice anyway)... But I don't know much about politics yet so I shouldn't try to talk about this much more.  Maybe you are right that government intervention is for the best in some cases involving natural resources--I need to learn more about it.

You should stop disparaging your own knowledge -- you are discussing these issues intelligently and eloquently.  My brother is throwing me off the computer right now, so I will only give you one quick answer, namely that I don't think libertarian views do in fact account for solutions to these problems.  Libertarianism seems to posit that the market is intelligent and will take all these sorts of things into account, but I personally don't think that markets are far-sighted enough to avert serious future disasters.  (In fact, I think markets tend to be very short-sighted.)  I will give more substantive response eventually.

i disagree with your remark that govenrment messes everything up, prior to government intervention in the American economy, there existed a powerful undemocratic elite (Carnegie (who wanted to help society), Morgan, and Vanderbilt) these men exercised enormous power and the working class had horrible working conditions and couldnt change them.  The country significantly improved with government regulation and interference.  Also when the people had poor working conditions party bosses were able to control elections and prevent rue democracy since they essentially bought votes.

I am afraid Communism cannot be established [i]democratically[/i]. It takes away liberty, and gives you equality, which I believe is more imporant. This democracy is ruining the world today.

communism can be established democratically like anything else what I think you mean is that a liberal democracy can not implement communism which depends on what liberties you consider essential (communism really only conflicts with rights to property/unfair searches and appropriation)

I did not mean it is against democracy. I meant it gives greater stress in social justice and economic equality.

[quote="Aunt Sally"]
Money is more than just green paper.  When you take somebody's money, you also take the time it took him to earn it, the energy it took him to earn it, and the happiness he could've gotten by spending it.  Your life consists primarily of your time, your energy, and your happiness.  Therefore, when you steal someone's money, you have stolen a part of his life.

Now that the 20th century has happened, and we've seen tens of millions of people die of starvation, or of murder by their own governments, in all the places where anything like communism was tried... And now that we have available to us such clear explanations of why communism is such an economic disaster in practice... It is surprising that some young people still become enamored with communism.  It was more understandable before the 20th century happened.  Now that the 20th century has happened, it is...less understandable.

A partial explanation is that people care more about morality than they do about practicality.  If they believe a system is the most moral, they may be willing to overlook the most horrible practical results that system has had in practice.

But when you consider that the "moral ideal" that supposedly is communism involves setting up a government that systematically uses force to control the way each citizen lives his life, instead of doing the peaceful thing, which would be to leave people alone...  Then, it becomes quite hard to understand why communism still appeals to some people.[/quote]
I agree you have a right to be left alone but do you have a right to allow people to starve or exploit other humans for your own profit.  The government shouldn't be pointing a gun at you and taking your money but they shouldnt have to.  Depending on how you make money but probably (not you specifically) you ultimately make your money from the exploitation of other peopel and that is wrong and in fact then you are as bad as the govenrment that points a gun at you because you ruin someones life.  Note that in most cases of governments killing people communism was not the reason it was typically dictators who were paranoid killing people they thought were threats (sometimes in the name of communism but rarely actually because of communism).  Most times when communists were involved in killing they were the ones dying beause of totalitarian right wing governements or so called communists (ie Stalin killed most of the old communist thinkers because they were competition).

[quote="Aunt Sally"][quote]I am afraid Communism cannot be established democratically. It takes away liberty, and gives you equality, which I believe is more imporant.[/quote]

Bubka, when you show up at my door with a gun and point it at me, and tell me that I must give you the wealth I've created, in the name of "equality", then I won't be able to defend myself----all I'll be able to do is fall on my knees and beg you to put the gun away, and leave me alone, and let me live my own life.  Perhaps at that moment you will feel pity for me and your view of the value of liberty will change.

Money is more than just green paper.  When you take somebody's money, you also take the time it took him to earn it, the energy it took him to earn it, and the happiness he could've gotten by spending it.  Your life consists primarily of your time, your energy, and your happiness.  Therefore, when you steal someone's money, you have stolen a part of his life.

Now that the 20th century has happened, and we've seen tens of millions of people die of starvation, or of murder by their own governments, in all the places where anything like communism was tried... And now that we have available to us such clear explanations of why communism is such an economic disaster in practice... It is surprising that some young people still become enamored with communism.  It was more understandable before the 20th century happened.  Now that the 20th century has happened, it is...less understandable.

