Gerhard Paseman
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 11h comment Is there any $ABCDS$ pyramid (where $ABCD$ is a rectangle) in which each 2 edges have different lengths and $|AS|+|CS|=|BS|+|DS|$? You can turn this into an algebra problem: Given two pairs of positive real numbers (a,b) and (c,d) such that a+b=c+d and a^2 + b^2 = c^2 +d^2, are the pairs distinct up to symmetry (so that (a,b) is not distinct from (b,a) )? I see this as a math.stackexchange question. Gerhard "Please Try The Other Forum" Paseman, 2015.10.09 11h comment Is there any $ABCDS$ pyramid (where $ABCD$ is a rectangle) in which each 2 edges have different lengths and $|AS|+|CS|=|BS|+|DS|$? I think he(?) means of the four edges in the equality. Note for degenerate pyramids that the sums of the squares (of the four quantities) are equal (and thus also of nondegenerate pyramids with rectangular base). I would be surprised if a nontrivial example existed. Gerhard "It's How I See It" Paseman, 2015.10.09 1d comment Using the multiverse approach to decide the law of the exluded middle? @Andrej, indeed you did speak on the universe front, and in my view quite well too. When I read the question today, I saw it as being about a multiverse of logics, and thought the points above worthy of mention. If you see a mathematical point in my post that requires amplification, correction, or deletion, please speak up: I defer to your experience and expertise, and would enjoy a post from you along the lines sketched above. Gerhard "At Least, In This Universe" Paseman, 2015.10.08 1d comment $n$ groups of $n$ queens on a toroidal chessboard If the problem were about rooks, you would be talking about Latin squares, of which there is much literature. Queen problems on various boards have been considered. Try a web search and tell us what you find. Gerhard "Studying Boards Made Of Hexagons" Paseman, 2015.10.08 1d answered Using the multiverse approach to decide the law of the exluded middle? 1d comment Is the Manickam-Miklós-Singhi Conjecture solved? Welcome to MathOverflow! Can you say anything to resolve the question as asked? Following your last sentence, the preferred answers would be something like "I have not fully read the ArXiv paper." or "I have read it and have an issue with point X raised on page Y. The issue with the mathematics is Z. This and other points need to be resolved before I consider the conjecture fully settled." Gerhard "Happy To Talk Math Here" Paseman, 2015.10.08 1d comment How does this small change in the Pollard Rho method affect its complexity? To get some serious attention, this needs some serious editing. For example, the first two sentences should go something like "In finding a large factor p of an input number n, the Pollard Rho method takes time bounded by a function in $O(\sqrt{p})$. (Did I get that right?)" . Further, what is the rational for looking for (p-2)? Are you going to add 2 to every number in the cycle and check if the result is a factor? Edit the question to include why your change makes sense. (It doesn't currently.) Gerhard "Can Still Improve The Question" Paseman, 2015.10.08 1d comment Does the hyperdeterminant calculate a quantity akin to the volume of a parallelepiped? Perhaps (for Joseph's definition of "somehow analogous") this could be a "Yes" answer. For those of us still struggling with the concepts, could you confirm/refute the assertion that "the classical case considered by Cayley" as you put it is the "second hyperdeterminant" as so-named in the Wikipedia article? Gerhard "Still Learning Geometry Of Hypermatrices" Paseman, 2015.10.08 1d comment How the idea of adjugate matrix has been designed? Indeed. He reworked his research several times, with posts in 1890 and 1906. A copy of one of them can be found here: archive.org/details/theoryofdetermin01muiruoft . People were saying to him "Inter-net? Whereof speakest thou?", to which Muir calmly replied "Just wait." Gerhard "Clearly Ahead Of His Time" Paseman, 2015.10.08 2d comment Does the hyperdeterminant calculate a quantity akin to the volume of a parallelepiped? In related questions on MathOverflow, including one Joseph asked some years ago, hypermatrices are (my interpretation) notational devices for tensors, and the formula for multiplication is derived and understood by students of hypermatrices, and bears some relation to composition of transformations. I think your statement about its having columns that are linearly independent in R^3 is wrong (I have a 0-1 