physics – Isochronic pendulum – Mathematics Stack Exchange

The figure shows a possible arrangement of an isochronic pendulum, that is, a pendulum that does not change in frequency with changes in ambient temperature. The three light vertical bars, of length $ L $, have coefficients of linear thermal expansion $ alpha_1 $ and the two dark vertical bars, of length $ L_2 $ have coefficient $ alpha_2 $. Determine $ L_1 $ and $ L_2 $ so that the length $ L $ remains constant. Suppose the joints and supports are fixed and rigid.

enter image description here

Can someone help me? I think it is possible to solve … but how? It does not make it very clear what is fixed and what is not

discrete mathematics – Distinguishable balls and urns

Suppose you have $n_1$ distinguishable red balls and $n_2$ indistinguishable yellow balls. A red ball can be distinguished from a yellow
ball.

How many ways are there to distribute these $n_1 + n_2$ balls into $m$ distinguishable urns so that there is no urn with only yellow balls?

Prove your answer.

mathematics – Given its direction, and another vector that begins at the same point, how can I find the magnitude of that vector using only vector math?

Fundamentally, you have a right-angle triangle

enter image description here

Your vector a is (conveniently) the adjacent side

You want to know the length of the hypotenuse which is coincident with b.

θ is the angle between your two vectors which we can get by calculating their dot product.

So I can’t see any simpler option than

magnitude(a)/cos(dot(a, b))

Note that in your question, you alternate which vector is extended and which is attached to a right-angle. The above assumes you know a and want to find the magnitude of b.

recreational mathematics – Why dividing negative numbers gives positive outcomes

How comes that when we divide two negative numbers we get a positive answer? Consider that negative numbers are nonexistent in real life, for instance there are no -10 apples. So when we divide -10 apples among -2 people, why should we get 5 apples?? Does it mean real substances can exist from nowhere?

gm.general mathematics – Control & Experimental Group Selection Methodology using STDEV and T-Test Methodology?

I would like to know if my methodology was ‘correct’:

I am trying to conduct an experiment on my stores.
I would like to find out the effect of a marketing campaign on the number of transactions.

Only about 20% of the stores are participating in the marketing campaign.

The original methodology was to use the entire 20% as the experimental group and the remaining 80% as the control group. Unfortunately, these two groups are incomparable in terms of number of transactions.
when plotted as box and whisker plots next to each other, their distributions are incomparable (mean, median, quartlies, min, max, etc).

So what I did was filter out the ‘outlier stores’ at each end until the box and whisker plots for each group were practically identical. I then ran a t-test on the filtered groups, we failed to rej the null (meaning that these groups are statistically the same prior to the promotion).

Now that we have 2 comparable groups for time -1, we run the promotion for a month.
after promotion month is over, we take the number of transactions from each group and run another t-test.
We Rejected the null in favor of the alternate Hypo, which is that these 2 groups are now statistically different with an alpha of 0.05.

My first question is: is this methodology okay ?

My second question is: alternatively from using box and whiskers and removing outliers until both groups’ descriptive stats are similar, can i use a normal distribution and STDEV to remove outliers and create comparable groups within my population ?
The box and whisker method worked to get a comparable groups as confirmed by the t-test, but is very manual. So i would like to create an automated method and was wondering if using a normal distribution and removing outliers by STDEV would be plausible ?

Sorry for the long read.
Thank you

discrete mathematics – Does this proof make sense? m | n if and only if m | n^2

Question and proof below. I’m a bit unsure if this proof make sense and want a second opinion. I could find a counter example I guess but I was trying to show that it is the case sometimes but not is not implied for all cases.

Either prove or find a counter-example to the statement $m :|: n$ if and only if $m :|: n^2$. (Here $n$ and $m$ are integers.)

Proof(Direct)
$m :|: n Leftrightarrow n=km$
$m :|: n^2 Leftrightarrow n^2=km$
$n^2 = m^2 k^2 = m (m k^2)$
For any Prime $P$ : $P :|: ab Rightarrow (P :|: a$ or $P :|: b)$
If $m$ is Prime $m :|: n^2 Rightarrow m :|: n$
$therefore :m :|: n notLeftrightarrow m :|: n^2$ $blacksquare$

mathematics – Helmet Mounted Display Math

I’m trying to create a helmet mounted display as a mod for a flight simulator. Unfortunately, I’ve been trying to figure out the math for this, but, to be honest, I’m stuck. I’ve done my best to look up the math related to this, but this kind of math really isn’t my thing, so I figured I’d ask here to see if anyone can provide some help.

The information I have access to is as follows:

  • The pilot’s position
  • The pilot’s head’s rotation
  • The location of the “thing” I’m trying to put a box on the HMD over top of

The HMD itself is drawn in screenspace as normalized 2D coordinates starting from the bottom left corner.

From what I can find, I can treat the pilot’s head like a camera in a scene, and the thing I’m trying to track as an object in that scene, which means I’d need to use a view matrix, followed by a projection matrix, which would get me all the way there aside from translating that to the screenspace coordinates; at least that’s what I can determine, but as there’s no real way to debug problems aside from guessing and trying to fix them, I’d really appreciate having someone confirm/deny that route and correct any misinformation.

How does bitcoin work from a Mathematics standpoint?

I would like to know how bitcoin works from a mathematics standpoint

mathematics – Predict future position of a moving body in Phaser arcade physics

I am looking for an equation for predicting the future position of a moving arcade physics body in Phaser 3. The body has drag applied and isDamping set to true. Phaser applies the drag using the following run on each update (showing x axis only but same applies to y).

//  Damping based deceleration
dragX = Math.pow(dragX, delta);

velocityX *= dragX;

Given the above, how would I write a kinetic equation to predict the future position?

I am currently using the method below, where I am iterating over frames to calculate the position accumulatively. But this is inefficient and inelegant so would prefer a solution which estimates the position without any looping.

public futurePosition(timeInSeconds: number): Phaser.Math.Vector2 {
  const DELTA = 1 / 60; // Assume we are running at 60fps
  const DRAG = 0.3; // Drag value
  const position = this.position.clone(); // Current position of body
  const velocity = this.body.velocity.clone(); // Current velocity

  // Inefficiently looping through frames
  for (let i = 0; i < timeInSeconds / (DELTA * 1000); i++) {
    velocity.x *= Math.pow(DRAG, DELTA);
    velocity.y *= Math.pow(DRAG, DELTA);
    position.x += velocity.x * DELTA;
    position.y += velocity.y * DELTA;
  }

  return position;
}

Any help appreciated. Thanks.

mathematics – How to project a Matcap correctly in Amplify Shader Editor

Unity’s Amplify Shader Editor comes with an example of matcap nodes setup, however, the matcap projection in this example is incorrect.
As you can see the texture becomes distorted when it’s on de edges of the screen, that shouldn’t happen with a correct matcap:

enter image description here

enter image description here

On the secoud image, this is how a matcap should look, without major distortions.

Does any expert in shaders knows the right way to project a matcap using the nodes? I’ve done my search and there is absolutelly no information about this around the internet, any response would add for this Shader documentation.
I’m extremelly in need that matcap works with this shader.