/sci/ - Science & Math

>>15571520
I'm very sorry, this image is not the OP image I intended to post.

>>15571520
Poincarre the GOAT.

Is this possible to solve?

>>15572093
There is a trick to proving whether these kinds of problems, for any amount of variables, are unsolvable where you make a graph and count the edge weights but I cannot remember the trick.

>>15572093
No. You can derive the bottom two equations from the top three. That means there are three non-redundant linear equations with four unknowns, which means c can have any value.

>>15572104
In this instance, does the Smith normal form method make a column matrix with infinite members?

i want to be a master of modern cryptographic methods what books for this?

I hate it when you spend ages working on a proof you are really proud of and then it turns out there is a small hole in it.

>>15572104
Interesting. The reason why I asked was related to picrelated which is a video that I was watching. I was wondering if you could solve the surface area for each individual variable.

Graduates, Diploma holders, PhDs, PostDocs etc who read this thread, however few you may be, has pursuing mathematics given you clarity and understanding or murkiness and confusion?

>>15572168
You can, but not with just the equations in >>15572093. You will need one more piece of geometry, and that is that a + a + c + c and b + b + d + d are each half of the total area of the square. Adding that equation to the set turns it into something you can solve for each variable.

>>15572184
Right, I wasn't thinking about that. Also I came up with a quick proof of what you said. If you think of "a+a" and "c+c" and two triangles, their heights (h and s-h) add up to the side length (s) of the square. And so (s*h)/2 + s*(s-h)/2 = (s^2)/2.

When you write a fraction do you squish it onto one line or do you put it over two lines (numerator on one line and the denominator on another line)?

Bros, does anyone know a good text or video guide to solving inhomogeneous second-order linear ODEs with non-constant coefficients? Emphasis on the non-constant coefficients, can't find that shit anywhere.

Here's what I'm specifically trying to solve:
[math] y'' - sin(x)y' + cos(x)y = tanh(x) [/math] with any initial data of your choice.

Are equivalence classes an algebraic notion or an analytic one?

>>15572304
algeblorg

So I can't make the set { 1, 2 3, 4 ... } from the superset { {0,1,2}, {1,2,3}, {2,3,4}, {3,4,5}, ... } without ACC?
Why can't I just do this in ZF? The function is computable, you just take the middle element of each subset.

>>15572345
Are you sure you can't do this in ZF? If you are given that the latter set you gave is actually a set, I believe you can invoke axiom of separation to build the former.

>>15572345
you don't need AC if you have an already defined choice function, as in your example. AC tells you there is a choice function even when it's undefinable.

>>15572217
to get an exact solution, try to manipulate the equation to be a derivative of (A(x)y'+B(x)y) on one side and a function C(x) on the other. Yours is the same as
[math](y'+(\cos x)y)'=\tanh x[/math]
so integrate and place a +C to account for the initial data. Then solve the first order ODE which necessitates multiplying by [math]e^{\sin x}[/math] and integrating again. I don't think it is an elementary function closed form.

>>15572126
Start with number theory and algebra, then explore https://arxiv.org/list/cs.CR/recent

>>15571520
Guys I am studying for the Putnam but in the Putnam and Beyond book I am stuck on the exercises. Is there no hope for me?

>>15572174
maths is useless outside academia

>>15572174
I have done much graduate work (taken about 15 grad quarter courses), currently a Senior of my BA in maths. I still legitimately don't understand why the Pythagorean Theorem is true. Yes I have read proofs (and discovered one myself). No they don't give enough intuition as to why this theorem is true. Yes I am investigating it further and am likely going to write a survey paper on it.

>>15572818
That's interesting. What do you see as the most insightful reason thus far, something about p-norms or something else?

>>15572589
>Yours is the same as [math](y′+(cosx)y)′=tanhx[/math]
I don't see how?

>>15572818
>They don't give enough intuition as to why this theorem is true.
How do you mean? Like do you mean how the law of cosine (generalized pythagoras) is true because intuitively if you fix the length of the two sides and the angle between them then there is only one possible line that connects the two ends together to form a triangle?

>>15572818
In the unlikely case that nobody has already brought this up to you, there are interesting generalizations of Pythag.T in Harmonic analysis.

>>15573073
Well, my current plan of action is to prove it using Quadratic Equations. I discovered the Pythagorean Theorem can prove a basic case of the Quadratic Formula and every other case of the latter is a generalisation of this basic case by linearity arguments. My trouble is doing the converse and to gain insight from it.
>>15573090
Yeah something like that. Look at the original proof by Euclid and see how utterly uninsightful it is to see my point. I haven't looked at all the proofs since I want to grapple the problem myself (if it is taught to highschoolers across the country it should have an insightful proof using highschool mathematics).
Also, perhaps paradoxically the Law of Cosines makes perfect sense to me. It is pretty much what you said if I recall, but it is a law of vectors if you simply use coordinate-based geometry to discover the answer. This however does prove the Pythagorean Theorem by accident yet it doesn't provide any special insight to this case that I could derive. It feels like magic in some ways and I want something reasonable for something so reasonably set as a bedrock of mathematics (the distance function!)
>>15573163
I'll check it out. I haven't discussed this with anyone until now since I don't want to talk to a prof. about it until I make significant headway.

What are Ext and Tor functors and why do we need them?

