Here is a silly little question that my friend Ori and I ran into. Let be a decreasing sequence of positive real numbers which tends to 0. Perhaps in a more decent universe, the sum would always equal some finite number, but already in kindergarten we learn that that’s not true: For example, the partial … More Converging Bernoulli Series
Still probably the best 70$ I ever spent.
Last time I introduced a neat little question: Is it true that every infinite path in contains arbitrarily long arithmetic progressions? For example, in the following path, the squares colored in red have coordinates , which form an arithmetic progression of length 3, where every two consecutive points differ by . As it turns out, … More Arithmetic progressions in space!
Here is a neat question that I stumbled upon during my academic random walk. A sequence of vectors is called an arithmetic progression of length k if there exists a vector such that . In other words, an arithmetic progression in is exactly the generalization you would expect of your good ol’ high-school arithmetic progression … More Arithmetic progressions in space?
What has been will be again, what has been done will be done again, and there is nothing new under the sun. A while ago I wrote a piece of spoken word, relating (in a contorted way) the multitude of cheeses found in the modern fromagerie to the multitude of gods found in the not-so-modern … More Nothing new under the sun
This is part three in a three-part series about recurrence and transience on the integer lattices. Here is part one. Here is part two. Weary and tired, we are approaching the end of our coin-flipping, lattice-exploring adventures, but with one aching question still burning deep within our hearts: Is the random walk on the three-dimensional … More There and back again, part 3
This is part two in a three-part series about recurrence and transience on the integer lattices. Here is part one. Here is part three. In the previous post, we saw that the integer lattice is recurrent. This means that a simple random walk, when starting at the origin , will return to the origin with … More There and back again, part 2