1:17 AM
What's an affine Hilbert space? Can't really pin down a good definition.
1:27 AM
Nevermind. I think the wikipedia article on affine vector spaces cleared things up.
arxiv.org/abs/1911.04272 (my first google result) has a definition on page 7 and a citation suggesting that this definition or something equivalent to it is found in a textbook.
no idea as to whether that is a "good" (i.e. useful or appropriate for your purpose) definition
oh, yeah. This looks pretty nice. Thanks!
@user193319 It's a kind of space, which happens to be both Hilbert and affine.

5 hours later…
6:55 AM
7:07 AM
Do you think this course is doable for a student with mathematical background only?
wandering the cosmos of that image for legible text
soumik: impossible to know from a course description but it sounds like it could be pretty mathy
@leslietownes hehe
Okay, I am checking if there are any list of recommended books

3 hours later…
10:24 AM
Hi all, may I ask help with solution here? I know the "end"-answer, but idk how to solve one specific case in solution.

[context free langs][pumping lemma]
So basically, i have $L := {b^i a^j b^k | 0 <= i < j and 0 <= k < j}$. I know about Ogden's Lemma but for this case I would like to use only Pumping Lemma.

So i choose w = a^p b^(p+1) a^p. |w| = 3p+1 > p and w = uv^i xw^i y in L for i in N >= 0.

I see 5 cases in total (but 3 general ones):
- vx contains only a's - pump till e.g. i = 10 => either first or second condition is breached;
- vs contains only b's - downpump (i = 0) => #a(w) >= #b(w) => not in L.
- vs contains a's and b's. What do in this case?

2 hours later…
12:05 PM
this: eprint.iacr.org/2018/421.pdf paper that I'm reading gives a multivariate Gaussian function as $p_{\sigma, c}(x)=e^{-||x-c||^2/2\sigma^2}$ where $x$ and $c$ are vectors in $\mathbb{R}^k$. I have never seen such a definition. Where can I read up on this?
I mean is the measure of this even $1$? It looks like its maybe missing a factor of $1/\sigma\sqrt{2\pi}$.
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated...
By measure of it, you mean the integral?
Yes
Its definitely missing some factor to integrate to $1$, but not sure if it was supposed to
Aren't all probability functions supposed to?
They consider those restricted Gaussian distributions which do integrate to $1$
I suppose its just that it didn't matter to write a constant term so they didn't
They never consider $\rho_{c, \sigma}$ as a probability distribution function most likely
12:19 PM
@Jakobian Oh okay that's highly plausible I'm not very far ahead in the article, its just the general mathematical background part. Thanks a lot!
The phrasing here is triggering me.
Specifically "if $S$ is measurable".
Haha try looking at the rest of the article
I've been reading crypto papers like this for the last two weeks and some part of me just wants to die
@Jakobian Although what is wrong with this statement I can't tell with my non-existent measure theory background
oh okay okay I forgot the parts before that phrase
@ephe that it needs to be of positive measure
12:50 PM
> Theorem 3.3. Let $X$ be a random variable whose moment generating function $\psi_X(t)$ exists for $|t|<h$ for some $h>0$. Then all moments exist, that is $E|X|^r<\infty$ for all $r>0$.
> Proof Since the exponential function grows more rapidly than a polynomial, we have $|x|^r\leq e^{|tx|}$ as soon as $|x|>x_1$ (say). On ther other hand, for $|x|<x_1$, trivially $|x|^r\leq Ce^{|tx|}$ for some constant $C$. Thus for all $x$, $|x|^r\leq (C+1) e^{|tx|}$ and hence $$E|X|^r\leq (C+1)Ee^{|tX|}<\infty,\quad |t|<h.$$
Why is $Ee^{|tX|}$ finite? The assumption is that $Ee^{tX}$ exists, not $Ee^{|tX|}$.
@psie note that e^{|tX|} ≤ e^{tX} + e^{(-t)X}, both terms on the RHS have finite integral by assumption
@s.harp ah, yeah, thanks! This explains it.

2 hours later…
2:56 PM
What are yourthoughts on this paper:
@Q_p almost every day there's someone who claims to have proven/disproven RH
3:14 PM
@Q_p Having read the first four words of the title: it is nonsense.
