Mathematics - 2023-06-24

linear_combinatori_probabi

03:19

Hi, could anyone help me with some simple math formulas?

I'm not the author of the math formula so forgiving me that in some cases it might not be well-defined.

Here is the description of my problem:

Given two positive variables $i,j$, and $i < j < n$, where $n$ is a fixed positive integer.

Now define a function $T(i)$ as follow:

$$ T(i)=\min_{i<j<n}( \max{(d_i+T(j))}, \max_{k}{((\sum_{s=i}^{k})+g_k+g_{k+1}} )) $$

What I want to do is that I want to prove that $T(i)$ is a decreasing function. (those $d_i, g_k$, given any $i,k$ are all positive integers.)

my apologies that this formula might not be well-defined (I'm not the one who first wrote it) and you might have questions like "what is $d_i, g_k$" all I can say is that this is a formula from Machine Learning and $d,g$ mean data, gradient respectively. I cannot provide more details.

D.C. the III

03:35

what is the summation summing over? You have a summation but there is nothing the index is attached to.

linear_combinatori_probabi

Lol, you're right I made a typo

there should be a $d_s$ inside the parentheses of the only summation.

D.C. the III

I would rewrite it out so others can see it in a cohesive sense

linear_combinatori_probabi

Here is the fixed formula:

$$ T(i)=\min_{i<j<n}{( \max{(d_i + T(j))} , \max_k {( (\sum_{s=i}^k{d_s}) + g_k + g_{k+1} )} )} $$

my bad eyes typing these formulas with a small font size.

D.C. the III

unless somebody here is familiar with the nuances of Machine Learning, all of $d_i, g_k, T(j)$ are just symbols so without any definition of the behaviour of those objects I'm not sure how you can prove said claim.

linear_combinatori_probabi

We can assume that $d_i = g_i$ (let me omit the context), and for simplicity we can assume that $d_i$ are all small integers like $1<d_i<100$. $T(i)$ is a function for the calculation of memory usage. Thanks for your reply anyway.

But from my current understanding, the formula itself depends on zero knowledge of ML. There is no hidden relation between $T(i),d_i,d_s,g_k$. Except for the function $T(i)$, all these variables can be thought of as just some positive integers from a log file.

D.C. the III

03:52

Ok, then how is the calculation for memory usage done? Because it seems $T(i)$ is going to be important in determining that the function is a decreasing one

linear_combinatori_probabi

The formula is the definition of $T(i)$. The calculation is done reversely. The base case is $T(n)$, and finally it will reach $T(0)$ and that's the answer.

D.C. the III

so induction perhaps?

the cavalry has arrived

linear_combinatori_probabi

but yes, when I said I want to prove that $T(i)$ is a decreasing function I do mean: $T(i)$ decreases when $i$ increases. (not saying you didn't get it, just want to put emphasis on this)

Yes :) I know Ted he helped me a lot years ago. (I also subscribed to his lectures on YouTube, but that's off-topic :P)

D.C. the III

We are in common as I'm working through his youtube lectures now. lol

linear_combinatori_probabi

I used to like math seriously then realized that I'm really bad at math so now I'm doing computer science and realize... it is still about math, lol.

D.C. the III

03:59

math is the foundation of it all.....you cannot hide from it... lol

linear_combinatori_probabi

exactly.

D.C. the III

but back to this. so overall is it the case that $T(0) < T(n)$?

linear_combinatori_probabi

the other way, it's $T(0) > T(n)$. The ML context is that we're doing backward propagation.

D.C. the III

or actually it is the cse that $T(n) < T(0)$ after reading your description

linear_combinatori_probabi

indeed a good question, and as you said these are just objects so without a context it's hard to reason about. apologies for this.

So it just stated as a lemma that $T(i)$ is decreasing function and the proof is just one-liner: by observation from the definition of $T(i)$. Lol

Ted Shifrin

04:04

Too yucky for me.

D.C. the III

I don't see why induction could still not be used though.....I may be wrong.

@TedShifrin not enough geometry.

