#linear system solution

[peeterjoot]

Geometric algebra, exact and least squares solutions of two variable linear system

New video (on Google’s CensorshipTube): Click here to watch the video on the Odysee platform. [Click here for a PDF version of this post (and supplementary notes for the video.)] Look below for a wordpress version of this post (and supplementary notes for the video.) Exact system. Recall that we can use the wedge product to solve linear systems. For example, assuming that \( \Ba, \Bb \) are not…

[hsmagnet]

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[liteunite]

Wireless Control System Singapore – Providing Smart Solutions

Thanks to the progressing trend of intelligent buildings in the contemporary world, the development of wireless control systems and smart solutions such as the KNX control system has become the key focus. Such systems entail immense ease, adaptability, and effectiveness in domestic, business, and manufacturing use.

Such types of wireless control system Singapore provide compatibility with many types of building functions including lighting, heating, security, and most other systems in a centralized wireless control system. One wonderful aspect of mind with wireless technology is that it’s very easy to install compared to wiring the whole facility.

These features are especially useful in retrofitting applications or when relatively small changes are needed in existing structures. As through applications or control panels, constant control of conditions and regulation of one’s environment and needs is possible, with energy efficiency. Wireless control system Singapore can be equipped with sensors and all the smart features necessary for managing wireless occupancy or daylight levels, making sure that the energy is needed only at the certain moment, thus helping to reduce utility bills and support sustainable endeavours.

KNX control system Singapore overcomes these challenges further by providing smart building’s automation system on a platform with an international protocol. Due to its robustness, it is able to control and operate lighting, HVAC, security, and other building functions effectively.

KNX Control System Singapore whose communications can be via wired or wireless depending on the given project. Its major attribute is the Interconnectivity whereby various brands and devices can easily share information and function with one another without conflict. This makes it ideal use in projects which involve the use of multiple vendors in the project.

Integrating a KNX control system Singapore into a wireless Control System Singapore provides the best of both worlds: the benefits of wireless communication and the solid, clearly deterministic environment what KNX stands for. This way the building managers can design effective energy-wise environments which are ready for the future, with full parametric control, automatic scheduling, and real time monitoring.

[golzarrahman66-blog]

https://mathgr.com/grpost.php?grtid=3

ম্যাট্রিক্স ও নির্ণায়ক এর মধ্যে পার্থক্য কী?, ব্যাতিক্রমী ও অব্যাতিক্রমী ম্যাট্রিক্স কী?, বর্গ ম্যাট্রিক্সের বিপরীত ম্যাট্রিক্স কী?, বর্গ ম্যাট্রিক্সের অ্যাডজয়েন্ট কী?, বিপরীত ম্যাট্রিক্সের বৈশিষ্ট কী?, অর্থোগোনাল ম্যাট্রিক্স কী?, একঘাত সমীকরণ জোট ও এর সমাধান কী?, ক্রেমারের নিয়মে একঘাত সমীকরণ জোটের সমাধান কী?, তিন চলকবিশিষ্ট একঘাত সমীকরণ জোটের সমাধান বিপরীত ম্যাট্রিক্সের সাহায্যে একঘাত সমীকরণ জোটের সমাধান কী?, বর্গ ম্যাট্রিক্সের ট্রেস কী?, Defference between Matrix and Determinant, Singular and Non-Singular Matrix, Inverse Matrix of a Square Matrix, Adjoint of a Square Matrix, Properties of Inverse Matrix, Orthogonal Matrix, System of linear equations and it's solution, Solution of system of linear equations using cramer's rule, Solution of system of linear equations using inverse myatrix, Trace of Square Matrix

[thydungeongal]

Actually the "combat as random unavoidable obstacles that the party needs to overcome because we need to have combat in this game or else it'll be boring" model of encounter design actually runs counter to modern D&D as well.

Okay so D&D, regardless of edition, is ultimately a resource management game. This is why even 5e has multiple different resources for characters to manage, each with different refresh rates (some refresh on a short rest, some on a long rest), and while whether the system is actually balanced in any meaningful way (it isn't) is a completely different discussion, it's clear that there is an intent to have characters manage these resources through an extended period of time. If you the GM are throwing a prescripted combat encounter at the party every once in a while with no telegraphing you are not really allowing them to exercise agency over how they go about their resource management.

