set theory – Detecting uncountable cardinals in $(mathbb{R};+,times,mathbb{N})$

For a structure $mathcal{X}=(X;…)$, say that a cardinal $kappa$ is $mathcal{X}$-detectable iff there is some sentence $varphi$ in the language of $mathcal{X}$ together with a fresh unary predicate symbol $U$ such that an expansion of $mathcal{X}$ gotten by interpreting $U$ as $Asubseteq X$ satisfies $varphi$ iff $vert Avertgekappa$.

For example, $omega_1$ is $(omega_1;<)$-detectable since a subset of $omega_1$ is countable iff it is bounded above. By contrast, it turns out that $omega_1$ is not $mathcal{R}=(mathbb{R};+,times)$-detectable.

I’m interested in the expansion $mathcal{R}_mathbb{N}:=(mathbb{R};+,times,mathbb{N})$ of $mathcal{R}$ gotten by adding a predicate naming the natural numbers (equivalently, adding all projective functions and relations). Since we can talk about one real enumerating a list of other reals, $omega_1$ is $mathcal{R}_mathbb{N}$-detectable (“there is no real enumerating all elements of $U$“). More pathologically, if $mathfrak{c}=2^omega$ is regular and there is a projective well-ordering of the continuum of length $mathfrak{c}$ then $mathfrak{c}$ is $mathcal{R}_mathbb{N}$-detectable. So for example it is consistent with $mathsf{ZFC}$ that $omega_2$ is $mathcal{R}_mathbb{N}$-detectable.

I’m curious whether this type of situation is the only way to get $mathcal{R}_mathbb{N}$-detectability past $omega_1$. There are multiple ways to make this precise, of course. At present the following two seem most natural to me:

  • Is it consistent with $mathsf{ZFC}$ that there are at least two distinct regular cardinals $>omega_1$ which are $mathcal{R}_mathbb{N}$-detectable?

  • Is it consistent with $mathsf{ZFC}$ that there is a singular cardinal which is $mathcal{R}_mathbb{N}$-detectable?

Note that an affirmative answer to either question requires a large continuum, namely $geomega_3$ and $geomega_{omega+1}$ respectively. Although my primary interest is in first-order definability, I’d also be interested in answers for other logics which aren’t too powerful (e.g. $mathcal{L}_{omega_1,omega}$).

algebraic number theory – non-Archimedean normed space

Definition Let $(F,|cdot|)$ be a field equipped with an absolute value. A normed vector space over $F$ is a pair $(V,|cdot|)$ consisting of an $F$-vector space $V$ and a map $|cdot| : V → mathbb{R}$ satisfying

  • $|v|geq 0$ with equality if and only if $v = 0$,
  • $|cv| = |c|cdot|v|$ for all $cin F$ and $vin V$,
  • $|v + w| leq |v| + |w|$ for all $v,w in V$.

How to prove that if $F$ is non-archimedean then there is a constant $C > 0$ (depending on the normed vector space) such that $|v+w| leq Cmax(|v|,|w|)$ for all $v,w in V$.

probability theory – Expected Value of Nonnegative Identically Distributed Random Variables

Let $X_1, X_2 geq 0$ be two non-negative identically distributed random variables. I wonder if the following equation holds.

$$
mathbb{E}[X_1X_2] = mathbb{E}[Y^2]
$$

where $Y$ is a random variable having the common distribution of ${X_1, X_2}$.

My attempt: We know that, in general, $mathbb{E}[X_1X_2] neq mathbb{E}[X_1^2]$ and/or $mathbb{E}[X_1X_2] neq mathbb{E}[X_2^2]$. One can take $X_1 in {0,1}$ with probability $1/2$ and take $X_2 := -X_1$. However, the nonnegativity excludes this case.