Ted Shifrin

the final max is largest for $i=0$, but so what.

D.C. the III

it would just go in $T(0) > T(1) >T(2) > \dots$ still inductionable

Thomas Finley

07:37

Our professor gave this to us, and I am pretty sure that this question has too many typos!

First of all, the recurrence relation in all probability should be $u_{n+1}=\sqrt{2u_n}$ and there is an extra phrase repetition of "and $u=\sqrt 2$ " and thirdly, the part $u=\sqrt 2$ should've been $u_1=\sqrt 2$ , right?

What d'ya all feel?

Am I the one mistaken ?

Imho, this is a too popular question.

leslie townes

07:52

i think that's a reasonable interpretation of what's given, but ask your professor to be sure. there are enough weird things about that that they may have intended something else.

Jakobian

08:30

This is a simple geometric series

$u_n = 2^{\sum_{k=1}^n 2^{-k}}\to 2^{\sum_{k=1}^\infty 2^{-k}} = 2^1 = 2$

Thomas Finley

08:56

@Jakobian Yeah

@leslietownes I'll give it a try, but prolly that's what they had in mind. Typo can often lead to ambiguous decisions.

I had one more question.

It's a very popular theorem that : A sequence (x_n) is convergent iff limsupx_n=liminfx_n

Strangely, I found a two pages proof of this! I feel it can be done in 3-4 lines.

If (x_n) is convergent, then all the subsequences converges to the same limit and hence, liminfx_n=limsupx_n immediately follows.

Jakobian

depends on your definition of limsup and liminf

Thomas Finley

@Jakobian hmm, that's correct, but lemme write the converse part and I'll add the defn after I finish...just a sec

PNDas

09:11

Wiki says if $X$ is independent of $\sigma(Y,\mathcal H)$ then $E(XY|\mathcal H)=E(X)E(Y|\mathcal H)$. And it's not necessarily true if $X$ is only independent of $Y,\mathcal H$.

I want to find a counterexample.

While proving this I'm using $X$ is independent of $Y \chi_A$.

PNDas

09:25

And in the next statement it says suppose $X,Y$- independent, $\mathcal{G,H}$- independent, $X,\mathcal H$- independent and $Y,\mathcal G$- independent then $E(E(XY|\mathcal G)|\mathcal H)=E(X)E(Y)=E(E(XY|\mathcal H)|\mathcal G)$

Now how to show that $E(XY|\mathcal G)=E(X)E(Y)$?

Thomas Finley

I was cut off in between.

Reposting the whole thing together.

Jakobian

I'd rather you write it in LaTeX

Thomas Finley

Okk

Latexing...

Jakobian

csd.uwo.ca/~watt/pub/reprints/2008-casa-symdecomp.pdf

here's something interesting, it's a problem I was interested in high school and remember solving it for small cases

Thomas Finley

I had one more question.

It's a very popular theorem that : A sequence $(x_n)$ is convergent iff $limsupx_n=liminfx_n$

Strangely, I found a two pages proof of this! I feel it can be done in 3-4 lines.

If $(x_n)$ is convergent, then all the subsequences converges to the same limit and hence, $liminfx_n=limsupx_n$ immediately follows.
Conversely if $x_n$ is such that $liminfx_n=limsupx_n$ then, let us conisder an arbitrary subequence $(x_{n_k})$ that is convergent. Now, $limx_{n_k}=l$(say) then, $l<=limsupx_n$ and $l>=liminfx_n$ and it follows that, $l=limsupx_n=liminfx_n$. Again, since $x…

@Jakobian Here it is, latexed...

Jakobian

09:35

alright. Now fix the all the limsups, liminfs, inequalities, lims

$\liminf$

if I add a backslash in the front as here, I get a pretty liminf

for inequality I can just type geq or leq with backslash in front

writing <= is ugly

same thing for >=

it's passable in text but here we have MathJaX

Thomas Finley

I had one more question.

It's a very popular theorem that : A sequence $(x_n)$ is convergent iff $\limsupx_n=\liminfx_n$

Strangely, I found a two pages proof of this! I feel it can be done in 3-4 lines.