"Ah, I see," you say, having turned into a D&D 3e era adventure designer for Dungeon magazine, "the solution is a combat gauntlet set within an ultimately linear dungeon." Well, no. While it is true that WotC D&D is the most opinionated about combat, all the WotC editions (yes, even 4e, you actually never read the dang rules did you) actually have support for noncombat challenges, often in ways in which they have meaningful interactions with the combat part of resource management: spell slots in 3e and 5e cover both the use of combat and utility spells, 4e allows for draining character combat resources like healing surges outside of combat, and all three games have lots of ways built into them in which characters can be challenged outside of combat.

But alternating between combat and noncombat encounters isn't enough: you need to allow players to weaponize agency to truly make that resource management shine. All potential encounters are heavily telegraphed. Some encounters look like noncombat encounters that can turn into combat encounters if failed. Characters always have the option not to engage and can often look for various routes. There are shortcuts that can be unlocked through the clever use of character abilities and resources.

That's right: even WotC D&D thrives in a dungeon. It's a very different kind of dungeon than a classic old-school dungeon (WotC D&D is more opinionated about combat than TSR D&D, so combat becomes even more of an expectation; WotC D&D doesn't really work that well with wandering monsters, so there need to be other ways to model time pressure), but the structure of the dungeon is still very much the optimal model to allow players to engage with resource management while allowing for agency.

And once you knock down those illusory walls of a dungeon and put a different coat of paint on them, your "dungeon" can be transformed into any location-based adventure. A fey forest? A dungeon. Breaking into the evil lord's castle? A dungeon. The sewers under the city? A dungeon. The streets of the city after the portal to the Hell Dimension has been opened and there's Pandemonium on the streets? A dungeon.

[pommenomnom]

"its taking so long" i havent even had my period twice since the 1.5 update, yet some ppl r acting like its been 83 yrs. is patience a word in their vocabulary or do they just think infinikki is made by independent fairies who also dont need to eat or sleep

"they want our money, they posted new banners right after the announcepology" literally it is on schedule for those banners to b running, meaning they utilised every minute they could before then to analyse solutions until they had to make a statement, cause the scheduled post were going to happen

"this isnt enough" its not, but can u imagine the process all of these decisions have to go through to b approved? how every solution needs programming and compensation? theres literally paid bundles for the dye system currency. how r they going to compensate players who already bought those if they change the dye system? and if they have ideas, will higher ups approve them?

"they didnt address all this stuff" its in the statement that they have other issues they r aware of that they r going to b fixing in due time. it even says they cant talk abt it yet, probably cuz the exact decisions havent been made yet or the optimal solutions havent been figured out.

"they said that the retcon was accident, what a lie!" they did not say that, they said they didnt know how to implement this new stuff to integrate it into the story and in their rush to publish the new update, this is the decision they arrived on. they said theyll think of a way to fix this. its not easy, cuz how do u make new players who had the sea of stars tutorial experience the beginning of the game w/o messing up their accounts with non-linear quests. programming is hard and takes time. let those poor employees sleep, theyre not the ceos and investors u r actually mad at.

i know the 1.5 update is a fucking disaster, but with that in mind, of course its gonna take time and delicate work to tackle. you want them to respond and say what? "we hear you and were working on it"? if they have to make that statement every week without being able to disclose what theyre even working on (cuz they themselves arent sure what exactly will be implemented or how, due to needing a green light from higher ups) ppl will just throw a "youre lying!" fit instead. its a lose-lose situation.

u should definitely apply pressure, u should definitely express what it is exactly u want fixed, but complaining that difficult things take time is just not okay. istg ppl have completely lost their ability to patiently wait at this point.

[niteshade925]

April 14, Xi'an, China, Shaanxi History Museum, Qin and Han Dynasties Branch (Part 3 – Innovations and Philosophies):

(Edit: sorry this post came out so late, I got hit by the truck named life and had to get some rest, and this post in itself took some effort to research. But anyway it's finally up, please enjoy!)

