# All Questions

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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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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!
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 ...
94 views

### An inequality in the optimality of Bayes' theorem

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### 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 ...
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 ...
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 ...
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 ...
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 $\overline{\mathbb{F}_p}((t))$ ...
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### 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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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\...
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 ...
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### 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 ...
$\DeclareMathOperator\ord{ord}$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 $n$ ...