## Equivalence of strong convergence

Is this Equivalence is true?

$$u_n rightarrow u$$ strongly in $$L^{infty}left( 0,T ;L^2left (Omegaright)right) iff u_n rightarrow u$$ strongly in $$L^2left (Omegaright)$$

## Draw all the non-isomorphic strong simple digraphs on 4 vertices and 5 arcs

The questions are.

1. Draw all the non-isomorphic strong simple digraphs on 4 vertices and 5 arcs.
2)If D is a digraph in which every vertex has out-degree 1, then show that D has
exactly one directed cycle.
3)A simple digraph D is called k-regular if outdeg(v) = indeg(v) = k, for every
vertex v ∈ V (D).
(a) Draw a 2-regular digraph on 5 vertices.
(b) Let k and n be integers such that 0 ≤ k < n. Describe a construction to
obtain a k-regular digraph on n vertices.
4)If D is a weakly connected digraph, then show that m ≥ n − 1.

## complexity theory – how to prove that \$NP cap co – NP\$ = { S | S such that there exist a Strong Deciding Algorithm for S}?

i need to prove that and i find it struggle:
given:

for deciding problem S:

a non deterministic algorithm $$A(x)$$ is strong deciding algorithm if:

1. $$x in S =>$$ fo every run of $$A(x)$$ returns “Yes” or “Maybe”, and there exist run which $$A(x)$$ returns “Yes”

2. $$x notin S =>$$ for every run of $$A(x)$$ returns “No” or “Maybe”, and there exist run which $$A(x)$$ returns “No”

now we need to prove i assume with equality between two sets that

$$NP cap co – NP$$ = { S | S such that there exist a Strong Deciding Algorithm for S}

thank you,

## Is this (slightly revised) conversion of a 5e homebrew race in line with the power of “strong” (but still LA+0) 3.5e races?

I found a 5e “Anime Girl” race on dandwiki.com. Realistically most “anime girls” are just humans with plenty of class levels, but for the sake of general memery with some friends, I wanted to try converting the first listed subrace to 3.5e.

## The Preface

I have come to understand that if you follow the link and read the name of the subrace I’m trying to convert, it’s a word that may raise some flags. It is not–and was never–my intention to sexualize anyone with the content of this question, nor the plans behind it. My friend group uses that term entirely wholesomely, unlike what (from a quick Google search) appears to be the case for the world at large. Because I use it and hear it used in a non-sexual context, I had forgotten that it even had such a disgusting connotation.

If I had specced more into Wisdom then maybe I would’ve looked more into the word and confirmed its inappropriateness for general use, but it didn’t even cross my mind–however, now I know. I apologize for using it in the previous version of this post, and for any feathers I may have ruffled by doing so. For the rest of the question, I’ll instead refer to the 5e subrace–and, by extension, 3.5e version of the race–as “Short Anime Girl”.

Link to the race/subrace

## The Goal

Despite how it may seem not-so-serious, I’d still like for the race to be reasonably well-balanced and completely usable as an actual race–preferably closer to the high-tier options like dwarves and humans, but without being too powerful to be considered LA+0. If it falls a little short, that’s fine, but if it’s like kobolds or half-orcs which receive very little benefit compared to their drawbacks (especially relative to other races), then it could probably stand to be buffed.

Ideally, I want to keep the race as close to the source as possible, so as to just make a “3.5e version” of the 5e race rather than inventing anything too new. The original has seen numerous revisions, so I took the page’s entire history as valid inspiration, but I tried not to add or alter anything unless it was based on something that the 5e version of the race has actually mentioned at some point.

## The Changes

The 5e version currently gives +1 to Charisma from the base race, and +1 to three other ability scores of the player’s choice from the subrace. Previous versions of the subrace only gave Charisma bonuses of varying sizes. Early versions of the race came with a Strength penalty but very large bonuses to other scores. Most versions of the subrace also imposed disadvantage on Strength rolls. All this in mind, I gave the race +2 Charisma, -2 Strength. The DMG says this is a highly-unfavorable set of modifiers, but the general consensus among players seems to be that such weighting is incorrect. The 5e race seems like one of its major benefits is having above-average total ability score bonuses, but aside from stuff like Lesser Planetouched, I think every 3.5e LA+0 race has net-zero ability score modifiers, so that’s the convention I followed.