A little background first, because this naming might lead to some confusions.....when you see location adjectives like "eastern", "western", "northern", "southern" added to the front of Zhou dynasty, Han dynasty, Song dynasty, and Jin/晋 dynasty, it just means the location of the capital city has changed. For example Han dynasty had its capital at Chang'an (Xi'an today) in the beginning, but after the very brief but not officially recognized "Xin dynasty" (9 - 23 AD; not officially recognized in traditional Chinese historiography, it's usually seen as a part of Han dynasty), Luoyang became the new capital. Because Chang'an is geographically to the west of Luoyang, the Han dynasty pre-Xin is called Western Han dynasty (202 BC - 8 AD), and the Han dynasty post-Xin is called Eastern Han dynasty (25 - 220 AD). As you can see here, in these cases this sort of adjective is simply used to indicate different time periods in the same dynasty.

Model of a dragonbone water lift/龙骨水车, Eastern Han dynasty. This is mainly used to push water up to higher elevations for the purpose of irrigation:

Model of a water-powered bellows/冶铁水排, Eastern Han dynasty. Just as the name implies, as flowing water pushes the water wheel around, the parts connected to the axle will pull and push on the bellows alternately, delivering more air to the furnace for the purpose of casting iron.

The Nine Chapters on the Mathematical Art/《九章算术》, Fangcheng/方程 chapter. It’s a compilation of the work of many scholars from 10 th century BC until 2 nd century AD, and while the earliest authors are unknown, it has been edited and supplemented by known scholars during Western Han dynasty (also when the final version of this book was compiled), then commented on by scholars during Three Kingdoms period (Kingdom of Wei) and Tang dynasty. The final version contains 246 example problems and solutions that focus on practical applications, for example measuring land, surveying land, construction, trading, and distributing taxes. This focus on practicality is because it has been used as a textbook to train civil servants. Note that during Han dynasty, fangcheng means the method of solving systems of linear equations; today, fangcheng simply means equation. For anyone who wants to know a little more about this book and math in ancient China, here’s an article about it. (link goes to pdf)

Diagram of a circle in a right triangle (called “勾股容圆” in Chinese), from the book Ceyuan Haijing/《测圆海镜》 by Yuan-era mathematician Li Ye/李冶 (his name was originally Li Zhi/李治) in 1248.  Note that Pythagorean Theorem was known by the name Gougu Theorem/勾股定理 in ancient China, where gou/勾 and gu/股 mean the shorter and longer legs of the right triangle respectively, and the hypotenuse is named xian/弦 (unlike what the above linked article suggests, this naming has more to do with the ancient Chinese percussion instrument qing/磬, which is shaped similar to a right triangle).  Gougu Theorem was recorded in the ancient Chinese mathematical work Zhoubi Suanjing/《周髀算经》, and the name Gougu Theorem is still used in China today.

Diagram of the proof for Gougu Theorem in Zhoubi Suanjing. The sentence on the left translates to "gou (shorter leg) squared and gu (longer leg) squared makes up xian (hypotenuse) squared", which is basically the equation a² + b² = c². Note that the character for "squared" here (mi/幂) means "power" today.

This is a diagram of Zhang Heng’s seismoscope, called houfeng didong yi/候风地动仪 (lit. “instrument that measures the winds and the movements of the earth”).  It was invented during Eastern Han dynasty, but no artifact of houfeng didong yi has been discovered yet, this is presumably due to constant wars at the end of Eastern Han dynasty.  All models and diagrams that exist right now are what historians and seismologists think it should look like based on descriptions from Eastern Han dynasty. This diagram is based on the most popular model by Wang Zhenduo that has an inverted column at the center, but this model has been widely criticized for its ability to actually detect earthquakes. A newer model that came out in 2005 with a swinging column pendulum in the center has shown the ability to detect earthquakes, but has yet to demonstrate ability to reliably detect the direction where the waves originate, and is also inconsistent with the descriptions recorded in ancient texts. What houfeng didong yi really looks like and how it really works remains a mystery.

Xin dynasty bronze calipers, the earliest sliding caliper found as of now (not the earliest caliper btw). This diagram is the line drawing of the actual artifact (right).

Ancient Chinese "Jacquard" loom (called 提花机 or simply 花机 in Chinese, lit. "raise pattern machine"), which first appeared no later than 1st century BC. The illustration here is from the Ming-era (1368 - 1644) encyclopedia Tiangong Kaiwu/《天工开物》. Basically it's a giant loom operated by two people, the person below is the weaver, and the person sitting atop is the one who controls which warp threads should be lifted at what time (all already determined at the designing stage before any weaving begins), which creates patterns woven into the fabric. Here is a video that briefly shows how this type of loom works (start from around 1:00). For Hanfu lovers, this is how zhuanghua/妆花 fabric used to be woven, and how traditional silk fabrics like yunjin/云锦 continue to be woven. Because it is so labor intensive, real jacquard silk brocade woven this way are extremely expensive, so the vast majority of zhuanghua hanfu on the market are made from machine woven synthetic materials.

