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12:32 AM
How can I show subgroups of order 6 are conjugate in a group of order 42?
one way is outlined in math.stackexchange.com/questions/3010207/… some of the answer is scattered across the comments
yeah I actually saw that post but it only shows the number of conjugates. But I don't know the number of subgroups of order 6
2 hours later…
2:27 AM
OOP sucks
2:39 AM
1 hour later…
3:46 AM
@Euler2 I disagree, but this is highly opinion-based
OOP can be useful, so many people like it
nothing wrong with that
4:06 AM
what what
I think I proved it
Does exponential decay have to be related to $ln()$? Can't $r^{-x}$ be considered exponential decay? And in the inverse square case, x is a constant ($x=2$) ie. $r^{-2}$?
What do you people think? Actually the post @leslietownes linked was very helpful.
4:43 AM
oop is no panacea but it is useful. i liked dylan's (and lisp, or course) multi methods.
1 hour later…
5:43 AM
@antimony for the first part, yes, and in applications it’s inevitably so. One speaks of exponential decay having some time constant T, such that $y=y_0 e^{-t/T}$. But that can equally well be written as $(e^{1/T})^{-t}=r^{-t}$
For the second, tho, not really. $r^{-x}$ is an exponential function of $x$, but it’s a power-law function of $r$
It being exponential in $x$ is irrelevant if $x$ isn’t changing, eg when $x=2$ to get inverse-square behavior in $r$
@Semiclassical thanks i've revised my view in terms of the definition of 'exponential decay'. however re. power law i think this seems to be a preferred terminology
checking books.google.com seems to show using that cambridge definition of 'exponential' (rather than 'power law') is going out of fashion
and was more popular back in the 1950s etc
*seems to be a matter of preferred terminology rather than incorrect terminology
i would not use a general purpose dictionary for 'exponential,' the word in common english usage has no meaning
or perhaps, too many distinct meanings
exponential decay is as semiclassical describes it
@leslietownes you will note it says "Mathematics Specialised' its not a general definition
but i agree for 'exponential decay' i was wrong i think
it is a general purpose dictionary
5:59 AM
i'm not sure i agree that it is incorrect about one being able to apply 'exponential' to refer to power law type forms
you can stuff logs in if you want. b = e^(ln(b)) is often used to write base-b expressions in terms of base-e expressions
if you have a preferred source of definitions please direct me to it
i don't think dictionaries are very useful in math. the best source of definitions for anything is the materials in front of you, which might not agree with dictionaries or wikipedia or even what someone with a phd in math says is in common use
i think it is the case that exponential to refer to non-natural-exponential power-law type things is going out of fashion, rather than being "incorrect"
there is too much variation in how people use terminology and it isn't practical to list all of them. wikipedia is OK but not always best at signaling when there are multiple definitions
6:01 AM
right, agreed
wikipedia is funny: "The exponent is usually shown as a superscript to the right of the base. In that case, b^n is called "b raised to the nth power", "b raised to the power of n",[1] "the nth power of b", "b to the nth power",[3] or most briefly as "b to the nth"."
weirdly none of those are how i would say it. "b to the n"
yes i think there is a language relic here since n is (still) referred to as the exponent
the page reads like it was written by a chatterbot. fairly common problem with topics applicable to a broad range of contexts. every editor adds or edits one the sentences they are interested in and you get this frankenstein monster of facts
> It being exponential in $x$ is irrelevant if $x$ isn’t changing
@Semiclassical do you pls know where i can read more about that ^ so i can learn what the distinction is?
haha yes @leslietownes
i was wondering where the complex plots were for complex exponents. whenever anything can be graphed in the complex plane, it should be graphed in the complex plane, according to wikipedia. even if those pictures might not tell you anything, or at least not be one of the first 10 things you learn about a concept. but they're stored separately, in he page for "exponential function"
no offense if anyone around here is like the world curator-in-chief of these wikipedia pages
6:12 AM
i think there's a business opportunity for us to publish a Mathematical Dictionary
though i don't think i should be the one to write it lol
i will take a pay cheque, however :)
vandalism in wikpedia is intense
i'm surprised that wasn't auto rolled back
6:44 AM
A: (From Axler)Why does the basis satisfy the required conditions?

KoroThe idea is this: If $T\in L(V,W)$. Let $B=\{u_1,...,u_m\}$ be a basis of $V$ and let $C=\{w_1,...,w_n\}$ the given basis of $W$. Matrix of $T$ with respect to $B$ and $C$ has $Tu_i$'s as columns. Applying elementary column operations on the matrix won't change the column space of $T$ and the co...

