Mathematics - 2023-04-13

Xander Henderson

00:12

Another place where this really frustrates me is in the discussion of "piecewise functions".

D.C. the III

If I have the function $f(x,y) = \|(x,y) - (0,0)\|$ and I need to integrate it over the ball at origin, i.e $\int_\Omega f(x,y)dxdy$. But because I don't want to deal with square roots and deal with the square of the distance instead, is there no way I can retrieve the solution from $\int_\Omega \|(x,y)- (0,0)\|^2$ and find the solution of $\int_\Omega \|(x,y)- (0,0)\|$?

Xander Henderson

It is not really right to say that a function is piecewise. Being "piecewise" is a property of how the function is presented. For example $$ x \mapsto \begin{cases} x^2 & \text{if $x \ge 0$, and} \\ x^2 & \text{if $x < 0$.} \end{cases} $$ is a piecewise definition of the function $x \mapsto x^2$. These two functions are the same---hence either everything is piecewise (since we can always carve up the domain), or it doesn't make sense to talk about a function being piecewise.

So it is better to talk about piecewise definitions of functions, or piecewise defined functions.

@D.C.theIII Convert to polar coordinates.

D.C. the III

Yes I did all that, that's why the question is coming up. When I converted to polar I ended up with an extra $r$ between the squared version and non squared. Specifically: $\int_\Omega r^2 dr d\theta$( the non squared version) and $\int_\Omega r^3 dr d\theta$ (the squared version)...........this is after all the right conversions and such.

I got the right answer when I used the non-squared version, but I originally had squared my distance function. So I was curious if there was a way to get the right solution doing what I had done.

Ted Shifrin

No.

D.C. the III

Terse....but I guess there isn't more to it.....lol

Xander Henderson

00:22

@D.C.theIII Check your arithmetic.

You are integrating a radially symmetric function over a ball. You want to be in polar coordinates. Period.

Ted Shifrin

00:37

@JackBosma What platform?

I don’t see the need for another layer for me personally at this time.

Ted Shifrin

00:58

Nice of you not to mention that it’s $33 a month with an annual commitment. Good grief.

D.C. the III

🤣🤣🤣

geocalc33

$s\Gamma(s)^2\zeta(s)=\int_0^\infty t^{s-1}\sum_{\alpha \in \Bbb N} \Phi_{\alpha}(t)~ dt$

I wonder if this can be extended to a functional equation on the critical strip by introducing a branch cut for $t^{s-1}$

so far I've been operating on the assumption that yes it can

but it would be good to prove

D.C. the III

01:38

for the average distance from a point on the boundary of the ball of radius $a$ in $\mathbb{R}^2$ to points inside the ball. I stated my distance function as $f(x,y) = \|(x-b_1,y-b_2)\|$, then using polar coordinates the set up is $\int_0^{2\pi}\int_0^a \sqrt{(r \cos(\theta) - b_1)^2 + (r \sin(\theta)- b_2)^2}r dr d \theta$.

I've done a bunch of algebra to try and simplify things using the quadratic formula, but I'm still ending up with a complex expression and not getting rid of the square root. Is there a trick I'm missing or is this just straight brute force?

@TedShifrin

Ted Shifrin

No brute force required if you use your brain first. You have to start thinking pro-actively particularly with applications .

D.C. the III

I thought initially of relating it to the same sort of question but in terms of the origin....

Ted Shifrin

Aha.

So what does that entail?

D.C. the III

mainly because I could put my arbitrary fixed point on the boundary as being the origin

Ted Shifrin

Yes, that’s using your brain. So where is the disk?

D.C. the III

01:45

ah...this allows me to use the idea of the triangle being inscribed in the circle with diameter as one of the sides must be a right triangle

Ted Shifrin

Right.

D.C. the III

This is what I signed up for, getting to the point of "applying" the ideas........this will be fun.

geocalc33

02:03

desmos supports higher order derivatives now i.e >4

one potato two potato

03:12

My original plan for this semester was to focus on real analysis course and relatively less focus on algebra course, but because of my ambitious algebra professor, my attention is almost focused on algebra.

Ted Shifrin

Don't demanding professors just suck.

one potato two potato

I mean it's good. I like those kinds of courses.

So I decided to take algebra 2 course next semester which wasn't planned.

Ted Shifrin

What will it cover?

geocalc33

$$ \frac{\partial^2}{\partial t^2}\Phi(t,x)=-\frac{t}{x} \frac{\partial}{\partial x}\Phi(t,x) $$ does anyone recognize this PDE?