Also, if we look at
$$
mathbb{E}[Y^2] = int y^2, dF_{Y} = int y^2 , dF_{X_1,X_2}
$$

where the last equality hold since $Y$ has the common joint distribution of ${X_1,X_2}$. But I get stuck to see if this is equal to $
mathbb{E}[X_1X_2] = int x_1x_2 ,dF_{X_1,X_2}$
. Any comment is appreciated.

graph theory – Centrality measures in a network with negative correlations

I have a bidirectional network where the weights of edges are based on partial correlation matrix. I have both positive and negative values as weights. Now, I want to compute centrality measures as degree, closeness, betweenness and eigenvector. How can I handle the negative values? Would I get correct values for these measures, if I keep the negatives? Should I use absolute value or take (1-absolute value)?

Basically, I am confused about if these values would affect the outcome in any way. I have not found any resources that would discuss this. Please recommend, if you know any.

nt.number theory – Robin’s criterion, Goldbach’s conjecture and upper bound for $r_{0}(n)$

This question is a follow-up to both About Goldbach’s conjecture and Question in Proof of Hardy Ramanujan theorem about $omega(n) =sum_{p|n} 1$.

Can one derive from Robin’s criterion for RH an inequality of the form $sigma(n)ll nomega(n)$? Moreover, as Carl Pomerance told me in a private communication that the quantity $r_{0}(n):=inf{r>0,(n-r,n+r)inmathbb{P}^{2}}$ is expected to fulfill $r_{0}(n)lllog^{2}nloglog n$, can one get from this criterion an upper bound for $r_{0}(n)$ of the form $r_{0}(n)llfrac{sigma(n)}{N_{2}(n)}$ where $N_{2}(n)=sharp{0<r<n-sqrt{2n-3},(n-r,n+r)inmathbb{P}^{2}}$ under RH?

number theory – A question in calculating a constant in section square free values of quadratic polynomials

I am studying square free values of quadratic polynomials from class notes and I am struct on a deduction.

Consider this conjecture, I have no problem in understanding it:

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Consider this theorem, I have no problem in understanding it.
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But I have problem in calculating $c_f$

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I don’t understand , how author wrote $rho(p^2) = 1+ (-1/p) $.

Kindly consider giving any hints.

nt.number theory – Modern treatment of Delange’s Tauberian Theorem

Tauberian theorems abound in the literature. One of the most general, powerful, and versatile ones I know of is due to Delange, and appears as Theorem I of the paper:

H. Delange – Généralisation du théorème de Ikehara, Annales scientifiques de l’École Normale Supérieure, Série 3, Tome 71 (1954) no. 3, pp. 213-242.

Now this result was published in the 1950’s and my understanding is that it was very influential. But this was a long time ago. Is there a modern treatment of Delange’s result anywhere in the literature? For example whether the proof or some of his long list of technical assumptions have been simplified with hindsight. Or maybe it has been generalised?

I’m ultimately interested in applying this result to a Dirichlet series to deduce an asymptotic formula for the partial sums of the coefficients. Delange’s result is stated in terms of Laplace transforms, but there is a standard trick to apply this to Dirichlet series.

Please note that I am not looking for results with significantly weakened assumptions; in particular I do not want to assume that my Dirichlet series admits a holomorphic extension to some line, only that it admits a continuous extension to the line. For me this is part of the power of Delange’s result, as is also the case for the Wiener-Ikehara Tauberian theorem.

measure theory – $int_Gg(x)f(x) ~ mathrm{d}mu(x)=0 ~ text{for all }gin L^1(G)$ implies $f=0$

My Definitions. Let $G$ be a locally compact Hausdorff group and $mu$ be a Haar measure on $G$. We have defined “Haar” measure as follows:

(Haar measure) It’s a nonzero left invariant outer Radon measure on $G$. An outer Radon measure is a locally finite Borel measure on $G$ which is outer regular on Borel sets and inner regular on open sets

(Assuming this definition) the author of the text (I think?) have used the following statement without proof. But I’m not very sure how exactly to prove this:

Statement. Let $f in L^1(G)$ such that
$$
int_Gg(x)f(x) ~ mathrm{d}mu(x)=0 ~ text{for all }gin L^1(G)
$$

Then $f=0$.

