# All Questions

**3**

votes

**1**answer

373 views

### Use of infinitude of primes in the Green-Tao theorem [on hold]

In a video I watched last night on nuking mathematical mosquitos, Matt Parker gave the following proof of the infinitude of primes: suppose there are finitely many primes. The Green-Tao theorem says ...

**-2**

votes

**0**answers

38 views

### Finding topological properties under a metric on set of composition operators of L2 [on hold]

We define a new metric on all composition operators in $L^2$:
$$ d_R (A,B)= \sqrt{\Vert P_{R(A)}- P_{R(B)} \Vert^2+\Vert A-B\Vert^2 }. $$
Now we would like to find ...

**4**

votes

**1**answer

211 views

### Can we always attain another prime via inserting digits between the digits of a fixed prime?

The sequence OEIS A080437 is
For n > 10, let m = n-th prime. If m is a k-digit prime then a(n) = smallest prime obtained by inserting digits between every pair of digits of m.
I don't see why ...

**10**

votes

**2**answers

413 views

### Blinking graphs

For any simple graph $G$, assign its nodes a weight/bit of $0$ or $1$.
Call this a bit assignment for $G$.
Now, generate a new bit assignment as follows:
Each node $x$'s bit is replaced by $1$ if the ...

**3**

votes

**2**answers

149 views

### Definition of the differential of the Cone of a morphism of complexes [on hold]

Let $(F^\bullet,d_F)$ and $(G^\bullet,d_G)$ be two complexes in an abelian category $\mathbf{A}$.
The complex cone $Cone(\varphi)^\bullet$ of a morphism of complexes $\varphi:F^\bullet \to G^\bullet$ ...

**2**

votes

**0**answers

86 views

### For which fields are the 1-dimensional algebraic groups known?

Given an algebraically closed field, or even a perfect one, a connected 1-dimensional algebraic group $G$ over the field $K$ is isomorphic to either $\mathbf G_a$ or $\mathbf G_m$.
For which fields ...

**5**

votes

**1**answer

192 views

### How does pseudoconvexity restrict the topology?

A domain of holomorphy in $\mathbb{C}^n$ has vanishing de rham cohomology in real dimensions greater than $n$ - half of it's cohomology is missing. Are there any other restrictions? If I give you a ...

**53**

votes

**8**answers

7k views

### Why is Lebesgue integration taught using positive and negative parts of functions?

Background: When I first took measure theory/integration, I was bothered by the idea that the integral of a real-valued function w.r.t. a measure was defined first for nonnegative functions and only ...

**21**

votes

**3**answers

805 views

### Is the Rado graph a Cayley graph? If so, what is the group like? (And other questions…)

The countable random graph, also known as the Rado graph, is characterized as the unique countable graph in which every two disjoint finite sets $A$ and $B$ of vertices admit a vertex $p$ connected to ...

**2**

votes

**2**answers

509 views

### Can we compute every definable number with knowledge of the halting problem?

Suppose we knew the answer to the halting problem, and the halting problem for this new system with the old halting problem solved. And so on. Would this allow us to compute every definable number?

**1**

vote

**0**answers

47 views

### On the sum of digits of primes in binary form [duplicate]

Let $s_2(m)$ be the sum of digits of $m$ in binary form.
I would like to ask the following question:
Is it true that for every $n\in \mathbb{N}$ there is at least one
prime $p$ which has ...

**1**

vote

**0**answers

42 views

### Extra-Lorentzian Kac-Moody algebras

My question is about Kac-Moody (KM) algebras of finite rank with symmetrized Cartan matrices $B = C A$ ($A$ is Cartan matrix) of signatures $(-,-,+,...,+)$, $(-,-,-,+,...,+)$, etc. i.e. with ...

**2**

votes

**3**answers

133 views

### Surgery of $S^3$

I have been troubled by this seemingly simple question recently.
How do we easily visualize the statement:
Surgery of $S^3$ over a trivial unknot gives $S^1 \times S^2$?
All I can think of for ...