Chinese purple is a synthetic pigment with the chemical formula BaCuSi2O6. There's also a Chinese blue pigment. If anyone is interested in the chemistry of these two compounds, here's a paper on the topic. (link goes to pdf)

A list of common colors used in Qin and Han dynasties and the pigments involved. White pigment comes from chalk, lead compounds, and powdered sea shells; green pigment comes from malachite mineral; blue pigment usually comes from azurite mineral; black comes from pine soot and graphite; red comes from cinnabar; ochre comes from hematite; and yellow comes from realgar and orpiment minerals.

Also here are names of different colors and shades during Han dynasty. It's worth noting that qing/青 can mean green (ex: 青草, "green grass"), blue (ex: 青天, "blue sky"), any shade between green and blue, or even black (ex: 青丝, "black hair") in ancient Chinese depending on the context. Today 青 can mean green, blue, and everything in between.

Western Han-era bronze lamp shaped like a goose holding a fish in its beak. This lamp is interesting as the whole thing is hollow, so the smoke from the fire in the lamp (the fish shaped part) will go up into the neck of the goose, then go down into the body of the goose where there's water to catch the smoke, this way the smoke will not be released to the surrounding environment. There are also other lamps from around the same time designed like this, for example the famous gilt bronze lamp that's shaped like a kneeling person holding a lamp.

Part of a Qin-era (?) clay drainage pipe system:

A list of canals that was dug during Warring States period, Qin dynasty, and pre-Emperor Wu of Han Han dynasty (475 - 141 BC). Their purposes vary from transportation to irrigation. The name of the first canal on the list, Hong Gou/鸿沟, has already become a word in Chinese language, a metaphor for a clear separation that cannot be crossed (ex: 不可逾越的鸿沟, meaning "a gulf that cannot be crossed").

Han-era wooden boat. This boat is special in that its construction has clear inspirations from the ancient Romans, another indication of the amount of information exchange that took place along the Silk Road:

A model that shows how the Great Wall was constructed in Qin dynasty. Laborers would use bamboo to construct a scaffold (bamboo scaffolding is still used in construction today btw, though it's being gradually phased out) so people and materials (stone bricks and dirt) can get up onto the wall. Then the dirt in the middle of the wall would be compressed into rammed earth, called hangtu/夯土. A layer of stone bricks may be added to the outside of the hangtu wall to protect it from the elements. This was also the method of construction for many city walls in ancient China.

A list of the schools of thought that existed during Warring States period, their most influential figures, their scholars, and their most famous works. These include Confucianism (called Ru Jia/儒家 in Chinese; usually the suffix "家" at the end denotes a school of thought, not a religion; the suffix "教" is that one that denotes a religion), Daoism/道家, Legalism (Fa Jia/法家), Mohism/墨家, etc.

The "Five Classics" (五经) in the "Four Books and Five Classics" (四书五经) associated with the Confucian tradition, they are Shijing/《诗经》 (Classic of Poetry), Yijing/《易经》 (also known as I Ching), Shangshu/《尚书》 (Classic of History), Liji/《礼记》 (Book of Rites), and Chunqiu/《春秋》 (Spring and Autumn Annals). The "Four Books" (四书) are Daxue/《大学》 (Great Learning), Zhongyong/《中庸》 (Doctrine of the Mean), Lunyu/《论语》 (Analects), and Mengzi/《孟子》 (known as Mencius).

And finally the souvenir shop! Here's a Chinese chess (xiangqi/象棋) set where the pieces are fashioned like Western chess, in that they actually look like the things they are supposed to represent, compared to traditional Chinese chess pieces where each one is just a round wooden piece with the Chinese character for the piece on top:

A blind box set of small figurines that are supposed to mimic Shang and Zhou era animal-shaped bronze vessels. Fun fact, in Shang dynasty people revered owls, and there was a female general named Fu Hao/妇好 who was buried with an owl-shaped bronze vessel, so that's why this set has three different owls (top left, top right, and middle). I got one of these owls (I love birds so yay!)

