# All Questions

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### Catenarity of monoid algebras

Let $R$ be a commutative ring, let $M$ be a commutative monoid, and let $R[M]$ denote the corresponding monoid algebra. Suppose further that $R$ is universally catenary. One may ask for conditions on ...
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### Reducing join-incomplete lattice homomorphisms to homomorphisms with co-domain ${\bf 2}$

Let $L$ be an complete lattice. A lattice homomorphism $f: L\to L$ is said to be join-incomplete if there is an infinite set $S \subseteq L$ such that $f(\bigvee_L S) > \bigvee_L f(S).$ Is it true ...
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### Minimum number of people such that 2 can be expected to sit next to each other

We are given a large, round table with $n$ seats. It is easy to see that whenever $p\geq \text{int}(\frac{n}{2}) + 1$ people are seated, at least $2$ people will sit next to each other (here ...
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### integrability of Brownian motion stopped at some stopping time

Let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion starting at zero and denote by $S=(S_t)_{t\ge 0}$ its running maximum, i.e. $S_t=\sup_{0\le s\le t}B_s$. Given a fixed number $p>1$, define the ...
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### Bounded metric spaces with non-surjective self-isometry

A metric space $(X,d)$ is said to be bounded if there is $r\in\mathbb{R}$ such that for all $x,y\in X$ we have $d(x,y) \leq r$. A self-isometry is a map $\iota:X\to X$ such that for all $x,y\in X$ we ...
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### How to find PDF of ordered random variables?

Assumpion: Let $X_1, X_2, \ldots, X_L$ be $L$ independent and identical random variables (RVs). Let $F_{X_i}(x_i)$ and $f_{X_i}(x_i)$ be CDF and PDF of $X_i$. Suppose that $F_{X_i}(x_i) = F_X(x_i)$ ...
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### probability of reaching a point in a 2d grid in a certain number of steps [on hold]

I have a random walk process in a 2d grid with N steps where N is small. How can I calculate the probability that any given cell was reached in N, N-1, N-2 ... 1 steps. That is, I would like to be ...
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### Are there infinitely many $k$ for which $\sigma(k)/k$ is a square?

I would like to know if there are infinitely many $k$ for which $$\sigma(k)/k=n^2$$ such that: $m=kn$ with $m,n>0$ ? Note: $\sigma(\frac{m}{ n})$ is the sum of divisors function of ...
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### Is there a way to find an efficient set of relations for presenting the subgroup generated by two matrices in $SL(2, q)$?

Given two elements $a, b \in SL(2, \mathbb{F}_q)$, is there a way to find an efficient presentation $$\langle x, y \mid \text{relations}\rangle$$ of the subgroup $\langle a, b \rangle$? My intention ...
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### What am I missing in this highly oscillatory integral?

I want to numerically integrate this equation (in python): $\int_{0}^{\infty}{\rm d}k f(k) J_v(r k)J_v(s k)$, where f(k) is a non-smooth function, and $J_v$ are the Bessel function of the fist ...
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### Bundles over Grassmanian with given top Chern class

So, I have been working on Chern classes for my master thesis and apparently (My proofs could be wrong and a few things are still vague) I was able to give a construction method and exhibit, via ...
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### Partitioning graph for Graph Isomorphism

Motivation: I am studying graph isomorphism problem. I am trying to construct a partitioning method to reduce search cases required to get solution of graph isomorphism. Construction: $G$ is a $r$ ...
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### Weierstrass form of genus one $y^{10} z^{30} - 8000 y^{4} z^{20} + 12800000 z^{20} + 1600 y^{2} z^{10} - 64=0$

Related to the n-conjecture. We are looking for Weierstrass form and map from it of the genus one curve: $$y^{10} z^{30} - 8000 y^{4} z^{20} + 12800000 z^{20} + 1600 y^{2} z^{10} - 64=0$$ It is ...
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### Asymptotics of a Bivariate Generating Function

I have the following generating function, $$G(x,y)=\sum_{n,k \geq 0}a(n,k)x^ny^k = \frac{(y^2-y)x+1}{(y-y^3)x^2-(y+1)x+1}$$ and I am interested in obtaining an asymptotic for the sequence $a(n,k)$ ...
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### Fibrations on blow-ups of $\mathbb{P}^2$

Let $X_n = Bl_{p_1,...,p_n}\mathbb{P}^2$ be the blow-up of $\mathbb{P}^2$ in $n$ general points $p_1,...,p_n\in\mathbb{P}^2$. Let $f_i:\mathbb{P}^{2}\dashrightarrow\mathbb{P}^1$ be the linear ...
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### binomial coefficients and irrationals

The following, probably either currently impossible to deal with, or having a negative solution, arose from an ergodic theory question, presumably itself currently intractible. I am not a number ...
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### Polynomials with Unique Critical Value

My question is extremely simple to state: I am looking for a characterization of multivariate complex polynomials $f$ such that $f(Sing(f))=\{0\}$. My motivation is that I recently read somewhere that ...
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### A question about tensor product [on hold]

For every $f, g$ in $L^1(G)$, we know the function $[(x,y)↦f(x)g(y)]$ belongs to $L^1(G\times G)$ where here $\times$ means cartesian product. Why are these functions dense in $L^1(G\times G)$?
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### Statistical test for the independence of components in a N-dimension Gaussian random variable

Assuming that there is a N-dimenson Gaussian randon variable $\mathbf{X}=\left[\mathbf{x}_1,\mathbf{x}_2,...,\mathbf{x}_N\right]^T$ and we have K observations ($K\ll N$) of $\mathbf{X}$(i.e., we have ...
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### Reference for the Banach Manifold structure of $C^k(M,N)$

I'm completely new to the subject of banach manifolds and I'm looking for a reference of the following: Let $M$,$N$ be smooth (=$C^\infty$) finite-dimensional compact manifolds. Consider the set ...
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### Runge-Kutta convergence [on hold]

I am facing a problem solving a ODE with a Runge-Kutta 4th order method: The expression in order to solve is : $$Ay^{''}+By^{'}+Cy= Cu$$ y =OUTPUT ...
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### Derivative in terms of finite differences

Consider expanding the differentiation operator in terms of the forward difference operator as $f' = \log(1 + \Delta)f = \displaystyle \sum_{n = 1}^{\infty} \frac{(-1)^{n + 1}\Delta^n f}{n}$. For some ...
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### Intuition behind the Morse inequalities?

Forgive me if this is sort of a vague question, but can someone supply me with their intuition behind the Morse inequalities?
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### Does every Lawvere theory arise in this way?

By a Lawvere theory, I mean a finite-product category $\mathsf{T}$ equipped with a distinguished object, such that every object of $\mathsf{T}$ can be expressed as a finite product of the ...
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### Relation between moduli spaces and classifying spaces

I hope this question is suitable to be posted here on MO. I wonder if there is a systematic relation between the notation of a classifying space, and the notion of a moduli space. I don't consider ...
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### Is there a generalization of homotopy groups to fractional dimensions

Does there exist a reasonable candidate for such an object as $\pi_{\frac12}(X)$?