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How Large Diameter Pipes are Used in Irrigation and Agriculture
Large diameter pipes play a pivotal role in modern irrigation and agriculture systems. As the demand for efficient water management solutions grows, these pipes have become indispensable in ensuring sustainable farming practices and addressing water scarcity challenges. For those in need of reliable piping solutions, Tube Trading stands out as a prominent large dia pipe distributor in Gujarat and a trusted large dia pipe dealer in Vadodara. In this blog, we will explore the applications, benefits, and importance of large diameter pipes in irrigation and agriculture.
The Importance of Water Management in Agriculture
Agriculture is highly water-intensive, and efficient water delivery systems are crucial for sustainable farming. Traditional irrigation methods, such as open canals, often lead to significant water loss due to evaporation and seepage. Large diameter pipes offer a modern solution to these challenges, enabling efficient water transportation and distribution with minimal waste.
For farmers and agricultural businesses, investing in high-quality piping systems is essential. Companies like Tube Trading, a leading large diameter pipe supplier in Gujarat, provide durable and reliable solutions tailored to meet diverse agricultural needs.
Applications of Large Diameter Pipes in Irrigation and Agriculture
1. Main Water Supply Lines
Large diameter pipes are widely used as main supply lines to transport water from reservoirs, rivers, or wells to agricultural fields. Their large capacity ensures that substantial volumes of water can be delivered over long distances efficiently.
2. Drip Irrigation Systems
Drip irrigation is one of the most efficient irrigation methods, delivering water directly to plant roots. Large diameter pipes serve as primary conduits, supplying water to smaller pipes or emitters in these systems. This ensures consistent water flow and reduces the chances of blockages.
3. Sprinkler Systems
In sprinkler irrigation, water is distributed through a network of pipes connected to sprinklers. Large diameter pipes form the backbone of these systems, ensuring uniform pressure and adequate water flow to all sprinklers.
4. Flood Irrigation
In flood irrigation systems, water is delivered to fields in large volumes. Large diameter pipes make it possible to supply significant quantities of water quickly, covering extensive agricultural areas efficiently.
5. Drainage Systems
Proper drainage is essential in agriculture to prevent waterlogging and soil erosion. Large diameter pipes are used in constructing effective drainage systems that channel excess water away from fields, ensuring optimal soil conditions for crop growth.
Benefits of Using Large Diameter Pipes in Agriculture
1. High Flow Capacity
Large diameter pipes can transport vast amounts of water, making them ideal for agricultural applications where high volumes are required.
2. Durability and Longevity
Made from materials like steel, HDPE, or PVC, these pipes are designed to withstand harsh environmental conditions, including UV exposure, chemical exposure, and mechanical stress. This durability ensures long-term performance, reducing maintenance and replacement costs.
3. Reduced Water Loss
Unlike traditional irrigation channels, large diameter pipes minimize water loss due to seepage and evaporation. This contributes to water conservation, a critical factor in sustainable farming.
4. Energy Efficiency
Large diameter pipes reduce the energy required to pump water over long distances, thanks to their lower frictional resistance. This translates to cost savings for farmers.
5. Easy Installation and Scalability
Modern large diameter pipes are designed for ease of installation. Their modular nature allows for scalability, enabling farmers to expand their irrigation systems as needed.
Why Choose Large Diameter Pipes from Tube Trading?
When it comes to sourcing large diameter pipes for irrigation and agriculture, Tube Trading is a trusted name. As a renowned large diameter pipe dealer in Vadodara, we understand the unique requirements of farmers and agricultural businesses in Gujarat. Here is why you should consider us:
Wide Range of Products
We offer a diverse selection of large diameter pipes made from high-quality materials, ensuring durability and reliability.
Expert Guidance
Our team of experts provides personalized assistance to help you choose the right piping solution based on your specific needs.
Competitive Pricing
We are committed to offering top-quality products at competitive prices, making us a preferred large dia pipe supplier in Gujarat.
On-Time Delivery
With a robust distribution network, we ensure timely delivery of products to locations across Gujarat and beyond.