And that concludes the museums I visited while in Xi'an!

[thewertsearch]

-- grimAuxiliatrix [GA] began trolling gardenGnostic [GG] -- GA: Are You Feeling Any More Cooperative In This Timeframe GG: password! GA: I See No Of Course Not GA: Please GA: I Know You Were Talking To Her Shortly Before GA: What Did You Say To Her GG: I SAID PASSWORD FUCKASS!!!!!

This is a pretty clever idea, actually. Usually when you message someone on Trollian, it lives up to its name by serving up a completely random timeframe, confusing your conversations with what I'm almost convinced is deliberate malice.

Employing a password system to keep things linear is a very creative countermeasure – although it’s not exactly foolproof, since Kanaya could request future passwords from her own older self.

GA: Past You Doesnt Care About Passwords What Happened […] GG: it depends, do you want to have a silly conversation or a serious conversation?

I’m really liking this no-bullshit incarnation of Jade. It’s an attitude we haven’t seen from any of these other jokesters – and as we've recently discussed, it’s particularly striking to see it from Jade Harley, who’s been silly since day one.

GG: you have to give me that password to start our next conversation GG: this ensures that past you cant jump ahead into the conversation and mess everything up, like you are trying to do now!

Like I said, it's not fully secure, since she could get passwords from her future self, or use Trollian’s viewport to snoop on Jade’s future conversations.

Luckily, this is Kanaya we're talking about. She's sensible enough to play fair.

I love that the Iron Man suit appears to be fully functional.

This is a much more versatile flying solution than the jetpack - but it pales in comparison to the majesty of Unreal Air.

GA: Im Appreciating Our Conversations From This Timeframe More And More GA: Past You Is Much Less Of A Taskmaster Than Future You Or Pre Blackout Rose […] GG: […] what do you mean by blackout??? GA: I Guess Youll Find Out Soon GA: And Then Report It To Me Under Extremely Specific Circumstances GA: Which Is Good Because I Sure Dont Know GG: hmm @_@ […] GA: Your Eyes Are Right To Be Swirled Letters

Love Kanaya’s continued fascination with ASCII typography. It might just be my favorite running gag in the comic.

[plurapony]

like whispers in the wind, the trees know where i have been. when i am one with the earth - there are no expectations, there is no more "me"; there is only the earth which now i am a part of too.

It's been hard to post to Tumblr recently, I feel like I have assumed the part of a role model to the system community and I cannot live up to my self-made title. Though I can understand that the so called pressure I have on myself is entirely in my mind, that is only the first step of many to come.

Recovery is hard, we all know that. It's not linear and as you climb the ladder, time and time again you will fall. The higher you climb, the higher you can fall and sometimes after so many times of falling you want to just give up and instead live with the fact you'll never recover. But you need to. I don't know personally what awaits at the top of the ladder but I know it is something worth fighting for.

For me? I sit at the bottom of the ladder to recovery, I haven't started climbing again yet but I know that I will. I'm taking my time - perhaps a little bit too much, I feel as though I tried to climb too fast and too hard and I now am exhausted. Yeah, being a system is fucking exhausting. I love my parts, and I do appreciate being a system but God it's so overtly complicated and I wish it was simple. I wish I could find a solution to the problems in my head that were easy - but alas that's not the way things work.

I think sometimes we forget to humanize our opponents and of course the people we look up to. When we are separated by screens and identified only by display names it's only natural to forget the human behind it all. But it's important to remember the human behind the post, we all have our own struggles, our own personal demons and a lot of it probably isn't shared online. I think that's why kindness is so important - you don't know what is going on in an internet stranger's life and a little bit of kindness can go a really long way.

No matter what stage of recovery someone is at, whether they're already at the top, whether they're still climbing or like me, catching their breath and waiting to start climbing again, we all deserve kindness and we should be treated with respect and care. The golden rule? Treat others like you wish to be treated. And be kind to others as you wish others would be kind to you.