Show that a Riemann hypothesis paper is nonsense any%

2 hours later…
5:10 PM
> Show that if $X_1\in\Gamma(p_1,a)$ and $X_2\in\Gamma(p_2,a)$ are independent random variables, then $X_1+X_2\in\Gamma(p_1+p_2,a)$.
I know the moment generating function (mgf) of $Y\in \Gamma(p,a)$ is $$m_Y(t)=\frac{1}{(1-at)^p},\quad t<\frac1a.$$ I also know of the theorem that states that if $Y_1,\ldots,Y_n$ are independent, then the mgf of $S_n=Y_1+\ldots+Y_n$ is simply $m_{S_n}(t)=\prod_1^n m_{Y_i}(t)$. So...is this simply an application of that theorem? Seems too simple to be true.
@psie this is the power of moment generating/characteristic functions
they've got a lot of power indeed
once you know what form is a moment generating function/characteristic function of a distribution, you can deduce distribution of things like $S_n$
yeah, it's a piece of cake

1 hour later…
6:19 PM
@Jakobian my apologies, i am been (and still am) very busy and have not had time to process your question.
atm i only make trite remarks to easy questions as a mental relief to the daily grind.
Its okay, to be honest I completely forgot about it
7:03 PM
@XanderHenderson i'm only here for productive mathematical discussions about the paper. Doesn't sound like that's your aim here.
what do u think a 100 level course called "mathematics of money" teaches?
money-algebras?
@Obliv 100 level? Is this some kind of rpg
I put too much points in general topology
it's the system for U.S. colleges they sort courses numerically by the hundreds usually 100 to mean first year/freshman, 200 for sophomore/2nd year etc
@Jakobian lol
have to reset skill tree and redistribute points
thinking about what "resetting a skill tree" would mean in real life has lead me to some dark conclusions
:( all the classes I might wanna take are filled/bad section times
i just have to wait and apply for closed sections in august rip
7:14 PM
i wanted to take set theory but it meets at like 830 in the morning which is too early to be doing math especially since it takes me an hour to drive to school
I think its nonsense since I've googled the author and year ago he seemed to be posting similar things and people call it nonsense
Hadamard already proved that the zeros of the zeta are in the strip $0<Re(s)\color{red}<1$
I guess I'll take differential geometry or real analysis if that opens up.
@Obliv you don't
7:17 PM
@Q_p If RH could be resolved in five pages, someone would have done it already. It is not credible that anyone could use existing techniques to resolve RH in five pages.
@Jakobian you don't :P
> If RH could be resolved in five pages, someone would have done it already.
When Fermat fell, it was only after Wiles created entirely new mathematics. I don't see that anywhere in the document you've sited.
I don't know about the thing I quoted, its something likely but not certain maybe
there is this slim chance someone can do it in five pages with usual techniques
@Q_p are you the author of the paper?
7:21 PM
Wiles contribution wasn't just Fermat's big theorem, no one really cares about this I'm sure. Its that he contributed a lot of methods, I'm sure
@Jakobian Exactly this. Resolving FLT was cool, but it is the new mathematics that were really interesting.
Ditto the ABC conjecture, and why it has taken so long for anyone to really understand what is going on with that (the author's reluctance to actually explain things to anyone being a contributing factor).
What is the current consensus on that proof, by the way?
I should go look at that...
Mochizuki's
Yes. And I see that it is still considered unresolved by most of the mainstream.
I assume Mochizuki's proof is still considered to be incomplete, and they try to fix it without success while telling people there is nothing wrong with it
i.e. nothing changed
That is what the internet says, after a very quick Google.
7:24 PM
@jakobian do u have a good beginner text for diff geo?
@Obliv I don't know differential geometry. Try Ted Shifrin
Oh right I bookmarked it. Thank you
his books are supposed to be for beginners I'm pretty sure
yeah they looked really promising
if anything I'm going to learn differential topology one day and not differential geometry
7:31 PM
investing all the points into topology. respect
@Jakobian Lee is a fairly standard text for that.