The subrace has a size of Small, which I kept. Some versions have a walk speed of 25 feet, while others do not modify the race’s 30-foot speed. Specific beats general, and small/slow races in 3.5e tend to have a land speed of 20 feet, so I went with that. Most versions of the subrace have a carrying capacity increased to that of a Large creature, though. This seems like quite a huge jump, considering (or perhaps to counter) the reduced Strength, so I made the target carrying limit that of a Medium creature. 3.5e’s Powerful Build seems to do everything but increase carry limits, so I instead translated this carrying boost into allowing weapons and lifting/carrying limits as if the race were Medium.

Almost every version of the 5e race has had proficiency in Charisma (Persuasion) checks. Most versions of the subrace also gained proficiency in Charisma (Deception) checks as well as advantage on “Charisma and Persuasion checks”, despite having disadvantage on Charisma (Intimidation) checks. The current version replaces all these advantage/disadvantage traits with a limited-uses-per-day advantage on Charisma (Persuasion) checks. All in all, I took the idea of advantage on “Charisma…checks” and went with a +2 racial bonus on Charisma-based skills that involve interaction with intelligent creatures, with the exception of Intimidation, which gets a -2 racial penalty. I don’t know of any other races with racial skill penalties, but +2 to skills is quite common.

Another feature which has been universal among versions of the 5e race is awkward fall. In most versions this has been an action which damages the user and knocks her prone, while forcing the target to make an (untyped) saving throw or also take damage and fall prone. The most recent version clears up the appropriately-awkward wording, but attaches the feature to an optional part of the Attack action. I’m not aware of anything in 3.5e that causes you to take damage based on your own Strength modifier (which all versions of the 5e feature have done, despite the fact that that’s a lot of damage in a system where falling 10 feet deals just 1d6), but there is the bull rush special attack. So I just made an alternative to bull rushing, which knocks the user prone and–if successful–knocks the target prone, as well.

Almost all versions of the 5e race have had a trait which wanted slight reflavoring for the subrace. When the character is dealt bludgeoning damage, the attack “bounces off” and deals damage to the attacker. It’s unclear whether this is meant to negate the attack, but that would be extremely powerful. Instead, I simply gave the race DR 2/piercing or slashing (where 2 is based on the 1d4 recoil damage of the original)

Lastly, many versions of the 5e subrace have had a trait which has some implications that aren’t relevant to the 3.5e short anime girl race–and, if read literally, is far too powerful for any LA+0 racial trait, anyway. The name of the trait has to do with the idea of other creatures breaking rules in a not-okay way, but I figured it might work if I flipped it entirely, so that the trait instead allows the short anime girl herself to “break some rules” in a way that makes sense for the race:

3.5e (unlike 5e, AFAIK) generally requires characters to be adults (which, as with the 5e anime girl race at large, happens at the same age as for humans). However, there are lots of examples of characters–never mind just short girls–in anime becoming adventurers (even in “difficult” professions like wizard or monk) at any age, so it made sense to allow class levels regardless of age. It’s also particularly common for anime characters (short girls included) to live for hundreds or thousands of years. The current version of the 5e race imposes the same lifespan as humans, but earlier versions of the race said they “will live for however long they will”, which isn’t very clear but seems to imply that they can either will themselves to live, or are able to live until killed. Considering either reading of this phrase, getting rid of the maximum age made sense–in which case, it seemed too powerful to allow mental scores to accrue bonuses without penalties (but immortal characters in media seem none the weaker for their age, so physical age penalties didn’t make sense either), so I removed age-related ability score changes entirely, à la the Endless trait from Dragon Magazine. D&D Wiki has another race with many similarities to short anime girls (but which is pretty overpowered and wouldn’t lend itself to being converted to a LA+0 3.5e race) that has a trait which also gives an indefinite lifespan, so I took the new name from that, discarding any of the actual effects/flavor of the 5e subrace’s trait.

