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5
votes
0answers
97 views

Bi/tricategorical coherence in terms of surface diagrams

Is there a typed-up version of the coherence theorem for bicategories in terms of surface diagrams? What about the GPS tricategorical coherence theorem in terms of 'volume diagrams'? I'm aware of ...
3
votes
1answer
220 views

Compact hyperkahler manifold as algebraic variety in weighted projective space?

Many examples of Calabi-Yau manifolds are constructed as algebraic varieties in weighted projective space, or more generally as complete intersection Calabi-Yau (CICY) manifolds. Are there such ...
1
vote
0answers
58 views

Is the following algebra 1-syzygy finite?

$\Lambda$ is a finite dimensional algebra given by $$\begin{array}{rcccl} 1 \rightrightarrows 2 \stackrel{\alpha}{\circlearrowright} \ \ \ \alpha^{3}=0\\ \end{array}$$ Is $\Lambda$ is ...
10
votes
1answer
205 views

Divergent series summation beyond natural boundaries

I'm hoping to investigate the effects of divergent summation methods on series which cannot be analytically continued due to a dense set of singularities. At least a priori, it doesn't seem that a ...
5
votes
1answer
307 views

Asymptotics of degree of $\textrm{SO}_n$?

(This is an offshoot of Degree of parametrization of $\textrm{SO}_n$?) Consider $G=\textrm{SO}_n$ as an affine subvariety of the affine space of $N$-by-$N$ matrices. There is an explicit expression ...
0
votes
0answers
43 views

Johnson-Lindenstrauss with Orthogonalization

I have been looking at constructions satisfying the Johnson-Lindenstrauss Lemma (e.g., projections onto random subspaces, random Gaussian matrices, random Rademacher matrices, etc.). It seems that ...
3
votes
0answers
123 views

Anti-canonical divisor of hypersurfaces

Consider a normal hypersurface $X\subset\mathbb{P}^2_1\times\mathbb{P}^2_2$ of bidegree $(2,k)$. The $X$ is defined by an equation of degree $2$ (in the variables of $\mathbb{P}^2_1$) whose ...
1
vote
0answers
104 views

Mirzakhani's length function integration formulas and representation varieties

Mirzakhani develops a method to integrate geodesic length functions on moduli space by considering circle bundles over moduli space given by level sets of these functions. There are natural circle ...
0
votes
0answers
28 views

Construction of isosceles trapezoid given its bases sum and one angle [closed]

I can use just a ruler and a compass. Given that I know one angle I know every angle. But I can't think of anything else. Any help would be appreciated!
8
votes
1answer
439 views

VC dimension of standard topology on the reals

Can there be an uncountable set $S\subseteq\mathbb R$ such that for each subset $D\subseteq S$, there is an open set $U$ with $D=S\cap U$? I'm asking merely out of curiosity, but I'll mention that ...
2
votes
1answer
94 views

An inequality in the optimality of Bayes' theorem

$\DeclareMathOperator\Ent{Ent}\newcommand{\prior}{\mathrm{prior}}\newcommand\Data{\mathrm{Data}}$I came across this paper on the optimality of Bayes' theorem https://sinews.siam.org/Portals/Sinews2/...
0
votes
1answer
35 views

Example(s) where replacing a multivariate, discrete RV with a single, univariate RV fail

Let $X_1,\ldots,X_n,Y,Z$ be $n+2$ binary random variables and define $X=(X_1,\ldots,X_n)$. In most problems, instead of treating $X$ as $n$ distinct binary random variables, there is no loss of ...
12
votes
0answers
647 views

A question on Fargues-Scholze

As far as I understand it, the main goal of the recent work of Fargues and Scholze on the geometrization conjecture is to show that the local Langlands conjecture of a local field is equivalent to the ...
2
votes
0answers
35 views

Sum of number of perfect matchings and a constant constuction

Suppose we have two bipartite graphs $G_1$ and $G_2$ with perfect matching count $P_1$ and $P_2$ respectively then their disjoint union gives a bipartite graph with perfect matching $P_1P_2$. Is ...
2
votes
1answer
95 views

"Lagrange inversion" around an extremum

Cross-posted from Math Stackexchange. In an older question to which I provided an answer it was asked how to compute a particular limit involving the roots of a transcedental function around its ...
1
vote
0answers
151 views

Why is $\overline{\mathbb{F}_p}((t))$ transcendental over $\mathbb{F}_p((t))$?

