MWresearch

I had a hunch that since the derivative of the inverse of the function is the reciprocated derivative in the original function as a function of the inverse that, if you had a function that you didn't know the inverse precisely, you could calculate the taylor series for its inverse using that derivative property. Upon researching this, the only conclusive thing I found was something called the Lagrange Inversion theorem. Coincidentally, the way it's described makes it seem like it does exactly what I want it to do, finding the inverse of a function which can only be described analytically. However, I'm not good at reading mathematics from scratch and whenever I look at the wiki page for it, it still looks like a mess of variables that aren't defined in much detail. Could someone work through an example with something simple showing how I use the theorem?

But a cycle takes time, so on an individual scale, it must be affected by time dilation even though the final result after all calculations should show it isn't. Not everyone can agree that a cycle occurs at the "same time" as I'm sure you know and they also can't agree that a cycle takes the same amount of time to happen. But that's beside the point, I already know it's not suppose to change between frames, I'm just trying to show it mathematically in a particular circumstance. Can you address the point about the temperature?

I also think k is the "same" but I still think it's worth showing mathematically. Off the top of my head, I would guess that the gamma transformations cancel out with some being in the numerator and some being in the denominator. As for the temperature increase, is there anything I could introduce that would stop the temperature from increasing that wouldn't affect the dimensions and density of Uranium that much? Otherwise, I'm neglecting a lot of things. If I neglect the density expansion due to the energy released in fission events, is it scientific still? Just how much neglecting can I get away with?

If I'm interpreting that information correctly, a critical mass piece of U235 in a critical spherical volume will explode on its own but it will take a little time to ramp up since the number of fission events follows an exponential growth? But, wouldn't the neutrons released by the first fission increase the number of fission events more than just delayed neutrons? Or does that equation consider that delayed neutrons initiate the reaction and then the fast and slow neutrons resulting from critical events start the take over and sustain the chain reaction?

I still can't figure out if the mechanism is set up right though, john. So if I have critical mass in say, a bare sphere of 17cm diameter with 52kg of assuming pure U235 (but without knowledge of the actual grade used), the delayed neutrons alone in that sphere will start a chain reaction themselves? Because if that's true, all I need to do is show that the effective neutron multiplication factor stays less than one in both frames, but if I need some initiator like a neutron gun or plutonium pit, that complicates things a lot more than is necessary to show the point of the research.

3 semesters of calculus and one of statistics. Though, I see this discussion being wrapped up sooner rather than later, it seems trigonometric functions with complex arguments have rather straight forward identities. Besides inverse trig complex argument identities, the rest are details, perhaps for another thread if they are discussed more in depth.

The only question I see is "do you understand the difference between an equation and an identity?" After looking up the difference, I do not understand complex analysis or the gamma function any better, so I chose to refrain from commenting on an element of the discussion that would lead it in an unproductive direction. Otherwise, what other questions did you have? At this point I think I'm just missing is inverse trig identities with complex arguments, the rest is just practice with using those trig identities in equations. Although I don't completely understand how a Fourier transformation decomposes trigonometric functions using these imaginary/complex trig and Euler identities, but the integral itself looks like a hybrid between the gamma function and Euler's identity.

That integral seems too specific to be useful, could you expand more on the generalized process by which it is used? Could it be used anytime you have trigonometric functions or any time you have exponents with opposite signs? Would the gamma function ever be useful in equations which aren't restricted to being represented as integrals?

Hmm. So it seems like there's nothing really tricky about it, you just do a direct substitution using those particular complex identities and use logarithms to solve for a numerical answer in the end. It might be useful if there were also identities harnessing the inverse sine and inverse hyperbolic functions. As something else to explore, the gamma function wouldn't happen to be connected to complex analysis because of its identity gamma(x)(gamma(1-x)=pi*csc(pi*x) would it? I am not seeing much about the gamma function on google related to all of this, but I would imagine someone explored it. I also can't find much on the identities of the inverse functions involving complex arguments in google. Where is a list of identities with things like arcsin(ix) and arccosh(ix)?

