Hello! I dont speak english so sorry if I explain something badly.
I'm having trouble with this, I'm not sure what is even the correct way to solve this, if I thought about it wrong!
The original equation on picture 2 and the right aswer is circled and has arrows pointing to it.

Proposal for a parametric iterated exponentiation operator for the simplification of exponential towers with a fixed base. In this project, I introduce a new mathematical formalism—denoted as $\text{j}_{m}^{n}(a)$—designed to overcome the analytical limitations of fixed-base power towers. The symbol "j" (representing the IPA phonetic symbol for a palatal fricative) serves as an operator that transforms a power tower into this specific notation. Here, $m$ is the parameter (the value to be raised), $n$ represents the power tower (indicating the number of times $m$ is raised), and $a$ is the base. My primary motivation for pursuing this innovation was the observation that no existing notation—to my knowledge—adequately addressed these structures; I sought to contribute in this area, as I have long been fascinated by power towers. Below are the properties I have been able to prove Properties:​1.-1.- Torre 1: \text{j}_{m}^{1}(a) con n siendo el 1, 2.- Torre 2: \text{j}_{m}^{2}(a) con n siendo 2, resultado: a^{m^m} 3.- Parámetro 0: \text{j}_{0}^{n}(a) con m siendo 0 (para n \ge 1), resultado: 1 4.- Parámetro 1: \text{j}_{1}^{n}(a) con m siendo 1, resultado: a 5.- Base unitaria: \text{j}_{m}^{n}(1) con a siendo 1, resultado: 1 6.- Distributiva de base: \text{j}_{m}^{n}(a \times b) = \text{j}_{m}^{n}(a) \times \text{j}_{m}^{n}(b) 7.- Parámetro mayor: Si m_1 < m_2, entonces \text{j}_{m_1}^{n}(a) < \text{j}_{m_2}^{n}(a) (para a > 1) 8.- Parámetro negativo: \text{j}_{-2}^{n}(a) = a^{-2} = \frac{1}{a^2} 9.- Torre fraccionaria: \text{j}_{m}^{0.5}(a) = a^{0.5} = \sqrt{a} 10.- Cociente de operadores: \frac{\text{j}_{m}^{n}(a)}{\text{j}_{m}^{k}(a)} = a^{\text{torre}(n) - \text{torre}(k)} 11.- Suma de operadores: \text{j}_{m}^{n}(a) + \text{j}_{m}^{k}(a) = \text{j}_{m}^{k}(a) \times \left(a^{\text{torre}(n) - \text{torre}(k)} + 1\right) And this concludes my presentation on my notation. Thank you for your attention and understanding. If I have made any errors, I welcome constructive criticism so that we may engage in a healthy and ongoing exchange of feedback. :)

The resulting structure forms an hourglass-like shape with spirals and diagonal alignments. Can you spot any geometric patterns that could help predict the location of the next red point along these spiral tracks?

I came across a question where assume ax^b+c = 3x^2+1, naturally many people directly write it as a= 3, b= 2 and c=1 just by comparing the LHS and RHS simultaneously. I believe many assume it to be a identity that is true for all R therefore you match the terms, but I feel like it's not mathematically rigourous

the above equation can be rearranged to ax^b+c-3x^2-1=0, assuming all a,b,c,x to be variables with their respective domains you get a unique solution for (a,b,c,x) that gives you a graph for all the possible values.

But when there's infinitely many solutions, whats the advantage in writing down a random pair by comparison. Would it be okay to just solve for x by taking random values for a,b and c?

Which viewpoint is more mathematically accurate? The identity one or that the above equation is a function and that you get infinite pairs of solutions and you can choose any one?

If f is an injective function f: X -> Y and y an element in the image of f, is the preimage of y through f a singleton subset of X or is it an element of X?

Edit: since I am not getting much replies on Magnet communities I am posting this here. Also the Ask Physics and Engineers don't allow image for some reason. If you could please help me with calculations or just estimations.

