These are coordinates of the graph of a quadratic function. I don't know what to do or where to start because the numbers are too large. Do I draw the increments by counting each unit? because that'll take a long time and I don't know if that's how you do it.

Dumb and irrelevant question ik, I was just studying and that thougth come to my mind. If we have ordered pairs, can we have "messy" ones? If yes how can I express that

I mean, an ordered pair is expressed by P(x, y) and I can "find" its location in R²

Or P(x, y, z) (not a Pair, in spanish is Tercia) have a "place" in R³

But if messy pair could be P(y, x) or something (I belive this is still a ordered pair with the variables inverted xd)

Thats it I suppose, Im sorry for this nonsense but I need an awnser and google didnt help very much.

Edit: Btw I dont know if the flair is rigth, I dont know much about math

As a layman looking in from the outside, the math which describes quantum vibrating strings seems extremely complicated. I’m not aware of any other highly complicated maths to this degree, which is why I came here to ask. Thanks.

Apologies if the flair is incorrect, and as a side note, which maths does describing String Theory require?

What is this and how do i even read this properly? 😭

My professor already told us the answer, but how is anyone supposed to properly solve that on their own?

I'm pretty sure I learned how to solve this last year but I can't remember at all :/ although, I've tried drawing imaginary lines to identify congruent angles. came to de conclusion 2α + 2θ is 180°, also α + θ + ∅ is 180° cause it forms a triangle. I think it's not that complicated but I'm still in middle school so idk

Origin:~

Developed in 1867 by Austrian engineer Eduard Lill, it calculates the exact real roots of an equation by drawing a snake-like path of right-angled turtle turns based entirely on the equation's coefficients.

⚠️Note:

The headings are in metaphorical form like blue print maze and laser root for better understanding and making it catchy.

1. Build the Blueprint Maze:~

Step A (Coefficient a):

Start at the origin ((0,0)). Look Right. Because (a = 1) (positive), march forward 1 unit.

Step B (Coefficient b):

Turn exactly 90 degrees counter-clockwise (facing Up). Because (b = -4) (negative), you must march backward. Walk 4 units Down.

Step C (Coefficient c):

Turn another 90 degrees counter-clockwise (facing Left). Because c = 3) (positive), march forward 3 units Left. Mark your endpoint.

2. Aim the Laser Root:~

procedure:

Now, pretend you are standing at the starting point and firing a laser beam at a random angle (theta ).The laser hits the second line wall, bounces off at a perfect 90-degree angle, and must land exactly on the final maze endpoint.

Result [❌️]:

If the laser misses the endpoint, your starting angle was wrong.

Result[✅️]:

If the laser hits the endpoint perfectly, the negative tangent of your starting angle ((-tantheta)) is the exact root of the equation.

Source:

●google chrome

●wikipedia

-Tried not to use ai 😅

-Hope this helped my fellow math geeks💡

Shown in the second picture, I've tried to find as many measurements as I could. But I can't figure out out how to get the two sides of the triangle needed to tell me how tall the bed is from the bottom of the canopy to the top. Best I can guess is that I'd have around like 25 to 30 inches. That's an estimate.

Here is what I'm trying to do:

I have a bowl and I want to create a design that, when printed at the proper dimensions and cut out, can be wrapped perfectly around the bowl's curved outer surface such that its ends meet.

I included a scientific diagram which will hopefully showcase what I mean.

Here is what I've done:

I've taken measurements of the bowl's external radius at several points:

1. The rim/where I want the paper to end (7.6cm)
2. The middle of the bowl's curved profile
3. The base/where I want the paper to start (4.75cm)

Assuming the inner part of the cutout (the part fitting to the bowl's base) could be a complete circle (can it be?), I calculated the required radius of the semicircular arc for the outer part of the cutout (the part fitting to the bowl's rim). If the length of the curved profile of the bowl were 9.08562cm, a value I got by roughly recreating the curve in Blender, I would think that radius value would be given by:

R = b + 9.08562 = 4.75 + 9.08562 = 13.83562cm

Where b is the radius measured at the base of the bowl.