can anyone please take a look at my answer here?
3 hours later…
9:32 AM
@antimony How was Euler 2 was able to edit the wikipedia like that?
10:08 AM
Anyone can check my proof who's in interest chat.stackexchange.com/transcript/message/59226872#59226872
10:35 AM
@leslietownes I am against all generalizations.
10:55 AM
can anyone give hint regarding this question math.stackexchange.com/questions/4259698/…
1 hour later…
12:04 PM
Let $N_2 \rtimes G \le N_1 \rtimes G$ be internal semidirect products, where $N_2 \le N_1$. Is there a nice formula for the index $|N_1 \rtimes G : N_2 \rtimes G|$? I think it's equal to $|N_1 : N_2|$; is that right?
I want to check if the following is true : If $n$ is a number of distinct subgroup of order $6$ of some large group $G$, then there are at least $6n-3n+3$ distinct elements in $G$.
suppose the constants a, b, c, d, e, and f are positive. does mean that a,b,c,d,e,f >=0 or a,b,c,d,e,f >0
@Mohcine The second one.
thank you so much
@jasmine I think you should also mention what $(13)$ is.
1 hour later…
1:18 PM
@Prithubiswas anyone can edit wikipedia pages (unless the page is locked), that page is probably locked right now since it was recently edited for mischievous means
@antimony Can you edit the html?
2:04 PM
If all eigenvalues of a matrix $A$ over a field $\Bbb C$ are identical, then $A$ is a constant multiple of identity matrix?
I'll never make this mistake: For any $x\in \mathbb R, x-\sqrt {x^2}$ is equal to $0$.
2:26 PM
the vandalism happened in april 2012
3:28 PM
One question please:
If we have
\mathcal{T}(2^{\log_2 z})=\frac z2+\frac 12\mathcal{T}\left(2^{\log_2{\frac z2}}\right)

now calling $\mathbb{T}(\cdot)=\mathcal{T}\left(2^{\log_2{(\cdot)}}\right)$ and $u=\log_2 z$ we follow with

\mathbb{T}(u)=2^{u-1}+\frac 12 \mathbb{T}(u-1)
Thank you very clear. Last question please, should not we have $\mathbb{T}(2^u)=2^{u-1}+\frac 12 \mathbb{T}(2^{u-1})$ instead of $
\mathbb{T}(u)=2^{u-1}+\frac 12 \mathbb{T}(u-1)$ as I did substitutions based on values you wrote for $u=\log{z}$
4:15 PM
Has anyone seen this definition before: math.stackexchange.com/questions/4260001/…
Literally everyone I asked has not seen this definition of an affine space before, so I'm not able to get help with that problem lol
Hopefully it's just linear algebra
Could someone help?
4:42 PM
it's a definition chase more than it is linear algebra. someone should fill it in pretty quickly. it might help to prove at the outset that d(a,b) = -d(b,a) for any a, b; can you prove that?
I'm confused at affine subset of a vector space V; and affine parallel subset of V?
yeah, the T in the title should be X
I think that both are the same. That is if U is a subspace of V then for any v in V, left coset v+U is affine subset of V
And "affine subset of V" and "affine parallel subset of V" are the same thing.
Background: definition 3.81 in chapter 3.E LADR
actually, something's goofy here.
4:55 PM
what is x + V, if V is a subspace of T
what is T?
I am referring to this definition: 1) An affine subset of V is a subset of V of the form v+U for some v in V and subspace U of V.
x is a "set of points," T is a vector space as in the definition of the problem
how do you add an element of X to an element of T
i'm not saying that there isn't some obvious way to do this, but my internal type checker is thrown by this
2) For v in V, and U a subspace of V , the affine subset v+U is said to be parallel to this. My confusion is: are (1) and (2) different? If not, why were they stated in two different points?
oh, i see, it's defined in terms of this bijection thingy
@leslietownes I think you're referring to some other question. I have not used T here. Are you referring to the question that I asked the other way about finding a basis of U?
4:58 PM
the conditions (1) and (2) to be proved are different, one talks only about T, the other requires the restriction to X' to be bijective (when the definition only gives you the restriction to X being bijective)
i'm thinking out loud about epsilon emperor's question, koro
not whatever you're thinking about
what a misfortune to have X and (1) and (2) having independent meaning in two problems at the same time
Ah, I see. My question has nothing to do with @epsilon's question. I had confusion in definition 3.81 in LADR. I was trying to understand that.
that's a really weird coincidence. his question also has the word affine in it
@shintuku what was the solution to your $y-y^3$ with [-1,1]?
koro, in your setting, (1) defines the term "affine subset", and (2) defines what "parallel to" means, so yes, they're different? they involve the same set of notions, but define different things?
but then what is the difference between "affine subset" and "parallel affine subset"?