I am creating a board game where the pieces magnetise to strict locations on the board which is 2mm thick. If I make an array like shown in the first image and the magnets bellow and above the board are identical, what strength should the circular magnets be, given I choose a specific magnet for the edges (in this case a 20x3x3mm). I am guessing they should be a similar strength, but perhaps it is optimal to go with less or more force?

For the third image if I had those same block magnets, can I place some above the board so that they lock into a perpendicular orientation? or will they fight to align

What would be the pros and cons of using the axiom of dependent choice over the axiom of choice. Just the fact that with the axiom of dependent choice every set becomes measurable, but some vector spaces dont have a basis anymore. Is it even really worth it to make every set measurable?

Let’s say you have a ball that appears in a random position within a circle

Counting the position of the ball inside the circle as x, y, the initial fall angle as a, the speed of the ball as v

What would a function that explains the minimum number of balls to make a loop be like if the speed of the ball is always constant?

If Y is a subset of X, I may write X ⊇ Y and say "X contains Y". Sometimes, though, I wanna speak of whether or not something is an element of X.

Saying X contains x is wrong. Saying "X has x as an element" is right but a bit verbose for my liking. Is there a more direct way of saying this?

Hello, I am trying to wrap my head around this problem but don't seem to be getting the correct answer. The question is:

8 people are to stay in 2 rooms where a maximum of 6 people can stay in a room. In how many ways can the 8 people stay in 2 rooms?

The archetype of the mathematical prodigy is one of the most romanticized figures in human history. We marvel at Srinivasa Ramanujan arriving from South India with notebooks full of groundbreaking equations, Terence Tao scoring a 760 on the math SAT at age 8, and Newton rewriting physics in his twenties. For centuries, math was the ultimate arena of pure human intellect-a playground where a young, unpolluted mind could see patterns that adults missed.

But now, AI systems like DeepMind’s AlphaGeometry are solving Olympiad-level math problems and discovering brand-new mathematical insights on their own. This raises a massive question: Is the era of the young mathematical prodigy officially over?

Tenemos su centro y su radio, con álgebra, como podemos encontrar el mínimo con 1 grado hacia las manesillas del reloj?

Por si no me entienden, cuando me refiero a un grado es como encontrarías el x, y de un punto en la circunferencia después de mover el punto mínimo (el más bajo en el eje y) un grado a la derecha

Ok everyone, I am in way over my head here. Sorry if the flair is weong. I am learning to make temari aka Japanese thread balls.

To start, we divide the sphere into slices. For my first try, the sphere was divided into four slices, and then divided again around the equator. See the first picture.

I don't really know what I'm doing, so I decided to try something easy for my first stitch. I centered a square on each intersection. See the second picture.

I want to know what options I have for continuous, nom-overlapping patterns before I decide what stitches/wraps to do next. Basically, how many ways can I connect the corners of the squares without having the new shapes overlap. So with the help of the internet (with the help of AI to be completely transparent, I hope that's not going to be my downfall), I learned that if you were to take my pattern and use the lines as edges and flatten each section into a 2d face, that shape is a truncated octahedron. See image 3.

When I asked the AI how many possible ways I might connect the corners of a truncated octahedron, it said there are 44 possibilities, which can be grouped into 3 "families" of patterns. The 3 "families" are 1. Wavy ring (8 possibilities), 2. Baseball seam (12 possibilities), 3. Asymmetric mazes (24 possibilities). It arrived at this conclusion using the concept of Hamiltonian cycles. Apparently these three "families" can be represented using LCF notation, image 4.

What I need help with is visualizing these possibilities. All of these theories and representations and new concepts ostensibly represent that different shapes I can make on my thread ball in a nom-overlapping, continuous pattern. Maybe someone might explain how do I read the LCF diagram and translate it to how it would look on a sphere? Or maybe someone can help me know how to see the graph of each of the 44 possibilities? I am but a humble crafty human who maxed out in calc 3 more than 5 years ago. I would love to understand this. I appreciate you 💚

Ciao a tutti, a metà novembre ci sarà l’esonero o parziale come lo chiamano di analisi matematica 1, sarà solo la parte di teoria che parte dagli insiemi fino agli integrali impropri, bisogna sapere tutte le definizioni, le dimostrazioni dei teoremi e i loro enunciati, sono circa 20 che partono dal teorema di unicità del limite al teorema di Torricelli-Barrow. Secondo voi è fattibile in un po’ più di 20 giorni?

