Community RSS Feed https://community.wolfram.com RSS Feed for Wolfram Community showing any discussions in tag Algebra sorted by active Unable to find roots for a simple equation https://community.wolfram.com/groups/-/m/t/2455951 I tried to find a solution to this simple equation using Mathematica&#039;s Solve: In:= Solve[(2 Sqrt V (m \[Epsilon]f)^(3/2))/( 3 \[Pi]^2 \[HBar]^3) == Ntot, \[Epsilon]f] Out= {{\[Epsilon]f -&gt; Root[-9 Ntot^2 \[Pi]^4 \[HBar]^6 + 8 m^3 V^2 #1^3 &amp;, 1]}} This does not give an expression for my fermi energy. So I tried reduced with some more result In:= Reduce[-9 Ntot^2 \[Pi]^4 \[HBar]^6 + 8 m^3 V^2 \[Epsilon]f^3 == 0, \[Epsilon]f] // FullSimplify Out= 2 m \[Epsilon]f == 3^(2/3) \[Pi]^(4/3) (Ntot/V)^(2/3) \[HBar]^2 I don&#039;t understand why the 2m is in front of the fermi energy but at least I can read off the solution. But if even now Mathematica cannot handle my equation: In:= Solve[ 2 m \[Epsilon]f == 3^(2/3) \[Pi]^(4/3) (Ntot/V)^(2/3) \[HBar]^2, \[Epsilon]f] Out= {{\[Epsilon]f -&gt; Root[-9 Ntot^2 \[Pi]^4 \[HBar]^6 + 8 m^3 V^2 #1^3 &amp;, 1]} Why is it that it cannot find an explicit expression for \[Epsilon]f? And how to get it in another way?&amp;[Wolfram Notebook] : https://www.wolframcloud.com/obj/3b5188c1-603a-4302-a07e-a94fab9fe750 Willem Verheijen 2022-01-26T07:45:36Z An asymptotically closed loop of tetrahedra https://community.wolfram.com/groups/-/m/t/2456465 *WOLFRAM MATERIALS for the ARTICLE:* &gt; Elgersma, M., Wagon, S. An Asymptotically Closed Loop of Tetrahedra. &gt; Math Intelligencer 39, 40&#x2013;45 (2017). &gt; https://doi.org/10.1007/s00283-016-9696-4 ![enter image description here] &amp;[Wolfram Notebook] : https://community.wolfram.com//c/portal/getImageAttachment?filename=sdafq34dfagasfsdc4.jpg&amp;userId=11733 : https://www.wolframcloud.com/obj/fa551709-69ef-4eac-82be-e93ecd7869d3 Stan Wagon 2022-01-26T17:55:36Z Compute the prime implicants of a Boolean function? https://community.wolfram.com/groups/-/m/t/2454010 Hello. Is there a function in the Wolfram Language to compute the prime implicants of a Boolean function? If yes, which one? If no, can I find one in a function repository or elsewhere? Many thanks, any help highly appreciated. Francisco Francisco Gutierrez 2022-01-23T23:24:26Z Discovering symbolic square roots? https://community.wolfram.com/groups/-/m/t/2453684 Is there any technology in Mathematica (or even as pure research), that would let me discover formulas for square root of a matrix specified as a formula? For example for any vector with positive entries adding up to 1, the following factorization seems to hold $$A=\text{diag}(p)-pp&#039;=B^T B$$ where $$B=\text{diag}(\sqrt{p}) - (\sqrt{p}) p^T$$ As you can check with the code below:  p = {1, 10, 3}; p = With[{q = Abs[p]}, q/Total[q]]; (* positive + normalized *) A = DiagonalMatrix[p] - Outer[Times, p, p]; B = DiagonalMatrix[Sqrt[p]] - Outer[Times, Sqrt[p], p]; A == Transpose[B] . B  I got this formula by expanding the components and staring at them for a while. This obviously doesn&#039;t scale. This was a factorization of cross-entropy hessian, and there are at least 20 more [loss functions](https://pytorch.org/docs/stable/nn.html#loss-functions) I&#039;d like to do .... can anyone think of a way such formulas could be discovered automatically? This begs the question of what &#034;formula&#034; is....if we restrict attention to factorizations of Hessians, a symbolic way to derive them is [this](https://arxiv.org/abs/2010.03313) . You can see that if your original loss was specified in terms of pointwise ops and some tensor contractions, then its Hessian will consist of a sum of terms with some pointwise ops and some tensor contractions. The author posted formula for 3rd derivative of cross-entropy loss on Mathematica [stack-exchange](https://mathematica.stackexchange.com/a/262193/217) so you can see that small formulas remain small after differentiation. Yaroslav Bulatov 2022-01-23T20:08:55Z Wolfram|Alpha doesn't understand query about multivariable problem https://community.wolfram.com/groups/-/m/t/2454874 Hello. I&#039;m new to Wolfram and I&#039;ve been trying to run this multivariable problem. Solve for b, B and k in terms of a, A, c, C, f. F, alpha: b = lsin(alpha)+a, B = lcos(alpha)+A, d = a+k(b-a), D = A+k(B-A), e = b+k(c-b), E = B+k(C-B), f = e+k(d-e), F = E+k(D-E) Joseph Choi 2022-01-24T21:57:31Z