Commitment to Quality
We adhere to strict quality standards, ensuring that every pipe we supply meets or exceeds industry benchmarks.
Case Studies: Successful Applications of Large Diameter Pipes
Case Study 1: Drip Irrigation for a Large Farm in Vadodara
A progressive farmer in Vadodara approached us for a reliable irrigation system for their 100-acre farm. After assessing their needs, we supplied high-quality large diameter HDPE pipes, which served as the main water supply line. The result was a significant improvement in water distribution efficiency, leading to higher crop yields and reduced water consumption.
Case Study 2: Flood Irrigation for Sugarcane Cultivation in Gujarat
A sugarcane farmer in Gujarat required a robust solution for flood irrigation. Our durable steel pipes ensured quick water delivery to the fields, covering vast areas in minimal time. This not only saved the farmer time and labor but also optimized water usage.
How to Maintain Large Diameter Pipes in Agricultural Systems
While large diameter pipes are durable, regular maintenance is essential to ensure optimal performance. Here are some tips:
Inspect Regularly: Check for leaks, cracks, or blockages periodically.
Clean Internally: Remove sediment buildup to maintain consistent water flow.
Protect Against Corrosion: For metal pipes, use anti-corrosion coatings or linings.
Monitor Pressure Levels: Maintain appropriate water pressure to prevent pipe damage.
Store Properly: If pipes are not in use, store them in a shaded area to avoid UV degradation.
Future Trends in Large Diameter Pipes for Agriculture
The agriculture sector is continuously evolving, and so are the technologies associated with it. Here are some trends to watch:
1. Smart Irrigation Systems
The integration of IoT sensors with large diameter pipes allows real-time monitoring of water flow, pressure, and quality, enhancing efficiency.
2. Sustainable Materials
Eco-friendly materials, such as recycled plastics, are being used to manufacture large diameter pipes, reducing environmental impact.
3. Customizable Solutions
Manufacturers are offering customized pipe solutions tailored to specific agricultural requirements, ensuring better performance.
4. Hybrid Systems
Combining large diameter pipes with renewable energy sources, such as solar-powered pumps, is gaining popularity among eco-conscious farmers.
Conclusion
Large diameter pipes are revolutionizing the way water is managed in irrigation and agriculture. From ensuring efficient water transportation to reducing wastage, these pipes contribute significantly to sustainable farming practices.
At Tube Trading, we take pride in being a trusted large diameter pipe distributor in Gujarat and a reliable large diameter pipe dealer in Vadodara. Whether you are looking to set up a new irrigation system or upgrade your existing one, we are here to provide high-quality solutions tailored to your needs.
For more information or to explore our product range, contact Tube Trading today and take a step towards efficient and sustainable agriculture.
#Large dia pipe distributor in Gujarat#Large dia pipe dealer in Vadodara#Large dia pipe supplier in Gujarat#steel industry#steel companies#seamless pipes#piping material#plumbing industry#construction structure#construction sites#stainless steel pipes#Gujarat#Rajasthan#Madhya Pradesh#Maharashtra#Uttar Pradesh#Daman#India
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being aromantic is like. hey btw you're going to live a life that is the culmination of most of society's worst nightmares. sorry lol ✌️ but then you turn around and take a really good hard look at it and it turns out that living in that nightmare is fucking awesome and you get to wake up every day and take that fear that other people have and laugh and hold it close until it's a great joy for you instead. and being happy is a radical act that you define instead of someone else. and you're sexy as fuck that's just a fact of life i don't make the rules on that one
#aromantic people are just sexy i'm not making the decisions here it's just facts#course ur hot as fuck. it came free with the aromanticism#being sexy is just default settings for aromantic people 👍#hope this all helps. anyway i'm on my 'i hope i die alone <3 i can't wait to die alone <3' kick rn#i think the existential fear that people have of Not Partnering specifically is so. well.#obviously that shit is strong and it is SO awesome to be free of it.#realizing you're aro and you don't Want a partner can be such a hit to the solar plexus#cause society says that's the only thing that'll make you happy. so either you go without that thing or you force yourself#into doing something you don't want which would make you unhappy anyway.