If $(x_n)$ is convergent, then all the subsequences converges to the same limit and hence, $\liminf x_n=\limsup x_n$ immediately follows.
Conversely if $x_n$ is such that $\liminf x_n=\limsup x_n$ then, let us conisder an arbitrary subequence $(x_{n_k})$ that is convergent. Now, $\limx_{n_k}=l$(say) then, $l\leq \limsupx_n$ and $l\geq\liminfx_n$ and it follows that, $l=\limsupx_n=\liminfx…

Jakobian

looking better but MathJax interprets what you wrote as liminfx and limsupx instead of liminf and limsup

Thomas Finley

@Jakobian I know, but text here don't render in mathjax in my browser, that's why I had the mess

leslie townes

clearly it can't be done in 3-4 lines if you keep spamming it into the chat

Jakobian

there needs to be a space there

that's hilarious

Thomas Finley

09:39

@leslietownes You know why I had to spam:)

Jakobian

are you on a mobile phone?

Thomas Finley

Enough of it, now what about my proof. Jokes apart

@Jakobian yes

Jakobian

if not you can bookmark MathJax tinyurl.com/cfqcvpc

Thomas Finley

@Jakobian Is my proof valid?

Ofc if you can read it.

@leslietownes Comment if you can decipher it. Thanks

Jakobian

It's not perfect as I'd like it but I can somewhat read it

leslie townes

09:42

the validity of any of that kind of stuff depends on the definitions, like jakobian said a long while ago

if the only reason you're doing this is that someone somewhere wrote too long of a proof, maybe consider just not doing it

Jakobian

you know, it's important to latexify your questions if you want others to answer them - pleasure for eyes means more people are willing to read it

Thomas Finley

@Jakobian i have a solutionwhich don't spam things. If you like u can see it, in some time.

Jakobian

alright so your definition of limsup and liminf seems to be using least/greatest limit of subsequence

Thomas Finley

@Jakobian yup

Jakobian

I mean your proof sounds alright, though it will probably have more luggage

in that for the definitions to make sense you need to define and prove more

we're taking those things for granted now

Thomas Finley

10:05

0

Q: Can this proof be done in $3-4$ lines? Or did I miss anything?

It's a very popular theorem that, " A sequence $(x_n)$ is convergent iff $\limsup x_n=\liminf x_n.$" The definition of $\limsup$ and $\liminf$ of a sequence that I am using are as follows: If $(x_n)$ is a sequence in real number, then we define $\limsup x_n $ as the greatest subsequential limit ...

real-analysis

@Jakobian No need to take things granted, as I took the trouble and made it as pleasing and formal as possible.

Less spammable @leslietownes as well.

@Jakobian Now, pass the long-awaited remarks.

Phew, I am tired. Maybe, should've done this earlier.

But, as they better late than never!

Jakobian

@ThomasFinley uh... I think you misunderstood what I said. It's up to me and everyone that reads what you wrote to decide if you probably know these things already or if it's an assumption that you never encountered and we need to check you from it

so yes there is a need to take things for granted because I don't sit in someones head

leslie townes

"if x_n is a sequence with the property that every convergent subsequence of x_n converges to L, then x_n is itself convergent and converges to L" is generally false, and it seems like maybe you're implicitly using it in your proof

please don't ping me with 20 replies to this, that's the last i'll be saying about it

Jakobian

Well, I was thinking in extended reals

leslie townes

which comes back to all of this stuff really depending more on the definitions than anything else

Jakobian

I think this is a good example of why mathematics isn't about the formalism but the results and their interpretations

Thomas Finley

10:19

@leslietownes I wont ping you with 20 replies to that. But just one reply to it as this. I don't know why that assumption is wrong. If you clarified this, it would be helpful, but again as you mentioned that was the last and first time you'll say it, so I don't know what to say, anything else.

leslie townes

@Jakobian another meta-theme is, while generally, concise is better than verbose, and we can often recognize bad organization when we see it, 'length of a proof' isn't that useful of a concept

some of those arguments in rudin's PMA come to mind

Jakobian

To me what comes to mind is that one article by Conway :)

