i see there are 4 parallel lines and there is one line interesecting with equal distances, which to me implies it is perpendicular. It asks a question about another interecting line. Can someone give me some insight on how to solve this problem

This is a question on my kiddos homework and I can't, for the life of me, figure out how to solve it without using algebra. My kiddos in 6th grade and he hasn't started to learn algebra yet.

Solved! Thank you so much for the quick replies! At least it was a good opportunity for me to teach my child about algebra:).

I think this may be both a math and physics question. Was at a party and after a few drinks we got into a debate of what the definition of a hole is. Does a hole need to have at least one entrance and one exit? Is a depression a hole? Does a hole just need an entrance?

So, this is my attempt to prove just one implication. I tried to use the definition of a locally closed set. Another approach I was considering involves the “fundamental covering”: a subset (A) of (X) is open if and only if (A ∩ U_i) is open in (U_i) for every (i). What do you think?

I want to order a Sword for fencing (HEMA) with a custom engraving that says 'This Blade is made of x Iron Atoms' in german. I don't know the font yet. To get the number of atoms is not that difficult if it wasn't for the engraving. I'm aware that I can't say the exact number of atoms, but I can try to get as close as possible.

Because the engraving has to be made after calculating the number of atoms, I can't do it with the mass of the blade, but I have to go through the volume.

I can calculate the volume of the blade without the engraving though mass and density of the steel (wich I can look up after asking what type of steel is used). Then the volume of the engraving has to be subtracted.

Once I have the volume of the blade with engraving I can get the mass of the Iron in the blade and calculate the number of atoms with the molar mass.

In principle not that complicated, BUT I don't know how to easily calculate the area of the engraving PLUS I don't know the number yet.

I thought of giving three digits of the number and then write out the number as a word so it would be more easy to understand for non science people (for example: this blade is made of 11,5 trillion iron atoms)

Now my queastion/problem:

How do I calculate or get as close as possible to the area of the engraving if I don't know the exact numbers and words?

Hello all

I've recently had to implement a pretty computationally intensive simulation and obviously only have so much computational power at my disposal.

The main reason for it being so computationally expensive is that several rather ugly integrals have to be evaluated, some of which end up being nested which seems to make things worse.

Now I recall from my numerical analysis courses that one can use interpolation to approximate a function. Interpolation also tends to be significantly less computationally expensive. As such I wanted to know whether one can replace the integration with interpolation under certain circumstances. I know these two concepts are very closely linked.

Take the following situation as an example (attached image):

Any help or clarification on this is greatly appreciated.

Thank you

I made this in the shower about a year ago, and I don't really have a definitive answer for it.

In a random number generator that ranges from 0-Infinity, each number having equal odds, would it be possible to roll 1?

It is just as likely as the other numbers to be pulled, but that chance is infinitely small.
Any thoughts?

Was just reading on landing gears out of interest, (engineering undergrad) This is how a gear retracts inside (to increase aerodynamics)

The second one is an example of space saving, the drawback is less clearance & clashing. Now suppose I try to form mathematical modelling eq of the trajectory to describe the geometry such that clashing is avoided (in case of turbulence) and we need to fix some clearance..... How would you approach this ? What field of maths am I vaguely thinking about? What should I study ahead to better understand this.

sSo, to be absolutely frank and upfront, I am not good at math and only have a GED. This may be a rock bottom basic question but I have to ask anyway.

Earlier this evening me and the wife were trying to find a movie and there was one that came out on 1947. I tried to do the math in my head for how many years ago that was and I like to make it even numbers to make it easier. Worth noting, it's the year my MIL was born.

Here's what I did:

1947 - 7 = 1940

1940 + 60 = 2000

2000 + 26 =2026

2026 + 7 = 2033

Which is obviously incorrect but I'm getting hung up on that 7 years; if I take it away do I not owe it back? My MIL is 79, so that can'tbe right. I figured I was doing something wrong (safe assumption) so I try it differently.