#so you think it's a lose lose situation and you have to come to terms with what amatonormativity presents as the worst possible situation#but then! whoa! turns out personhood is inherently valuable in and of itself and romantic partnering is just a construct!#and that nightmare is now your life to do with as you please... define as you will... structure as you want...#best case scenario. is what i'm saying.#every day i wake up ready to spit all that amatonormative rhetoric back in life's teeth by being alone and being happy#and it's so fucking satisfying. every day.#fucking JUBILANT being by myself. and i love being a living breathing 'fuck you' to the romantic system#you need a partner to be happy? oh that's sooo fucking crazy guess i'll go be miserable then. in my perfect fucking dream life lmao#yeah obviously it's the worst possible outcome on earth to die without a partner. so terrible. can't wait for it :)#aromantic#aromanticism#aro positivity#aroace#arospec#sorry to bitches who are sad about not having a partner. i could not give a fuck though get better soon#you couldn't EVER pay me enough to go back to a mindset in which my inherent value wasn't enough by myself.#FUCK that shit. absolutely miserable and a bad life outlook in general. like genuinely do the work w/ amatonormativity and get better#life is something that can be so fulfilling whether someone wants to kiss you or whatever or not#i'm on antidepressants and i have people i care deeply about. what the fuck would i need a partner for lmao
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A protection that becomes more creepy
Azul in my heart. You can see the original art here and read the monster list here @lustlovehart
[Alt under the cut]
My first concept, since my style could not simulate the texture of slime in its purest state
It is quite thick so water can not enter or wet. Only small puddles where you can accumulate
It is a monster and that, magic, but I can imagine that it can only reach a height by the pressure, it can come out expelled sometimes
#When I read that I had some kind of armor so it did not expand everywhere#I imagine it’s some kind of spell that must carry with you to stay a little more solid#It would also be interesting that to have shape must have a skeleton. As any construction with cement. Found this and clean it#it lacks some parts but otherwise is a good structure#It’s fun to think that your vision distorts by the drastic change in size so you should wear lenses#I wonder what function their hearts will have if there is no blood for two of them#He would be perfect to experiment scientifically#I thought two hearts set in the head until you see the original art#it would be funny that you try to kill him in the chest but only become a little silly#twisted wonderland#twst#fanart#digitalart#drawing#monster boy#creature design#azul ashengrotto#twst azul#honneydraws ⊹⃬۫🍜̸᩠໋࣪꣹۫
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Cathedral of Zagreb Located: Zagreb, Croatia
#Cathedral of Zagreb#cathedral#construction#photography#scenery#structure#architecture#Zagreb#Croatia
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MAYUMI MIYAWAKI MATSUKAWA BOX, 1978 Tokyo, Japan Image © Shinkenchiku-sha
#architecture#design#architect#art#designer#archdaily#artwork#photography#juliaknz#form#material#space#artist#architectural form#dezeen#construction#structure#residential#house#housing#japanese architecture#japanese houses
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one of the wild things about people’s stubborn insistence on misunderstanding The Ones Who Walk Away From Omelas is that the narrator anticipates an audience that won’t engage with the text, just in the opposite direction. Throughout the story are little asides asking what the reader is willing to believe in. Can you believe in a utopia? What if I told you this? What about this? Can you believe in the festivals? The towers by the sea? Can we believe that they have no king? Can we believe that they are joyful? Does your utopia have technology, luxury, sex, temples, drugs? The story is consulting you as it’s being told, framed as a dialogue. It literally asks you directly: do you only believe joy is possible with suffering? And, implicitly, why?
the question isn’t just “what would you personally do about the kid.” It isn’t just an intricate trolley problem. It’s an interrogation of the limits of imagination. How do we make suffering compulsory? Why? What futures (or pasts) are we capable of imagining? How do we rationalize suffering as necessary? And so on. In all of the conversations I’ve seen or had about this story, no one has mentioned the fact that it’s actively breaking the fourth wall. The narrator is building a world in front of your eyes and challenging you to participate. “I would free the kid” and then what? What does the Omelas you’ve constructed look like, and why? And what does that say about the worlds you’re building in real life?
