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Creo que RUclips permite traducir automáticamente los subtítulos. Puedes intentar usar esta función =)

Not a big fan of those abbreviations, and also even within those abbreviations, the definitions are often different among math, physics, engineering, and other communities (I try not to get hung up on semantics). In terms of the domain of definition, it often comes down to convergence type (pointwise v. uniform v. etc) and stability of the solution (unconditional stable v. conditional stable v. unstable) and the phase plane behavior.

When you say "sums you don't know how to calculate", do you mean that you don't know if they converge, or ones that you don't know if it converges to? If the former, the calc tests (comparison, dominated convergence, etc.) can be used. If the later, the probability that a random series converges to a number that one can represent is 0 (almost never); this is a result of the denseness and complexity of the transcendentals in the reals. Although once you introduce special functions (like poly-logarithms and such), you can of course "represent" the solutions more often than sticking with the original "elementary" function sets.

Moreso a matter of preference. One could just label with the color label only, but some people also want the percentage, the sub sample size, or potentially other information on their graph.

Oh, yes! I second guessed myself, got lost in the moment. Good catch!

Math can be quite an adventure! Sometimes scary and intimidating like a dragon, but quite rewarding when we make it to a goal =)

The parabola y = (x-3)^2 will have symmetry about the lime x = 3, which is a horizontal shifted x = 0, and can be viewed as a shifted y axis, yes. Note a lot of teachers and books will still refer to this as not having x, y, or origin symmetry (sadly), but I think for the long term view of math, making a relation be symmetric about curves is a useful perspective, and comes especially in handy with physics and probability later on.

I mention that the tied rank is the average of the ranks they span. Since I only have duplicates, then it's just the median of the two, so usually I assume that is simple by the time we mention non parametric tests. For more than two, it would still be the average. For example, if a number has ranks spanning positions 5,6,7,8,9, the all five of the numbers get a rank of 7. Since the set of consecutive ranks is always consecutive integers, the tied rank (the average) will (also) always be the median of the data set, so not much mental math (once comfortable with median calculations); this is one advantage of working with discrete uniform distributions. I try to use previous courses as prereqs, else the video length will go beyond an hour if the calculation is drawn out. By this stage, most will usually be using R, Python, or Excel for calculations, especially once you have a data set that is in the hundreds or more. But to recap, the mean and median of a discrete uniform data set are always equal, so we can rapidly get the tied rank by calculating the median. Ex1) 1 2 3 4 5 6 7 will each map to 4 Ex2) 1 2 3 4 5 6 will each map to 3.5

Thank you =) quite peaceful~

Be familiar with various types of operators and their properties. Be able to answer questions about their boundedness, continuity, smoothness, invertibility, behavior at singularities, behavior at 0, and what their associated eigenfunctions are. Numerical approximation of them is also a plus.

Thank you sir

Hello =) Thank you for your feedback. I definitely understand your point. Reading from a .ppt (or something similar) is something I personally don't like (bad experiences from my education experience), but your recommendation does give me an idea that I may try out in the winter and see how I feel with it. Side note: back when I first started, I was writing on a glass tablet, and the handwriting was much worse 😅 but I've noticed that it depends on the app I use also; so definitely will work on an improvement strategy. Much appreciated though =)

Definitely on my list of things to add in the coming months =) hope you are doing well.

Yes, indeed.

Great question. When it comes to domain, it honestly depends on what you allow and accept, but the surface level answer is no, they won't have the same domain, but they can if you allow domain extensions and analytic continuations in the transformations. But a more interesting point is that depending on how you approach a particular derivative (in the order sense), you could technically approach a "different function", with the domain being the first thing in question to be different.

Correct ^_^

Thank you so much! Comments like these help motivate me to continue trying to better the content I enjoy sharing with others. I've recently been on a break, but plan to push out a few months of content a couple weeks later, just currently deciding which topics to include. If you have a recommendations of things you would like to see, feel free to reach out and let me know!

You're very welcome =) hope you are doing well.

Yea, that's correct, W(0) = 0 :)

Currently, Microsoft OneNote :)

Yea, family wise is alright. But it's nothing compared to what Bonferroni's got.