**6**

votes

**1**answer

188 views

### Does “$|{\cal P}_2(X)| = |X|$ for $X$ infinite” imply ${\sf (AC)}$?

This comes from a comment made by user bof in this thread.
Let $X$ be a set, define ${\cal P}_2(X) = \big\{\{a, b\}: a\neq b\in X\big\}$.
Consider the statement
${\sf (S)}$ If $X$ is an ...

**2**

votes

**1**answer

197 views

### Moduli of stable bundles - analytic approach

Consider a compact Riemann surface $X$ of genus $\ge 2$, and consider the set $M$ of stable holomorphic vector bundles of rank $n$ and degree $d$ on $X$, up to isomorphism.
At that point, one states ...

**4**

votes

**0**answers

135 views

### Dynamical Mordell-Lang on Kahler manifolds?

Suppose that $X$ is a smooth projective variety over $\mathbb C$ and $\phi : X \to X$ is an endomorphism. Let $p \in V$ be a point and $V \subset X$ a subvariety. The dynamical Mordell-lang ...

**48**

votes

**3**answers

3k views

### What are the implications of the new quasi-polynomial time solution for the Graph Isomorphism problem?

This week, news came out that Laszlo Babai has found a quasi-polynomial time algorithm to solve the Graph Isomorphism problem (that is: $O(\exp(polylog(n)))$). He is giving a series of talks this ...

**2**

votes

**0**answers

58 views

### Detecting Negative Cycles in Undirected Graphs

I recently faced the problem of quickly detecting negative cycles in undirected, weighted graphs. Resorting to the Bellman-Ford Algorithm, as commonly suggested, turned out to be very inefficient and ...

**0**

votes

**1**answer

73 views

### Example distance metric that is not conditionally negative definite

Theorem 4.1 of this paper says that there exist distance matrices that are not conditionally negative definite (CND). How do I construct an example of a distance matrix that is not CND? Do you know an ...

**0**

votes

**0**answers

41 views

### Inapproximability of Combinatorial Optimization Problems

I've been reading the "Inapproximability of Combinatorial Optimization Problems" by Luca Trevisan (see: link). On pages 3-4 it mentions that a polynomial time algorithm for 3SAT would exist if there ...

**0**

votes

**1**answer

55 views

### About $c(A)$ in $c(A)|A|\leq |A^{-1}A|$

Let $G$ be a finite group, $\emptyset\neq A\subseteq G$, $A^{-1}:=\{ a^{-1}:a\in A\}$, and put $$c(A):=\max\{t\in \mathbb{Z}: t|A|\leq |A^{-1}A|\}$$
It is clear that $1\leq c(A)\leq ...

**3**

votes

**1**answer

51 views

### Converging to moments obeying Carleman's condition

I believe that the following is true, and I'd like to make sure that it is and to have a reference. Suppose that $\mu_N$ are a sequence of measures on $\mathbb{R}$. Let $m_{N,k}$ be the $k$-th ...

**9**

votes

**3**answers

228 views

### Random links and $3$-manifolds

In Jeffrey Weeks book "The Shape of Space" he explaines at the end of Chapter 18 (on page 255) the following about the geometrization conjecture:
A non-trivial connected sum $M_1\# M_2$ admits a ...

**-1**

votes

**0**answers

119 views

### The eigenvalue of operator $-\Delta$

Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. We know the eigenfunction of Laplacian operator $-\Delta$ is an orthonormal basis of $L^2$. Let $\{\omega_n\}$ denote the ...

**24**

votes

**9**answers

5k views

### What are the reasons for considering rings without identity?

I think a major reason is because Lie algebras don't have an identity, but I'm not really sure.

**29**

votes

**2**answers

3k views

### Everywhere differentiable function that is nowhere monotonic

It is well known that there are functions $f \colon \mathbb{R} \to \mathbb{R}$ that are everywhere continuous but nowhere monotonic (i.e. the restriction of $f$ to any non-trivial interval $[a,b]$ is ...

