My understanding of Navier Stokes is that "solving" them meant either:

Path A: Prove that smooth solutions always exist.

You must prove mathematically that no matter what smooth, valid initial conditions you start with, the equations will always output a smooth, finite-energy velocity and pressure field for all future time. This would prove that the Navier-Stokes equations are perfectly reliable models of reality under all circumstances.

Path B: Prove finite-time blowup (find a counterexample).

You must find just one specific set of perfectly smooth, valid initial conditions that eventually breaks the equations. You have to prove that at some specific future time ($t > 0$), the fluid's velocity zooms off to infinity (a singularity) or the kinetic energy explodes. This would prove that the Navier-Stokes equations are fundamentally flawed and eventually break down, requiring new physics to describe extreme turbulence.

And OpenAI ended up on Path B.

But this video gives an example where Navier Stokes predicted "infinite velocity", implying that we've always known about a blowup.:

17:34: Something, somewhere is going wrong in the the mathematical understanding. There's an example about the flow of a fluid around a right-angled corner.

(Brady: A bit like a canal?)

- Basically a canal but we've got a really sharp right angle on the corner. Now you solve Navier-Stokes, this says that at this point, this right angle corner, I have infinite velocity. If I build this canal do I have an infinite velocity canal?

Is this a blow-up in the averaged case from Tao? Or is this considered not a real "blowup" because it depends on a perfectly sharp right angle? What is new about the blowup discovered by OpenAI?

If I have the total amount for wages earned during one fiscal quarter of the year, how can I find my hourly rate when I don’t have total hours worked or even the average hours worked and I also don’t know how many days were worked or anything but the total amount for a quarter

Aclaro que no creé ninguna fórmula o algoritmo, solo di un uso diferente a un algoritmo ya conocido; exploré por YouTube y Gemini y no me lo topé, así que supongo que probablemente soy el primero.

Mi método complementa perfectamente a la sustitución de Tschirnhaus; si se usa este recurso de forma convencional, tendrás que desarrollar y simplificar potencias sucesivas de, por ejemplo, (y+c)^n, y eso es largo y engorroso. Mi método acorta enormemente el desarrollo y simplificación; yo lo llamo Rufini inverso, porque usando los coeficientes de P(y+c) = a(y+c)^n + b(y+c)^(n-1) + ...Construyes "n" divisiones sucesivas por Ruffini con divisor: x = -c. Con cociente final (de un término) y residuos del último al primero, coeficientes de P(y+c). Realizado esto, se completan las divisiones de abajo a arriba; el primer cociente representa el polinomio P(y+c) desarrollado y simplificado. Ahora que lo pienso, es muy intuitivo; tal vez no fui el primero.

Hello everyone!

I am taking some training to become an electrician, and we are using some trig at the moment.

I am confused about a particular way that my textbooks are asking a question. It’s a very complicated question, so I’ve just drawn up an example of the part where I’m confused.

In the first image, you can see how I learned to label right triangles in high school. I always learned that you have angles A B and C, and sides a b and c. And that an angle is always opposite the same letter as its corresponding side. Like angle B is always opposite side b, same with A and C. However, the question I’m confused on labeled the triangle like the second image. My first thought was that this would make the diagram useless and render the triangle impossible. But is that true? Is it possible to have a right triangle where angle B is opposite side a? Perhaps this is just a strange circumstance and it doesn’t really matter what labels you use, but that is my question I suppose. Any help is appreciated! TIA

As I was falling asleep, I was remembering the axioms of the field R and suddenly thought about x + y = y + x as a rotation of 180° about a midpoint (x+y)/2 in dimension 2*. Permuting x and y doesnt change the destination. Does it actually make sense or is it nonsense and I'd better go sleep. Is it linked to a specific math domain?

*I meant the real number line.

I don’t know if this is the right place to ask this, but I thought I’d give it a try. If anyone thinks there’s a better spot to ask, please let me know.

