# All Questions

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2 views

### Approximation of quadratic variation

Here $M$ and $N$ are two bounded continuous martingales with respect to some filtration $(\mathcal F_t)_t$. I found this claim in a paper I was reading: $t$ being fixed, then a.s.
$$\lim_{h\rightarrow ...

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4 views

### Maximum Entropy for Dirichlet with Constrained Expectation

How would I find the maximum entropy distribution of a Dirichlet with a known expectation (i.e if I know the expected value, a multinomial, how would I find the concentration parameter that maximizes ...

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36 views

### It is possible to prove that there is an immersion of the $K3$ surface in $R^8$ using the Mayer integrality theorem?

I will apply the following theorem
In the case of the $K3$ surface the first Pontryagin class is given by
$p_{{1}} \left( {\it K3} \right) =-12\,{h}^{2}$
and
$M \left( TK3\right) ...

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34 views

### On degrees of polynomials with matching zeros in a subset

Let $S\subsetneq \Bbb R^n$ such that $|S|<\infty$ and for all partitions $S_1$ and $S_2$ of $S$ there exits a multilinear polynomial $h$ such that $h(s)=1-h(s'),\mbox{ }\forall s\in S_1\mbox{ and ...

**2**

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25 views

### spectral sequence of a bicomplex equipped with a group action

Let $(A_1^\bullet,\partial_1)$ and $(A_2^\bullet,\partial_2)$ be complexes of $\mathbb{C}$-vector spaces.
We suppose that $(A_1^\bullet,\partial_1)$ and $(A_2^\bullet,\partial_2)$ are equipped with ...

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9 views

### Can the conservative form of the advection equation be re-written by replacing the velocity term with an integral over all other points in space?

Suppose I have a 1D advection equation in conservation (divergence) form
$\partial_t u(x,t) = -\partial_x [v(x)u(x,t)],$
where $u$ is a conserved quantity in space, and $v$ gives the velocity of the ...

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39 views

### Are compact complete geodesics closed?

note: I find this question In stackexchange math, I would be interest to know how I could be answer this kind of question,I pasted it here as I see it appropriate For MO.
Let $(M,g)$ be a compact ...

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29 views

### Characterizing subgroups $H$ of $\Bbb T$ with $\{(f(x),g(x))\mid x\in H\}$ dense in $\Bbb T^2$

Let $\Bbb T$ be the circle group with Euclidean topology. Is there a way to determine all $H\le \Bbb T$ such that there are $f,g\in Aut(H)$ with $\{(f(x),g(x))\mid x\in H\}$ dense in $\Bbb T\times ...

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12 views

### matrix summation and uniform convergence of matrix sequence [on hold]

I start studying summation theory and as we know a matrix is called a Schur matrix if i)lim (a)nk= x when n goes to infinity and ii)the sum of (a)nk that means the convergence is uniform in n. can ...

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37 views

### On prime ideals in regular local rings

Let $(R, \mathfrak{m})$ be a regular local ring of dimension $d$ and $\mathfrak{m} = (x_1, ..., x_d)$. It is well known that $(x_1, ..., x_i)$ is prime for all $i \le d$.
Question. Let $I$ be a ...

**2**

votes

**1**answer

26 views

### Finding nodes with a particular weight in a graph

Say that an edge $e$ hits a node $v$ if one of its two extremes is $v$.
Then we can also say that $v$ is hit by $e$.
Given a weighted graph $G$ (see below for all assumptions and definitions) and ...

**2**

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**1**answer

74 views

### On Serre's problem regarding the injectivity of Albert-Algebra cohomological invariants

In these Lecture Notes http://molle.fernuni-hagen.de/~loos/jordan/archive/cohinv/cohinv.pdf from 2006 by Garibaldi on page 21. 7.5 there is the following open problem mentioned:
Is the map
$g_3 ...

**3**

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**1**answer

145 views

### Intersection of nonzero prime ideals is zero — does it have a name?

The Rabinowitch trick (in Eisenbud's Commutative Algebra with a view toward Algebraic Geometry, page 132) says that $R$ (commutative unital ring) is Jacobson if and only if for every prime ideal $P ...

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29 views

### Criterion for global dimension of subring

All rings are assumed to be associative and unital.
If $B$ is a commutative sub-ring of $A$ (which itself needs not be commutative) then what properties of $B$ are both necessary and sufficient for ...

**4**

votes

**1**answer

101 views

### Biquadratic reciprocity for $p\equiv 1\pmod 4$ and $q\equiv 3\pmod 4$

For a prime $p\equiv 1\pmod 4$, let $\left(\frac{\cdot}{p}\right)_4$ denote the rational biquadratic residue symbol; that is,
$$ \left(\frac{a}{p}\right)_4 =
\begin{cases}
\ \ \ ...

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40 views

### Residual Intersections of a complete intersection

Let $R$ be a Cohen-Macaulay local ring and $I=(b_1,\dots,b_s)$ be a complete intersection generated by a regular sequence $\underline{b}$. Let $\mathfrak{a}\subseteq I$ such that ...