A partial explanation is that people care more about morality than they do about practicality.  If they believe a system is the most moral, they may be willing to overlook the most horrible practical results that system has had in practice.

But when you consider that the "moral ideal" that supposedly is communism involves setting up a government that systematically uses force to control the way each citizen lives his life, instead of doing the peaceful thing, which would be to leave people alone...  Then, it becomes quite hard to understand why communism still appeals to some people.[/quote]

Aunt Sally, I am of course taking away a part of your life, but I am giving it to people who have not have a life at all. You must remember that you did not get the same opportunities to make money as the proletariat. It is all very well of us to sit here AoPSing and thinking the money we earn is ours by right, but it is actually people who work much harder for nobler causes who should get them.

[quote]Depending on how you make money but probably (not you specifically) you ultimately make your money from the exploitation of other peopel and that is wrong and in fact then you are as bad as the govenrment that points a gun at you because you ruin someones life[/quote]

Under laissez-faire capitalism, the worst that you can do to someone is ignore him, or make him an offer that isn't very good.  You can't exploit someone by making him an offer.  You can't exploit someone by giving him an option that he didn't previously have.  You can't exploit someone by ignoring him.  And you can't ruin somebody's life by doing any of those things either.  You can't ruin somebody's life by interacting with him on a voluntary basis.

(Perhaps his external conditions have already ruined his life; perhaps his oppressive government has ruined his life; but that is not your fault and you did not do it to him by interacting with him on a voluntary basis.)

On the subject of the "exploitation" that is supposedly a result of capitalism, here's a passage from The Capitalist Manifesto by Andrew Bernstein:

***************************************************************

"The close correlations in the world today between freedom and prosperity--and between statism and destitution--were discussed above.  Still, capitalism's enemies currently assail it as a system oppressive to the workers and the poor.  They attack Asian factories affiliated with such American companies as Nike and The Gap as "sweatshops."  The anti-capitalist agitators criticize Nike for paying Vietnamese workers an average daily wage of 1.60.  [All money amounts here are in dollars.] Such an amount of money is truly miserable by the standards of modern capitalist countries.  But what is it relative to the alternatives open to these workers elsewhere in their Communist, socialist or feudal economies?

According to the World Bank, in 1998 more than 278 million people in East Asian countries subsisted on less than a dollar a day.  Communist Vietnam is down at the bottom of the list regarding prosperity.  If a 260-day work year is assumed, then 1.60 per day adds to 416.00 per year, significantly above the 250.00 per year income that many in Bangladesh, Vietnam and other countries subsist on.  No wonder Nike has no difficulty attracting workers into its employ.  It is paying wages quite a bit higher than what is available to these laborers elsewhere in the oppressive feudal or Communist societies in which they live.

But no such hypothetical calculation needs to be taken as the final word.  The actual facts of Vietnamese living conditions redound even more greatly to capitalism's credit--for the minimum wage there averages out to 134 annually while Nike's workers receive on average 670.  In Indonesia, Nike's subsidiaries pay an average of 720 annually in contrast to the minimum wage which totals 241.  'In the poorest developing countries, someone working for an American employer draws no less than eight times the average national wage.'  Further, foreign companies in the poorest countries pay their workers, on average, twice what the corresponding native firms pay.

The difference, of course, is the technology made possible by the greater capital invested by American and other Western firms, which raises the productivity of each worker.  Similarly, the more modernized plants of the American companies translate to better and safer working conditions.  Technology and industrial mechanization--the achievements of the minds of men made possible by freedom--are raising living standards in 21st century Asia just as they did in 18th century Europe.

The real problem in Third World countries is not that Western companies 'exploit' their workers--they do not; it is that indigenous dictatorial regimes--whether communist, socialist, theocratic, feudal or military--oppress their own citizens.  The moral imperative is not to pressure Nike, et. al., into 'better treatment' of its employees; it is to overthrow the communist, theocratic or military despots and establish capitalism, the only system that respects the rights of the individual."

**********************************************************

It is cynical to give up on the ideal of voluntary interaction and assume that the systematic use of force against everyone is a necessary part of life.  Perhaps some small amount of government force is a necessary part of life, but not very much.  Laissez-faire works very well; it is lack of laissez-faire that causes the terrible poverty in the world.

