Ag. Algebraic Geometry – Is There Clear Evidence for the Main Theorem of Dimensional Theory?

The main theorem of the theory of dimension in commutative algebra states that a module is given $ M $ over a noetherian local ring $ A $, we have $ s (M) = text {dim} (M) = d (M) $ (from where $ s (M) $ is the infimum of integers $ k $ so that it exists $ x_1 … x_k $ so that $ M / (x_1 … x_k) M $ is of finite length and $ d (M) $ is the degree of the Samuel polynomial of $ M $)

I know the standard proof, as written down in Serres Local algebraor in the book Homological methods for commutative algebra,

Question 1

I was wondering if there is a short / smooth / simple / more conceptual proof by applying
a little more technology, maybe with homologous algebra or
algebraic geometric methods or whatever.

Question 2

What are good references to study the dimension set apart from the books above? Please give some details.

Why has nobody ever found credible evidence of a large-scale electoral fraud?

Why was California forced by court order to remove 1.5 million voters from the list?
The registration of motor voters in California can not guarantee the sanctity of the vote and Foreign Minister Padilla says, "Mistakes in the DMV threaten to undermine confidence in the program." LOL

"Today, the Public Interest Legal Foundation has issued documented evidence that in just 138 districts and cities, Virginia's voters have tacitly canceled 5,556 non-national voters over the past few years, and one-third of them have cast ballots duplicated in many other states, which is why a national investigation is inevitable. "

LOL

Update to comment: Apparently, the courts have taken them seriously ….. LOL

What is the evidence complexity of E-KRHyper (E-Hyper Tableau Calculus)?

Before the question, let me better explain what E-KRHyper is:

E-KRHyper is a sentence checking and model generation system for equivalent first-order logic. It is an implementation of the E-Hypertableau calculus that integrates an overlay-based equality treatment into the hypertableau calculus (Source: System Description: E-KRHyper).

I am interested in the complexity of the E-KRHyper system, as it is used in the question-answer system Log-Answer (LogAnswer – A deduction-based question
Answering machine (system description)
).

I found a partial answer:

Our calculation is a non-trivial decision-making process
Fragment (with equality) that captures the complexity class nexptime (Source: Hyper Tableaux with Equality).

I do not understand much complexity theory, so my question is:

What is the complexity of a theorem to prove in terms of the number of axioms in the database and in terms of some parameters of the question to be answered?

Adcash.com (cash) is a big scam with evidence (warning)

This partner network & # 039; Adcash.com & # 039; Just cheat me with 500 Euro, close my account and do not pay me for my work. If I ask you, he does not want to give me an answer. I have a support staff in MSN Live Chat under the name & # 039; s (maxim) and then I ask here why he did that and the answer was to block me.

See Appendix 70282

and here my account is blocked without giving reasons :(

See Appendix 70283

I send them in about 5 messages and without an answer :(

Warning: do not trust that …

Adcash.com (cash) is a big scam with evidence (warning)

discrete mathematics – properties of relationship evidence

I practice some characteristics of relationships and I can not seem to figure out a specific question. It follows

Consider the relation R on Z+(positive integers) as: For all m,n belonging to Z+, mRn means m|n. 
Is R reflexsive, symmetric or transitive?
Provide a complete proof or counterexample for each property.
You may only use the definition of divides

The definition of divisions according to my special textbook is as follows.

let n,d ∈ ℤ+ and d≠0.
n is divisible by d if and only if ∃ k ∈ ℤ such that n = dk

How would I go about doing that? Every help is appreciated. Many thanks

Is there any evidence that the Clinton killed anyone?

No, but over the years, many people around them have died under very suspicious circumstances.

It is also terribly strange how many of these cases have been handled by the authorities and what strange causes of death have become known in these cases. You have a case that, according to the findings, is a pretty obvious murder case, and then it's suicide. Does anyone really believe that people shoot themselves twice in the back of the head to kill themselves? People even blame the Clintons for killing themselves, since such things have happened so often.

bitcoin core – Does the evidence of work directly contribute to avoid duplication?

I know that similar questions have already been asked (eg how a job statement prevents double spending), but I have trouble imagining the solution for duplicate issues, which depends directly on the evidence of employment.

Imagine a miner, Bob, trying to spend 1 BTC twice by sending to Alice and Jim. Either he can record both transactions for Alice and Jim into a single bad block that he breaks down. In this case, the block is also rejected with a valid permission. Or he transfers his first (valid) transaction to Alice and has her deal with someone else, and later on with the transaction, he breaks down Jim's own faulty block, which in turn is rejected, regardless of whether or not there is a valid proof of work. Are duplicate issues really protected from the transaction validation that each node is doing on the network? This means that PoW does not immediately stop duplicate issues – what stops duplicate issues is simply that the nodes know the previously confirmed transactions. And the reason why we can trust that all transactions in the blockchain are "confirmed" is PoW?

If that's true and PoW is more of a high-level system-wide solution, here's a final question that I hope someone can clarify, why we can not replace PoW with an automatic 10-minute block addition based on a simple timestamp? Is it because it is such a mission-critical task (which involves the trust of the entire system) that, unlike the difficulty calculation (where we rely on timestamps), we have to find an alternative that can not fake time value ? (even if this were the case, some nodes might reject it at the wrong times). Is it wrong to think of proof-of-work as a substitute for trust in a (hackable / spoofable) time stamp server that could regulate "block every 10 minutes"?

General Topology – Torus $ mathbb {R} ^ n / mathbb {Z} ^ n $ is Hausdorff, an evidence check.

I want to prove that $ T ^ n = mathbb {R} ^ n / mathbb {Z} ^ n $ is Hausdorff.

We define in $ mathbb {R} ^ n $ the equivalence relation $ sim $ by $ x sim y $ then and only if $ y = z + x $ from where $ z in mathbb {Z} ^ n $, Equip the room $ mathbb {R} ^ n / _ sim $ with the quotient topology the projection map $ pi: mathbb {R} ^ n to mathbb {R} ^ n / _ sim $ is open. In fact, if $ U subset mathbb {R} ^ n $ is so open $ pi ^ {- 1} ( pi (U)) = bigcup_ {z in mathbb {Z} ^ n} U + z $This is a union of open subsets. Now that's given $ mathbb {R} ^ n $ is Hausdorff and $ pi $ is open to prove that $ T ^ n $ Is Hausdorff, we can prove that the graph of $ sim $ defined by $ R_ sim = {(x, y) in mathbb {R} ^ n times mathbb {R} ^ n: x sim y } $ is closed in $ mathbb {R} ^ n times mathbb {R} ^ n $,

I would like to know if my proof is correct:

To let $ f: mathbb {R} ^ n times mathbb {R} ^ n to mathbb {R} $ defined by $ f (x, y) = f (x_1, ldots, x_n, y_1, ldots, y_n) = sum_ {i = 1} ^ n sin ( pi (y_i-x_i)) ^ 2 $, Then clearly $ f $ is steady, though $ y = x + z $ With $ z in mathbb {Z} ^ n $ that implies that $ y_i = x_i + z_i $ for all $ 1 leq i leq n $ and thus $ f (x, x + z) = sum_ {i = 1} ^ n sin ( pi z_i) ^ 2 = 0 $, Conversely, if $ f (x, y) = 0 $ then $ sin ( pi (y_i-x_i)) = 0 $ for all $ i $, That's why $ y_i-x_i in mathbb {Z} $ for all $ i $ and thus $ x sim y $, Finally $ f ^ {- 1} (0) = R_ sim $ which is closed.

Thanks a lot!