Ted Shifrin

03:28

Never in a million years.

one potato two potato

Algebra 2? It slightly depends on who teaches that class but if it's the same professor, past syllabus tells that basic field theory and commutative algebra (with a view towards algebraic geometry) and some number theory if time permits (he's a number theorist). But since this professor talks about Coxeter group and a finite group of Lie type and so on this time, I don't think it will be really 'basic'

Oh only 6 people took this class before usually about 15 people take this class

Franklin

04:06

@SouravGhosh Now, that you mention it, I searched thoroghly and found it.

leslie townes

haha. [course title] 2, aka, topics in whatever the instructor likes. those can be good courses.

one potato two potato

How about [course title] 3?

leslie townes

at some point they just call it 'topics in __' or have it be 'seminar' or something

Franklin

@SouravGhosh I think you meant the theorem in the picture I posted above. But I guess this is not equivalent to my version of Picard's theorem, I stated previously i.e chat.stackexchange.com/transcript/36?m=63361541#63361541 jn this comment

11 hours ago, by Franklin

@SouravGhosh But Strange, I dont find Picard's theorem in SL Ross. My version of Picards theorem is : The differential equation $\frac {dy}{dx}$, $f(x,y)$ , $y(x_0)=y_0$ has a unique solution on $|x-x_0|\leq h$ $(h>0)$ if both $f$ and $\frac{\Delta y}{\Delta y}$ are continuous on the domain $D=\{(x,y)| (x-x_0)\leq h, |y-y_0|\leq k, h,k>0\}$. Is this what your version of Picard's theorem as well ?

Koro

05:45

Leslie, I figured out yesterday's question [k, K].

leslie townes

i'm trying to read [k, K] like one of those horizontal emoticons ^_^ and still lost on what expression in the eyes [k, K] is supposed to convey

looking wistfully off to the lower left, tired of field theory

Koro

[k:K]

😅

This one.

one potato two potato

So if a function $f$ is continuous on a compact interval, in particular Riemann integrable, Riemann integral of $f$ equals Lebesgue integral. So I can freely use Riemann integration theory? like change of variable

Koro

Here is what I did: K is finite extension of k so finitely generated extension i.e. there are $a_i$'s such that K=k(a_1,...,a_n)

Then I considered min polynomials of a_i's

which must split in N as N/k is normal.

So a hom. in Hom_k(K:N) maps a_i to roots of p_i.

etc.

@onepotatotwopotato yes?

Franklin

06:25

@SouravGhosh I did a typo $\frac{\Delta y}{\Delta y}$ this is a wrong latex code for partial differentiation.

It should have been a partial diff of f wrt y. But nevertheless, I think there is not a mention of this theorem in the book....

Franklin

08:52

0

Q: Let $x_n$ be the nth non- square positive integer. Then $x_1=2,x_2=3,x_3=5,x_4=6,$ etc. For a positive real number $x,$ denote the integer closest t

Let $x_n$ be the nth non- square positive integer. Then $x_1=2,x_2=3,x_3=5,x_4=6,$ etc. For a positive real number $x,$ denote the integer closest to it by $\langle x\rangle.$ If $x=m+0.5,$ where $m$ is an integer, then define $\langle x\rangle=m.$ For example, $\langle 1.2\rangle=1,$ $\langle 2....

number-theory

Can anyone please help me with this ?

geocalc33

10:09

0

Q: Solutions for this partial differential equation?

What are the solutions to this PDE? $$ \frac{\partial^2}{\partial t^2}\Phi(t,x)=-\frac{t}{x} \frac{\partial}{\partial x}\Phi(t,x) $$ $(x,t) \in (0,1)\times (0,\infty)$ and $\Phi(x,0)=1.$ I found the heat equation and wave equation but they are slightly different than this equation. I found this...

analysis partial-differential-equations reference-request heat-equation

one potato two potato

11:01

6

Q: Group $G$ is nilpotent if and only if $G^n = 1$ for some $n \geq 0$

This is from the book Abstract Algebra, $3$rd edition, by Dummit & Foote; theorem $8$ on page $194$. Definition (upper central series): For any group $G$ define the following subgroups inductively: $$Z_0(G) = 1, \qquad Z_1(G) = Z(G)$$ and $Z_{i+1}(G)$ is the subgroup of $G$ containing $Z_i(G)$ s...

abstract-algebra group-theory reference-request

My professor referred to this (false) D&F proposition...

Proof: Straightforward induction

Koro

11:19

Aug 11, 2022 at 17:51, by Koro

has anyone faced the issue: macbook not connecting to wifi?

@robjohn: the issue seems to have resolved now after 1 year.

I'm very grateful to this nice answer here

1

A: MacBook Pro can't see my android phone WiFi hotspot

Same issue, I changed my Android hotspot settings to the 5GHz band and to WPA3 Personal, but the issue was that I had an extra blank space in my password.

:-)

geocalc33

wow

Koro

my device was connecting to wifi but not to hotspot created by android.