I just think simple but with enough details to be comprehensible - that's whats the best

but definitely there's some skill in putting words in such a way that they're easy to understand

Rudin is a special case, it's supposed to be a teaching material but I don't think Rudin entirely thought of the book that way

he definitely did a sloppy job with a lot of the proofs - for better or the worse

leslie townes

i mostly meant, while a lot of the time a shorter proof might be clearer or closer to the fundamental ideas, sometimes it's not. if you optimize almost purely for length (which i do think rudin was doing at least some of the time, for whatever reason) you don't really get "simplest"

Jakobian

but this will also keep on repeating I guess, it's more important to understand what author tried to imply rather than what they wrote

ah okay so you're arguing that short proofs don't necessarily give us the reason for why some things hold

from Rudin I only read his complex and real analysis book, I don't really find it exciting

leslie townes

10:37

you don't see it too much on MSE because most questions are pretty close to the surface, but sometimes people will say things like "oh, why don't you avoid that complicated theory and JUST do blahblahblah, then you don't need theorem X and you save four lines" and they save the space by reproving the case of theorem X that they need with two or three "non-obvious" ideas that come out of unrolling some proof that makes more sense and snipping out the concepts

and its like.. okay, if it doesn't apply to anything new, all it does is obscure the general thing it came from, it's not a better argument, even if it's shorter

ink and paper are basically free, pixels are even more free

the other thing i meant was, you can make a proof of anything short if you make the definitions long, or vice versa :D there's a principle of conservation of difficulty that you can't really get around just with clever argument

Jakobian

I'm thinking of teaching myself how to cook

but it sounds hard

leslie townes

it depends on your goals i think :D if you want to make a really broad range of stuff well, that takes a ton of time and work

if you just want the skill of being able to create meals for X people using Y ingredients and Z time, it's very doable

but maybe also less fun

Jakobian

I can't even make a pancake

I tried it before, and failed

I think my error was I put too much oil? Or too little? I don't know

cooking is complete black magic to me

leslie townes

yeah somehting like that sounds maybe like just an error in the recipe you were using

Jakobian

I wasn't using a recipe, my dad was standing there and telling me how much oil to put and I still failed

leslie townes

10:49

haha. can he make them?

Jakobian

Yeah. He was making them in the first place, I just got invited to try

leslie townes

i do think of baking as a whole lot harder to do well than other kinds of cooking, and i'd put pancakes in that category, but anybody should be able to make a passable pancake

Jakobian

I mean those aren't American pancakes more like thin ones

leslie townes

and it really might be as simple as 'less oil' or something like that

where you just change a single variable once, try again, and then suddenly you can do it

by 'baking' i mean like cooking with starch i guess and turning it into something that isn't burnt

not making any particular style of pancake

Jakobian

I find it hard to see how much oil I'm adding in the first place, so it's hard for me to even compare them between tries

it probably comes with experience

user 858770

10:53

Try a measuring cup :-)

leslie townes

^^

but yeah there's some amount of practice in it which will involve messing food up, but it's significantly less iterative than you might think

unless you're ambitious

user 858770

Like the French.

They make great crepes.

user 858770

11:16

Pancake

A pancake (or hotcake, griddlecake, or flapjack) is a flat cake, often thin and round, prepared from a starch-based batter that may contain eggs, milk and butter and cooked on a hot surface such as a griddle or frying pan, often frying with oil or butter. It is a type of batter bread. Archaeological evidence suggests that pancakes were probably eaten in prehistoric societies.The pancake's shape and structure varies worldwide. In the United Kingdom, pancakes are often unleavened and resemble a crêpe. In North America, a leavening agent is used (typically baking powder) creating a thick fluffy pancake...

Wow, huge article :O

Sha Vuklia

14:32

I think MSE should incorporate the option to compile tikzcd code

Thomas Finley

Guys! Check out my new experiment:

0

Q: An Alternative Proof of Bolzano-Weierstrass Theorem(?)