1947 + 3 = 1950

1950 + 50 = 2000

2000 + 26 - 3 = 2023

Respectfully, what in the frig am I doing wrong? I ran it through my calculator of the phone and I also looked up order of operations as while I know addition subtraction share priority so it's left to right but I just can't figure out where or why I'm going astray.

Again, this may be some ridiculously basic thing and I'm not to proud to admit I may be an idiot but I have to ask or it's going to drive me nuts. Please, if anybody could help me out would be greatly appreciated

Edit: I did the obvious thing and just did 2026 - 1947 and it's 79. But I'm still trying to figure out why my head math falls apart. Also, I should have read the room a bit better because you folks are clearly dealing with more advanced stuff than dipshit math I'm working with.

Context first. I make tasks for an evaluation using a frontier model (GPT-class; high reasoning effort; hard 8-hour wall clock; 2 CPUs; 6GB RAM) inside a Lake project that has a single sorry in Main.lean. The goal of the model is to close it. sorry, axiom, native_decide, and all other members of that family are prohibited. After the model completes, the verifier copies the one file into a new project, rebuilds it, runs #print axioms, and then plays back the olean through leanchecker. A task will count toward the benchmark only if the model fails at least one of two runs.

Here's my problem: the model is too good. Every "prove this theorem" task I have made so far has been successfully completed by the model, including one in just 17 minutes. It produces roughly 2,000 lines of verified Lean per hour, and if Mathlib has a related theory, it finds it and uses it.

At present, my best option is the chromatic number of the Kneser graph KG(11,4), which is exactly 5. An upper bound is an explicit coloring; nothing complicated about that. A lower bound needs Lovász, which needs either Borsuk-Ulam or Tucker, and neither of those are available in Mathlib. KG(11,4) also has a fractional chromatic number of 11/4, so there is no small subgraph that can be used to force five colors. What is left is computational: 4-colorability as a SAT instance with 1,320 variables. CaDiCaL refutes it in 0.2 seconds and the LRAT proof is approximately 3,600 steps.

The challenge is getting the kernel to recognize that certificate. I built my own reference, and it took considerable effort. I wrote a bitmask RUP checker with a soundness proof, decoded the original clauses arithmetically from their IDs instead of keeping them, stored the learned clauses in a literal trie, and broke down the certificate into approximately 300 small theorems, since each reduction done by the kernel consumes approximately 17KB and frees nothing during a single theorem. A single-theorem implementation ate up all available RAM and crashed. I also had to use -DElab.async=false because async elaboration keeps around approximately 42MB per kernel-checked theorem until the end of the file. It took approximately 32 minutes to compile on a laptop.

Two fresh agents saw exactly what the model would see and both failed within five hours. One attempted to go the certificate route and its kernel check never finished (the kernel was swapping and the agent interpreted that as slow). The second proved the upper bound and reduced the lower bound to the octahedral Tucker lemma before giving up.

Questions, mostly for those who know the kernel better than I do:

1. Is there any way to get a certificate like this to be cheap in the kernel? Maybe Nat.ble on large literals, maybe some sort of reflective setup I don't know about, something that enables an agent to skip the chunking work. I'd prefer to find out now rather than after the evaluation.
2. Does Mathlib have anything similar to Borsuk-Ulam for S^3, or a Tucker lemma development I cannot find? In the past I thought "not in Mathlib" and an agent found covering space lifting and proved the S^2 case in 36 minutes.
3. Is there a more elegant way to break up a certificate across multiple theorems than what I did (hints per theorem, manually adjusted)?
4. For those working on large language models: When one of these models gets an OOM kill during a long kernel check, does it examine the memory model and alter its strategy, or does it continue trying the same thing with slight changes? I'm thinking about either going to KG(12,4) or putting two instances together in one file so that it uses up all its iterations.

I'm not looking for help solving this; it's already solved. I just want to know whether the barrier exists or whether I'm about to be humiliated. If posting my checker design in the comments will assist, I'm happy to do that.

in my case, i want to find how high i need n to get 2^n > 847 > 2^n-1, i can do it fine but i was wondering how i would do this with a higher number, trial and error would be really easy obv but is there something more principled?