>>15573715
There are weaker versions of inner product spaces where the parallelogram identity is true but the pythagorean theorem isn't. I remember reading that the Parseval identity is involved somehow before I gave up and studied number theory instead.

I hate computer scientists who think they can do math

>>15573076
I misread on my phone
The technique I mentioned works in this case though if you do work it out.
In general, also, if you can solve the =0 version then you can solve the =tanhx (or whatever) version
You can also try factoring the operator (or a multiple) as (a d/dx + b).(c d/dx + d) in worse scenarios

>>15572174
Clarity and understanding. Especially in the applied math sense. The real beauty of math is NOT in the solving of the problem, it's not in the theory, it's not in the proofs or any of that. The real absolute beauty of math is in the translation of reality into usable symbols, it's very hard to teach this and typically completely ignored in math education, but easily the strongest skill in mathematics.

Consolidating what you see, hear, read, and think about, into usable symbols fit for manipulation. Whether it's algebraic, graphical, machine like diagrams, etc. the very act of reasonably translating something into a mathematical artifact that can then be USED to learn more about true and real thing is the beauty of math. Imo, anyone can solve a problem and high level autists can further understanding of mathematical objects and fields, but the truth masters of math know how to translate reality INTO math.

>>15573928
Pretty sure 99% of Mathematicians do not understand how a computer works.....

>>15573732
>What are Ext and Tor functors
The derived functors of the Hom and tensor product functors.
[math]\mathrm{Ext}^1 [/math] classifies extensions of modules, and [math]\mathrm{Ext}^n [/math] classifies [math]n [/math]-extensions, see https://en.wikipedia.org/wiki/Ext_functor#Construction_of_Ext_in_abelian_categories
[math]\mathrm{Tor}_n [/math] has less of an interpretation, its name comes from a special case of [math]\mathrm{Tor}_1 [/math], where it is used to detect the torsion of modules.
Not that these interpretations really matter, anyway
>why do we need them?
We use them to cope when we tensor by a module which is not flat, or when take [math]\mathrm{Hom}(-,C) [/math] and
[math]C [/math] is not injective, or when we take [math]\mathrm{Hom}(C,-) [/math] and [math]C [/math] is not projective.
Example: Let [display]0\to A'\to A\to A'' [/display] be a short exact sequence, and tensor by [math]B [/math]. We get the exact sequence
[display]A'\otimes B\to A\otimes B\to A''\otimes B\to 0, [/display]
which can be extended to the long exact sequence
[display]\cdots\to\mathrm{Tor}_1(A,B)\to\mathrm{Tor}_1(A'',B)\to A'\otimes B\to A\otimes B\to A''\otimes B\to 0. [/display]
But now, if instead [math]A'' [/math] is flat, we still get a short exact sequence, because [math]\mathrm{Tor}_1(A'',B)=0 [/math]. Or more generally, we still get a short exact sequence if [math]\mathrm{Tor}_1(A,B)\to\mathrm{Tor}_1(A'',B) [/math] is surjective.
So basically, you've turned the qualitative problem "I want my short sequences to be exact" into a quantitative problem "I want my Tor functors to vanish", and so you can start a computation (whether in theory or in practice), and even when it turns out that they don't vanish, you at least know how much left-exactness fails.

>>15575096
I forgot that /sci/ uses eqn and not display, I'll report the equations below
[eqn]0\to A'\to A\to A'' [/eqn]

[eqn]A'\otimes B\to A\otimes B\to A''\otimes B\to 0 [/eqn]

[eqn]\cdots\to\mathrm{Tor}_1(A,B)\to\mathrm{Tor}_1(A'',B)\to A'\otimes B\to A\otimes B\to A''\otimes B\to 0 [/eqn]

>>15574729
Neither do computer scientists.

I cannot prove this theorem ahhhhhhhh

>>15575426
You will fren. Have patience.

>>15573715
If it's so unintuitive how did people discover 2000+ years ago?
>euclid's proof
The reason euclid proved it that way was to use the other propositions in book 1. There have always been simpler proofs.

Why don't you dumb down the conic sections? That's an entire branch of geometry thats pretty much been lost to modern people.

>>15571524
it made me laugh pretty hard so thanks for your mistake

>>15572093
it's 4
>please provide an ex-
no
it's 4

For proving the convergence of an infinite series can I just use the ratio test on everything? Is there any point in memorizing the other complicated ones?

>>15575649
The ratio test is inconclusive if

[eqn]\limsup_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \geq 1[/eqn]
and
[eqn]\liminf_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \leq 1[/eqn]

>>15575493
>Why don't you dumb down the conic sections? That's an entire branch of geometry thats pretty much been lost to modern people.
I think the problem would be where to put it in the uni curriculum. My linear algebra course treated conics at the end, and you can recover most elementary properties this way. But, what are we missing?
>the locus of points definition
Just some algebraic manupulations
>Pascal's theorem, and projective stuff
You would need an intro to algebraic geometry
>Chevalley-Warning for conics, i.e. conics over finite fields have a rational point
Eh, you only need some commutative algebra, but students wouldn't get why it's important. Serre puts it at the start of a course in arithmetic, and I think it fits well (but where should this material be covered in a curriculum? I just read the book by myself one summer)
>conics as Severi-Brauer varieties
you'd need some more algebraic geometry (or at least algebraic number theory, if you're coming from that side of the aisle) to appreciate this

What I'm trying to say is that one could very well write a good book on conics, providing a motivation for introducing many different topics, stratified along different levels of reading. But your question wasn't this, you were looking for a way to collectively teach conics to people, and idk where to put that in a curriculum

>>15575671
I'm in calc II I don't know what sup and inf mean.

