TITLE: Bounded Convergent Sequence
QUESTION [0 upvotes]: I am trying to put together a step of an implication problem:
L-1 < Sn < L+1 implies |Sn| < |L| + 1
I am not following how to get this conclusion.

REPLY [2 votes]: $L-1 < S_n < L+1$ is equivalent to $|S_n-L|<1$. From the triangle inequality obtain $$|S_n|-|L|\leq |S_n-L|<1,$$ from which you get $|S_n|<|L|+1$.


text:
\section{Examples}
\label{sec:app}
In this section, we will discuss several examples of $\tp{k}$ systems, which we use to illustrate our theory as well as to point out future directions and relationships to recent related work.

\subsection{Neuronal networks}
Physiological neuronal networks are often modelled as a parallel interconnection of externally positive subsystems of type
\begin{equation*}
G_a(z) = \frac{k_1}{z-p_1}, \quad G_b(z)= \frac{k_2}{z-p_2} - \frac{k_3}{z-p_3},
\end{equation*}
followed by a non-decreasing activation function such as the sigmoid function. \textcolor{blue}{Rodolphe: Can you add a reference and perhaps background on ion channels/fading memory?} We will now see that such a systems can have various degrees of local extrema diminishment from past input to future output.

To see this, we first note that non-decreasing non-linearities are local extrema diminishing by the chain rule. Further, Hankel variation diminishing systems are due to linearity also local extrema diminishing, which is why we only need to analyse the variation diminishing property of the linear system part. If all subsystems are of type $G_a(z)$, it follows from Proposition~\ref{prop:toep_inf_pos} that the neuronal network is Hankel totally \ovd. In many other cases, however, the network can be at most \ovd{k} for finite $k$ by Theorem~\ref{prop:con_minor} and \ref{thm:decomp_tpk}.

For example, let $G(z) = G_a(z) + G_b(z)$ with all $k_i,p_i > 0$, $p_i$ distinct, $k_2 \geq k_3$ and $p_2 > p_3.$ Then, $G(z)$ is Hankel $\tp{1}$ as a sum of externally positive systems, but at most $\tp{2}$ by Theorem~\ref{thm:decomp_tpk}. To decide the latter, we need to check by Theorem~\ref{prop:con_minor} and \ref{thm:tpk_poles_coeff} if
\begin{equation*}
G_{[2]}(z) = \frac{r_1r_2 (p_1-p_2)^2}{z-p_1p_2}- \frac{r_1r_3 (p_1-p_3)^2}{z-p_1p_3} -  \frac{r_2 r_3 (p_2-p_3)^2}{z-p_2p_3}
\end{equation*}
\normalsize
is externally positive. It is easy to see that this is the case if and only if $p_1p_2 \geq p_1p_3$ and $r_1 r_2 \geq r_1r_3 + r_2 r_3$.

As the role of the Hankel \ovd{k} property seems to be neglected in the analysis of such networks, our work can be considered a first step towards this novel understanding.

\subsection{Heavy-ball method}
The so-called heavy-ball method for (convex) optimization was designed by Polyak to damp the possibly high number of local extrema in the iterates of the well-known gradient descent approach \cite{polyak19641methods,ghadimi2015global}. For a convex function $f: \mathbb{R} \to \mathbb{R}$, the iterates $x(k)$ of this method are given by the closed-loop system
\begin{align*}
x({k+1}) &= x(k) + \alpha u(k) + \beta (x(k) - x({k-1}))\\
y(k) &= \left . \frac{d}{dx}f(x) \right |_{x = x_k} \\
u(k) &= -y
\end{align*}
for constant $\alpha,\beta > 0$. Since the linear system from $u$ to $x$ reads $G(z) = \frac{\alpha z}{(z-1) (z - \beta)}$, it is Toeplitz totally \ovd{} by Proposition~\ref{prop:toep_inf_pos}. Further, as the derivatives of convex functions are non-decreasing \cite{Boyd2004}, it follows as above that the entire open loop system from $u$ to $y$ is local extrema diminishing.