[catgirltoofies]

i think saints row 4 did something extremely good but in the wrong genre.

so, basically, saints row is a non-linear gangster open world game, similar to grand theft auto, mafia, and red dead redemption. it takes itself a lot less seriously, being full of jokes and gags, the quality of which are up to interpretation. more importantly, though, it was still mostly pretty sane with its broad theme - it was silly, sure, but it was still taking place in believable, grounded cities with gun stores and lots of cars and what have you.

in saints row 4, however, most of the game takes place in a simulation of the city saints row 3 takes place in, and one of the major abilities is just straight-up superpowers. you get to run super fast, jump across blocks at a time, and even fly, along with the myriad combat abilities. for the purposes of this argument, though, i want to hone in on one single ability: super-speed.

saints row, for all its silliness, is still a game based on driving stolen cars and making them look pretty. this is not the case for saints row 4, because while those systems are still in place, there's no reason to ever drive when you can just super-run and fly wherever you want, for free. this is why i think it's in the wrong genre; because of the super-speed, you're discouraged from using the major system of "steal cars and pimp them out".

which brings me to where i think it WOULD work. and that's the open-world rpg genre. of particular note, I'll call out the elder scrolls and assassin's creed.

first, elder scrolls: fast traveling is a good option to have, of course, but it almost completely negates exploring on foot. when's the last time you used a horse in skyrim? have you ever done mounted combat? both of them SUCK. horses are slow and clunky and mounted combat is finicky at best. enter super-speed. naturally, this would be disabled during combat, to avoid totally trivializing any encounter with just super-speed and a bow, as well as while overencumbered, but with super-speed, you no longer have to slowly trudge everywhere that doesn't have a fast travel marker. you get a shortcut to travel that doesn't disincentivize exploration, and it's not like this idea doesn't have precedent. look at morrowind, where you can down a dozen skooma and more than double your movement speed. this was how people got around! just bring back a method to boost speed, perhaps arbitrarily capping it to avoid clipping through walls.

next, assassin's creed. there's no kind of super-speed in any assassins creed game that I'm aware of, but it would WORK. horses, of course, give you the means to travel around with speed, and they're at least faster and easier to use than the horses in the elder scrolls, but once you get into combat you have to dismount, use overpowered horse archery, or use the godawful mounted melee combat. if there's a location only 300m away, summoning my horse is a chore just for a short jaunt, but running that distance on foot is agonizingly slow. simple solution: replace the mount with an Animus hyper-speed mode that makes time speed up until you disable it or get into combat.

please tell me any problems with this idea because i know my truth

[hsmagnet]

Permanent Magnet Rotor Inspection Solutions

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[nokillbananashelter]

A HUGE portion of the criticisms for BOTW and TOTK boil down to “there isnt (xyz) in this game like other games!” And the answer to it is almost always “yes there is, you are just ignoring it or actively avoiding it or willfully engaging in it in a way that is breaking it and then getting mad that it broke”

Like, if you want a tight 50 hour 3d zelda with a full ACTIVE PRESENT story and linear story/mechanical progression & big dungeons….just listen to what the game is telling you to do and inviting you to see when you see it and you will literally get a classic linear 3d zelda experience complete with “yes, we did infact have ONE PARTICULAR solution in mind for this puzzle!”, deep lore implications, extremely melodic and complex orchestral music, progression, & a zany cast of characters.

Both of these games are so much more than that though, and contains so much more content that people don’t see that meat and potatoes run because of their own choices and desire to break the game wins out over wanting to meet the game where its at. And to be fair, that is a major component to what these games are daring you to do. And you should engage with those, but if you really did want the classic experience, it should be obvious you should treat it like the OOT era games and it will LITERALLY provide that for you in neon blinking lights and guardrails.

I really do feel like these two games are sort of an exercise in discipline /listening /spotting subtly /recognizing things in their new shape as much as they are an invitation to explore experiment and fuck off, and people are mad that the option to break from the path exists at all because they dont have the control or desire or sight to focus in on whats being asked of them and they don’t want to accept the shape of what the game is working with.

I’m not saying these people are playing the game wrong, but the attitude and closed mindedness is the problem. it’s almost always the source of these kinds criticisms that want to present these games as empty repetitive experiences, when these two games give the entire franchise a run for its money on uniqueness and novelty and story depth and profoundness.

You went in for a bad time, didnt want to play, and got exactly what you worked for- a lame, monotonous level skip that jumps ahead of itself, disrespects the intention of the developers, and misses the point of very obviously and enticingly marked gametunnels they made for you.