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48 views

### Precise density estimates for Cantor sets

Let $C_\lambda$ be the classical Cantor set associated to a real number $0<\lambda<\frac{1}{2}$, as defined for example in the book of K. J. Falconer The geometry of fractal sets. I recall ...

**17**

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**1**answer

339 views

### What is an (automorphic) L-function?

To preface, I am a student of automorphic representation theory, and I know full well the definition of the L-function attached to an automorphic representation.
I am intending to give a talk on the ...

**1**

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**1**answer

177 views

### When is an holomorphy ring a PID?

I posted this question on mathstackexchange but I realized it is probably more suitable for mathoverflow.
I will use the notation and language of Stichtenoth, Algebraic Function Fields and Codes.
...

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18 views

### What's the probability that at least one of k top data still in top k positions of a data set with error?

Let {${d_1,d_2,d_3,..., d_k,...d_n}$} is a descendingly sorted data set.
Now we suppose that each data in the set has a probability $p_e$ to go wrong.
The problem is what's the probability that at ...

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**2**answers

148 views

### Decomposing adelic points using torsors

Let $k$ be a number field and $X$ be a $k$-scheme. Let $G$ be a linear algebraic group over $k$ and let $f: Z \to X$ be a $G_X$-torsor ($G_X = G \times_k X)$. We can twist the torsor $f$ by 1-cocycles ...

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53 views

### Finding inverse of a function [on hold]

I have a function:
$$h(x) = \frac{1-\sqrt{x}}{1+\sqrt{x}}$$
With just pen and paper, how can I determine if there exists an inverse function? Am I supposed to sketch it on paper to see if it can ...

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36 views

### Almost sure convergence of a sequence of Markov chains

Consider for each $n \in \mathbb{N}$ a continuous-time Markov chain $(X^{(n)}_t)_{t \geq 0}$ with $2$ states $\{0, 1\}$, generator $Q^{(n)} = \begin{pmatrix} -n & n \\ n & -n \end{pmatrix}$ ...

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52 views

### Extending integrable almost-complex structure

Suppose $(X,I)$ is an almost-complex real analytic manifold where $I$ is a real-analytic almost complex structure. Suppose there exists an $I$-almost complex submanifold $M\subset X$ where this means ...

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**2**answers

314 views

### When is a topological space the homotopy colimit of an open covering?

Suppose that $X$ is a topological space and $\left(U_i \to X\right)$ is an open cover. We can associate to it the Cech diagram of this cover $$C_U:\Delta^{op} \to Top.$$ I know that for many good ...

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40 views

### Unitary factor in polar decompositions

Let $A, B$ be $n$-square (Hermitian) positive definite matrices. Let $AB=U|AB|$ be the polar decomposition of $AB$. So $U$ is unitary (called the unitary factor of $AB$). What is the optimal constant ...

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30 views

### Estimating polynomial approximation error in high dimension

Question
Let $x \in [-1, 1]^d \subset \mathbb{R}^d$ be a $d$-dimensional variable and assume that -- given $n$ -- I have a way of computing a polynomial $p_n(x)$ of degree $n$ that approximates a ...

**3**

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**1**answer

122 views

### Determinant of the oriented adjacency matrix of a tree

Let $(V,E)$ be a finite oriented directed graph, with vertices and edges ordered, and $M$ the $|V|\times |E|$ matrix with entries
$$ m_{ve} = \begin{cases} 1 &\text{if $e$ points at $v$}\\
-1 ...

**2**

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24 views

### Name for class of contracting permutations

Let $S_n$ be the symmetric group. For any sequence of numbers $y=[y_1,y_2,\cdots,y_k]$, define the flattening operation as $\mbox{flat}_{k}(y)$ as a relabeling of $y_1,y_2,\cdots,y_k$ in terms of ...

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70 views

### Calculating the quotient group $\mathbb{Z}\times\mathbb{Z}/<(1,1),(1,-1)>$ [on hold]

Let $G$ be the group $\mathbb{Z}\times\mathbb{Z}$ and $H$ be the subgroup of $G$ generated by $(1,1)$ and $(-1,1)$. I am trying to calculate the quotient group $G/H.$

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49 views

### System of linear ODEs with hypergeometric coefficients

For quite some time I have been trying to solve the following system of differential equations for the two functions $G$ and $H$ defined on the interval $[0,1]$:
$$
\begin{align}x ...

**2**

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**1**answer

96 views

### Simplest form for sum of Binomial Expressions

How difficult is the problem of reducing the number of terms in a sum of binomial expressions? Formally:
Given $a_1, a_2, a_3, … a_n$, and $b_1, b_2, b_3, ... , b_n$, where $a_i, b_i \in \mathbb{Z}$, ...