I found it fairly approachable, but I haven't read other books in the field to compare it to.
that said, I won't be going into differential topology in the near future except for learning it by osmosis
as a side note, I did find a new-found appreciation for the concept of partitions of unity, and even a paper that mixes this with things I am interested in
7:50 PM
I feel silly, but the author of the text I'm currently reading shows that the log normal distribution has all moments existing, but the moment generating function does not exist for any $t>0$. There's this estimation that is done that I just don't comprehend. Why is $$\log t+\mu+\frac12\sigma^2n\geq \frac14\sigma^2n?$$
Couldn't $\log t$ be crazy negative? And $\mu$ likewise?
it says t is fixed for purposes of that remark. i think this is just a + bn eventually being less than c + dn as n goes to infinity if b, d are positive numbers and b < d (here the key part is just 1/2 being larger than 1/4)
if enough things aren't fixed that you aren't just comparing linear functions of n there, i dunno
ok, so $a+bn\geq cn$ eventually if $b\geq c$
@SineoftheTime The fact that zeta(sigma+it) doesn't have any zeros at sigma=1, doesn't rule out the existence of zeros arbitrarily near 1. For example, there could be zeros with s
*with sigma = 1.
*with sigma=1-(log log t)^{-1}
$\log t+\mu+\frac12{\sigma}^2n\ge \frac14{\sigma}^2n \iff \frac14 {\sigma}^2n \ge -\log t-\mu\iff n\ge -\frac4{{\sigma}^2}(\log t+\mu)$
@psie yes. note there's a "by choosing n sufficiently large" right before it. [also, it's a pretty natural interpretation of "[statement of inequality] as n -> infty," where if you meant that the inequality held literally all the time, you probably wouldn't qualify it with an "as n -> infty"]
8:01 PM
as $n\to +\infty$
too many logs. What is this, a sawmill?
@psie well, if $b > c$ at least
"2) The union of the elements of any subcollection of T is in T." does this mean the same thing as $\bigcup\limits_i A_i \in T$ where $A_i \in T$
yeah, that's more likely. $b\geq c$ was gibberish
It means that if $S\subseteq T$ then $\bigcup S\in T$
@Obliv yes
I guess I'm confused because it seems very obvious?
8:08 PM
what does
what is
I thought this condition was always satisfied but it clearly isn't.
@XanderHenderson I understand your skepticism. However, you have to acknowledge that those are mere beliefs, not facts. For example, before Erdos-Selberg, it was long believed that a proof of the PNT was impossible without complex analysis.
what condition
that the union of any elements of a set is an element of the set.
this is always satisfied if the set is the power set of a set
yes, it is
do you mean that its not satisfied for all families $T\subseteq \mathcal{P}(X)$
okay so you are fine with throwing cryptic remarks but not with explaining what you mean
8:14 PM
@Jakobian I did not mean this, but this makes sense
I was simply skimming this definition in munkres because I was curious
and $T \subset \mathcal{P}(X)$ seems like an important property
@jakobian I'm having a hard time comprehending this definition. A topology on a set $X$ is a set of subsets $T$ of $X$ such that $\varnothing \in T$ and $X \in T$ so already this means for any $a \in T, a \in X$ and since $X \in T$ and $\varnothing \in T$, we have $\varnothing \in X$ and $X \in X$?
i should say from only 1) and above in that definition
8:24 PM
I don't know if I feel patient to spend few hours going over how $\in$ and $\subseteq$ are different again
I know the difference, simply saying that is the mistake is enough
and how $\in$ isn't transitive
"A topology on a set X is a collection T of subsets of X having the following properties:" means $T = \{ A \mid A \subseteq X\}$ right?
No
it means $T\subseteq \mathcal{P}(X)$
that is a clear difference, but I don't get how I would have interpreted that from "subsets of X"
8:28 PM
It means that every element of $T$ is a subset of $X$
@Jakobian I thought not every $A \in \mathcal{P}(X)$ is a subset of $X$ though
then you thought wrong
$\mathcal{P}(X)$ is literally the set of all subsets of $X$
if $A\subseteq X$ then $A\in \mathcal{P}(X)$ and conversely
@Jakobian okay.. now i see why you said this
thx i get it now
i mistakenly thought for $X = \{\varnothing,1\}$ that $\mathcal{P}(X) = \{\varnothing,\{\varnothing\},\{\varnothing,1\},\{1\},1\}$
but it's just $\{\varnothing,1,\{\varnothing,1\}\}$ I think
no, it has four elements
since $X \subseteq X$
8:34 PM
The generalized Gamma distribution has pdf $$f(x)=Cx^{\beta-1}e^{-x^{\alpha}},\quad x>0,$$where $C$ is a normalization constant, $\beta>-1$ and $0<\alpha<1$. Is it "obvious" that the moments of this distribution exist? According to my book it is, but I don't really see it.
