Number Theory – How does the Chinese Remainder Theorem help prove $ phi (ab) = phi (a) phi (b) $?

I understand that the one-to-one correspondence refers to an element of set A, for example by mapping a single element distinct from set B. How does this one-to-one correspondence appear after applying the remainder theorem? Chinese to the system of equations of the referred image? ?

Theory of Elementary Numbers from Quantization and Quantum Information of Nielsen and Chuang, Appendix 4

complex analysis – Since Riemann maps $ f: Omega to mathbb D $, prove that $ inf_ {z in partial Omega} | za | frac1 {f) (a)} z in partial Omega} | za | $.

I look at the following problem from an old qualifying exam that I found:

Let $ Omega $ to be an open subset just connected $ Bbb C $, let $ a in Omega $ and suppose that we are given an analytical bijection $ f: Omega to mathbb D: = {z: | z | <1 } $ satisfactory $ f (a) = $ 0 and $ f (a)> $ 0. Prove $$ inf_ {z in partial Omega} | z-a | frac1 {f) (a)} z in partial Omega} | z-a |. $$

To be honest, I do not know where to start. my first thought was that $ 1 / f (a) = (f ^ {- 1}) & # 39; (0) $, and hoped to use this to try to apply the maximum module principle to a function involving $ f $ or $ (f ^ {- 1}) & # 39; $ and $ z-a $but the first problem that I got even before trying to write a function was to realize that $ f $ and $ (f ^ {- 1}) & # 39; $ may not even be defined on the respective limits, so it does not appear that I can use the maximum module to determine anything about the limit points in relation to these functions.

I have also assumed, seeing this, that it was simply something proven and used in the demonstration of Riemann's mapping theorem, but that does not really seem to be to be found the.

scriptsig – Can I prove a double-spending attempt with a signed transmission with the same entry?

I design an inter-channel atomic exchange in a custom blockchain – Protoblock. I think this can be done as long as Protoblock knows how to analyze p2pkh bitcoin transactions. The critical part is to be able to "reverse" things when he sees a double expense.

  • Alice has 1 Bitcoin
  • Bob has 100 Fantasybit
  • Alice will exchange 1 BTC for Bobs 100 FB

Step 1) Bob signs a Protoblock transaction "Swap" (TX1a) containing a Unsigned bitcoin transaction (TX1b) which must have 1. ScriptSig entry with Alice p2pkh (bitcoin-address) and 2. p2pkh output to Bob (bitcoin-address)

Step 2) Bob's 100 FB is locked for 24 hours or until …

Step 3) We see a Protoblock transaction (TX2a) containing the signature for TX1b and TXID (TX1a)

Step 4) 100 FBs are transferred to Alice and are locked for 24 hours or until …

Step 5) We see a Protoblock transaction (TX3a) containing a sign bitcoin transaction (TX2b) comprising 1. the same ScriptSig entry with p2pkh from Alice (bitcoin address) as TX1b and 2. an output that is NOT an output p2pkh to bob (bitcoin address)

Step 6) 100 FBs are returned to Bob in the same state as before. Step 1

The reason for step 5 is that a signed bitcoin transaction is not enough to prove that Bob has received Alice's bitcoin because Alice could simply sign the transaction, but not the spend in bitcoin and double the expense.

However, Bob will see that Alice has passed the entry and will create TX3a which makes her 100 Fantasybit her. In fact, anyone can look at bitcoins and create TX3a as a service for Bob.

Question: Is there a way for Alice to spend the same TX1b entries, which will be undetectable with the algorithm above?

Note: The bitcoin transaction with the outputs for the TX1b entries will be created by the portfolio software performing the atomic swap. So, Alice will first have to send her 1 Bitcoin to make sure it's a simple p2pkh. Here is the Protoblock code that creates the p2pkh and bitcoin tx outputs:–skillSale/blob/4e59f50b7555527046804418d6483b0df1933637/share/fantasybit-core/FantasyAgent.cpp#L444–skillSale/blob/4e59f50b7555527046804418d6483b0df1933637/share/fantasybit-core/FantasyAgent.cpp#L525

How to prove an implication on an upper bound mentioned in the proof of the main theorem?

