transaction fees – Will Bitcoin eventually disappear spontaneously when mining becomes meaningless?

Is Bitcoin Mining Coming to an end?

What Happens to Bitcoin After All 21 Million Are Mined?

The Supply of Bitcoin Is Limited to 21 Million In fact, there are only 21 million bitcoins that can be mined in total. Once miners have unlocked this number of bitcoins, the supply will be exhausted. However, it’s possible that bitcoin’s protocol will be changed to allow for a larger supply. What will happen when the global supply of bitcoin reaches its limit?
Once all Bitcoin has been mined the miners will still be incentivized to process transactions with a much much higher fees.

Will Bitcoin eventually disappear spontaneously when mining becomes meaningless?

set theory – How to compare three supremums of ordinals eventually writable by Ordinal Turing Machines?

This question implies that we have fixed: (i) a particular enumeration of Ordinal Turing machines; (ii) a particular way to encode an ordinal by an infinite binary sequence.

The class of $(1)$-machines is defined as the $1$st iteration of the strong jump operator for Ordinal Turing Machines. That is, a machine is equipped with an oracle able to answer any question of the following form (note that $(0)$-machines are Ordinal Turing Machines with no oracles):

Does an $i$-th $(0)$-machine halt given an infinite binary sequence $x$ as the input?

The ordinal $alpha_1$ is defined as the supremum of ordinals eventually writable by $(1)$-machines with empty input.

Let $m_i(x)$ denote a computation performed by an $i$-th $(0)$-machine, assuming that the input is $x$. If $m_i(x)$ eventually writes a countable ordinal $alpha$, then $M_i(x) = alpha$. Otherwise, $M_i(x) = 0$.

Then the function $F(i)$ is defined as follows: if the value of $sup {M_i(x) | x in mathbb{R}}$ is a countable ordinal, then $$F(i) = sup {M_i(x) | x in mathbb{R}};$$ otherwise, $F(i) = 0$. Here “$x in mathbb{R}$” implies that we take into account all infinite binary sequences.

The ordinal $alpha_2$ is defined as follows: $$alpha_2 = sup {F(i) | i in mathbb{N}}.$$.

The ordinal $eta$ is defined as the least ordinal $gamma$ such that $L_gamma$ and $L$ have the same $Sigma_2$-theory (see part 3 of Lemma 3.11 in the paper “Recognizable sets and Woodin cardinals: Computation beyond the constructible universe”). As far as I understand, $eta$ is equal to the supremum of ordinals eventually writable by Ordinal Turing Machines ($(0)$-machines) with empty input.

Question: which ordinal is larger, $alpha_1$ or $alpha_2$? Is any of these two ordinals larger than $eta$?

abstract algebra – Proving that ${{X}_{m}}={Pin {{k}^{n}}left| {{f}_{1}}(P)=…={{f}_{m-1}}(P)=0 ;;hbox{and} ;; {{f}_{m}}(P)=1} right. $ is eventually empty.

We consider a field $k$, $n in mathbb{N}$ and ${{f}_{1}},{{f}_{2}},…in k({{x}_{1}},…,{{x}_{n}})$. For every integer $mge 2$, we define ${{X}_{m}}={Pin {{k}^{n}}left| {{f}_{1}}(P)=…={{f}_{m-1}}(P)=0 ;;hbox{and} ;; {{f}_{m}}(P)=1}. right.$
I need to show that there exists an integer $N$ such that ${{X}_{m}}=varnothing $ for every $m ge N$.
I encountered this problem in a course of commutative algebra, in the chapter of Noetherian rings.
First, I suspected that I should use the property “every increasing sequence of ideals is eventually constant”, which characterizes a Noetherian Ring, but I am not very sure.
Any help would be greatly appreciated.

architecture – Idempotent Kafka consumer with an “eventually consistent” database

What are my options if I want to make my Kafka consumer idempotent (due to Kafka’s at-least-once semantics) if my project’s only database is DynamoDB, which is eventually consistent?

I’m afraid setting the “consistent read” flag might slow down and overload the leader node;

I have neither control over the topics I consume nor over the Kafka cluster.

Using an authentication provider to keep a user permanently logged in eventually causes session problems

Because I got fed up of having to log in to my local development sites every however long, I tried making a simple custom module that just has an Authentication Provider service that always returns user 1, like this:

   * {@inheritdoc}
  public function applies(Request $request) {
    return TRUE;

   * {@inheritdoc}
  public function authenticate(Request $request) {
    return $this->entityTypeManager->getStorage('user')->load(1);

The idea being that with this module enabled, every page load considers the authenticated user to be user 1. In theory, I’d be logged in for ever!

