In math, groups are the building block of algebras. Groups are required to do any sort of sensible translation or mapping. Groups are what makes addition possible. Groups allow shapes to be rotated and flipped. Groups are what makes modular arithmetic, the modulo operation, and cryptography possible.
A group requires a set. A set is just a collection of well-defined elements. It could be the set of integers, the set of real numbers, the set of dogs, or the set of purple plastic toys manufactured in 1992.
A group requires an operation. An operation is a function that maps a series of inputs to a single, well-defined output. The inputs are called the domain. The outputs are called the codomain. An example of this is addition. Addition is an binary operation. That is, it accepts two inputs. It then spits out a well-defined output which we call the sum. For a group, it doesn't matter what this operation actually is. It could be addition. It could be multiplication. It could be an operator with a shooting star shape. It doesn't matter what it looks like or what it does. All that matters for the purposes of a group is that the operation is binary and allows you to combine, in some fashion, any two elements in the group's set.
A group is a closure. It is closed under its operation. That is to say, when performing its operation on any two elements in the group, it will always return a member of the group.
A group must have an identity element. This is an element that when combined (read: operated upon) with any other element, you get that element. For example, The additive identity of the group of integers is 0. 0 plus any integer results in that integer.
Each element in a group must have an inverse. An element combined with its inverse will return the identity element.
Lastly, a group is associative. This means that the order of operations doesn't matter so long as the sequence is preserved. E.g., (1 + 2) + 3 = 1 + (2 + 3).
This is all that is required for a group. These rules allow groups to be useful constructs for various mathematical applications. Indeed, you may find that every bit of math you can think of might very well have been with a group.
While all groups require that the order of operations doesn't matter, they do not require that the sequence of operations also not matter. This is because there are many instances of constructs that behave very much in a group-like fashion that would be excluded from the definition if sequence wasn't required to be maintained (such as the symmetry group) If sequence indeed does not matter (also known as being commutative) then we call that type of group an Abelian group.
So what's the point of rigorously defining things as groups? They expose behaviors that we otherwise wouldn't see. Modern cryptography would be impossible without the notion of a group. A key component is based on the notion that the set of integers can be mapped as quotient groups in to residue classes. As an example, we can take every number evenly divisible by 3 and stick them in a group. We can take every number that leaves a remainder of 1 when divided by 3 and stick them in their own group. Finally, we can take every number that leaves a remainder of 2 when divided by 3 and stick them in a group. We've now mapped all of the integers in to three quotient groups. The cool part is if you add any number from the first group to any number of the second group, you'll always get a number in the third group. In fact, if you pick any number from one group and any number from any other group, your result will always be a number from the same group as any other result.
We've translated the integers in to three residue classes and those classes together are a group. This is the basis of modular arithmetic (sometimes called "clock arithmetic") Now, I can construct a function that takes in a set of integers, sums them together, and spits out the modulo. Thus, I've taken an arbitrary-length input and mapped it to a fixed-length output. This is an example of a simple modular hashing function. This and similar concepts are extended in applied usage, for example, to verify a password is correct without having to store or even know what the password is. Such a cryptographic hash requires a number of properties not provided in our naive example but at the heart of it all are groups.
Say you wanted to draw a picture without your pencil leaving contact with your paper. Say you wanted to give that picture a mathematical formula. That formula would be a Fourier series.
A Fourier series can be thought of as the sum of a number of vectors which each rotate at a constant integer frequency. That frequency will start at zero and will go off in either direction. Each of those vectors can have whatever magnitude and starting direction you wish. This type of sum is known as a harmonic.
A Fourier series can also be thought of as the infinite sum of sine and cosine functions which expand some periodic function. That is to say, if you have, say, a square wave, you can represent that square wave as the infinite sum of sine waves of various frequencies, magnitudes, and phases.
Let's construct the formula we're looking for with our drawing.