#ursula k le guin#omelas#There are so many ideas in this story that simply do not get engaged with!#I’ve heard it argued that a central element of anarchism#as a political philosophy#is the expansion of the imagination: what is truly possible if we forget the structures we are raised in?#if we forget what we have been told is or is not possible?#le guin wasn’t an anarchist but her work is heavily inspired by anarchist thought#Also the idea that the compassion of citizens of omelas is possible only because they are able to see themselves in relation and contrast t#the kid#very interesting stuff there#arguably a searing critique of moderate liberals#who feel compassion from on high but rationalize the ways in which those who suffer cosmically deserve it#in order to maintain structures of suffering#This short story is breaking the fourth wall Constantly to grab you by the collar and ask#what do you think is possible in the world and what do you think is good and what do you think is necessary#If you want to free the kid then what! What does that mean!#ALSO if omelas is a place being constructed as an idea#are the ones who walk away meant to be literally deserting a place#or are they rejecting an idea#hmmm much to think about
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we get a lot of really great stuff in system collapse about murderbot's relationships with ART and ratthi, which makes sense, because it spends almost the entire book with them. but i also love how even though mensah isn't there for most of the story, other people keep reminding mb of her:
chapter 2, page 25: “From ART’s personnel file, she [Karime] was older than Mensah and she didn’t look like an intrepid space explorer, either, even in the protective environmental suit.”
2, 27: “It took Karime three seconds to process the abrupt statement. (She was almost as good at not looking annoyed as Mensah was.) She kept her expression neutral and patient.”
2, 28: “In the underground colony room, Karime lifted her brows. ‘Another occupied site?’ I thought she was being careful not to show too much reaction. It was the way Mensah would have played it.”
4, 70: “Iris looked at me and I saw her hesitate, because her hesitation looked a lot like Dr. Mensah’s hesitation. And I realized I really didn’t want to go down there.”
5, 104: "Iris has that same thing as Dr. Mensah, the thing where she’s able to look and sound calm under circumstances where shit is possibly about to go down.”
it's spent so much time with her and it knows her so well and respects her so much that she's the model against which it compares all other humans. it thinks about her when they're not together. it's protective of her. it has such total faith in her competence. it (non-romantically) loves her and doesn't want to not see her again. idk man, it just gets to me! and they were teammates (oh my god they were teammates!!)
bonus:
I said, aloud, "You have to be kidding me." (ch. 2, p. 28)
seven pages later, in reaction to the same thing:
Mensah had had time to review the feed video. She muttered, "Oh, you have to be kidding me." Yeah.
twinsies 🥰
#something about their relationship is just sooo potent because of the structural power that mensah has over mb#and how mb has always been trying to navigate how much it cares about her and respects her with how afraid it is#of being a thing that she owns#because even if she doesn't exercise that power over it the fact remains that the context in which they live makes that possible#like! socially constructed ideas of personhood creating inherent violence in the human-construct relation!#even if she means well. even if it's necessary for mb to have a legal owner so that some corporate doesn't claim it as property.#it's not that mensah is an unsafe person it's that the power imbalance is intrinsically baked in because of#the broader social context of constructs as nonpersons. as property and equipment#all of that is true! AND it respects her and loves her. AND they're teammates! they are both trying so hard#to create sth just and beautiful out of sth really messed up. and trusting each other to collaborate in that way!#it was SO necessary for mb to leave at the end of ASR and SO necessary for it to get proof at the end of ES#that she would not try to stop it from leaving. that it was free. that it was autonomous#idk idk i'm rambling i just. i care them#the murderbot diaries#mb teammates#mb sc#system collapse spoilers#system collapse#my posts#murderbot
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i somehow feel like most public bathrooms probably use the floor toilets bc they're... probably cheaper....
i pondered about it for a bit so here are my offerings
#ask#askart#i don't think someone has to manualy clean the poo like the old days and theres a powerhose system that can sweep it down a connected drain#yes i think about these 24/7 im actually crazy#i know nothing about construction but does it cost more to make the same structure but to put half of it into empty walls? guess it does#but i belive in fluffverse tax dollars are used in these things as tax money is intended
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i hate men who want girlfriends who are basically mother figures that also suck dick. hop on the incest train and leave the bad bitches alone.
