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

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.

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.

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?

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?

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.

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…)

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

I cant understand this its so wierd can i have help

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?

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.

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

I work in software so its relatively easy to understand why a base 2 system is preferred: because it can be represented by a simple on/off signal

I just don’t really understand the relevance of other bases. Like why is base 10 inherently good compared to base 12 or even base 60? Why is it seen as such a big deal in the history of math? Are there any cases where using a different base number is better?

Can you use anything for a base? Can you do math across 2 different bases? Like could I do math with a base 10 number system and a base 2 number system, or would that simply be translating one to the other?

(any admin can change the flair, I didn’t know what the appropriate discipline is)

I've been trying to find info on complex p-adic numbers, and while I have gotten lots of stuff of how they are constructed, I haven't seen many examples of like... just an example of one. Like, what they look like in practice. Like how you can write 1 + 5i and that's a numerical example of a complex number, are there any decent resources and examples of what a complex p-adic actually looks like and how to find them?

Also keep in mind I am not like, a super high knowledge of mathematics. Like I would say I know more than the average person but I don't know a lot of notation for like, set theory or field theory or what not. I have just been interested in number theory lately and have wanted to learn more about it.

Hello everyone, my first midterm in abstract is just about a week away, and he said we can have 2 sheets of open notes for the midterm. Is there a fundamental way or best way to optimize an open note sheet, especially for a class as odd and unordinary as abstract algebra? We have only mainly covered up to infinite and finite cyclic groups as of now, so anything a little beyond that is probably where the exam will cap at information-wise.

Should I keep it to example proofs or pure definitions, any feedback from anyone more experienced would be awesome! Thanks Yall <3

I'm not sure how to find or compare the rate of growth of 2ⁿ with any positive decreasing sequence, in order to find the exact cutoff point where some positive decreasing sequence decreases slower than 2ⁿ increases.

How do I find that exact cutoff point in this situation?

The series in the title only converges when the all positive decreasing sequence (a_n) decreases faster than 2ⁿ increases.

Hello. I love the compactness theorem and I’m looking for all sorts of problems to solve with it. I’m aware of some that involve graph theory, binary trees, etc. but would appreciate some more. Feel free to copy-paste any that you remember. Higher order compactness is fine, but I especially like the simplicity of the propositional version. I look forward to trying to solve your problems!

Hi everyone, I wanted to share a mathematical relationship I derived regarding the coordinates of a parabola's vertex, which I call the Kumar Theorem of Balanced Vertices. This theorem establishes the exact algebraic condition required to constrain the vertex of a vertical parabola to the identity lines y = x or y = -x.

Let a standard vertical parabola be defined by the quadratic equation:

y = ax^2 + bx + c

The x-coordinate of the vertex represents the axis of symmetry and the average value of any two symmetric points on a horizontal line intersecting the curve. This is given by the formula:

x_avg = -b / 2a

The Kumar Theorem of Balanced Vertices states that if we impose the strict geometric constraint where the magnitude of the y-coordinate at the vertex equals the value of this average axis of symmetry (y_vertex = ± x_avg), the vertex is forced to balance perfectly on the lines y = x or y = -x.

We can derive this using the standard formula for the vertex y-coordinate:

y_vertex = (4ac - b^2) / 4a

Applying the theorem's condition (y_vertex = ± x_avg), we set them equal to each other:

(4ac - b^2) / 4a = ± (-b / 2a)

To simplify the expression, we can multiply the right side by 2/2 to establish a common denominator:

(4ac - b^2) / 4a = ∓ 2b / 4a

Since the denominators are now identical on both sides, we can eliminate them completely:

4ac - b^2 = ∓ 2b

Rearranging the terms to isolate the c variable gives:

4ac = b^2 ∓ 2b

Finally, solving explicitly for the constant c yields the final formula for the theorem:

c = (b^2 ± 2b) / 4a

In conclusion, according to the Kumar Theorem of Balanced Vertices, for any vertical parabola where you choose the coefficients 'a' and 'b', this single consolidated formula calculates the precise value of 'c' needed to lock the vertex exactly onto the diagonal identity lines.

I would love to hear your thoughts on this derivation or if this specific constraint is utilized under a different name in advanced coordinate geometry frameworks!

Suppose the Jacobian conjecture was undecidable. It would then have no counterexample, as that would make it decidable.

But if it has no counterexample it is true.

Therefore any proof that makes it undecidable is also a proof that it is true, which I think would make it decidable? Am I making a mistake here?

Edit: thanks for the help all, i think I see my mistake now!