# All Questions

152,086
questions

5
votes

2
answers

263
views

### What would you do with a new model of linear logic?

I have been working for some time with collaborators developing some models of linear logic which we are confident are new. However, none of us is deep enough in the field to answer the sceptic's ...

-3
votes

0
answers

61
views

### $a \cos(x) + ib \sin(x)$ reduced to $\cos(x+iy) $form [closed]

As the titled suggests, is there is formula remotely near this expression?
$a \cos(x) + ib \sin(x)$ reduced to $\cos(x+iy)$.
Contextual basis, I'm required to convert a Laurent polynomial to a trig ...

0
votes

0
answers

53
views

### Optimality condition for strongly convex function under sparsity constraint

Let $f: \mathbb{R}^p \to \mathbb{R}$ be a $2s$-sparse strongly smooth, $2s$-sparse strongly convex and twice differentiable function. In other words, there exists positive constants $\alpha, L >0$ ...

0
votes

0
answers

54
views

### asymptotic expansions for $C^{1+\epsilon}$operators

I want to understand the asymptotic expansion for $C^{1+\epsilon}$ operators.
More precisely, if a complex operator $T(z)=\sum_{n \ge 1}T_n z^n$ is defined on a closed unit disk, and it is $C^{1+\...

4
votes

1
answer

188
views

### Height of a conductor ideal

We say an extension of Noetherian rings $R\subset S$ is elementary subintegral if $S=R[b]$ for some $b\in S$ with $b^2,b^3\in R$. The conductor ideal is defined to be $\operatorname{Ann}_R(S/R)$. What ...

7
votes

1
answer

236
views

### Some fusion rings/categories I don't recognize

Recently (what I believe are) all multiplicity-free fusion categories up to rank 7 have been posted on the AnyonWiki. Most of the fusion rings belonging to these categories belong to one of the ...

9
votes

1
answer

156
views

### Matrix ring isomorphisms of different sizes

Do there exist (unital, associative, noncommutative) rings $R$ and $S$, where $\mathbb{M}_2(R)\cong \mathbb{M}_3(S)$, but these matrix rings are not isomorphic to $\mathbb{M}_6(T)$ for any ring $T$?

5
votes

0
answers

167
views

### Reference request: Automorphisms of $\mathbb C\{x,y\}$ which preserve the equation of the cusp, $x^3 - y^2$

In my research I encountered automorphisms of the ring of convergent power series
$$\varphi: \mathbb C\{x,y\} \to \mathbb C\{x,y\},$$
which preserve $f = x^3 - y^2$, i.e. $\varphi(f) = f$. I'm ...

2
votes

1
answer

83
views

### Does a polynomial $P(X,Y)$ that specializes to a polynomial $P(x_0,Y)$ with distinct roots in $\overline{k}$ have distincts roots in $\overline{k(X)}$

Let $k$ be a field, $P\in k[X][Y]$ be a monic polynomial of degree $n$ in $Y$.
I am looking for a simple proof of the following fact.
"If there exists $x_0\in k$ such that $P(x_0,Y) \in k[Y]$ has ...

2
votes

1
answer

79
views

### Is this form of replacement suitable for ZF - Powerset + well-ordering principle?

The following scheme can be understood as a form of replacement. Axiomatizing $\sf ZF$ with it instead of the usual replacement schema renders it immune to removal of extensionality; see here.
In an ...

2
votes

0
answers

118
views

### Question about pro-representable Automorphism functor in Sernesi's "Deformations of algebraic schemes" with trivial relative Tangent space

Let $k$ be an algebraically closed field of arbitrary characteristic, $R$ a complete local noetherian, but non Artinian (see #Edit below for justification) $k$-algebra with residue field $k$ and $S$ a ...

3
votes

0
answers

183
views

### A problem about the series $\sin(n^p)$

Prove that when $p>0,$ the series $$\sum_{n=1}^\infty \sin(n^p)$$
is divergent

0
votes

0
answers

21
views

### Building hypercubes from the bottom up

let $H^k$ denote a $k$-dimensional hypercube in a complete symmetric graph $G(V,E)$ without self-loops and parallel edges; let $|V|=2^n$ be the number of vertices.
setting
$\mathbb{H}^0 := V$, i.e. ...