>>15575767
lim sup is limit superior, i.e. the upper limit of the sequence
lim inf is limit inferior, which is the opposite

How many India anons are here?
Are you ISI, CMI or TIFR? IISER chuds need not reply.

This is not homework, Im self-studying and I've been stuck with the following problem about differential geometry for some time now:
>Prove that the stereographic projection from the lower half of the sphere with center (0, 0, 1) to the plane (carries points from the sphere to the plane like in pic rel) carries geodesics to geodesics.
This problem appears in a section dedicated to the exponential map, so it may use the exponential map.
Thanks

>>15575426
>>15575467
I did it

How do you solve x from this?

When you guys are learning from a book how many exercises do you do, and whats your general strategy for studying? Im doing this because i want to learn obviously but im having trouble gauging how well im actually learning. Grades arent the end all be all but its weird to me not having some kind of benchmark.

Answering >>15550657
> Is there any categorical treatment of sheaf theory?
Well, a priori you can define sheaf with values in any category. In Hartshorne, conditions (3) and (4) for a sheaf are replaced by the exactness of pic related (which is a screencap of my own notes); so I guess the minimal condition one would ask is that the category posses products and a final object.
For the existence of stalks you need colimits to exist, or actually filtered colimits which may be a little weaker.

Now as you already guessed, these two questions are the hard part:
> Can properties of morphisms still be determined by the corresponding properties at the stalk level?
> Does the sheafification still make sense?
The stalk verifications and sheafification make some relatively strong use of the fact that groups are sets of some kind, so your category has to be something a bit stronger than just "having products, filtered colimits and final object". Here my advice is to follow Stacks chapter on sheafs (https://stacks.math.columbia.edu/tag/006A)) which uses "type of algebraic structures" (cf., https://stacks.math.columbia.edu/tag/007M)) which is a sufficiently good category for our purposes.

> Are there nice enough categories for which the notion of a sheaf can be generalised?
Apart from the previous talk, you can notice that a presheaf is a totally general definition, since is no more than a contravariant functor. The "gluing condition", which relies on exactness and coverings is a bit more demanding, but the solution is to define a "category with coverings", known as a site! (The family of coverings is known as a Grothendieck (pre)topology; the analogues are the terminology: topology/topological space.) There is plenty of literature on Grothendieck topologies: Stacks itself, SGA 4, SGA 4 1/2, Mac Lane and Moerdijk's Sheafs, etc.; but I was reading it on Kashiwara-Shapira's book "Categories and Sheaves".

>>15575776
So the ratio test is always inconclusive?

>>15576720
"and", not "or".
If both the limit superior and limit inferior of the absolute value of the ratio of successive terms are greater than 1, then it conclusively diverges.
If they are both less than 1, it conclusively converges.
The test is inconclusive only if that value is at least 1 for the limit superior and no greater than 1 for the limit inferior.

>>15576748
>limit superior and limit inferior
I get it now, this must be a European thing. In the states we just say "lim" and it refers to both the left and right side limit.

>>15577015
no, it's very much not a European thing, and it's not the same thing as a right-side/left-side limit
for a basic example consider sin(x)
this has no limit as x tends to infinity, but its limit superior is 1 and its limit inferior is -1

>>15577015
>>15577038
better yet let me just rip the image from Wikipedia
there is no singular limit, but if you consider the local maxima, the limit as they tend to infinity is the limit superior.
likewise for the local minima and the limit inferior

>>15576348
Expand the brackets and match terms. This will give you expressions for a and b in terms of the other, then substitute and solve for x.

>>15577015
Ehhh no.

I've only encountered this in my real analysis class btw, definetely not calc II

a limit superior and limit inferior are basically what the local Infimum and supremums tend towards as the input of the function go to infinity.

[math]\text{White woman's orgasm}=\text{Penis}_{bbc}⋅\overrightarrow{thrust}[/math]
Is this mathematically accurate?

>>15572174
>>15574641
this. the real benefit of practicing math is the ability to work in abstraction

>>15572217
Best bet would be some kind of asymptotic/perturbation series.
If you want to solve around 0, just represent y as a taylor series and solve term by term.
This will likely only converge for |x|<pi/2
>Here's what I'm specifically trying to solve
where the hell did it come from?

If [math]\mathbf{F}: \mathbb{R^n} \times \mathbb{R^m} \to \mathbb{R} [/math] is differentiable, what does its gradient look like?

Is the gradient a function [math]\mathbb{R^n} \times \mathbb{R^m} \to \mathbb{R^n} \times \mathbb{R^m}[/math] defined by

[math]\nabla \mathbf{F}(\mathbf{x}, \mathbf{y}) = \left(\dfrac{\partial \mathbf{F}}{\partial \mathbf{x}}, \dfrac{\partial \mathbf{F}}{\partial \mathbf{y}}\right)[/math]?

