# Visualizing cycles in row-major transposition encodings

2021-01-23, post № 239

mathematics, programming, c++, shell, #matrix, #encoding, #permutation, #rainbow

A matrix $A\in \mathcal{D}^{h\times w}$ of discretely representable entries $\mathcal{D}$ may be linearly layed out in memory using row-major order, concatenating successive rows into a contiguous $(h\cdot w \cdot\texttt{sizeof}\,\mathcal{D})$-bytes long array. Such a representation, however, is disruptive to the matrix’ two-dimensional nature: whilst horizontally neighboring elements remain neighbors, vertically neighboring entries are torn apart by insertion of $w$ non-neighboring elements. As such, on matrices naturally defined operations get distorted by this encoding.
One such inherently two-dimensional operation is matrix transposition. In the realm of matrices, $\mathcal{D}^{h\times w}\neq\mathcal{D}^{w\times h}$ are for nonsquare dimensions semantically different, being mapped to one another by transposition. Projecting onto their encoding, this semantic is lost and one is left with a permutation on memory $\bullet^\top\in\mathrm{Sym}(\{1,\dots,hw\})$.
To visualize this permutation, its cycle decomposition is computed of which each cycle is given a color of the rainbow dyeing this cycle’s corresponding two-dimensional pixels when interpreting its path on the underlying array in the semantics of the original matrix.

## Initial transposition cycles

Above listed are all visualizations for $1\leq h\leq 8,1\leq w\leq 16$, shuffled. Whilst some behave extremely regularly — for example square matrices’ transposition permutations decompose into transpositions —, others are wildly intricate. Each of them adheres to a rotational symmetry; the top left and bottom right are fixed points.

# Factoids #2

2020-12-26, post № 238

mathematics, #bijection, #calculus, #Lipschitz, #naturals

## VII) Cardinality coercion: $\mathbb{R}\hookleftarrow\{\mathbb{N}\to\mathbb{N}\}\twoheadrightarrow\mathbb{R}$

Claim. There exist both $\iota:\{\mathbb{N}\to\mathbb{N}\}\hookrightarrow\mathbb{R}$ together with $\pi:\{\mathbb{N}\to\mathbb{N}\}\twoheadrightarrow\mathbb{R}$.
Proof. Iota. Define $\iota:\{\mathbb{N}\to\mathbb{N}\}\hookrightarrow\mathbb{R}$ via
and observe any sequence’s reconstructibility by dyadic expansion.
Pi. Define $\pi:\{\mathbb{N}\to\mathbb{N}\}\twoheadrightarrow\mathbb{R}$ via
and observe any real’s constructibility by dyadic expansion.

Thus, $\{\mathbb{N}\to\mathbb{N}\}\cong\mathbb{R}$ in $\mathbf{(SET)}$ is shown.

# A Month of Pining

2020-11-28, post № 237

hardware review, #opinion, #diary, #freedom

Precisely one month ago, I opened a long awaited parcel shipped directly from Hong Kong: my very own Pinebook Pro had arrived. Eager to further my grasp on both software and hardware freedom, I unwrapped my mail and began to incorporate pine64’s flagship laptop into my workflow. Accompanying this journey, I wrote a diary which I share in this post.

Whilst a Pinebook may not boast with specs of performance, novel input methods or included licenses, it is part of a niche computing sector which values and upholds principal human rights even in the alien realm of microprocessor-based technology. Following my steadily growing discomfort regarding my computing equipment over the past months, I hoped for the Pinebook Pro to introduce me to a world of productive, in part minimalist and most importantly truly free computing experience. After a month of usage, I can confidently proclaim it to have delivered on this front more satisfactory than any other of my multitude of computing devices I have amassed over the years.

Especially in current times, it is calming to interact with an intricately mingled system of physical machinery and abstract commandments in which I can at least to a certain extent trust, lifting an ever-present veil of fear cloaking most boxes of bits. A Pinebook comes with software switches to turn off the built-in camera, microphone and WiFi card as well as Manjaro KDE Plasma running out of the box. Whilst I would prefer not having a camera or microphone built into my device at all and have a real physical switch to turn both off, as well as a more minimalist desktop environment, I am capable to entrust it my presence infront of it; in contrast to e. g. a proprietary phone lying on a tabletop nearby.

I am unaffiliated with pine64 [1] and purchased my Pinebook Pro myself. I wrote the entirety of this blog post including the diary seen below using my Pinebook [2].

A detailed account of my experience with my Pinebook is given by a diary I wrote over the course of November 2020:

# Halloween MMXX

2020-10-31, post № 236

imagery, poem, #halloween, #spooky

The sea is full of dangers,

size being only one.

Alluring depths and gruesome beasts

have intertwined in majesty.

The humble one left all alone

sought guidance in a spark of light.

# Crashing GCC with 63 Bytes

2020-10-15, post № 235

C++, error, #GCC, #g++, #internal compiler error

Enthusiastically following C++20’s new compile-time capabilities as well as awaiting more powerful ones being promised in upcoming standards, yet being stuck on C++17 [1], I try to take full advantage of the limited C++17-constexpr features. However, without static storage promotion, a constexpr std::string is all but a pipe dream, thereby necessitating the use of a wrapped static pointer; std::string_view.

One does not program against one’s compiler; one programs against the abstract machine defined by the standard.