1
vote

1
answer

65
views

### Convexity property of an equivalent norm on $\ell_2$

Let us consider the space $\ell_2$ with an equivalent norm defined by
$$
\Vert x \Vert = \max \{ \Vert x^{'} \Vert_2, \Vert x^{''} \Vert_2 \},
$$
where $x^{'}=(0, x_2, x_3, \cdots)$, $x^{''} = (x_1, 0,...

1
vote

0
answers

58
views

### Representing geodesic compactifications of $S^1\times \Bbb R$ as analytic sections over base (analytic) foliations

Given a smooth nested set of "partial" foliations $\mathcal F_{\alpha}=\big\lbrace e^{\frac{\alpha}{\log x}}: \alpha \in (1/k,k), x\in(0,1),k\in [1,\infty) \big\rbrace$ of $X^2=(0,1)^2$ with ...

1
vote

1
answer

94
views

### Problem in understanding maximum principle for subharmonic functions

I am reading subharmonic functions and their properties from the book From Holomorphic Functions to Complex Manifolds by Grauert and Fritzsche. Let me first define what a subharmonic function is.
...

0
votes

0
answers

73
views

### Classification of principally polarized abelian surfaces - reference request

I found in Encyclopedia of math
https://encyclopediaofmath.org/wiki/Abelian_surface
there is a claim that:
"A principally polarized Abelian surface $(A,λ)$ is either the Jacobi variety $J(H)$ of ...

0
votes

1
answer

27
views

### Regular maps on hyperbolic plane for large number of vertices

I want to generate large regular maps of a tiling on hyperbolic space. How I can start doing that?

0
votes

0
answers

112
views

### Noether's formula for real algebraic surfaces

Is there a version of Noether's formula for the Euler characteristic of a surface for Real algebraic surfaces?
Specifically, given $X$ a real algebraic compact smooth surface, what is the relationship ...

3
votes

0
answers

158
views

### What should be unipotent de Rham homotopy group?

What exactly should unipotent $\pi_1^\text{dR}$ be conceptually? What formal properties should it satisfy? This seems to be answered by Chen's theorem, which is stated in Corollary 3.269 of Multiple ...

3
votes

0
answers

75
views

### Composition of Frobenius $n$-homomorphisms, characteristic-free?

This question is, as so often, a crossbreed of curiosity and laziness. The
former has led me to an interesting, but somewhat unsatisfactory paper by
Khudaverdian and Voronov
(arXiv:2002.02395v2) and, ...

2
votes

0
answers

49
views

### Reference request: Fréchet embedding

Given a separable metric space $(X,d)$, we have an isometric embedding $X\to\ell^\infty$ given by taking $x_n$ the countable dense subset and sending $x\mapsto\lvert(x,x_n)-d(x,x_0)\rvert$.
This ...

1
vote

0
answers

43
views

### Reverse mathematics on lightface $\Pi^1_1$-uniformization for unary relation

It is known that the following form of $\Pi^1_1$-uniformization is equivalent to $\Pi^1_1$-Comprehension over $\mathsf{ATR}_0$ (cf. VI.2.6 of Simpson's book) :
(Kondo's uniformization theorem) For ...

2
votes

1
answer

149
views

### Estimating ${\left(\sum_{i=j}^k {x_i}\right)^2} \times \left\lvert\sum_{i=j}^k {a_i}\right\rvert$

Given two sets; $X = \{x_i : x_i \geq 0; i \in [\sqrt{n}]\}$ and $A = \{a_i : |a_i| \leq 1; i \in [\sqrt{n}]\}$ of size $n^{\frac{1}{2}}$ each, with the following properties
\begin{equation}\label{...

2
votes

0
answers

41
views

### Perron-Frobenius theory for operators on matrices

Let $A$ be a Hermitian linear operator on the space of $n\times n$ complex matrices. Let's suppose $A$ is "non-negative" in the sense that it preserves the cone of non-negative definite (...