We can start with Euler's identity:
Why start there? Well, remember that we're looking for a sum of rotating vectors. If a vector rotates completely back to its starting position, we have a circle. Euler's identity gives us that starting point. e is the base of the natural logarithms. A logarithm is the inverse of exponentiation, just as division is the inverse of multiplication and subtraction is the inverse of addition. When we mutiply by e, we see a rate of growth.
i is the imaginary unit. If we want to break out of one-dimensional space and in to two-dimensional space, we need to utilize i since, of course, all real numbers reside in a single dimension: the real number line. If we raise e to a real power, we get continuous growth. If we raise e to an imaginary power, we get continuous rotation. Hence, e is raised to the i.
π is the ratio of a circle's circumference to its diameter. e^i gets us moving along a circle. Why does e^(πi) equal -1? The diameter of a circle is 2πr. If I'm moving along e^i for π seconds, I will have travelled half the circumference of a circle. I will have started at the real number 1, rotated halfway along the unit circle, and reached the real number -1.
It's a simple matter to travel the complete circle:
Now we have a vector rotating in a circle. However, remember we want periodicity. We want to travel this circle over and over again. So, let's add a time component:
Next, we want to be able to adjust the magnitude and direction of our vector in the circle. So, we can add a constant:
Now, we have a vector rotating in a circle, at whatever frequency we want, with whatever magnitude we want, and whatever starting direction we want.
We want a whole bunch of these vector rotations. In fact, we want one for every integer frequency. So, we can sum together the expressions:
...where n is an integer from -∞ to ∞. Great! We can call this summation, the function: f(t). The only hiccup is that we currently do not have a way to determine our cn values in the function.
If we consider c0, it is easy to calculate since its frequency is 0. It's our one vector that doesn't rotate. If we plot the output of f(t), we'll see that the outputs approach c0. So, if we run f(2π), we see that:
That gives us c0 but we want to generalize this for all of cn. So, we need to find something to multiply f(t)dt by that will effectively stop the rotation of each c so that we can measure its direction and magnitude. If we multiply f(t) by e^(-n2πit), that accomplishes our goal.
That gives us our final formula:
where
If we were to deconstruct a function in to a Fourier series, that would be called a Fourier transformation.
Back in the Pentium 4 era, there was a big hub-bub about how processor frequency didn't matter. The argument that was trying to be made was that processor frequency wasn't the only thing that mattered when trying to measure computational output. But, in making such a brazen statement, the argument was lost on people who intuited that frequency obviously matters to some degree.
If folks want to convince others of an argument, then they need to employ rhetorical skills. Logic is great. It gets to what is true. Logic doesn't care to convince, though. That's the purview of rhetoric.
If you make a sound or cogent argument but do so in a way which is incomplete, vague, ambiguous, or off-putting, then you may find yourself in danger of failing to convince your opponents.
The act and purpose of critique has been lost a bit on people. People believe their criticism of a work is the same as critiquing a work. It is not.
Critique is the act of employing your expertise to break down a work in to its constituent parts and judge them through the lens of the author's intent. In the example of a movie: if the author intends to frighten, then, did the script work toward that goal? How about the lighting? The framing? The pacing? The editing? The scene? The acting?
Critique requires expertise. That's not to say you need to be an expert in movies to criqitue movies. Maybe you're a horror buff. That's expertise. You can rely on that expertise to critique a horror movie. A horror buff with no experience or interest in comedies will have a difficult time critiquing a comedy, though. The horror movie buff who doesn't play video games will have a hard time critiquing a horror video game. Critique requires expertise.
Critique then takes that expertise and presents both a problem and a solution. E.g., "In your horror movie, the slasher is meant to startle the audience in this scene; however, the lighting is too bright and you can see him approaching in the background. It would be more impactful if either the lighting was toned down, the shot was reframed to keep the slasher out of the point of interest, or the slasher is obscured from view until the moment of his reveal."
