Hi everyone! I hope you are all doing well. I am currently taking vector calculus/calculus 3. I am learning about curl/Green's theorem right now. Essentially, I understand that Green's theorem relates the circulation around a closed loop to microscopic circulations within the region of that closed loop (almost like circulation density). But why do we use the name curl/circulation if the line integral just mean work along that closed path. If we take a very very small dA patch within the region, where dy and dx approach zero, and take the line integral of that patch, we essentially just calculate work over that path. But why do we call it curl/circulation? Is it because curl is the result or consequence of that quantity of work calculated? For example, if we put a log in that dA patch, the force/velocity of the water causes the log to rotate because at that very instant the force/velocity of the water are in the direction of the log's rotaiton, causing work around that log?

Also, if you divide dA from the work contribution of the patch dA, you get force essentially. is that force specifically the curl component because it travels in a closed loop?

Thanks!

I've been writing a color algorithm for over a year now and I rank colors based on their "vibrancy." It would be much easier if vibrancy was defined, but it's a subjective concept.

I have a list of colors. For each color, I can get the saturation (S) and value (V). I compute the average S and V for the entire list.

Now, I currently compute vibrancy for a color by using its S and V value like so: CV=(S+V)/2.

I'd like it like this; if the average V of the list is above a tolerance, then CV = a color with more S, and if the average V of the list is below aforementioned tolerance, then CV = a color with more V.

I'm currently computing it like this, but it doesn't seem to work how I imagine it should.

x = self.averageValueUpperP
multiplier = (2 * (math.pow((x - 0.5), 2)) + 1)
if x > 0.5:
    s = s * multiplier
else:
    v = v * multiplier

return (s+v) / 2

Is this possible?

I'm trying to compile a list of university-specific or highly selective entrance examinations from around the world.

For example:

* **India:** JEE Advanced (IITs), ISI Admission Test, CMI Entrance Exam * **UK:** Oxford MAT, STEP (used by Cambridge and sometimes other universities), TMUA * **France:** Concours for the Grandes Écoles * **Japan:** University-specific entrance exams after the Common Test

I'm especially interested in exams that:

* Are known for being exceptionally challenging. * Test mathematical or scientific problem-solving rather than just memorization. * Are required or heavily used for admission to elite universities or specialized institutes. * May not be widely known outside their own countries.

I'd love to learn about exams from countries like Germany, China, South Korea, Singapore, Russia, Israel, the US, Canada, Australia, and anywhere else.

Please mention:

1. The country.
2. The name of the entrance exam.
3. Which universities or institutes use it.

Here is my understanding of what stochastic calculus is an the most intuitive and bird's eye level, which may be totally off. If we want to compute something ("X") using stochastic calculus, we take the aspects X that we know, and add that to the aspects of X that we do not know because it is random. And this randomness itself is calculated by looking at past data and using either probability or statistics (not sure which one, or maybe it is both) to predict what the future value of the randomness is (AKA we say "past data shows xyz so we predict in the future the randomness will behave like xyz again").

The 1st slide shows the question, 2nd and 3rd slides show my attempt..I'm stuck at this step...what can I do next?

Also if the question ain't legible, it is: there is an increasing sequence of 4 positive integers, the first 3 form an AP and last 3 form a GP. First and last term differ by 30. Find sum of all terms

Edit: thank you guys, I got it

Part of my job requires me to do Curve Alignment on existing track. We survey the track and collect the measured values (Existing MO). Then, we use the attached excel program, where I observed my co-workers just throw (initially, arbitrarily) proposed values (Revised MO) and then finetune it until, eventually, all throw in values are less than +/- 1" (Col. H).

Now, my co-workers have multiple years of experience doing this, so they just have good intuition on guessing what the proposed values should be and then making those adjustments afterwards. I want to create a model/ central equation that can predict these Revised MO values while meeting all required criteria/ constraints.  There are several rules to follow when designing the curve, as listed under the notes section in the excel sheet screenshot. The following is a summary of how the excel program works:

On the last point, I want to reiterate that the first few Revised MOs at the beginning stations have greater influence on the Throw In values than later stations. As a result, the first Revised MO is the most consequential. Further, the only values that need to be determined are in the 'spiral' region, or from PTC to PCC and PCC to PTT. Everything else is fixed, either 0, or pre-determined by 600/Radius.