Like, seriously here, both but totk especially is quite literally an on-rails CINEMATIC GAMEPLAY experience if you want it to be, it literally has a path that is like 99% laid out in such a way that you are directly on mine cart rails and jumping up and down on direct vertical pathways. The regions-to-temple pathways are about as much of a bee-line as you can get. The skylands & cave systems are effectively old oot era compartmentalized zones. Same with shrines. There are so many ways they have created pathways and say “GO THIS WAY” that if you ignore those and get lost its kinda on you for not being observant….. which has always been one of the top asks of the player required to beat a Zelda game.

The biggest difference between these two games and other 3-D Zelda’s is that it just has taken the roof and some walls off the design. It Lets you experience more in context to your journey at any given time and that gives you a lot of freedom and agency to break from the path or stay on your goals, and both enrich your experience and inform what links journey is going to be and how narratively/mechanically/progression cohesive it will be.

[lenzoli]

tried making a gquuuuuux themed physics problem. when i tried solving it i ended up in the depths of algebra hell. here's the final version of the problem. good luck

Machu lives in a space colony, the interior of a cylinder 6.4km in diameter. In order to simulate gravity, it spins once every 113.5 seconds, generating a centrifugal force that is extremely close to Earth's gravity. One day, the system malfunctions and the rotation slows down massively. Machu jumps vertically (perpendicular to the ground) with an initial velocity of 1m/s, and finds that she has shifted exactly 10cm when she lands. How long does a rotation of the colony take now? Neglect air resistance and approximate to the first 3 significant figures. Online tools may be used.

i'm going to make an easier problem after this. notes on this one under the cut, and a solution at the end

when i say online tools may be used i mean online tools MUST be used do NOT DO THIS MANUALLY FOR GOD'S SAKE

there are two approaches to this problem. both of them suck major ass

the first, the route i was first planning to go down, was using coriolis and centrifugal forces. instant roadblock, before i even got to the coriolis part the centrifugal force hit me with a stupid differential equation that requires the error function to solve

the second, the backup route, was linear. it involves trigonometry and stuff, but that's not the bad part. the bad part is that the final equation you end up with for the angular velocity looks like absolute shit, has no easy aproximations, and gets even worse when you plug it into wolframalpha for the exact solution. like.

at least it's closed form i GUESS

anyway just. approximate it. use newton-raphson or something. i don't care

here's my full solution. enjoy

[henrysjohn]

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[yaldabaothadeez]

How fast does a wave move?

According to my poll 9/11 people (never forget) didn't already know this, so here's my attempt to explain the difference between group and phase velocities of waves with minimal (but not no) background.

Honestly, you're better off reading the wikipedia page "group velocity" from which I am stealing all my pictures, but I'm doing this anyway as an exercise in pop-science writing.

Summary: a small ripple on the surface of water actually looks something like this

i.e. not a fixed pattern of peaks and troughs moving together as a block, but a bunch of wiggles moving more slowly than the peaks of the waves.

The rest of this post is an attempt to explain why this happens, and that in the particular case of water surface waves the ripple is traveling at half the apparent speed of the waves that make it up.

Waves and dispersion

First, let's think about waves in general.

An idealized uniform plane wave (in one dimension) looks like this

the y-axis is amplitude (e.g. the height of the surface for a water wave[1], pressure for a sound wave, field strength for radio waves etc.)

The x-axis is, well, it could be either space or time, right? The wave varies left-to-right, and the amplitude of any fixed point wobbles up and down, something like:

y = h(x,t) = sin(k x - w t)

for some constants k and w. w is the frequency of the wave, 2π/ (the time between successive peaks) , k is the wavenumber, 2π/ (the distance between successive peaks) [2]

The physics of the situation, whatever they happen to be, are going to have solutions with certain frequencies at certain wavenumbers. So we can treat w as a (currently undetermined) function of k

w = w(k)

this is the "dispersion relation", and is all we'll need to work out the rest of this.

Phase velocity

How fast do the peaks of the wave move?

The peaks occur when sin(k x - w t) = 1, so we need (k x - w t) to be a constant. That is, the peaks move at a speed

v_p = w/k

This is called the "phase velocity" of the wave, and is given by the ratio of frequency to wavenumber.

In general, this will vary with k, unless the dispersion relation is linear.

Ripples

Now, we want to think about something more realistic - not a uniform sine wave extending out to infinity, but a small localized disturbance to an otherwise flat medium (caused by a duck landing in a pond or something).

