## Mining theory: Does AsicBoost not reduce the search space for a valid block header hash?

ASICBoost is a new method to extract bitcoins with a 20% acceleration, as covered in this article:

Overview of the covert AsicBoost allegation

Now, consider the following diagram:

Apparently there are two versions of AsicBoost: covert and open. Overt AsicBoost involves changing the Chunk 1 Version section only, while Chunk 2 remains unchanged (except for the nonce, which still increases). This is faster than the traditional method of increasing the nonce and then adding the additional nonce.

However, does Overt AsicBoost not reduce search space? The current difficulty of the Bitcoin network is 15,546,745,765,529, which means that a miner has to iterate through approximately 2 ^ 76 hashes before finding a valid block header hash. However, the Version section of the block header has only 4 bytes (32 bits), and the nonce also has 4 bytes (32 bits). Therefore, if a miner implements the Overt AsicBoost technique, the search space is only 2 ^ 32 * 3 ^ 32 = 2 ^ 64 hashes, which is much smaller than 2 ^ 76. Wouldn't this be completely insufficient to find a valid block header hash?

## complexity theory: simple question about epsilon and estimation machines

I am getting very confused. I reached a point where I had to calculate the lim when $$n rightarrow infty$$ for an optimization problem, and I got to the point where I had to calculate a fairly simple limit: $$lim_ {n rightarrow infty} {3- frac {7} {n}}$$.

now used to $$3 – epsilon$$ and I'm trying to prove that there can't be $$epsilon> 0$$ so that the algorithm estimate is $$3- epsilon$$, because there is a "bigger estimate", and this is the part of which I am not sure, what is the correct direction of inequality? $$3- frac {7} {n}> 3 – epsilon$$ or the opposite? I am trying to show that the estimation ratio is close to 3.

I think what I wrote is the right way, but I'm not sure. I would appreciate knowing what is right in this case. Thank you.

## java – Coupling theory vs reality

I will not put it so black and white. A bit of coupling is necessary. In addition, some indirect ways to get rid of the coupling may introduce too much overhead. In particular for applications that need to respond in real time. They are compensations.

Yes, you want to avoid high coupling. However, if the introduction of patterns in an attempt to reduce coupling prevents you from first meeting your requirements (which may include response time, budgets, etc.), then it is not worth it.

In Java, when A creates an object B and calls one of its methods, it is said to be closely coupled. But if we create an IB interface, used from A where B implements IB, it is said to be flexibly coupled. I don't see why, since a change in the interface would have an impact on A and B. And a change on B would have an impact on IB and A. They still seem to be coupled to the call.

Yes, they are still docked calls. Well, it depends on how you define the metric. However, interfaces are not the right tool if you want to deal with it.

Regardless, with the interfaces, classes would be freely coupled, since A does not directly link to B. It could have other implementations of the interface.

A common antipatron is to create an interface that matches what the consumed class offers. You must create an interface that matches what the consumer class needs. See interface segregation. The interface becomes a requirement for the class consumed.

If you conceptualize the interface that way, the interface would only change when necessary for your consumers. When changing A, you can decide to change the interface or use a new one.

If we decide to change the interface when changing A, the change would spread from A to B. Instead of propagating from B to A. However, we don't have to decide to change the interface, or we can introduce an adapter that implements the interface and the wraps. B. That is, we have opportunities to stop the spread of change. That's what we want. And that is what the loose coupling buys us. We design to have more options, not less.

Again, that is not solving the call coupling.

The same reason applies to the Facade GoF design pattern. It is said to promote low coupling, since we put an intermediary between a subsystem and the customer code. In this case, it seems that we transfer the customer code problem to the Facade. Since a change in the subsystem would have an impact on the Facade instead of the customer code. The client code is no longer coupled to the subsystem, but the Facade is.

The facade hides everything you do behind it. It is similar to how an object encapsulates its state. The facade encapsulates a (sub) system. The consumer code only needs to talk to the facade and ignores the details behind it.

Of course, it is still docked. And yes, you have moved the problem to the Facade. However, thanks to the Facade, the consumer code does not have to change due to changes in what is behind the facade.

But the answers are not specific enough. First, since encapsulation is also about treating objects such as black boxes, the first response does not specify any gain when using interfaces compared to regular classes (= closed coupling, in the case of black boxes)

If you use the class directly, the class must implement what the consumer code needs. If the consumer code uses an interface instead, then it does not have to be implemented by any particular class. You can change the class without the consumer code knowing. The consumer code has less knowledge, therefore, it is less coupled.

What the interfaces provide is the decoupling between the interface and the implementation when there are multiple implementations. But it does not solve anything related to call coupling.

Correct, it's not about call coupling. You are the one who reduces the discussion to call the link and the interfaces. And then wondering why they provide nothing.

Interfaces are not the right tool to deal with call coupling. Instead, you want an event driven architecture, a consumer subscriber pattern or the like. That way, there may not even be an implementation on the other side. Of course, some infrastructure may be required, if not provided by the language and runtime. Oh, this is Java, yes, some infrastructure is required.