I'm losing my mind rn

>>15578118
Yep, that looks right. [math]\mathbb{R}^n\times \mathbb{R}^m[/math] is just [math]\mathbb{R}^{n+m}[/math]

this is a mathematical problem. how am i supposed to deduce from the text here
https://codeforces.com/contest/1409/problem/A
that the answer is (abs(a - b) + 9 ) / 10?
i couldn't figure it out but this outputs correct answer somehow

How could one find the following limit?
[eqn]
\lim_{n \to \infty} \frac{1}{n} \sum_{k=0}^n \binom{n}{k} p^{n-k} {(1-p)}^k \log \binom{n}{k}
[/eqn]

>>15578511
[math]p \in [0,1][/math]

>>15577041
But isn't that bounded monotonic? Does it really converge to 2 different values?

>>15577041
>>15578696
Actually, is that the difference between conditional and absolute convergance?

>>15571524
just offer me a job already

>>15578118
What does [math](\frac{\partial \textbf{F}}{\partial \textbf{x}}, \frac{\partial \textbf{F}}{\partial \textbf{y}})[/math] even mean? Is it the inner product? Weird notation.

>>15578720
Partial derivative of F with respect to x is x value and partial derivative of F with respect to y is y value

>>15578726
Oh yeah obviously. I confused the input dimension for [math]n\times m[/math] instead of [math]n + m[/math].
Even though you've probably understood it by now, for a sanity check you can consider the function [math]G[/math] defined by
[eqn]G(x) = F([x_1\dots x_n], [x_{n+1} \dots x_{n+m}])[/eqn]
Then [math]G[/math] is equivalent to [math]F[/math] and so [math]\nabla G[/math] should reveal the same information as [math]\nabla F[/math].

Here's a problem I invented by myself in MS Paint. Can you solve an exact analytical solution?

>>15578696
Just so we're clear, the function in question is that marked by the blue dots, not the red curves, which represent the bounds of the function beyond the point you're looking at. The upper and lower one will converge to the limit superior and inferior, respectively.
By definition, a function cannot converge to two different values. That's still divergence.

>>15578769
So it doesn't have a general limit. Because the general limit is the only thing I deal with in calc II

I hated doing math when I was a kid.
However, I have recently realized that math is God, so I want to learn it.
How practically applicable is "higher" math, and what free resources do you guys recommend?
I know of openstax and that's about it.

>>15578511
1. show k near (1-p)n contributes all but a limit 0 amount
2. (1/n)log (n choose k) is near the binary entropy function at p
3. sum the probability distribution to get a factor of 1
there is also an entropy interpretation. Write the formula for entropy of Bin(n,p) and rearrange, and use that the entropy of Bin(n,p) is only size around log n (versus the division by n)

>>15578904
>How practically applicable is "higher" math
I suppose this depends on what you mean by higher math, if you count linear algebra and calculus, then it's very practical, the world runs on linear algebra. Higher than that, not so much practical, but incredibly interesting.
>free resources
Textbooks are free. If you want to learn linear algebra and calculus, you can also find 100s of lecture series on youtube, I recommend khan academy.

>>15579364
The world runs on fast fourier transform you and that requires complex analysis

Just learned about the isomorphism between the cech cohomology and the de Rham cohomology. This shit's blowing my mind.

>>15578511
approximate (n C k) with (2^n)/sqrt[n*pi/2] * exp[-[n-2k]^2 / [2n]]
This is good for large n and k near n/2.
Just substitute it in the log.
I'm getting log(2) - 1/2 + 2(p)(1-p)
This seems good for p away from 0 and 1.

Today, in my exam, when I got my questions, I was elated because the problems were very similar to what I've practiced and glossed over right before the exam started.
But the moment I put my pen on the paper, my mind simply went blank. What I thought I'd ace, I bombed.
I'm so fucking done, I am unironically what you'd call a "not math person", I can't take this anymore

>>15579809
happens to the best of us
if I had... really any coin at all... for every time I've had to reinvent 95% of mathematics on the fly at an exam, I would be a rich man

>>15579811
I'm not sure if its because I haven't practiced enough or I'm just retarded. I admit I could have worked a bit harder but there's no way I should bomb just because I didn't do a few extra practice problems. I'm starting to think I shouldn't have gone into STEM while sucking at math.

I fucking hate acing exams while not getting the whole theory. I fucking hate math professors who don't take the time to explain proofs nor theory. University is a scam for mathematics.

>>15579809
I do doodles with my pencil and try to calm myself down when that happens.

>>15579863
its not that I'm super stressed, although that happens. Its just that my whole mind goes blank

>>15571520
>degenerate
>deviant
>trans
>dilate
yet another incel thread
>/mg/
oh it's just /mg/
oh wait it's /mg/

>>15579834
Why don't you study on your own (which you're supposed to and expected by your professors to do) instead of waiting to get spoon fed? No system can teach you everything, it's your responsibility to fill those gaps.

>>15579809
Happened to me multiple times. It means that either you didn't study enough, or the idea of the question is too complicated to ask on an exam. I have encountered the latter but most of the time the reason was the former.

>>15579937
yeah, I really should study more. I basically didn't care about studying at all in school. But, I decided to get my stuff together and be serious in college. Unfortunately, I'd practice a few problems, think I've understood the concept and move on quickly. I'm still new to studying hard consistently.

