u/Solonarv

the artillery turret also outranges worms

your 1700 life total is abysmal and is the number one reason you keep getting exploded. you want to double or triple that at minimum, depending on what other defenses you have. 

you can do this by grabbing a few more life nodes on the tree (you're right next to a few clusters you haven't taken), and getting good life rolls on more of your gear.

a 6link is easily within reach: you can buy a 6link base item with high ilvl (84+) and just throw some dense fossils at it. it should only take a few tries to hit something decent. take the life mastery for "increased life if no life mods on body armour".

on your passive tree you have three jewel sockets and I don't see any jewels socketed in them. those aren't doing anything, either respec them or fill them. 

your flasks are not doing much for you. i would run something like this: - one instant divine life flask - a second life flask if you want - jade flask - some combination of diamond, quartz, quicksilver, silver

use instilling orbs to automate your utility flasks if you don't want to press them yourself. I play a similar build and my flasks are instant life, quicksilver, diamond, jade, quartz. most of them have "gain charge on crit" and instilling to use when full, but honestly any prefix that gives you more flask uptime will do. use alterations to get the right flask prefix. make sure you have a flask to remove bleed, you can beastcraft this.

you do not need a mana flask on this kind of build. get the Clever Thief cluster and the leech mastery for 10% instant leech; together with the mana cluster outside ranger (which you are already taking) this solves all your mana issues.

link faster attacks to your whirling blades.

get an immortal call linked to cast when damage taken. don't level those gems up too high, you want them to actually trigger often. level 2 or 3 on CWDT should be fine, immortal call as high as that allows.

it's very important to be freeze immune. you can get this on a flask, or you can upgrade the brine king pantheon power. i recommend brine king because it has 100% uptime. there are other methods as well but this is the cheapest.

I'm not sure which of your many socketed auras you're actually turning on, but I would run wrath and grace here. you don't need any of the rest.

there are more things that could be improved but at this point I'm running out of space. if you just want to look at a build guide, the pinned thread for this event has a list with several lightning strike POBs and some have videos as well.

There are two lightning ailments: shock and sap (sap reduces damage dealt). Anything that affects "lightning ailments" will affect both of those. You can't normally inflict sap and need specific uniques in order to do so.

Maximum shock is not affected by % increased effect of shock.

The new leading term is the sum of the two previous leading terms

Only if the addends have the same degree! Otherwise the leading term will just be the leading term of the higher-degree addend.

Wither roses are a simple way of preventing everything except wither skeletons from spawning but cannot be placed on nether bricks. Preventing other mobs (blazes and zombified piglins) from spawning is beneficial because it means they don't take up the mobcap.

I built ianxofour's simple design in my SP world, which indeed uses nether bricks inside the larger bounding box. It outputs about 135 skulls/hour, more than enough for typical singleplayer needs.

they will be on equal ground with the martials (except fighters of course, they always have a higher hit bonus)

martials get fundamental runes on their weapons, which puts them ahead of prof+abilitymod - AFAIK casters don't get anything equivalent. As a result casters will be behind martials (in terms of hit chance) even when targeting AC.

Are you looking for the definition, an intuition, examples, or something else?

Two functors F, G are adjoint if Hom(F(-),=) is naturally isomorphic to Hom(-,G(=)). F is the left adjoint and G is the right adjoint. The natural isomorphism is called an adjunction.

I see someone doesn't like Stone spaces!

(a Stone space has a base of clopen sets)

Yes. This should be expected since coequalizers generalize quotients. Proof sketch follows.

A coequalizer for f,g: X -> Y in Set is the quotient Y / ~, where ~ is the smallest equivalence relation with f(x)~g(x) forall x in X.

If we want to recover arbitrary quotients as coequalizers then for a given Y and ~ we need to find X, f, and g such that the property above holds.

Write Y as a union of equivalence classes E_i. For each class E_i, let X_i = E_i^2. Define X = union_i X_i, f(<a,b>)=a, and g(<a,b>)=b.

Now for some y~y' in Y, we have <y,y'> in X, and therefore f(<y,y'>)=y and f(<y,y'>)=y' are mapped to the same element of the coequalizer. Verifying that unrelated y,y' aren't mapped to the same element seems more tedious to prove, so I'll omit it.

As far as I'm aware there is no concise term for this; I imagine it's because the concept is rarely needed.