## complexity theory: find an algorithm that, after removing k edges, we obtain an acyclic graph

Assuming there is an algorithm that can decide to belong to ACYCLIC in polynomial time. How can I use this algorithm in another algorithm that at the entrance of a directed graph and a positive number k, returns k edges that after eliminating them, the graph becomes acyclic (it has no cycles). There are no weights on the chart, just regular directed chart.

ACYCLIC is for a directed graph that is acyclic.

What I am trying to do is something like this:
For a directed graph G = (V, E), assuming that there is an isacrylic algorithm that returns true or false if the given graph is acyclic or not:

1) select a vertex and start traversing the graph

2) while the number of edges> 3 and we do not finish crossing the graph (3 edges can form a cycle, and I need at least one more because k should be positive, that is, number of edges that by removing them I will get an acyclic graph )

2.1) if (number of traversed edges – 3) greater than k

2.2) if isacyclic returns false:

2.3) try to delete the last border and re-run isacyclic, if it returns true: add the border to a list / stack and continue browsing without it

if at the end of the algorithm there are no k borders in the list / stack – reject

if you don't accept

The algorithm I wrote was a general idea of ​​what I am looking for. I suppose there is an algorithm that can decide to belong to ACYCLIC in a polynomial time for the sake of the questions.

What I am looking for is an algorithm that, for a directed graph g and a positive number k, can give me k edges that, when I delete them, I get an acyclic graph

The algorithm must be polynomial in regards to its input size.

Thank you!

## Computability theory: randomness of Kurtz and supermartingales with infinity * limit *

Suppose you replace the usual success conditions for a supermartingale (lim sup is infinite) with the requirement that the real limit be infinite, p. a supermartingale $$B$$ succeeds in $$X in 2 ^ omega$$ only if

$$lim_ {n to infty} B (X restriction_n) = infty$$

I'm 90% sure that you don't get a notion of randomness & # 39; valid & # 39; because any real generic enough should qualify. However, if I had to guess, I would think that this notion turns out to be equivalent to Kurtz's "randomness" (avoid all $$Pi ^ 0_1$$ null sets). As I suppose this is a known result and my brain feels very cloudy, I thought about asking

## probability theory – Decomposition of mutual information

I found a book where the author uses the following property of mutual information:

Leave $$X$$,$$Y$$,$$Z$$ be arbitrary discrete random variables and leave $$W$$ Be a random indicator variable.

$$(1) I (X: Y mid Z) = Pr (W = 0) I (X: Y mid Z, W = 0) + Pr (W = 1) I (X: Y mid Z, W = 1)$$

I do not understand why this property is maintained in general.
To show this, I was thinking of proceeding as follows:
begin {align} I (X: Y mid Z) & = E_z (I (X: Y mid Z = z)) \ & = E_w (E_z (I (X: Y mid Z = z) | W = w)) \ & = Pr (W = 0) E_z (I (X: Y mid Z = z) | W = 0) \ & + Pr (W = 1) E_z (I (X: Y mid Z = z) | W = 1). end {align}
where the second line follows the law of total expectation.
However, this does not seem to be the right approach since it is not clear to me that
$$E_z (I (X: Y mid Z = z) | W = w) = I (X: Y mid Z, W = w)$$
holds.

What is the correct way to show (1)?

## Theory of the ct category: characterization of the left / right classes of factorization systems (weak) in locally presentable categories

Leave $$mathcal M subseteq Mor ( mathcal C)$$ be a class of morphisms in a locally presentable category.

1. It is well known that $$mathcal M$$ it is the left half of an accessible orthogonal factorization system iff $$mathcal M$$ is accessible (as a complete subcategory of the morphism category $$mathcal C ^ (1)}$$), and is closed under colimits in $$mathcal C ^ (1)}$$, change of base, composition and isomorphisms.

2. Similarly (although perhaps this is less known), $$mathcal M$$ is he Right kind of an accessible orthogonal factorization system if it is accessible and embedded in an accessible way $$mathcal C ^ (1)}$$and closed under limits in $$mathcal C ^ (1)}$$, change of base, composition and isomorphisms. The test, of course, is not dual: it is observed that under these conditions, $$mathcal M$$ it is accessible reflective in $$mathcal C ^ (1)}$$, shows that a section of each map unit must be an isomorphism, so that the reflector provides factorizations and then verifies some things.

In the case of weak factorization systems (wfs), the situation may not be so simple. On the one hand, not all accessible wfs in a locally presentable category are co-generated, so any "small generation" argument will be more delicate.

