Here we give a short video on how to return a boolean correctly (this case in Python). Many beginners make this mistake! Thanks to the following supporters of the channel for helping support this video. If you want to contribute, links are below. Dolev Abuhazira, Josh Hibschman, Micah Wood, Morgan Jones, Patrik Keinonen, Simone Glinz, Tao Su, Timothy Gorden, unit220, Valentine Eben Easy Theory Website: https://www.easytheory.org Become a member: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg/join Donation (appears on streams): https://streamlabs.com/easytheory1/tip Paypal: https://paypal.me/easytheory Patreon: https://www.patreon.com/easytheory Discord: https://discord.gg/SD4U3hs Merch: Language Hierarchy Apparel: https://teespring.com/language-hierarchy?pid=2&cid=2122 Pumping Lemma Apparel: https://teespring.com/pumping-lemma-for-regular-lang If you like this content, please consider subscribing to my channel: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg?sub_confirmation=1 ▶ABOUT ME◀ I am a professor of Computer Science, and am passionate about it. I have taught many courses at several different universities, including several sections of undergraduate and graduate theory-level classes. The views expressed in this video are not reflective of any of my current or former employers. ... https://www.youtube.com/watch?v=-mEQ_SWc2GE

Original covering arrays video: https://www.youtube.com/watch?v=m1j4OBs1wsY&ab_channel=EasyTheory arXiv preprint: https://arxiv.org/pdf/2211.01209.pdf Here we discuss a recent proof of mine of a conjecture in my research area, which has to do with sizes of covering arrays. I first go over what the problem, which is determining the asymptotic sizes (# of rows) of covering arrays of "higher index", which is to say each interaction must appear at least a certain number of times, instead of just once. I give "easy" upper and lower bounds; unlike the prior video's scenario, there is a "gap" between them. Next I give some upper bounds that have been published and are "better", but there is still a small gap. My proof this year is that there is no gap between the "easy" lower bound and the new upper bound I prove, which is logarithmic in the number of columns, and additively linear in the index. This proof also uses the probabilistic method, but with a new ingredient: the Lambert W function. My proof first uses the Cauchy-Schwarz inequality on the equation coming from the probabilistic method. The C-S inequality relates cross correlations of sums (hard to understand) to self-correlations (easy to understand), but paying a penalty; in other words, sum(a_i * b_i) changes to sum(a_i^2) * sum(b_i^2) * penalty. Then we use simple known upper bounds on the two sums that come out. The final "trick" is to use the Lambert W function, which is the inverse of f(W) = W * e^W. One would have to analyze the equation that comes out, and notice where in the Lambert W function one is; in this case, it is in the negative region. To finish everything off, we use *lower* bounds on W since in this region, W is negative (which translates to *upper* bounds on covering array sizes). Easy Theory Website: https://www.easytheory.org Discord: https://discord.gg/SD4U3hs If you like this content, please consider subscribing to my channel: https://www.youtube.com/channel/UC3VY6RTXegnoSD_q446oBdg?sub_confirmation=1 ▶ABOUT ME◀ I am a professor of Computer Science, and am passionate about it. I have taught many courses at several different universities, including several sections of undergraduate and graduate theory-level classes. The views expressed in this video are not reflective of any of my current or former employers. ... https://www.youtube.com/watch?v=D3H8yEuCsSk