Yeah, when I was your age, my mind was a bit more math-agile than it is these days. Sigh...
Here's one you'll appreciate: I was a junior undergrad, and my best buddy (who was a year ahead of me and played lead guitar in our band) was giving me grief about not knowing how to do contour integrals. I responded that I could use conventional methods, and felt I was pretty damned good at doing integrals. He gave me a problem which I spent about a day attacking. I ended up with a series of solvable terms, and an integral (which was integral from 0 to infinity of log x/(1+x^2)). OK, I was decomposing that one using integration by parts and chewing through it slowly while I was at the library. My faculty advisor walked by and said (loudly), "HEY STU, WHATCHA DOING?" "Oh, I'm trying to solve this last integral. He stared at it for about 3 seconds, and said, loudly, "IT'S ZERO," grinned, and walked away.
An hour later, I had it figured out. Yes, it was zero.
Gotta hand it to you egghead. You're the only person I know that can take such an uninteresting subject and make it that funny.
You much teach young Grasshopper your ways.
😀 (Appreciating all parts)
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😀 (Appreciating all parts)
Do you want to know how he did it? As usual, it all looked so easy once it was explained...
A quick trig sub (which isn't so quick right now 😀), but I'm guessing there's a shortcut to it all that I'm missing--is the whole thing imaginary for x>0? That'd do it.
Nope, it's a real function, so its integral over a real axis would have to be real, right?
Here's the trick that my prof used: split the integral into the sum of an integral from 0 to 1, where log x is negative, and 1 to infinity, where log x is positive. Then for either term, substitute y = 1/x. The arguments become equal (since dy = -1/x^2 dx) and the integration limits are reversed for the one getting the substitution. So... they cancel. He instantly saw the symmetry that I missed.
Here's the trick that my prof used: split the integral into the sum of an integral from 0 to 1, where log x is negative, and 1 to infinity, where log x is positive. Then for either term, substitute y = 1/x. The arguments become equal (since dy = -1/x^2 dx) and the integration limits are reversed for the one getting the substitution. So... they cancel. He instantly saw the symmetry that I missed.
Clever. I can get there with the trig substitution. Using the (odd) symmetry of log(x) around [0,1] and [1, infinity), you can let x = tan(t), dx = sec^2(t)dt and your limits of integration are 0->0, 1->pi/4, infinity -> pi/2. dx/(1+x^2) goes to sec^2(t)dt/(1+tan^2(t)).... which is simply dt.
Areas of your two subintegrals are both pi/4...there's your cancellation.
Edit to add: yes, real function. I was wandering over to the negative values of x. (Open mouth, insert foot)
Areas of your two subintegrals are both pi/4...there's your cancellation.
Edit to add: yes, real function. I was wandering over to the negative values of x. (Open mouth, insert foot)
I will try to get that novel. To be honest I am not much of a reader. But Thanks there are couple of books which were recommended that I must read.I've only heard of something like this in fiction (and how would I ever hear about a "real" one? I'm not a telepath, like SOME people in this thread!). "The Demolished Man" is a classic novel and (IMHO) a good read.
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Mathematics : I was/is very bad at maths. But I remember the movie 'Contact'. What a nice movie ! It had a sequence when first contact was made by transmitting signal by maths. Some say mathematics has answers to some unsolved mysteries of nature and universe. Leonardo Fibonacci.
Regards.
Derf, that's a bit harder than my prof's method, but far, far easier than my brute force approach.
SY, don't get me wrong--I really like the elegance of your prof's solution. More so than my kludge.
Hiten: math and physics are so intertwined that it's nigh impossible to separate, which is why some of the greatest minds in both overlap so much. I think there's something pretty cool about that.
Hiten: math and physics are so intertwined that it's nigh impossible to separate, which is why some of the greatest minds in both overlap so much. I think there's something pretty cool about that.
I thought the only difference between physicists and mathematicians is that mathematicians get upset when a practical use is found for their beautiful theory, thereby sullying it with reality.
Note: real math types may not resemble this stereotype 🙂
Note: real math types may not resemble this stereotype 🙂
Speaking of calculus:SY, don't get me wrong--I really like the elegance of your prof's solution. More so than my kludge.
Hiten: math and physics are so intertwined that it's nigh impossible to separate, which is why some of the greatest minds in both overlap so much. I think there's something pretty cool about that.
https://www.youtube.com/watch?v=danYFxGnFxQ
Again, for the hundredth time, for the thousandth time... all they've done in this (below shared article), is prove it (telepathy) to blowhards who incorrectly believe that science somehow defines realty. For science does not define reality, it never did. The root of science is to attempt to describe reality, science is not independent of the frame.
Reality existed for a long time before science and it exists and is of all things. So no, it did not become real because NIST made it real. But we do run into the psychosis of dogma in people who think it is not real until the misapplied and misunderstood religious parody surrounding science 'makes it real'.
The idea of telepathy in and of the people has been around since time immemorial, this gives credence to the investigations into telepathy. Scientific investigations are only as good as the people who design the given investigation.
Quantum science says that reality is an emergent phenomenon.
Reality as humans generally think of it, does not occur until it emerges from quantum considerations and as those components are compared to one another, and the differential found, is our 'reality'.
In that 'reality' AS differential in quantum associated values, we find that coupling of distant objects does indeed occur. In scientific terms, as found repeatedly, undeniably so.
NIST team proves 'spooky action at a distance' is really real.
For those who think that 'telepathy is not real', you can now officially go pound sand or eat worms, whatever you want.
This subject is dead for you, as the scientific basis for it being 100% real is now established by the very mechanisms and levers you try and pull and manipulate to say that telepathy is not real.
Please understand that I'm not angry with naysayers, per se, but I will interject at times, to curb stomp persistent pernicious pouty projected illiteracy.
Reality existed for a long time before science and it exists and is of all things. So no, it did not become real because NIST made it real. But we do run into the psychosis of dogma in people who think it is not real until the misapplied and misunderstood religious parody surrounding science 'makes it real'.
The idea of telepathy in and of the people has been around since time immemorial, this gives credence to the investigations into telepathy. Scientific investigations are only as good as the people who design the given investigation.
Quantum science says that reality is an emergent phenomenon.
Reality as humans generally think of it, does not occur until it emerges from quantum considerations and as those components are compared to one another, and the differential found, is our 'reality'.
In that 'reality' AS differential in quantum associated values, we find that coupling of distant objects does indeed occur. In scientific terms, as found repeatedly, undeniably so.
NIST team proves 'spooky action at a distance' is really real.
For those who think that 'telepathy is not real', you can now officially go pound sand or eat worms, whatever you want.
This subject is dead for you, as the scientific basis for it being 100% real is now established by the very mechanisms and levers you try and pull and manipulate to say that telepathy is not real.
Please understand that I'm not angry with naysayers, per se, but I will interject at times, to curb stomp persistent pernicious pouty projected illiteracy.
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Yes, you've been very clear that you're unable to solve even the simplest problems in QM, so can't understand why one thing has nothing to do with the other. But if it makes you feel better to expound and create poetry, more power to you. Does it help you move product?
Once was an Orn from Nantucket.
Head oddly shaped like a bucket.
Said with chagrin as scalpel slid in,
Die Earth Cow! Die! Die! Die!
You're Welcome...
Head oddly shaped like a bucket.
Said with chagrin as scalpel slid in,
Die Earth Cow! Die! Die! Die!
You're Welcome...
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A profound meaning-of-life question I've been struggling with for a long time:
What if the hokey pokey IS what it's all about?
What if the hokey pokey IS what it's all about?
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