## proof scaling property for gamma distribution

How do I prove the following property for the gamma distribution?

$$Xsim Gamma(k,Theta) Rightarrow cXsim Gamma(k,cTheta)$$

for $$c>0$$.

## proof of work – Could BTC be changed so that it limited the amount of computation power a given miner used?

The amount of transactions are already limited. Each Bitcoin block can contain exactly 4,000,000 weight units of data, and there are about 144 blocks per day as the network regulates itself to a ten-minute block interval.

Your query kinda misses the issue, though:
The primary purpose of mining is to secure the Bitcoin network. It does so by committing to the updates to the network’s ledger and making the transaction history immutable. This process secures balances amounting currently to about \$1T of value. Each block takes the same amount of work to produce whether the block includes one transaction or 3,500 transactions. While one can obviously create a ratio between transaction count and energy consumption, it is not meaningful. More or less transactions don’t cost more energy.

The main misunderstanding seems to be that Bitcoin doesn’t do anything useful. This is apparent when Bitcoin transactions are compared with payments on Visa or PayPal. A PayPal transaction is an update of a custodian’s internal account balances: “move some funds from Alice’s account balance to Bob’s account balance.” We can do that with bitcoins too: a transaction from one Coinbase user to another Coinbase user is just as cheap and quick.

But Bitcoin is not book money like PayPal or Venmo balances. Bitcoins are a digital bearer instrument. An on-chain Bitcoin transaction is an irreversible transfer of a bearer instrument directly from sender to receiver that finalizes in minutes. Consider being able to directly hand someone cash or a bar of gold on another continent without any intermediaries. Turns out that lots of people find a system like that useful, e.g. to gain access to banking-like services, for cheaper remittances, to facilitate international business deals, to make donations, or to hedge against the rampant inflation of their national currency.

And that’s just the tip of the iceberg. Being digital, bitcoins can be used in smart contracts enabling more complex systems. One such system is the Lightning Network which is a decentral payment system that has the potential to scale to millions of transaction per second.

So, people think Bitcoin isn’t useful and thus object to its energy consumption. Let’s ignore for a moment that hundreds of millions of people disagree with that assessment.
About 160,000 TWh are produced globally per year, of which about 50,000 TWh are wasted. Bitcoin miners maximize their profits by minimizing their energy costs and they are arbitrarily mobile—they only need a roof, power, and an internet connection. The cheapest energy is renewable or stranded—energy that has no other use. Even as the bidder of last resort, Bitcoin mining uses 120 TWh—which amounts to 0.24% of the energy wasted every year. And if energy becomes more expensive, they’ll use less. I’d like to suggest a new target for your frown: how about we reduce wasted energy and stop subsidizing fossil fuels left and right.

## stochastic processes – Proof of the link between the Fokker Planck equation and SDE?

I know the link between the Fokker Planck equation and SDE given by the Feynman-Kac theorem is as follow:
$$d X_{t}=muleft(X_{t}, tright) d t+sigmaleft(X_{t}, tright) d W_{t}$$
$$frac{partial}{partial t} p(x, t)=-frac{partial}{partial x}(mu(x, t) p(x, t))+frac{partial^{2}}{partial x^{2}}(D(x, t) p(x, t))$$ with $$sigma = sqrt{2D}$$.
Are there any clues(or reference) to prove this Theorem? I found a version of the proof, here is the link, however the Fokker–Planck equation is used with problems where the initial distribution is known, but the problem here is to know the distribution at previous times, which means that it is necessary to prove the Keynman-Kac theorem.
By the way, how do you find a proof for a specific theorem.

## dg.differential geometry – Question in the proof of Hilbert’s theorem

I’m researching about Hilbert’s theorem which says that there isn’t isometric immersion of a complete surface with constant negative Gaussian curvature in $$mathbb{R}^3$$. I’m taking as a reference the book “Differential Geometry of Curves and Surfaces by Manfredo P. do Carmo” and also “A Comprehensive Introduction to Differential Geometry, Vol. 3 by Michael Spivak”. I have only two specific doubts:

In the attached document, the Spivak’s proof is presented first in English and then the do Carmo’s proof in Spanish: click here

• The first is that it must be shown that an asymptotic curve on a complete surface with constant negative Gaussian curvature can be defined in all $$mathbb{R}$$, I have framed it in red in the attached document. The arguments mentioned in the documents I know are valid for compact surfaces but not for complete surfaces and I have not been able to come up with a clear proof of this result.
• The second is about the injectivity of $$X(s, t)$$ in the first edition of do Carmo’s book, he uses two lemmas within which he makes cases to arrive at the result, while in the current edition (2016) he mentions how to get there To that result of a faster one using coating applications but I haven’t been able to find a clear proof, I haven’t gotten stuck and I cann’t get out of that, I have framed it in orange in the document.

