30
votes
Is there a mathematical and information theoretic explanation for this cube packing phenomenon?
• Concerning question 2, you might want to take a look at Simulation of cubical particle packing under mechanical vibration (2016). The precise effect mentioned in the 2017 paper is not considered in ...
20
votes
Is there a mathematical and information theoretic explanation for this cube packing phenomenon?
I doubt that a mathematically rigorous explanation of the phenomenon discovered in that paper exists using today's technology. While mathematical statistical mechanics is a well-developed field of ...
14
votes
Is there a mathematical and information theoretic explanation for this cube packing phenomenon?
The physical reason is that the cubically packed state has lower gravitational potential energy than the jammed random state. The overall process is analogous to annealing, although the reduction in ...
14
votes
Symmetric polynomials that detect positivity
The elementary symmetric functions $1=e_0,e_1,\dots,e_n$ will
suffice. Clearly $e_j>0$ if each $a_i>0$. Conversely, let $P(x)=
\prod_{i=1}^n(x+a_i)$. If some $a_i<0$ and all $e_j>0$ then
$...
6
votes
Accepted
How do solutions of a PDE depend on parameters?
If $p=2$ and $\sigma_1, \sigma_2 \in L^\infty(\Omega)$ one has the estimate
$$\|\nabla u_1 - \nabla u_2\|_{L^2(\Omega)} \leq C \|f\|_{H^{1/2}(\partial\Omega)} \|\sigma_1 -\sigma_2\|_{L^\infty(\Omega)}$...
6
votes
Accepted
When does the maximal root of this polynomial have unit magnitude? Prove an inverse linear relation between parameters
Solutions with $|x|=1$ are given by $x=e^{i\theta}$ and $\alpha=2\sin(\theta/2)$ with $\theta=\pi/(2\tau+1)$. From this we get $1/\alpha=(2\tau+1)/\pi + O(1/\tau)$.
5
votes
How to study the global stability for this 3D system?
This is more a long comment than an answer, but I know that a similar problem, also originated from (macromolecular) biology, was studied and solved by Gaetano Fichera, Maria Adelaide Sneider and ...
5
votes
Accepted
Is the convergence of $\dot{x}=2A(t)x$ faster than that of $\dot{x}=A(t)x$?
A simple counterexample can be constructed as follows: take
$$ A(t) = \pmatrix{0&\tfrac{\pi}{2}\\-\tfrac{\pi}{2}&0} \qquad \text{for } t \in [2n, 2n+1) ,$$
and
$$ A(t) = \pmatrix{-\alpha&0\...
5
votes
Stability of nonsmooth, Lipschitz continuous, autonomous system of differential equations
I suggest looking at Lyapunov's direct method and possibly LaSalle's principle and their nonsmooth extensions.
http://www4.ncsu.edu/~schecter/ma_532_fa12/lasalle.pdf
http://ieeexplore.ieee.org/...
5
votes
Decay estimates for wave and Klein-Gordon equation in "generic" curved backgrounds
A longish bunch of remarks: there's a big leap that doesn't make sense between your question and your motivation.
On the maximally extended Schwawrzschild solution there is no decay to the wave ...
5
votes
Is there a mathematical and information theoretic explanation for this cube packing phenomenon?
This should be seen as a comment, but it is too long for a comment:
One aspect, that has not been addressed in the answers, is the influence of the container geometry. In the experiment, a ...
5
votes
What is the current status on methods to find limit cycles?
There are many principles to show the existence of periodic orbits in high- and infinite-dimensional systems, in particular, there are generalizations of the Poincaré-Bendixson theorem. I mention here ...
5
votes
Accepted
Exact solution to a periodic linear ODE sought
Rather incredibly, your (corrected) system does have a closed-form solution, which I found with Maple's help.
$$ x(t) = 1+4\,\cos \left( 2\,t \right) +3\,\sqrt {8\,\cos \left( 2\,t \right) +
17}$$
$y(...
4
votes
What's "bad" about unstable sheaves?
The standard explanation is that if we want to work on the level of the coarse moduli space and not on the level of the stack itself, we need to tame the automorphism groups of objects (as Peter ...
4
votes
Accepted
Does gravity constant affect boundedness of solution?
$\newcommand\la\lambda$No. E.g., if $n=1$, $x_0\ne0$, and $f(u)=-u^2/2$ for all real $u$, then the solution
$$x(t)=\frac{x_0}{2} \, \Big(\frac{e^{\la_+ t}-e^{\la_- t}}{\sqrt{4 g+1}}+e^{\la_- t}+e^{\...