One of the solutions I got had those imaginary exponents that look like they were transformed by some Euler identity involving imaginary exponents. I don't really know what all those imaginary exponents are used for exactly, but they come up a lot when I plug complicated equations into wolframalpha and I want to know how to use them by hand because I do a lot of algebraic manipulation, and I don't always need to come up with solutions the traditional way. Is solving complicated trig equations even the primary purpose of Euler's identity and imaginary hyperbolic functions? As far as I can tell, all of those equations involving trigonometric identities on the complex plane are part of something called "complex analysis," and I'm told that kind of stuff will come up in engineering, but I have yet to see how. I also do some audio engineering and I am loosely familiar with a fourier transformation" being related to trginometric functions over time which use complex exponents in some way.

Ok, thanks.

It would be nice to use Wikipedia in research papers because its quality has increased a lot since it was first developed. However, because of its history of inconsistencies and letting anyone edit any article, I am not sure if it is accepted as a credible source of information by the scientific community.

I assume you are thinking I am mistakenly using a classical Bhor planetary model. But, even though electrons propagate as waves, that does not mean that oscillations or measurements of changes in position are incapable of yielding near-light velocities in atoms. Thus, we have a corrected form of chemistry called "relativistic chemistry."

You are familiar with chemistry I see, then you are aware that some circumstances in chemistry change as a result of relativity, like the fact that the electrons are traveling close to the speed of light around massive nuclei. The principal of this shouldn't be too much different than that. Things like the specific activity and the rate at which neutrons are given off will be slowed, and I think the atomic radii will contract along the direction of motion which will probably affect some cross section variable. The same principal is used to explain muon decay in the atmosphere wherein muons reach Earth's surface before decay because of the time dilation lengthening the amount of time before they decay. If you're not familiar with it which would be completely understandable since industry standards might not warrant a relativistic model, you can google "relativistic chemistry" and the first few links should show what I'm talking about. You will see than even Planck's constant is affected by relativity. EDIT: After rereading the thread, I realized "affects Planck's constant" might be misleading. Rather, relativity affects equations involving Planck's constant like relative mass and energy. At this point though I think I see what Swansont is saying about the probability. Since a chain reaction would never occur when the Neutron Multiplication Factor is less than one, I simply have to show that k (the NMF) is less than one in both frames. Is that correct Swansont?

The entire discussion could have been simplified if you simply said "my bad, I did not notice you were not asking for a real numerical solution" but instead you for some reason go into denial over basic facts because you seem to have an irrational fear of appearing wrong in front of others. I admitted I made a mistake in a couple other threads, but the world didn't end. So, let's go over the facts: Fact #1, apparently the most important fact: Nowhere did I ask for only numerical solution and only in terms of real numbers Fact #2: You repeatedly stated I was asking for a real solution and ignored my multiple statements that I in fact was not looking for a real numerical solution and I specifically stated I wanted generalized solutions and the process by which they were obtained. Fact #3: I never stated hyperbolic trigonometric functions were in the first post, but rather I specifically stated that imaatfaal brought them up first but is still connected to information in the first post, which you are in denial over. Conclusion: Nothing remotely resembling "productive" or "helpful" has resulted from your presence on this thread and it is being prolonged in a negative direction as a result of your irrationally aggressive attitude, therefore the best course of action for the site and thread members is for you to leave the thread. At this point, you do not have to state any fault, you just have to stop posting and everyone can move on.

When you say neutron yield...do you mean the 2.07 to 2.47 neutrons emitted per fission event that you can find on google?

It's not about the critical mass of U235, but apparently it's how the neutron multiplication factor must stay the same regardless of the frame of reference. It is essentially what everyone has been saying and assuming I didn't understand, except in addition to just stating the obvious like that the number of atoms is intrinsic, I'm trying to actually show it mathematically in conjunction with the conditions of a nuclear detonation. Even though a chunk of U235 can be accelerated to a speed such that the length contracts and mass appears to increase in such a way that the density appears to be critical, that doesn't necessarily mean that k is greater than or equal to 1. But, the mathematics to actually show that involves both nuclear physics as well as relativity. You said no one bothered transforming such values into other frames, well perfect, that will make the paper even more unique.