>>15579884
Might as well draw if nothing comes to your mind.

bump

>>15579834
It's okay not to grasp the whole theory until later. Knowledge is a lot like cement, it has to take time to cure. Sometimes it wont make sense until it bumps with other things in your brain.
"Human knowledge is not (or does not follow) a straight line, but a curve, which endlessly approximates a series of circles, a spiral. Any fragment, segment, section of this curve can be transformed (transformed one-sidedly) into an independent, complete, straight line, which then (if one does not see the wood for the trees) leads into the quagmire, into clerical obscurantism." - Lenin

>>15574641
That's physics

>>15579002
I'm not sure I understand steps 2 and 3. Are these steps meant to find an approximation or the exact figure? Either way, what is the use in finding the sum of the probability distribution when the other terms are multiplied?

Anyway, I am glad you noticed the entropy interpretation! I posted this question wondering if anyone would spot it. You can use this to go even further, as I will explain.

>>15579691
That is a good approximation.

Computing the Shannon entropy of the nth convolution of [math]\mu \sim Bernoulli(p)[/math], I found
[eqn]
H\left(\mu^{*n}\right)
= - n \left[ p \log p + (1-p) \log (1-p) \right]
- \sum_{k=0}^n \binom{n}{k} p^{n-k} {(1-p}^k \log \binom{n}{k}
[/eqn]
and the Avez entropy is then
[eqn]
\lim_{n \to \infty} \frac{H\left(\mu^{*n}\right)}{n}
= - \left[ p \log p + (1-p) \log (1-p) \right]
- \lim_{n \to \infty} \frac{1}{n} \sum_{k=0}^n \binom{n}{k} p^{n-k} {(1-p}^k \log \binom{n}{k}.
[/eqn]
Since [math]\mu[/math] has finite support, it is pretty easy to show that the Avez entropy is zero, from we find
[eqn]
\lim_{n \to \infty} \frac{1}{n} \sum_{k=0}^n \binom{n}{k} p^{n-k} {(1-p}^k \log \binom{n}{k}
= p \log p + (1-p) \log (1-p).
[/eqn]
If I was wrong in my calculation of the Shannon entropy, obviously all of this is wrong, but I'm pretty sure what I have is correct. I thought it was neat as this limit on its own looks quite difficult to calculate.

have you guys ever mentored a young lad? like, college freshman. how did it go? was it hard? did you learn anything from it? do you think you helped him or would it have been better if he had learned on its own?

>>15572216
>put it over two lines

proper way

other way is for typists, bad to do.

when i was studying math i noticed that a lot of people were able to memorize proofs and also come up with small proofs for lemmas and what not but do you guys actually have any intuition about your topics? starting by just finding examples for the concepts that you deal with. some concepts in math get so esoteric that its almost impossible to find an example. like can you write down a closed formula for a wiener process? i think math is heading in a weird direction and id like to see more done in applied math research.

Mathbros please help identify this math from a 17th century ship captain's journal.
>>15581556

Mathlet here. Can someone explain Lang's proof for N5?
From what I understand, -(a+b) is the additive inverse of (a+b), so if -a(a+b) = (-a-b), then (a+b)+(-a-b) must equal 0. Is this right?

>>15582066
>a(a+b)
-(a+b)
Typo

>>15579809
I'm this guy. The exam I bombed was an EE course. On the other hand, today I had math exam, which I was dreading because I suck at maths. Somehow, I inexplicably managed to ace this.

>>15581150
approximation but the total error goes to 0 in the limit so it's valid

>>15572103
Sounds retarded. There are no 'tricks' in mathematics. The ONLY way to solve this is to use a matrix algebra.

>>15582066
For every integer [math]x[/math], [math]-x[/math] is the unique integer [math]y[/math] for which [math]x+y=0[/math]. I.e. [math]\forall x\forall y. -x=y \leftrightarrow x+y=0[/math]. (This is what Lang explained after "Remember that...")

Now, if you instantiate the above with [math]x=a+b,y=(-a)+(-b)[/math] then you get the equivalence [math]-(a+b)=(-a)+(-b)\leftrightarrow (a+b)+((-a)+(-b))=0[/math].

>so if -(a+b) = (-a-b), then (a+b)+(-a-b) must equal 0
That is correct and is the forward direction of the aforementioned equivalence. But that is not the direction we're interested in for this theorem! We want to prove [math]-(a+b)=(-a)+(-b)[/math], so by the reverse direction of the equivalence we might instead show that [math](a+b)+((-a)+(-b))=0[/math]. (This is what Lang means by "Thus to prove our assertion...")

>>15571520
Do irrational numbers exist?

>>15582277
The square root of two is irrational and exists.

>>15582214
Thank you, anon. I think I understand now. But I am a litte confused as to what you mean by "reverse direction."

are any of you hobbyists? how do you internalize complicated formulas and concepts without having to use them all day or for a test? I like interesting math problems but when they just require me to plug in a prelearned algorithm I just kinda tune out...

I understand proofs about the Fundamental theorem of Calculus, yet I don't feel convinced. Am I a fucking retard? I tried looking at James Gregory's proof, yet he already assumes there's some sort of connection. How the fuck did these mathematicians know there was a connection with derivatives and integrals all along?

>>15583141
It is literally just telescoping series.

