Mathematics - 2017-06-03 (page 1 of 2)

Meow Mix

01:05

Hi

Eric Stucky

g'morning

Ivan Ivanov

01:34

Hello

Is it possible CFE to have terminals like S->P , P->x

I know it can be written as S->x directly but is this wrong?

Zee

02:07

The smell of rotten dreams in the air...

arctic tern

04:05

Call a 2-subset of {1,...,2n} a "duad." Call a collection of n disjoint duads (i.e. a partition of {1,...,2n} into duads) a "syntheme." Call a partition of (2-subsets of {1,...,2n}) into 2n-1 disjoint synthemes a "synthematic total," or "total" for short. How many totals are there, as a function of n? For n=3, the standard answer is 6, which is related to the construction of the (unique) outer automorphism of S_6.

Semiclassical

Hmm. Now I want to make some pun on 'duad-enom'

SBM

@Semiclassical no; intestinal issue?

Semiclassical

I didn't say it would be a good pun.

SBM

@Semiclassical oh

Automorphism?

Semiclassical

shrug

Abhishekstudent

04:29

I am a bit confused with this limt

$$ \lim_{x\to 0}[\frac{tanx}{x}] $

$$ \lim_{x\to 0}[\frac{tanx}{x}] $$

where [] r5epresents gif

or greatest integer function

I am unable to understand how the answer comes out to be 1

I think that the answer should be 0 and 1

@Semiclassical

@SBM

SBM

@Abhishekstudent

At x=0 tan x ≈ x right?

Abhishekstudent

yes

SBM

So what happens if you divide two things that approach the same value and take limits

Abhishekstudent

1

SBM

You get the limit is one, right?

Abhishekstudent

04:36

yes

BUT, WHY ISNT THAT THE CASE WITH SINX

arctic tern

for x close to 0, tan(x)/x is in between 1 and 2, so [tan(x)/x]=1

Abhishekstudent

when I am doing this by replacing tanx with sinx

arctic tern

for x close to 0, sin(x)/x is in between 0 and 1, so [sin(X)/x]=0

BAYMAX

1

Q: Which of the following sets in $\mathbb R^2 $ have the positive lebesgue measure?

Which of the following sets in $\mathbb R^2 $ have the positive lebesgue measure? For twe sets $A,B \subset \mathbb R^2$,$A+B$={$a+b:a\in A,b\in B $} 1)S={$(x,y):x^2 +y^2=1$}. 2)S={$(x,y):x^2 +y^2<1$}. 3)S={$(x,y):x=y$}+{$(x,y):x=-y$}. 4)S={$(x,y):x=y$}+{$(x,y):x=y$}. Since lebesgue measure...

Abhishekstudent

ok

when, x<0 tanx < x

arctic tern

04:38

and both are negative

so tan(x)<x divided by x (when x<0) gives tan(x)/x>1

I mean, tan(x) and x are odd functions, so tan(x)/x is even.
if you know tan(x)/x>1 for x>0, then you know it for x<0 too.

Abhishekstudent

op

ok

Leaky Nun

04:52

Is $S_\omega$ generated by 2 elements? What about $S_{\omega_1}$?

arctic tern

$S_{\omega}$ is uncountable isn't it?

(bijections $\Bbb N\to\Bbb N$, effectively)

unless it's supposed to be permutations with finite support,
in which case no it's not generated by 2 elements

BAYMAX

05:17

I think in the above question I posted in, the answers both have differet answers for 3 rd question ,

also how the measure of a circle is positive

it can be thought of as a curve (closed curve)of finite length

which must have measure zero

what is the Lebsegue measure of a unit circle

$$ ZERO $$

Leaky Nun

@arctictern no finite support; why not?

oh, simply because it is uncountable...

arctic tern

yes

Leaky Nun

05:33

that's... an interesting proof

arctic tern

heh

Leaky Nun

it's cheating

SBM

hmm

arctic tern

Why isn't a mouse an elephant? You could look at body hair or tail color, but the most obvious distinguishing feature is size.

BAYMAX

Now I remember movie SING and Ratatouille

Balarka Sen

05:45

@arctictern lol

Daminark

Hallo

SBM

Hello

BAYMAX

hololulu @Daminark

Daminark

(Kek @Baymax) How's it going everyone?

BAYMAX

gong rough and smooth for me :)

Daminark

05:56

Smooth, but what about analytic?

SBM

Hope things get better soon @BAYMAX

Daminark

Lol for real though good luck!

BAYMAX

thanks

SBM

When is a function analytic?

