# All Questions

152,023
questions

-1
votes

0
answers

14
views

### Braches and the Radius of Convergence

I have the impression that Newton's Polygon makes precise the intuitive notion of the branches of an algebraic finction and gives a convergent series corresponding to each branch. Is my impression ...

0
votes

0
answers

14
views

### Conditional convergence of exponential sums related to a Hecke modular form

Definition
Consider the Fourier coefficients $\psi(n)$ of the modular form $\eta^4(6\tau)$,
which are defined in terms of $q=\exp(i2\pi\tau)$ by the identity:
$$\eta^4(6\tau) = q \prod_1^\infty (1-q^{...

-1
votes

0
answers

33
views

### Newton's Parallelogram

George Chrystal discusses Newton's Parallelogram. (Page 386, Part 2, Algebra: An Elementary Text-book.) Why is Newton's Polygon also called Newton's Parallelogram?

0
votes

0
answers

7
views

### Under which condition, such that all second-order critical points satisfy $\sum_j\cos(\theta_i-\theta_j)>0$ for all $i\in[n]$?

Consider the following non-convex function
$$E(\theta):=-\sum_{i,j}A_{ij}\cos(\theta_i-\theta_j)$$
and its second-order critical point, i.e., the points $\theta$ satisfying
(1) $\sum_{j}A_{ij}\sin(\...

5
votes

0
answers

112
views

### Relations in a certain Lie algebra

Let ${\mathfrak g}$ be the (real) Lie algebra generated by infinitely many generators $D_i, E_i$ with $i=1,2,3,\dots$ subject to the following relations for any natural numbers $i,j$:
$$ [D_i, D_j] = [...

1
vote

0
answers

15
views

### Commutant of rank one operators over Hilbert module

Let $\mathcal{E}$ be a right Hilbert $C^*$-module over the $C^*$-algebra $A$. Consider for $\xi, \eta\in \mathcal{E}$ the rank-one operator
$$\theta_{\xi, \eta}: \mathcal{E}\to \mathcal{E}: \zeta \...

2
votes

1
answer

105
views

### Kuratowski's 14 theorem and universal algebra

For a tuple of functions $\overline{p}$ on a set $Y$, let $cl_{\overline{p}}$ be the associated closure operation: $cl_{\overline{p}}(Z)$ is the smallest subset of $Y$ containing $Z$ and closed under ...

0
votes

0
answers

14
views

### Exponential-like function equivalent for the Dixonian Elliptics

Is there some exponential-like function that acts as partner for the intricate Dixonian Elliptics, in a similar way that the Exponential function acts as a partner for the trigonometric functions ?

1
vote

1
answer

91
views

### Rescaling Fourier coefficients of a continuous function by a bounded sequence

This question stems out of: which sequences $(a_n)_{n\in\mathbb{Z}}$ of complex numbers have the property that if there exists a continuous function $f$ on the circle with Fourier coefficients $b_n$, ...

2
votes

0
answers

37
views

### $L^2$ space of Hilbert-Schmidt operator valued functions

Let $\mathscr{S}$ denote the space of all Hilbert-Schmidt operators on $L^2(\mathbb R)$. Consider the Hilbert space $L^2(\mathbb R, \mathscr S)$ of square-integrable $\mathscr S$-valued function, that ...

7
votes

0
answers

52
views

### Large V-categories admitting the construction of V-presheaves

By a result of Foltz, and Freyd and Street, a category $C$ is essentially small (i.e. equivalent to a small category) if and only if both $C$ and $[C^{\text{op}}, \mathrm{Set}]$ are locally small. I ...

1
vote

0
answers

31
views

### When is $B^G\backslash(B/A)^G$ finite?

Let $G$ be a locally compact group, let $A,B$ be (not necessarily abelian) connected reductive complex groups equipped with continuous actions of $G$ via algebraic automorphisms. Let $\phi:A\to B$ be ...

2
votes

0
answers

47
views

### Riemannian structure on Hilbert manifolds

The infinite-dimensional separable Hilbert space $H$ has the unusual property that it is diffeomorphic to its unit sphere. Therefore, $H$ admits the round metric as a complete and bounded Riemannian ...

