Recent content by renormalize

Must be a Rorschach test! I see Africa, the Middle East and Madagascar in 1972 and Australia(?) in 2026.

I boosted the brightness and contrast a bit on your Artemis photo and voilà! Earth in 2026 actually looks as good or even better than it did in 1972! My point is simply that you cannot draw scientifically valid conclusions without referencing your data to properly calibrated standards.

Your step:$$\lim_{x\to0}\frac{\frac{\tan(\tan x)}{tan(x)}\cdot\frac{1}{sin(x)}-\frac{\sin(\sin x)}{sin(x)}\cdot\frac{1}{tan(x)}}{\frac{1}{sin(x)}-\frac{1}{tan(x)}}=\lim_{x\to0}\frac{1\cdot\frac{1}{sin(x)}-1\cdot\frac{1}{tan(x)}}{\frac{1}{sin(x)}-\frac{1}{tan(x)}}\tag{1}$$is wrong. Focus on the...

Can you cite a credible textbook reference that supports those claims?

So you assert that the metals mercury at room temperature and aluminum at 660°C are not soft materials?

The observational data on the fine-structure constant has since been extended out to 13 billion light years (as of 2020) and so far shows no evidence of variation: https://www.physicsforums.com/threads/can-physical-constants-be-different-in-other-galaxies.1046458/#post-6811957

I've done no complaining nor conceding because my very first post in this thread is #20 just above. Do you even bother to check who you're responding to before you shoot off your post?

Maybe because mass and weight are different in both Imperial units (slugs vs. pounds) and metric units (kilograms vs. newtons)? So why did you bring up the metric system in the first place when discussing mass vs. weight?

Can you better describe the points you're asking about by making reference to the diagram below? (from Shadow of the Moon and general relativity: Einstein, Dyson, Eddington and the 1919 light deflection)

But how much effort have you expended to understand and appreciate the already existing technology? For example, it took me less than a minute online to find this article from 2023: Water quality analysis using mmWave radars Abstract: Water quality and drinkability assessment are of high...

I'm unaware of practical examples of your function:$$f_{k}\left(x\right)\equiv x^{k/x}$$but a transformation of that function:$$\log\left(f_{k}\left(u^{-1}\right)\right)=-k\,u\log u$$with the specific value ##k=-1## is quite common: Stirling's Approximation:$$\log n!\approx n\log...

Perhaps you can gain intuition regarding the kind of beams that can be formed from a phased array of antennas by playing with on-line calculators. For example, here's one that allows you to adjust the number of antenna elements, their spacing and relative phase shifts...

Spectacular results! I no longer even have to refresh the page occasionally, as I used to, to get proper previewing. Kudos to Greg.

In Physics Forums you should use ## to wrap in-line LaTeX equations and $$ to wrap display equations. I've reformatted your post below: And to make our lives easier, please try using LaTeX to post your solution attempt here, rather than making us all open your PDF attachment.

Yes, the spreading of single-atom wave packets has been observed experimentally and agrees with theory: In-situ Imaging of a Single-Atom Wave Packet in Continuous Space