But it seems to have been fixed now. :-)

I dunno how I landed on the answer :-). But I recall it wasn't connecting even to iphone hotspot. Today it was connecting. I guess that is also due to a similar reason.

Koro

11:41

Suppose $x^n=a$ (n). Given that $(a,n)=1$, can we say that $(x,n)=1$?

0

Q: Proving that $a$ is an $n$th power residue mod $m$ iff $a^{\phi(m)/d}\equiv 1 (m)$, where $d = (n,\phi(m))$.

If $m\in \mathbb Z+$ possesses primitive roots and $(a, m) = 1$, then $a$ is an $n$th power residue mod $m$ iff $a^{\phi(m)/d}\equiv 1 (m)$, where $d = (n,\phi(m))$. Its proof in the book goes along the following lines: Let $g$ be a primitive root mod $m$ and $a = g^b, x=g^y$.Then the congruence ...

number-theory proof-explanation

Most likely, this question will face heavy downvotes. I don't have much to say about my question in this case.

Jake

Hi guys, quite a simple question here... If I have $\frac{cos^4x - sin^4x}{cos^2x}$ the nice way of simplifying is to recognise the difference of two squares, but why can't we cancel the $cos^2x$ and get to the answer that way?

Koro

12:06

nvm, I got it. $d=(x,m)\implies d|x^n, d|m\implies d|(x^n-a)\implies d|a\implies d=1$ as $(a,m)=1$.

geocalc33

yes

I'm heating up... $\frac{\partial^2}{\partial t^2}\Phi(t,x)=-\frac{x}{t}\frac{\partial}{\partial x}\Phi(t,x)$

geocalc33

12:39

Is the Grothendieck group $K(S)$ of the Abelian semigroup $(S,\times)$ where $S= \Bbb R \cap (0,1)$ equal to $K(S)=\Bbb Q \cap (0,1)$?

nvm can't be that...

one potato two potato

Why $PSL_2(\Bbb F_q)$ can be view as a subgroup of symmetric group $S_{q+1}$? What is the action?

geocalc33

I have to take $S \times S /\sim$

where $(a,b)\sim (c,d)$ if $a\cdot d\cdot k=b\cdot c\cdot k$

which is $(0,1)^2$ quotiented by that equivalence relation.

$k \in S$ also.

I believe the answer is $K(S)=(\Bbb Q^+,\times)$

Lucas Henrique

14:22

is it just me or this question feels AI-generated?

0

Q: Proving $a \leq b$ for all $\epsilon \in (0,1)$: need help with inequality involving real numbers $a$, $b$, and $c$.

I'm currently working on a math problem that involves proving an inequality involving real numbers $a$, $b$, and $c$. The given inequality is $a + c \epsilon \leq b$, where $\epsilon$ is a positive number in the open interval $(0,1)$. The complete math question: Let $a$, $b$, and $c$ be real numb...

real-analysis real-numbers

Ted Shifrin

14:32

@onepotatotwopotato Think automorphisms of the projective line.

Xander Henderson

@TedShifrin Sounds dirty.

AMDG

15:40

Huh. Wat I tell ya. I'm stupid for not thinking to just check mathworld for everything I intend to immediately study.

There is some sort of relation between the integral of $\int_0^1 \left\{\frac 1 x\right\} dx$, $\ln x$, and $H_n$ or $\psi(n)$.

mathworld.wolfram.com/FractionalPart.html

Which... I already knew there's a relationship between $\{\ln x\}$ and $x$ itself, so this is kind of a remote observation relative to what I'm after, but it's interesting nonetheless. Let's see what the rest of the page says...

shintuku

15:58

@Koro what were the important theorems in group theory you've needed for algebraic topology so far?

leslie townes

@LucasHenrique the last paragraph certainly could have been written by an AI, but it might just also be a new user trying to sound 'formal' :) the slight misstatement of the question, missing hypotheses, and nearly content-free discussion of possible strategies sounds a bit like they've asked somewhere and already been given 'hints,' or maybe even that the problem as stated contained hints which they "put in their own words" so it sounds like an attempt.

one potato two potato

@TedShifrin Thanks

leslie townes

i'd guess non-AI, if only because a chatbot might have asked a more interesting question

Koro

@shintuku none.

But from modules, yes.

shintuku

@Koro are modules from commutative algebra?

Koro

16:10

I don't know. But think of them as generalized vector spaces.

vector spaces are defined over fields, modules are defined over rings.

shintuku

i see thanks for the answer

Koro

we had modules last semester in Algebra 1 course.

This sem, we have algebra 2 wherein we have group theory, fields and Galois theory.

shintuku

oh cool

no group theory in algebra 1?

Koro

next sem, we have a a course called 'commutative algebra' which I won't take. I am not sure what will be the content of that course.

@shintuku no, it was rings and modules.