5:06 PM
he doesn't define 'parallel affine subset', he defines a relation ("parallel to") between an affine subset and a vector subspace
by his definitions, every affine subset is parallel to a subspace, if that's what you're asking, so "being parallel to a subspace" is not a new property of affine subspaces but one that every affine subspace has by definition
but that's not what he's defining
I ask because just after the above stated definition. There's a section "parallel affine subsets" in which there are two examples.
@leslietownes hmm, I understand now.
Thanks a lot, Leslie. :)
i agree that these definitions aren't "doing" very much, but you do sometimes see this. keeping definitions as simple as possible so they are easier to verify
@leslietownes. Are you familiar with recurrence relations please?
I have one question about some algebric manipulations done in one question
I am very close to get my hands on them, but still they are not easy to work with
Is there a mathematical proof that is not verifiable by any human?
I am just wondering here where did $p$ go inside $2f(m-1)$?
5:20 PM
How can I choose the parametric coordinates of a parabola?
@Wolgwang For $y^2=4ax$, you could choose $x=at^2$ and $y=2at$.
@Koro I have to take $\theta$ as a parameter.
and what is $\theta$?
> The parametric coordinates of any point on the parabola $y^2 = x$ can be
> (A) $\left(\sin ^{2} \theta, \sin \theta\right)$
(B) $\left(\cos ^{2} \theta, \cos \theta\right)$
(C) $\left(\sec ^{2} \theta, \sec \theta\right)$
(D) none of these
Ok. So here, since x can be any non-negative number. The first two are ruled out. (Note that (100, 10) lies on the parabola but sine and cosine have maximum value 1)
and similarly, what can you say about $C$ ?@Wolgwang
5:36 PM
@Koro Its range is $(– \infty, -1] ∪ [1 , \infty)$
@Wolgwang right.
I meant to say any point with abscissa in $[0,1)$
@Koro Thanks :-)
@Wolgwang yeah, so take any such point on the parabola. Is it possible to write that point using $(C)$?
5:55 PM
@Koro Yeah, I didn't notice that
6:46 PM
@Wolgwang The language on that is confusing. From the options, I assume they are looking for a parameterization of the whole parabola, not just points that lie on the parabola (it would have been more confusing if "all of the above" were a choice). In which case, $(0,0)$ is only in (A) and (B), whereas $(4,2)$ is only in (C).
$\left(\tan^2(\theta),\tan(\theta)\right)$ would do.
7:06 PM
Hello, how are you? Are you well now ? @robjohn
Getting better, but still not all that great.
great. I hope you get well soon, professor Rob.
the language in that parabola question is indeed confusing. I think the question asker's intention was: "the parametric coordinates of all points" on the parabola...
Hyello everyone!
Let $I$ be a nonempty directed set, and $H_i \le G_i$ a directed sequence of groups living in some ambient group. Does the following index formula hold: $$|\bigcup_{i \in I} G_i : \bigcup_{i \in I} H_i| = \lim_{i} |G_i : H_i|?$$
For one-one and onto functions can i bypass the usual methods, by saying for any y=f(x) if there exist dy/dx=0 then the function is not onto and one-one? And also I ensure that the point where dy/dx=0 is a local minima/maxima?
7:18 PM
@Koro that was my guess, but it's not good to be guessing.
@AdilMohammed is that a mellow hyello?
@AdilMohammed think of $y=x^3$...
that is a bijection from $\mathbb{R}$ to $\mathbb{R}$.
yet $y'(0)=0$
@robjohn Nah its better, yellow h-yello (we are talking about the pronunciation right lol)
@AdilMohammed no... Mellow Yellow
quite rightly
@leslietownes electrical banana is bound to be the very next phase
7:35 PM
@robjohn ooh the 60s songs... I always picture myself in a hammock when I am listening to them
@robjohn Oh yes true but for one-one, i expanded to see if its local minima/maxima (searching for turning points basically)
here is something i find mildly amusing
for reasons that i will not disclose at the moment, consider function $y=\cos(x)(\sin(x)+\sqrt{\sin(x)^2+a})$ with $a\geq 0$.