Do u guys know some BASED analysis books that are not Zorich and Amman & Escher? Gimme recs

>>15583494
Real analysis? Kolmogorov & Fomin
Intro Analysis grad course? Folland

>>15583282
elaborate

>>15583141
>Am I a fucking retard?
No, you're just lacking the basic explanation.
Suppose you have a data sequence:
1 3 6 2 3 7
first differences are
2 3 -4 1 4
with the initial offset of 1, the cumulative sum of your first differences gives you the original sequence.
But the cumulative sum is the area, and you'd use it as the estimate for the area if you're working with something of which you only have discrete estimates
(think dead reckoning, we walked 10000 paces today and 20000 paces yesterday, how far have we gone?)
so this is how someone like gregory would be familiar with this idea pre infinitesimals.
and the infinitesimal/limit/continuous extension follows easily.
see how simple it is?
isn't it wonderful?

posting 2 pictures because i don't know how to stack 2 images on top of eachother on gimp. (1/2)

I don't see how "trichotomy" makes strict total orders special in relation to total orders, despite the fact that one is irreflexive. Why does the "or x=y" change? In any binary relation, any x will be paired with itself as many times as the set has elements, then if it's always true and it has nothing to do with R, why is it included? Am I /misunderstanding/?

>>15583622
You can see this better with an example, "[math]\leq[/math]" is a total order on real numbers, because two real numbers are "comparable" with respect to [math]\leq[/math]; it is either [math]x \leq y[/math] or [math]y \leq x[/math] (the case [math]x = y[/math] is already included in those cases.)

But [math]<[/math] is a strict total order; it is either [math] x < y [/math], [math] y < x [/math], or [math] x = y [/math].

So if you have a non-strict total order, [math](x \leq y) \wedge (y \leq x) \implies x = y[/math]. In a strict total order, [math]x < y \wedge y < x[/math] is always false.

>>15573732
Some of various weird things in traditional algebra that are a result of it just being a shadow of higher algebra.

https://mathweb.ucsd.edu/~asalehig/brian-problems.pdf
This is incorrect right? if [math]n = 3, x = F[/math] then [math]R^{-1} = R^{2}[/math] but [math]RF \neq R^{2}F = R^{-1}F[/math]

>>15584498
you're right, as is most easily shown by multiplying through by the inverse of x on the right.
My immediate assumption would be that they wanted xR^{-1}, if only because the alternative of xR would seem to be a bit too easy to derive from the definition (even if that wouldn't be much harder)

Speaking of James Gregory, he was the first to discover Taylor series and the Fundamental Theorem of Calculus. Yet, nobody seems to give him much recognition. Why is that ?

>>15584522
some people just get unlucky and end up having their contributions forgotten or usurped.
Just look at Oresme for another example.

>>15583557
[math]Let\ \Sigma(a)_N = \sum\limits_{n=0}^{N}a_n,\ \Delta(a)_N= a_{N+1}-a_N.\\
\Sigma(\Delta(a))_N=\sum\limits_{n=0}^{N}a_{n+1}-a_n=a_{N+1}-a_0.\\
\Delta(\Sigma(a))_N=\sum\limits_{n=0}^{N+1}a_n-\sum\limits_{n=0}^{N}a_n = a_{N+1}.\\
Let\ 0<\delta\ small,\ a_n=f(n\delta),\ \int=\delta\Sigma,\ \partial={\Delta \over \delta}, N = {x \over \delta}.\\
Clearly \int(\partial)=\Sigma(\Delta)\ and\ \partial(\int)=\Delta(\Sigma).\\
Plugging\ in\ N = {x \over \delta}\ and\ using\ a_n=f(n\delta)\ gives\ the\ result.
[/math]
Starting from there is where the inspiration comes from.
There is even a finite analog for integration by parts.
https://en.wikipedia.org/wiki/Summation_by_parts
Afterwards, people were worried about definition of the integration operation since there was a lot of freedom in how you could choose to chop up interval which could give limits that differ from uniform sampling for some pathological f. That is where Riemann came up with his definition of the integral (which basically just discards the f that give ambiguous values with respect to the partition of the interval).

>>15584498
You forgot to switch sides.
You want RF = FR^(-1)
not RF = R^(-1)F

>>15584587
you mean as in >>15583569 then.
I see

saw this on ebay for $4.50 + free shipping and I couldn't help myself. I haven't started reading it yet, but does anyone have experience with this book? It seems to be well regarded

>>15585902
I've been reading it and it seems pretty good so far

>>15585902
This might be on that 150 gigabyte math book collection that some anon posted a link to a while ago

>>15586209
just checked: it is not, actually

>>15586217
There's quite a bit of the same subject though

Is somebody familiar with the math behind neurosciences ? A bit of algebraic topology, information theory and maybe something more ?

I was looking at Pascals triangle and stuff waddaya think

>>15586217

He won.

Why is this true? I can understand that q*qT is symmetric, but the Lambda in the middle messes the whole thing up.

>>15587377
Lambda is a diagonal matrix consisting of the eigenvalues of A. Who tf writes a whole text on a diagonalization without once saying the word "diagonal"??

>>15587418
Ah, of course, if Lambda is diagonal then it's symmetric. I somehow forgot Lambda is the Eigenvalues, not the Eigenvectors. Thank you!

It's from Gilbert Strang's Linear Algebra (MIT OCW), a summary of the video lecture (in general it's very good).

The section on sphere bundles in picrel is killing me, Plox help.

>>15583494
Real Analysis by N.L. Carothers
Real Analysis by Shakarchi and Stein

This is the equation of a parabola that passes through the points (x1, y1), (x2, y2) and (x3, y3).