Balarka Sen

if there is a convergent power series expansion at every point

SBM

06:01

oh

BAYMAX

or may be differentiable at that point and there exists some neighborhood in which the function is differentiable at all the points in that neighbothood !

Balarka Sen

that's not equivalent to being analytic

you're thinking of complex functions and complex differentiability

Daminark

Baymax, I imagine you're thinking of complex differentiable

Balarka Sen

sniped

Daminark

Wai does this happen to me?

BAYMAX

06:04

yeah daminark

Daminark

I think you need to be careful about real analytic and complex analytic

Also @Balarka met Farb today

He's very fun

BAYMAX

yes

Daminark

(Also very bouncy)

Balarka Sen

heh

BAYMAX

so we cannot tell real analytic at a point in terms of differentiability?

like is there only power series criterion?

Daminark

06:07

Yeah, real differentiability is a very weak condition

You don't know as much as smoothness

SBM

smooth? differentiable infinitely everywhere?

arctic tern

differentiable k times does not imply differentiable k+1 times (any more than continuous implies differentiable), and being differentiable k times for all k (smooth) does not imply analytic (for real functions)

Balarka Sen

real differentiability is very not the same as complex differentiability

the latter, despite similarities, is secretly a PDE condition

BAYMAX

he he Cauchy Riemann equation

Zee

It's just simply the result of the CR equation

SBM

06:09

@BalarkaSen Cauchy Riemann equations

Balarka Sen

The complex differentiability => analytic proof is secretly a bootstrapping proof

Daminark

Sniping left and right

@Balarka that's via the Cauchy integral formula, right?

BAYMAX

hits all secretly with a LAW 80 :)

Balarka Sen

@Daminark Yeah

BAYMAX

google.co.in/…

Who watched Wonder Woman?

Zee

06:12

What is the weakest condition on a complex function for it to be complex differentible?

Daminark

It's tricky to say which is weakest

Zee

Well, continues implies diff in complex functions

Daminark

Continuous? That's not really true

Zee

sure its true

Balarka Sen

lol

Daminark

06:14

Wat?

No

arctic tern

sigh

walked into that

Balarka Sen

#trolled

Daminark

is confuzzled

arctic tern

the weakest condition would be those conditions equivalent to the condition, including the condition itself.

any weaker and it wouldn't imply the condition

#helpful

Daminark

That is a true statement

So we'll say analyticity

:P

Zee

06:18

en.m.wikipedia.org/wiki/Looman–Menchoff_theorem

Daminark

The hell?

Zee

I thought the word "complex" implied CR equation

To rephrase , a CR complex function that is continues is analytic

Looman menchof theorem

Daminark

Oh well if we're assuming it satisfies CR that's another story entirely

I thought you were saying any continuous function period, and was like whoa hold on a second wat?

Balarka Sen

complex function just means f : C --> C

SBM

yes

Zee

06:23

I realize that, I just wasn't aware I need to spell every detail

Balarka Sen

you don't, but you do need to use standard terminology

Daminark

Like, your statement as stated was wrong

Zee

Sure but this is an informal discussion

Anyway, my original question holds

Daminark

Yeah but you need to be clear about what you're saying

Zee

How far can we weaken a CR function to imply smoothness?

Daminark

06:28

Hmm

Zee

You can go as far as distributions but I don't if that's the weakest condition

Alessandro Codenotti

Isn't a smooth complex function also analytic?

SBM

should be

Zee

Ya

Alessandro Codenotti

So you can't weaken CR to imply smoothness because that'd imply analyticity which implies CR or am I missing something?

Zee

06:33

Am asking the question assuming CR holds

You can't weaken CR since being differentible (even once) implies CR

Alessandro Codenotti

I don't understand the question, assuming CR you already have smoothness

Zee

Only if the function is continues , the continuity can be weakened, the question is by how much?

Leaky Nun

@arctictern what about with finite support?

arctic tern

@LeakyNun Given any finite number of permutations with finite support, every element in the subgroup they generate will have support contained in the union of the originals' support, so the subgroup generated must be finite.

Leaky Nun

@BAYMAX 1:0, 2:pi, 3:infty, 4:0

@arctictern thanks

Might I say that the 3 group axioms are too strong?

BAYMAX

06:50

why 4th is zero having only the line $y = x$ but 3rd is infity ?

@LeakyNun

Leaky Nun

For example, I only need existence of left-identity and left-inverse

@BAYMAX 3 is the whole plane

arctic tern

@LeakyNun too strong for what?