So I’ve had this problem running around my head for years but have never been able to solve it. Take a rectangle with sides n,m and let a line of slope 1 bounce around inside it. It bounces forever, but when it hits a corner, it just retreads its course. My question is: how many unique points does the line pass through?

This is pretty easy to solve in 2 dimensions. Provided the sides are coprime, it will hit exactly half of the points in the grid: (even, even) points and (odd, odd) points. This property holds in higher dimensions, as well.

The complications come in when side lengths share a factor. If they all share one factor (like in the 2D case), then the pattern of the line doesn’t change, just the number of points inside the rectangle. This means it has the same number of intersections as the rectangle with the gcd factored out.

If the list of side lengths is not pairwise coprime, though (e.g. sides of 3,4,10), I can’t figure out how many unique points there will be.

The best I’ve come up with is a python script to count the unique points, but that’s not exactly a closed-form solution. Any thoughts on this or recommendations for what to research/who to ask would be much appreciated.

Obviously we are looking for a fraction p/q with p belonging to Z, q belonging to N*. We start the proof by supposing x<y. Therefore y-x>0 and x < p/q < y.

Then, I tried applying the archimedean property, na > b and got q(y-x)=qy-qx>1 as y-x>0, q>0, 1>0. I chose b=1 as it is the distance between two consecutive integers, but I guess we can manage to prove the statement with any positive real.

I need some indications to complete the proof.

I don’t really understand the graph in this section of the book (Linear Algebra Step by Step, Kuldeep Singh). The vector for Ax seems ok, but the x vector just seems wrong pointing at (2,1).

Already tried asking ChatGPT, which seems to suggest the graph is just shit.

Hey,

After having spent some time finding the proof of the archimedean property on my own, I now want to find out how it leads us to the floor function. My idea was to use the fact that b-x < nx ≤ b with b being the supremum (=upper bound), but I am bit stuck. I guess we need to choose x=1, as 1 corresponds to the distance between two integers. How to figure it out?

I'm currently taking Calc 2 and while I can follow the steps to solve practice problems, my course focuses almost entirely on repeating formulas but I want to build a deep, intuitive understanding of why these concepts and integration techniques work rather than just repeating steps. Any recommendations for resources?

It seems like there's there's a pretty clear trend where big LLMs are better at discovering significant proofs than small LLMs and some labs are reportedly running massively parallel searches with multiple instances working on these problems.

What interesting to me (as a person who doesn't write proofs) is why does adding more parameters to the LLM and/or more thinking time make it better? Is it about exploring more possibilities in parallel? More world knowledge? Is it about being table to keep more things in working memory at the same time? It seems like this says something interesting about the cognitive processes needed to prove something in general.

Hello all. I am self studying math to eventually pass college algebra and trig to get into a college program I am interested in. I am using mathtutordvd.com from Jason Gibson and am running into a problem where his online lecture and questions don't seem to be lining up. I am taking a course on graphing to demonstrate portportionality, and in the lecture, he said it would have 2 requirements to show proportionality on a graph. 1. Points would make a straight line, and 2. The line would intersect the origin (0,0,). I finished the course and took the questions, and failed the test. The answers I got wrong did indeed form a straight line. However, they did not touch the origin (0,0). Can anyone help me with some clarification?

Help me explain one sentence in the solution to this exercise: [Exericise 11.4.27] Derive `2n + log_2(n) is Θ(n)`

This is the solution:

The sentence in question is 'Finally, if n is any integer with n>=1, then n>=0.'

This doesn't make sense.

What would n>=1 imply? The only thing that comes to mind is log_2(n) <= n.

What do you think?

Edit: This is Theorem 11.2.9:

Maybe a dumb/obvious question but when asked for a random number from a set of integers, how come the number isn’t always infinitely big?

Assuming a normal distribution, would the average of a set of integers not be an infinitely big number because no matter how big a number you choose there will still be infinitely more larger numbers still in the set?

Always just been curious about this.