If $a=0$, this simplifies to $y=\cos x\sin x$ with max value $y(x=\pi/4)=1/2$
But at a glance, that looks incredibly painful to maximize for arbitrary $a$
And yet, the solution is way nicer than you'd expect: the critical point occurs when $x=\tan^{-1}(1/\sqrt{1+a})$ and yields a max value of $y=\sqrt{1+a}$
i can show that in various ways, including without use of calculus at all. and yet it still just baffles me how the answer can be as simple as it is, starting from such an ugly equation
8:21 PM
how to define operatorname in mathjax?
as in, how to define it so you don't have to keep doing \newoperator{} over and over?
i know i've seen it done
though only on the main site. i think on chatjax it'd be impossible
i was in the middle of posting one question. I wanted to define an operatorname for null as in null T
I got it.
the fix i'm seeing is that \newoperator needs to be done in math mode
so dollar signs around it
8:27 PM
\newcommand{\name} {\operatorname {\name}}
then calling \name works
another option is \DeclareMathOperator
e.g. \DeclareMathOperator{\End}{End}
so that \End would give $\operatorname{End}$
that's specific to making new operators rather than new commands
But somehow, that's showing an error: Mathjax internal buffer size exceeded, is there a recursive macro call?
no idea
operatorname should just use name, not \name
yeah, I noted that. Now there's a strange problem that I just encountered.
when I do this and try calling \null, it prints twice (concatenation) happens
that is, nullnull
8:33 PM
why is it so?
i'm not seeing that error
@Koro you shouldn't have the backslash inside the \operatorname
just as Semiclassical said
yeah, I fixed that but still that concatenation is happening
try reloading the page first
2 mins ago, by Koro
8:34 PM
it looks like \newcommands in mathjax are persistent until the page is reloaded
yeah working now :)
thanks a lot :)
$$u\frac{\partial}{\partial u}\left(u\frac{\partial f}{\partial u}\right) + v\frac{\partial}{\partial v}\left(v\frac{\partial f}{\partial v}\right)=\frac{\partial^2 f}{\partial u^2} + \frac{\partial^2 f}{\partial v^2}\tag{1}$$
is that supposed to be a differential equation?
it's not an identity
yeah it's asking when the two differential equations are equal
if ever
8:47 PM
is it possible to define many operatornames in one go?
mathematica's DSolve command shrugs its hands at it
not with one function, i suspect @koro
what're you trying to define, tho
can you construct a smooth manifold which is a surface built by taking a rotating homotopy that projects onto planar homotopic curves?
plane homotopy has two fixed endpoints
I can only do this for a simple case using rotation matrices to rotate the curve by a certain angle and then projecting down onto the planar curve. then repeating the process
the hard part is if the plane homotopy is say analytic chiral, you have to "deform/twist" the curve s.t. it projects precisely onto the planar curve, probably sacrificing analyticity of the surface
9:04 PM
@Semiclassical in chat, they may be. On the main site, they should only be active for the post in which they were executed.
@Koro you need multiple \newcommands
what i meant specifically was: suppose you define and use a new command in an answer. the wysiwyg editor on the main site will properly format that command
if you delete the definition from the post, the use of the command will persist in the display
but upon reloading the page it won't
yo joe
does anybody know what the probability measure $\mathbb{P}^X$ mean?
9:07 PM
@Koro it works as it should for me.
so if you make an error in creating a mathjax definition, it can persist until you reload the page
ah I see you got it working
I see so multiple \newcommands will be required.
again, what specifically are you trying to do?
I wanted to define operatornames span, null, range etc.
@Semiclassical: I was trying to solve one problem on linear maps. I discussed that problem here also the other day. I have tried to present my understanding here in this answer math.stackexchange.com/questions/1924952/…
I request you to please take a look at it. Thanks.
9:12 PM
ah. yeah, if they're all different things, then gotta define each seperately
I had imagined a matrix and on that used the fact that column operations on a matrix won't affect its column space.
to be clear, i'm only looking at the mathjax :)
ah, okay :)
but the question being: if two matrices have the same null space, then they're equivalent up to an invertible linear transformation?
"they are equivalent up to " ?
9:18 PM
as in, $T_1=ST_2$
I don't know about this terminology yet.
ah, okay. That's the question that I posted today.
koro i don't think axler has anything about row or column equivalence of matrices. a more matrix theoretic book would.
oh, i saw that you added a question and thought that was what you meant
i think he does talk a little bit about change of basis and matrices in a later chapter, but maybe only in the context of operators on a single vector space? memory is hazy about this.
nothing explicit about it, probably
9:20 PM
yes, nothing explicit. every linear algebra book has this under the hood but only some of them will have theorems expressly talking about it.
@Semiclassical math.stackexchange.com/questions/1924952/… I request a review of this answer of mine please.