To also analysis the closed-loop system, let us assume that $f$ is quadratic. Then, $y(k) = ax - b$ with $a > 0$, i.e., $y$ is the response of a linear closed-loop system to a step with height $b$. It is easy to show that the closed-loop system is Toeplitz totally \ovd{} if and only if $\beta \geq (\sqrt{a \alpha}+1)^2$. This observation verifies the intuition that the required amount of momentum (damping) for variation diminishment scales with the the gain of the gradient, which in the optimization literature is associated with the Lipschitz constant of the gradient \cite{ghadimi2015global}. Then, choosing $x(0) = 0$ results in iterates without local extrema. In particular, local extrema from noisy gradients (which may be modelled as time-varying $b$) can be diminished independently of the noise distribution. Since suggested choices for $\alpha$ and $\beta$ \cite{ghadimi2015global,polyak19641methods} do not necessarily result in closed-loop total \ovd{}, an investigation of the effects of Toeplitz $\tp{k}$ designs to convergence rates and robustness seems highly desirable.

This application demonstrates the importance of variation diminishing theory in optimization and motivates the extension of our theory to non-linear negative feedback interconnections. Note that in case of multi-dimensional iterates, the algorithm as well as our analysis applies component-wise. Further, since $G(z)$ permits a minimal realization $(A,b,c)$ with totally positive $A$, it becomes evident how our analysis complements the study of autonomous $k$-positive systems \cite{weiss2019generalization,margaliot2018revisiting}.

\subsection{Monotone Cyclic Feedback Systems}
\emph{Monotone cyclic feedback systems} are autonomous systems with the properties
\begin{align*}
& x_i(k+1) = f_i(x_i(k),x_{i-1}(k)), \; i = 1,\dots,n, \; x_0 := x_n, \\
& \delta_i \frac{\partial f_i(x_i(k),x_{i-1}(k))}{\partial x_{i-1}(k)} > 0,
\end{align*}
where $|\delta_i| = 1$ \cite{MalletParet1990}. In other words, these systems are closed-loop systems, whose open loop dynamics consists of the serial interconnection of non-linear first order systems, where each non-linearity $f_i$ is increasing in $x_{i-1}$. System of this type occur frequently in biology, e.g., if each first order system is the firing-rate model of a neuron, then $f_i(x_i,x_{i-1}) = p_i x_i + b_i s(x_{i-1})$, where $p_i, b_i > 0$ and $s$ is an increasing activation function.

While a Poincar{\'e}-Bendixon theorem for these system has been known since~\cite{MalletParet1990}, its simplified Lyapunov theory based on sign-changes has only been noted recently \cite{avraham2020dynamical}. The present work opens up the complementary route towards an open loop analysis, which as noted in previous examples invokes Toeplitz totally positivity. As in the discussion of the heavy-ball methods, this path would make it easy to incorporate exogenous inputs such as references and disturbance signals as well as to extend the open-loop dynamics to Toeplitz $k$-positivity.

config: arxiv


TITLE: Question about complex path integral
QUESTION [1 upvotes]: Let $$f(z) = \int_{z_0}^z \frac{d\zeta}{\zeta^{1/2}} $$
where the integral taken over any path in $\mathbb{C}-\{iy :y \le 0\}$
(Since it is simply connected and $\zeta^{1/2}=e^{\frac12\log{z}}$ is holomorphic in that region.) In my book(Stein) the following is stated but I cannot understand why. Could anyone explain it?

Since $\zeta^{-1/2}$ is integrable near $0$, $f$ is continuous at
  $0$, and this implies that the integral can be taken along any path in
  $\mathbb{C}-\{iy :y \lt 0\}$.

REPLY [1 votes]: If you have a path that includes $0$, you can do an improper integral, and integrate along the path, excluding a small $\epsilon$-neighbourhood around $0$.  You will find that the limit as $\epsilon\to 0$ will exist.


TITLE: Em radiation incident on an atom
QUESTION [0 upvotes]: What happens when the EM wave is incident on a atom? Do the de-localized electrons in the atoms or the valence band electons move in the direction of the electric field? Also when the electrons move, does the magnetic field in the EM wave also exert a force on the moving electrons?

REPLY [0 votes]: Note: this is a copy of my answer to OP's question on electronics.stackexchange.com. This might use notation that is not very typical in a physicist's context. It looks like the equations I've learned in my EE studies.