I know how you feel upon hearing this: group theory before ring theory.

😅

shintuku

huh yeah i thought group theory is usually before that

Koro

16:15

exactly, I thought that too.

And I agree that it should come before ring theory.

shintuku

did you find some theorems in ring theory/modules were better understood starting from groups first?

Koro

but some say that (here in chat also, if I remember correctly) rings have two binary operations, groups have only 1 so rings feel more intuitive.

@shintuku first group theory and then ring theory worked fine for me. In group theory, we have 'isomorphism theorems' for example, we have those in ring theory also and the proofs are more or less the same.

and the same theorems have analogues in modules and in vector spaces too!!

shintuku

very cool

Curio

If X is a discrete random variable, is f(X) a discrete random variable?
And viceversa?

If X is an absolutely continuous random variable, is f(X) an absolutely continuous random variable?
And viceversa?

shintuku

16:34

what is $f$? @Curio

Curio

@shintuku A generic function

shintuku

@Curio then no, we need the image of $X$ to be a probability space

geocalc33

17:26

Did I just solve the backwards heat equation and get a downvote?

Tough crowd

AMDG

Upvote/downvote has regressed to cringe subjectivism. I blame Jeff Atwood.

Says it in its own description. "I think this question is (not) helpful."

As opposed to, "This question is (not) helpful."

geocalc33

$$\frac{\partial^2}{\partial s^2}\Phi(s,x)=-\frac{x}{s}\frac{\partial}{\partial x}\Phi(s,x)$$

that's what I solved by the way

AMDG

Wait, hold on. Sorry. I stand corrected. Either it's changed (at least for math SE), or it's always been like this.

Yep, seems changed compared to before.

"This question (does not) show(s) (any) research effort; it is (not) useful and(or) (un)clear"

Ok, but... what is Stack Exchange in essence?

A Q&A site*

*questions must be useful and clear; questions must show research effort at all?

Also it probably wasn't "I think..." before, but it may as well have said so.

Koro

17:42

now they close library by 6 pm evening. Tell me a place in this entire universe where such nonsense happens.

AMDG

Your place, apparently

Nah but fr I think I remember the libraries I've been to had such closing times or the like.

Koro

at the amazing college where I did my UG from, this was not the case. Library there closed at 12 (mid night) and opened at 8 am.

every day

AMDG

RIP all-nighter study seshns

Koro

and apart from that, we had departmental library which was open 24*7.

Xander Henderson

@AMDG They were never good for anyone, anyway. :P

Koro

17:45

and apart from that, hostel library which was also open 24*7.

AMDG

Literally. I don't get why anyone thinks all-nighters are productive. Sleep deprivation is a serious problem these days making us perform poorly in the day.

Xander Henderson

But, yeah, my undergraduate and graduate institutions had libraries which were open past midnight.

AMDG

hostel library

Next, you'll tell me to rush B

Xander Henderson

But staffing a library is expensive, and it has been difficult to hire people in the last couple of years. Even students.

(And I would find it hard to make a good case for hiring a student to work the swing shift at the library---they are supposed to be students first, and those work hours are not going to be conducive to their studies).

AMDG

It doesn't help that we've got all these weird, arbitrary systems named after people no one remembers like "the Dewey decimal system" as opposed to a strict lexicographical perfect hashing function.

Xander Henderson

17:48

@AMDG Yeah, I think maybe this is a you problem.

Koro

Have you ever turned off lights in a library and slept on a sofa there?

AMDG

;-;

Xander Henderson

The Dewey system organizes things by topic, which is incredibly useful when you have actual books on the shelf.

I would often go into the stacks to find a particular text, and end up with three or four of its neighbors, instead.

AMDG

Eh... you can still do it by alphanumerical lexicography that doesn't give me a number and possibly something that can be figured out on-the-fly by just reading the title and author names.

Xander Henderson

@AMDG I have no idea what that is supposed to mean.

AMDG

17:50

Then determine the topic (which can't be determined per se from the title and author) and once again sort in-place lexicographically.

Lexicography seems to be something entirely different to what I meant in essence.

Lucas Henrique

hi there. do y'all have any good references on algebroid curves? I've been reading about them for my master's but I don't really have a good intuiton on how algebroid curves should look or why we use the ring of power series beside being the completion of the associated poly ring

Koro

If it were under my control, I would fire the entire staff at my college and rehire people.

Xander Henderson

I still don't understand what you are proposing... the idea of Dewey's system is the you first sort books by topic. Closely related books will be next to each other on the shelf. Within topics, sort by author and title.

AMDG

You have an alphanumeric string. It is a sequence in a finite tree of depth equal in length to the string where each node is just the latin alphabet.

The string itself is a path in this tree.

Xander Henderson

That sounds an awful lot like Dewey's system...