As you can see, critique focuses on a specific problem and provides possible solutions generated from the critic's expertise. It is focused on the work, breaking its components down, and identifying both what works toward the author's intent and what does not (It's important to note that this means critique often also involves presenting positive aspects of the work)
Criticism, on the other hand, is the act of chastizing both a work and its author. It is generally ad hominem. E.g., "The makers of this game were so stupid when they..." Criticism doesn't rely on expertise because it doesn't view a work through the lens of an author's intentions. Criticism views a work through the lens of the criticizer. Criticism allows a person to say, "This TV show is made for toddlers. I'm not a toddler. So, this TV show is awful."
Nothing constructive comes from criticism. Often, nothing is actionable from a criticism. If it is, it generally takes the form of the criticizer declaring that the work should simply be more to the criticizer's taste. All this would do is help to narrow the body of work down to a subset of what it can be: the subset that corresponds to the criticizer's taste.
Critique requires a modernist view. It requires study and deliberation. It requires an acknowledgement of objectivity. There are classes which teach critique and are often a part of certain cirriculums.
Critique aims to improve a work. Criticism aims to destroy a work. Be a critic who wants to improve things. Don't be one who wants to destroy things.
People have lost their minds when it comes to both technology and finance. Gamestop and AMC stocks. Cryptocurrency. NFTs. Decentralized Autonomous Organizations. It's all nonsense and nonvalue.
I think people are tired of the old guard and are looking for a way to install a new guard. One which, they hope, they will be a part of. The problem is they seem so desperate for this that they're handing their would-be power right back over to the old guard.
Inflating Gamestop's stock price does absolutely nothing for anyone except those who are already wealthy. Trading fees are still being paid. Gamestop's value isn't commensurate with its stock price. So, new value isn't being willed in to existence. Inevitably, when the connivers invested in these meme stocks feel they can't get the stock price any higher, they will pull the rip cord and leave the vast majority of folks taking a wash, folks who thought they were part of a legitimate movement to overthrow the man.
The madness of it all is when trading platforms have protections in place, these folks want those protections removed. These folks seem to genuinely want the ability to go broke.
They cheer when a hedge fund takes a wash on some short stocks, never understanding that the people involved in the hedge fund will be, and in fact are, perfectly fine. It's you, as a part of a meme stock movement, who is ultimately going to get royally screwed.
The same can be said about cryptocurrencies. They are abject failures as currency. They are either an asset, a security (such as with a legitimate ICO), or pure fraud (such as with most ICOs) Their value is so volatile that, outside of a handful of organizations grandstanding, they are never used as currency. They can't be used as a currency.
So, where does their value come from? Pure speculation. And, as with meme stocks, conmen attempt to drum up value by inventing pure nonsense. Things that tap in to people's desperation for change. Jargon such as Smart Contract, web3, NFT, and DAO get tossed around sounding like a bright new future if you can only be convinced that you are smart enough to see it. Meanwhile, no real value is being created. Massive infrastructure built around a blockchain simply cannot work as it stands. And things of value today, like credit card protections, theft prevention, fraud protections, computationally efficient systems, and quick transaction clearing, are nowhere to be seen in this blockchain utopia being touted by conmen. That's because, of course, the conmen plan to run off with the money from those they've duped.
Is there a bright future of wealth in store for people in to meme stocks or cryptocurrency? Well sure, if you're one of the conmen. If you're not, then all you're doing is helping to dig your existing hole bigger while being convinced that you're building a ladder to climb out of it.
People are being duped in to believing that they are going to be part of a great financial upheaval. People think that the current wealthy will get toppled and, because these folks got in on some dubious mechanism early, they're going to be part of the New Wealth. They know absolutely nothing about finance, economics, or technology. Yet, they believe they're going to be part of the new group that will be in control.
All rebels are closet aristocrats. Good luck to you, I guess.
Shin Megami Tensei V is everything that made me fall in love with JRPGs in the first place.
There's minimal dialog. Just enough to get the plot across. Plot is generally what interests me in a story, not characters. I don't want reams of character interaction. I want a compelling plot. Megaten has that.
Combat is highly technical. You are required to understand the mechanics of the game and work within them. You are required to find solutions to hard problems. This is a game about achievement. The goal is to excel at combat rather than combat just being a series of roadblocks to the next story beat.
