ct.category theory – Is the typical category/groupoid zigzag-connected?

Following this answer and comments, I have checked a list of 139 named categories, and I think that they fall into one of five cases:

  • Finite categories used in diagram/shape work
  • Zigzag-connected categories, including those with initial/terminal/zero objects
  • Disconnected groupoids like $mathbb{P}$, whose objects are sets and arrows are bijections
  • Quillen model categories and homotopy categories
  • The quirky disconnected category $mathbf{Field}$, whose objects are fields and arrows are field maps

I don’t think there’s an underlying pattern here, but I’m curious: Is the typical category or groupoid connected? I would understand if the question were only sensible or tractable for (in)finite collections of arrows.

functional inequalities – Szegö’s inequality in approximation theory

Let $T_n$ be the space of all real-valued trigonometric polynomials on $(0,1)$ of degree at most $nin mathbb{N}_0$ and $pin T_n.$ Then
$$
left|p'(x)right|leq 2pi nsqrt{|p|^2_infty -|p(x)|^2}. (*)
$$

The author proves this by contradiction. He assumes that there is a point $x_0$ and a $pin T_n$ such that $|p|_infty <1$ and $left|p'(x_0)right|= 2pi nsqrt{1 -|p(x_0)|^2}.$ (**)

The proof uses mainly the condition of the norm.

My question isn’t about the proof, rather about the logic. How does the fact that there are no such functions (**) imply the required inequality (*)?

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. $mathcal{L}_{omega_1,omega}$).

logic – judgmental and propositional statements in homotopy type theory

In homotopy type theory one has to distinguish between judgmental and propositional statements, eg
in case of $a: A$ (“$a$ has type $A$“) and equalities $a =_p b, a=_A b$. That is, are a judgemental and propositional “$x$ has type $X$” statement, und there are also a judgmental and propositional equality.

I not know how can I evolve an intuition on how to distinguish them. What I read is that generally a statement is propositional if it is “internal” in the theory, while judgmental means that the
statement is “external” in the sense that it is “about” the theory itself and therefore cannot be derived (or say better “constructed”) in the theory. On the other hand what does it concretely mean that a statment is “internal” to the theory?

If we try to think about it in case of our “toy examples” $a: A$ and the two equalities, that’s sounds strange. How can $a: A$ be simultaneously a internal and external statement? How it can be an internal and external statement (that is “within” the theory and “about”)? The same question arises on judgmental and propositional equalities.

Is there a conventional way how one should distinguish between them? Sorry, if the question is too broad and indifferent, I think it’s hard to find a key to approach these two crucial concepts.

ct.category theory – Weak descent and effective equivalence relations

I want to prove that weak descent of a $1$-category implies the classical Giraud axioms.

More precisely, let $mathsf{C}$ be a cocomplete, finitely complete $1$-category. We say that $mathsf{C}$ satisfies weak descent if the following conditions are satisfied:

  • $(mathbf{D1}a)$-(Universal coproducts): Given a collection of objects ${ Y_i }_{i in I}$, let $Y = coprod_i Y_i$. Let $f: X to Y$ be a morphism, and let $X_i = Y_i times_Y X$. Then the induced map $coprod_i X_i to X$ is an isomorphism,
  • $(mathbf{D1}b)$-(Universal pushouts): Given a span $Y_0 leftarrow Y_1 to Y_2$, let $Y = Y_0 coprod_{Y_1} Y_2$. Let $f: X to Y$ be a morphism and let $X_i = Y_i times_{Y} X$. Then the induced map $X_0 coprod_{X_1} X_2 to X$ is an isomorphism.
  • $(mathbf{D2}a)$-(Effective coproducts): Given a collection of maps ${ f_i: X_i to Y_i }$, let $X = coprod_i X_i$, and $Y = coprod_i Y_i$, and let $f: X to Y$ be the coproduct $coprod_i f_i$. Then the natural maps $X_i to Y_i times_Y X$ are isomorphisms for each $i$.
  • $(mathbf{D2}b)$-(Weak effective pushouts): Given a map of spans:
    $require{AMScd}$
    begin{CD}
    X_0 @<<< X_1 @>>> X_2\
    @Vf_0VV @Vf_1VV @Vf_2VV\
    Y_0 @<<< Y_1 @>>> Y_2
    end{CD}

    let $X = X_0 coprod_{X_1} X_2$ and $Y = Y_0 coprod_{Y_1} Y_2$, and let $f:X to Y$ denote the induced map of pushouts. Then the natural maps $X_i to Y_i times_Y X$ are regular epimorphisms.

Condition $(mathbf{D2}b)$ is the real difference between $1$-topoi and $infty$-topoi, and I am trying to better understand this comparison. Now recall the classical Giraud Axioms:

  • $(mathbf{G1})$ Coproducts are disjoint, namely $A times_{A coprod B} B cong varnothing$,
  • $(mathbf{G2})$ For any morphism $f: X to Y$, the base change functor $f^*: mathsf{C}_{/Y} to mathsf{C}_{/X}$ preserves colimits,
  • $(mathbf{G3})$ Equivalence relations are effective.

Rezk sketches how to prove that $(mathbf{D1}) = (mathbf{D1}a) wedge (mathbf{D1}b)$ is equivalent to $(mathbf{G2})$, and that $(mathbf{D2}a) implies (mathbf{G1})$.

My suspicion is that $(mathbf{D2}b) implies (mathbf{G3})$ or is possibly equivalent to it, but I can’t quite see how to prove it. I also suspect that the way one can prove it by is proving that $(mathbf{D2}b)$ is equivalent to the coequalizer defining equivalence relations being effective. Namely if $R xrightarrow{(s,t)} X times X$ is an equivalence relation, then $R rightrightarrows X to X/R$ is a coequalizer iff $X/R cong R coprod_{R coprod R} X$, and my idea is to show that having condition $(mathbf{D2}b)$ hold, but this time up to isomorphism rather than regular epi but only for pushouts of equivalence relations as above, and this would be equivalent to $(mathbf{D2}b)$ and from this somehow it would be easier to see that it implies $(mathbf{G3})$, but I’ve had no luck with this either.

Any ideas or insight would be appreciated.

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 – Upper bound suggested by the first generalized Hardy-Littlewood conjecture

As mentioned by Terrance Tao, the first Hardy-Littlewood Conjecture in quantitive form is:

Conjecture 2 (Prime tuples conjecture, quantitative form) Let ${k_0 geq 1}$ be a fixed natural number, and let ${{mathcal H}}$ be a fixed admissible ${k_0}$-tuple. Then the number of natural numbers ${n < x}$ such that ${n+{mathcal H}}$ consists entirely of primes is ${({mathfrak G} + o(1)) frac{x}{log^{k_0} x}}$.

Based on the conjecture, is there any way one can find an upper bound for $n$ such that ${n+{mathcal H}}$ consist entirely of primes? i.e. when can we assert that $o(1)$ term is smaller than $mathfrak G$?

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.

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.