Thank you for your kind words! And yes, you are correct. Retrospectively, I would prefer to say that a relation R is reflexive, provided that for any x chosen from S, (x,x) must belong to R; perhaps this is a safer way.

Thank you ^_^

Thank you :) let me know if there is anything you have curiosities about or questions!

Thank you =) hope you are doing well. Feel free to let me know if you have any questions!

Thank you four kind words =) and yes, some will define fisher in terms of what you mention, which becomes intuitive once one develops a geometric sense of what we are doing. Shannon entropy (and other measures like KL divergence) also closely relates to these mechanisms also, but hope to investigate these at a later time when I get the chance. Hope you are well!

Thank you! Maybe one day =) For now, I just enjoy making content even just for one person to help at a time

Will try to have new ones in the month of July =)

Thank you =)

Yes! Thank you for point this out. I've pinned a comment above with a correction (two possible ways to correct, one being the one you mentioned).

Possibly ^_^

Thanks for your kind words :) Technically yes, but a bit more complex since we usually look at phase portraits of 2d or higher for their solutions, but we can look at each individual variable over time for both solutions; that is if we have an x' = ... y' = ... system, solve using both methods graph x v. t and y v. t for both methods to compare solutions; if the time graph comparisons are similar, then the corresponding phase portraits will be similiar; this will also work for systems with 3 or more equations also.

Digamma is defined (at the least) on the positive reals. Extensions can be made on a larger subset of C though, just like the gamma function :)

Thank you :) feel free to let me know if there are any topics you want to see or if you have any feedback so that I can better help your learning journey!

Hello :) what I've realized actually is the app I'm drawing in (onenote) and the tablet I have aren't ideally compatible. I haven't yet found a drawing app to replace it yet, but if you have any recommendations, would love to hear. Thank you for the feedback though.

A great number for a great person ^_^

Great question. In short its a transformation of coordinates from polynomial to exponential to polynomial with two u-substitutions between; I omitted here since it would add a bit more time than I wanted, but I think exploring different ways to get to the end result is just as magical, as long as (as you mention) things converge in the same way (pointwisely at the least) as h tends to 0.

@@LetsLearnNemo Intersting. Thanks for the feedback. I'll try playing around with potential transformations. BTW, your videos on fractional calculus are the best I've found, by far!

Thank you so much :) I've been meaning to add and update the series but have been to busy, but this summer I have some definitive additions I plan to add. So I'm excited about that

You mean the dominated convergence theorem? :) I leave most theory + proof things for real analysis, but yea DCT is amazing. Triangle inequality and/or Cauchy-Schwartz makes it a bit more easier though.

Thank you :)

It's moreso an awareness type of thing and sharing your realization with others. Sometimes things get lost in history for a variety of reasons with lack of interest being one of them, then later people are interested again. So although not a major discovery per se, but an interesting and potentially useful result, just like your 41 divisibility rule :)

@@LetsLearnNemo I discovered to identify a multiple of 41 you can substract 4 times the last digit by the rest of the number

That's pretty neat. I find it interesting that divisibility by (a lot of, maybe all?) primes can be equivalent to divisibility by shifts of primes (composites).

@@LetsLearnNemo usually divisibility rules of prime numbers that end in 1 require substracting the last digit by the rest of the number

@@matiaspereira9382 This is super neat actually. Number theory has quite a few results I find quite fascinating!

Hello :) At the moment, I do not have an answer bank built for the eov questions, but do plan on having it built sometime later this year.

I plan to, yes, in terms of release times, I'm not sure. But possible two more time series vids across April :)

Thank you for your kind words :) Helping a lot of people is my goal, but just helping a few people is still okay for me :) Feel free to reach out if there are any topics you would like to see or need help on~

Thank you :) Hope you are doing well!

Hello! So if KW states there is some difference but pairwise isn't practical for large g (say 1000), partition into smaller sets, say 5 sets of 200, do KW on each set of 200. Let's say that only 1 set of 200 let's KW reject equality; then you can do pairwise on the 200 rather than the total 1000. Note that the partitions need to be chosen at random. This is one way around it :)