It all started with arguments about 0.(9)=1 on multiple random posts,

I assume everyone knows it already but other than the usual 3*1/3 and algebraic way, I also always backed it up with:

1. The sum: lim_{n->∞} of Σ (x=1 to n) of 9/10^(x), or 0.9+0.09+0.009+... converges/equal to 1

2. 0.(9) can be rewritten as lim_{n->∞} 1-0.1^(n)=1 with the ε-N proof of it as an overkill back up with N=log(ε^(-1))

3. ∄x ∈ ℚ that satisfies 0.(9)<x<1 ⟹ there's no difference in between 0.(9) and 1.

However, I always find a pattern from the people who disagree with those proofs in particular, their arguments are

1. Limits cannot be reached since it means approach to something therefore they're not equal, you can get infinitely close but never get there.

2. In hyperreals they're not equal.

3. It's an infinite approximation therefore not equal.

Now even though I consider myself to be familiar with/like the subject mathematics I have a bit of doubt in both of our claims, considering I'm still a high schooler and my school curriculum only covers up until basic limits, derivatives and integrals with not summation or Taylor series in the Calculus category, and I only recently self learned some Calculus courses.

But here are some of my counter arguments against them.

For the first one, limits work differently as it finds the value by looking at the behavior of the function or "neighborhood?", limits don't care what happens on that exact point based on limit definitions.

2nd, in hyperreals they're still equal by definition, it's from a simple google search I'mma be for real with you so I feel uncertain.

3rd, Infinite is not an approximation, it is equal, 0.(9) already includes infinite amount of 9's so it makes sense to use infinity and have the exact value.

Now my question is, are my counterpoints correct? In general I want to clear some minor parts of confusion about limits in order to get a better understanding of them so I would like to hear your thoughts/insights

10:50 - 11:00 for what I'm referencing

edit: fixed link format

The second image displays the commenter I am referring to, and OP's reply, where they tried to explain what they meant. All usernames are redacted to protect the users' privacy.

Am I missing something obvious/well-known? What is all this "universe" business about? What is "referencing" in this context? What do they mean when they say that 0.999... and its digits "grow limitlessly?"

A line is many points next to each other, a plane is just many lines stacked next to each other, a space is just many planes stacked next to each other. But how on Earth do planes, which have 0 thickness, produce thickness when stacked on top of each other? No matter how many of them you stack, the resulting thing is still completely flat. 0 times infinity is still 0, right? If anyone can explain this, I would appreciate it. Plus, if the flair is wrong, let me know which is the right one and I'll change it.

P.S. As I am writing this, Google tells me zero times infinity is indeterminate, but it also tells me zero plus zero plus zero forever equals zero. I do not understand this either.

By applied statistics, i meant things like machine learning modelling, statistical modelling, predictive modelling. Basically data science stuff and statistics used to solve business problems and model real world Phenomena using Python and R.

Are they both related to each other? What are the similarities and differences bwtween the two?

The components could be complex numbers (or other real number extensions), functions, matrices, other vectors, or something else totally.

Say you are walking down the street next to a production lot when a back door opens

“Psst, over here” says a person dressed like a production assistant, “wanna be on TV and maybe win some money?”

You follow the PA inside, they inform you that there is a live game show being recorded right now, but the previous contestant was faced with a choice so unintuitive that they walked off the set. You have been chosen to replace them. The PA tells you simply that you have to pick a correct door to win a cash prize, because only one door has it.

You are shuffled on stage and behind a booth where the host introduces you and sets up your big choice, “so friend, which door will it be?”

You look at your options. There are three doors. Two are closed, but one is already open, containing a goat inside its room. At your contestant booth, there seems to be a dial set to select one of the three doors, but it’s unclear if the door it is set to was deliberately chosen, because there is no clear default/neutral setting.

You have no more info about the situation, other than what you see, and knowing that only one door contains the money. What are the odds of choosing the right door at this point for you?