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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 ...
himanee's user avatar
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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^{...
Christopher-Lloyd Simon's user avatar
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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?
himanee's user avatar
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(\...
happyle's user avatar
  • 29
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] = [...
Terry Tao's user avatar
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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 \...
Andromeda's user avatar
  • 169
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 ...
Noah Schweber's user avatar
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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 ?
Jaime Yerbabuena's user avatar
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 $b_n$, ...
Logan Hyslop's user avatar
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 ...
Ribhu's user avatar
  • 271
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 ...
varkor's user avatar
  • 8,477
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 ...
user449595's user avatar
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 ...
Zerox's user avatar
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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^...
Vladimir's user avatar
  • 359
-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 ...
Zuhair Al-Johar's user avatar
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 ...
graphtheory123's user avatar
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 ...
dm82424's user avatar
  • 371
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} \...
Tardis's user avatar
  • 1,015
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(\...
Obrad Kasum's user avatar
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 $$ ...
Pavel Gubkin's user avatar
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 ...
Botwinnik's user avatar
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 ...
Morgan Rogers's user avatar
-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 ...
dereuodeum's user avatar
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 $\alpha, L >0$ ...
De vinci's user avatar
  • 329
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+\...
super's user avatar
  • 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 ...
Varadharajan R's user avatar
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 ...
Gert's user avatar
  • 213
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 $T$?
Pace Nielsen's user avatar
  • 17.9k
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 ...
red_trumpet's user avatar
  • 1,051
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 ...
Oblomov's user avatar
  • 2,491
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 ...
Zuhair Al-Johar's user avatar
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 ...
user267839's user avatar
  • 5,786
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
adobereader's user avatar
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. ...
Manfred Weis's user avatar
  • 12.6k
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,...
PPB's user avatar
  • 71
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 -...
António Borges Santos's user avatar
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 ...
53Demonslayer's user avatar
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\...
nervxxx's user avatar
  • 205
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. ...
Anacardium's user avatar
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 ...
finiteness's user avatar
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?
Zahid Malik's user avatar
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 ...
Serge the Toaster's user avatar
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 ...
W. Zhan's user avatar
  • 333
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, ...
darij grinberg's user avatar
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 ...
Gesh's user avatar
  • 73
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 ...
Hanul Jeon's user avatar
  • 2,754
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{...
Krish's user avatar
  • 21
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 (...
AdamNie's user avatar
  • 53
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 ...
stupid_question_bot's user avatar
-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 ...
blark's user avatar
  • 97

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