# All Questions

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### Is $[u_1,u_2]$ an edge of the polytope $conv(F)$?

Here's the problem. I have a finite set of vectors $F \subset \mathbb{Z}_{\geq 0}^d$. I define $P$ to be the convex polytope $conv(F)$ i.e. the convex hull of $F$. Given $u_1, u_2 \in F$, is the ...
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### Modular polynomials for elliptic curves point counting

The Schoof-Elkies-Atkin (SEA) algorithm (for counting points on elliptic curves over a finite field) performs computations over polynomials modulo some modular polynomials. Originally the "classical" ...
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### On monotonicity of the roots of polynomials

I feel that the question that I am posting is not a general research level question but probably some special form of it where the behaviors of the roots of polynomials are studied. Here is my ...
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### Colon ideal and Artin-Rees lemma

Let $a$ and $b$ be ideals of a Noetherian integral domain $R$. Prove that there exists a natural number $r$ such that $(a^n:b) = a^{n-r}(a^r:b)$ for $n > r$.
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### Reference for holder estimate on parabolic equation with neumann boundary condition

I saw a type of holder estimate in Friedman's book: partial differential equations of parabolic type(page 200 3.24) as following: Suppose we have a uniformly parabolic equation with holder ...
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### Two Questions on $\pi(x)$

I have recently came to know about this conjecture. The questions that naturally came to my mind are, $\forall$ $x,y \geq2$ and $x,y \in \mathbb{N}$ prove that $\pi(x) \pi(y)\geq \pi(xy)$ ...
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### Is the identification between symmetric tensors and homogeneous polynomials useful?

The general question: Given an $n$-dimensional vector space $V$ over a field $k$, there exists an identification $$\mathrm{Sym}^d(V) \sim k[x_1, \dots, x_n]_d$$ between the space of symmetric order ...
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Given a Lie group $\mathfrak{G}$ with finite centre and with Lie algebra $\mathfrak{g}$, I am looking at a simple proof that negative definite Killing form implies compactness. This proof is given ...
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### Is there an integral point in the group generated by an rational point?

Let $E$ be an elliptic curve over rational field. Let $P=(a/d^2,b/d^3)\in E(\mathbb{Q})$ and $$G=\{P,2P,3P,4P,\cdots\}.$$ Is there an integral point $Q\in G?$
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### Can this simple integral be zero for a Jordan curve?

The following simple problem came up while doing some unrelated research. Does there exist a Jordan curve $\gamma : [0,2\pi] \to \mathbb{C}$ of positive orientation, lets say $C^1$-smooth (just to ...
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### extension of function in an abstract metric space

my question is the following.(Maybe my title is not quite proper for this question): Let $(E,d)$ be a Polish space (or a separable metric space), let $\xi: E\to R_+$ be a Lipschitz function. Now set ...
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### Standard form for partitions of $Z_n$

Let $A$ and $B$ be partitions of $Z_n$. Suppose we say that $A$ and $B$ are equivalent if $A=Bx+y$ for some $x\in\{1, -1\}$ and $y\in Z_n$. In other words, two partitions are equivalent if one can be ...
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### Number of double cosets of a Young subgroup

Let $\lambda\vdash n$ be a partition of $n$ with $k$ parts and $S_\lambda$ be a Young subgroup of $S_n$. Further let $S_\lambda\backslash S_n/ S_\lambda$ be the set of double-cosets. Now I would like ...
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### Strongly regular graphs with strongly regular edge colorings

Given a positive integer $c>1$, for what parameters $(v,k,\lambda,\mu)$ does there exist a $(v,c k, c \lambda, c \mu)$ (simple) strongly regular graph that can be given an edge coloring with $c$ ...
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### Probability of that the sum is even [on hold]

There are $2n$ letters which are randomly placed into $2n$ envelopes (each envelope can have only one letter). The letters and the envelopes are numbered from $1$ to $2n$. What is the probability that ...
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### Not too classical group characters

When teaching group theory, one has too speak of characters, which are morphisms $\chi:G\rightarrow k^\times$ into the multiplicative group of a field $k$ (mind that I don't mean representation ...
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### Characterization of a subset of [0,1] $III$

I have a question related to the previous one. Characterization of a subset of [0,1] $II$ Let $T\subseteq [0,1]$ be some subset closed under lower limit topology, i.e. $t_n$ is said to converge to ...
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### Number of possible wall positions in the game Quoridor

I'm trying to figure out how many possible board positions there are for the game Quoridor. I think sorting out the legal positions from the illegal positions will be difficult, so to start I'm trying ...
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### Dubins car shortest paths: Decidable?

A Dubins car follows a Dubins path in $\mathbb{R}^2$, with constant wheel speed and limited turning radius. It is known that the shortest Dubins path in the absence of obstacles follows circular arcs ...
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### Lifting a quadratic system to a non vanishing vector field on $S^{3}$

Let $P:S^{3}\to S^{2}$ be the Hopf fibration. For a vector field $X$ on $S^{2}$ there is a non vanishing vector field $\tilde{X}$ on $S^{3}$ such that $DP(\tilde{X})=X$. It is constructed in ...
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### Explicit non observability example for the wave equation?

Is there a simple (one dimensional, radial, by separation of variable...) example of non observability for the linear wave for a non constant in space velocity, in a simple domain? By this I mean: ...
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### Existence of partitions

Good morning everybody. I would like to know if anybody is aware of nontrivial results of the following form : if a family $\mathcal I$ of subsets of $\mathbb N$ satisfies such and such assumption, ...
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### Mean and variance of a general multivariate skew normal distribution

I have a problem about a general multivariate skew normal distribution. There is a $p\times 1$ vector, $\mathbf{y}=(\mathbf{y}_1',\mathbf{y}_2',\ldots,\mathbf{y}_n')',p>n$, which has the density as ...
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### What is the *smallest* number used in a mathematical paper? [on hold]

Numbers such as Skewe's Number and Graham's Number and a few others are relatively well-known in mathematical folklore as being among the largest, if not the largest, numbers encountered in proper ...
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### A simple question in Hitchin's paper “The Geometry of Three-forms in Six Dimensions”

I am reading Hitchin's beautiful paper "The Geometry of Three-forms in Six Dimensions". Everything goes smooth up to now except for a tiny problem in Section 6.2, which can be formulated as follows. ...
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### Classification of countable posets?

Is there a classification of countable posets where between each two comparable elements there is a third element between them?
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### A question about some notation involving the exclamation mark

What does the symbol ‘!’ signify? Is it $\text{argmin}$? For example, $\| A x - y \| = \min!$.
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