Why is $\overline{\mathbb{F}_p}((t))$ transcendental over $\mathbb{F}_p((t))$ ? I guess $\overline{\mathbb{F}_p}((t))$ is not unramified over $\mathbb{F}_p((t))$ because $\overline{\mathbb{F}_p}((t))$ ...
2
votes
0answers
21 views

Reduction of the general Lauricella hypergeometric function $F_B$ for identical parameters and variables

The Lauricella function $F_B^{n}$ of $n$ variables is defined as $$F_B^{(n)}(a_1, \ldots, a_n, b_1, \ldots, b_n, c; x_1, \ldots x_n) = \sum_{k_1, \ldots, k_n = 0}^\infty \frac{1}{(c)_{k_1 + \ldots + ...
3
votes
0answers
128 views

Dominance of $w\mu$ for dominant cocharacter $\mu$

Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$ of rank $n$ and a Borel $B \supset T$ defining a set of simple roots $\Delta$. Let $W$ be the Weyl ...
1
vote
0answers
64 views

"Tails" of a multinomial distribution

Let $X_1,\dots,X_N$ denote a collection of independent samples of a uniform multinomial random variable in $\mathbb{Z}^k$, with the number of trials equal to $n\ll k$. (By "uniform", I mean ...
5
votes
1answer
150 views

Number of distinct normalized vectors from the center of a hexagon in a hexagonal grid

Consider an infinite hexagonal grid composed of regular hexagons. Choose any hex to be the origin hex. Let n be a natural number. Find an expression, in terms of n, for the number of distinct ...
2
votes
0answers
84 views

Quasi-crystaline generalization of elliptic functions

I came across some meromorphic function, call it $f(z)$, which is "quasicrystalline" in the following sense: one can write $f$ as: $$ f(z)=\frac{\sum_i a_i e^{i(q_{i,x}x+q_{i,y}y)}}{\sum_i ...
0
votes
0answers
58 views

LLN of random nearest neighbor function

There are two samples of iid random variates: $X=\{X_1,X_2,...,X_n\}$ and $Y=\{Y_1,Y_2,...,Y_n\}$. Further, $\forall i,j: X_i$ is independent of $Y_j$. The probability distributions $P,Q$ are unknown ...
1
vote
0answers
77 views

A random process with conserved momentum: 'particle decay'?

Consider a particle $p_1$ moving at unit speed along a straight line in $\mathbf{R}^2$, directed by some vector $v_1 \in \mathbf{S}^1$. Equid this particle with a Poisson clock $\tau_1$, with ...
8
votes
3answers
1k views

Should every modern day mathematician care about category theory? [closed]

As far as I know, category theory is used mainly in topology. I have a dislike towards category theory, similar to my dislike of Bourbakism, and want to avoid it as much as I can. However, the head of ...
3
votes
1answer
218 views

Bounding integrals involving $\operatorname{li}(x)-\pi(x)$

Let $x >0$. How can one find good $O$ bounds on the integrals $$\int_0^x\frac{\operatorname{li}(t)-\pi(t)}{t}dt$$ and $$\int_x^\infty\frac{\operatorname{li}(t)-\pi(t)}{t^2}dt$$ where $\pi(x)$ is ...
0
votes
1answer
95 views