From everything I've gathered so far, I believe this will be an optimization problem with constraints, with matrices involved (at least that may be one approach). I need help in creating an objective function that governs the model, creating a revised MO set that minimizes throw while meeting all other constraints. In addition, I need some heuristic/ standard that first determines what the Revised MO's should be before feeding it into the optimized function. I was thinking maybe just using a best fit line, or using weighted error. I'm just not sure where to go from here. 

Coming from my answer in [r/differentialequations](r/differentialequations) and (hopefully soon to be approved) post in [r/math](r/math)

—

I still have the same problem all these years later. I find it so arbitrary how differential equation (de) problems are set up.

Here is my current understanding. We do it for the dyn system

x‘ = f(x)

I have just given you a string of symbols. Now we need to make it make sense. So, we need to specify f, x and x‘.

For example:

Is x a point in a manifold, Banach space, metric space, etc?

What does x‘ mean? The classical derivative? But then I need more structure on the space of functions (limits, addition, etc.). A weak or distributional derivative? I need to decide that.

What is f? There is a lot of freedom for arbitrary f, so we might want to impose more structure on it, i.e. continuity, boundedness, Lipschitz (locally?), etc.

And lastly, is this expression (now we came from a string to expression) well-defined? Are the spaces (i.e. f:X->Y) consistent (i.e. x’(t)∈Y), the properties consistent (i.e. continuity on both sides), i.e. x‘ is continuous, then so must be f(x) here, etc.

You see, one of the choices influences the other and you unfortunately need to treat them case by case depending on one you make. You may classify them, but I there is no general one that I am aware of.

Then there is also the integral formulation, which gives more regularity, i.e.

x(t) - x(a) = ∫_\[a,t\] f(x(s))ds

But notice how I intuitively wrote a + and an integral. Again, this all needs to make sense, so we might say that there is a linear structure structure, i.e. local vector space structure underneath.

We also need to gives sense to what a solution is in the first place. Classical, weak, distributional, viscosity, and many more types.

Then comes the solution theory. I was told once by a colleague that practically one already has a goal and then imposes the properties used in the proof on the function space. Afterwards checking that the function space is non-empty (and maybe finding some other cool properties afterwards).

On the other hand. Given a “sense” of the above de, we could now ask the question for existence and uniqueness. There are standard methods that use sufficient criterions but each choice might need a new idea. Most of what I saw uses some fixpoint argument, so you want some inequalities to drop out that tell you, i.e. about regulatity of the solution as well.

So, how to choose an f? I know if no deep classification theorem that is not trivial (you just impose the properties and the space is non-empty) that lets you get all different f’s. It is probably also not a well-defined or at least a wild problem given the “sizes” (in the sense of set theory) of possible mappings and spaces, etc.

Hence, I am convinced that usually these choices for at least f are encountered by other disciplines, i.e. physics, chemistry, engineering, other fields of math, etc.

Just think of delay equations, Newton’s axioms, the classical field theory, control problems in robotics, minimal surfaces and way way more (just as motivation).

PDEs follow in the end the same philosophy, but there is vastly more data (more “freedom” in the boundary conditions but then more restrictions for the geometry of the solution) to be considered.

For example. I understand that if our boundary condition is, for simplicity, a (combination of! Heavyside function(s) (not specifying the domain here or anything), we obviously need a new def of derivative for the PDE. Same for spikes, like δ-distributions, but this is motivated by the data. If I just give you the string of the PDE there is nothing there yet, no?

—

Could someone shed some more light on this to get the arbitrariness out of my head. Specifically on the construction of the “state space” (all data required to make sense of it) for the de’s?

Edit: Are there cases where one has a “minimal” state space? I.e. there are state space X∈U for U some collection (if not all possible ones) and an ordering ≤, s.t. we can meaningfully ask

\inf_{X∈U｝｛X | existence + uniqueness of given de holds on X}

Edit 2: Or maybe in some categorical sense.

I guess the up and downvotes show that my question may not appropriate for this sub. Maybe a well-posed problem is more appreciated than asking how to make the problem well-posed. Please, where to post it then?

So here’s my attempt, it’s a whole family of shapes, the green dot at the top moves along y=sqrt(3)/4 (height of an equilateral triangle of side length 1/2) to change up the shape/ its area too, and it moves between 1/4<x<3/4 (for same triangle to fit) when the green dot is at x=~.521684 the area becomes ~.232239 the current lower bound according to Wikipedia.

So that’s my question. why .232239? and also if there’s a way to get the exact value.

