I graduated from La Sorbonne in Paris, France (Pantheon) well over 20 years ago. There I received my masters. (Ateliers de conversation-L’Anglais en master-Master BI) I studied English, German and Spanish. But that was a long time ago.

I was informed today by a student who is in university studying English that in “modern English” the subject word is one word these days and is to be spelt as “proofreader”. This is not what I was taught so very long ago.

So what do you brilliant writers have to say about it? How is it to be spelt, is the American way different than the British or Canadian? What say you, how do you spell it? As a non native speaker of English I would really like to know what is what with the spelling/writing of this word(s). One ought to never want to stop learning…true?

P.S. Yet another form of this word is a hyphenated version which is written as “proof-reader”.

## mathematical philosophy – Gödel’s ontological proof & Benzmüller’s work

For a decade or so, Christoph Benzmüller from Berlin has explored Gödel’s ontological proof (and variants) of existence of God. He uses the proof assistant Isabelle/HOL. I recently posted a preprint, which was highlighted by the cover of the French magazine Science et Vie.

Well, I am not familiar with AI, yet even less with applications to metaphysics. But many practitioners of MO must be. I should like to know how serious is Benzmüller’s work considered in this community. Is it controversial or is this considered a respectable research activity ?

## soft question – What is the best way to ask for comments on your proposed proof of some known theorem on MathOverflow?

Firstly, i apologize in advance if this post in not suitable for MO, but in my personal circumstances, i have no other place where i could post it.

Suppose one is fairly convinced that they have found a short proof of some famous theorem. But the poblem is, this individual has no connections to any specialist of the area (due to geographical location or otherwise), such that they have no one to discuss their wok with. This lack of professional connections would also mean that they are going to find it difficult to find endorsements to publish on arxiv, hence their only option would be a place like vixra, where one should really never post their work. And if they try to find some specialist of the field, they are most certainly going to be dismissed as a ”crank”, which may not necessarily be so.

So, considering the really difficult circumstances that this young aspiring and passionate mathematician is in, what is the best way for them to ask for comments on MathOverflow, bearing in mind that MO has a policy against such posts ?

Maybe the indiviaual in question could be given the option to post their proposed proof using a polite tone like $$“$$the following proof must be certainly wrong, but where is the mistake ?” And as soon as any mistake is found, have their account banned definitely or indefinitely ? Just a thought.

## Complete proof of PAC learning of axis-aligned rectangles

I have already read PAC learning of axis-aligned rectangles and understand every other part of the example.

From Foundations of Machine Learning by Mohri, 2nd ed., p. 13 (book) or p. 30 (PDF), I am struggling to understand the following sentence of Example 2.4, which is apparently the result of a contrapositive argument:

… if $$R(text{R}_S) > epsilon$$, then $$text{R}_S$$ must miss at least one of the regions $$r_i$$, $$i in (4)$$.

i.e., $$i = 1, 2, 3, 4$$. Could someone please explain why this is the case?

The way I see it is this: given $$epsilon > 0$$, if $$R(text{R}_S) > epsilon$$, then $$mathbb{P}_{x sim D}(text{R}setminus text{R}_S) > epsilon$$. We also know from this stage of the proof that $$mathbb{P}_{x sim D}(text{R}) > epsilon$$ as well. Beyond this, I’m not sure how the sentence above is reached.

## Proof that NL is contained in P

I do not succeed to prove that.

How can I do that by using st-conn, or Savitch’s theorem?

I tried to use these but I did not succeed.

Thank you.

## proof explanation – A function $f: mathbb{Z} rightarrow mathbb{Z}$ is defined by $f(n)$ = $2n+1$. Determine whether $f$ is injective and surjective.

$$f: mathbb{Z} rightarrow mathbb{Z}$$ is defined by $$f(n)$$ = $$2n+1$$

I proved injective first:

$$f(a) = f(b)$$ therefore it is $$a = b$$

$$2a + 1$$ = $$2b+1$$

$$2a = 2b$$

a = b , therefore $$f(n) = 2n+1$$ is injective

now for surjective, I proved this by

Let $$r in mathbb{Z}$$ so there exist $$nin mathbb{Z}$$ such that $$f(n) = r$$

$$2n+1 = r$$ , $$n = frac{r-1}{2}$$ , $$f(frac{r-1}{2}) = 2(frac{r-1}{2}) + 1$$ and that equals $$r$$. Therefore it is surjective.

I looked the answer is says: the function is injective but there is no surjective because there is no $$n in mathbb{Z}$$ such that $$f(2) = 2$$. I am trying to figure out, why I am wrong?

## mathematical proof about prime number question

If for all positive integers $$n$$, there is a conclusion holds that $$p_{1} p_{2} ldots p_{n}+1$$ is a prime number.

Here $$p_{n}$$ denotes the nth prime number.

## inequality – Proof that if one divisor is smaller than square root of the dividend, then other divisor is greater than the dividend.

Thanks for contributing an answer to Mathematics Stack Exchange!

But avoid

• Making statements based on opinion; back them up with references or personal experience.

Use MathJax to format equations. MathJax reference.

## Question on the given proof of $(det A)(det B) leq [(tr AB)/n]^n$

I was reading through this paper and on page 6, there is a lemma proving
(det $$A$$)(det $$B$$) $$leq$$ ((tr $$AB$$)/n)$$^n$$ for two positive semideifinite matrices $$A$$ and $$B$$.

I get every single line until the part that concluded tr $$AB$$ = $$sum lambda_l mu_l$$. Especially, what’s bugging me is the very last equation. Before the equality sign we have
$$sum S_{li} (S^T)_{il} (D_1)_{ii} (D_2)_{ll}$$
and after the equality sign we have
$$sum delta^{il} (D_1)_{ii} (D_2)_{ll}$$
Since S is a symmetric orthonormal matrix, $$S_{li} = (S^T)_{il}$$, so I believe the first term should just be $$s^2_{li} (D_1)_{ii} (D_2)_{ll}$$. But it doesn’t make sense that $$s^2_{li} = delta^{il}$$, since the orthonormal matrix just confirms that the 2-norm of any ‘row’ or ‘column’ is 1, not that the single $$s^2_{li}$$ is $$1$$ for $$l=i$$ and $$0$$ otherwise.

There must be something I am missing. Can anyone help me filling the gap between those two line, please?

P.S. The paper is about elliptic equation, but the lemma is for general positive semidefinite matrices, so I put tag just for these type of questions.