nt.number theory – A generalization of strong primes

In this post we denote the sequence of prime numbers as $p_k$ for integers $kgeq 1$. I don’t know if the following definition is in the literature.

Definition. We define the $theta$-strong primes, or strong primes at level $theta$, as the sequence of those prime numbers $p_n$ that satisfy the inequality $$p_n>theta, p_{n-1}+(1-theta)p_{n+1}tag{1}$$
for some fixed real number $0<theta<1$.

Remark. My definition arises as a generalization of the known as strong primes. See the definition in number theory of strong primes from the Wikipedia Strong prime.

Question. I would like to know, if my question is interesting, if it is possible to prove that there exists some $hattheta$ for which the corresponding sequence of prime numbers has finitely/infinitely many terms. Many thanks.

Here I emphasize that $0<theta<1$.

I have no intuition if my question is interesting, I don’t know what can be a more interesting question at research level about the study of these sequences $(1)$ and their corresponding $theta$. I did some calculations for $thetaneqfrac{1}{2}$ a rational number.

ct.category theory – Left Kan extensions of “strong” monoidal functors

Consider the 2-category $mathsf{MonCat}$ where objects are monoidal categories,
1-cells are strong monoidal functors, and 2-cells are monoidal natural transformations.

Given arrows $f: mathsf{C} to mathsf{D}, g: mathsf{C} to mathsf{E}$ in this category,
we can obviously ask for a left Kan extension $Lan_fg : mathsf{D} to mathsf{E}$.

If we instead work in $mathsf{MonCat}^mathrm{lax}$, the 2-category of monoidal categories, lax monoidal functors and monoidal natural transformations, this left Kan extension is known to exist under some reasonable assumptions.


  1. Under reasonable assumptions on the categories involved, does the “strong monoidal Kan extension” exist?
  2. Does the “lax monoidal Kan extension” recover the strong version in the case where the two input functors are strong monoidal? In other words, are the subcategories of strong monoidal functors stable under left Kan extension along strong monoidal functors?

At the highest level of generality, we can consider lax or strong $mathcal{O}$-monoidal functors between $mathcal{O}$-monoidal $infty$-categories, in the sense of Lurie.
In that context, the lax left kan extensions seem to correspond to Lurie’s “operadic Kan extensions”, but again there does not seem to be a strong version of this in the literature.

algorithms – Doubt regarding strong component in a graph

I know that strong component in a graph means between any 2 vertices there should be bi-directional path.

My doubt is cycle is always a strong component. can there be any other subgraph with some property P so that it is strong component ie., P implies strong component.

fa.functional analysis – Strong topology on a topological vector space

I’m not sure this is an appropriate question for this site but I’ve tried math stack exchange and got no answers. Also, this problem arose in one of my research problems, so I’m stating it here.

The strong operator topology is defined on Simon’s book as follows. It is the weakest topology on $mathcal{L}(X,Y)$ such all the maps $E_{x}: mathcal{L}(X,Y) to Y$ defined by:
$$E_{x}(T) := Tx $$
are continuous for all $x in X$. Here, $X$ and $Y$ are supposed to be Banach spaces and $mathcal{L}(X,Y)$ is the space of all bounded linear operators from $X$ to $Y$. A neighboorhood basis for this topology, in Simon’s words, is given by the sets of the form:
$$ {S: hspace{0.1cm} Sin mathcal{L}(X,Y), hspace{0.1cm} ||Sx_{i}||_{Y}<epsilon, hspace{0.2cm} i=1,…n}$$
where $x_{1},…,x_{n}$ is any finite collection of elements of $X$ and $epsilon > 0$.

I know the notion of strong topology can be extended to more general spaces such as topological vector spaces, but I don’t want to get too deep into the theory. However, I’m interested in the case where $X$ is not Banach but $Y = mathbb{C}$ is Banach.

My question is: In my setup, if $X$ is a Fréchet space and $Y=mathbb{C}$ is Banach, the above definition seems to work just fine if I replace $mathcal{L}(X,Y)$ the space of bounded linear operators to its analogue, the space of all continuous linear maps. The same properties seem to hold in this case. Is it a correct definition of a strong topology to my particular case? In other words, if I was to consider $X$ as a topological vector space and $X^{*}$ its topological dual, would the strong topology defined on $X$ be the same topology I’m proposing?

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?”

proof techniques – Lambda calculus without free variables is as strong as lambda calculus?

First question: How would one prove that by removing free (unbound) variables from lambda calculus, and allowing only bound variables, its power is not reduced (it is still Turing-complete) ?

Second question: Is the proposition given above really true? Is lambda calculus sans free variables really Turing-complete?

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Does this strong form of being almost 1 for 1 involve injectivity?

Let $ pi colon (X, T) to (Y, T) $ be a factorial map between the minimum sub-shifts. Suppose there is $ tilde {Y} subseteq Y $ such as

  1. $ # pi ^ {- 1} (y) = 1 $ for everyone $ y in tilde {Y} $.
  2. $ tilde {Y} $ is a residual subset of $ Y $ that is to say. $ overline {Y setminus tilde {Y}} $ has an empty interior.
  3. $ mu ( tilde {Y}) = 1 $ for each $ T $– invariant probability measure in $ Y $.

Is it true that $ pi $ is injective?