1. More specifically, the left class of a wfs can be accessible without the wfs being accessible, and conversely (at least under anti-large cardinal hypotheses) the left class of an accessible wfs does not need to be accessible. So, although one might guess that closure under co-products, change of basis, isomorphisms, composition, transfinite composition and retractions should almost characterize the left wfs classes accessible in locally presentable categories, it is not clear what kind of "accessibility" hypothesis to add to get a characterization

2. However, there may be more hope to characterize the Right wfs classes in locally presentable categories. In particular, the following assumption seems reasonable:

Question:
Leave $$mathcal M subseteq Mor ( mathcal C)$$ be a class of morphisms in a locally presentable category. Suppose that $$mathcal M$$ is accessible and embedded in $$mathcal C ^ (1)}$$, and closed under products, change of base, isomorphisms, composition, co-transfinite composition and retractions. Does it follow that $$mathcal M$$ it is the correct kind of a weak factorization system accessible in $$math C$$? If not, is there a characterization in similar lines?

Presumably, the proof of any characterization will proceed using some form of Garner's small object argument, but beyond that it is not clear to me.

## geometric measurement theory – Generalization of approximate tangent spaces to subsets of arbitrary varieties?

I am anything but an expert in Geometric Measurement Theory, so please forgive me if I am asking a trivial question.

Leave $$(M ^ n, g)$$ be a soft Riemannian collector, $$d geq 0$$ Y $$A ⊂ M$$ be a subset that is measurable w.r.t. the Hausdorff$$d$$ measure $$D ^ d$$ coming from the metric $$g$$. Embed $$(M, g)$$ isometrically in some $$N ^ N$$. by $$D ^ d$$-measurable sets of $$N ^ N$$ (I like $$A$$) there is the notion of approximate tangent spaces.

So let's suppose $$A ⊂ M ⊂ ℝ ^ N$$ It has an approximate tangent space $$T_x A$$ to $$x ∈ A$$. Are the following statements true?

1. $$T_x A$$ it's a subspace of $$T_x M$$.

2. $$T_x A$$ It is independent of isometric embedding.

3. (Assuming 2 is true), $$T_x A$$ is independent of the choice of $$g$$. (This requires that $$D ^ d$$-medibility of $$A$$ is independent of the choice of $$g$$ which should be the case, unless I am wrong).

In other words, the question I ask is: To what extent is the approximate tangent space of $$A ⊂ M$$ to $$x$$ an intrinsic concept, which depends only on the structure of $$A$$ as a subset of $$M$$Y $$M$$ as a differentiable variety? The main reason why I ask is that I often need to switch between metrics (and, therefore, isometric inlays) and I don't want to worry about changing approximate tangent spaces when I do that.

## nt.number theory – A homogeneous polynomial with zeros in a geometric progression.

Leave $$a_1, …, a_n in overline { mathbb {Q}}$$ be algebraic numbers, so that $$frac {a_i} {a_j}$$ it is not a unit root for $$i neq j$$. Also leave $$m in mathbb {N}$$.

Question: Is there a homogeneous polynomial $$P in overline { mathbb {Q}} (X_1, …, X_n)$$ in $$n$$ grade variables $$k$$such that $$P (a_1 ^ {m ^ k}, …, a_n ^ {m ^ k}) = 0$$ for all $$k in mathbb {N} _0$$?

## rt.representation theory – Magnitude of ADR algebras

Leave $$A$$ be a quiver algebra connected $$n$$ simple and radical Jacobson modules $$J$$ and length Loewy $$n + 1$$ (that is to say $$J n + 1 = 0$$ Y $$n$$ It is minimal with this property).
ADR algebra $$B_A$$ from $$A$$ is defined as $$B_A: = End_A (A oplus A / J oplus A / J ^ 2 oplus … oplus A / J ^ n)$$.
Remember that the magnitude (see http://www.tac.mta.ca/tac/volumes/31/3/31-03.pdf) of a finite global dimension algebra and Cartan matrix $$C$$ is defined as the sum of all the entries of the inverse of $$C$$.

Question 1: is the magnitude of $$B_A$$ equal to $$n$$?

In the special case when $$A$$ is a commutative algebra of Frobenius,
we have to Cartan matrix $$C$$ from $$B_A$$ has tickets $$C_ {k, t} = dim (Hom_A (A / J ^ k, A / J ^ t))$$, which should be equal to $$dim (J max (0, t-k)} / J ^ t)$$ In case you have not made any mistakes, then you have an elementary interpretation.

Question 2: is the magnitude of $$B_A$$ equal to one in case $$A$$ What is a commutative algebra of Frobenius?

The questions are tested for many algebras with the computer, but, of course, there are too many algebras to have really good tests.

(Question 2 is, of course, a special case of question 1, but here the Cartan matrix has a more elementary way to calculate explicitly).

For example, leave $$A = K (x, y) / (x ^ 2, y ^ 2)$$, then the Cartan matrix of $$B_A$$ is given by $$C$$= ((1, 2, 1), (1, 3, 3), (1, 3, 4)).
The inverse matrix is
((3, -5, 3), (-1, 3, -2), (0, -1, 1)), where the sum of all entries is equal to 1.