if we use von Neumann definitition of natural numbers then $1 = \{\emptyset\}$ and above you wrote the same set twice
I was just explicitly counting the empty set, without doing so it's just $X = \{1\}$ and we have $2^1 = 2$ elements for $\mathcal{P}(X) = \{1,\{1\}\}$?
oh thats wrong as well(?)
@psie near $0$ this functions is integrable, and at infinity the exponential takes care of things
its "obvious" that the moments exist for the same reason that its "obvious" that $f$ is integrable
ok, we are integrating $f(x)x^r$ where $r>0$ though, or?
we can go for $r\geq 0$
8:38 PM
true :)
either way, this will have the exact same form as $f$ but with different $\beta$ constant
the $\beta > -1$ is there just to assure this is integrable near $0$
ok, I have to keep staring some more at that density to convince myself that it is
this is another example of a distribution that does not have a mgf
I don't think the restriction $\alpha < 1$ is needed
@psie actually here it should be $f(x) = x^\beta e^{-x^\alpha}$, or we should take $\beta > 0$
that's the expectation, isn't it?
@psie you are misunderstanding what I'm commenting on
we can't take $\beta > -1$ and $f(x) = x^{\beta-1}e^{-x^\alpha}$
this is not integrable near $0$ for $\beta \leq 0$
@Obliv what do you mean by explicitly counting empty set
again, $\in$ and $\subseteq$ are not the same
$\emptyset\subseteq X$ for every set $X$, but that doesn't mean e.g. $\{\emptyset, 1\} = \{1\}$
no, those are different sets
8:52 PM
it doesn't matter, my confusion came from forgetting the definition of $\mathcal{P}(X)$ which is just the set of subsets of $X$. I'm still not entirely sure property 2) doesn't follow already from $T \subseteq \mathcal{P}(X)$
@Jakobian really?
@Obliv it doesn't
I thought $\varnothing$ was synonymous with $\{\}$
and?
that's true
but however that helped you achieve your conclusion, you're wrong either way
okay so $\varnothing$ doesn't represent the "nothingness" inside of the set $\varnothing = \{\}$
it represents the set itself
you could say it that way, sure
8:55 PM
i.e., $\varnothing \subseteq X$ for all $X$ but there can exist $A$ for which $\varnothing \notin A$
I don't know what that means
why introduce new language for this
its confusing for no reason
I am not trying to introduce new language, I just don't have a solid foundation of set theory (which is why I wanted to take that course)
I know operationally the differences between $\subset$ and $\in$ but idk how I'd explain them
you were introducing new language, regardless of intentions
how to explain the differences between $\subseteq$ and $\in$? Why explain differences?
no. Explain whats $\subseteq$ in relation to $\in$
its explaining how things relate to each other that does
$\subseteq$ in relation to $\in$ is like the quantified version?
like $\subseteq$ is $\in$ but applied to all of something
how to explain what $A\subseteq B$ means? Simply write it in terms of $\in$
9:01 PM
for all $a \in A( a \in B)$
here you go, you explained it
lol thank you
and $\in$ needs no explanation
and an indexed union $\bigcup\limits_i A_i$ is just the set $\{a \mid a \in A_i\}$ for any $i$ in the indexing set or whatever
no
$\bigcup_i A_i = \{a : \exists_{i\in I} a\in A_i\}$
adding words to things generally speaking just makes everything confusing and explains nothing
simplicity is where the real explanation lies
why do politicians say so many words?