0
votes

1
answer

38
views

### On polynomial equation of fourth order depending on two parameters and bound on a maximal root

I would like to apologize in advance for a too technical question. Let us consider the following fourth order polynomial equation in $x$:
\begin{eqnarray}
F(x) \equiv (2 a p +2)x^4+ (6a(1-a)p^...

-1
votes

0
answers

79
views

### Can this form of reflection be consistent?

Is this form of reflection consistent?
First I'll begin by clarifying the notation I'm using here:
By a quantifier being relativized or bounded it means that the first occurrence of the quantified ...

0
votes

0
answers

26
views

### Max-flow modeling with unified vehicle and commodity variables

I am working on a network flow problem that involves routing through a time-space network. The network consists of:
A single source node and a single demand node.
A fleet of vehicles with specified ...

2
votes

0
answers

27
views

### Brauer pairs on GAP

In "Representations of groups: a computational approach" by K. Lux and H. Pahlings the ten smallest Brauer pairs of order $2^8$ are given with their GAP IDs. However, many more Brauer pairs ...

2
votes

1
answer

57
views

### What does the matrix-valued solution of $X = A X A^T + \operatorname{Id}$ look like?

Let $A \in \mathbb{R}^{n \times n}$ be an invertible contraction, i.e. all singular values are in $(0,1)$. By reformulating the equation
\begin{align*}
& X = A X A^T + \operatorname{Id} \tag{1}
\...

3
votes

0
answers

32
views

### Closure of a pointclass under universal real quantification

Let us assume $\mathsf{AD}^+$ and let $\Gamma$ be a pointclass such that $P(\mathbb{R})\cap L(\Gamma)=\Gamma$ and $L(\Gamma)\models\mathsf{AD}_\mathbb{R}+\mathsf{DC}$. Since the cofinality of $o(\...

1
vote

0
answers

33
views

### Integral inequality related to the (mixed?) moments of two functions

For $k\in \mathbb{Z}_+$ and $t\in [0,1]$ we set
$$
S_{k}(t) = \{(t_1,\ldots, t_k)\colon 1\ge t\ge t_1\ge\ldots\ge t_k\ge 0\}.
$$
Let $a$ and $b$ be two continuous functions on $[0,1]$. Introduce
$$
...

0
votes

0
answers

26
views

### Growth of cocycles in higher degrees

Let $G$ be a group with finite symmetric generating set $S$ and
let $\pi:G\rightarrow\mathcal{U}(\mathcal{H})$ be a unitary representation
of $G$ on a Hilbert space $\mathcal{H}$. A 1-cocycle with ...

4
votes

1
answer

161
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

52
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

31
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

21
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+\...

2
votes

0
answers

52
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 $b\in B$ with $b^2,b^3\in R$. The conductor ideal is defined to be $\operatorname{Ann}_R(S/R)$. What ...

3
votes

0
answers

58
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 ...

5
votes

0
answers

58
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 $T$?

4
votes

0
answers

122
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

76
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

65
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

90
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

162
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

18
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

48
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

47
views

### Generalized HLS inequality

The HLS inequality implies that if $p, q > 1$, $0 < \lambda < d$ are such that
$$
\frac{1}{p} + \frac{1}{q} + \frac{\lambda}{d} = 2,
$$
then
$$ \int_{\mathbb{R}^d}\int_{\mathbb{R}^d} f(x) |x -...

1
vote

0
answers

29
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 ...

0
votes

0
answers

69
views

### Haar averaging over symplectic orthogonal group

I am trying to achieve a concentration statement in the following setting.
Let \begin{align}
E = O (\lambda \mathbb{I}_{n\times n} \oplus \lambda^{-1} \mathbb{I}_{n\times n})O^T\end{align}
be a $2n\...

1
vote

1
answer

79
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

54
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

16
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

97
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 ...

2
votes

0
answers

127
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 ...

2
votes

0
answers

59
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, ...

1
vote

0
answers

43
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

34
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

0
answers

67
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

35
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

64
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

89
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 ...