No, you just replace 0 by 1 in the statement of the axiom.

You obviously need to modify the PA a little but you can start the naturals at 1 just fine. You just end up with axioms like n+1=Sn and n×1=n instead of n+0=n and n×0=0.

The foundations don't really matter in most fields of math, so swapping out ZFC for some other (equally powerful) system doesn't really change anything. The only change is for fields that actually examine the foundations, but I'm not qualified to say what that changes would be.

Any time an entity goes through a nether portal, the 3x3 chunks on the other side get loaded for 15 seconds (also more chunks get loaded but won't process entities). So yes, if you throw an ender pearl through a portal it will teleport you once it hits something after returning to the same dimension.

The portal cooldown is also 15 seconds, so you will probably have to do some extra chunkloading on the nether side. You'd have to do that anyway if your nether portals are more than 31 blocks apart.

use pistons (w/ blocks on them) to push the items into specific spots, let them fall on trapdoors and come to a rest

Yes, I mentioned the crops not growing.

To answer the question in the title: You built this in the non-entity-processing border of the spawn chunks. The green box is the entity-processing part of the spawn chunks. If you built your breeder there, it would still work while you're not present, except for the crops not growing.

I'm not sure how viable "mill my opponents" is when everyone is running Thassa Oracle

It's fine. Thoracle is a sorcery speed play usually, so if you mill out your opponents they'll die on their draw step before they can cast thoracle. Occasionally the opponents might have emergence zone or something for instant speed thoracle, but you can see most of those lines coming easily.

Aside from portal linking you can also shoot them up using slimeblock launchers, not sure if that's actually faster than rails though. If nothing else it'll look cooler than a big rail spiral.

Functions are probably a much better fit for this than command blocks.

Here's a sketch:

Set up an objective called time_elapsed and an objective called increment_interval, both with criteria dummy.

# mypack:increment_time
# goes in #minecraft:tick
scoreboard players add Steve time_elapsed 1
execute if score Steve time_elapsed > Steve increment_interval run function mypack:grow_border

# mypack:grow_border
scoreboard players set Steve time_elapsed 0
scoreboard players add Steve increment_interval 7200
worldborder add 1

The conjecture is that specific sequences converge to some specific values. OP is asking for a proof of this; rewriting the conjectured limits in terms of the gamma function isn't a proof of anything about the original sequences.

The conjecture is that specific sequences converge to some specific values. OP is asking for a proof of this; rewriting the conjectured limits in terms of the gamma function isn't a proof.

There isn't a clear maximum among farms people have actually build, so here's the theoretical maximum:

Random ticks (i.e. sugarcane growing) can happen in any chunk whose center is within 128 blocks (horizontal only) of a player. That's about 200 chunks, I'm not going to count the exact number.

This gives you a total growing area of 200*256≃50000 m^2. Only 4/5 of those can have sugarcane on them, so 40000 plants per layer.

You can stack sugarcane farms on top of each other, and if you use flying machines carrying hopper minecarts to harvest, you don't need anything between layers. So you can have a layer every 4 meters vertically. The world is 384 m tall (from -64 to 320), allowing for 384/4=96 layers.

End result: 4 million sugarcane plants. This would result in about 12 million items/hour.

I'm confident in saying nobody has actually built and run this, because the number of items entities that would be generated is completely insane and would crash the server for sure.

How big a sugarcane farm you (or anyone) builds, then, is not determined by design limitations or game mechanics, but simply by a combination of how much the server can handle and how much effort the builder is willing to put in.

Without this foundation, exponentiation could only be defined for rational exponents even with a restriction to the reals.

Not so. The map q -> a^q, for rational q and a fixed positive rational a, is continuous. Since the rationals are dense in the reals, the map has a unique continuous extension to real q, which yields the usual notion of real exponentiation.

If you destroy 40 leaf blocks, the expected number of saplings you get is 1. That means you might get 0, or 1, or more than 1.

The probability of getting 0 saplings after destroying N leaf blocks is (39/40)^N. For N=40, that's about 36.3%.

If you want a 90% chance of getting at least one sapling, you need to break 91 leaf blocks. 99%, 182. No matter how many leaf blocks you break, it remaining possible to get no saplings, and no matter how many leaf blocks you've already broken, the probability of getting a sapling in the next 40 leaf blocks is still just 73.7%.