As for the last bits that all 3.5e races require: Common and “Senpai”(Japanese) are Automatic Languages, just like the 5e race. In 5e, Senpai can be traded out depending on the character’s appearance, but I don’t know of any 3.5e races that allow such language options. The race has any normal Bonus Languages to match the 5e version’s freely-chosen extra language. Also, any class can be a Favored Class, since 5e doesn’t have multiclass penalties.

## The Result

Short Anime Girl Traits

• +2 Charisma, -2 Strength.

• Small: As a Small creature, a short anime girl gains a +1 size bonus to Armor Class, a +1 size bonus on attack rolls, and a +4 size bonus on Hide checks. However, short anime girls can use weapons designed for creatures one size larger without penalty, and their lifting and carrying limits are treated as if they were one size larger.

• Short anime girl base land speed is 20 feet.

• +2 racial bonus on Bluff, Diplomacy, Disguise, Gather Information, and Perform checks.

• -2 racial penalty on Intimidate checks.

• Awkward Fall: When a short anime girl performs a bull rush, she can choose to turn the action into an awkward fall, gaining a +4 bonus on the Strength check. If the awkward fall attempt is successful, the defender moves back a maximum of 5 feet and falls prone. Regardless of the result, the short anime girl then falls prone.

• Protective Softness: Short anime girl have damage reduction 2/piercing or slashing.

• Limited Immortality: Short anime girls can take class levels at any age. They receive no bonuses or penalties due to age, and won’t die of old age.

• Automatic Languages: Common and Senpai. Bonus Languages: Any (other than secret languages, such as Druidic).

• Favored Class: Any. When determining whether a multiclass short anime girl takes an experience point penalty, her highest-level class does not count.

## The… Problems?

Most of these features seem relatively minor to me, but many of them are things which don’t really have much precedent that I’m aware of when it comes to LA+0 races.

As such, I can’t decide if the race is super weak (the ability score modifiers are supposedly quite poor and no number of years will further increase Charisma, awkward fall is largely inconsequential–especially for a Small race, which also has to deal with reduced speed, and the race doesn’t have too many useful features going for it) or super strong (Small size but with Medium weapons/carrying gives powerful bonuses at little cost, the wide range of skill bonuses are very useful in urban adventures, inflicting prone every turn is too dominant or synergizes too well with some particular Feat, damage reduction increases survivability too much or is out-of-line with anything any other race receives, or there’s some unforeseen consequence to being able to adventure at any age).

So, as boring a phrase as I’m sure this may sound here… “Is this race balanced?”

## Is this conversion of a 5e homebrew race in line with the power of “strong” (but still LA+0) 3.5e races?

I found this 5e “Anime Girl” race. Realistically most “anime girls” are just humans with plenty of class levels, but for the sake of general memery, I wanted to try converting the loli subrace to 3.5e.

## The Goal

Despite how it may seem not-so-serious, I’d still like for the race to be reasonably well-balanced and completely usable as an actual race–preferably closer to the high-tier options like dwarves and humans, but without being too powerful to be considered LA+0. If it falls a little short, that’s fine, but if it’s like kobolds or half-orcs which receive very little benefit compared to their drawbacks (especially relative to other races), then it could probably stand to be buffed.

Ideally, I want to keep the race as close to the source as possible, so as to just make a “3.5e version” of the 5e race rather than inventing anything too new. The original has seen numerous revisions, so I took the page’s entire history as valid inspiration, but I tried not to add or alter anything unless it was based on something that the 5e version of the race has actually mentioned at some point.