BAYMAX

oh

Leaky Nun

in other words, span{(1,1),(1,-1)} = R^2, if you are familiar with linear algebra

@arctictern well, itself.

BAYMAX

yes

4th is zero

Leaky Nun

06:52

yes

BAYMAX

?

why he took the sme set added with itself?

Leaky Nun

yes, 4th is 0.

arctic tern

@LeakyNun logical strength of hypotheses is not the only consideration in a definition - it should also match our intuitive idea of what is being defined.

Leaky Nun

@BAYMAX I would have no idea.

arctic tern

groups distill the abstract properties of symmetry groups and permutation groups

BAYMAX

06:53

but the last set means only the line $y = x$

Leaky Nun

@arctictern sure. Can we make it weaker still?

BAYMAX

?

Leaky Nun

@BAYMAX yes.

arctic tern

@LeakyNun I recall reading you can ensure a binary operation defines a group with a single axiom.

BAYMAX

so measure of the line being zero area so measure is zero

Leaky Nun

06:54

What is the dimension of the trivial vector space?

@arctictern yes, that ab' in G for each a and b in G.

@BAYMAX yes

BAYMAX

Thanks@leaky

Daminark

Well, this is ensuring that a given set is a subgroup, but you would still need to know that there's some identity lurking somewhere, and that inverses exist

arctic tern

@LeakyNun uh, no, that doesn't imply associativity

Daminark

So I'm not sure that this is really saying less

Also yeah associativity :P

Leaky Nun

@arctictern then what is it?

arctic tern

06:56

@LeakyNun you mean what is the algebraic structure with a binary operation having identity and two-sided inverses?

Leaky Nun

@arctictern no, I mean what is the single axiom

arctic tern

trying to find it again

Leaky Nun

Is there any result regarding the solutions of x^2 + y^2 = 1 in a ring, say Z_m?

arctic tern

break it into parts with CRT, lift from Z_p with Hensel maybe

@LeakyNun If a magma M satisfies a=b((((dd)a)c)(((dd)b)c)) for all a,b,c,d then a((bb)b) is a group operation on M.

(And there is a converse, so it is an equivalent characterization of groups.)

Leaky Nun

07:16

@arctictern wat

arctic tern

yeah, exactly

SoumyoB

07:35

soundcloud.com/soumyo-biswas/hot-alkali-metal-beginning-only

^I just made this song

anyone care to tell me if it has any errors?

Samuel Schlesinger

07:55

hello, I'm curious if anybody is aware of any first order statements one can make (say about graphs, but any vocabulary suffices) which is a tautology in all finite models but is not in all infinite models

for instance "every DAG has a sink"

this is clearly true in all finite DAGs but not in infinite ones

user147690

08:22

@SoumyoB Sounds good to me. Everything felt like it fit together well, so I imagine there were no 'errors' (although I'm not entirely sure in regard to that, what I should have been looking for).

Mats Granvik

08:43

Associated with Math.SE; for both general discussion & math questions alike. Just ask; don't ask to ask. Rarely if ever expressible as a ratio of integers. Participants are required to keep it real at all times. Chat guidelines: tinyurl.com/hzl2955 | $\LaTeX$ in chat: tinyurl.com/cfqcvpc

There are other rooms, with 71 users currently talking in 58 rooms. Wink, wink, nudge, nudge.

Astyx

09:19

Is the preimage theorem the same thing as the implicit function theorem ? Or am I missing something ?

Alessandro Codenotti

What's the preimage theorem?

Wiki says it's a variation of the implicit function theorem en.m.wikipedia.org/wiki/Preimage_theorem

Balarka Sen

@Astyx The two are equivalent. Preimage theorem says $f^{-1}(0) \subset \Bbb R^n$ is a $(n-m)$-dimensional submanifold where $f : \Bbb R^n \to \Bbb R^m$ is a smooth map with $Df(0)$ is surjective.

This is equivalent to saying for any $p$ with $f(p) = 0$ you can write $f = 0$ locally in a nbhd $U \subset \Bbb R^n$ of $p$ as a graph.

Astyx

09:37

Right, that's what I thought

Thanks

Alessandro Codenotti

10:01

Is there a way to use wolframalpha to calculate the residue of a function at a point? Or any other software that can calculate it?

SBM

try it

should work probably

Alessandro Codenotti

Oh, lol, it just worked, I must have been half asleep when I couldn't get wolframalpha to calculate it yesterday evening

SBM

hmm

False Promise

10:24

Guys can you help me with one integral? I know the solution, but i can't prove it strict enough.

