We characterize them into types like we do with integrals but the list is ultimately endless.
Mathematics is a creation of the human brain; the Earth may have revolved around the sun before mankind but the mathematical constructs of gravity, in my humble opinion, did not. And of course the...
I have something to add based on my own experience: the road to the right one is littered on both sides with wrong ones. Embrace this principle and learn to work with it. Hall (a PF member) once said, "you don't just stare at the problem and wait for the answer to pop into your head...
I know firsthand some admirable qualities Marcus exhibited on this forum. Perhaps if the members wanted, we could offer an award yearly to someone that exemplified them.
I do not believe that's true.
Look no further than “At Home in the Universe,” by Stuart Kauffman:
And if I may be allowed to make a comment in the interest that some reading this may be stimulated to pursue this further, Kauffman mentions "phase-transition." That phenomenon is very common...
The dynamics necessary for the origin and evolution of life existed before life and are independent of life. The inconsequential trappings of biology and chemistry instantiating life on Earth are just happenstance and convenient for our particular planet. Consider a concrete example: termite...
Here's the best primate tree I can find. See, right there in that little nook near proconsul. That is, using best available evidence, the common ancestor...
There is a discussion of traffic flow and a PDE describing it in the textbook "Basic Partial Differential Equations" by David Bleecker and George Csordas. Should be available in any university library and may help you better understand the dynamics which involves a cusp catastrophe (a wreck).
Is that not correct in the sense that (chemical) survival and reproductive success were relevant, crucial, and necessary for the origin of life on Earth even during pre-biotic evolution, even before any signs of life existed on earth, and therefore the fundamental tenant of evolution, survival...
I do not feel this is an unreasonable question to ask. We know the Universe is massively non-linear and further, that non-linear dynamics can create infinite diversity. Consider the Lorenz attractor (the owl-eyes icon of Chaos Theory), a set of three coupled non-linear differential equations...
Well, I don't wish to be cold and callous but it is the mechanism of infection that most fascinates me: how (exactly) does the virus cause microcephaly? Someone above mentioned an autoimmune response. Are the neurons in microcephaly being destroyed soon after they are created or are they...
Hi guys,
Been keeping up with your comments. This is the first reference I've seen to suggest some are convinced the virus is causing microcephaly:
http://www.foxnews.com/health/2016/02/01/brazils-zika-virus-outbreak-worse-than-believed-health-minister-says.html?intcmp=hphz02
What evidence is...
And sorry if I misled you. No, what I think would be nice is for someone to take interest in this and generate the equilibrium surface and then add something: draw little bacteria and single-celled organisms on the top surface (the pre-Cambrian) and then trajecting through the catastrophe...
But I want someone else to do it. Well, I've done it plenty of times, even the swallow-tail for that matter but I digress. No, I want someone to be taken by this discourse and change history forever. :) Butterfly effect right? :)
Yes. Thanks for that. It's certainly the surface but it's the "how" part, precisely, "exactly how" part. That is, given y'=-y^3-by+c, generate the catastrophe surface preferably in Mathematica in an elegantly-pleasing format (a nice-looking graph that clearly shows the cusp). Can we plot it...
Ok. Thank you kindly for that. Yes. I'm having problems with it. Also, I think perhaps the one I'm most familiar with is:
\frac{dy}{dt}=-y^3+by+c
I was wondering then since I'm having problems with it, would someone be so kind to generate the catastrophe surface, identifying the...