# Book review: A Tour of C++

2020-10-03, post № 234

C++, book review, programming, #text, #opinion

Avidly writing C as well as golfing in it for over two years, I decided to venture out further in the realm of C-descendent languages, trying to get a foothold in C++.
However, contrary to many other languages with their own closed communities, style guides, best practices and general higher machine abstraction induced ease of use, C++ is a different language. With a rich history spanning multiple decades, millions of users from varying backgrounds and the necessity of enabling low-level hardware communication influencing its design and ability to be used effectively, it is a daunting task to acquire enough knowledge and experience to successfully write a standards-conforming C++ program which accomplishes a desired task.
Adding to this the plethora of textbooks supposedly correctly presenting their selected corner of the language and with that the immense danger of a poorly written textbook supplying a misleading first impression thus wrecking the fundamental knowledge of the language as well as the path ahead, I hoped for and succeeded in finding a thought-through, holistic and internally consistent as well as surely correct presentation of C++ in A Tour of C++, written by the inventor of and therefore a reasonable authority on C++.

What follows are my thoughts on the book A Tour of C++ [1], having begun to read it after only sporadically adjusting single lines in C++ golfs and having used it to facilitate a two thousand line C++ project [2] simulating a ficticious 32-bit architecture [3].
I am in no way affiliated with either the author Bjarne Stroustrup nor the publisher Pearson (Addison-Wesley) and purchased the book on my own.

# jblog has moved.

2020-09-05, post № 233

internal, #blog movement, #freedom, #politics

Having read Free as in Freedom [1] and being appalled at the digitalization-driven annihilation of once fought for values paired with de-facto government-mandated acceptance of this new brutal reality, I concluded to no longer be morally capable of supporting the web in its current form and thus began work on my own blogging software in April of 2020.
Now, beginning of September 2020, this project has reached a presentable state; having transferred all 232 jblog posts, reformatting and annotating them as well as improving minor syntactically or grammaticallly uneven sections, I can announce my blog’s movement:

https://www.jfrech.com/blog

If you choose to visit my blog, I can assure you that no trackers, cookies or malicious scripts will be employed against you. I will not try to collect a lot of data about your visit and the few inherently technically required data points in form of non-aggregated system logs will not be shared with third parties.

This 233rd post is the last to be published on https://jonathanfrech.wordpress.com, which will be shut down on the 31st of December 2020.
If you are interested in my blog, you can use my RSS feed to get a full list of my blog posts, updated as new are written. RSS is not like other subscription models — there is no server which knows your details and eagerly sends traffic your way to coerce you to a click. RSS requires you, the user, to request a feed and only then will transmit said feed.

Note that this blog’s content is subject to a clearly defined set of licenses specifying the terms and conditions with which it can be used.

# Setting Up a HTTPS Backend Using Go and Let’s Encrypt

2020-08-08, post № 232

Go, programming, #backend, #SSL, #systems administration, #VPS, #web

I recently needed to set up a dynamic web backend to both serve dynamically generated files as well as HTTP POST forms. Thus, I thought long and hard to find a solution that is both modern and versatile yet hassle-free to set up — since my systems administration skills are rather sparse. The solution I found is to write a Go webserver; implicitly concurrent and enabling a high level of control.

Of course, nowadays you cannot run a (esp. dynamic) website without HTTPS; every browser worth its salt will display potential visitors a message framing you as a ruthless criminal. Fortunately, Let’s Encrypt gifts you the required bits: an SSL certificate!
Since I need a URL for the upcoming snippet and am personally thinking of switching my own homepage from a webserver to a self-hosted VPS setup, I chose to use www.jfrech.com as an example domain herein.

% # ... installing certbot (for example via apt update && apt install certbot) ...
% certbot certonly --standalone --preferred-challenges http -d www.jfrech.com
% # ... cli certbot interaction ...

% # ... installing Go ... (for example via apt install golang) ...

% # ... periodically renewing certificates ...

Once certbot has issued a certificate (the URL has to have a DNS entry linked to the current VPS and port 80 has to be unoccupied), incorporating it into the server is made straight forward by http.ListenAndServeTLS:

func handle(w http.ResponseWriter, r *http.Request) {
fmt.Fprintln(w, <!doctype html><html><body><p>ip <code> + template.HTMLEscapeString(r.RemoteAddr) + </code> requesting <code> + template.HTMLEscapeString(r.URL.Path) + </code></p></body></html>)
}
func main() {
http.HandleFunc("/", handle)
log.Fatal(http.ListenAndServeTLS(":443", "/etc/letsencrypt/live/www.jfrech.com/fullchain.pem", "/etc/letsencrypt/live/www.jfrech.com/privkey.pem", nil))
}

Since now the server only listens on port 443, one can add a HTTP-to-HTTPS redirection on port 80:

func handleHTTP(w http.ResponseWriter, r *http.Request) {
target := "https://" + r.Host + r.URL.Path
if len(r.URL.RawQuery) > 0 {
target += "?" + r.URL.RawQuery
}
http.Redirect(w, r, target, http.StatusTemporaryRedirect)
}
func main() {
http.HandleFunc("/", handle)
go http.ListenAndServe(":80", http.HandlerFunc(handleHTTP))
log.Fatal(http.ListenAndServeTLS(":443", "/etc/letsencrypt/live/www.jfrech.com/fullchain.pem", "/etc/letsencrypt/live/www.jfrech.com/privkey.pem", nil))
}

One advantage of writing one’s one web server is full control of how the website behaves. One implication, which can be seen as a downside, is that you have to do everything; even the most basic of logging. However, writing your own log files enables you to confidently follow the logging policy you employ.
From a security standpoint, you can exactly control which files are served and which result in a 404 response; avoiding accidently exposing the whole server’s directory structure for the world to see.

Jonathan Frech's blog; built 2024/05/27 06:43:58 CEST