Koro

17:53

Also here, I almost never find the books I am looking for.

AMDG

We can generalize this to all permutations of the alphabet, then extend every finite string such that it has infinite length, then this string is a unique path in the infinite tree.

geocalc33

you're re-inventing the Dewey

Koro

They do not have Frietag and Busam for example.

PNDas

Suppose $B$ is a two dimensional Brownian motion and $D_r=\{x\in\mathbb R^2: ||x||\leq r\}$.How to compute $P(B_t\in D_r:B_0=0)$ for any $t\geq0$?

AMDG

Ok, but... where's number 626.25? Indeed, they must necessarily be the same in essence because any arbitrary association reduces to a tree of some kind.

Lucas Henrique

17:54

maybe you know something about this, @Ted? I'm currently trying to understand the construction of the value semigroup of a space curve singularity but the choice for a complete ring here seems so arbitrary... any tips on visualizing something like $k[[x_1, ..., x_n]]/\mathfrak{p}$ versus $k[x_1, ..., x_n]/\mathfrak{p}$?

PNDas

I was thinking of using $B_t\sim N(0;t)$. But don't know how to proceed. Any hints?

Koro

I miss my UG college so much.

Xander Henderson

@AMDG 626 is an unused part of engineering.

Specifically: 600 (technology) -> 620 (engineering) -> 626 (unused).

Ted Shifrin

@LucasHenrique Absolutely nothing!

Lucas Henrique

@TedShifrin damn. either way, thanks!

AMDG

17:59

@XanderHenderson My complaint is specifically that I have to use a number to find a book instead of some alphanumeric string that can easily be recalled, easily computed (especially in my head), and then its physical location easily found (which is a different matter and really beyond scope here unless you can build a library from scratch and design the building).

geocalc33

My complaint is that the Dewey decimal system to too good

I can find books too easily

AMDG

In other words, my gripe is with the interface, not the system itself per se.

Xander Henderson

@AMDG Which basically means that you want to replace "600" with "tech", "20" with "eng", and so on, leading to a complicated scheme like "tech.eng.whatever", which is unlikely to fit on the spine of a physical book very well.

AMDG

Poor user experience.

Xander Henderson

The point of Dewey's system is that it does tell you how to find a physical book on an actual shelf, organized into a section with related books, in a way that doesn't take up too much space on the spine so that you can, you know, still read the spine.

geocalc33

18:02

The point is that you can't outdo the Dew

Xander Henderson

Readability is a secondary concern.

Of course, if you use Dewey enough, you start to grok the system, and it becomes second nature.

Ted Shifrin

@XanderHenderson When I was in school and college, I was pretty good at it. These days ….

AMDG

@XanderHenderson Basically. As for that stuff, well, a number on the spine is an implementation detail. Ideally, you can just know where it belongs from the book itself, thus no need for a spine tag. If you're a human, you have no trouble categorizing. The problem is finding a perfect hashing function that is simple enough that you could do so just by title and author (excepting collisions in the case that you have two different books with different authors of the same name having the same title).

Yeah, well, I was never one to be so acquainted with books as to constantly look for them on the shelf. Even so, if you get a book, how long does it take you to read it cover to cover? Are you going to really remember that 600 is technology? Easy enough solution is to put lookup tables everywhere, but still...

I'm a man of quality. I don't seek a solution that merely works. I seek solutions that work; easy to use; convenient; etc.

geocalc33

So if you were to solve the Riemann hypothesis you would solve it very elegantly? @AMDG

Xander Henderson

@AMDG You seem to be missing the point.

AMDG

18:10

@geocalc33 lol

@XanderHenderson So what's your point then?

Xander Henderson

Dewey is meant to help organize books in a physical space, where the physical size of the "hash" is important. It does a damn fine job of this. What you are proposing might be more "human readable", but it isn't going to work well in a physical space.

The intended use of Dewey, for the vast majority of people, is "look up a book in the catalog -> get a set of coordinates for that book -> walk to the physical shelf".

It is like using latitude and longitude (or UTM coordinates) to identify a spot on Earth.

AMDG

~~I mean they all take up the same volume no matter how you permute them in space.~~

@XanderHenderson Ok, but everyone uses Google maps, not a GPS, ironically.

It's funny because it uses GPS for its implementation

If you have two unique perfect hashing functions, then they necessarily have an implicit mapping between them.

Xander Henderson

@AMDG And real people use card catalogs (and their modern, digital analogs).

AMDG

You could use Dewey for organization, and the better, human-readable one as the human interface.

Xander Henderson

@AMDG Again... the card catalog.

AMDG

18:16

I just don't like flipping bits in my hard drives using a microscope and magnetized needle

Xander Henderson

The Dewey code is like the lat/long position of a book; the catalog is like the nicely formatted Google map.