Once again I find your assumptions irrational. Though it is true I did not answer the first question because it seemed to me like you were being purposely troublesome as your question can not possibly be rooted in an understanding of my post, I did answer the question about the relationship of hyperbolic trig functions and Euler's identity. I'm not quite sure why you'd lie so blatantly when it can easily be seen. If x is imaginary, then ix would yield i^2 as a coefficient of the variable, giving a negative answer. I also stated my situation clearly but perhaps repeating it a different way will allow you to see what I am saying. Imatfaal originally brought up hyperbolic functions and after I saw that, I was reminded of how hyperbolic functions are connected to Euler's identity and thus how I should have included them into the criteria of what I was searching for, which by no means excludes the usage of Euler's identity. I said it before and I will say it again. I did not, and was not looking for solutions limited to real answers. You made the mistake of wrongfully assuming I desired numerical solutions which were also limited to real numbers. The word "real" does not exist in my original post and I made no effort whatsoever to state I was looking for only real numerical solutions. You are simply wrong.

Why is an infinite medium a poor assumption in the vacuum of space? Also, did you read the paragraphs where I explained the chunk of Uranium is traveling considerably close to the speed of light? That's where relativity comes in. At the very least, I know for a fact that time dilation would affect the half life and specific activity when measuring it from Earth. If you can help me with this last hurdle, I could reference you in the paper unless you wish to refrain from that. Although I don't think it is formally permissible to omit a source of information. If I were to reference you as an advisor (or something else??), how would you want it?

That's not really a personal attack, it's an objective assessment of your involvement in this thread. I don't think you read the first post, nothing about that is an attack, just a statement of negligence. Mentioning hyperbolic trig functions also isn't a contradiction because it does not exclude the usage of Euler's identity. Hyperbolic trig functions were mentioned originally by imaatfaal and then later incorporated into my search criteria due to the fact that hyperbolic trig functions and complex exponents can be related to each other to find solutions to complicated trginometric equations. If you use the same website that you used before, Wikipedia, and search "Hyperbolic Trig Functions," you will find an article which shows the identity between hyperbolic trginometric functions and Euler's identity, such as cosh(ix)=(1/2)(e^(ix)+e^(-ix))=cos(x) Furthermore, I did not use the word "real" in my first post. You made a mistake in assuming that I was only looking for real solutions and continue to make mistakes with other irrational assumptions.

To be honest I don't think you actually read the first post. My question isn't anything to do with whether or not a solution is complex or imaginary, it's how to algebraically derive any solution in general using the identity e^(ix)=cos(x)+isin(x) and sinh and cosh.

Hmm. Maybe instead of focusing on probability, I could focus on conditions for criticality to show that critical conditions still aren't met even though it appears as though it's at a higher density. Could I use a 4-factor formula an in infinite medium to accomplish that, with all necessary factors affected by time dilation, length contraction and relative mass increase taken into account with that equation?

No I actually phrased it just fine, you are the one who assumed I wanted real solutions instead of complex ones, not me. Wolfram gives an answer using those identities, so there must be a way. I really can't think of a reason for you to be so obtuse over an inaccurate assumption.

I'm fine with complex/imaginary solutions, I just want to know how to calculate them at all. How would I solve the equation I listed for the variable? I can't think of any conventional method that would work, and Wolfram once again gave me complex answers based off of those Euler exponents and things that seem like they were derived from sinh and cosh.

Please read my text more thoroughly. I already stated that a bomb that does not detonate in one frame will not detonate in another. As for sensai, I keep trying to tell you that yeah, I agree you still have the same number of atoms as you had at rest, but the coordinate system of an object still rotates, and it rotates in such a way that the effects of time dilation and length contraction are the same as in a gravitational field, so it appears as though mass increases, even though the number of atoms stays the same.