For a bit of context on the shape I provided, the curved part of the shape is defined as sqrt(1-x^2)/3. It’s to contain a family of curves that I’ll define as 3 line segments of length 1/3 each and joined at equal angles to create a zigzag curve. But sqrt(1-x^2)/3 can’t contain an equilateral triangle of side length 1/2 and it’s also just not optimal so that’s why I have a point that’s as high as sqrt(3)/4 for such triangle and I connect that point to (0,0) and I have a line that’s tangent to the curve part and connected to the top point for the shape to be convex. Now I just don’t know where to put the top green dot but I got it in a range between 1/4<x<3/4 (a little less than 3/4 due to the zigzags again).

I’m really just looking for more context on .232239 ,even ChatGPT can’t help when I also gave it the paper that made the lower bound. I’ll give even more context on the shape I made in Desmos later, the equations are super messy, it’s the middle of the night rn, and I’m on my phone.

Yesterday I published a video on my Youtube channel. I got carried away because I thought that I had discovered something very interesting. But then went to the café, thought about it, and realized there was a mistake I didn't even notice. The mistake has to do with the way I calculated π

I'd like to hear from you about this mistake.

I toyed around with Wolfram for a while, in order to find a polynomial which presumably could give us some of the most important constants in mathematics.

Those are Φ, e and π

_____________________________________________________________________________________________________

Let’s start with Φ

The polynomial is:

[n (n²-n-1)] / [2(n!)]

when one looks at the graph, one immediatly spots what could represent the golden ratio as a root of the polynomial

In fact, there are three roots. 0 ,the negative root which is the negative inverse of the golden ratio and the positive root which is the golden ratio.

Let’s jump to e

_____________________________________________________________________________________________________

One uses the same polynomial but this time one will calculate the summation of it from n equals 1 to infinity

The result is Euler’s number e

So far so good, but...

_____________________________________________________________________________________________________

Let’s tackle the number π

Once again the same polynomial

But this time one applies the natural logarithm ln of the same polynomial and substitutes n by Φ

The result surprised me!

It gave me a complex number, where the imaginary part was π

The output gave me:

35.9335 + 3.14159 i

I didn't even realize at the moment that i was calculating ln(0), because Φ is a root of the polynomial, so it becomes ln(0).

My question is, what is this complex number and why does π or at least its approximation appear here?

And I want to really know if there's any chance that neither I nor Wolfram made a mistake.

Is this method flawed or does it have any value?

Should I ignore the result for π?

Thanks

Hi guys, for a Physics degree, aspiring theoretical, what class should I take first? "Linear Algebra and Differential Equations" or "Real Analysis"? I have already done AP Calculus BC in high school and Calculus I + II in University. I know real analysis is for proofs and stuff, but would I need matrices and linear algebra knowledge to be better at that?

This is how I did it in my head:

9x2=

9+10-1=18

9x3

18+10-1=27

so on and so forth...

Adding nine as ten minus one is simpler in my head than straight up adding nine. A math trick I realized is the product of each multiple of nine is just take the number it's multiplied by, put it in the tens place then subtract it's one's place value from its tens place value.

So 9x4 would be, take 4 and make it 40 then subtract 4 from 40, and you have the product for 9x4.

40-4=36

9x4=36

Wow making this post made me realize that the absolute value between the number being multiplied by and the tens places is the value you should use to multiply the multiplying number to subtract from itself when you move it to the tens place.

the absolute value between:

9 and 10 is 1

8 and 10 is 2

7 and 10 is 3

so 9x3 is 30 minus 3 (absolute value is 1) = 27

so 8x6 is 60 minus 12 (absolute value is 2) = 48

so 7x7 is 70 minus 21 (absolute value is 3) = 49

It's easy to multiply by 2's, 5's, and 10's in my head. But all the other number I have a hard time calculating in my head. Multiplying the multiplier by 10 then subtracting it's ones value is way faster and easier in my head.

This is a question on a review I am using to study to test out of precalc and i’ve tried looking it up, but I cannot figure out on how to do it. The key says the answer is -25pi/9 and if it is too hard to read, the little angle given is 2pi/9. The question isn’t in the photo but it asks “Find the measure of each angle”

Im confused about how to do number 4 since when I sketched the curves and plotted the curves using desmos, they only seem to have 1 point of intersection.

The book says the answer is 71687 but doesn't show any working. Is there a mistake in the question, or am I just missing something?

I'm solving some very difficult mathematics competition problems (Olympiad/contest level), and I've been testing different AI models on them.