What I do when I listen to this politicians is I just turn off. And does it matter what they say? No. Because people look at this from the perspective of belonging to a tribe. It doesn't matter what they say as long as they are kept in the belief that those people who represent their "tribe" don't say something that would make them consider them not a part of it
9:09 PM
hmm I don't have much practice with pure 1st order logic but I often read $\exists$ as almost like a declaration rather than something to connect a "such that"
@Jakobian there is no room to misinterpret that definition?
i.e., I might read $: \exists$ as "such that there exists" or as "such that exists" almost as if it's an action to bring something into existence
If $\{a : \varphi(a)\}$ is some set then $\varphi$ can be anything. This only becomes meaningful when we are saying $x\in \{a : \varphi(a)\}$ at which points we are declaring that $\varphi(x)$
If I said $x\in \bigcup_i A_i$ that would be such "declaration" that something exists, namely $i\in I$, such that $x\in A_i$
boils down to understanding of set-builder notation
@Jakobian yeah but how can we know the precise number of words/information to get a point across? It's probably my lack of experience in logic/math but not only is it encouraged, but it seems necessary to incorporate "natural language" into proofs and writing.
I thought maybe the language of 1st order logic somehow bypassed this but it doesn't seem to be the case? How can you define literally anything useful with just 1st order logic (i think leslie linked the FTC in some version of logic for computer based proofs)
sorry I got off topic as always
9:31 PM
@Obliv inexperience in communicating
Don't excuse yourself with being inexperienced with math and logic because that's not it
I wasn't talking about replacing symbols with their equivalents in English
2 mins ago, by Jakobian
@Obliv inexperience in communicating
And creating comprehensible strings of thoughts
You can probably solve the former if you solve the latter
I am not mature enough to learn about coq it seems.
Lets do a thought experiment. How do you think I get convinced of something I read in math
@Jakobian it makes sense to you
No
Before it makes sense to me I need to convince myself
How do I do that
I thought that you were convinced by what you read, now you are saying you must convince yourself?
I have no idea
9:40 PM
...
I am talking about the process of coming from not convinced to convinced
Are you with me on that
Oh, i guess you would have to read and interpret the words, internalize and compare to what you know, then come to a conclusion
That's part of it. But the important take away is that I must explain it to myself
Yes, so you must communicate with yourself
Not really a math question more so than a convention question but, if you have 2 angles, you might represent them as Theta and Phi, but if you have 3, you have Theta, Phi, and then what?
Yes. So to be good at learning math you need to be able to explain things to yourself, and others
9:45 PM
@Jakobian never thought about it like that actually, that makes sense.
ive always just assumed if it made sense to me, then I could communicate it to someone if need be
but never actually thought about how i'd do so
@CPlus psi
@CPlus there are various Greek letters used for angles for example α, β, γ
@jakobian also, just to reiterate because sometimes I forget it myself, but i'm primarily a physics student I just like math because it's very interesting to me (even though widely considered useless)
I am very much not a traditional math guy who writes and reads proofs for fun, so that is why I tend to explore the stuff that is most accessible to me and my limited knowledge.
Math isn't useless
That's a vast generalization
i wholeheartedly agree. It is like the vessel in which any other natural science can navigate with
I guess I consider logic and math to be in the same boat
9:52 PM
What is true that a lot of math is not able to be applied to some field science
yea, it's like the world of ideas which we can draw from to apply to the real world.
Will it change anything if a $T_1$ compact space with countable pseudocharacter but uncountable character exists? No. Does that make math useless to science? No
Maybe all math can be used in some capacity. if nothing else you can use it in sci-fi novels
No it can't. Wishful thinking
@SineoftheTime @Jakobian Thanks.
10:05 PM
Besides who cares if it can be used or not somewhere?
People who say math is useless, maybe they're just insecure about something
something that kind of gets overlooked in a lot of these conversations is that it should go without saying that most people never 'use' physics or chemistry or whatever else they learn in college
money and profits, typically
@leslietownes that too
even if the textbook is in principle about something closer to real life, that doesn't mean you're ever going to be tasked with "using" it in any way
and students' failure to do that is not necessarily a problem with the educational system but more with one's expectations around what formalized education can/should and cannot/should not do
People are the problem
"When am I ever going to use this?"