2
votes

1
answer

74
views

### Simple curves on hyperbolic tori

In the paper "Simple curves on hyperbolic tori" by McShane and Rivin, they show that if $T$ is a hyperbolic once punctured torus, one can define a norm on the homology $H_1(T,\mathbb{Z})$ by ...

-2
votes

0
answers

96
views

### Extending a map between $A$ and $B$ to a map between $L(A)$ and $L(B)$

Are any known results about extending a map $\phi:A\to B$ to a map $\overline{\phi}:L(A)\to L(B)$ or $\phi':HOD(A)\to HOD(B)$? This seems like something that would have been investigated already, and ...

1
vote

0
answers

66
views

### Unique polarization on a very general curve with Mumford-Tate

I try to understand why a very general curve (smooth, projective over $\mathbb{C}$) has an unique polarization up to scalar on the $H^1(X,\mathbb{Q})$.
I was advised to look at the maximality of the ...

7
votes

0
answers

120
views

### Are there any interesting classes of limits containing finite limits?

Let $\Phi$ be a class of limit diagrams that contains all finite diagrams. Some examples include the classes $\Phi_{\kappa}$ of all diagrams of size bounded by a cardinal $\kappa$... but are there any ...

-4
votes

0
answers

46
views

### Calculate the rotational volume that arises when the area enclosed by the curve 𝑦 = 𝑒^−𝑥^2 and the x-axis can rotate around the y-axis? [closed]

so my question for thiscalculus homework is : Calculate the rotational volume that arises when the area enclosed by the curve 𝑦 = 𝑒^−𝑥^2 and the x-axis can rotate around the y-axis?
now am i ...

1
vote

0
answers

21
views

### Multivariate delta method gradient calculation with mixed moments

Here's a slightly simplified version of my problem (using fewer dimensions than what I'm actually solving).
Take $X \in \mathbb{R}^2$. We already know that $\sqrt{n}(X - \mu) \overset{d}{\to} N_2(0,\...

0
votes

0
answers

20
views

### How to control the angles of Kuramoto model by controlling its order parameter?

Consider Homogenous Kuramoto model in this paper. In theorem 3.1, the author derive condition on $A$ such that all second-order critical points of $E(\theta)$ are in two opposite quadrants, by saying ...

2
votes

1
answer

331
views

### The Fourier transform of the Liouville function?

The Liouville function in number theory is defined as:
$$\lambda(n) := (-1)^{\Omega(n)} \text{ where } \Omega(n) := \sum_{p|n} v_p(n)$$
Taking the discrete time Fourier transform and then taking the ...

2
votes

1
answer

63
views

### Generating sets for a module and scalar extension

Let $k$ be an algebraically closed field and $K/k$ a (transcendental) field extension. Let $A$ be a finite dimensional $k$-algebra, and $M$ an $A$-module. Suppose that the $K \otimes_k A$-module $K \...

1
vote

1
answer

70
views

### Mittag-Leffler expansions converging to bounded function

Is it true that
$$\lim_{N\to\infty}\left\langle\sum_{n=-N^2}^{N^2}\frac1{(Nx-n)^2}\right\rangle_N=\pi^2$$
for some suitable definition of "minima smoothing" such as $\langle f(x)\rangle_N\...

6
votes

2
answers

223
views

### If a Banach / Fréchet manifold $M$ happens to be a topological vector space, is $M$ just a Banach / Fréchet space?

In finite dimensions, if $M$ is a smooth manifold that happens to be a vector space, then it is indeed just the Euclidean space.
I wonder if the same result holds valid in infinite dimensions. More ...

0
votes

0
answers

69
views

### Hyperbolicity in sense of Koszul

Question :
I have a question about the notion of hyperbolicity in the sense of Koszul.
Koszul asserts that locally flat compact hyperbolic manifolds do not contain a parallel vector field.
Do you have ...

14
votes

2
answers

2k
views

### Is Freyd's thesis available online anywhere?

Peter Freyd is a great category theorist. His PhD dissertation, Functor Theory, dates from Princeton in 1960. It's cited as [14] in Mitchell's book Theory of categories. In fact, Google scholar says ...