Another quote from The Capitalist Manifesto:

"The parallel between the pre-capitalist European economies of the 18th century and the pre-capitalist Asian economies of the 20th and 21st centuries is striking.  In both, state-dominated economies led to destitute conditions or worse.  In both, capitalists opened factories to which workers flocked because the wages offered were significantly higher than what was otherwise available to them.  In both, the factory system was denounced by anti-capitalist writers as 'exploitative.'  Regarding both, capitalism's critics are guilty of the logical fallacy of context-dropping.  If they were truly concerned with helping the poor, they would criticize the statist political systems responsible for causing the poverty.  They would not attack the germinating capitalism that begins to relieve it."

I accept that it is posssible to pay foreigners less and still pay them sufficeintly (ie wages can be based on what is required to live in their homes).  However there are still two problems

First why the standard of living is lower there.  Some of this is natural created by tariffs (Japan pays more for rice the US pays more for sugar so if we base standard of living on food we need more than a country with no tariff if we decided sugar and rice are required foods (they are for me)), supply and demand (either supply of workers or materials), and natural resources (or other easily available things).  but there are other causes: some are the fault of the governments suppressing the people and some are the fault of capitalists of old (ie imperialists).  the imperialists stripped many countries of their natural wealth and hindered development of many areas (although some were also the vehicles of good and radically helped areas).  the governments that were corrupting were usually capitalist with the wealthy elite forming the government while this may be communist (elite form government control wealth) it is not in the spirit of communism but is actually the spirit of capitalism.

Second ignoring the argument about other countries, i dont believe you can say you really earned your wealth without exploitation until we have a redistribution of hte wealth in a way that is fair because otherwise you are exploiting the people who were born poor, didnt have any opertunites, now need to eat, and therefore must accept your offer (unless a better one is made but without resources to start with finding better offers maybe much ahrder than for you or me).  We live off the exploitation of others that will not be undone unless we give everyone equal oppertunities and this doesnt happen without government aid.  I know there are many problems with this idea but I think that we should take all the worlds wealth (including land) and divide it evenly amongst all the people than we let people do what they want (still have laws) but now all your wealth is really yours.  I know that idea is full of flaws (such as stock redistribution isnt as easy because microsoft stock is much better than its worth in toothpicks etc how to deal with people who have lived in this world for a while and hence will still have advantages and disadvantages)

It's important to distinguish between "pure" communism and "hyperestablished" communism.

"Pure" communism is socialism willingly entered into by all members of the society involved. In it, all resources are willingly shared among everyone as needed.  it is essentally anarchy (but not in the negative sense). I like to call this an "agapic (selfless) anarchy", and it is important to note that it can exist within a higher governmental system (though it does not require one). The only example of this form of communism that I can provide is the early Christian Church at Jerusalem (see the record in the book of Acts in the Bible). I consider the agapic anarchy to be the ideal form of government, but it is of course unfeasible where people look out for themselves first, before others.

"Hyperestablished" communism is the form seen in the former USSR etc. It is establishled by force from above (hence the newly-coined term), and is often not completely voluntary. It often becomes totalitarian etc. I am not sure of whether it has ever been established democracily.

You must surely understand that the bourgeoisie will not enter a communist system wilfully. At least the majority of them won't (which a glance at this forum is enough to convince anyone). The Paris commune did the mistake of pardoning all the class enemies. After the counter revolution, these very people slaughtere fourteen thousand communists in the streets of Paris. If the bourgeois legacy remains, the dream of anarchy will never come true. The force is not to establish communism, but to attack the class enemies. Once they are taken care of for good, communism will come automatically through conscious evolution. My father rejects this attitutde of ours as 'elitist', but it can't be helped.


- If $ a,b \in (0,1)$,prove the inequality:

$ \log_a \dfrac{2ab}{a\plus{}b}*\log_b \dfrac{2ab}{a\plus{}b}\ge1$

[hide="solution"]WLOG $ a \ge b$.