#i hate little emotional boys i would line them up and shoot them if i could#idc what social structures it enforces put them in hvac or construction! go build smthn
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It's a late Friday evening in Midorijima, and that means that it's time for the weekly Benishigure meetup at the Black Needle. It's always a good opportunity to meet new members, get drunk, and have some fun with the other guys you'd usually miss due to conflicting schedules. Today's meeting is a bit more special- Aoba decided to join in, which he doesn't always do. He's the boss' boyfriend, but he's not officially a member himself- he has, however, gotten close to the gang in the time they've been together, so everyone just treats him like an honorary Benishigure, which both makes him happy, in a way, and annoys him to no end, given all the attention he recieves and the fact that he's a subject of never-ending jokes and bits (in a friendly way, of course).
Everyone's a bit drunk by now, laughing and toasting for anything that comes to their heads, no matter how small and insignificant. Koujaku is smoking a cigarette and sipping on his sake, taking it all in and quietly enjoying seeing the people he brought together. Aoba had a bit to drink too; He rarely does, but something tempted him to drink today- perhaps a desire to join in and fully enjoy the atmosphere was stronger than his usual convictions.
The conversation turns to relationships, as it always does at some point when a group of drunks congregates.
"Wait, how long have you two been together? Like, a year?" One of the Benishigures asks as he turns to Aoba and Koujaku. "That's pretty impressive, I gotta admit- it's probably the longest anyone stayed with Koujaku"
He barks a laugh, like he was suprised at how funny he found his own joke. Aoba looks at Koujaku with suspicion, but he only responds with a grin and a sly look in his eye- completely unmoving, cigarette still held in his teeth.
"You thought about tying the knot yet? I'm telling ya, if you managed to keep him around for a year, that means you gotta have something special going on."
"I doubt any shrine would be willing to do the ceremony for two guys." Aoba rolls his eyes. He's mostly trying to convince himself; Both he and Koujaku are respected in their local community, so he has no doubt that their potential union would be honored. He thinks that this whole "being in a relationship with a man" had sunken into his mind by now, but in moments like these, he really worries that he will never internalize it fully.
"Since when do you need a shrine?" The Benishigure snorts dismissively and waves his hand around with no grace at all; His eyes have a spark in them that already tells Aoba that he came up with something only a drunk could make up. "You're already with your friends and family, right...? And everyone's wasted on sake...? All we need now is for the lovebirds to smooch." He grins. "I can even be your official, if you want"
He raises his mug, beer and froth sloshing with his uncoordinated movements and begins to chant "KISS! KISS! KISS!", his yell echoing through the bar. All the other members at the table quickly abandon whatever they were talking about and join in, not knowing what they're cheering for, but knowing that it's paramount to see their leader and his boyfriend make out at the table this instant.
This unrest finally rouses Koujaku. He knows that Aoba doesn't enjoy being the center of attention, and especially not when he's being goaded into doing something by a bunch of rowdy, drunk dudes. He usually trusts his men to know when to cool it when it comes to teasing Aoba, but it seems like alcohol might've clouded their judgement.
He grabs his cigarette between his slender fingers and opens his mouth to speak. Before he manages to make a single sound, Aoba is pressing his lips against his, tasting the smoky flavor of the kiss. The room erupts in cheers, and the clinking of glasses and mugs raised in toast adds to his overwhelmed state.
