All Questions

0
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0answers
14 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
1answer
18 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
votes
1answer
54 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
votes
1answer
67 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 ...
0
votes
0answers
22 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
1answer
82 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} \ \ \ ...
0
votes
0answers
34 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 ...
3
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0answers
27 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 ...
12
votes
1answer
248 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
vote
1answer
119 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. ...
-1
votes
0answers
17 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 ...
1
vote
2answers
114 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 ...
-4
votes
0answers
52 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 ...
0
votes
0answers
31 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 $Q^{(n)} = \begin{pmatrix} -n & n \\ n & -n \end{pmatrix}$ ...
1
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0answers
50 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 ...
14
votes
2answers
256 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 ...
1
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0answers
37 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 ...
1
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0answers
24 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
votes
1answer
119 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 ...
1
vote
0answers
22 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 ...
-1
votes
0answers
67 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 $G/H.$
3
votes
0answers
46 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 ...
1
vote
0answers
56 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 $a_i, b_i \in \mathbb{Z}$, ...
-2
votes
0answers
42 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 $\{ \lambda_i , e_i \}_{i=1}^n$. ...
0
votes
0answers
19 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
votes
3answers
402 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
votes
0answers
51 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. ...
1
vote
1answer
52 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 ...
-2
votes
0answers
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 ...
1
vote
0answers
25 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 ...
5
votes
0answers
39 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 ...
0
votes
0answers
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 ...
0
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0answers
61 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: ...
1
vote
1answer
89 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 ...
-7
votes
0answers
50 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?
4
votes
0answers
52 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 $\operatorname{dim}_{\mathbb C} V = n$, ...
1
vote
0answers
67 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 ...
0
votes
0answers
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
vote
0answers
25 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
votes
0answers
113 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 ...
6
votes
1answer
513 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 ...
0
votes
0answers
17 views

cohomology of a variation of wreath product

Let $C$ the space of points that looks like $(z_1,z_2,\ldots,z_n,z_{\sigma(1)},z_{\sigma(2)},\ldots,z_{\sigma(n)})$ with $z_i\in \mathbb{C}$ and $\sigma$ runs over all the permutations of $S_n$. Is ...
-4
votes
0answers
23 views

Problem regarding heat equation partial differential equation [on hold]

A metal bar of 100m long has ends x=0 and x=100 kept at zero degrees initially half of the bar is at 60 degrees while the other half is at 40 degrees. Assuming a thermal diffusivity of 0.16 egs units ...
0
votes
0answers
36 views

Matrix equation

Let $A$ be $k\times n$ matrix i.e., $A=(a_{1},\ldots, a_{n})$ where $a_{j} \in \mathbb{R}^{k}$, $rank(A)=k$ and $1\leq k \leq n$. Let $q=(q_{1},\ldots, q_{n})\in\mathbb{R}^{n}$ be such that ...
2
votes
0answers
72 views

Eigenvalue problem

I am studying torsional Alfven waves in spicules. In this concern I have encountered the following equation: $ \left(1-m^2 e^{-αz}\right)y''(z)+\left(4π i m ...
2
votes
2answers
149 views

Is there an Oka-Grauert principle for homogeneous spaces?

Suppose we have a fibration over the punctured disc (i.e., a deformation of complex manifolds) such that each fiber is a homogeneous space. Is the total space a product of a fiber with the punctured ...
4
votes
0answers
440 views

Question on Atiyah-Patodi-Singer on $T^3$

I wanted to test my understanding of the Atiyah-Patodi-Singer theorem by studying flat bundles on $T^3$ explicitly, and miserably failed. Namely, I computed the eta invariant explicitly for flat ...
1
vote
1answer
155 views

Monodromy of a punctured disc

I meet the following problem which I think related to the monodromy: Let $D: = \{z \mid |z|<1 \}$ be a disc, and $U \to D$ be a variety fibred over $D$. For each point $t \in D \backslash \{0\}$, ...
1
vote
1answer
69 views

Do the sequences with divergent associated $\zeta$-function form a vector space?

Let $V$ be the set of sequences $a \in\mathbb{R}^\mathbb{N}$ such that $\lim_{n\to\infty} a_n = 0$. The set $V$ can be seen as a real vector space, with pointwise addition and scalar multiplication. ...
12
votes
3answers
482 views

Conjecture regarding closest point inside a discrete ball to a line

I'm a PhD student in image processing, where I've stumbled into a problem that seems to be essentially number theory. I've hunted around online and while I've found many results on similar problems, ...

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