**39**

votes

**4**answers

10k views

### Non-Borel sets without axiom of choice

This is a simple doubt of mine about the basics of measure theory, which should be easy for the logicians to answer. The example I know of non Borel sets would be a Hamel basis, which needs axiom of ...

**84**

votes

**4**answers

3k views

### What do the stable homotopy groups of spheres say about the combinatorics of finite sets?

The Barratt-Priddy-Quillen(-Segal) theorem says that the following spaces are homotopy equivalent in an (essentially) canonical way:
$\Omega^\infty S^\infty:=\varinjlim~ \Omega^nS^n$
...

**69**

votes

**5**answers

6k views

### What do epimorphisms of (commutative) rings look like?

(Background: In any category, an epimorphism is a morphism $f:X\to Y$ which is "surjective" in the following sense: for any two morphisms $g,h:Y\to Z$, if $g\circ f=h\circ f$, then $g=h$. Roughly, ...

**6**

votes

**1**answer

364 views

### Itô's article “A measure-theoretic approach to Malliavin calculus”

Apart from citations all over the internet, the following paper appears to be off-the-grid.
K. Itô, A measure-theoretic approach to Malliavin calculus, in 'New Trends in Stochastic Analysis', Proc. ...

**7**

votes

**1**answer

340 views

### Is the fundamental group of $II_{1}$ factors invariant under a relation?

In order to define the equivalence relation, let's first recall the Tomita-Takesaki modular theory and conditional expectation for von Neumann algebras.
Let $H$ be a separable Hilbert space and ...

**3**

votes

**1**answer

158 views

### Braid wiring diagrams and matroids

recently I started reading some articles about the presentation of the fundamental group of lines arrangements in $\mathbb{C}^{2}$ via Wiring diagrams.
I also found some relation with matroid theory. ...

**3**

votes

**1**answer

155 views

### A decreasing sequence involving the divisor function?

Define $N_k \geq 6$ to be the $k-th$ primorial number and let $\sigma(n)$ be the divisor function.
It seems that $u_k = \dfrac{\sigma(N_k)}{N_k \log\log N_k}$ is a decreasing function ?
By ...

**21**

votes

**8**answers

2k views

### Is there a compact group of countably infinite cardinality?

Apologies for the very simple question, but I can't seem to find a reference one way or the other, and it's been bugging me for a while now.
Is there a compact (Hausdorff, or even T1) (topological) ...

**9**

votes

**4**answers

1k views

### Basic questions on the homotopy category

I apologize in advance if this the answer to this question is standard or well-known. I am not in any way an algebraic topologist.
$\newcommand{\s}{\mathscr}$Let $\s T$ be the category of topological ...

**4**

votes

**3**answers

1k views

### Finding f such that f(f(x))=g(x) given g

Suppose $g(x)$ is a smooth increasing function defined for $x \ge 0$ such that $g(x) \ge x$ for all $x$. Does there exist a function $f$ with similar properties such that $f(f(x))=g(x)$ for all $x \ge ...

**6**

votes

**2**answers

566 views

### Is assigning the endomorphism object in some sense functorial?

Let $\mathcal V$ be a monoidal category and let $\mathcal C$ be a $\mathcal V$-category. Let's denote the $\mathcal V$-valued hom-functor $[-,-]$. Now for every object $X\in\mathcal C$ we have it's ...

**6**

votes

**1**answer

452 views

### Repeated Second Eigenvalue of the Adjacency Matrix of a Graph

This question is motivated by a talk I went to earlier today.
Suppose we have a $d$-regular graph $G$ with $n$ vertices, with adjacency matrix $A$.
Let $$\lambda_1\geq \lambda_2 \geq\dots \geq ...

