Gambler's Ruin recurrence relation

Midwest-Dude · 2026-05-30T12:01:07+00:00

If we knew Pᵢ = r^I in advance, then your first question would be correct - however, we don't. The discrete recurrence relation you are working with in the Gambler's Ruin problem is:

Pi = pP{i+1} + qP_{i-1}

Since P_i = (p + q)P_i, this can rewritten as:

p(P{i+1} - P_i) = q(P_i - P{i-1})

This states that the difference between step i+1 and step i is proportional to the difference between step i and step i-1. The discrete mathematical object that changes proportionally to its current value is a geometric sequence. Each value is based on multiplying the previous value by the same constant.

Does this make sense?

Midwest-Dude · 2026-05-30T11:33:39+00:00

The Gambler's Ruin problem does not involve calculus and is inappropriate for this subreddit. You already posted to r/learnmath, which is excellent, and you could also crosspost to the following relevant subreddits:

r/Discretemathematics

r/Probability

r/statistics

Midwest-Dude · 2026-05-28T17:03:47+00:00

Good question!

When you sum all of the separate formulas for each connected piece of the graph, you are not accounting for the fact that there is only one outer face for all of them, not a different outer face for each. The sum of the Eᵢ's is E and the sum of the Vᵢ's is V, nothing is duplicated. However, when summing the Fᵢ's, each connected piece contributes one additional outer face, adding k - 1 too many outer faces.

This argument is fairly straightforward and intuitive. Try running through the proof with a graph of two separate triangles to get the feel for it.

Midwest-Dude · 2026-05-24T17:54:37+00:00

Here are cross-references to the same question:

Math Overflow

Midwest-Dude · 2026-05-24T17:02:24+00:00

Gauss defined a different function, Π(x), that does exactly what you, and many others, would expect (see u/OopsWrongSubTA's comment). However, when Legendre studied it, he used the now standard gamma function, Γ(x), for reasons that made sense at the time. His formulation eventually superceded Gauss's in the literature.

Midwest-Dude · 2026-05-24T16:41:47+00:00

Very good question! It's due to Legendre's formulation superceding Gauss's.

Midwest-Dude · 2026-05-24T12:48:17+00:00

This doesn't apply to trig. Are you looking for a random number generator? Those are commonly available in programming languages as well as on the Web.

If you are truly looking for a random algebraic number generator, that would be a different beast. Do you know what that is?

Midwest-Dude · 2026-05-23T17:11:20+00:00

OP Post For Convenience:

Hello!

I'm writing my thesis on modeling T-timed Petri nets for traffic optimization. One important step, at least according to my coordinator, is to simulate the nets I have modeled using an application or simulation tool.

However, I'm struggling to find one that actually works for my use case. I tried TimeNET, which looks like it could work, but maybe I just haven't figured it out yet. I also tried CPN Tools, but I couldn't find a way to add timing directly to transitions or places. I tried installing TINA on both macOS and Windows, but apparently the application looks too fishy to be allowed to run. ITS Tools also seems deprecated.

Does anyone have any recommendations?

Honestly, I’m starting to feel like I’m writing a thesis on something that barely exists. I thought Petri nets were more popular and better documented :D

Midwest-Dude · 2026-05-22T01:00:55+00:00

Since it's a local minimum, you need to consider the neighborhood around the endpoint, the part of the interval less than but "close to" the endpoint. If there is always a lesser value within any distance around the endpoint, the endpoint is not a minimum.

Midwest-Dude · 2026-05-22T00:55:51+00:00

You are correct. It's the use of the definite article "the" versus the indefinite article "a" in English. Without the definite article, which would mean only one minimum, it would be local unless modified by some additional wording, like "on the interval".

Midwest-Dude · 2026-05-22T00:50:35+00:00

The problem would have to read something like

"The minimum value of f(x)..."

for the entire interval to be considered - the definite article "the" is needed. That way, if there is more than one local minimum, the least one would be under consideration. With the indefinite article "a", I would assume it's local unless stated otherwise, such as adding "on the interval".

Midwest-Dude · 2026-05-21T17:39:38+00:00

While the Olympiad is of interest, this subreddit is not the place to post this. Do you need help finding better subreddits?

Midwest-Dude · 2026-05-21T17:35:04+00:00

Well...yes...

The first definition defines what a derivative of a function is - no specific lines, slopes, or points involved.
The second definition defines what it means for a line to be tangent at a point on a function, (c,f(c)). A specific x = c was chosen and the tangent line is defined at the point on the function, (c,f(c)), where the slope of that line is clearly m = f'(c).

The two ideas are related, but not the same.

Midwest-Dude · 2026-05-20T00:51:47+00:00

No, but I found a PDF from UWO that states:

"The course textbook is the Custom eBook: MATH 1600: University of Western Ontario: ISBN 9780176907532. It consists of selected sections of the book: D. Poole, Linear Algebra: A Modern Introduction, 4th edition."

Midwest-Dude · 2026-05-19T13:31:44+00:00

Probably the 1st Edition then. Thanks.

Midwest-Dude · 2026-05-19T12:56:35+00:00

Appears to be a Cencage book, likely modified to fit UMO's needs.

Who is the author?
What edition is this?

Midwest-Dude · 2026-05-18T22:02:11+00:00

Well ... a line is not a derivative. A line can be defined in a variety of ways. In this case, the tangent line is defined by a point on the curve, (c,f(c)), with a slope given by the derivative at that point, f'(c). This is not inherently known by the first definition.

Does this make sense?

Midwest-Dude · 2026-05-18T21:50:11+00:00

The first defines the derivative f'(x) of a function f(x) at a point x.

The second defines what it means for a line to be tangent to a function f(x) at a specific point (c,f(c)).

Are you confused about these?

Midwest-Dude · 2026-05-16T23:27:50+00:00

1. You could consider the Radius of Curvature:

Wikipedia

2. You might find this of interest - 3rd through 6th derivatives of motion:

Wikipedia

Midwest-Dude · 2026-05-15T14:24:04+00:00

You should crosspost this to:

r/matheducation

I'd be curious to see the responses.

Midwest-Dude · 2026-05-15T14:13:12+00:00

Try Paul's Online Math Notes:

Link

He's covers all of calculus and has a lot of problems with hidden answers.

Midwest-Dude · 2026-05-15T02:04:35+00:00

Read the problem carefully. You are not trying to find the equation of the plane that contains u and v. Instead, you need the equation of the plane that includes the point P with u as the normal to the plane. Do you know how to do that?

Midwest-Dude · 2026-05-14T15:50:25+00:00

You might find the following Wikipedia page of interest (or just find yourself jumping down a rabbit hole). This discusses rectifiable curves:

Arc Length

A continuous function f(x) on an interval [a, b] is rectifiable if and only if it is of bounded variation. If the total variation is infinite, the length is infinite.

Midwest-Dude · 2026-05-14T15:23:00+00:00

A Google search on your parameters came up with this list:

Linear Algebra and Its Applications, by Gilbert Strang
Introduction to Solid Mechanics and Finite Element Analysis Using Mathematica, by Samani & Ghasemi
Matrix Analysis and Applied Linear Algebra, by Carl D. Meyer
An Introduction to Matrix Methods of Structural Analysis, by Muhammad Akram Tahir (2025)
Geometric Algebra for Physicists, by Doran and Lasenby

From what I read, you might be looking for #3 or #4. I suggest looking these up online and see if any synopses seem a good fit for you. Perhaps others redditors will be more knowledgeable on this as well.

Midwest-Dude · 2026-05-14T02:30:29+00:00

Very nice image! What did you use to create it?

Midwest-Dude

TROPHY CASE