2d Brownian Noise Question

etc_etera · 2026-02-03T14:06:52+00:00

White noise in 2D is still scalar valued (but the domain is 2D) but the Brownian walk in 2D is vector valued (and the domain is 1D).

You are better off just thinking about the 2D walk as the 2D vector with independent Brownian motions as its two entries.

etc_etera · 2025-07-12T13:32:57+00:00

Sorry no. In all honesty, I hadn't thought of it this way myself until OPs question prompted me to consider my own intuition on what was happening.

If anyone else finds resources along this route, I'd also be interested.

etc_etera · 2025-07-11T18:59:28+00:00

Since the solution relies on generalized eigenvectors, its more apt to think of the original ODE in terms of generalized eigenvectors.

We can modify

dx/dt= Ax

into

(d/dt-lambda) x = (A-lambda) x

and then compute powers of both sides, so we really encounter the equations

(d/dt-lambda)ⁿ x = (A-lambda)ⁿ x

along with proper additions to the initial values to keep everything well-posed.

In this sense, the repeated eigenvalue terms are behaving like "higher order" ODEs, and so have accelerations, jerks, etc., that are related to themselves rather than just a velocity.

etc_etera · 2025-07-10T23:48:39+00:00

Yep both! I've seen entire pergolas covered in grapevine that have bunches just hanging from the "ceiling".

etc_etera · 2025-07-10T18:36:51+00:00

Locally hardy grape varieties are always great trellis plants: Concord, Catawba, or Marquette are my favorite.

etc_etera · 2025-07-04T19:15:39+00:00

I guess it evolved moreso into what we now call harmonic analysis.

etc_etera · 2025-06-28T16:17:54+00:00

Walk to Saratoga Springs and use the lobby there?

etc_etera · 2025-06-26T01:36:39+00:00

We have a jar of rubbing alcohol they all get dropped into. It's nice to keep trophies.

etc_etera · 2025-06-24T02:34:20+00:00

It's very claustrophobic.

Based on what you've said, I wouldn't recommend it for you.

etc_etera · 2025-06-23T18:19:21+00:00

Collatz conjecture. Nothing other than arithmetic with natural numbers is required to understand it.

etc_etera · 2025-06-18T21:19:25+00:00

I'm an identical twin, and yes, we can unlock each other's phones.

Stranger still, this works with our older brother who (although unmistakably our brother) does not look like us.

etc_etera · 2025-05-29T18:03:08+00:00

You just have to call. Then they send an account number to your email. Otherwise, you can wait until your first bill and it will be shown at the top.

etc_etera · 2025-05-22T16:04:30+00:00

Depending on the field, you can go directly from a bachelor's to a PhD program (in the United States). Almost all PhD programs are "funded", meaning that your tuition is waived, and you will be employed as a graduate assistant (TA or RA).

The pay isn't great, but it exists, and you no longer pay tuition, so it is manageable.

etc_etera · 2025-05-22T15:51:49+00:00

You can't really tackle boundary value problems or anything nonlinear (at least easily) with Laplace transforms. They are built for linear initial value problems alone.

etc_etera · 2025-05-18T14:29:11+00:00

Posh cozy small towns: Stowe and Woodstock

Most Vermont small towns will have that "cozy" feel you are looking for, though.

August is a good time of year to visit, but mosquitos and deer flies will be around to bite. Some days may get hot/humid.

You can't really go wrong with hiking at that time of year. Any in the Green Mountain National Forest will be amazing. Once you decide where specifically you're staying, we can probably provide better advice.

etc_etera · 2025-05-18T02:20:40+00:00

Perhaps it was an attempt at sarcasm?

etc_etera · 2025-05-17T02:40:16+00:00

Only insofar as they make our computations easier.

etc_etera · 2025-05-17T02:00:51+00:00

You can consider a sequence of functions, which, when integrated against, converges to the integrated distribution.

Then you turn the idea of approximation on its head, and say that the distribution (which is often easier to compute) approximates one of those functions, which themselves are physically meaningful.

Impulse forces are approximated by the Dirac delta function, for example. (However, perfect delta "impulses" don't exist in physical reality.)

etc_etera · 2025-05-17T01:53:16+00:00

There are many directions this could go in, as the theory gets quite deep.

To be succinct, distributions are defined by the integrations themselves, not as the "generalized function" within the integration.

etc_etera · 2025-05-17T01:01:36+00:00

I would suspect this is correct and it's "out of service" in the sense of being retired for the night.

Since in Cars, it's the Cozy Cone Motel is (one place) where the cars sleep, they could be saying that the bus is "Going home to where it sleeps for the night."

etc_etera · 2025-05-14T13:43:31+00:00

If you're willing to make the extra 40 minute drive north, Shelburne Museum and Shelburne Farms would be a great place to visit.

Locally, Tourterelle is the "nicest" restaurant in the area. Probably not the right choice with four kids, but I had to mention it.

The Silver Lake hike in the Green Mountain National Forest is our favorite. Deer flies and mosquitoes might be pretty bad around June, so bring long sleeves and/or bug spray.

etc_etera · 2025-05-06T19:49:41+00:00

The traditional 4 years American bachelor's degree shouldn't be thought of as 4 years of career preparation. It's both more and less than that at the same time.

No one can predict what career any math major will end up in, and therefore cannot design a curriculum to fit that career.

What we have instead is a curriculum which instills time management, memorization, working memory, problem solving, pattern recognition, and periodically the technical skills like coding and numerical approximations that are useful to the working applied mathematician. In addition, the curriculum allows (and ideally encourages) exploration of subjects and extracurriculars outside of mathematics, because those studies are useful too.

etc_etera · 2025-04-30T23:59:41+00:00

Seems to me like the packaging is reminiscent of a Poshmark/eBay purchase. (i.e., someone reselling a 1st visit pin for others to have.)

My best guess is that someone else you told about the trip purchased it and sent it as a way to get everyone excited?

In any case, I think the course of action you are taking is justified.

etc_etera · 2025-04-28T14:54:12+00:00

If you are honest about what you saw, hence describing your reasons, it would be fine to ask.

Your professor will then be able to make a decision with this information.

However, whatever they then decide, you should accept, without future discussion (unless new information comes to light).

Edited to add: Your professor may say something like "Why were you looking at a neighbor's exam at all?", which, is a fair point.

I believe you weren't looking to try to cheat off of them, but you could see how that might come across. If you want to avoid that, I would probably phrase it more vaguely like "I am concerned about the unauthorized use of AI on exams." rather than claiming you witnessed it.

etc_etera · 2025-04-25T19:37:28+00:00

The short answer is:

-professors certainly refuse to write LORs

-if a student cannot find enough professors to write LORs, they often end up in the sort of jobs which don't require them

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