10:11 PM
a sometimes useful framing is, what can you be expected to get out of your formal education, like, does the stuff in a book contribute to things in your head that you end up drawing from later
and not like oh god i never learned a situation where i need to solve this PDE, so it's useless
@leslietownes numerical analysis :( but yeah the only thing I see an undergrad physics/applied math degree being useful for is teaching systematic thinking principles and problem solving so you have the "mind" for working in the industry, whatever that is
well numerical analysis is one example of that, about ten zillion people are out there computing stuff and most of them have never taken a numerical analysis class. does that mean the classes are useless? no. but it doesn't mean the classes are "useful" in the sense of "oh you're going to be analyzing error propagation in euler's method in real life"
when will it be useful to me
the closer you get to what people do in real life, you also get closer to stuff that is so tied up with the details of implementation, and so application-specific, where there's almost no value in learning the specifics of how something is done at any one time, because all of the specifics of best practices at that level of granularity are going to change by the time the textbook is printed
pyoor math
10:14 PM
which isn't saying "oh, everything changes so fast i can just be a blank slate armed with wikipedia and chatgpt" but it should put the "theory" vs "applied" distinction in perspective
People have a very egoistic self-titled worldviews
sometimes a textbook chunk of theory can be more useful than a textbook chunk of specific application, because it will have a longer shelf life
Blame me for being cynical all you want. World made me like this
but that's useful in the sense of what you derive from what's in the book, not 'useful' in the sense of 'oh people are running this textbook algorithm right now'
It will be useful to you on the next page(s).
10:18 PM
@user85795 lol
so let us move on please :)
time is money
@leslietownes or because it is immediately useful (say, for example, the theory of reducing carbon emissions or something :P)
No one is paying me for my time so...
obliv: well yeah but i was thinking in terms of the stuff people are likely to see or produce in a formalized system of education
well, idk the requisite bio/chem(or both) of that topic but I'm sure there is more general theory underlaying it
10:21 PM
you might be making exactly the leap which i'm sort of cautioning against
which is "oh, carbon emissions relate to chemistry, so a chemistry major must be being trained in that from chemistry textbooks"
That's different from mathematical theory
"an applied mathematics class must be more useful than a pure mathematics class because applied mathematics at least relates to the real world"
Semantics
well, maybe? maybe not. most people in universities just stare at textbooks and muddle through and don't address anything in the external world
a lot of an undergrad curriculum is just "textbook fundamentals" and does not engage with any core things that you might think of as driving useful research
I don't read that fast I was still talking about Obliv's use of the word theory
10:23 PM
possible exceptions would be the social sciences where profs sometimes go to great lengths to say, this stuff we're doing in our grad seminar is happening out on the streets right now and your papers can be a part of this
what I meant was that the "chunk of theory" can be more useful than the "chunk of specific application" if the theory is more aligned with the "real world"'s needs at the given time, not just because theory is more widely applicable.
which is just people getting high on their own supply
law^ of the street broken
yeah, its like the eleventh commandment not to do that
thou shalt not get high on thine own supply?
10:25 PM
@Obliv yeah i think the key is just taking care in assessing when people make claims about what is "useful," like, what is the actual "use" they have in mind
idk, I never studied business or whatever real people do in the real world.
more often than not its some vague thing
So what useful relates to. Its a relative word
I mean it's psychologically related too, like the markets even
How I think about it is that there are some words which become meaningful when conditions against some y. Say x is useful for y.
That's what I mean by relative. People throw around such words with no further explanation
10:29 PM
@robjohn I don't get it? Explain please.
15 mins ago, by robjohn
pyoor math
What people think are objective truths, are nothing but meaningless assesments
@user85795 pure math to sound like poor math? idk
I was commenting on leslie's previous comment: "pure math"
thnx
@Jakobian I wouldn't say meaningless? What happened to your whole thing about communicating :P
10:32 PM
You probably think you're being snarky
But I do agree that meaning is very vague
What is the meaning of meaning?
¯\_(ツ)_/¯
:D
The Meaning of Meaning: A Study of the Influence of Language upon Thought and of the Science of Symbolism (1923) is a book by C. K. Ogden and I. A. Richards. It is accompanied by two supplementary essays by Bronisław Malinowski and F. G. Crookshank. The conception of the book arose during a two-hour conversation between Ogden and Richards held on a staircase in a house next to the Cavendish Laboratories at 11 pm on Armistice Day, 1918. The original text was published in 1923 and has been used as a textbook in many fields including linguistics, philosophy, language, cognitive science and mo...
> what is meant by a word, text, concept, or action.
10:37 PM
meaning is a personal determination, i think, which can be influenced by others but ultimately it's what you feel/believe