## The Changes

The 5e version currently gives +1 to Charisma from the base race, and +1 to three other ability scores of the player’s choice from the subrace. Previous versions of the subrace only gave Charisma bonuses of varying sizes. Early versions of the race came with a Strength penalty but very large bonuses to other scores. Most versions of the subrace also imposed disadvantage on Strength rolls. All this in mind, I gave the race +2 Charisma, -2 Strength. The DMG says this is a highly-unfavorable set of modifiers, but the general consensus among players seems to be that such weighting is incorrect. The 5e race seems like one of its major benefits is having above-average total ability score bonuses, but aside from stuff like Lesser Planetouched, I think every 3.5e LA+0 race has net-zero ability score modifiers, so that’s the convention I followed.

The subrace has a size of Small, which I kept. Some versions have a walk speed of 25 feet, while others do not modify the race’s 30-foot speed. Specific beats general, and small races in 3.5e generally have a land speed of 20 feet, so I went with that. Most versions of the subrace have a carrying capacity increased to that of a Large creature, though. This seems like quite a huge jump, considering (or perhaps to counter) the reduced Strength, so I made the target carrying limit that of a Medium creature. 3.5e’s Powerful Build seems to do everything but increase carry limits, so I instead translated this carrying boost into allowing weapons and lifting/carrying limits as if the race were Medium.

Almost every version of the 5e race has had proficiency in Charisma (Persuasion) checks. Most versions of the subrace also gained proficiency in Charisma (Deception) checks as well as advantage on “Charisma and Persuasion checks”, despite having disadvantage on Charisma (Intimidation) checks. The current version replaces all these advantage/disadvantage traits with a limited-uses-per-day advantage on Charisma (Persuasion) checks. All in all, I took the idea of advantage on “Charisma…checks” and went with a +2 racial bonus on Charisma-based skills that involve interaction with intelligent creatures, with the exception of Intimidation, which gets a -2 racial penalty.

Another feature which has been universal among versions of the 5e race is awkward fall. In most versions this has been an action which damages the user and knocks her prone, while forcing the target to make an (untyped) saving throw or also take damage and fall prone. The most recent version clears up the appropriately-awkward wording, but attaches the feature to an optional part of the Attack action. I’m not aware of anything in 3.5e that causes you to take damage based on your own Strength modifier (which all versions of the 5e feature have done, despite the fact that that’s a lot of damage in a system where falling 10 feet deals just 1d6), but there is the bull rush special attack. So I just made an alternative to bull rushing, which knocks the user prone and–if successful–knocks the target prone, as well.

Almost all versions of the 5e race have had a trait which wanted slight reflavoring for the subrace. When the character is dealt bludgeoning damage, the attack “bounces off” and deals damage to the attacker. It’s unclear whether this is meant to negate the attack, but that would be extremely powerful. Instead, I simply gave the race DR 2/piercing or slashing (where 2 is based on the 1d4 recoil damage of the original)

Lastly, many versions of the subrace have had a trait which probably existed mostly as a joke, but has the potential to be incredibly powerful if played straight: while in a city, any creature that feels sexually attracted to the character is “instantly” arrested. Gating in a zelekhut would be pretty funny, but doesn’t seem reasonable for a racial trait. Seeing as how the trait is called “Illegal”, I retooled its effects, so as to bypass a couple 3.5e systems in a way that I thought made sense: The character can ignore most things in the Random Starting Ages and Aging Effects tables. 3.5e (unlike 5e, AFAIK) generally requires characters to be adults, but lolis in media are frequently not adults, so it made sense to waive that restriction. The hundred-(or thousand-)year-old loli is also a common trope, and earlier versions of the 5e race said they “will live for however long they will”, which isn’t very clear but seems to imply that they can either will themselves to live, or are able to live until killed. Either way, getting rid of the maximum age made sense for that–in which case, it seemed too powerful to allow mental scores to accrue bonuses without penalties (but immortal lolis in media seem none the weaker for their age, so physical age penalties didn’t make sense either), so I removed age-related ability score changes entirely, à la the Endless trait from Dragon Magazine.

As for the last bits that all 3.5e races require: Common and “Senpai”(Japanese) are Automatic Languages, just like the 5e race. In 5e, Senpai can be traded out depending on the character’s appearance, but I don’t know of any 3.5e races that allow such language options. The race has any normal Bonus Languages to match the 5e version’s freely-chosen extra language. Also, any class can be a Favored Class, since 5e doesn’t have multiclass penalties.