Fourier transform on lattice strip

I am working with a triangular lattice $L=\{n_1 a_2 + n_2 a_2 : n\in\mathbb{Z}^2 \}$ and $a_1 = \pmatrix{1 \\ 0}$ and $a_2 = \frac{1}{2} \pmatrix{-1 \\ \sqrt{3}}$, and I want to compute the Pontryagin ...
3
votes
1answer
195 views

Pull back of Spin$^{\mathbb{C}}$ bundle

Let $M$ be a closed $4$-d Riemannian manifold and $Z$ be its twistor space of $M$, i.e., the bundle of almost complex structures on $M$. Let $V$ be a Spin$^{\mathbb{C}}$ bundle, $V_+$ denote the ...
6
votes
1answer
184 views

A strong Borel selection theorem for equivalence relations

In Kechris' book "Classical Descriptive Set Theory" there is the following theorem (12.16): Let $X$ be a Polish space and $E$ an equivalence relation such that every equivalence class is ...
1
vote
0answers
59 views

Uniqueness of decomposition for positive-definite integral bilinear forms?

Let $\Lambda$ be a lattice, that is a free finitely generated abelian group with a symmetric bilinear form. In general, decomposition of lattices into indecomposable orthogonal sublattices is not ...
4
votes
1answer
184 views

Adelization for any classical arithmetic subgroup

In the classical setting, we can define automorphic forms on $\text{SL}_n(\mathbb{R})$ with respect to any lattice $\Gamma$. In fact, for $n \geq 3$, all lattices are arithmetic subgroups. I have ...
1
vote
0answers
28 views

Isotopy of open book supporting same contact structure

In dimension 3, the Giroux correspondence gives us a bijection between contact structures (up to isotopy) and open book decompositions (up to positive stabilisation). Moreover, Giroux shows that two ...
2
votes
0answers
35 views

Interior derivative of meromorphic 3-form w.r.t complex null vector on twistor space

In this paper one finds the following derivation involving twistor space. On page 4, the following $(3,0)$-form $\Omega$ on twistor space \begin{align} \label{add:1.1} \Omega=\mathrm{D}^3 Z\...
0
votes
0answers
45 views

A counterexample of a theorem about matching extendable

$M$ is perfect if $M$ covers all vertices of $G$, and $M$ is extendable if $G$ has a perfect matching containing $M$. Moreover, a graph $G$ with at least $2k + 2$ vertices is said to be $k$-extendable ...
4
votes
0answers
118 views

Does the tropical semiring admit a universal property?

Each of the rings $\mathbf{Q}$, $\mathbf{Z}_p$, $\mathbf{Q}_p$, $\mathbf{R}$ admits a universal construction: the rationals are the field of fractions of the integers: $\mathbf{Q}:=\mathrm{Frac}(\...
3
votes
1answer
287 views

What is known about constructively irrational numbers?

Intuitively, a constructively irrational number is one for which we can effectively separate it from any rational number in terms of the latter's denominator. More formally, a constructively ...
1
vote
1answer
90 views

Epi-mono factorisations in schemes via scheme-theoretic image

Suppose that $f : X \rightarrow Y$ is a morphism of schemes. Let $Z \hookrightarrow Y$ the scheme-thereotic image of $f$. Under what conditions is the morphism $X \rightarrow Z$ an epimorphism? If ...
3
votes
1answer
58 views

Divergence-free Gaussian vector field with given mean magnitude and correlation function

My general question is how to construct an isotropic random vector field $\vec f: \mathbb{R}^3 \to \mathbb{R}^3$ with a given mean magnitude $\mathbb{E}[\|\vec f(\vec x)\|]=\mu$ and with vector ...
11
votes
2answers
732 views

A variation of the Ryll-Nardzewski fixed point theorem

Is there a fixed-point theorem that implies the following result? Let $F$ be a nonempty convex set of functions on a discrete group with values in $[0,1]$. Suppose $F$ is invariant with respect to ...
0
votes
0answers
105 views

Subalgebras of $B(H)$ consisting of all operators leaving a given finite dimensional space invariant

Let $H$ be an infinite dimensional separable Hilbert space. Let $V$ be a finite dimensional subspace of $H$. Put $$A=\{T\in B(H)\mid T(V)\subseteq V\}.$$ So $A$ is a Banach algebra. Can we equip $A$ ...
3
votes
1answer
185 views

Is the automorphism group of a normal affine scheme a group scheme or an algebraic space?