Remember the Schrödinger equation from which you know what a "delocalized electron" is?
$$ j\hbar\frac{\partial}{\partial t} \Psi(x,t) = \left [ - \frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x,t)\right ] \Psi(x,t) $$
Exactly in that you insert your time-dependent (vector) potential $V(x,t)$. You'll see that the derivative of the \$\Psi\$ now has a harmonic component.

Do the de-localized electrons in the atoms … move

Um, we really need to be careful when we say that something without a location moves... but, yes, the probability mass $\lvert\Psi(x,t)\rvert$ is time-dependent under a changing potential at a fixed point $x$. So, the electrons' loci "move" (with spoken-out quotation marks).

Also when the electrons move, does the magnetic field in the EM wave also exert a force on the moving electrons?

Err. Now I'm not sure how we can interpret "moving": The electron's probability of occurrence "move"; the electrons themselves "stochastically interact"; this is quantum regime, you can't first call something "delocalized" and in the next sentence act as if it was a classical (i.e., non-quantum) particle.
To me, this sounds like you're trying to apply Maxwell's equations to quantum things. Don't. Schrödinger's equation is what you'll need deal with.


TITLE: Why having the loss matrix to be 1 - identity function yields a loss of $1-p(C_l|x)$ ? (Bishop exercise 1.24)
QUESTION [0 upvotes]: In the solution for exercise 1.24 in Bishop's PRML solutions book I see the following statement:

For a loss matrix $L_{kj}=1-I_{kj}$ we have
  $\sum_kL_{kl}P(C_k|x)=1-p(C_l|x)$

Now I understand where the $1$ is coming from:
$\sum_kL_{kl}P(C_k|x)=\underbrace{\sum_kp(C_k|x)} - \sum_kp(C_k|x)$
The item in the underbrace equals $1$ because it's just the sum of the probabilities. But how is the second item equal to $p(C_l|x)$ ?

REPLY [1 votes]: The second term is actually $\sum_k\color{red}{I_{kl}}p(C_k|x)$ which equals $p(C_l|x)$ as $I_{kl}=1$ only if $k=l$, else $I_{kl}$ is zero.


TITLE: Prove that a maximal planar graph/triangulation of $V>3$ has minimum degree = $3$
QUESTION [4 upvotes]: I'm trying to prove that any maximal planar graph that has more than 3 vertices will have a minimum degree of 3, that is any vertex in such a graph has greater or equal to 3 edges connected to itself. So it doesn't have to have a degree 3 vertex. I want to prove that when there's a degree 2 vertex in a planar graph G (say the rest of G is already maximal), why it is always possible to add at least another edge to the degree 2 vertex to form a maximal planar graph (without crossing).
I can visualize a worst case where a new vertex (u) is added to a maximal planar graph (G) and lands in a triangle formed by three vertices that are previously added to the graph. The only option for u is to connect with the three vertices of the triangle since a fourth edge trying to connect u and the "outside" will cross at least one edge, resulting in a non-planar graph. If u lands outside G, then it should be able to connect to at least three vertices without crossing any edges since G's outer face should also be bounded by three edges, which means that at least three vertices on the verge of G can be connected to u?
How to formally prove the minimum degree of a maximal planar graph G (with more than 3 vertices) is 3?

REPLY [5 votes]: Given a planar graph, embedded in the plane, you want to add a number of edges to get a maximal planar graph, and you want to be sure that resulting maximal planar graph won't have vertices with degree $=2$.
From this Wikipedia page:

A simple graph is called maximal planar if it is planar but adding any edge (on the given vertex set) would destroy that property...

The process of adding edges is straightforward - you need to find all faces, bounded by more than $3$ edges (including the external face), and triangulate these faces by adding new edges.
Let's consider the case, which looks problematic to you - a vertex $u$ with degree $2$, for which it's impossible to add a new edge, incident to the $u$. You have two vertices $v$ and $w$, adjacent to the vertex $u$, and two faces, partially bounded by edges $\{v,u\}$ and $\{u,w\}$. If a new edge, incident to the vertex $u$, can't be added, then both these faces should have an edge $\{v,w\}$, which prevents this addition. So, the graph contains two parallel edges $\{v,w\}$, which is not possible, because we consider simple graphs only.