I have been justifying the steps that both authors leave without proof, in order to fully understand this result but I cannot understand those two points that I mention, I hope they can help me, I thank you in advance.

## proof of work – Decreasing Energy Consumption of Bitcoin’s PoW with paired mining rounds

I found the following whitepaper on an algorithm called “Green-PoW” by the authors and i’m quite sure this would be a perfacet solution and also would get acceptance by the miners and the community.

You can find the white paper here:
https://arxiv.org/pdf/2007.04086.pdf

Key point of the paper here as a quote:

“In order to reduce the energy consumption in PoW, Green-PoW considers
mining rounds in pairs and adjusts the election mechanism during the
first mining round in the pair, to allow the election of a small
subset of miners that will exclusively mine in the second round.
Assuming the total network hashing power is equally distributed among
miners, a significant energy saving up to 50% could thus be achieved”

What is your opinion on this approach, are there any problems with this?

## Expert Eyes needed to proof read an ER-Diagram and relationship Schedule

Myself and my fellow students are working on a covid-19, ER-diagram, and relational databaseschedule. As it is the first time we’re working with UML, we are a tad bit unsure if our work is lacking in any way or form.

Would a kind soul take a look at what’ve made.

Relational Database Schedule:
Green: Primary Key
Yellow: Foreign Key

## eigenvalues – Proof (or reference) about \$lambda_i(A+epsilon e_je_j^*) = lambda_i(A) + epsilon |v_{i,j}|^2 + O(epsilon^2).\$

I’m looking for a proof (or a reference in a textbook) about the fact that
$$lambda_i(A+epsilon e_je_j^*) =_{epsilon to 0} lambda_i(A) + epsilon |v_{i,j}|^2 + O(epsilon^2),$$
where $$A$$ is an hermitian matrix, $$lambda_i(A)$$ is an eigenvalue of $$A$$, $$e_j in bf{R}^n$$ is defined by $$(e_j)_i = delta_{i,j}$$, $$v_{i,j}$$ is the $$j-$$th component of a unit eigenvector of $$lambda_i(A)$$.

This theorem is from perturbation theory, a field I’m not very familiar with.

This is used in : Peter B. Denton, Stephen J. Parke, Terence Tao and Xining Zhang. $$textit{Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra}, 2021;$$
arXiv:1908.03795 (page $$13$$).

## Success stories / Payment proof butterflyclixx. com

Hello members,

Here will be referenced the payment proofs of the GPT website butterflyclixx. com

Good earnings to all!

Best regards,

## proof of work – How does bitcoin implement checkpoint mechanism to finalize the blockchain history?

Checkpoints are a relatively misunderstood part bitcoin – Bitcoin does have a few checkpoints, but they are only used in one very specific case. That case is just to ignore forks from the chain early on, before the most recently seen checkpoint. When a node has seen a block it recognises as a checkpoint, any further blocks received below that height will be ignored.

You can see this here: https://github.com/bitcoin/bitcoin/blob/0dfc25f82a01d9fec26380d95915df31e1fe2c02/src/validation.cpp#L3107

It is a long term goal of removing the checkpoints entirely, because they are a source of confusion over the security model and power the developers have. But currently the role they serve is to prevent low difficulty header flooding attacks, and there has been no alternative solution proposed yet (that I know of)

## ag.algebraic geometry – Seek for a algebro-geometric proof: \$mathrm{SL}(2,mathbb{Z}) rightarrow mathrm{SL}(2,mathbb{Z}/Nmathbb{Z})\$ is surjective

It is a well-known fact that $$mathrm{SL}(2,mathbb{Z}) rightarrow mathrm{SL}(2,mathbb{Z}/Nmathbb{Z})$$ is surjective.

What I want is a proof by method of algebraic geometry. For example, realise an element in $$mathrm{SL}(2,mathbb{Z})$$ as a point in certain scheme.$$mathrm{SL}(2,mathbb{Z}/Nmathbb{Z})$$ may corresponding to a subscheme or something else. And we may argue by saying if a morphism between proper scheme is proper and its image contains the generic point then it is surjective.

But all such realisation seems difficult.