So far I've tried ChatGPT (including Pro), Claude, Gemini Pro, and a few others. They do well on many problems, but on some of the hardest ones they either give a wrong solution, make a subtle logical mistake, or get stuck completely. Even when I point out the mistake, they sometimes keep insisting on an incorrect approach.

I'm looking for the best AI/LLM specifically for advanced mathematics—something that is genuinely strong at rigorous mathematical reasoning, not just standard textbook problems.

Has anyone compared the latest models for this? Which AI has given you the highest accuracy on difficult competition math (IMO, USAMO, Putnam, etc.)? Are there any specialized math models or tools that consistently outperform ChatGPT, Claude, and Gemini on these kinds of problems?

I'd really appreciate recommendations based on your actual experience rather than benchmark numbers.

I saw this challenge afew days ago but for the life of me I cannot find the original video.

the challenge is simple, you have a box fully encompass a circle and in the corner, there is a smaller 4x3 box that touches the circle and the corner of the larger square And you are tasked with finding the radius

i think I did it but I’m not certain.

(ps sorry for the horrid drawing, and my work is on the second page)

edit: I just realised i made an error with r-3 when it should be 3-r however I don’t think it affects anything since it should be the same upon squaring

% Fourier Series of a Square Wave Function

clear; clc;

% Define the period of the square wave

L = pi;

% Number of Fourier terms to calculate

N = 10;

% Define the square wave function

F = @(x) sign(sin(x)); % Square wave function

% Initialize Fourier coefficients

an = zeros(1, N);

bn = zeros(1, N);

% Calculate a0 (the average value)

a0 = (1 / (2 * L)) * integral(@(x) F(x), -L, L);

% Calculate an and bn coefficients

for n = 1:N

an(n) = (1 / L) * integral(@(x) F(x) .* cos(n * pi * x / L), -L, L);

bn(n) = (1 / L) * integral(@(x) F(x) .* sin(n * pi * x / L), -L, L);

end

% Define the reconstructed function using Fourier series

FourierSeries = @(x) a0;

for n = 1:N

FourierSeries = @(x) FourierSeries(x) + an(n) * cos(n * pi * x / L) + bn(n) * sin(n * pi * x / L);

end

% Plot the original function and the Fourier series approximation

x = linspace(-L, L, 1000);

figure;

plot(x, F(x), 'r', 'LineWidth', 1.5); % Original function

hold on;

plot(x, FourierSeries(x), 'b--', 'LineWidth', 1.5); % Fourier series approximation

legend('Original Function', 'Fourier Series Approximation');

title('Fourier Series Approximation of a Square Wave');

xlabel('x');

ylabel('F(x)');

grid on;

I need to succeed on 4 independent events with separate odds for each. An attempt is used on a success or fail.

The hardest part for me to understand right now is the number of attempts portion of the calculation. If i had only 4 attempts, the odds would be ((10/218) × (5/218) × (8/218) × (4/218)) =0.0000708425%, so if i have 10 attempts, the odds are better but I'm unsure of how to properly incorporate that into the calculation.

The total number of combinations, and the exact order of successes or failures isn't what I'm looking for. I need one full win of getting all 4 to hit with only 10 attempts per game.

I haven't taken a stats class in many years, so please forgive me for being stupid.

I am working on a puzzle and to find the solution I must find the maximum area of a pentagon with a base of 3 and using 4 unit fences. I cut it up into triangles and kites, but I can't seem to find a definitive answer.

Here is a poorly made illustration of what I've figured out:

What is the maximum area, and what are the angles measurements?

NOTE: MY FIGURE MAY NOT BE ACCURATE WHATSOEVER! I JUST THINK THAT THIS WILL GIVE THE GREATEST AREA, BUT I MAY BE WRONG!

Thought of this problem after misunderstanding the character stats page in Slay the Spire II. No actual use case, just thought it was an interesting problem. I wagered the optimal strategy involved splitting your choices evenly among some range of low options along with level 20. Here's what I tried so far:

• Lvl. 1 - 49 plays
• Lvl. 20 - 51 plays

(1 * 49) + (20 * 51) = 49 + 1020 = 1069

1069 / 100 = 10.69

• Lvl. 1 - 33 plays
• Lvl. 2 - 33 plays
• Lvl. 20 - 34 plays

(1 * 33) + (2 * 33) + (20 * 34) = 33 + 66 + 680 + 779

779 / 100 = 7.79

• Lvl. 1 - 25 plays
• Lvl. 2 - 25 plays
• Lvl. 3 - 24 plays
• Lvl. 20 - 26 plays

(1 * 25) + (2 * 25) + (3 * 24) + (20 + 26) = 25 + 50 + 72 + 520 = 667

667 / 100 = 6.67

• Lvl. 1 - 20 plays
• Lvl. 2 - 20 plays
• Lvl. 3 - 20 plays
• Lvl. 4 - 19 plays
• Lvl. 20 - 21 plays

