Recent content by aheight

Perhaps spoon feeding was too harsh a word. Also should have realized someone would ask for references and that just creates tension which I try to avoid in here so I'd rather not identify any particular ones. I still however think it would be a good idea to more stress the importance of...

I find it a bit counter-productive to support requests asking for ideas to work on at the university level. Shouldn't we rather encourage the idea of being more responsible doing so on your own? Wouldn't that be better than spoon-feeding? At some point in life, you must go by yourself...

Phinds, no offense but I think your friend did a poor analysis of this study: The cited reference refers to a new 2020 study of the original 1970 World3 results done by Gaya Herrington who is an analyst at KPMG, a professional service company providing ESG (environmental, social and governance)...

The article mentions the Ennis-Weir Critical Thinking Essay Test. The test involves analyzing a letter sent to the Moorburg newspaper about parking on streets in Moorburg: In case readers missed it: The Moorburg letter. Interesting I thought.

Could perhaps be an interesting (diophantine) minimization problem: Given a set of points ##\{x_i,y_i\}##, find ##(A,B,C)\in\mathbb{Z}## such that some error metric such as: $$ \int_a^b \big|g(x)-h(x)\big|dx $$ is minimum where say ##g(x)## is a (relatively) good or good enough fit of the data...

Mathematica's ##\texttt{NSolve}## for ##\alpha(x)## returns a set of solutions for ##a=1/2## and ##b=3/2## one of which is $$ \alpha(x)=\frac{1}{2 \left(\log \left(\frac{\sqrt[3]{\frac{2}{3}}}{\sqrt[3]{\sqrt{3} \sqrt{27 e^{2 x}-4}-9 e^x}}+\frac{\sqrt[3]{\sqrt{3} \sqrt{27 e^{2 x}-4}-9...

Ok, thanks for that. Sorry I missed it. I'm not a Number theorist but am personally skeptical we can predict the future behavior of diophantine expressions based on the previous results of those expressions. My understanding is that integer arithmetic does not work this way. Rather it is...

Regarding the Wikipedia reference: Results of computer search for a perfect cuboid: The author concludes: This seems naive to suggest after only checking a basically insignificant number of cases, i.e., there may be one that has extremely large faces. Why would he make what I consider to...

See: https://repository.asu.edu/attachments/190173/content/Specialized%20odorant%20receptors_2017.pdf Basically it's a beautiful example of neuro-biochemistry. The following (Olfactory receptor neurons) is a description of the process in the human nose but I believe it's very similar to how...

I recently watched "Black Holes: The edge of everything we know" on Netflix: Netflix trailer. It's the story of how a group used the Event Horizon Telescope (EHT) to image a black hole. What's fascinating for me is how the EHT works. It's actually 8 telescopes around the globe that...

Mathematica is giving a complex answer because without explicitly defining the domain of L, x, and y, Mathematica assumes they are complex. To restrict L, a, and b to Reals, use the code: ClearAll[theL, a, b]; Integrate[ v^2 ArcTan[(theL u)/(v Sqrt[u^2 + v^2 + theL^2])], {v, -b, b}, {u, -a...

I feel I did not adequately explain (label) my plot above and can no longer edit it. Here it is with labels. As "t" is adjusted, the point J move around the circle and as "a" increases, the circle grows larger up to JM=0.49 as the points K and L are adjusted so that at all times...

If it were my test question, I'd check all of them as I agree the angle can be any from 0 to 180 if the point M is anywhere along the segment KL. If you have access to Mathematica, perhaps you can study the code below with KL=1 and as the parameters "t" (angle of MJ on the circle -- the red...

I'd like to continue a bit to solve this somewhat manually. We have the general expression: $$ \frac{dy}{dx}=\sqrt{a e^y+b e^{2y}+c} $$ Separating variables again and integrating (I did the integration with Mathematica): $$ \begin{align*} \int\frac{dy}{\sqrt{a e^y+b e^{2y}+c}}&=x+c_2 \\...

(* edited: noticed DE in p easily solved by separation of variables *) I suggest start by writing it as $$ y''-a e^y-b e^{2y}=0 $$ Then let ##y'=p## as the standard notation. Then ##y''=p\frac{dp}{dy}## leaving $$ p\frac{dp}{dy}-a e^y-b e^{2 y}=0 $$ or $$ pdp=(ae^y+be^{2y})dy $$ It's at...