TITLE: Modifying a function $X \to \mathbf{C}$ into a nowhere vanishing map.
QUESTION [2 upvotes]: Suppose $X$ is a normal space (compact, if this would help here), and $f : X \to \mathbf{C}$ is continuous. Set $K = f^{-1}(\{0\})$ and let $E \supset K$ be some open neighborhood of $K$ such that $f$ is bounded away from $0$ on $X \backslash E$.
My question is the following. Is it possible to construct (or prove the mere existence of) a continuous map $g: X \to \mathbf{C}$ such that $g$ vanishes nowhere and it coincides with $f$ on $X \backslash E$? My intuition tells me yes.
However, I have difficulties constructing such a $g$.
If $u : X \to [0,1]$ is an Urysohn function which vanishes outside $E$ and equals $1$ on $K$, setting $g := f + u$ will work when $f$ is real-valued and positive on $E$, but otherwise it might brake down. This can be avoided by writing $f(x) = |f(x)| e^{ i \arg f(x) }$, and then setting $g(x) := ( |f(x)| + u(x) ) e^{i \arg f(x) }$, however this is ill-defined whenever $f(x) = 0$, so then the question would be if $x \mapsto e^{i \arg f(x) }$ can be continuously extended from $X \backslash K$ to $X$, in such a way that this extension does not vanish. If we think of this as a map from $\overline{X \backslash K}$ (continuously extended to $\partial K$) into $\mathbf{C}$, then Tietze's extension theorem gives us an extension, but this might vanish on $K$. This seems like a dead end to me.
There are probably some other constructions that I'm overlooking. Or maybe there exists some cunning counter-example?
Any advice is welcome. :-)

REPLY [2 votes]: Here is a counterexample: Let $X = \mathbb{C}$, $f(z) = z$, and $E= \{ z \in \mathbb{C} : |z| <1\}$. If such a $g$ would exist, then it would be a map of the closed unit disk into the punctured plane with $g(z)=z$ for $|z|=1$. Then $h(z) = \frac{g(z)}{|g(z)|}$ would be a continuous retraction of the closed unit disk onto its boundary, and we know that such a map does not exist.


TITLE: What is the difference between Noise and chaos?
QUESTION [0 upvotes]: Two terms are mixed me , I heard about noise from stochastic process phenomena and i heard about chaos from dynamical system , then Is there someone who can help me to get difference between noise and chaos ?

REPLY [0 votes]: The first distinction you have to make is between chaotic and deterministic dynamic systems.
Even though this may be surprising at first, these attributes are best applied to model systems, and not to reality or subsystems thereof.
A model dynamics is stochastic if randomness (or noise) influences its temporal evolution.
Otherwise it’s deterministic: The initial conditions completely determine the system’s future.
Some deterministic dynamical systems can be chaotic, which means that their temporal evolution strongly depends on initial conditions.
By contrast other deterministic dynamical systems (fixed-point dynamics or periodic dynamics) do not have this property.
With other words:

For regular deterministic systems, identical initial conditions lead to an identical outcome and similar initial conditions lead to a similar outcome.
For chaotic (deterministic) systems, identical initial conditions lead to an identical outcome and similar initial conditions (usually) lead to a drastically different outcome.
For stochastic systems, identical (and similar) initial conditions (usually) lead to a drastically different outcome.

Now, what about reality?
Every real system is inevitably subject to phenomena like thermal noise, air turbulence, or similar, which we can only treat as noise¹.
Thus the most accurate model for every real system would be stochastic² – but this model is also unreachable and infeasible.
Instead, when choosing a model we have to make a trade-off between feasibility and accuracy, where we usually go for the simplest model that reproduces whatever phenomenon we are interested in with sufficient accuracy that we can understand or sufficiently predict it.
Depending on the system and our needs, we may arrive at a deterministic or stochastic model.
For example, you would use a deterministic model for a pendulum in most cases, while the most successful model for reality at the microscopic level (Copenhagen-interpretation quantum mechanics) is stochastic.
A real system is usually called chaotic if a reasonable model of it is chaotic, for example the double pendulum.

¹ Whether these phenomena are actually deterministic or stochastic doesn’t really matter, since they are extremely high-dimensional and complex and every sufficiently complex deterministic system cannot be practically distinguished from a stochastic one.
² A prominent exception may be the entire universe, but we cannot possibly decide that one. We only know some restrictions of a possible determinism from quantum mechanics.


TITLE: Find $\sum_{i=[1,\ldots,n], j=[1,\ldots,m]: i\neq j }\frac{1}{j}$
QUESTION [3 upvotes]: I have the following double   $\sum_{i=[1,\ldots,n], j=[1,\ldots,m]: i\neq j }\frac{1}{j}$.
If I ignore restriction $i \neq j$ then
$$
\sum_{i=[1,\ldots,n], j=[1,\ldots,m] }\frac{1}{j}=n \sum_{ j=[1,\ldots,m] }\frac{1}{j} \le n(1+\ln(m))
$$
Now, I am not sure how to it with this restiction?

REPLY [6 votes]: With the restriction you have
$$\begin{align*}\sum_{j=1}^m\;\sum_{i\in[1..n],i\ne j}\frac1j&=\begin{cases}
\sum_{j=1}^m\frac{n-1}j,&\text{if }n\ge m\\\\
\sum_{j=1}^m\frac{n-1}j+\sum_{j=n+1}^m\frac1j,&\text{if }n<m
\end{cases}\\\\
&=\begin{cases}
(n-1)H_m,&\text{if }n\ge m\\\\
(n-1)H_m+(H_m-H_n),&\text{if }n<m
\end{cases}\\\\
&=\begin{cases}
(n-1)H_m,&\text{if }n\ge m\\\\
nH_m-H_n,&\text{if }n<m\;,
\end{cases}
\end{align*}$$
where $H_n$ is the $n$-the harmonic number.