My version of Bolzano-Weierstrass theorem states that: "A bounded sequence in $\Bbb R$ always have a convergent subsequence." Till now, I only knew one or two standard proofs that are always cited in books. But suddenly when I , out of just sheer curiosity tried proving it, I found that there is ...

real-analysis alternative-proof

Sha Vuklia

4

Q: Embedding tikzcd diagrams

I know some tags are not supported, and some of them will likely never be (I have the <table> tag in mind). However, it is sometimes useful to embbed commutative diagrams in some question/answers. Currently, the "life hack" that does not require third-party tools is somewhat dirty, and the beauti...

feature-request mathjax

I'm surprised this post has so few upvotes.

Thomas Finley

Lately chat has become less active. Dunno where everyone went :/

3

Soumik Mukherjee

15:05

@ThomasFinley Try taking lim sup instead. Just supremum won't work as they may be isolated.

Thomas Finley

@SoumikMukherjee yeah, that's the mistake I made. The thing is, I assumed implicitly that supremum is always a limit point of a set, but is not true.

Jakobian

15:53

@ThomasFinley they're dodging all the noise coming from being mentioned by another user

@AlessandroCodenotti I found a really nice book about topology, geometric aspects of general topology by Katsuro Sakai

in the sense, that I like the authors style of writing (and the subject too of course)

shintuku

this is the end of mathSE chat, it was predicted a long time ago

user 726941

Just in time to see Moscow fall.

shintuku

wtf how did i miss this

a civil war has broken out in russia

Jakobian

What are you currently learning?

shintuku

integral elements of a number field form a ring and the head of wagner group rebelled

user 726941

16:04

Snooze you lose.

Jakobian

Sounds very number fields

user 726941

To go with the battlefields.

Thorgott

Sakai's book is great, Alessandro recommended it to me before

I learned some ANR theory from it

Jakobian

@Thorgott it has a sequel

he seems to be going into the infinite-dimensional manifolds direction

I already read a bit from a similar book by van Mill, though I kinda dropped it after a while

well, I'm definitely going to read it at some point, both of the ones by Sakai

Thomas Finley

16:41

@Jakobian Not a good joke. Today's comedies have turned into "roast videos". This is in general, I don't understand what sort of fun, one gets by "roasting" someone. Maybe, that's a new sense of comedy, I have to cope up with. Next time, you give a hand at writing jokes, be sure tomaintain quality standards.

Or better, you might take tips from me, just like I on some occasions take tips from you on topology ;)

@user726941 Oh, I missed it too!

(Idk if that's real)

Jakobian

16:56

It wasn't a joke

shintuku

jakobian the master joker

what do you call it when you think you're being roasted but you're not actually being roasted. are you actually roasted or not? i think you are because you feel roasted

Ted Shifrin

17:13

Salut @Asytx

Soumik Mukherjee

@shintuku I feel roasted by the scorching summer heat

Thomas Finley

17:31

@Jakobian I know this is a joke as well , but not a good one. Try hard! Practice makes a man perfect.

I will consider you to be a success when you will be able to make me laugh out loud, but that won't be too easy;)

shintuku

The Solipsistically Roasted Man

a story in seven acts

Jakobian

17:55

imagine doing something, being completely aware of it, and then getting passive aggressive when someone points it out

I find it very strange to say the least

robjohn

@SoumikMukherjee make sure to salt yourself well. No sense in roasting without it.

@Jakobian Some of us can't be here all the time, you know.

;-p

shintuku

i would strongly recommend against any attempts to roast yourself, or any other form of preparing oneself as a main dish of any sort

Jakobian

@robjohn I don't get it. What do you mean?

oh, I get it

robjohn

@Jakobian I was trying to be passive aggressive about something

Jakobian

oh. I didn't get that part

Alessandro Codenotti

18:06

@Jakobian yes it's a great book

I didn't know it has a sequel, what's the title?

Jakobian

topology of infinite-dimensional manifolds

it's from 2020

Xander Henderson

18:38

@shintuku What about as a side?