So,
$ \log_{a}\frac{2ab}{a\plus{}b}*\log_{b}\frac{2ab}{a\plus{}b} \ge {(\log_{a}\frac{2ab}{a\plus{}b})}^2$

But $ \frac{2ab}{a\plus{}b} \ge b$.

which implies
$ {(\log_{a}\frac{2ab}{a\plus{}b})}^2 \ge {(\log_a b)}^2$

and since $ a \ge b$
$ {(\log_a b)}^2 \ge 1$

I hope I have not made any mistake.[/hide]

[hide="my solution featuring wayy more length"]
Let $ m,n\geq 1$.
We have $ \frac {\log m + \log n}{2}\geq \sqrt {\log m \log n}$ by AM-GM. Then
\[ \frac {\log mn}{2}\geq \sqrt {\log m\log n}
\]
Further, we have again by AM-GM that $ \left(\frac {m + n}{2}\right)^2\geq mn$, and that $ mn\geq 1$; therefore
\[ \log \left(\frac {m + n}{2}\right)^2\geq\log mn
\]
so
\[ \frac {\log\left(\frac {m + n}{2}\right)^2}{2}\geq \frac {\log mn}{2}\geq\sqrt {\log m\log n}
\]
\[ \frac {2\log\left(\frac {m + n}{2}\right)}{2}\geq \frac {\log mn}{2}\geq\sqrt {\log m\log n}
\]
or
\[ \left(\log\frac {m + n}{2}\right)^2\geq \log m\log n
\]
Dividing both sides by $ \log m\log n$, we have
\[ \frac {\left(\log \frac {m + n}{2}\right)^2}{\log m\log n}\geq 1
\]

\[{ \left(\frac {\log \frac {m + n}{2}}{ - \log m}\right)\left(\frac {\log \frac {m + n}{2}}{ - \log n}\right)}\geq 1
\]

\[{ \left(\frac {\log \frac {m + n}{2}}{\log \frac 1m}\right)\left(\frac {\log \frac {m + n}{2}}{\log \frac 1n}\right)}\geq 1
\]
By change-of-base,
\[{ \left(\log_{\frac 1m} \frac {m + n}{2}\right)\left(\log_{\frac 1n} \frac {m + n}{2}\right)}\geq 1
\]

\[{ \left( - \log_{\frac 1m}\frac 2{m + n}\right)\left( - \log_{\frac 1n} \frac {2}{m + n}\right)}\geq 1
\]

\[{ \left(\log_{\frac 1m}\frac {\frac {2}{mn}}{\frac {m + n}{mn}}\right)\left(\log_{\frac 1n} \frac {\frac {2}{mn}}{\frac {m + n}{mn}}\right)}\geq 1
\]

\[{ \left(\log_{\frac 1m}\frac {2\frac 1m \frac 1n}{\frac 1m + \frac 1n}\right)\left(\log_{\frac 1n} \frac {2\frac 1m \frac 1n}{\frac 1m + \frac 1n}\right)}\geq 1
\]
Now let $ m = \frac 1a$, $ n = \frac 1b$. We can do this since $ a,b$ are between 0 and 1 exclusive; thus $ \frac 1a ,\frac 1b \geq 1$. We then have:
\[{ \left(\log_{\frac a}\frac {2ab}{a + b}\right)\left(\log_{\frac b} \frac {2ab}{a + b}\right)}\geq 1
\]
Which is the inequality we were asked to prove, as desired. $ W^5$.
[/hide]

[quote="solafidefarms"][hide="a problem with your solution"]
Let a=9/16,b=27/64; we have then in your last statement $ (\log_{9/16} 27/64)^2\ge 1$, but $ \log_{9/16} {27/64} \equal{} \frac 23$.
[/hide]

I haven't actually solved the problem myself, sorry.[/quote]

The log you wrote actually amounts to $ {3\over 2}$


topology
advanced fields
advanced fields unsolved

This seems so easy and intuitive, but I can't seem to put my hands on a proof of this: If $(X\times Y, T_{1}\Pi T_{2})$ is a product space (defined usually), show that $\{U\times V: U\in T_{1}\text{and}V\in T_{2}\}$ is a basis for this space.

I tried to work with the basis we know is the case $T_{1}\Pi T_{2}=<\{\pi_{1}^{-1}(U_{0})\cap \pi_{2}^{-1}(V_{0}): U_{0}\in T_{1}, ~V_{0}\in T_{2}\}>$, then picking an open set $E\in T_{1}\Pi T_{2}$ and a point in this set $x_{1}\times x_{2}$ and showing that there exists a $U \times V$ containing $x$ that sits inside $E$. This may be intuitively trivial, but I can't think of a convincing way to say that.


