Hello,
Given the three maps x_{n+1}=Ax_n with
A_1=\begin{pmatrix} 1&-1\\1&1 \end{pmatrix}, A_2=\begin{pmatrix} 1/2&1/2\\-1&1 \end{pmatrix}, A_3=\begin{pmatrix} 3&2\\5/2&2 \end{pmatrix},
describe the dynamics, and say whether or not the dynamics is hyperbolic.
Finding eigenvalues...
The A-formula gives me an equation containing cos^{-1} terms, which neither me or maple can solve. The integral-method is likewise not solveable for me.
A friend of mine told me that the problem has no simple analytic solution (squre root, fraction, etc.) to describe the relation of r and R...
A friend asked me the following question: Two circles with radii R and r are placed so that the one with radius r has its center on the circumference of the circle with radius R. How big should r be, so that the area of the overlap is exactly \pi R^2/2.
The simple solution would be to insert...
According to Wikipedia the the s-G equation is the Euler-Lagrange equation of the following lagrangian
\mathcal{L}(\phi) = \frac{1}{2}(\phi_t^2 - \phi_x^2) + \cos\phi.
Thus i suppose my question is simply how to derive this lagrangian for the mentioned mechanical system.
Hello,
I'd like to verify that the elastic ribbon model [ depicted here: http://www.math.h.kyoto-u.ac.jp/~takasaki/soliton-lab/gallery/solitons/sg-e.html ] is governed by the sine-Gordon equation. I suppose this can be shown by writing the lagrangian L = T - V and looking at the variation. The...
Yes; But isent this exactly what the Hopf Theorem states? I mean, if you find that it is possible to have purely imaginary eigenvalues to J with a positive derivative of the real part at some point I_0, then this I_0 is a Hopf bifurcation point, and i have thus proven that such a point exists...
Hello all,
i was given the following assignment: FitzHugh proposed the dynamical system
\dot{x}=-x(x-a)(x-1)-y+I
\dot{y}=b(x-\gamma y)
to model neurons. I is the input signal, x is the activity, y is the recovery and 0<a<1, \gamma>0, b>0 are constants. Prove that there is a critical...
=> or is this to be understood as a less strict requirement, because \frac{\gamma \beta}{\theta (\beta + \theta)} < \frac{\gamma \beta}{\theta^2} ? Therefore if \frac{\gamma \beta}{\theta^2} < 4 then also \frac{\gamma \beta}{\theta (\beta + \theta)} < 4 ?
The largest possible value of the second term, minus the (abs.) smallest possible value of the first is
\frac{\gamma \beta}{\theta^2} - 4 < 0 because \vline \ \max_{0<z<1} \ \frac{1}{z(z-1)} \ \vline = 4
which leads to \frac{\gamma \beta}{\theta^2} < 4 and not the expected...
The second term is largest when z \rightarrow 0, where it takes the values \frac{\gamma \beta}{\theta^2}. But as i see it, the first term is a problem since it goes toward \pm \infty (or undef.?) when z \rightarrow 0 \ \wedge \ z \rightarrow 1 respectively.? So can this term be bounded in...
Hello,
given is the function h(y_s) = \ln (1-y_s) - \ln y_s - \gamma + \frac{\gamma}{\theta + \beta (1-y_s)}
my job is now to show that h'(y_s) < 0, \forall y_s \in ]0,1[ when
\frac{\gamma \beta}{\theta (\beta + \theta)} < 4
I guess that all constants can be assumed to be real and...
I can't seem to solve this integral,
\int \frac{x^2+x+1}{(x^2+1)(x+1)}dx
Maple, however, solves is exact quiet easily, and i'd really like to see how this can be done "by hand".
Best regards.
Hello,
I'd like some ideas where to look in solving these two questions:
1) Are You presently sitting in an Electrical field? If yes, what's the size of the E-field?
2)A environmental requirement states that children arent supposed to stay in areas with electrical fields that exceed 600...
Hello,
i've been asked a hypothetical question about Keplers three laws: What if the gravitational force was proprotional to 1/r^3 instead of 1/r^2? And for one of the laws it apparently "isent easy to decide". My thoughts:
keplers 1. : my first thought was that this law was the "not easy to...
Alright, then if i got it right, you mean to say that
\sum_{n=1}^\infty \frac 1{(2n-1)^2(2n+1)^2} = -2 \sum_{n=1}^\infty \frac{1}{(2n-1)(2n+1)} + \sum_{n=1}^\infty \frac 1{(2n-1)^2}+\sum_{n=1}^\infty \frac 1{(2n+1)^2}
How how do you evaluate the two inverse squared sums... seems like i just...