example in mind), but I can't be sure since we haven't agreed on what it is you mean. Gerhard "Again, What Is A Column?" Paseman, 2015.10.07 2d comment Does the hyperdeterminant calculate a quantity akin to the volume of a parallelepiped? I'm afraid slice is also not useful to me. For n=3, let there be a hypermatrix of 3x3x3=27 entries. I pick 3 of them at a time to embed in three space. I partition the hypermatrix entries into 9 such groups which I call "vectors". What do I do with these 9 vectors in 3-space that allows me to talk about linear independence? Take the vectors 3 at a time? Gerhard "Really, What Is A Slice?" Paseman, 2015.10.07 2d comment Does the hyperdeterminant calculate a quantity akin to the volume of a parallelepiped? I'm sorry: what are you saying? I have trouble coming up with a notion of column that would embed in 3-space. Even if you embed it in n-space, I don't see that the n^2 many columns are linearly independent for n > 1. Gerhard "Please, What Is A Column?" Paseman, 2015.10.07 2d comment Does the hyperdeterminant calculate a quantity akin to the volume of a parallelepiped? From the first paragraph on the Wikipedia article for hyperdeterminant: "Many other properties of determinants generalize in some way to hyperdeterminants, but unlike a determinant, the hyperdeterminant does not have a simple geometric interpretation in terms of volumes." I read that as a "No", but you may be looking for something more. Gerhard "It Is Wikipedia, After All" Paseman, 2015.10.07 2d comment Does the hyperdeterminant calculate a quantity akin to the volume of a parallelepiped? Umm, Wikipedia has an entry (which on my reading says no to your question), and there have been some ArXiv articles computing hyperdeterminants for small cases which might also help with your question. Perhaps you mean to ask something else? Gerhard "Question Not Ready For Primetime?" Paseman, 2015.10.07 2d comment How the idea of adjugate matrix has been designed? I think the Wikipedia entry of Cramer's rule is quite accessible and well motivated (and relevant here). Also, Thomas Muir posted a history of determinants (which existed before matrix notation!) and their development. You can find both resources on the web. Also, the question could use an example of what shape is wanted for the answer: pointers to the literature, a copy of the Wikipedia article, a category theory or foundational approach, or something understood by someone with only one or two linear algebra courses behind them. Gerhard "Falls Into The Last Category" Paseman, 2015.10.07 Oct 6 comment Non-principal prime ideals in infinite distributive lattices You might be able to prove existence by considering the associated lattices IDL(L) and FIL(L) for a given infinite distributive lattice L. I would not be surprised if this dealt with choice issues, e.g. that in a variant of set theory without choice one finds a model of this theory which believes it has an infinite distributive lattice all of whose prime ideals are principal. Gerhard "Still Capable Of Showing Surprise" Paseman, 2015.10.06 Oct 6 comment Proposals for polymath projects What happens if, during the running of this algorithm, the "none is a substring of another" condition is violated by $s_{i1},...,s_{ik}, b_{n-k+1}$, where the $b$ string is the newly formed big string? Or is that the point? Gerhard "Sometimes Not Quick On Uptake" Paseman, 2015.10.06 Oct 6 answered What is the term for combining functions $f_1,f_2,\dots,f_n$ into a tuple $(f_1,\dots,f_n)$? Oct 5 comment Factorial Sums over Compositions or Unlabeled Permutations" If the c_i's are not ordered, then you might find multinomial more useful than partition in searching. I would still look at partition as some sources might talk about the partition case being easier/different than the composition case (and give you something on compositions). Also, I read your post and (despite your worthy attempt) I still think partition. For the more dense among us, you might make it amazingly clear that Euler's partition function is not part of your picture. Gerhard "Not Quite 'MathOverflow For Dummies'" Paseman, 2015.10.05 Oct 5 comment Factorial Sums over Compositions or Unlabeled Permutations" I think you might find "integer partition" a more useful search term than "integer composition". (I am guessing that $c_i \leq c_j$ iff $i \leq j$.) You might find "multinomial" in combination with some other word also useful. Gerhard "Don't Know About The Sequence" Paseman, 2015.10.05