Let $x_0=7$ and
$x_1=\sqrt{x_0+\sqrt{x_0+\sqrt{x_0+\sqrt{x_0+...}}}}$
and
$x_{k+1}=\sqrt{x_k+\sqrt{x_k+\sqrt{x_k+\sqrt{x_k+...}}}}$
What is
$\lim_{k\rightarrow{\infty}}{x_k}$
?

[quote="amirhtlusa"]$\frac{1+\sqrt{29}}{2}$? ;)[/quote]

well, i am not sure at all but i think that it is smaller than that because $\frac{1+\sqrt{29}}{2}$ is just $x_1$. Then if you take nested square roots of that, you'll end up with something smaller...

[hide]
$x_{k+1} = \sqrt{x_k + x_{k+1}}$.  Let $l$ be the limit.
$l = \sqrt{l+l} = \sqrt{2l}$.
$l^2 = 2l$
$l = 2$.

So the limit is 2. Note that it is independent of $x_0$.[/hide]

Nice problem. :)

You have to be a little careful though. By setting something that's infinite (for example a continued radical, or the limit of a sequence, etc.) to something else, you're assuming a priori that it converges.

But you really need to establish at some point that it does in deed converge. Otherwise all you've shown is that [i]if[/i] it converges, then it converges to that particular value. Proving convergence can sometimes be a much more difficult problem though.

[quote="Treething"]Erm, it has to be more than 3 :P .[/quote]

I don't know where you got that from, but the term $x_2$ is already less than 3.  The term $x_6$ is approximately 2.00415, rather much closer to 2 than 3.


[quote="Suh Joon-Choi."]:D

When $ \frac{1}{2} \leq x,y,z\leq1$  ,

prove that

$ (x\plus{}y\plus{}z)^2  ( xy\plus{}yz\plus{}zx)^2 \plus{} 11(x\plus{}y\plus{}z)(xy\plus{}yz\plus{}zx)xyz \geq  6xyz (x\plus{}y\plus{}z)^3   \plus{} 18x^2y^2z^2$[/quote]
$ (x\plus{}y\plus{}z)^2  ( xy\plus{}yz\plus{}zx)^2 \plus{} 11(x\plus{}y\plus{}z)(xy\plus{}yz\plus{}zx)xyz \geq$
$ \geq6xyz (x\plus{}y\plus{}z)^3   \plus{} 18x^2y^2z^2\Leftrightarrow$
$ \Leftrightarrow\sum_{cyc}(x^4y^2\plus{}x^4z^2\plus{}2x^3y^3\minus{}4x^4yz\plus{}x^3y^2z\plus{}x^3z^2y\minus{}2x^2y^2z^2)\geq0\Leftrightarrow$
$ \Leftrightarrow\sum_{cyc}(x\minus{}y)^2(2z^4\minus{}x^2y^2\plus{}z^2xy)\geq0.$
Let $ x\equal{}b\plus{}c,$ $ y\equal{}a\plus{}c$ and $ z\equal{}a\plus{}b.$
Then $ a,$ $ b$ and $ c$ are non-negative numbers and use the SOS's reasonings.
It's ugly but works. :wink:

[quote="Suh Joon-Choi."]:D

When 1/2 <= x,y,z, <=1  ,

prove that

(x+y+z)^2  ( xy+yz+zx)^2 + 11(x+y+z)(xy+yz+zx)xyz  >=  6xyz (x+y+z)^3   + 18x^2y^2z^2[/quote]
I think  $ x,y,z$ be positive real numbers  :D
ineq$ \Leftrightarrow (x \minus{} y)^2S_{z} \plus{} (y \minus{} z)^2S_{x} \plus{} (z \minus{} x)^2S_{y}\ge 0$
in there
$ S_{z} \equal{} z^2(x \plus{} y \plus{} z)^2 \minus{} \frac {xyz(z \plus{} 3x \plus{} 3y)}{2}\ge z(x \plus{} y \plus{} z)(z(x \plus{} y \plus{} z) \minus{} \frac {3xy}{2})$ (Because $ 3(x \plus{} y \plus{} z)\ge z \plus{} 3x \plus{} 3y$)
$ S_{x}\ge x(x \plus{} y \plus{} z)(x(x \plus{} y \plus{} z) \minus{} \frac {3zy}{2})$
$ S_{y}\ge y(x \plus{} y \plus{} z)(y(x \plus{} y \plus{} z) \minus{} \frac {3xz}{2})$
$ x \equal{} Max(x,y,z)$
Easy to prove
$ S_{x},S_{x} \plus{} S_{y},S_{x} \plus{} S_{z}\ge 0$
So ineq was proved  :blush:


trigonometry
inequalities
AMC
USA(J)MO
USAMO
inequalities unsolved

Hello Again!