SBM

@FalsePromise Which integral?

False Promise

@SBM $int_0^1 \frac{\sin (1/x)}{(sqrt{x} - x)^k}$ and i want to investigate in on convergence absolute convergence.

I'll upload the picture

of what i've done so far

so I think i have to use Abel's and Dirichlet's tests here

So i think i want to calculate the power of y herr

*here

and then say something about its asymptotics

also here alpha is any real number

so in the denominator the max power of y is alpha/2

so together with numerator we get alpha/2 -2

Secret

[Random, made it up just for fun]
The mathematics of circle glyphs:

$\bigcirc\bigcirc=\large{\bigcirc}$

$\bigcirc^2=\small{\bigcirc}$

${\Large\circledcirc}=\circ$

$\circ^\circ=\bigcirc$

${}^{\bigcirc}\circ=\circledcirc$

$\circledcirc\bigcirc=\mathbf{O}$

${}^{\bigcirc}\circ\bigcirc={\Huge\bigcirc}$

${\huge\bigcirc}{\large\bigcirc}=\circledcirc^2$

Fargle

10:40

Ya lost me

Secret

(hence the "made it up just for fun") It's not supposed to mean anything. I just happened to play with some falling sand like simulation game and got a pretty circlular pattern and thus I expressed that emotions by starting to draw and type circles randomly

I really like round things. It might have something to do with how round things tend to tie with the concept of infinity and recursions

but of course, if I do put an effort on that, I may be able to came up with an algebraic structure where it is expressed entirely using circles as the operators and symbols, but this is not my current priority

Abdullah UYU

11:44

$x^2+4xy+ky^2+2x-2y-5=0$ indicates a parabola. $k$ has to be 4. why? i tried to reform the equation to liken $y^2=4cx$ or something like that. but it has a term $4xy$. it is impossible. i plotted it in geogebra. it is not a parabola that is parallel to $x$ or $y$ axis.

SBM

A parabola doesn't always need to be parallel to the x or y axes

Abdullah UYU

yeah, i observed it :)

user379685

Hello, could someone help me calculate the volume between: plane z=0, cylinder $x^2+y^2=8x$ and cone $2z=sqrt(x^2+y^2)$ ?

SBM

hmm

user379685

12:14

hello, anyone?

Astyx

What have you tried ?

user379685

i just remembered i should try to convert into cylindrical coordinates

Astyx

What do you get ?

user379685

12:30

64pi?

Astyx

Perhaps, I haven't done the calculations

I was asking what you get once you use cylindrical coordinates

SBM

hi chat

Astyx

Hi

SBM

cylindrical coordinates?

Astyx

$(r, \theta, z)$

BAYMAX

12:32

0

Q: dimension of subspace 8

Let $W_1 ,W_2 ,W_3 $ be three distinct subspaces of $\mathbb R^{10}$such that each has dimension 9. $W=W_1\cap W_2\cap W_3 $.What will be dimension of W? I think dimension of $W$ will be 7 since $W_1 ,W_2 ,W_3 $ are distinct.But answer is dimension of $W$ could be 8 as well. According to my reas...

matrices vector-spaces

$W_{1}$ is of dimension2

$W_{2}$ is of dimension 2

$W_{3}$ is of dimension 2

in the answer

Astyx

Yeah ...

SBM

@Astyx polar with the $z$ from cartesian?

Astyx

@SBM Yeah, that's one way to see it

BAYMAX

And dim$(W_{1}\cap W_{2}\cap W_{3} ) = 1$

but what do we conclude from this regarding the question ?

are we generalizing this

?

SteamyRoot

It's an example to show how you get can dimension $n-2$

BAYMAX

12:36

oh

ok so we will generalize this

SteamyRoot

Woops, yeah

BAYMAX

to 10 dimension

like we can get intersection to that as 8

SteamyRoot

We will indeed, but that's pretty much trivial.

Astyx

You can add as many dimensions as you want to these subspaces

BAYMAX

yes

Is manifold a surface in 3D or is there something more to it?

:)

Astyx

12:38

A 2-manifold yeah

A curve is a 1-manifold

BAYMAX

oh like if we are talking about n dimensional space then that maniflod is $n-1$ manifold

?

Fargle

@BAYMAX It's not about the space it's in, it's about the topology of the manifold itself.