In a way though, Koujaku is oddly gleeful about it all; He never expected to get married, or at least not like that- when he was younger, he'd probably imagine a serious, grim ceremony where he's forced to marry a girl he never saw before, one that was chosen by his father in order to strenghten their family's position in the criminal underword. Nowdays he has no family he could invite to such a ceremony, but the Benishigure are the closest thing he has, and he loves them like he would his own blood, so getting married while listening to their howls and cheers is probably how he'd like it to go anyway. As for Aoba... Being able to be around him in any capacity was a dream come true; Koujaku knew that he himself is never going anywhere anyway. But thinking about Aoba pledging his loyalty to him with such fervor, and in front of so many other people... He's just happy he's not a crying kind of drunk. Instead, he chooses to cup Aoba's head with his free hand, gently rustling the short, stiff hair right at his hairline.
On the next day, Aoba doesn't talk about the kiss, but when Koujaku tries to talk to him about it, he can see that Aoba remembers it all happening by the way he stammers, badly puts on a facade and downplays the whole event. Koujaku just laughs; He knows that Aoba needs to process it- he knows him well enough to know how he behaves after events like this, and Koujaku is, if nothing, a patient man. Besides, Aoba was always adorable when he was in that part of processing something, so he really doesn't mind.
Benishigure who are "in the know" sometimes call Aoba "bride" to mess with him, or ask Koujaku to say hi to his "wifey" from them. Mizuki finds out from them through rumors that the Black Needle hosted their very first wedding/reception when he happened to have a night off and he can't live that down. As for Aoba and Koujaku, they don't really consider themselves an officially married couple, but the bit is nicer to carry out than they thought- jokingly talking to eachother like an old married couple comes to them more naturally and is more fun than either has expected.
#dmmd#dramatical murder#kouao#aoba seragaki#koujaku#hatter blathers#ahhhhhh im kinda nervous to post it 😖 i fought with myself for two days about this#but you know what? it is the site to get sillay about your fictional favs. devils sacrament and all that#so i apologize if this is cringy or terribly written lol i just had a need to do it#i took a break from writing due to burnout and this is a nice way to exercise that muscle you know?#short and sweet#anyway i saw that a lot of people liked my last aoba and benishigure post#so i figured i can explore it a bit more#idk why everyone in benishigure is a fujo for their leader. just that kind of universe i guess lmao#i think theyd call aoba bride or wifey bc i feel like it just suits them more than groom and hubby#like idk. they see koujaku as the epitome of masculinity and the way gender roles and sexuality is constructed in this universe#feels a bit vague. which is good for headcanons but it can be a bit tricky to construct#they just mean it as a joke. theyre not doing this maliciously.#i like aoba becoming more ingraied into the benishigure structures once he starts dating koujaku#he never becomes a member but hes kind of an honorary second in command and hes liked and respected despite the teasing#IM SORRY IF MY WRITING IS ASS i was trying to be concise and add some flourish without going overboard#my non-dmmd mutuals: you didnt see anything 😶 im sorry if my perception of me has forever shifted#or something. im baring my soul to everyone#the sake thing is a part of japanese wedding tradition where the married couple sips sake from eachothers cups#it has more meaning and is more complex ofc but. you know. nothing in this scene is traditional lol
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Tube Trading is a leading stainless steel plates dealer in Vadodara and Gujarat, offering top-quality stainless steel plates for various industrial applications. As a trusted stainless steel plates supplier in Vadodara and Gujarat, we provide durable and precision-engineered plates that meet the highest standards of quality. Our commitment to excellence and reliable delivery makes us the preferred choice for stainless steel solutions in the region. With a strong focus on customer satisfaction, we ensure that every order is handled efficiently and delivered on time.
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this year my challenge for everyone is to unlearn the association between love and morality. love is not something that is inherently morally good, and the absence of love is not something that is inherently bad. sex without love isn't morally bankrupt, it's just an action. people without love aren't less kind or less good, they're just people. when we can get past this false (and often unnoticed) dichotomy of good love/evil lovelessness then i think we are going to be able to take leaps and bounds in sex positivity, aro advocacy, certain discussions of mental health...