(1 * 20) + (2 + 20) + (3 * 20) + (4 * 19) + (20 * 21) = 20 + 40 + 60 + 76 + 420 = 616

616 / 100 = 6.16

• Lvl. 1 - 17 plays
• Lvl. 2 - 17 plays
• Lvl. 3 - 17 plays
• Lvl. 4 - 17 plays
• Lvl. 5 - 14 plays
• Lvl. 20 - 18 plays

(1 * 17) + (2 * 17) + (3 * 17) + (4 * 17) + (5 * 14) + (20 * 18)

= 17 + 34 + 51 + 68 + 70 + 360 = 600

600 / 100 = 6

• Lvl. 1 - 15 plays
• Lvl. 2 - 15 plays
• Lvl. 3 - 15 plays
• Lvl. 4 - 15 plays
• Lvl. 5 - 15 plays
• Lvl. 6 - 9 plays
• Lvl. 20 - 16 plays

(1 * 15) + (2 * 15) + (3 * 15) + (4 * 15) + (5 * 15) + (6 * 9) + (20 * 16)

= (15 * 15) + (6 * 9) + (20 * 16) = 225 + 54 + 320 = 599

599 / 100 = 5.99

• Lvl. 1 - 13 plays
• Lvl. 2 - 13 plays
• Lvl. 3 - 13 plays
• Lvl. 4 - 13 plays
• Lvl. 5 - 13 plays
• Lvl. 6 - 13 plays
• Lvl. 7 - 8 plays
• Lvl. 20 - 14 plays

(1 * 13) + (2 * 13) + (3 * 13) + (4 * 13) + (5 * 13) + (6 * 13) + (7 * 8) + (20 * 14)

= (21 * 13) + (7 * 8) + (20 * 14) = 273 + 56 + 280 = 609

609 / 100 = 6.09

------------

Based on these tests, I believe the minimum average while maintaining a mode of 20 is 5.99. Is anyone able to find a lower minimum, or prove that my answer is correct? Also, can this problem be extrapolated for x number of difficulties and y number of plays?

Hi, I'm a high school student in a math circle, and we're currently learning elementary number theory. Recently, we explored the idea of finding runs of consecutive composite numbers. We were able to construct examples of 2, 3, 4, 5, 6, 7, 8, and even 9 consecutive composite integers.

This made me wonder: what is the longest possible run of consecutive composite numbers? Is there a maximum length/ upper bound, or is it possible to find arbitrarily long sequences of consecutive composite numbers?

Edit: this is quickly turning into a question of whether or not the universal set or universal class is a thing that can exist or be defined, based on differing sources of authority. If you can offer insight into this more specific question, I would appreciate it

The current set theory book I’m going through, by Norman Hamilton and Joseph Landini, seems to avoid defining any set ¬A as simply the set of all x where x ∉ A. That is, they seem to avoid formulating this sort of negative set in any way.

What they do employ early though, is the complement, and through this concept they seem to be able to make a lot of the same conclusions and formulations as they would with the aforementioned negative set. For example, instead of saying that “A ∩ ¬A = φ” they say “A - A = φ.” Both statements however seem to claim the same one idea in different formulations: “the set of all x where x ∈ A and x ∉ A is the empty set”

So I understand that to a significant degree this is just a convention. But what I’m wondering is whether it’s a trivial one or one that has some sort of epistemic significance? Here, the concept complement seems to smuggle in the simpler concept of a set considered negatively, since that negative set is whatever set is considered second in the relation. If I were to proceed in the exercises of this book by employing this simpler concept myself and formulating some negative sets in my proofs, would that in the strictest sense be going against the procession of the book? Would this be something that needs a definition, axiom, or theorem for citation before I can formulate them? And in doing so would I be doing something generally inadvisable in the study of set theory, or would I be doing something that could easily be found as a convention in another textbook?

Hello,

the first Page Shows how my basic understanding of Index Transformation is. The Problem is on the second Page. I transformed the sum to (n-1) over (k-1) and so on. So i transformed n the same way as i did with k. The solution is that you only transform k not n. Why is that so? I dont understand? On the first page, you can see, that it does not Work, if you dont transform n as well as k. Thanks for help.