But, hey, if you think that you can do better, I would encourage you to pursue some coursework in library sciences, learn more about why things are the way they are, propose a better solution, and get that implemented.

AMDG

Ah ok. If only I had been introduced to card catalogs then.

Well, this is really just an information technology domain problem, so nothing new under the sun here for me to study.

Information here in more ways than one :P

Physical location? That's the bin packing problem

Or one of those...

subset or superset... the knapsack problem? idk

Xander Henderson

@AMDG How very dismissive of a field that you seem to know nothing about...

AMDG

I can never remember the names of the things I work with, but then again, I rarely have need of communicating them to others, so I forget easily.

@XanderHenderson If you can convince me it is something more than a physical database, then I will be glad to call it a field with you.

Tell me what a library is beyond that

Specifically as regards books

Xander Henderson

@AMDG Library sciences is a field. It includes understanding the physicality of a library, knowledge classification schemata, knowledge of the needs of different kinds of libraries (academic, public, etc), knowledge of abstract research methods, etc.

shintuku

18:23

there's also value theory: how do you choose what to catalog and how do you choose the ease of access of each thing in accordance to it's potential usefulness. useful for archival libraries and newspaper libraries (both problems arising from the limitedness of space/resources)

Xander Henderson

The problem of categorizing and cataloging actual physical texts is a problem which has been studied in that context. The reason that Dewey is still used is that it works remarkably well, and no one has managed to come up with a better solution. Institutional inertia likely plays a role (Dewey may be a local optimum, with great costs associated with getting away from that optimum towards a better one).

There is also the fact that you seem to believe that there exists an "easy to use, convenient, etc" solution. I think that if you actually spend time studying the problem, in context, you will find that these things often run counter to each other.

"Cheap, fast, good: pick two."

shintuku

so an application for value theory is e.g. suppose you're a bank and you have to catalog every paper element that might potentially be useful in a lawsuit

if you think about this for a few seconds you'll figure out that it isn't an easy thing to do

geocalc33

wow that's extremely difficult

AMDG

@XanderHenderson Can you provide a more specific definition of library sciences as a field?

I'm not fully understanding based on what you've said here.

Xander Henderson

Library and information science

Library and information science(s) or studies (LIS) is an interdisciplinary field of study that deals generally with organization, access, collection, and protection/regulation of information, whether in physical (e.g. art, legal proceedings, etc.) or digital forms. In spite of various trends to merge the two fields, some consider the two original disciplines, library science and information science, to be separate. However, it is common today to use the terms synonymously or to drop the term "library" and to speak about information departments or I-schools. There have also been attempts to revive...

AMDG

18:33

First paragraph is mentioning everything that belongs to databases

Xander Henderson

@AMDG So dismissive...

AMDG

I'm not sure what's wrong with that, though. If it is literally nothing more than the study of databases of a physical kind, then what more is there for me to say? What that ultimately means is that something like a SQL database belongs to library sciences, but more fundamentally, it's just a giant hash map, so what else would you prefer I say?

shintuku

10 mins ago, by shintuku

so an application for value theory is e.g. suppose you're a bank and you have to catalog every paper element that might potentially be useful in a lawsuit

do you think there is an easy solution to this

Xander Henderson

@AMDG Your approach here comes across as incredibly arrogant. You have a certain toolset with which you are familiar, and you regard this problem as just "an application" of that toolset, thereby dismissing all the hard work of actual people who have actually invested time into understanding how these things actually work.

It is like talking to a certain breed of category theorist: "Oh, hey, analysis is just category theory, in the category of sets where the morphisms are smooth functions, right? How hard can it actually be?"

shintuku

so, bank organizing its papertrails against lawsuits, first example. second example: historical economics databases. you want to keep an archive of the bank of england between 900 and 1200 that will have significance for research in the history of economics. what documents do you keep, which ones you don't, and how do you proceed?

historical databases in general have this problem

geocalc33

18:41

you take the derivative wrt. time maybe

that is an interesting question

is that like when do you shed documents and when do you obtain more?

gotta be some sort of partial differential equation

shintuku

@geocalc33 that's one of the problems you'd face

it's hard to fathom the sheer amount of information produced by economic entities on a daily basis

let alone the bank of england, for two centuries

in paper

geocalc33

maybe you try to make inputs equal outputs across time

if you want the document total to stay constant

shintuku

@geocalc33 you'd want value over quantity. what you're interested in is finding a measure of the value of documents

for very real and concrete potential uses

geocalc33

What a beautiful AND theoretically rich equation $$-\frac{t^3}{xyz} \frac{\partial^4}{\partial t^4} H(t,x,y,z) -\frac{3t^2}{xyz} \frac{\partial^3}{\partial t^3} H(t,x,y,z) -\frac{t}{xyz} \frac{\partial^2}{\partial t^2} H(t,x,y,z) = \frac{\partial}{\partial x}\frac{\partial}{\partial y} \frac{\partial}{\partial z} H(t,x,y,z)$$

shintuku

sometimes historical archives have a main catalog and another offsite one. the offsite one is absolutely MASSIVE, spanning kilometers, and the main catalog is a choice of stuff from the massive catalog that researchers think will be worthwhile to have easily accessible.