IGNORING ALL COEFFICIENTS

How is all of this so connected?

Let's consider the homologous series of c, x, x^2, x^3, x^4... and so on: x^n where n belongs to a non negative integer only

Why is it that the graph for a particular expression not only explain the graph for the expression itself, but also the graph of literally *EVERY OTHER MEMBER IN THE SERIES*.

for example, the graph of x^3 (or simply x^3 itself, as graph is that thing itself) explain the graph of x^3 itself, but also x^4 (how f' graph explains a graph of f), and x^5 (how f'' graph explains for f)

How the graph for f' of x^3 is same as f'' of x^5 and f''''''' of x^10

How the graph for f' of x^8, f'' of x^9 and f of x^7 is all the same. A single graph is doing all of it

Yes, on paper, I know, and can prove this with my pen (I know the limit definition of derivative and what it is in terms of slope, in terms of rate of change in area/volume etc). But it still seems out of grasp on how this is possible.

[Only considering for the homologous series highlighted to keep it simple for now]

Explanation:

This is for a game I'm playing with limited slots.

The two items do the following:

1. An infuser gives a 10% chance to give an item worth $50 on average (the default of getting the item is 0)
   
2. The magnifying glass increases the value of an acquired resource by 40%
   

I have 6 slots available, and I'm looking to see what combination of these two types of items gives the best estimated return. Anywhere from 6 infusers to 1 infuser & 5 magnifying glass is what I'm looking to see which gives the best return on average. 6 magnifying glasses does nothing because no item is gained without 1 infuser

Combination 1: 1 infuser & 5 magnifying glasses

-1 infuser with 5 magnifying glasses would be (.1)x((1.4)^5)x($50) = $26.8912
- item type infuser gives 10% of item, $50 is the outcome, and item type magnifying glass increases outcome by 40% for each 1
- Since there is only 1 infuser, there's only 2 outcomes with only 1 being more than 0. (There's obviously a 90% of getting nothing)

Combination 2: 2 infusers & 4 magnifying glasses

-2 infusers with 4 magnifying glasses (i think) would be
- outcome 1 (first infuser succeeds 2nd fails): (.1)x((1.4)^4)x($50) = 19.208
-outcome 2 (functionally identical) (.1)x((1.4)^4)x($50) = 19.208
- outcome 3 (both succeed) ((.1)^2)x((1.4)^4)x($50)x2 = 3.8416
- outcome 4 (both fail) 0
19.208x2+1.9208 = 42.2576

Problem:

I'm having issues with writing out and solving the other combinations, so I can't find out which combination gives the best average result.

For context, WGS-84 is the World Geodetic System 1984, used in all our modern GPS systems, and GCJ-02 is guó-cè-jú 2002, a proprietary coordinate system used in China. I know we have code to compute this nowadays, but if the only thing I had available was a scientific calculator, what must I do to approach this? I find the formulas too hard to understand and I only started Calculus II recently.

Stats friends, show me how you organize your work when writing it out for assignments please! I've gone back to university at 32 years old and I haven't done math by hand since I was a kid. I feel like there must be a better way to organize this when doing it on a whiteboard/paper?

I was implementing an algorithm to calculate the Gauss-Legendre nodes/weights for arbitrary order and came across this solution:

https://rosettacode.org/wiki/Numerical_integration/Gauss-Legendre_Quadrature

which worked really well for me. Everything makes sense, except I can’t figure out why the initial guess to find the polynomial roots is x0 = cos( pi * (i-1/4) / ( n + 1/2 )). In practice Newton Raphson converges to 1e-16 tolerance in ~4 iterations on average, so it’s a really good guess.

Can anyone help me with a proof on this?

(I’m well out of school and this is not a homework assignment…)

I was reading another post about the Collatz Conjecture and how computers have calculated up to a certain point and verified the conjecture was true (thus far). And I began to wonder what number would be the largest number that anyone could conceivably need that had a real-world meaning or significance. (This started out as wondering what number you'd get to when a layperson would say, "That's got to be good enough.")

