the tree

That's just how cosh is defined.

What? Even if the first statement, then still, what?

It would be helpful if you could include the Maple code rather than leaving us to transpose it. I figured it was something like: ( -2*exp( -3*f(t))*Diff(f(t),t)^2 +2*exp( -3*f(t))*Diff(f(t),t,t) +3*exp( -2*f(t))*Diff(f(t),t)^4 +2*sqrt( (-1+ exp(f(t))*(Diff(f(t),t)^2 ))*exp(-4*f(t)) ) *sqrt( -(exp(-f(t)) - Diff(f(t),t)^2*exp(-f(t)) )) *exp(-f(t)) )/ ( (exp(-f(t)) - (Diff(f(t),t))^2 ) *sqrt( (-1 + Diff(f(t),t)^2 )*exp(-4*f(t)) ) *sqrt(-(exp(f(t)) - Diff(f(t),t)^2 ) * exp(-3*f(t)) ) ) = 0; Which could easily be wrong. From there I would suggest manually simplifying the expression (simple things like [imath]\sqrt{a}\sqrt{b}=\sqrt{ab}[/imath]) and maybe replacing the [imath]e^{-n f(t))}[/imath] with a new function, then letting Maple solve the pair of them as a system of equations.

If you're trying to solve for a(t) then you just need dsolve([ec1,ec2],a(t));. Merged post follows: Consecutive posts mergedAlternatively, if you're trying to solve for all three functions at once (not that I'm sure you'd get a solution at all) then you use the PDEtools package to help with that. The relevant help file for doing so is called dsolve,system. PDEtools[declare]((A,a,f)(t), prime = t); ec1 := #whatever ec2 := #whatever dsolve([ec1,ec2]); Note that'll have to use diff when defining the equations this way, rather than the inert Diff.

Mathematical Biology is also an important field: for example competing populations can be treated as dynamical systems, which is a very broad topic, and many mathematicians study these in order to understand the effect that environmental changes may have on indicator species or to track the track the spread of a viral epidemic.

When you're trying to find a value for this, two multiplications and a square root is preferable to two multiplications, a square root and a division. Partially because any operation is going to take up computation time and cause a loss of accuracy and also because division in particular is very sensitive to small inaccuracies.

Or,you could read some relevant history. How do you not get the difference? Some evidence being suppressed is not the same as a giant absence of evidence being acknowledged. The issue it not that it is unprovable, it is that there is no reason whatsoever to suspect that there is a single grain of truth to it and many reasons to suspect otherwise.

That's a pretty interesting question, good luck with it. In case you want to read around it, the general form of a Fibonacci sequence is a called a Lucas sequence. Also, if you look at the Wikipedia article on the Fibonacci sequence you'll find an inductive proof for the limiting ratio and it'll confirm that it works with any two seed numbers.

@triclino Yes it is. Although maybe part (1) should state that (2k2+2k) is in N, just so your point is obvious. @timo Maybe your browser isn't rendering the character 'ε' but it is definitely there. Aside from that there is nothing wrong with their presentation. Don't be harsh just for the sake of it.

The time it takes to solve a problem (shortest path or whatever) is often expressed in terms of a differential.

I took the no-splashing as just a rephrasing of the standard no-funny-business clause for logic puzzles, so that it'd cover evaporation. Either way, I think it'll fill slightly faster. Since being kept stationary the distance from the tap will always be at the maximum ("bottom of the stream"), but being moved the average distance will be around the average of the minimum and maximum distance.

If you're solving a system of equations then they should be written as an array in the first parameter, the second parameter should be the function you're trying to solve for. (same as using the solve command for non-differential equations). e.g. ec1 := Diff(f(t),t)^2 - 4*f(t) = 0; ec2 := Diff(f(t),t,t) - 2 = 0; dsolve([ec1,ec2],f(t));

This. But you can use fairly simple maths to highlight various details, such as edge detection to spot major discontinuities and noise reduction to disregard less significant ones.

eep, yeah, of course. I'm not a massive fan of mixing d's and and flicks in notation, but each to their own I guess.

Just one more point: if a suggested algorithm is claimed to be able to do something that is definitely impossible - there is no reason to inspect the algorithm before stating that it cannot work.

The chain rule is [imath]\frac{d}{dx}y(u(x))=\frac{dy}{du}\frac{dy}{dx}[/imath] or, in function notation, [imath](f \circ g)' = (f'\circ g) \cdot g'[/imath]. There shouldn't be an addition there (there is one in the product rule - but you don't need that). Try finding functions [imath]f[/imath] and [imath]g[/imath] so that you can write [imath]w := f\circ g (t) = f(g(t))[/imath] (depending on your preferred way of writing functions). Tip: don't bother differentiating anything w.r.t. [imath]x[/imath] or [imath]y[/imath].

If you want to discuss the Vedic square properly then you could ask a mod to move this thread to the mathematics board, or start one yourself there. There's probably a whole load of algebraic and geometric properties that it's beyond my ability to notice just by inspection.

There's quite a few nice patterns that don't take adding any information. Taking either 0-9x0-9 or 1-8x1-8: Reflections about the vertical or horizontal centre lines interchanges pairs of numbers, specifically the ones that add up to nine. e.g. 2 and 7 switch places with a reflection, whereas the 9s are symmetric over those reflections. The shape formed by each member of one of those pairs of numbers is unique to that pair so the square forms 5 distinct shapes each with rotational symmetry of degree 4 (or 10 with degree 2). Quite a bit of information there already - little need to add more.

Subjectively speaking, maybe. But take a serious look at the gamma function over the complex plane and tell me there isn't a beauty about it. Not all fractals take complex numbers, nor are they frequently very complicated. Take the logistic map for instance - chaotic but from a very simple system and I would say the majority of fractals aren't based around polynomials - think Sierpiński gaskets, space filling curves, Lorenz attractors...

There are certainly more interesting plots out there. There's various fractals, population models, the Gamma Function even nearly random plots have their own charm.

G. H. Hardy? Oh yeah, of course. I was just saying that it's not massively important that the public should know about the abstract philisophical musings when there's all the all the everyday groundwork to be laid, which is not only useful to more people, but interesting to more people as well - same with all the sciences really. Oh, okay.

I'd hope that US types at least get to see it when it comes out on DVD - is this just corporations wanting to shy away from controversy? Even though there shouldn't be any, it's about a guys life.

Indeed - all the everyday applications are really pretty important, and worth learning about for anyone but the purely academic side is just that.

Okay, I'll bite. This post has been bugging me for a while and honestly I just want to know what on earth your point is. I think I see what you mean here, but I wouldn't agree with your choice of words. I find that lots of people see maths as something incomprehensible and detached from their natural understanding. Okay this bugs me, the philosophy of maths is fun - and the physchological aspects are interesting but really, I couldn't care less what other people know about them. If there are things that the vast majority of people should know about maths then it should be things like what the word average means, why breaking distance is proportional to the square of velocity and why casinos have upper limits to the amount you can gamble even though the odds are always in their favour.

Well, that much would be silly, but finding a problem where the 2d vector isn't on the (x,y,0) plane* seems quite natural (the first thing that comes to mind is an aircraft flying with a fixed (non-zero) altitude versus a surface to air missile). Which is why it'd need to be specified to avoid confusion. *osculating plane? Is that the word?