## The Result

Loli Traits

• +2 Charisma, -2 Strength.

• Small: As a Small creature, a loli gains a +1 size bonus to Armor Class, a +1 size bonus on attack rolls, and a +4 size bonus on Hide checks. However, lolis can use weapons designed for creatures one size larger without penalty, and their lifting and carrying limits are treated as if they were one size larger.

• Loli base land speed is 20 feet.

• +2 racial bonus on Bluff, Diplomacy, Disguise, Gather Information, and Perform checks.

• -2 racial penalty on Intimidate checks.

• Awkward Fall: When a loli performs a bull rush, she can choose to turn the action into an awkward fall, gaining a +4 bonus on the Strength check. If the awkward fall attempt is successful, the defender moves back a maximum of 5 feet and falls prone. Regardless of the result, the loli then falls prone.

• Protective Softness: Lolis have damage reduction 2/piercing or slashing.

• Illegal (or Not): Lolis can take class levels at any age, even before reaching adulthood. If a loli reaches adulthood, she becomes a legal loli and will receive no bonuses or penalties due to age, and won’t die of old age.

• Automatic Languages: Common and Senpai. Bonus Languages: Any (other than secret languages, such as Druidic).

• Favored Class: Any. When determining whether a multiclass loli takes an experience point penalty, her highest-level class does not count.

## The… Problems?

Most of these features seem relatively minor to me, but many of them are things which don’t really have much precedent that I’m aware of when it comes to LA+0 races.

As such, I can’t decide if the race is super weak (the ability score modifiers are supposedly quite poor and no number of years will further increase Charisma, awkward fall is largely inconsequential–especially for a Small race, which also has to deal with reduced speed, and the race doesn’t have too many useful features going for it) or super strong (Small size but with Medium weapons/carrying gives powerful bonuses at little cost, the wide range of skill bonuses are very useful in urban adventures, inflicting prone every turn is too dominant or synergizes too well with some particular Feat, damage reduction increases survivability too much or is out-of-line with anything any other race receives, or there’s some unforeseen consequence to being able to adventure at any age).

So, as boring a phrase as I’m sure this may sound here… “Is this race balanced?”

## Can anyone recommend US colos with strong DDOS protection

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## How strong is a set theory necessary for practical purposes in sheaf theory?

Is it known how much ZFC is really necessary for the basic and familiar constructions and theorems in sheaf theory, in line with section II.1 (and its exercises) in Hartshorne's "Algebraic Geometry" textbook?

I apologize for this strange question. Here's the motivation: I'm a mathematician who works automatically and implicitly at ZFC, like most users of this website. Last year, some members of the philosophy department at my university showed me some ancient questions in metaphysics that seemed to me as if they were axiomatically accessible. In fact they were, and some of the structures studied by the philosophers turn out to be (not obviously) equivalent to bundles of sets in a certain topological space. I was able to progress on those problems using some classical and elementary ideas in sheaf theory, particularly the equivalence of bundle sheaves and local homeomorphisms. My colleagues in philosophy support this, but suggest that the unrestricted use of ZFC axioms to solve questions in philosophy may open the door to arguments by philosophers who have various kinds of skepticism about set theory, and that it is better to offer the argument in a way that uses only a fragment of ZFC as weak as possible.

I spent some time reviewing the proofs for the sheaf theory theorems I'm using, and it seems to me that all the proofs work in ZAC (Zermelo set theory with choice), but perhaps one can be further weakened. The result will be better if this kind of work is done by someone who knows set theory more than I do, so I'd be happier if someone else has already figured out how strong a set theory is needed for elementary envelope theory. Is this already out there somewhere in the literature? I'm not even aware of a magazine that would feature research of this kind, but perhaps people working in some areas of the fundamentals of mathematics have a place to solve such things.

I apologize for my ignorance of the fundamentals of mathematics, and I also apologize again for this very strange question.