4
votes

1
answer

136
views

### Is the asymptotic rank of a tensor bounded by (naive) border rank?

Write $R(T)$ for the rank of an order-$3$ tensor $T \in \mathbb C^{m \times n \times p}$ over the complex numbers. If $T' \in \mathbb {C}^{m' \times n' \times p'}$ is another such tensor then let $T \...

2
votes

0
answers

44
views

### Degeneracy maps of Drinfeld modular curves

Over number fields, we have two natural degeneracy maps
$$X_0(N)\leftarrow X_0(pN) \rightarrow X_0(N)$$
between the (compactified) moduli space of elliptic curves with level $pN$ and $N$ respectively (...

3
votes

2
answers

119
views

### Are there outer models $V \subset W$ of $L$ such that $V$ is "far" from $L$ but $W$ is "not too far" from $V$?

In the following, whenever I say "$V_1$ is an outer model of $V_2$", I mean
$V_1, V_2$ are transitive models of $\mathsf{ZFC}$,
$V_2 \subset V_1$,and
$ORD^{V_1} = ORD^{V_2}$.
I am curious ...

1
vote

0
answers

39
views

### Defining properties of categories out of an indicial category

$\newcommand{\Hom}{\operatorname{Hom}}$Suppose we want to define the category of arrows of $S$. Below are two forms of doing it.
Definition 1: If $D$ is of the following type: $\bullet \to \bullet$, ...

4
votes

0
answers

117
views

### Permutation generation problem using swaps

This is motivated by Aaronson's post, Probability of generating a desired permutation by random swaps. I am interested in a related problem where the swaps are given in the input.
We're given as input ...

16
votes

3
answers

2k
views

### What's the earliest result (outside of logic) that cannot be proven constructively?

Although mathematicians usually do not work in constructive mathematics per se, their results often are constructively valid (even if the original proof isn't).
An obvious counter-example is the law ...

0
votes

0
answers

33
views

### complexity of membership problem in finite general linear group

Suppose $G$ is a subgroup of $GL(n,q)$ given by a list of generators. What is known about the complexity of the corresponding "membership problem", that is, the problem of deciding whether a ...

1
vote

0
answers

21
views

### How to extract 'top k' multiple solutions from a quadratic optimization problem?

Imagine we are interested in the following problem:
$$
\min_{w} \left( w^T V w + \lambda \|w\| \right) \\
\text{s.t. } w^T R \geq c
$$
Where 𝑤 is an $N \times 1$ vector, $V$ is an $N \times N$ ...

1
vote

1
answer

208
views

### A possible ${\sf (ZF)}$-theorem in the spirit of the $3$-set-lemma

The number $3$ plays an interesting role in the following statement:
$\newcommand{\S}{\sf(S_3)}\S$ Let $X$ be a non-empty set and let $f:X \to X$ be fixpoint-free (that is $f(x) \neq x$ for all $x\in ...

2
votes

0
answers

32
views

### Geometric explanation of Fueter-Sce-Qian Theorem and similar situations

In Clifford analysis there is a fundamental theorem due to Fueter and extended by Sce and Qian that says (in modern terminology) that the given a slice regular function $f:\mathbb{R}^{m+1}\to\mathbb{R}...

0
votes

0
answers

68
views

### Upper-bound of the tail of a weighted sum of iid random variables

I have a question related to this one. $X_i$ are n iid random variables with CDF $1_{[0,+\infty[}(x) \Phi(x)$, i.e. it is a mixture between a folded Gaussian and a delta in $0$, both with weight $1/2$....

1
vote

0
answers

51
views

### Is the adjoint action of $\mathrm{SU}(2)$ on the Schwartz space proper and free?

$\DeclareMathOperator\SU{SU}$Let $t_1, t_2, t_3$ be generators of the Lie algebra $\mathfrak{su}(2)$.
Let us consider a Schwartz space $\mathcal{S}$ defined as
\begin{equation}
\mathcal{S}:= \Bigl\{ \...