$O(1)$ time, $O(1)$ state random access Brownian motion?

I would like to generate discrete samples $ 0 = B(0), B(1), \ldots, B(T)$ of a Brownian motion $ B : [0,T] \to \mathbb{R}^d$ . It is possible to get $ O(\log T)$ time random access into a consistent sequence of samples $ B(k)$ by constructing the path in tree fashion. We choose $ B(T) = \sqrt{T} Z_0$ where $ Z_0$ is a standard Gaussian, then let $ B(T/2) = B(T)/2 + \frac{\sqrt{T}}{2}Z_1$ where $ Z_1$ is an independent standard Gaussian, construct $ B$ for the intervals $ [0,T/2]$ and $ [T/2,T]$ independently conditional on $ B(0), B(T/2), B(T)$ , and so on recursively.

Since any particular $ B(k)$ constructed in this fashion only depends on $ O(\log T)$ Gaussian random values, this gives us $ O(\log T)$ time random access into a consistently sampled sequence of Brownian motion as long as we have random access random numbers. The only state we need is the random seed. As long as we use the same seed, evaluations of $ B(k)$ for different $ k$ will be consistent, in that their joint statistics will be the same as if we had sampled Brownian motion in sequential fashion.

Question: Is it possible to do better than $ O(\log T)$ , while preserving the $ O(1)$ state requirement (only the seed must be stored)?

My sense is no, and that it would easy to prove if I found the right formalization of the question, but I don’t know how to do that.

Can I measure time by counting frames and trusting on the ‘240 FPS’ that my iPhone 7+ slow motion camera is capable of record?

I’m using a slow motion video recorded using an iPhone 7+ to track something but would like to avoid recording a chronometer to know the time the process is taking. I need to measure about 10 seconds with an uncertainty of at most 0.1 s… Is this possible by just counting 2400 frames of my homemade video?

Leap Motion : Leap service running, “Leap Service not connected” in Unity3D, connected in SteamVR

I’m having trouble to use Leap Motion with Unity3D on one of my machines.

Unity doesn’t see Leap Service running : Leap Service not connected

Both computers run the same version of the software: Unity 2018.2.15.f1 and Orion 3.2.1

  • The Leap Service is running fine on the PC – I tried also to start and stop it manually, it didn’t affect the results

  • I installed and re-installed the drivers below – in the SteamVR window, I see the controllers, and they do stop blinking when the hands are in front of the sensor

    • https://github.com/cbuchner1/driver_leap (original one)
    • https://github.com/SDraw/driver_leap (updated one)
  • When opening the Diagnostic Visualiser, the hands show just fine

  • I tried connecting the Leap to the headset and to different USB ports, didn’t change anything

  • When running the same project on the other PC, leap motion hands are displayed in Unity normally

Any leads?

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Projectile motion function [on hold]

An artillery crew is trying to hit multiple different targets over the course of a battle. Where and when these targets show up is initially unknown but they all appear at different ranges (x) and different heights (h). The crew needs to calculate the angle ( Θ ) to aim the gun at to hit a target. Assume the velocity (v) is 100m/s and gravity (g) is 7.5.

Find the equation for Θ where x and h can remain variables so that the gun crew can use them to deduce the angle they need to aim their gun at.

I really need this to be cookie cutter, plug in the x and h and it rolls out theta if you crunch the numbers.

Does D&D physics enable a perpetual motion machine?

For various reasons perpetual motion machines are impossible in real life. They violate the laws of thermodynamics and cannot be created. It has been established, however, that D&D is not a physics simulator and the normal rules of physics do not apply.

Inspired by Shalvenay and MikeQ in chat. I was wondering, can a perpetual motion machine be made in D&D?

Imagine the following scenario:

Timmy the Tinkerer is going to a tinkering competition. The premise is simple; build the most magical device without using magic. The rules are as follows:

  • Magic cannot be used to create the item, and the item itself must not be magical.
  • It must be permanent, any item that relies on a once/day ability to sustain it will not count.
  • It must obey the physics of the material plane. No planar trickery is allowed.

Timmy decides he would love to make a perpetual motion machine.

Given these rules and using any officially available content; is it possible for Timmy to do this? For bonus points, how can he do it?

Maxima of Brownian motion

It is well-known that Brownian motion attains infinitely many maxima in each time interval $ [0,T]$ a.s..

From a physics perspective it seems reasonable that when the disorder of the path of a particle decreases and the motion becomes more deterministic, then the number of maxima should decrease.

But I could not find anything on that. Now, there were two natural things to look at:

Is there a way to quantify that a Brownian motion with large variance (large disorder) has more maxima than one with little disorder?

Or is there a way to say that a diffusion process

$ dX_t = \mu (X_t) \ dt + \alpha dB_t $

has “less” maxima when $ \alpha $ is small compared to $ \alpha$ large?

I guess it is hard to make this question more precise, since this is not a question of cardinality of maxima but more about finding a suitably chosen measure that could capture such an effect.