Astyx

Formally it's something in $\Bbb R^n$ that locally looks like $\Bbb R^{n-1}$

Fargle

The circle is a 1-manifold, whether you're in $\Bbb R^2$ or $\Bbb R^{20}$

BAYMAX

ha nice one there

Astyx

12:41

I guess intuitively speaking a $n-1$-dimensional manifold in $\Bbb R^n$ is a "hull"

Abdullah UYU

when $\Delta$ of a conic less than zero, can it refer to a hyperbola? i guess i misspell it on my notebook.

BAYMAX

hey @Pjotr5 I like Pandas :)

No @AbdullahUYU I think that should be ellipse

Hey@Astyx @Fargle @SteamyRoot you heard of Slow Manifolds?

Astyx

No, I've barely started reading about them

SBM

@AbdullahUYU $\Delta$ of a conic?

Astyx

By the way is the open unit ball a manifold ?

Abdullah UYU

12:52

yeah, that was sloppy.

SBM

@Astyx ball? interesting.

BAYMAX

If the equation of the conic is $ax^2 + 2bxy + cy^2 + dx + ey + f = 0$ then $\triangle = b^2 -ac$

@SBM

If $\triangle < 0$ then the conic represents an ellipse

SBM

oh that

yes

BAYMAX

If $\triangle > 0 $ then the conic represents Hyperbola

Abdullah UYU

i was going to write it, baymax did it

BAYMAX

12:54

$If \triangle = 0 $ then the conic represents Parabola

SBM

$\Delta > 0$ hyperbola

yes

Alessandro Codenotti

@Astyx isn't that $\Bbb R^n$?

BAYMAX

ok @Astyx if you are reading slow manifolds, can we discuss it together sometime?

Astyx

@AlessandroCodenotti Probably, I'm brain checking. But the closed ball isn't one right ?

@BAYMAX Sure. It might not be anytime soon though

BAYMAX

oh ok

Alessandro Codenotti

12:57

@Astyx the closed one is a manifold with boundary

With boundary means that some points have nbhds that look like the closed half space of $\Bbb R^n$ rather than $\Bbb R^n$

Astyx

Oh right, fair enough, I didn't know about those

Makes sense

And if $H$ is a hyperplane of $\Bbb R^n$, then $\Bbb R^n\setminus H$ is a manifold (something like $\Bbb R^n\times\{0,1\}$) ?

BAYMAX

I have a ps file , any one knows how to open it ?

like .ps extension

google search gives me some pdf creator then I can view

anyone encountered before?

Alessandro Codenotti

@Astyx Some people require manifolds to be connected, but I don't think that's very popular? Anyway I agree, that should be the disjoint union of 2 $\Bbb R^n$

Isn't that a photoshop file?

Astyx

No, postscript

Photoshop is psd

I don't know about postscript though

Pjotr5

@BAYMAX Me too :)

BAYMAX

13:08

:)

SteamyRoot

13:36

ps is a bit like pdf's predecessor

BAYMAX

so like how can I open it ?

you know @SteamyRoot

SteamyRoot

Just like pdf is djvu's predecessor (/wishful thinking)

Well, google for a "ps to pdf converter" or a "ps reader", I guess

SoumyoB

@AlexClark thanks for the review, I was just looking for notes that sounded bad in my song

BAYMAX

ok

Alessandro Codenotti

14:03

Another student in Trento?! Hi @Gabriele

Vrouvrou

14:22

Hello

how are you ?

Martin Sleziak

@BAYMAX GSview is probably the best known software for opening PostScript files.

GabrieleCaselli

@AlessandroCodenotti Hi! Yes, another one here :) Master or Bachelor's?

Martin Sleziak

@SteamyRoot I do not know much about the two formats. From my experience it seems that djvu is much better for scanned content, but for other purposes I would probably prefer pdf.

Alessandro Codenotti

@GabrieleCaselli just finished the second year of my Bachelor's (as far as lectures as concerned, now there a few exams...)

I see you're a Master's student instead

GabrieleCaselli

@AlessandroCodenotti Yes, right! Just finished my Master's first year, which is also my first year here in Trento

Vrouvrou

14:41

no one say to me Hello !!!

>_<

Astyx

14:53

Hi @Vrouvrou

Vrouvrou

hi @Astyx

how are you

Astyx

Good and you ?

I'm glad the weather cooled down a bit

Vrouvrou

good

Astyx

Although the rain isn't too good either

Vrouvrou

En Algérie il fait chaud

Astyx

14:59

Ah tu n'es pas en France ?

Je croyais

Oui j'imagine que c'est pire qu'ici

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$Mathematics$ Mathematics