#and also. not the direct focus. but love doesn't make things good. you can be in love and do terrible terrible things.#people do bad things in the name of love and in despite of love all the time.#but!! imagine a world where people could exist as people and not be demonized.#sex positivity means being cool about All sex. reexamine your internal systems of moral judgement.#this goes for sex workers. for aroallo people. especially aroallo men. for aro people in general who might enjoy sex.#and frankly i think it can easily bleed into discussions about mental health disorders around 'not feeling' certain things#especially demonizing ppl who don't feel as much empathy. i think there's definitely a correlation between that and the emphasis on love.#our support needs to go out to Everybody and i think these things are all structured together in one way or another!!#it might not be immediately obvious but when i tell you it all leads back to amatonormativity..... little bit wild.... large bit wild....#anyway. horror movie psychopath 'oh he can't feel emotions or love' damn alright. well. let's take a closer look at that.#silly that there's an association between lack of love and Murdering. feel like that might affect some stuff.#love is just an emotion/a feeling it doesn't mean anything about you one way or another#same with empathy. you can feel it all you want but it doesn't inherently change the actions you choose to take#anyway. thesis statement. there is a socially constructed link between love and morality. unlearn that.#kiss kiss (<— lovelessly)#aromantic#aromanticism#arospec#talking#aroace#aspec#sex positivity
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Okay so to get the additive group of integers we just take the free (abelian) group on one generator. Perfectly natural. But given this group, how do we get the multiplication operation that makes it into the ring of integers, without just defining it to be what we already know the answer should be? Actually, we can leverage the fact that the underlying group is free on one generator.
So if you have two abelian groups A,B, then the set of group homorphisms A -> B can be equipped with the structure of an abelian group. If the values of homorphisms f and g at a group element a are f(a) and g(a), then the value of f + g at a is f(a) + g(a). Note that for this sum function to be a homomorphism in general, you do need B to be abelian. This abelian group structure is natural in the sense that Hom(A ⊗ B,C) is isomorphic in a natural way to Hom(A,Hom(B,C)) for all abelian groups A,B,C, where ⊗ denotes the tensor product of abelian groups. In jargon, this says that these constructions make the category of abelian groups into a monoidal closed category.
In particular, the set End(A) = Hom(A,A) of endomorphisms of A is itself an abelian group. What's more, we get an entirely new operation on End(A) for free: function composition! For f,g: A -> A, define f ∘ g to map a onto f(g(a)). Because the elements of End(A) are group homorphisms, we can derive a few identities that relate its addition to composition. If f,g,h are endomorphisms, then for all a in A we have [f ∘ (g + h)](a) = f(g(a) + h(a)) = f(g(a)) + f(h(a)) = [(f ∘ g) + (f ∘ h)](a), so f ∘ (g + h) = (f ∘ g) + (f ∘ h). In other words, composition distributes over addition on the left. We can similarly show that it distributes on the right. Because composition is associative and the identity function A -> A is always a homomorphism, we find that we have equipped End(A) with the structure of a unital ring.
Here's the punchline: because ℤ is the free group on one generator, a group homomorphism out of ℤ is completely determined by where it maps the generator 1, and every choice of image of 1 gives you a homomorphism. This means that we can identify the elements of ℤ with those of End(ℤ) bijectively; a non-negative number n corresponds to the endomorphism [n]: ℤ -> ℤ that maps k onto k added to itself n times, and a negative number n gives the endomorphism [n] that maps k onto -k added together -n times. Going from endomorphisms to integers is even simpler: evaluate the endomorphism at 1. Note that because (f + g)(1) = f(1) + g(1), this bijection is actually an isomorphism of abelian groups
This means that we can transfer the multiplication (i.e. composition) on End(ℤ) to ℤ. What's this ring structure on ℤ? Well if you have the endomorphism that maps 1 onto 2, and you then compose it with the one that maps 1 onto 3, then the resulting endomorphism maps 1 onto 2 added together 3 times, which among other names is known as 6. The multiplication is exactly the standard multiplication on ℤ!