@leslietownes I think knowing a thing or two about matrices helps immensely to think in terms of linear maps.
some books go the route of, the following are equivalent for a linear map T: V to V. [list of about 20 things that are equivalent to "T is invertible"]. then another one for T: V to W possibly of different dimensions with stuff about rank, nullity, left invertibility, right invertibility.
i'm a physicist, so working in bases is pretty common to me
$\require{begingroup}\begingroup\newcommand{\test}{\operatorname{blah}}\test\endgroup\test$ I used the same macro, but limited its scope.
a favorite trick in QM is basically just notation for changing bases
9:23 PM
oh semi what do you work in I never asked
i did my phd in statistical physics/qm
since then i've been doing quantum foundations stuff
koro yeah. note that the other answer solves the problem very quickly and directly. the approach about column operations is a little more 'conceptual,' in that it suggests a whole family of exercises on the same subject and hints at a larger point on the kinds of matrices that can be chosen to represent a given operator.
trying to move over to quantum computing, at least in terms of teaching it
then do you know what the notation $\mathbb{P}^X$ refers to?
not the distribution, by chance?
or just the probability measure associated with the rv $X$? is that it?
9:24 PM
i saw you ask about it earlier, yeah. i would need context
it doesn't resemble any notation i have seen, but notation is far from standardized. i would expect a book or resource to define it somewhere.
@leslietownes yeah, I agree. But I'm glad that you reviewed my answer. Thanks a lot.
not much context unfortunately, I was just hoping it was standard
well, you had to have gotten it from somewhere :P
Calculate $\mathbb{E}(X)$ for the following probability measures $\mathbb{P}^X$...
if youd like
9:26 PM
one might be able to guess at a meaning from a number of examples. a guess still being a guess.
i'm not asking what problem you got it from
i'm asking what source
oh just a problem sheet
not a book
probably need to ask the prof then
I suspect Varadhan might define it somewhere
but I don't have my book on me
if only there were some way of infringing copyrights in books using the internet.
9:28 PM
perhaps just a way to insist on the fact that $\mathbb{P}$ is the probability measure of $X$? 🤔
nah varadhan ain't online
as far as I remember
google books has a preview but
@JoeShmo looks like it might be functions from $X$ to $\mathbb{P}$ which I assume is a probability space.
joe: the mathcal P is often used to denote a fixed probability measure on some space. we don't seem to have a notation for that space. here X is a random variable on that space. X does induce a measure on the real numbers: assign the subset E of the real numbers the measure P(X in E).
so maybe P^X is that measure. this is just a guess.
if my guess is right, it strikes me as weird to use a superscript for that purpose. robjohn is interpreting the superscript in a more standard way.
and i'm interpreting it as "not my problem" :P
i think we should come up with a slate of two or three more guesses and decide the issue by majority vote.
semiclassical has provided a third option. i think it would win the vote.
9:36 PM
Hope patient is doing well.
Off to San Jose if I know the way.
sounds like fun!
@robjohn no unfortunately $\mathbb{P}^X$ is a probability measure
the suggestion is that $B^A$ is common notation for functions from $A$ to $B$
in which case being an element of $\mathbb{P}^X$ might be plausible
but that doesn't seem to quite fit i think
I understood that
oh, mystery solved (?)
he writes here explicitly (..blah blah compute expectation for...)
$\mathbb{P}^X = p \delta_a + (1-p) \delta_b$ and
$\mathbb{P}^X(\{n\}) = e^{-\lambda} \dfrac{\lambda^n}{n!}$
looks like they're just measures on the reals? but then i don't understand what X is or why you would decorate P with X
9:51 PM
if X is the identity function on the reals, you could compute E(X) for each of those measures, but the measure doesn't depend on X
but then I guess he didn't define the probability space explicitly, and he wants you to compute $\mathbb{E}(X)$
so he wants you to know that $\mathbb{P}$ is is the probability measure associated with $X$
10:03 PM
Any ideas on how to integrate this beast (a,b are real>0)?:
I have a quick question about logic - how can two things be tautologically equivalent but not equal? The example given in the slides for my CS class were [ not (P and Q) ] and [ (not P) or (not Q) ] - firstly with De Morgan's laws I think these are equal, but even if we're not allowed to use that in this specific field, how can two formulas be equivalent but not equal?
This is the beast in question, (so you dont have to follow the URL):
int exp(-b*x)*sqrt(x(x+a)) dx, from x = 0 to infinity
oh @leslietownes I didn't see your last comment. No, $X$ is a random variable
10:48 PM
joe: i was suggesting that the underlying space might be the reals and X: R -> R the identity function X(t) = t
in this guise, X is a random variable
again, all of this is guessing, i'm not wedded to any of it

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