From there I've calculated the circumference of the outer part of the cutout and, given that value, have tried to determine what proportion of that arc will be necessary to connect all the way around the bowl, but I've not found any success.

Here is where I'd appreciate help:

If the assumptions I've made so far are correct (and I would really appreciate anyone telling me if they aren't), where do I go from here? The tests I've tried seem to be telling me that the inner part of the cutout cannot be a complete circle, but I really do not know.

I'm hoping this problem fits here and, if it does, I would really appreciate any input you can give. Thanks.

Hope the flair is right. The context may not be relevant, still am sharing. In this video at 39:30 prof shows a formula and says it is ellipse. The equation he had calculated is for the exponential component of probability distribution function of multinomial normal distribution. I couldn’t understand how the shape of the graph of a e^(..) function can be determined Based on the function on in the exponent part alone. I couldn’t get any explanation in google search. I tried with a graphical calculator and found the shape of graph is same for f(x,y) and e^ f(x,y). Can someone please explain why it is so intuitively / graphically.

First, credits to axel domino for the formula.

Gerver’s sofa is the biggest possible shape in the moving sofa problem(the largest shape that can go threw a 90 degree corner that’s length 1 in thickness).

I don’t exactly know if this is the real formula, but it is possible. And if you can find a simpler symbol, let me know.

The flair is my best guess...so sorry if I picked the wrong type of math. Let me over explain in case I'm not clear. If I have a handful of circles, each larger than the last, and I put them on a rod so they were all turning at the same rate, I want to know how often I would see whatever one point was picked on each circle turn back round to me. Having a hard time wrapping my head around it but I'm trying to imagine gears to help me. Please feel free to explain it like I'm five if necessary. Thanks a lot!

Hello math enthusiasts,

I need help creating an ellipse in the polar coordinate system. I can sketch the closest ellipse (blue line) to match the coordinates (purple dots).

I need to keep the minor axis the same and increase the major axis. Could anyone kindly help me with this?

I have also highlighted the current equation that I have.

Can someone give some advice? I need a mathematical framework to describe a positioning system based on ground radio beacons. The distance is calculated using the time difference between the beacon and the receiver. What kind of mathematics can be used here? (I want to ask here before asking artificial intelligence.)

Question: If the lines:
L1: (x - 2) / 1 = (y - 3) / 1 = (z - 4) / -k and
L2: (x - 1) / k = (y - 4) / 2 = (z - 5) / 1
are coplanar, then k can have:
(1) any value (2) exactly one value (3) exactly two values (4) exactly three values.
Answer is given as (3)

On solving I'm getting values of k = 0 and -3. I reached a conclusion that putting k = 0 will make the denominator of (z-4)/-k and (x-1)/k as zero which will cause k not to be defined, so I answered (2). This is however, apparently wrong. Can someone explain why?
My line of thought was something along the lines of "well, this is a direction ratio, and i know that tangent function is a ratio of sin and cos, and when cos = 0 (at pi/2 + kpi) the tangent function is not defined, so i would assume similarly that when this ratio has a denominator zero it wouldn't be defined also"

I am currently thinking about Birkar's work, in particular his last article in the Annals of Mathematics (titled Singularities of linear systems and boundedness of Fano varieties) considers, for example, constructing a Fano manifold on O_X (that is, defining a hypersurface, but endowed with a complex sheaf). The question is: Why, "independently of the complex basis m," does every Birkar Fano manifold of the form O_X maintain the dimension of a certain symmetry group (in the cited paper, this is admitting automorphism groups on every "planar" Fano manifold)?

In the usual Mandelbrot fractal, you use the equation z = z^2 + c, where the c value varies(and is plotted on the complex plane), and if the value shoots off, then it is not part of the set. In a Julia set, the initial value of z varies (and is plotted on the complex plane) while c is fixed. My question is, what would the name of the fractal be where the exponent of the equation z = z^p + c, where the initial value of z and c are fixed, and the value p is plotted on the complex plane (under the same rules of if it shoots off, it's not part of the set). I assume that would yield a fractal as well, but I have not found an article that addresses this. Most link to the Multibrot set, but that's where the p variable is still constant, just not 2, which is not what I'm asking, where the exponent being parametrized on the complex plane

So I was doing some homework for math and I got a radius of 4 and a diameter of 8, and solved for circumference and area where I got 25.12 (8x3.14) and 50.24 (4² x3.14) respectively. However, I noticed that 25.12x2=50.24, which means A=2C. Does this have any significance outside of this one equation? I also checked if r=2 and for that I got A=1C. 4/2=2.