A lot of things had to line up for this to work. For instance, the pointwise sum of endomorphisms needs to be itself an endomorphism. This is why we can't play the same game again; the free commutative ring on one generator is the integer polynomial ring ℤ[X], and indeed the set of ring endomorphisms ℤ[X] -> ℤ[X] correspond naturally to elements of ℤ[X], but because the pointwise product of ring endomorphisms does not generally respect addition, the pointwise operations do not equip End(ℤ[X]) with a ring structure (and in fact, no ring structure on Hom(R,S) can make the category of commutative rings monoidal closed for the tensor product of rings (this is because the monoidal unit is initial)). We can relax the rules slightly, though.
Who says we need the multiplication (or addition, for that matter) on End(ℤ[X])? We still have the bijection ℤ[X] ↔ End(ℤ[X]), so we can just give ℤ[X] the composition operation by transfering along the correspondence anyway. If p and q are polynomials in ℤ[X], then p ∘ q is the polynomial you get by substituting q for every instance of X in p. By construction, this satisfies (p + q) ∘ r = (p ∘ r) + (q ∘ r) and (p × q) ∘ r = (p ∘ r) × (q ∘ r), but we no longer have left-distributivity. Furthermore, composition is associative and the monomial X serves as its unit element. The resulting structure is an example of a composition ring!
The composition rings, like the commutative unital rings, and the abelian groups, form an equational class of algebraic structures, so they too have free objects. For sanity's sake, let's restrict ourselves to composition rings whose multiplication is commutative and unital, and whose composition is unital as well. Let C be the free composition ring with these restrictions on one generator. The elements of this ring will look like polynomials with integers coefficients, but with expressions in terms of X and a new indeterminate g (thought of as an 'unexpandable' polynomial), with various possible arrangements of multiplication, summation, and composition. It's a weird complicated object!
But again, the set of composition ring endomorphisms C -> C (that is, ring endomorphisms which respect composition) will have a bijective correspondence with elements of C, and we can transfer the composition operation to C. This gets us a fourth operation on C, which is associative with unit element g, and which distributes on the right over addition, multiplication, and composition.
This continues: every time you have a new equational class of algebraic structures with two extra operations (one binary operation for the new composition and one constant, i.e. a nullary operation, for the new unit element), and a new distributivity identity for every previous operation, as well as a unit identity and an associativity identity. We thus have an increasing countably infinite tower of algebraic structures.
Actually, taking the union of all of these equational classes still gives you an equational class, with countably infinitely many operations. This too has a free object on one generator, which has an endomorphism algebra, which is an object of a larger equational class of algebras, and so on. In this way, starting from any equational class, we construct a transfinite tower of algebraic structures indexed by the ordinal numbers with a truly senseless amount of associative unital operations, each of which distributes on the right over every previous operation.
#math#the ongoing effort of valiantly constructing complicated mathematical structures with 0 applications#i know i owe you guys that paraconsistency effortpost still#it's coming! just hard to articulate so far#so if you start with the equational class with empty signature your algebras are just sets#the first iteration of the construction gets you the class of monoids#but after that it's what i guess you could call 'near-semirings'?
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School in Vésinet, France - Lambert Lénak
#Lambert Lénak#architecture#design#building#modern architecture#interiors#minimal#modern#timber#timber frame#structure#wood#detail#construction#extension#school#interior design#hall#dining hall#windows#beautiful buildings#cool architecture#cool design#design blog#architectural#architectural photography#france#french architecture#school design#old and new
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Here take this meme I made the first time I listened to TMA s5
#the magnus archives#Tma#tma shitpost#Content warning blatant disregard for the actual tragic arc structure in tma#no this is not a well constructed image but knowing me I probably made it in <2 minutes
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#time capsule#flashback#retro#vintage#nostalgia#nostalgic#photo#photography#photograph#building#architecture#construction#structure#interior design#inside#indoors#light#lights#lighting#unusual#liminal space#liminal#mundane#surreal#mood#atmosphere#aesthetic#cool#awesome#interesting
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