## written test – Question on how to prove

So I came across an issue in a book that says, "Prove that in any area, if $$ax_1 ^ 2 + bx_1 + c = 0$$ and $$a 0$$then $$x_2 = – ( frac {b} {a} + x_1)$$ satisfied $$ax_2 ^ 2 + bx_2 + c = 0$$".

My question is, would simply substitute $$x_2 = – ( frac {b} {a} + x_1)$$ in the last equation be enough to complete the proof, or would I need to drift somehow the expression of $$x_2$$ of the first and the last equation?

## Authentication – Prove (disprove) that a document was created with a specific software version

I create software that essentially applies a specific process to create specific documents. Then the file is signed with the user's key (USB stick not linked to the software). Recently, stakeholders suggested signing it with the software key to prove that the process had been followed.

But no matter who can remove the key from the software and sign anything with it. So, how could I include something in the file that confirms (or disproves that) that it was created with an authorized version of the software?

It does not matter whether the copy is legitimate or not, just that it has not been modified.

I've limited the threat model to two cases:

• A. Someone (average solo developer) changes the software or creates a replacement to save time, then blames us when things go wrong. I need a way to show "This was not our construction".
• B. A lawyer says the document does not make sense because everyone could have done it A. The right guys have to show that it's not trivial and that it existed some protections against that.

The software must work mainly offline. Therefore, the online authentication of each signature is not acceptable. The project is far too small for anything with the monstrosity of Denuvo. But I still think that there must be a simple solution that I am too blind to see.

I have planned to use the user token to encrypt / decrypt the signing certificate, but it must be provided in an appropriate version and can not be too tedious (we already have a problem with a longer process than desirable). Thought a bit what the code signing could do, but that does not seem to do anything for this case. What do I miss?

## Reduction to prove the undecidability of the problem: machines M and N accept an infinity of words

I am struggling with the following problem:
Decide if this problem is decidable or not: For two data M and N machines of Turing, there is an infinity of words accepted by the M and N machines. In other words, is the language `{encodedMachine (M) #encodedMachine (N) | the intersection of the language of M and the language of N is infinite}` decidable?

Intuitively, one gets the impression that the problem is undecidable and stopping the reduction could serve to prove it, but I do not know at all how to start this reduction.

## complex analysis – Prove a generalized theorem of Goursat

I read Steins and Shakarchi Complex analysis and I am looking at Exercise 6 of Chapter 2. He says:

Let $$Omega$$ to be an open subset of $$mathbb {C}$$ and let $$T subset Omega$$ to be a triangle whose inside is also contained in $$Omega$$. assume $$f$$ is holomorphic in $$Omega$$ except possibly in one point $$w$$ in the interior $$T$$. Prove that if $$f$$ is bounded by $$w$$then $$int_T f (z) , dz = 0$$

I am aware of this question but I do not manage to analyze the geometric construction mentioned in the comments. I imagine that the first step probably consists of two subdivisions $$T$$ in three new triangles $$T_1, T_2, T_3$$ by connecting each vertex of $$T$$ at $$w$$ and observing that $$int_T f , dz = int_ {T_1} f + int_ {T_2} f + int_ {T_3} f$$ since we should have a cancellation. That's where it gets a little cloudy for me:

(1.) Is this annulment even justifiable since we have no information about what is happening at $$w$$? How can we be sure that the integral is well defined if we integrate on a point that is potentially a singularity?

(2.) Assuming that this is correct, how can we show that each of the three integrals is equal to zero?

Keeping in mind that it is an introductory manual, I would therefore like my solution to remain relatively untechnical.

From the question that I've linked, it seems the way forward is to define an auxiliary contour type and look for a type of boundary, but I'm not sure of the details.

## theoretical measure – Prove \$ {A cap E: A in mathcal A } \$ is a \$ sigma-algebra \$ of \$ E \$

Is my proof correct?