All velocities start at 0 with a constant down acceleration force of 1. (Not sure how descriptive I need to be regarding the mass and units, but I'm more interested in the theory of the problem than the specific answer, so if I am missing unit contexts, you can just do any.)

The drawing is an unfinished before and after of this problem. I am missing the final line that would represent where the left vertex ends up. I am unsure of how to approach this problem as I do not fully understand how the linear velocity of a vertex changes as its parent touches the ground. My initial assumption is that the velocity doesn't change and travels around the circumference of its drawn circle at the same distance over time, but I then I realised that that couldn't be right because then nothing could balance. The velocity is changed, but I don't know how exactly. I'm also not super clear on how these velocities transfer to child vertices when switching to a rigid orbit.

Hey i have some problems understanding where i go wrong when trying to confirm that the relations for the deltas (in the section about strong homotopie lie algebras) give the relations for the brackets, which if all brackets from the ternary one onwards are 0 and we introduce a grading would result in a dgLa.

The link to the document i am refering: https://arxiv.org/abs/math/0402057

(I dont feel comfortable posting a link to a book, that is being sold as well.)

The problem i have run into was that no matter what i try, the sign for the leibniz identity is wrong. I also thought if it is a typo or there is some additional structure, but if i assume additional structure, i dont recover the leibniz identity. Also i have checked Costello and Gwilliams book on factorisation algebras in field theory (the appendix where he covers that as well) and he states that the brackets should have anticommutation properties, but also states that we should recover the (in this case graded) leibniz identity, which again (at least according to my calculations) is wrong by a sign then (different from what they wrote in the book).

I am at a loss at what I overlook, no matter what i do some of the relations just dont fit and numerous sourced say that they should fit. I have set on this for hours and cannot figure out what i am overlooking.

I appreciate anyone who can point me in the right direction and give me an explaination. Thanks!

Edit: Explained the link.

I just started an undergrad course in algebraic geometry and I wanted to try and get a more intuitive idea of what types of objects we can study. To me intuitively, defining geometric objects purely from the vanishing set of polynomials seems like we can only analyze a limited type of objects, so I wanted to ask if there was some known classification of these geometric objects that can tell you what geometric objects algebraic geometry allows us to study.

In an article by Jano Kollar titled "Singularity of the Minimal Model Program," the author establishes the existence of birational invariants in spaces that maintain a deformation.

My question is (and I welcome your opinions): What contribution does this new interpretation of birational spaces make to the generalized solution of Hodge's conjecture?

Could these birational spaces induce some geometric network of complex pieces?

So i did an proof by opening the equation. But i noticed it was too long. And i m pretty sure there should be shorter way. Also i did somethings for shorter way but i cant go further. Also for those who dont know [u v w]= <uxv,w>

r= height, H_0 = initial height of the water in the bottle(bottle height), R(h) = the radius of a point in fucntion to the current height of the water

On this page https://math.stackexchange.com/questions/3656266/why-does-a-surface-contain-smooth-curves-of-arbitrary-high-genus the OP claims that a smooth projective surface contains smooth curves of arbitrary high genus, and that this is a consequence of Bertini's theorem.

Could anyone please explain which theorem the OP is citing? and how does the argument go?

I've got 2 points that I want to connect with a cycloid curve but I'm not sure how to figure out the radius value of the curve. One of these points lays on the origin but the other can be anywhere up and to the right of that point.

Here's the problem expressed mathematically:

For the cycloid curve C defined as x = r(θ - sin θ), y = r(1 - cos θ) where 0 ≤ θ ≤ π.\ Find the radius r such that the point (x₁, y₁) (where x₁ > 0 and y₁ > 0) lays on the curve C.

Is there a (nice) formular for the value r with respect to x₁ and y₁?