We'll assume that at a time t=0, we have

h(x,0) = f(x)

for some localized function f(x). We want to know how this little bump propagates over time.

Now f(x) is obviously not a plane wave, but if it is well behaved we can approximate it by a sum of plane waves of different amplitudes, phases and wavenumbers (we'd like to do this, as we understand how the plane waves move). In the continuous limit, we get the fourier transform

h(x,0) = ʃ H(k) sin( k x + b(k) ) dk

for some function H(k) [3].

If this perturbation is small enough that the system can be treated as linear, then solutions just add together and

h(x,t) = ʃ H(k) sin( k x - w(k) t+ b(k) ) dk

this is true but not very helpful - how does our ripple actually move?

Group velocity

From the stuff with the phase velocity, we can guess that this is going to depend on k.

So let's think about wavenumbers close to some fixed k0 and ignore the rest (this is fine because everything's linear). For convenience, I'll assume k0 is a local maximum of the function H(k), so I can ignore its derivative.

We'll write k = k0 + δk and expand the argument of the sine above to first order in the small value δk

k x - w(k) t+ b(k) = k0 x -w(k0) t + b(k0) + δk(x - w' t + b') +...

where w'(k0) = dw/dk is the first derivative (i.e gradient) of w(k) at k0

substituting that back into our expression for h(x,t), dropping any higher powers of δk and doing some basic trigonometry, we get

h(x,t) ~ cos[k0 x -w(k0) t + b(k0)] ʃ H(k) sin[δk(x - w' t + b')] dk +sin[k0 x -w(k0) t + b(k0)] ʃ H(k) cos[δk(x - w' t + b')] dk

The sine and cosine terms outside the integral that I've colored purple are just a plane wave at k0. The terms inside the integral in red show us how the pattern propagates: it moves at a velocity

v_g = dw/dk

This is the "group velocity" of the wave, and is given by the gradient of the frequency with wavenumber.

Of course, this is also generally a function of k, and so the ripple will gradually separate out into its constituent frequencies (hence the name "dispersion relation") [4]

Sound and light

For most types of wave you have probably encountered before (e.g. sound, electromagnetic waves in a vacuum), the dispersion relation is

w = c k

for some constant c. These types of waves are called non-dispersive, as v_p = c = v_g, for all values of k. Sound really does travel as fast as the pressure wave crests.

Waves on the surface of water

Okay, finally getting to the point. We're going to think about waves on the surface of a fluid (e.g. water) under gravity.

Assumptions: the waves are very small, the fluid is very deep (and of a uniform depth), surface tension and viscosity can be ignored.

Now, the "right" way to do this is to solve an appropriate limit of the Navier-Stokes equations, but that's too much like hard work:

k has units 1/length, w has units 1/time, we've assumed away everything in the problem except gravity so up to a constant

w = √(gk)

Where g (~9.81 m/s^2) is the acceleration due to gravity.

So, immediately, we can say

v_p = √(g/k)

these waves are dispersive, narrower ones go slower.

v_g =0.5 * √(g/k) = 0.5 * v_p

ripples travel at half the speed of waves that make them up.

footnotes

[1] called a "gravity wave" by fluids people, as that's what the restoring force is. Which is why LIGO and such are always careful to say they are measuring gravitational waves.

[2] my angles are in radians

[3] The function b is there because these waves might be out of phase with each other, if I were using complex numbers I could absorb that into H, but I'm trying to not use more types of math than I have to

[4] Clearly v_p and v_g can be different, but in more than one dimension, they might not even be pointing the same way.

[bubbloquacious]

So if your coordinates lie in an integral domain, the union of the solutions sets of two algebraic equations is itself the solution set of an algebraic equation, because xy = 0 if and only if x = 0 or y = 0. In this setting an algebraic set is a set that's the simultaneous solution set of a system of algebraic equations, which you can show pretty easily to be the same thing as the vanishing set of an ideal of polynomials. Algebraic sets are then closed under arbitrary intersections and finite unions, so they're the closed sets in a topology which we call the Zariski topology.

In other algebraic settings, though, this doesn't hold up! If you take two solutions sets of linear equations in some vector space, two lines through the origin for example, then their union is not generally the solution set of a linear equation, because we know that such sets are linear subspaces.