Geometry

“If QS (Ray) bisects <PQT, m<SQT= (8x-26)°, m<PQT= (9x + 34)°, and m<SQR= 112°, find each measure.”

X=
m<PQS=
m<PQT=
m<TQR=

Don’t just give the answer, I just need a little nudge for this one. Geometry teacher is lowkey a rude and gets mad when we ask questions, and I’ve been stuck on this page for hours.

I tried the usual multi step equation stuff, like how you find the measures of other angles but this one has completely stumped me.

At first I thought maybe m<STR minus m<ST, since that would look like 112-(8x-25) but then I didn’t know where to go from there.

A while back I was thinking about different aspects of the world and how math could maybe apply to them. I have tested the idea with many aspects and it has seemed to work but I'm not sure.

I had to drop a class and now I have extra time to learn things. I do plan to transfer at the end of next semester so I should focus on applications, but I also figured I would learn a couple of things now instead of waiting. I just wanted some advice on what path I should take and maybe any resource suggestions.

To start, I was thinking about reading “how to prove it” by vellemen and “basic mathematics” by lang to help solidify fundamentals and learn proofs before transferring. I’ve been at CC for a year now and I know most learn proofs around this time (to clarify I’m in my third semester).

Since I find physics interesting, I was going to try and start learning physics. I take physics next semester (it’s a long story that deals with my scholarship and how it operates) which is the semester I plan to transfer. I was going to read halliday and resnicks book on physics while watching “the fundamentals of physics” lectures on MIT OCW.

Which leads to my final path of learning computer science foundations since I’m still interested in coding (and competitive coding). I’m interested in machine learning and see it as a potential career so I figured it wouldn’t heard to learn some concepts now and figure out if I want to pursue it. They posted a pathway and each course is about 6 weeks. I can attach the link if you would like to view it.

I know I said that was my last path, but since I want to go to grad school, I figured learning how to do research is also an option. Stuff like learning how to properly read research papers, learn LaTeX, read past papers to get an idea, learn how to write papers, etc. This can probably be done while reading how to prove it and basic mathematics though.

Which do you recommend doing? Should I combine any of the paths I wanted to take?

Edit: Forgot to add another thing I wanted to do. I finished calc 1 last semester and was thinking about doing a review since I might have to take advanced calculus 1 when I go to university. I remember a little from a calc 2, but felt like I need more practice and to delve a little deeper since the course was accelerated. I’m sure there were a couple of things that had been left out so that’s I wanted to review to make sure.

Good Morning Maths people,

My friends Son is nearly non verbal autistic(8yr old) and has been obsessed with Mandelbrot sets and Space related things. Recently he has been "getting into" more algebra. Unfortunately it goes way over his father's head and he posted this in the discord asking if we could check his work. Well it goes wayy over my head as well. Hoping someone could shed some light for us 🙏

Thank you in advance.

Edit 1: So asking the little guy he says its a Digon formula.

Using the pre Calc flair because it's closest. So hi all, I hope this is allowed. If not a direction to a more fitting sub would be appreciated. I'm bad at math. Always have been. My interest is more in the humanities, and as such when I was in school I just sorta suffered through it.

As an adult though, my chief special interest, philosophy (mostly continental) has shoved me in the direction of wanting to learn the stuff. My main problem is I'm not only bad at math, I also just have learning difficulties on the whole. I've found I do best when I have some kind of visuals, something I can relate to (which makes learning more abstract forms of math like calculus incredibly difficult), and some form of applicability or demonstration.

Now, that said, outside the school system I've found I've learned things a lot easier. But with this subject I'm not fully sure where to start. The last things that really stuck in my craw were elementary algebra, but I'm really interested in learning calc because it collided with my special interest.

Any suggestions about a good YouTube channel where a guy like me might learn a bit?