Can it be simplified or does it have to look like this?

>>15572093
a = 4
b = 16
c = 0
d = 28
i am the smartest negro alive

Which is the more common spelling: counter-example or counterexample?

>>15589143
Counterexample

I've devalued the study of mathematics in my mind - to me, it's just a big pile of other people taking random walks, with some surface tension function, through a high dimensional phase space of knowledge, where the phase space is just there whether humans explore it or not.
The result is that whenever I start any work on math I start arguing with myself as to why it's pointless and then drop the work. It's been three weeks and I think I must have given myself a mind virus.
How do I deprogram?

>>15589188
Obviously the theorems are still there, but that doesn't mean it's not worth finding them! Would you told Colombus "don't bother with the expedition, you won't find anything that doesn't already exist"?

are books on math intuition a scam?
it seems like you can only learn it from practice

If A is a subspace of B, is H_1(A) a subgroup of H_1(B)?

>>15590567
Obviously not, R^2 has trivial homology, but S^1 doesn't.

>>15586713
Good thinking. Check out mathologers video on bernoulli numbers if you want to look into the connection between sum of powers and pascals triangle a bit further

>>15591174
Also should be n+2 ,n+1 but close enough.

>>15581150
You can simplify things without invoking the heavy machinery.
First multiply and divide in the log by p^(n-k)*(1-p)^k.
Use log rules to pull out the divided part.
Evaluate the sum for the divided part to get -p*log(p)-(1-p)*log(1-p). (use the power series for (1+x)^n and differentiate to do this).
The remaining part is just a negative entropy divided by n.
Extremize the entropy by letting each probability = 1/(1+n).
The |entropy term| is at most log(1+n).
log(n+1)/n goes to zero.
The limit is -p*log(p)-(1-p)*log(1-p).

>>15591174
I checked out his videos, now I have more questions

>>15579820
when it happens to me it's usually a mix of:
1. not practicing enough
2. test anxiety
3. lack of sleep
lack of sleep is the easiest to take care of. the first one is probably the most beneficial. the 2nd one is usually solved if you treat the 1st and 3rd issues.
too bad I have terrible discipline so I don't really put this to practice myself...

Is anyone familiar with this text? https://arxiv.org/abs/2006.01613
Would it serve as a good introduction to NBG set theory?
If not, are there any other set theory books that work in NBG (rather than ZF)?

>>15572677
Those aren't "exercises" in that book. They're "problems." You're suppose to be stuck on them. For each one set a timer for like 2hrs, grind on it, then when the timer goes off, look at the solution in the back.

>>15592418
Good. What are you wondering about

I have trouble understanding the solution here.
Can anyone help break this down? Assume I have no idea of improper integrals.

>>15594139
my friend told me you could prove that if the summation of a(n) from n=1 to n=k equaled k then a(n) =1. I hacked it with this proof that assumed a(n) was a continuous function, then used chain rule to show a(n) was constant, and so on. But that only applies to continuous functions. I know terms including (-1)^n are useful, but I dont know how to separate them from cop-out piece wise functions like (if n=1 then a=k, otherwise a=0). I fiddled around with a more general case and so far all I have is that if not all a =1, then there exists a maximum term >1 and a minimum term <1, which is pretty obvious from the get go, if you imagine a 1xK rectangle of constant area, then warping one of the K edges.

>>15596044
Suppose
a(1)=1
a(2)=-1
a(3) = k
Clearly the summation is equal to k but every term a(n) is not equal to k

I'm going into my senior year next month and I don't feel like I know a single thing about math yet. I've already forgotten how to do everything from my previous classes too. Is this a normal part of the process or is it a low IQ person thing (I'm retarded for real, not joking)

>>15572093
No. With 4 unknowns you need 4 independent equations. Let's name them 1 to 5. 1 and 4 yield 5 so 5 is linearly dependent on 1 and 4, hence we have 4 equations and 4 unknowns. Now 1 - 2 + 3 = 4, hence 4 is dependent on 1 , 2 and 3, hence now we have 3 equations and 4 unknowns. Therefore C can be any number.
Example: C is 100.
b + c = 16 -> b = -84
a + b = 20, 20 + 84 = 104, ->a = 104
c + d = 28. d = 28 - 100, d = -72
a = 104, b = -84, c = 100, d = -72
a + d = 104 - 72 = 32, OK
a + b + c + d = 104 -84 + 100 - 72 = 48 OK

>>15596259
This doesn't contradict anything, unless I wrote poorly.
a(2) = -1 is the min, and is less than 1
a(3) = k is the max, and is greater than 1
And not all terms equal a constant that is not 1

>>15596427
forgot image

>>15593416
>timer for 2hrs
Kek that's minimum time you should devote to an exercise before giving up in any good book. Problems requires days, even weeks of investment.

>>15575543
holy fuck it fits

To prove this you just use Taylor's theorem. When you're given any sequence of numbers, you can create a polynomial approximation of that number. This is called the Newton Gregory interpolation formula

Suppose we're given your summation as a sequence of numbers. We then take their first differences of this sequence to get the sequence of their differences
a1 a2 a3 ....... ak
Well if these numbers are all constant, their difference will be 0. It turns out the solution to that particular case is as you said all terms are equal in the first difference sequence.