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How can we rigorously prove the proposition "Suppose if in case 1 is true, the equation 4.23 is true"?
For the constants b and j, the implication in green is logical. If the upper limit of j has been set, the equation 4.23 follows directly. However, as n increases, the upper limit of j also increases, but is slower. This is where I find it hard to prove that there is always a value m> 0 such that for all n> = m, equation 4.23 is true.

Vector spaces – How to prove that if $ dim E = infty $, then | | E | = dim E $?

Let $ E $ to be a vector space with cardinality $ | E | $. $ dim E $ is the cardinality of a Hamel base $ E $. Assume that $ dim E = infty $. How to prove that $ | E | = dim E $?

The situation where $ E $ is a Banach space is already discussed in this article. Another article in German seems to answer the question about linear spaces in general, but I can not read it.

If the statement is not true for general vector spaces, can any one offer a counterexample?

topological groups – How can I prove the following isomorphism?

One of my teachers said the following: $ mathbb {A} _ { mathbb {Q}} $ to be the adele group of $ mathbb {Q} $. There is an isomorphism of topological groups $$ frac { mathbb {A} _ { mathbb {Q}} { mathbb {Q}} simeq varprojlim ( frac { mathbb {R}} {n mathbb {Z}} $ $, where the limit of the right is considered with $ mathbb {N} $ ordered by divisibility. I've tried to make a proof by using both spaces as compact and trying to build a continuous function. However, I had no success. Do you know how to prove it or do you have a clue?

Minimize the cost of recursive pair sums: how to prove that the greedy solution works?

The problem is in this other question.

Why does it always work? I do not see how we would use induction.

For $ n = $ 3, a quick calculation shows that it works, however, I think that generalizes well.

inequalities – Let three real numbers $ x, y, z $ so that $ 1 leqq x, y, z leqq8 $. Prove that $ sum limits_ {cyc} (! Frac {x} {y} – frac {2x} {y + z} !) Geqq0 $

Problem. Given three real numbers $ x, y, z $ so that $ 1 leqq x, y, z leqq $ 8. Prove it $ sum limits_ {cyc} (! dfrac {x} {y} – dfrac {2x} {y + z} !) geqq 0 $

Note. Using discriminant, I determined the maximum value of $ k so that the inequality holds for $ 1 leqq x, y, z leqq k $ (right here $ lceil $ $ rfloor $, you can see that this is the largest real root of irreducible polynomial, we can also find it involving SU Group (2)). And I really want to see $ uvw $ or $ pc $ right here !

Note: I think we can do it by finding $ l = $ constant so that the inequality is as follows:

Problème.2. Given three real numbers $ x, y, z> $ 0. Determine the value of $ l = $ constant so that : $$ sum limits_ {cyc} ( frac {x} {y} – frac {2x} {y + z}) geqq l. , frac {(8x-y) (8y-z) (8z-x) (8y-x) (8z-y) (8x-z)} {xyz (x + y) (y + z) (y + x)} $$

security – Prove that xpub is xprv?

I am looking for a paper wallet system where I generate an xpub / xprv HD
using an offline laptop. I write the xprv by hand. I capture the
xpub via QR code using my phone, to transfer to a complete online node.

A potential attack vector is to replace the receive addresses with
addresses under the control of the attacker. To fight against this, I plan to
to print the xpub as a QR code, occasionally using offline
laptop (which has no status), scan it and generate a new batch of
receive addresses that I can then manually check to match those created
by the complete knot.

But since the QR code printed xpub has necessarily passed through several
unreliable computers (eg online) before reaching the paper, how can I
to be sure that it was not falsified in transit?

Question: if I generate a single receive address (for example, m / 0/0), send
a small amount of BTC at this address, create a PSBT withdrawal
using the online node and then successfully sign that using xprv on
the offline node, is it enough to prove that the xpub is

In other words, is it possible that an attacker can edit the xpub file?
such that m / 0/0 corresponds to my xprv, but that (for example) m / 0/1 corresponds to the
xprv of the attacker? (And if that's feasible, how could I validate this
the printed xpub is the xprv for all possible addresses?)