digging for stuff in the offsite archive takes time and effort and money, so you want to find how to minimize a situation where you'd need to access the offsite archive (and minimize the cost of accessing the offsite archive)

geocalc33

18:51

Yeah

AMDG

@XanderHenderson My apologies if I come off that way. That is never my intent. What I am interested in is bringing about the greatest good, and I recognize perfectly well that they had to climb the mountain first before me, but I was born atop it after them, and so I see no value in looking at a primitive hand saw when I comprehend the chainsaw. By way of the chainsaw, I already know the hand saw, at least in principle.

Likewise, I know the hashmap, and by it, I know at least in principle how what anything that meets the specification of a database is; how it works; etc.

Xander Henderson

@AMDG It may not be your intent to come across as arrogant and condescending, but you continue to double down on that attitude. Dismissing the Dewey decimal system as a "handsaw" when you have seen a "chainsaw" seems profoundly ignorant and reductive, to me.

Stop dismissing an entire field of study when you haven't taken the time to at least learn the basics, in the context in which they are studied.

AMDG

Yeah, you're right. I have a lot to reevaluate as I've realized rather recently, and this is just one of those many things I have yet to reevaluate. The simple fact of the matter is that the very poor experience I've had in public school, and the things I was merely introduced to from kindergarten and onwards, were never really explored by said institutions in such a way as to give me appreciation for them by way of understanding, ...

...so I still blurt out unchecked beliefs about those things, causing me to appear arrogant and condescending.

The library is just one big part of that. It was only a place to get access to the computer for flash games after (middle) school, and the Wii for playing Smash

AMDG

19:14

So... just to reflect on what we discussed before, what I saw that I didn't like about the Dewey decimal system was strictly in regard to numerical identifiers, but I was not aware of card catalogs which you mentioned, and this would have ended the discussion there.

This I know well about identifiers because I've had to determine the correct manner of naming things in my own field since apparently no one can do that, and the fact that the half-joking phrase "the hardest part about programming is naming things" is a sort of evidence of this. What is easiest for man is what is natural to him. The numbers are not, but words are.

I'm not sure how discipline looks in mathematics, but what I can tell you is that in my own field, it's a complete dumpster fire of a disaster. There's an abundance of programming languages and conventions as multiplicitous as the number of Protestant denominations, but no One True Standard to unite them in discipline, hence my project.

PNDas

How to calculate $P[B_t\in K]$ for some set $K$?

shintuku

the power set? if it's finite you have to do it manually

PNDas

Probability

B= Brownian motion

Sha Vuklia

heya @Thorgott, can I ask an algtop question?

AMDG

@shintuku Wait, so if what Dr. Henderson was trying to tell me is that it is a matter of how databases themselves are applied as a science, then I wholly agree it is a true field. To be clear, every science has a material object and a formal object. The formal object is the end to which the science operates; the material object is what it operates on. If the definition can be given to me by defining both objects for me, then I will know what library sciences consists of.

Koro

19:26

how is customer care service of samsung?

apple's customer care service sucks.

shintuku

@AMDG it's organization of information following a variety of constraints, i gave you a couple of examples

AMDG

Reading the Wikipedia article, the formal object is given by "Humans becoming informed (constructing meaning) via intermediation between inquirers and instrumented records". Well that clears that up. Sorry, Shin, but your explanation is too vague to be a definition.

Its material object is information itself

shintuku

whatever the idea is to communicate it's not an obvious matter

and if it is obvious to you, you can make serious money out of it

AMDG

Well, generally speaking, I don't know what others mean when they say something is hard or difficult in essence, and I don't know what makes it to be so for others really.

I just know for myself that whatever I put my mind to is eventually solved no matter the difficulty, and I don't get much use out of those terms, so I don't really know what exactly they mean. I've forgotten if I did know at any time in the past.

Rain and wind will eventually wear a granite core from an ancient volcano.