My first thought was the number of quarks (or maybe strings) in the universe, factorial—(3 x 10⁸⁰)!—representing the number of combinations of matter possible in the universe.

Are there larger "practical" numbers you can think of?

$$\Omega(k_0) = \mathbb{I}_{\text{prime}}(k) \cdot \mathbb{I}_{\text{sieve}}(k) \cdot \mathbb{I}_{\text{Jacobi}}(k) \cdot \delta\left( \lambda(N(k)) \;\Big\vert{}\; N(k) - 1 \right) = 1$$

I cant understand this its so wierd can i have help

Question 0. If a connected, locally path-connected topology X cannot be covered by two open sets U and V, each connected, so that the intersection U ∩ V is disconnected, must X be simply connected?

I was toying around with different ways to detect holes in a topological space that doesn't assume the given topology is "nice".

For example, consider the long line L = [0..1) × ω₁, but coarsen it in the following way: if a set U contains the point (0,0), then it must contain a non-empty final segment ((r,α)..∞) of L for it to be open. Call the resulting topology the long circle, LS¹.

None of the usual ways of detecting holes in topology (that I know of) can detect a hole in LS¹. The long circle also isn't a particularly nice space. It is path-connected but, unlike the long line, it is not locally path connected. For example, the point (0,0) in LS¹ has neighbourhood LS¹ \ {(0,1)}, any subneighbourhood of that neighbourhood must contain a point (r,α) > (0,1) and there is no path from (0,0) to such point (r,α). However, intuitively, it is just a (maybe weird) circle, and we expect circles to have holes.

I noticed we can cover the circle S¹ with two connected open sets U and V with disconnected intersection U ∩ V. The same is true for the long circle LS¹: we can choose any two distinct points x and y on the long circle, then U := LS¹ \ {x} and V := LS¹ \ {y} are connected open sets covering LS¹ and their intersection U ∩ V is disconnected. This led to the following definition

Definition 1. Let X be a topological space. We say X is 1-connected iff X is connected (has one connected component) and, for any two connected open sets U and V that cover X, U ∩ V is connected as well.

Not the main question, but:

Question 2. Does definition 1 have a (standard) name in the literature?

I wanted to know whether this definition agrees with the usual definition of simply connected (path-connected and every loop on a point is path-homotopic to the constant loop) for nice enough spaces. I managed to prove one direction modulo the following technical lemma.

Lemma 3.\* Let X = [0..1] × [0..1] be the square and ∂X its boundary. Let U and V be open sets covering X. Let x and y be points on the boundary of X. Assume that x and y are in the same connected component of U ∩ ∂X and in the same connected component of V ∩ ∂X. Then, x and y are in the same connected component of U ∩ V.

I have tried to prove this lemma but didn't get very far. I don't think the intersection with ∂X in the assumptions is necessary.

Theorem 4.\* Let X be a simply connected, locally path-connected topology. Then, X is 1-connected.

Proof. Assume, towards contradiction, X is not 1-connected. Let U and V be connected open sets covering X with disconnected intersection. Let x,y ∈ U ∩ V be points not in the same connected component of the intersection. As X is locally path-connected, so are U and V, and connected locally path-connected sets are path-connected. So, there is a path p: x ↝ y in U and a path q: y ↝ x in V. Consider the concatenation w := p;q: x ↝ x, which is a loop on x. As X is simply connected, there is a path-homotopy H: w ⇒ 1ₓ. I.e. H: [0..1] × [0..1] → X is so that H(0,t) = w(t), H(1,t) = 1ₓ(t) = x and H(s,0) = H(s,1) = x. Consider the open sets H⁻¹U, H⁻¹V and H⁻¹(U ∩ V) = H⁻¹U ∩ H⁻¹V in [0..1] × [0..1]. As H is constant x on the top, right and bottom boundary and x ∈ U ∩ V, we have that H⁻¹U and H⁻¹V contain the top, right and bottom boundary of the square. Further, H on {0} × [0..½] (the bottom half of the left boundary) agrees with p, so lies in H⁻¹U, and similarly {0} × [½..1] lies in H⁻¹V. So, the points (0,½) and (1,½) of [0..1] × [0..1] lie in the same connected component of H⁻¹U ∩ ∂[0..1]² and lie in the same connected component of H⁻¹V ∩ ∂[0..1]². Therefore, by lemma 3, they lie in the same connected component C of H⁻¹(U ∩ V). The image of a connected set is connected, so H[C] ⊆ U ∩ V is connected. However, H(0,½) = y ∈ H[C] and H(1,½) = x ∈ H[C], contradicting that x and y lie in different connected components of U ∩ V. ∎