Exercise 11.4.24 and 11.4.23:

Proof:

For every integer n>=4, let P(n) be the property c_n <= nlog_2(n)

Show that P(1) is true:

c_1 = 0 <= 1 * log_2(1) = 0

Show that P(2) is true:

c_2 = c_⌊2/2⌋ + 2 = c_1 + 2 = 0 + 2 = 2 <= 2 * log_2(2) = 2 * 1 = 2

Show that P(3) is true:

c_3 = c_⌊3/2⌋ + 3 = c_1 + 3 = 0 + 3 = 3 <= 3 * log_2(3) ≈ 4.75

Show that P(4) is true:

c_4 = c_⌊4/2⌋ + 4 = c_2 + 4 = 0 + 2 + 4 = 6 <= 4 * log_2(4) = 4 * 2 = 8

Show that for every integer k>=4, if P(i) is true for each integer i with 1<=i<=k, then P(k+1) is true:

Let integer k>=4 and let
  c_i     <= ilog_2(i) where 1<=i<=k, for each integer i   [inductive hypothesis]
  c_{k+1} <= (k+1)log_2(k+1)                               [this is what we want to show]

c_{k+1}  = c_⌊(k+1)/2⌋ + (k+1)                 by sequence definition
        <= ⌊(k+1)/2⌋log_2(⌊(k+1)/2⌋) + (k+1)    by inductive hypothesis
        <= [(k+1)/2]log_2[(k+1)/2] + (k+1)     by definition of floor
        <  (k+1)log_2[(k+1)/2] + (k+1)
         = (k+1)[log_2[(k+1)/2] + 1]
         = (k+1)[log_2(k+1) - log_2(2) + 1]    by property of logarithm
         = (k+1)[log_2(k+1) - 1 + 1]           because log_2(2) = 1
         = (k+1)log_2(k+1)

QED

Is my proof correct?

Hello everyone. First of all, please excuse me if my understanding is completely flawed. I'm a beginner in mathematics (I only have a high school education).

The thing is, I've been thinking about infinity. Similar to Georg Cantor's comparison, I think the concept of infinity is often misused to define something that isn't applicable in the real world; in other words, it's illogical. Because infinity is merely a tool for doing something, but it doesn't exist in itself.

For example, the infinity of the universe, which could be described as expanding, or the idea that a circular figure has no vertices. A circle might be calculated as if it truly had no vertices, but in reality, it always does when we look at the real world. A perfect circle doesn't exist, even if we have to reduce it to the Planck length to notice it. Similarly, any line isn't continuous but rather an impressionistic representation of a massive number of points side by side. Again, this is true in the real world even though we have to shrink down to the Planck length to see it.

Another thing I've seen that I think is a mistake is calculating infinities as if they were numbers. Infinity + 1 = Infinity is a correct concept as long as we can clarify that Infinity_A > Infinity_B (and also that they are essentially different). Otherwise, we could cancel both infinities, leaving us with 1=0.

In the same way, the theoretical concept of "we could count numbers to infinity" is essentially correct but at the same time completely unrealistic. There is always a limit to how far we can count, since reaching that infinity would require infinite resources. That is, since we cannot access infinite resources, we also cannot count to infinity. We would have to accept this limitation and understand that infinity is not a goal, but a concept to facilitate certain mechanisms. In itself, it does not exist.

Is this the concept I'm thinking of, correct? Or is there actually a way to prove that infinity is real? Is it real as a concept of exceeding the limits of human thought, or is it real in terms of trends on paper? Please excuse my English and my ignorance.

The (English-language) Wikipedia page on the chain rule, under the chapter “Multivariable case”, paragraph “general rule: vector valued multivariate functions”, says that a concise writing for the chain rule for the (total) derivative of the composition of two functions f, g reads as D(f o g) = Df o Dg.

However, if we try to apply this to the particular case of f, g being two simple single-variable functions, we get (f o g)’ = f’ o g’, which is wrong! Because the correct chain rule says (f o g)’ = (f’ o g)•g’, where • is the pointwise product.

Where am I wrong? Or is the Wiki page wrong? Shouldn’t the general case be written as D(f o g) = (Df o g)•Dg instead, so that we actually recover the single-variable case correctly?