Let $$(X, mathcal A)$$ to be a measurement space and let $$E subseteq X$$. Prove it $$mathcal A_1 = {A cap E: A in mathcal A }$$ is a $$sigma-algebra$$ of $$E$$.

Evidence

1. $$clothes in mathcal A_1$$ because $$clothes in mathcal A.$$

2. $$E in mathcal A_1$$ because $$E subseteq X subseteq mathcal A.$$

3. assume $${A_n } subset mathcal A_1$$

then $$A_n cap E: A_n in mathcal A$$

then $$bigcup A_n subset mathcal A$$

Taking the union $$A_n cap E$$ returns $$bigcup A_n cap E$$ and as $$bigcup A_n subset mathcal A.$$

1. Let $$B in mathcal A_1$$

then $$B cap E: B in mathcal A$$

then $$B ^ c in mathcal A$$

So $$B ^ c cap E: B ^ c in mathcal A$$

$$so B ^ c in mathcal A_1$$

## How can I prove that no algorithm exists for a given problem?

Is there a general framework to show that a problem does not have an algorithm? For example, to show that two problems are as difficult one as the other, we use reduction.

## Prove the monotonicity of a function

I have this function here:
$$g (x) = (e ^ {H (x)} cdot H (x) ^ {({e ^ {H (x)}})}) ^ { dfrac {1} {e ^ gamma n ln ( gamma + nn) + frac {n} { ln ln n}}}}$$
or $$H (n)$$ is the harmonic series and $$gamma$$ is the constant of Euler. I was wondering if there was a concrete way to prove $$g (x)$$This is the monotony.

## Prove \$ forall k geq 4, log (1 + x_k) -x_k leq {-1 over6k} \$ where \$ x_k = {(- 1) ^ k over sqrt k} \$

The question:

Prove $$forall k geq 4, log (1 + x_k) -x_k leq {-1 over6k}$$ or $$x_k = {(- 1) ^ k over sqrt k}$$.

The inequality above is valid if and only if begin {align} & log (1 + x_k) leq x_k- {1 on 6k} \ & Leftrightarrow 1 + x_k leq exp (x_k- {1 over6k}) end {align}
Using $$x + 1 leq e ^ x$$we have $$x_k + 1 leq e ^ {x_k}$$

## Satisfiability – Prove that this language is NP-Hard

It seems like you are trying to prove that $$# mathrm {3SAT}$$ is in $$mathrm {NP}$$. Since $$# mathrm {3SAT}$$ is $$mathrm { # P}$$-complete, and $$mathrm { # P}$$ seems to be more difficult than anything in the polynomial hierarchy, it is very unlikely that $$# mathrm {3SAT} in mathrm {NP}$$.

The error in your proof attempt is that your certificate does not have polynomial length. A satisfying mission has length $$Theta (| x |)$$ (the formula can not have more variables than its length, but it could be of the form $$(x_1 lor x_2 lor x_3) land (x_4 lor x_5 lor x_6) land dots$$). The number $$y$$ could have a value between zero and $$2 ^ {| y |} -1$$Your certificate therefore has a length of about $$| x | 2 ^ {| y |}$$, which is exponential in the input size.

However, the question asks you to prove that it is $$mathrm {NP}$$-hard, not that it's $$mathrm {NP}$$-Achevée. You just have to prove that there is a reduction in the polynomial time of $$mathrm {NP}$$complete problem to $$# mathrm {3SAT}$$.

## to prove \$ dx ^ 2 + dy ^ 2 + dz ^ 2 = dr ^ 2 + r ^ 2 d ( theta) ^ 2 + r ^ 2 (sin theta) ^ 2 d phi ^ 2 \$

$$dx ^ 2 + dy ^ 2 + dz ^ 2 = dr ^ 2 + r ^ 2 d ( theta) ^ 2 + r ^ 2 (sin theta) ^ 2 d phi ^ 2$$

given $$x = rsin theta cos phi, y = rsin theta sin phi , z = r cos theta$$

How can I do this?