**0**

votes

**0**answers

41 views

### CLT for sums of an infinite sequence of rv with an asymptotic distribution

Excuse me if the question is ill-posed. I'll do my best to explain the problem.I have a vector $(x^{(n)}_1, x^{(n)}_2, \ldots x^{(n)}_n),$ whose individual components can be shown to be asymptotically ...

**7**

votes

**3**answers

264 views

+50

### Analytic continuation of the double sum $\sum_{n,m\ge0}x^ny^mt^{nm}$

Define function $f(x,y,t)$ as the analytic continuation of the series
$$f(x,y,t)=\sum_{n,m\ge0}x^ny^mt^{nm}$$
This series definitely converges when all the arguments are small enough. I would like to ...

**2**

votes

**1**answer

116 views

### Opposite of an E2-algebra

Suppose $C$ is the monoidal $\infty$-category of modules over an $\mathcal{E}_2$-ring spectrum $A$. Let $C' = C$ as a category, but with opposite monoidal structure to $C$. Is $C'$ the category of ...

**7**

votes

**0**answers

95 views

### The non-abelian Gauss-Manin connection; non-abelian M_dR; a Grothendieck lemma for cyrstals

I'm interested in understanding the non-abelian Gauss-Manin connection on Carlos Simpson's relative de Rham Moduli space $M_{dR}(X/S,n)$ for a smooth projective morphism of schemes $X/S$. The scheme ...

**5**

votes

**0**answers

97 views

### Has anyone seen these binary trees (Catalan-type related to the Gegenbauer polynomials and Motzkin paths)?

The OEIS entry A121448 enumerates binary trees with $n$ edges and $k$ vertices with outdegree 1.
Has anyone seen these trees?
The o.g.f. for this entry, $G(x,t)$, is essentially a discriminant ...

**0**

votes

**0**answers

107 views

### Additional condition to the Bollobas theorem (Sperner's therorem) in extremal set theory

The Bollobas'1965 theorem is the following:
If $A_1,...,A_n$ and $B_1,...,B_n$ are two sequences of subsets of $X=\{1,...,r\}$ such that $A_i\cap B_j = \emptyset$ if and only if $i=j$, then
...

**2**

votes

**0**answers

67 views

### Is it possible to write down the explicit expressions of some extensions of conformal vector fields on spheres?

Let X be a conformal vector field on the standard sphere $S^n$ with standard metric $g_{S^n}$, then there exists a unique conformal vector fields in the unit ball $B_1(0)\subset \mathbb{R}^{n+1}$, ...

**3**

votes

**1**answer

308 views

### Primary structures in $\mathbb Q$

I'll formulate a topic restricted here to the positive rational
numbers $\ \mathbb Q_{_{>0}},\ $, then will pose a question (Q2) plus some related, to which I don't know the answers nor reference. ...

**4**

votes

**2**answers

203 views

### Lefschetz on étale fundamental group for quasi-projective varieties

If $X$ is a smooth projective variety of dimension at least $3$ over $\mathbb{C}$, Lefschetz's Hyperplane theorem says that for every hyperplane section $H$
$$\pi^1(H)\to\pi^1(X)$$
is an isomorphism, ...

**1**

vote

**0**answers

60 views

### Counting growing tree trajectories

I am looking for help:
Beginning with a single node ($\circ$), at each discrete time step I can add a node/link pair to any node currently in the tree. Nodes are unlabelled and the tree is ...

**9**

votes

**3**answers

482 views

### Collection of dense subsets as a “fingerprint” for Hausdorff topologies?

Let $(X,\tau)$ be a Hausdorff space and let ${\cal D}$ denote the collection of dense subsets of $(X,\tau)$. Is it possible that there is another Hausdorff topology $\tau_1 \neq \tau$ on $X$ such that ...

**1**

vote

**2**answers

111 views

### Four Sphere Fibrations

Does there exist a manifold $M$ and a compact Lie group $H$ such that we have a fibration $H \to S^4 \to M$, where $S^4$ is the four sphere?