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44 views

### An analytic characterization of eigenvalues of a Hermitian matrix [on hold]

[..the following is trying to understand a certain argument of Terence Tao in a lecture notes of his..]
If $A$ is a Hermitian matrix with eigenvalues, eigenvectors $\{ \lambda_i , e_i \}_{i=1}^n$. ...

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22 views

### Probability of close approach for multivariate normal variables

The following problem comes from a physical model of two groups of particles in three dimensions. I need to know the probability that the two groups of particles approach each other within some ...

**2**

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**3**answers

465 views

### Koebe–Andreev–Thurston theorem - where can I find a proof?

Koebe–Andreev–Thurston theorem (known also as the circle packing theorem) says that any planar graph can be realized by a set of (interior-) disjoint disks corresponding to vertices, such that two ...

**3**

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55 views

### Series of swapping terms

Suppose $f(z)=a_0+a_1z+\cdots+a_nz^n+\cdots$ is defined in the unit disk, $\mathbb{D}$, and $\|f\|_{\infty}\leq 1.$ Lets form a series $f_k$ by interchanging $a_1$ and $a_k$ i.e. ...

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**1**answer

55 views

### Is ther a dense subset of parameter plane, which is not an interior?

In case of parameter plane of complex quadratic polynomial :
is it possible to find part of parameter plane, scanned with given limited precision ( rasterised) where :
every pixel contains part ...

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**0**answers

22 views

### why is $\frac {dy}{dx} dx = dy$? [migrated]

if $y$ is a function of $x$, why is $\frac {dy}{dx} dx = dy$? This has been bugging me. Why is it you can treat that as a fraction?
I would like the traditional calculus view first if possible, then ...

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27 views

### Solutions for n such that the path number to the n-th node is also n in a complete, rooted, ordered k-ary tree

The path to find the $n$-th node within a complete, rooted, ordered $k$-ary tree can be represented by a number to the base $(k+1)$ that contains no zero digits. See A245905 in OEIS as an example for ...

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**1**answer

52 views

### Number of median graphs?

What is the number of $n$-vertex median graphs? These graphs generalize hypercubes and trees, and have many applications. It seems unlikely that a closed form expression is known, so I would also be ...

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31 views

### Lifting $SHFC$ to non singular foliations

In this question we would like to complexify the idea in the following post.
We would like to lift a "singular holomorphic foliation by curves", briefly "SHFC", of $\mathbb{C}P^{2}$ to a ...

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63 views

### Integral of Weingarten Map / Shape Operator [on hold]

This Paper states that the Weingarten Map / The Shape operator $W_p$ of a two-dimensional surface $S\subset\mathbb{R}^3$ at a point $p$ can be expressed in the following way:
...

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**1**answer

95 views

### Normal Variation on Manifolds

Let $M$ be a smooth surface and let $x,y \in M$. Let $d_{M}(\cdot, \cdot)$ be the geodesic distance metric on $M$, that is the length of the shortest geodesic curve on $M$. Let $\kappa$ be the maximum ...

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52 views

### A true-false exam has five questions. Andy is completely ignorant and so he tosses a fair coin to answer each question [on hold]

A true-false exam has five questions.
Andy is completely ignorant and so he tosses a fair coin to decide his answer to each question.
What is the probability that he scores at least four correct?

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54 views

### Powers of traces, integrals over spheres and class functions

I asked this on math.StackExchange a while back but got no answers. I hope I'll be forgiven for the double post.
Let $V$ be a complex vector space of dimension $\operatorname{dim}_{\mathbb C} V = n$, ...

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68 views

### Identifying a cohomology class arising from a Postnikov decomposition of BU(2)

For various reasons I'm currently interested in Principal $U(n)$-bundles $U(n)\hookrightarrow E\xrightarrow{p} S^m\times X_g$ over the Cartesian product of an $m$-sphere and a closed oriented surface ...

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32 views

### Initial Value for an ODE Problem [on hold]

I have the following ODE
$\mathbf{A}\dfrac{d\vec{y}}{dt}+\mathbf{B}\vec{y}=\vec{x}$,
where $\vec{y}$ and $\vec{x}$ are $n\times1$ vectors and are functions of $t$, and $\mathbf{A}$ and $\mathbf{B}$ ...

**1**

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27 views

### Asymptotic variance for partial sum of a stationary process

Let $X = (X_1, \dots, X_n, \dots)$ be a sequence of random variables. We assume that the process X is stationary i.e. for any integer $k$, any set of indices $i_1 < \dots < i_k$ and any integer ...

**7**

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119 views

### Flat manifolds and irreducible representations

Let $M$ be a compact Riemannian manifold with vanishing curvature of Levi-Civita connection. Such manifolds were classified by Bieberbach; sometimes they are called Bieberbach manifolds. According to ...

**7**

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**1**answer

523 views

### How large can you draw an island on a map?

A cartographer friend asked me this question: could you classify (shapes of) islands by how much space they occupy on a map (comparatively to how much space is occupied by water) if you draw them as ...