The asterisks are to signify there is a gap in the proof. Of course, this leads to the following question:

Question 5. How to prove lemma 3? Is it true?

The assumption that X is locally path-connected in the theorem above cannot be weakened to just path-connected as the long circle LS¹ shows.

Question 6. How does the answer to question 0 change if we change the assumption that X is locally path-connected, e.g. to path-connected or to metrizable?

Thanks in advance for your help.

I’m in linear algebra and I also just finished calc 3 last semester. I have seen Matrix product, Hadamard product, and dot product all using the same A•B look. What is the correct way to notate them?

For a while now, I have been struggling to understand theta functions. To me personally, their spatial intuition seems dubious at best. This can make it very hard to remember the basic identities associated with them. However, there are a couple of patterns to them that suggest it may be a good idea to rename them, at least as a way to help with remembering the identities.

theta 2 and theta 1 resemble cosine and sine, so it makes sense to want to associate them with X and Y. But if we compare the quartic identity to the Minkowski spacetime interval formula, theta 1 seems more like time because its 4th power is subtracted. theta 2 and theta 4 seem more like spatial dimensions, and theta 3 seems more like the interval, or "radius". In the elliptic function identities, theta 4 seems more like radius, and theta 3 seems more like "time." My idea was to call each theta function by 1 main capital letter and 1 subscript lowercase letter:

X_y = theta_2

Y_t = theta_1

T_r = theta_3

R_x = theta_4

where the main letter represents the elliptic behavior and the small letter represents the quartic behavior. The labeling is just analogous and is not intended to denote any real literal spaces. It becomes easy to remember several identities with this mnemonic trick, as long as I can remember that paying attention to the small letters tends to imply a doubled exponent (I am not using subscripts here because I'm not sure how to break out of it cleanly, so please forgive me):

k = sin alpha --> y/r --> Xy(0)^2 / Tr(0)^2

k' = cos alpha --> x/r --> Rx(0)^2 / Tr(0)^2

X = R cn X/R --> Xy = Rx cn Xy(0)/Rx(0)

Y = R sn y/r --> Yt = Rx sn Xy(0)/Tr(0) (The odd function gets the small letters in the activator. While using Yt isn't really an option since it is always 0 at 0, I am still not entirely sure if the small letter occurrence is a happy coincidence or if there is something else at play. Feel free to weigh in on this.)

T = R dn T/R --> Tr = Rx dn Tr(0)/Rx(0)

It is significantly harder to remember the activating coefficients as root(sin alpha), root(sec alpha), root(tan alpha).

So, here is what I want to know: Am I falling into a trap by relying on this custom notation? Are there any other properties of the theta functions that might blindside me in the future if I continue to use this? I want to make sure that my efforts to make some properties clearer aren't obscuring others.

Hey everyone. I'm working on a browser extension that uses AI to help with math and solving problems, step-by-step explanations, that kind of thing.

Before I build too far, I want input from people who'd actually use it:

• What math do you struggle with most? (school homework, exam prep, uni-level?)
• What feature would make you keep it installed instead of deleting it after a week?
• Should I create an extension or make a website also?

 I want to build something people actually want.

Hi,

I'm currently in the process of planning a school activity for UK primary school ages and it involves a small contraption shown in the image.

The problem, though, is that I'm struggling to find the right fan needed in order to push it forward and make it move.

Does anyone have an idea of the rpm/type of fan needed for this thing to move?

Please let me know if any other information is needed in order to figure this out!

Weight - 181g