If $ \operatorname{Spec}(A) $ is a smooth affine scheme over an algebraically closed field $ k $, then is $ \operatorname{Aut}(\operatorname{Spec}(A)) $ a group scheme or an algebraic space? Please ...
3
votes
1answer
66 views

On the definition and an example of silting/tilting subcategories in a triangulated categories according to a paper by Aihara and Iyama

In the paper "Silting mutation in triangulated categories" by Aihara and Iyama, I stumbled upon this nice definition( Definition 2.1) of a tilting/silting subcategory of a triangulated ...
5
votes
2answers
412 views

Counting $\pm 1$ and $0$'s in the character tables of $\frak{S}_n$

Let $\chi_{\mu}^{\lambda}$ denote a value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu, \lambda\vdash n$. When $\mu=(n)$, then it's known that $$\sum_{\lambda\vdash n}\...
2
votes
0answers
128 views
+50

Sobolev (Triebel-Lizorkin) norm estimate for $F \circ u - F \circ v$

Let $F \in C^1(\mathbb R^d;\mathbb R)$ be such that $F(0) = 0$ and $$|F'(\tau v + (1 - \tau)w)| \leq \mu(\tau)(G(v) + G(w))$$ for some $\mu \in L^1([0,1])$ and some non-negative $G \in C^0(\mathbb R^d;...
8
votes
0answers
231 views

Degree of parametrization of $\textrm{SO}_n$?

Let $G=\mathrm{SO}_{2 n}$ (or $G=\mathrm{SO}_{2n+1}$, $G=\mathrm{Sp}_{2 n}$ …) defined over some field $K$. Consider $G$ as an affine subvariety of the space of matrices. (Warm-up question) What are ...
-2
votes
0answers
154 views

Legendre sieve, RH and Goldbach's conjecture

$\DeclareMathOperator\ord{ord}$This is a question that builds upon my question About Goldbach's conjecture, whose beginning I copy paste below: "let's consider a composite natural number $n$ ...
2
votes
0answers
132 views

The variety of $\mathbb{C}[t]_{< d}$-points on a variety

(This was posted to https://math.stackexchange.com/q/4244260/799193 where it did not receive an answer.) Let $X \subseteq \mathbb{C}^n$ be an affine variety defined by $f_i(x_1, \ldots, x_n)=0, 1 \le ...
0
votes
1answer
99 views

On ultraweak continuity

Let $A$ be a C*-algebra, $f$ be a representation of $A$, $F$ be the universal representation of $A$, and $g=f \circ F^{-1}$. For an ultraweakly continuous linear functional $w$ on $f(A)$, $w\circ g$ ...
3
votes
0answers
139 views

Polyhedrons and their centers of mass

Given a convex polyhedron, one considers 3 possibilities: wireframe - only the edges of the polyhedron have mass which is uniformly distributed. surface - only the surface is massive with uniform ...
8
votes
1answer
331 views

Finite group with squarefree order has periodic cohomology?

Is it true that a finite group with squarefree order has periodic group cohomology (with trivial coefficients)? I cannot see why this would be the case, but I'm looking at a paper which seems to ...
4
votes
0answers
119 views

Clarification of argument in "Elliptic curves over $\mathbb{Q}_{\infty}$ are modular"

In https://arxiv.org/abs/1505.04769 in the proof of Theorem 5 it is asserted that since $\rho_{E, l}:G_\mathbb{Q}\to\mathrm{GL}_2(\mathbb{Z}_l)$ is surjective then $E_{\mathbb{Q}_\infty}[l^\infty]=0$. ...

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