Maybe with some fava beans?

And a nice chianti?

shintuku

in case you can't get out of preparing yourself as a dish and must absolutely do it it is of the utmost importance to consult with your doctor about the specific modalities and procedures to ensure the diner does not contract kuru or some sort of prion disease

i absolutely did not mean you shouldn't prepare yourself as a dish because there is a significant chance of injury, this is a free country and anyone telling you you can't do something is a communist

Xander Henderson

19:03

@shintuku Hey, I'm not eating brain tissue. It'll be fine.

shintuku

truly an icon of freedom

Xander Henderson

Also, the risk of a prion disease is exceedingly small, particularly in the US and Europe.

shintuku

truly a well-informed icon of freedom

Xander Henderson

So even if I were eating brain, I don't think that there is anything to worry about.

robjohn

19:29

@XanderHenderson Yes, Clarisse...

PM 2Ring

19:50

@冥王Hades That looks very similar to the one I showed you late last year.

Dec 17, 2022 at 13:54, by PM 2Ring

@ThomasFinley Comedy roasts are not a new phenomenon. See en.wikipedia.org/wiki/Roast_(comedy) FWIW, I've never been a big fan of that style.

Jakobian

20:14

@PM2Ring I'm not a fan of this humour either, and it wasn't a joke, just a snarky remark.

push the world and the world pushes you back ¯_(ツ)_/¯

I wasn't even rude. This is just true

I'm not at fault for not acknowledging someone's insecurities

leslie townes

maybe everyone was banned by the mods for some reason and hasn't created new accounts yet

Daniel Donnelly

21:17

Got a good question for the room. Let $X = \{0\}, Y = \{1\}$ and $W = \{0, 1\}$. Then we can write $X\times Y + Y\times X$ for example and we cannot write a longer expression in falling in $\{0,1\}^2$. So the maximal complexity for homogeneous "polynomials" is two. + is union here. So for example $X\times Y \subset X\times W$ so if both appear in a sum, you reduce the expression to the larger term. So I'm looking for maximal complexity after reduction.

So what about homogeneity of degree $3$? I got $3$ again for max complexity. Does the same thing happen for degree $4$?

I say monomial ,so we can write $XYW \equiv X\times Y \times W$ etc.

Thus if you wanted to name the structure of such subsets it's simply a semiring or boolean algebra or something, but we don't really appeal to this structure at all in thinking about the complexity of a reduced expressionl

Ted Shifrin

22:08

@PM2Ring When I retired, the celebratory dinner was half roast and half sincere remarks that made me cry.

Soumik Mukherjee

22:53

@robjohn XD

The sweating part fulfills that as well:)

Carlos Beltran

Hi everybody

Jakobian

@TedShifrin math.stackexchange.com/questions/4724885/…

any advice for someone trying to learn GMT?

from Federer too

Ted Shifrin

23:10

I commented.

monoidaltransform

23:25

Hiya Ted

0

Q: Infinite Join homeomorphism property preserved

Problem: In the lecture notes the following is writen: Let $X_1,X_2,X_3,...$ be topological spaces. Put $X=\star_{i=1}^{\infty} X_i$ (join). Then assume $f_i:X_i\approx Y_i$ is a homeomorphism for each $i$, where $Y_i$ is a topological space. To showthat $\star X_i\approx \star Y_i$ (homeomorphi...

general-topology algebraic-topology solution-verification

eigenvalue

23:47

I got this sketch and it's not very clear what's going on here (see image via link below)...Does anyone have an idea what kind of manifold this spacetime is supposed to be? I am wondering in particular about the two pink areas? Are those boundaries (i.e. manifold with boundary)?

Globally hyperbolic spacetime with compact Cauchy hypersurfaces: https://www.dropbox.com/s/0yn4n5gstdq0kah/GH%20ST.png?dl=0

the metric could be $g=-t(dt)^{2}+dx^{2}$ for t>0 (with t=0 being the first foliation leaf)

Transcript for

$Mathematics$ Mathematics