However suppose the first difference sequence is not constant.
Then clearly at least two numbers in the sequence must be different to produce differences. I ended up proving this by drawing dots but don't feel like posting the proof atm.

>>15596754
OK im a nice guy so I will post the proof with dots. We know that base sequence which give us our answer is a row of 1's.
Suppose we increase one of the numbers in our sequence. Then to balance thing out, we have to subtract an equal amount from other numbers in our sequence.

In other words our maximum value for an will always be 1+some number, and our lowest value 1-some number.

>>15596770
This can be shown with simple algebra, I dunno what this interpolation formula stuff is

>>15596754
I'm interested but I can't read these samsung note scribbles

>>15586209
where to find it?

>>15596044
>d/df
>f is not any one single variable in the expression that is being differentiated
What sort of foul sorcery is this

>>15579809
learn how to study properly
at least use pomodoro method

>>15597147
Interpolation is simple algebra stuff. Notice you refer to your sequence of numbers as "a(n)". Well that doesn't tell us anything about the relationship between those numbers. Suppose i give you five random numbers 7 -4 3 5 -6
I ask you to find a relationship these numbers have. To do that we come up with a quartic function that generates all these points when x=0,1,2,3 and 4
One such function is given in pic related.

Astronomers before and during Newton's time would take orbital data and use interpolation to find where the planets would move next. You say you can prove your result using just algebra but if you can't do anything useful with it it's useless. Only modern academic frauds blab on about "this will be useful someday". If you can't apply your math to anything it's useless. Simple as.

Could 0 EQUAL the set of natural numbers itself, and have everything else follow normally?

>>15599110
I guess so. Why does it matter? The peano axioms don't say anything about what set a number is equal to.

math is just an artifact of the human mind and is not an objective description of the world

>>15599355
>x is just an artifact of the human mind and is not an objective description of the world

>>15597960
>>15597960
I considered the summation its own function of f, a S(f(x),k) type of situation, then applied chain rule differentiation, thus d/df(S) * d/dx(f). but it was a weird function to define, since it was continuous in x, but not continuous in k, an integer, so I just used the explicit summation

>>15587377
LPDU factorization with P = I => L = U^T where D is diagonal means the matrix is symmetric

See
https://agorism.dev/book/math/linear-algebra/groups-matrices-vector-spaces_james-carrell.pdf

Section 4.3.4

Vector calculus problems aren't that bad but I feel like a monkey regurgitating equations. Not really sure how to intuitively understand:
>Flux vs. Curl, how does Curl work for R3 surfaces?
>Is outward flux just the flux of the surface?
>Is circulation just the line integral of a vector field on a closed smooth surface?
>Green's Theorem and how it applies to Stokes' and Divergence
>how all that applies to the fundamental theorem of calculus

I get As, but rarely an A+. Am I good enough to prove Riemann's hypothesis, or it's over?

>>15600279
Unless you exclusively research all about
Riemann's hypothesis, have published a bunch
beforehand, and have a cogent response with other
respected mathematicians on the viability of it
all, an A grade simply doesn't cut it.

>>15596287
That usually happens with bad study habits. Try using a spaced repetition system.

whatever is the math used in social media when maximizing content engagement? especially those with 5 digits or more: how are the content quality or quantity, illustrations translated to sales and traffic? or if social credit included, what is the algorythm that connects between marketting and content, to sales and clout?

it seems like 'addition' is behaving like 'exclusive or', and 'multiplication' is behaving like 'and'. Why call them addition and multiplication at all, instead of just operation 1 and operation 2? It seems unnecessarily confusing to give them names of things that already have an expected behavior.

>>15601064
Because then they'd have to invent new notation for it. And it'd be hard to keep track of which was which. And it'd complicate all the examples that are subsets of the complex numbers, where addition and multiplication really are just normal addition and multiplication. But if you didn't care about all that you could do it differently.

>>15601064
Because it’s multiplication and addition in a GF(2) aka binary field?

>>15601051
Bump>>15601064

>>15601051
I mean this. Bump

https://www.youtube.com/watch?v=p2vadd_6550

>>15601064
big, obscure dataset (views, likes, sales, traffic, etc.) use stats/pattern recognition technique to determine hidden shakers and movers warranting further investigation. Then implement stuff based on what you learned.

Here's an answer with a bunch of keywords
https://datascience.stackexchange.com/a/1006

>>15601051
big, obscure dataset (views, likes, sales, traffic, etc.) use stats/pattern recognition technique to determine hidden shakers and movers warranting further investigation. Then implement stuff based on what you learned.

Here's an answer with a bunch of keywords
https://datascience.stackexchange.com/a/1006

>>15601424
...sorry if i find this confusing. How do i implement this to maybe online galleries, blogs, and however it should be curated based on the data then?like take example if i want to reach some hundred thousand people through my content.

How common is it to come back into math after a long hiatus? I did up to a master's degree (in physics, but my master's thesis was really in mathematical physics and I've kind of drifted more into math than physics) about 5 years ago and since then worked in jobs not super related to math. Now I want to go and get a PhD. Since my master's I've kept studying math in my own time and I still have contacts at universities. Is this feasible or will the fact I've been out of academia for so long instantly kill any application? Is there anything I can do to increase my chances?

>>15601632
I'm doing exactly that at the moment. Doing a part time masters in mathematics but my degree was in theoretical physics which I obtained a decade ago.

>>15601064
This is your brain on academics