Sha Vuklia

I guess I'll give the context: I want to show that for an $n\geq 1$, given a CW-complex $X$ of dimension $n$, we have $H^n(X,\mathbb Z)\cong [X,S^n]$. I know the result which says that
$$H^n(X,\mathbb Z)\cong\langle X,K(\mathbb Z,n)\rangle,$$
where $K(\mathbb Z,n)$ is the Eilenberg-Maclane space with $n$-th homotopy group equal to $\mathbb Z$. I believe we can take $K(\mathbb Z,n)=P_n M(\mathbb Z,n)=P_nS^n$, where $P_n$ stands for the $n$-th degree space of the Postnikov tower (and $M(\mathbb Z,n)$ is the Moore space with $\mathbb Z$ for the $n$-th homotopy group). So I would like to show that

AMDG

19:34

@shintuku What I can tell you, however, is what is clear to me as a programmer, though I don't find the problems obvious. In principle, it would appear, in the case of lawsuits, assuming a lawsuit is an object of investigation, then what is useful to store in principle is all those things which tie a primary cause to a final cause.

shintuku

yeah but what is that

AMDG

This must be so since investigation involves studying effects to determine causes.

shintuku

what things satisfy that condition

and if there are too many things satisfying that condition, how do you choose which ones to keep

AMDG

Well we can investigate it to find a specific rule that is universally applicable for any given circumstance, though said rule may be so general that it requires building rules using this rule in order to find a specific answer.

Ted Shifrin

@ShaVuklia This looks very false to me.

Try $n=2$, $X=S^3$.

AMDG

19:38

@shintuku You said this is a matter of banking, right? So what information is important? Is there anything besides that information which traces any sort of financial transaction or string of transactions to certain causative agents?

Sha Vuklia

@TedShifrin what exactly looks false? the initial statement that I try to prove?

$S^3$ is not 2-dimensional

note the dimension restriction on $X$

shintuku

@AMDG read Archives: Principles and Practices by Millar and Managing Archives: Foundations, Principles and Practice by Williams if this interests you, i'm busy with something else atm

Ted Shifrin

Oh, I didn't note it.

Sha Vuklia

understandable, there is a lot of text

I gave the full context in case an alternative argument works too

shintuku

@AMDG also Records and Information Management by Saffady

Ted Shifrin

19:40

So this is basically Hopf's Theorem, that for a (compact, oriented) $n$-manifold $X$, two maps $f,g\colon X\to S^n$ are homotopic if and only if they have the same degree.

AMDG

@shintuku I'll add it to a list of things to read in the future. I'm unfortunately too busy to do much, though I have to reevaluate my schedule itself too, so there may in fact be something I can do with that.

If it doesn't strictly pertain much to my profession, I shouldn't bother with it except as recreation.

Ted Shifrin

Hopf's Theorem was one of my favorite results to prove toward the end of the differential topology class I used to teach.

Back to your regularly scheduled programming.

Sha Vuklia

@TedShifrin that's a cool result!

Semiclassical

20:30

@leslietownes following up on the interval problem, I came across the following generalization in the literature:

Ted Shifrin

Minor third or augmented sixth?

Semiclassical

“for any $k < n$, the probability that in a family of $n$ random intervals there are at least k which intersect all others is $2^k/\binom{2k+1}{k}$”

(They don’t prove it, but they give a construction for the k=1 case which apparently gives this formula if one proceeds using “slightly more care”)

leslie townes

semi: i see you've made your peace with "intersect all others" :)

google suggests maybe we can blame "all others" on peter winkler

shintuku

21:02

♬late day coffee yeah yeah yeah♬

Thorgott

21:24

@Sha I think what you want for this is a fairly explicit model of $K(\mathbb{Z},n)$. Show that you can build a $K(\mathbb{Z},n)$ from a $S^n$ by only attaching cells of dimension $\ge(n+2)$. Then, $S^n$ is the $(n+1)$-skeleton of $K(\mathbb{Z},n)$ and the result you want is a consequence of cellular approximation (injectivity follows by applying cellular approximation to the homotopies, which requires that $S^n$ be the $(n+1)$- and not just the $n$-skeleton to work).

Sha Vuklia

@Thorgott o shoot, I had thought of this, but I thought it didn't work

because I'm aware indeed of that result about $\geq n+2$

great thanks, then I'll work that out

AncientSwordRage

Trying to figure out what to do with this question of mine

1

Q: Can I somehow simplify the perimetre of a rounded segment?

I set myself a challenge of calculating the perimetre of a rounded segment (that is, a segment with the 'sharp' edges rounded off). Here is a rough explanatory diagram: And with the perimetre shown in red. $$\theta=\sin^{-1}\left(\frac{d + r}{R-r}\right)$$ $$A=R\left(\frac{\pi}{2}-\theta\right)...

geometry trigonometry circles

I feel like I can phrase my actual question better if I delete and re-ask

Ted Shifrin

21:46

I totally have no idea what you’re talking about. A segment has no width, so cannot be rounded. Are you talking about rounding off the ends of a thin rectangle? Just add semicircles.

D Left Adjoint to U

21:59

Oh nvm, still has 0 width

Transcript for

$Mathematics$ Mathematics