3
votes
Accepted
Marginal stability of discrete linear time-invariant system
The matrix $\mathbf{A}$ is similar to the matrix
$\begin{bmatrix}
1 & 1\\
0 & 1
\end{bmatrix}$, which is not marginally stable. Hence the original matrix is not stable.
In general, one can ...
3
votes
Accepted
Almost sure stability of a scalar, nonautonomous, nonlinear SDE
Given a realization of the Ornstein-Uhlenbeck process $X_t$, the SDE $$
d Y_t = Y_t (1- Y_t) X_t (dt + d V_t) \tag{1}
$$ is scalar, nonautonomous, and nonlinear. Note that (1) has two fixed points at ...
3
votes
Accepted
One problem about tower stability
Stability
For all $k$ up to the total number of blocks $n$: the center of mass of the top $k$ blocks must lie above the surface of the $k+1$ block supporting them.
3
votes
Properties of matrix exponential without using Jordan normal forms
I do not know if this is what you are after. The following shows that the Jordan form can be replaced by the Schur form. This feels a bit like cheating. Is this enough? The trickier parts are the ...
3
votes
What is the current status on methods to find limit cycles?
This contains the question of the existence of periodic orbits (ot periodic solutions) in dynamical systems, a very wide question indeed. In dimensions >=3 it is much more complicated than in ...
3
votes
Why is the largest invariant set the following?
Here's how I interpret that paper.
Why (1,1,1,1) is special
First, to make things simpler, I'll rewrite the equations by collapsing some constants as follows.
$x' = x[a(\frac{1}{x} - z) + b(\frac{1}{x}...
3
votes
Stability analysis of equilibrium point of non-linear ODE system with Jacobian going to infinity
This is what (in my opinion) the general setup is and what should happen and why. Note that I haven't proved anything yet. What I offer is just a back of envelope computation at the physicist level of ...
2
votes
Properties of matrix exponential without using Jordan normal forms
You can use a weaker version of Jordan normal form, namely an upper triangularization. There is a very straightforward conceptual proof that every square matrix over an algebraically closed field $k$ ...
2
votes
Conditions for convergence to non-isolated fixed points
Yes, there do exist sufficient conditions for asymptotic stability when the Lyapunov function is negative semi-definite, which I describe below.
Krasovsky's Theorem
Given an autonomous ODE $\dot x ...
2
votes
Relation between controllability and stability of PDE
These two problems are fundamentally different, because their data are. As far as linear systems are concerned, controlability is a property for an ODE $\dot x=Ax+Bu$, where $u$ is the control, while ...
2
votes
Accepted
Is the kernel of a Fredholm operator stable under perturbation?
The kernel of a Fredholm operator is not continuous with respect to small norm perturbations: For $t\geq 0$, consider the operator $S_t:X\times Y \to X\times Y$ defined by $S(x,y)=(tx,y)$ where $X,Y$ ...
2
votes
Boundary controllability of the heat equation and observation
In many situations, the null controllability of a given system is equivalent to the observability inequality of the adjoint backward system, which explain the presence of the normal derivative as the ...
2
votes
Accepted
Stability of eigenvectors for diagonal perturbations
Simple eigenvalues depend smoothly on the matrix by the implicit function theorem
applied to the characteristic polynomial (parameterized by the symmetric matrix).
For the general situation see this ...
2
votes
Accepted
On local attractivity of a coupled non-linear differential equation
Since the RHS is $2 \pi$-periodic in all variables, one can consider it on the three-dimensional torus $(\mathbb{R}/2 \pi \mathbb{Z})^3$.
Assume $a_{21} = a_{31}$. Then the two-dimensional torus
$$
...
Only top scored, non community-wiki answers of a minimum length are eligible
Related Tags
stability × 154ds.dynamical-systems × 56
differential-equations × 27
ag.algebraic-geometry × 23
ca.classical-analysis-and-odes × 21
ap.analysis-of-pdes × 19
oc.optimization-and-control × 12
reference-request × 11
linear-algebra × 9
vector-bundles × 9
fa.functional-analysis × 6
complex-geometry × 6
stochastic-processes × 6
hyperbolic-pde × 6
moduli-spaces × 5
stochastic-differential-equations × 5
pr.probability × 4
matrices × 4
mp.mathematical-physics × 4
na.numerical-analysis × 4
asymptotics × 4
kahler-manifolds × 4
real-analysis × 3
polynomials × 3
elliptic-pde × 3