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Ilya Bogdanov

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 Name Ilya Bogdanov Member for 1 year Seen 14 hours ago Website Location Moscow Age 35
 2d comment Periodic tilings of the plane with fundamental domain given by $k$ squares of prescribed sizeFo you need a domain to be connected? Jun16 comment A sum-product estimate in Z/p^2ZSince the order of the multiplicative group of ${\mathbb Z}/p^2{\mathbb Z}$ is $p(p-1)$, you should replace $p$ by $p-1$ either in the group order or in the exponent og $g$.I assume the first, since the second is not interesting at all, Jun12 comment Does an origin-centered ellipse in the plane intersect each $L^p$-circle at most 8 times?Yes, I have had this dilemma in mind when I told `blowing up or away'. Sorry for being not that detailed... Jun12 accepted Does an origin-centered ellipse in the plane intersect each $L^p$-circle at most 8 times? Jun12 revised Does an origin-centered ellipse in the plane intersect each $L^p$-circle at most 8 times?added 234 characters in body Jun12 revised Does an origin-centered ellipse in the plane intersect each $L^p$-circle at most 8 times?added 184 characters in body Jun12 comment Does an origin-centered ellipse in the plane intersect each $L^p$-circle at most 8 times?I've added an explanation below the main text. Since I was to type something;), I have also added some words about a generalization. Jun12 revised Does an origin-centered ellipse in the plane intersect each $L^p$-circle at most 8 times?added 774 characters in body; added 36 characters in body Jun11 revised Does an origin-centered ellipse in the plane intersect each $L^p$-circle at most 8 times?added 473 characters in body Jun11 answered Does an origin-centered ellipse in the plane intersect each $L^p$-circle at most 8 times? Jun9 comment Group with 2 orbits on the nonnegative integers — description of the orbits1. Yes, I was looking for that, but it seems that the situation is harder here. 2. Yes, to find the density we need much more. Jun8 revised Group with 2 orbits on the nonnegative integers — description of the orbitsadded 6 characters in body Jun8 answered Group with 2 orbits on the nonnegative integers — description of the orbits May29 accepted Are roots of transcendental elements transcendental? May29 comment Are roots of transcendental elements transcendental?And yes, we may say that $k[t]={\mathbb F}[X,Y]/(Y^2)$, $t=\overline{X}$, and $k$ is a subring generated by $a=\overline{Y}$ and $b=\overline{X}^3\overline{Y}$. Then $at^3-b=0$, but $t^2$ is transcendental. May29 comment Are roots of transcendental elements transcendental?Yes, I was wrong about the term, sorry. The polynomial is simply $at^3-b$... May29 revised Are roots of transcendental elements transcendental?added 3 characters in body; deleted 92 characters in body May29 answered Are roots of transcendental elements transcendental? Apr13 comment “Box Nodes” in Directed Graphs with Paired IO SymmetryAs far as I undestand, the question may be formulated as follows. We have a digraph where each vertex has both in- and out-degrees equal to 2. $n$ independent vertices are marked (there are in/outputs of the black box), and for every permutation of the marked vertices there exists an automorphism of the graph inducing exactly this permutation. Is this what you meant? Apr9 comment The sum of same powers of all matrices modulo pThe approach is interesting. But it seems that you think there are $p^3$ matrices; in fact, there are $p^{p^2}$ of them. But you do not need so much, since there are less than $p^p$ distinct denominators (all costant terms are ones). This gives the estimate of about $2p^{p+1}$ coefficients to check. Unfortunately, now we have a half of this amount. Apr8 awarded ● Nice Answer Apr7 revised The sum of same powers of all matrices modulo pedited body Apr7 revised The sum of same powers of all matrices modulo pdeleted 25 characters in body Apr7 comment The sum of same powers of all matrices modulo pDoes it remain for larger $k$? Apr7 comment Citation for subset complement resultIt is so obvious that I do not think it deserves a citation. Apr7 answered The sum of same powers of all matrices modulo p Apr5 accepted Increasing sequence of normal magic squares Apr5 revised Increasing sequence of normal magic squaresadded 495 characters in body; deleted 8 characters in body Apr5 answered Increasing sequence of normal magic squares Apr3 accepted Lattice basis with Gram-Schmidt vectors of increasing length Apr3 answered Lattice basis with Gram-Schmidt vectors of increasing length Mar30 comment Area of a lattice polygon in terms of its widthIn fact, this bound is tight. Look at the triangle wth vertices $(0,0)$, $(2,1)$ and $(1,2)$. Its area is $3/2$, while its "integral width" is 2. To see that, first notice that all its altitude lengths are greater than 1, hence it is enough to check only the vectors of length less than 2. This check is straightforward. So, what is this bounty for? Mar30 comment Area of a lattice polygon in terms of its widthYou need to be a bit more careful, since the affine transforms do not preserve scalar product. So in fact you need to put the vector orthogonal to $v$ into $(-1,1)$. Mar28 comment Area of a lattice polygon in terms of its widthConsidering the quadrilateral $Q$ of maximal area inscribed into $M$. THen you may pass to the parallelogram with sides parallel to the diagonals of $Q$ --- its area is at most twice the area of $M$. Hence you may restrict yourself to the lattice parallelograms only. Mar24 comment Hobbled rook tour - Hamiltonian cycle on square gridIt is a well-known olympiad problem. Feb14 accepted On reducible polynomials with positive coefficients, $1$ as constant coefficient and certain bounds on coefficients Feb14 comment On reducible polynomials with positive coefficients, $1$ as constant coefficient and certain bounds on coefficientsWell, the construction blows up not the higher terms, but the terms in the middle... Feb14 comment On reducible polynomials with positive coefficients, $1$ as constant coefficient and certain bounds on coefficients@Gerry: I feel the same... Feb14 comment On reducible polynomials with positive coefficients, $1$ as constant coefficient and certain bounds on coefficientsI have added the way of constructing such an example. Feb14 revised On reducible polynomials with positive coefficients, $1$ as constant coefficient and certain bounds on coefficientsAn answer to the auxiliary question added; added 174 characters in body Feb13 answered On reducible polynomials with positive coefficients, $1$ as constant coefficient and certain bounds on coefficients Feb7 comment Rank of a matrix with missing entriesWhat does "undefined" mean? Do you need the lower bound for all matrices with some prescribed values? Jan27 accepted What is the most extreme set 4 or 5 nontransitive n-sided dice? Jan25 comment What is the most extreme set 4 or 5 nontransitive n-sided dice?Both numbers 6 and 8 are optimal. Jan25 revised What is the most extreme set 4 or 5 nontransitive n-sided dice?added 316 characters in body Jan25 answered What is the most extreme set 4 or 5 nontransitive n-sided dice? Jan25 accepted A quantitative version of Straszewicz’s theorem? Jan25 answered A quantitative version of Straszewicz’s theorem? Jan5 comment Covering all, but $k$ points with affine subspaces@domotorp: By a straightforward induction on $p$, the subset uncovered by $p$ hyperplanes, if nonempty, is an affine subspace of codimension at most $p$. Jan4 accepted Covering of a partial order by upwards convex sets