I tried to proceed with the same line of argument given here: since each compact set has finite measure, the assumption on $f$ implies that
$$
int_K f(x)~mathrm{d}mu(x)=0 ~text{for all compact }Ksubset G ~~~~~~~ (*)
$$

Case 1. $fin C_c(G)$ and $f$ is $Bbb{R}$-valued.

In this case I have managed to prove $f=0$.For this let $S:={fne 0}={f>0}cup{f<0}subset G$. With contrary if $mu(S)>0$ then WLOG $mu({f>0})=cup_n {f>frac{1}{n}}=:cup_n E_n>0$ hence there must exists $Nge 1$ s.t. $mu(E_N)>0$. Since $f$ is continuous so $E_N$ is open in $G$. Now I’ll use the (weak) inner regularity of $mu$ to arrive at a contradiction: if $K$ be a compact subset of $E_N$ with $mu(K)>0$ then we get
$$
0=int_K f(x) ~mathrm{d}mu(x)ge int_K frac{1}{N} ~mathrm{d}mu(x)=frac{1}{N}mu(K)>0
$$

a contradiction, hence $f=0$ in this case.

Case 2. $fin C_c(G)$

This follows from Case 2.

Case 3. $fin L^1(G)$.
Here I got stuck, I know $C_c(G)$ is dense in $L^1(G)$. So there would exists a sequence of members in $C_c(G)$ converging towards $f$. But the members of that sequence need not satisfy $(*)$. Also even such sequence exists I don’t know how to interchange the “limit” with “integral”??

Is the original statement valid only for continuous $f$?
Thank you.

complexity theory – What is wrong with this argument that if A is NP Complete, but B is in P, then AB is NP Complete and BA is NP Complete as well?

The following seems to me to be relevant to this question, but to me is an interesting exercise, especially since I have not formally worked with complexity before, but I want to learn more:

Suppose that A is NP Complete, but B is in P. I claim that AB is NP Complete and BA is NP Complete as well. To see this, assume first that AB is in P, and let X and Y be polynomial time algorithms for B and for AB, respectively. “Concatenating” X and Y as follows yields an algorithm Z for A:

Given L, test L using X;
if X outputs “yes”, test using Y;
if Y yields “yes”, output “no” and stop;
if X yields “no”, output “no” and stop; output “yes” otherwise and stop.

This algorithm Z runs in polynomial time, because if the (polynomial time) complexity exponent of X is k and the (polynomial time) complexity exponent of Y is n, then this algorithm clearly has (polynomial time) complexity exponent m=max(k,n). This would provide a proof that P=NP, so AB is NP Complete.

Now suppose that BA is in P. This time, let Y’ be a polynomial time algorithm for BA and let X be as above. We construct an algorithm Z’ for A, as follows:

Given L, test L using X;
if X outputs “yes”, test using Y’;
if Y yields “no”, output “yes” and stop;
if X yields “no”, test using Y’; if Y yields “no”, output “yes” and stop;
output “no” otherwise and stop.

This yields a polynomial time algorithm for A, and so again, this would entail that P=NP, so BA also is NP Complete.

+++++++++End of Example++++++++++++

While I don’t see anything wrong with the above at the moment, perhaps I have a mistake or complexity miscalculation? …because for a while as I was writing the second algorithm, I began to think it was odd and perhaps impossible that I can be right about BA also being NP Complete…

Like I said, I’m somewhat new to this area, so feedback would be appreciated.

gr.group theory – Length of commutators in the free group

Let $G=F_2$ be the free group of rank $2$. Is there a constant $c>0$ such that the word length $|(u,v)|$ of every commutator $(u,v)=uvu^{-1}v^{-1}$ where $u,vin G$, $|u|,|v|>c$ is at least $c(|u|+|v|)$ unless $(u,v)=(u_1,v_1)$ for some $u_1,v_1$ with $|u_1|+|v_1|<|u|+|v|$?