The game jumps right in to the action. There's a brief prologue and then you're just playing the game. I'm reminded of the original Final Fantasy where you were just dropped in the overworld and off you go.
The setting and theme are magical. Angels vs demons. Will you choose one side or the other? Perhaps you'll decide humanity itself deserves a seat at the table. The literal post-apocalyptic world is desolate and gloomy. The music is strange and off-putting. The monsters are jarring both in their appearance and their demeanor. It's brilliant.
I adore this game. This is a game where, if you have the right tempermant for it, you will die a thousand times and not once feel bad about it. It's in a class of its own and one of the best JRPGs out there.
Many people are so averse to losing that they will employ any tactic to avoid doing so, even when those tactics are outside the realm of playing the game. These are rules I require people to adhere to before playing a game (typically a tabletop game) with me. These are rules of sportsmanship and aimed squarely at preventing gamesmanship.
Narrate your turns. It should be crystal clear to all players what actions you are taking and what the results of those actions are. You should not be trying to obfuscate public information.
If you notice a mistake before it irreconcilably changes the game state, regardless of who made the mistake, bring it up and correct it. The goal is to be the better player of the game, not be the one who better takes advantage of rules violations due to mistakes.
If you made a mistake that has since changed the game state, deal with it. Once the game state has changed, a past mistake has to be suffered since it's difficult if not impossible to rewind the game state at that point.
If you're angry that you're losing, deal with it. Don't make everyone else miserable. Don't be an asshole.
If someone makes a decision that is within the rules and you feel that decision is unfair or over-powered, deal with it. It's fair play. Games are defined by their rules and these rules are implicitly agreed to by all players before the start of the game. If someone finds a mechanism that is over-powered or unfair, then they deserve to leverage it. Perhaps you should have found it first.
Play to the best of your ability until the game is over. Don't quit. That means don't leave the game as well as don't stop playing effectively. Always try your best, with each and every decision you make in the game. Most games are designed around the assumption that all players are rational. When a player stops behaving this way, it can break the game.
Always play to win. This is very similar to the previous rule. This rule is specifically to prevent a player from punishing someone if that punishment isn't in advancement of winning the game. For example saying, "If you attack me, I will spend the rest of the game making sure you lose" is disallowed.
Make decisions within a reasonable frame of time. You are not allowed to try to discern the entire decision tree before taking an action. Otherwise, games will take an inordinate amount of time to complete. I might even hold you to a Chess timer. If this bothers you then work on developing your decision making skills. This will help you to intuit better decisions. Anyone can pick the correct payoff from a list of decisions. Only good players can do so within a reasonable amount of time.
Play within the spirit of the rules rather than the letter of them. E.g., if the rules say you can do "anything" and clearly means you can take any in-game action, don't try to be clever and hit a player over the head with a frying pan or something.
Don't try to win by means other than playing the game at hand. E.g., don't try to stall until everyone quits. Don't try to bully someone in to not wanting to continue. Don't king-make simply to disincentivize a player from making choices in a future game. Etc.
What these rules boil down to is that when you sit down to play a game, you are expected to play the game. Play it in its entirety, honestly, fairly, and to the best of your ability.
Moving in to a new home sucks. There are endless additional expenses. There's the uncertainty of your new neighborhood. There's the loss of your patterns and stability. There's the fact that you can't find anything for weeks. There's the physical effort involved. There's the sense that your new home is foreign to you until you finally start getting things in place and art on the walls.
It's made worse when you're doing it alone. It is a lonely, depressing venture and honestly one of the worst "regular life" experiences there is.
Eventually you settle in and it starts to genuinely feel like home. However, until then, it sucks day after day.
In 2021, people who speak different languages interact through unique avatars to play, from across the globe, games with each other. They then share this experience with the rest of the world via a live broadcast.
Today, you can be whomever you want to be, even a cartoon character, and play games with anyone around the globe, regardless of language, geographic, or cultural barriers, and share that fun with everyone else.
That's just magical.
Hololive - 【SUPER MARIO PARTY】OCEAN PARTY! パーティータイム!