Prove that for angles $ A$, $ B$, and $ C$ in a triangle

$ (\sin{\frac{A}{2}}\plus{}\sin{\frac{B}{2}})^{2}\plus{}(\sin{\frac{B}{2}}\plus{}\sin{\frac{C}{2}})^{2}\plus{}(\sin{\frac{C}{2}}\plus{}\sin{\frac{A}{2}})^{2} \le 3$.

Thanks

[hide]By Rearrangement,

$ \sum\left(\sin\frac{A}{2}\plus{}\sin\frac{B}{2}\right)^2$

$ \equal{}2\sum\sin^2\frac{A}{2}\plus{}2\sum\sin\frac{A}{2}\sin\frac{B}{2}$

$ \le 2\sum\sin^2\frac{A}{2}\plus{}2\sum\sin^2\frac{A}{2}$

$ \equal{} 4\sum\sin^2\frac{A}{2}$

By Jensen's,

$ 4\sum\sin^2\frac{A}{2}\le4\cdot3\cdot\sin^2\frac{\pi}{6}\equal{}3$

[/hide]

i'm sorry to dissapoint you but $ \sum \sin^2\frac {A}{2}\geq \dfrac{3}{4}$... this inequality is equivalent to usamo 01 prob. 3... you can try this too:

perform the substitution $ \sin\frac {A}{2} = \sqrt {\dfrac{yz}{(x + y)(x + z)}}$... then, we have to prove that

$ \sum \dfrac{2yz}{(x + y)(x + z)} + \dfrac{2x\sqrt {yz}}{(y + z)\sqrt {(x + y)(x + z)}}\leq 3$.

note that
$ \dfrac{2x\sqrt {yz}}{(y + z)\sqrt {(x + y)(x + z)}} = \dfrac{2x\sqrt {y(x + z)z(x + y)}}{(x + y)(x + z)(y + z)}\leq\dfrac{x(xy + xz + 2yz)}{(x + y)(x + z)(y + z)}$

finally, note that $ \sum \dfrac{2yz}{(x + y)(x + z)} + \dfrac{x(xy + xz + 2yz)}{(x + y)(x + z)(y + z)} = 3$, and we're done  :D

[quote="mathgeniuse^ln(x)"]Prove that for angles $ A$, $ B$, and $ C$ in a triangle

$ \left(\sin{\frac {B}{2}} \plus{} \sin{\frac {C}{2}}\right)^{2} \plus{} \left (\sin{\frac {C}{2}} \plus{} \sin{\frac {A}{2}}\right)^{2} \plus{} \left(\sin{\frac {A}{2}} \plus{} \sin{\frac {B}{2}}\right)^{2} \le 3.$[/quote]$ 3 \minus{} \sum \left(\sin{\frac{B}{2}} \plus{} \sin{\frac{C}{2}}\right)^2 \equal{}\sum \tan{\frac{B}{2}}\tan{\frac{C}{2}}\left(\cos{\frac{B}{2}}\minus{}\cos{\frac{C}{2}}\right)^2\geq 0.$

This inequality was proposed 19 years ago :

Ji Chen, Zhen Wang, Problem 220, [url=http://menet.math.ecnu.edu.cn/teaching/index.html]Shuxue Jiaoxue[/url] (Chinese), 1990, No. 3, 38 and 1990, No. 4, 41-42.

See also: http://www.mathlinks.ro/viewtopic.php?t=242015


geometry
3D geometry
octahedron
probability
number theory
relatively prime

The numbers 1, 2, 3, 4, 5, 6, 7, and 8 are randomly written on the faces of a regular octahedron so that each face contains a different number. The probability that no two consecutive numbers, where 8 and 1 are considered to be consecutive, are written on faces that share an edge is m/n where m and n are relatively prime positive integers. Find m+n

Can someone help me with this?

Here are two threads on it.

[url=http://www.artofproblemsolving.com/Forum/viewtopic.php?highlight=number+adjacent+faces+octahedron&t=7001]Thread 1[/url]

[url=http://www.artofproblemsolving.com/Forum/viewtopic.php?highlight=number+adjacent+faces+octahedron&t=64603]Thread 2[/url]