However, after a few weeks of this working fine, all my form submissions broke! Every form submission fails with:

The form has become outdated. Press the back button, copy any unsaved work in the form, and then reload the page

This turns out to be because CsrfTokenGenerator::validate() fails. Specifically, the $seed in this bit is just NULL:

  public function validate($token, $value = '') {
    $seed = $this->sessionMetadata->getCsrfTokenSeed();
    if (empty($seed)) {
      return FALSE;

I presume it’s because the session has expired in some way? The time this has taken to stop working is roughly the amount of time it would have taken the site to log me out automatically.

How can I fix this?

pr.probability – If a Markov semigroup is eventually contractive, can we conclude that it admits a unique invariant measure?

Let $E$ be a separable $mathbb R$-Banach space, $rho$ be a complete separable metric on $E$, $operatorname W_rho$ denote the Wasserstein metric of order $1$ associated to $rho$, $mathcal M_1(E)$ denote the set of probability measures on $(E,mathcal B(E))$ and $(kappa_t)_{tge0}$ be a Markov semigroup on $(E,mathcal B(E))$ with $$operatorname W_rho(mukappa_t,nukappa_t)le ce^{-lambda t}operatorname W_d(mu,nu);;;text{for all }mu,nuinmathcal M_1(mu,nu)tag1$$ for some $cge0$ and $lambda>0$.

Are we able to conclude that $(kappa_t)_{tge0}$ has a unique invariant measure $mu_astinmathcal M_1(E)$?

I guess it’s true, since the Wasserstein space $$mathcal S^1(E,rho):=left{muinmathcal M_1(E):(muotimesdelta_0)rho<inftyright}$$ equipped with $operatorname W_rho$ is complete, but for some reason I struggle to conclude. I think it should somehow follow from the fixed-point theorem …

dnd 5e – If a Cleric uses “a sprinkling of holy water” in a spell, does the flask eventually get used up?

Supposing you are actually sprinkling a little bit of the water (since by the rules, it is sufficient for you to have the components in a pouch)…

Any rpg where the rule simulate some kind of reality have to draw the line of what they simulate somewhere. In modern D&D, simulation of moment-to-moment adventurous tasks is generally more detailed than longer-term consequences of said actions. Downtime and crafting rules are at a far higher level of abstraction, for example. We could say that maintaining equipment is generally abstracted away into lifestyle maintenance cost.

Likewise, characters can swing a sword or cast a cantrip as often as they get turns (or more); however, the rules do not say how long the characters can keep this up. If a character tries to continue doing this for hours on with no rest and end, the game master and the group are fully within their rights to say that there are consequences, such getting fatigue. The abstraction of the game does not handle such scenarios.

It is best to discuss such issues openly among the group, so that everyone can buy in into the decisions and rulings made. Otherwise it might feel like someone, maybe the game master, is arbitrarily adding restrictions into the game.

The game rules suggest the game master makes such rulings without consulting the group, and this can be best in some situation (dramatic situation, immature group but mature game master, players who enjoy character immersion a lot, players whose fun is spoiled by both discussing the rules and trying to act optimally within them). The following quote is from the basic rules and is part of the fundamental structure of the game:

The DM narrates the results of the adventurers’
actions. Describing the results often leads to another
decision point, which brings the flow of the game right
back to step 1.

The game assumes that the spell components are not spent, and whatever wear and time and minor issues equipment might have, the characters fix during downtime, or replace their things every now and then, etc. We can assume that replacing the sprinklings of holy water is a part of this.

The level of abstraction assumed by the rules works well when the characters occasionally are in civilized regions, have a few hours of downtime every now and then, and are not utterly broke. These are not restrictive conditions and probably also hold in your game.

Maybe your character or party is like a Robinson Crusoe on his island, or in some other situation of great scarcity. Maybe they use a small amount of holy water every now and then.

In this situation, the implicit assumption of the characters maintaining their equipment, including the hard-to-get parts of it, is no longer valid. Hence, if the character use holy water every now and then and have no means of creating them, the game master or the group is fully within the rules to declare that they run out at some point. The players have declared their actions and the game master declares consequences.

Due to reasons of realism and player buy-in, it is advisable to mention that half the water is now used, and now there are only a sprinklings left, and so on.

functional analysis – Density of tensor products of eventually zero sequences.

Let $c_{00}(mathbb{Z})$ denotes the space of eventually zero sequences over $mathbb{Z}$.

Let us define $$mathcal{P}c_{00}(mathbb{Z}^2)={(a_k)_{kin mathbb{Z}^2 } : a_{(k_1,k_2)}=b_{k_1}c_{k_2},(b_n)_{nin mathbb{Z}}, (c_n)_{nin mathbb{Z}} in c_{00}(mathbb{Z})} . $$
How to show that $mathcal{P}c_{00}(mathbb{Z}^2)$ is dense in $c_{00}(mathbb{Z}^2)$ in ‘supremum’ norm?

If a Cleric uses holy water in a spell, does the flask eventually get used up?

e.g. the Bless spell calls for “a sprinkling of holy water”

Is the flask essentially unlimited for Bless, or is there a conversion of X sprinklings in 1 flask?