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Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite 1 I would like to see a "metric proof" that if two metric spaces $X$ and $Y$ are CAT($kappa$) for some $kappa ge 0$, then so is their product. I would be satisfied to see a proof for $X=Y=S^2$. By "metric proof" I mean one which does not rely on Riemannian geometry, but rather only uses Alexandrov (metric) geometry. I already understand the case where $kappa le 0$ (where in fact the product will only be CAT($textmax(0,kappa)$). metric-geometry share | cite | improve this question asked Sep 6 at 11:04 Delfador Logalmier 11 2 What is $rm CAT[k]$ and their product ? – HK Lee Sep 6 at 11:36 A CAT($kappa$) space is a geodesic space all of whose geodesic triangles of perimeter less than $2D_kappa$ satisfy the CAT$(kappa)$ inequality. Here $D_kappa= +infty$ if $kappa leq 0$ or $fracpikappa$ if $kappa > 0$. One version of the CAT$(kappa$) inequal

Clash Royale CLAN TAG #URR8PPP up vote 0 down vote favorite The Problem Informal statement of the problem: consider a natural distribution $D_n$ of very sparse $ntimes n$ matrices over the binary field $mathbbF_2$ - that is, a matrix sampled from $D_n$ contains a constant number of $1$ per row. How likely is a matrix sampled from $D_n$ to contain a large invertible submatrix? More formally, consider the two "natural distributions" over $ntimes n$ matrices in $mathsfM_ntimes n(mathbbF_2)$: $D^0_n$ samples each entry of the matrix independently from the Bernouilli distribution with probability $p_n = d/n$, where $d$ is a small constant (that is, each entry of a sampled matrix is $1$ with probability $d/n$, and $0$ with probability $1-d/n$). $D^1_n$ samples each row of the matrix independently; to sample the $i$th row, pick a random size-$d$ subset $S_i$ of $[1,cdots, n]$. The subset $S_i$ denotes the position of the $1$s in the $i$th row (and $[1,cdots, n]setminus

Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite Evaluate $$dfrac1sin(2x) + dfrac1sin(4x) + dfrac1sin(8x) + dfrac1sin(16x)$$ It would be tough for us to solve it using trigonometric identities. There should be strictly an easy trick to proceed. Rewriting and using trigonometric identities $$dfrac1sin(2x) + dfrac1sin(2x) cos (2x) + dfrac1 2big [2sin (2x)cos (2x)cos (4x)big ] + dfrac1sin(16x)$$ What am I missing? Regards trigonometry share | cite | improve this question asked Sep 6 at 11:11 Busi 325 1 10 See math.stackexchange.com/questions/1591220/… – lab bhattacharjee Sep 6 at 11:27 add a comment  |  up vote 2 down vote favorite Evaluate $$dfrac1sin(2x) + dfrac1sin(4x) + dfrac1sin(8x) + dfrac1sin(16x)$$ It would be tough for us to solve it using trigonometric identities. There should be strictly an easy trick to proceed. Rewriting and using trigonometric identities $$dfrac1sin(2x) + dfrac