# All Questions

**3**

votes

**0**answers

93 views

+50

### 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**

votes

**1**answer

167 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

**0**answers

51 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

**0**answers

27 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

**3**answers

534 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

**0**answers

80 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. ...

**2**

votes

**2**answers

127 views

### Is there an (almost) dense set of quadratic polynomials which is not in the interior of the Mandelbrot set?

For the parameter plane of complex quadratic polynomials, $(z\mapsto z^2+c)_{c\in\mathbb{C}}$ :
Is it possible to find a part of the parameter plane, scanned with a given limited precision ...

**-2**

votes

**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 ...

**1**

vote

**0**answers

35 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

**1**answer

122 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

**0**answers

36 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 ...

**1**

vote

**1**answer

128 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 ...

**-8**

votes

**0**answers

66 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?

**8**

votes

**1**answer

139 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

**0**answers

81 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

**0**answers

36 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

**0**answers

48 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

**0**answers

138 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 ...

**8**

votes

**1**answer

586 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

**0**answers

36 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

**0**answers

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

**0**answers

47 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

**0**answers

85 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

**2**answers

218 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 ...

**5**

votes

**0**answers

806 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

**1**answer

181 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 $t \in D \backslash \{0\}$, ...

**2**

votes

**1**answer

77 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

**3**answers

569 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, ...

**4**

votes

**0**answers

81 views

### Many integral points on quartic models of elliptic curve via differences of squares

Pick fourth power free integer $n$ ($p^4$ doesn't divide $n$).
Represent $n$ as difference of possibly negative integer squares
$n=v_i^2-u_i^2$.
The goal is to find quadratic polynomial with integer ...

**-2**

votes

**0**answers

25 views

### mathematical modelling question [on hold]

You are asked to help with a sociological study on the survival of surnames.Consider a closed community with N individuals at time t=0 with K different surnames.Assume that all children of any ...

**0**

votes

**0**answers

17 views

### Solution to a system of linear equations containing some inequalities [on hold]

I have a system of equations as follows:
$a_{11}x_1 + a_{12}x_2 + a_{13}x_3 = b_1$
$a_{21}x_1 + a_{22}x_2 + a_{23}x_3 = b_1$
$a_{31}x_1 + a_{32}x_2 + a_{33}x_3 < b_1$
$a_{41}x_1 ...

**-1**

votes

**0**answers

78 views

### Fermat's little theorem question [on hold]

I'm studying Number theory (in my spare time) and I need to prove a lemma in order to prove the exercise. The topic is Fermat's little theorem.
Well the lemma goes like this:
Let's say we have ...

**2**

votes

**1**answer

155 views

### Simplicial version of the A-infinity operad

I am looking for a description of the $A_\infty$ operad in the category of simplicial sets. More specifically, I am looking for a formulation of the loop space recognition principle for simplicial ...

**-1**

votes

**0**answers

57 views

### A combinatorial and number theoretical problem [on hold]

Given N positive integers, not necessarily distinct, how many ways you can take 4 integers from the N numbers such that their GCD is 1.
For example,N=10 and the positive integers are ...

**-2**

votes

**0**answers

116 views

### Does any one know what this problem is called? [on hold]

We are given finite sets A and B and a set S⊆P(A). The members of S may have arbitrary intersections with one another and their union is not necessarily A. We wish to determine whether there is a ...

**-2**

votes

**0**answers

30 views

### How to calculate a sequence in Maple [on hold]

Please how can I using Maple obtain the following sequence defined successively
$$X_1=1,\quad X_k=\Big(\frac{1/k+\sum_{i+j=k}X_i X_j}{k^2}\Big)^{1/2}$$
here $i,j,k\in\mathbb {N}$
i wish to have ...

**-5**

votes

**0**answers

34 views

### Diffusion Equation [on hold]

Kindly give me suggestions on my following assignment of Simulations in Fluid Flow:
Solve the following differential equation for transport of f(x,y,z,t) by MS Excel
∂f/∂t+Ux ∂f/∂x+Uy ∂f/∂z+Uz ...

**-4**

votes

**0**answers

28 views

### Find the vector component of vector u orthogonal to vector a [on hold]

I have vector u = (-2, 3, 1) and vector a = (-2, 2, 2). How do I find the vector component of u orthogonal to a?
I've done the cross product and I get (-4,-2,-2), but I am assuming that this is also ...

**0**

votes

**0**answers

46 views

### Integrate Faddeeva function

I came across this integration in my studies.
$\int_{-\infty}^{\infty}|F((w_\textbf{_} - \hat{w_\textbf{_}})\tau) |^2 . d\tau$
It uses the Faddeeva function which is $F(z) = e^{-z^2}erfc(-iz)$. I ...

**2**

votes

**0**answers

57 views

### What is the computational complexity to compute the integral numerically?

Given $$\int_{\Delta}\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}$$ where $P_i$ is polynomial(that is $P_1(x_1,x_2,\dots,x_n), P_2(x_1,x_2,\dots,x_n)$ are polynomial) whose coefficients are ...

**2**

votes

**1**answer

116 views

### A perfect $(n,k)$ shuffle function

Suppose you have a deck of $n$ cards; e.g., $n{=}12$:
$$
(1,2,3,4,5,6,7,8,9,10,11,12) \;.
$$
Cut the deck into $k$ equal-sized pieces, where $k|n$;
e.g., for $k{=}4$, the $12$ cards are partitioned ...

**-2**

votes

**0**answers

124 views

### how to solve 3 6-degree polynomial equations for 3 variables? [on hold]

I am a physicist and need to solve three $6$-order polynomial equations for $3$ unknowns $(p, q, r)$. Here is the system of equations looks like:
$$\sum(A[n]*p^i*q^j*r^k) = 0,$$
...

**0**

votes

**0**answers

57 views

### Parallel topologies on a Prüfer group with the trivial group topology as the only group topology contained in both

Let $p$ be a prime number. A homomorphism $f:\Bbb Z_{p^\infty}\to \Bbb T$ induces a group topology $\mathcal T_f$ on $\Bbb Z_{p^\infty}$ with a base of neighborhoods $\mathcal N_f$ of $0$.
Are there ...

**5**

votes

**1**answer

268 views

### Groups with a unique composition series

Which finite groups $G$ have a unique composition series? I don't mean in the sense of the Jordan-Holder theorem, but rather actually unique.
Some examples are the cyclic groups $C_{p^n}$ and the ...

**6**

votes

**1**answer

87 views

### Injectivity of Rewrite Rule in a Free Lie Algebra

Let $L$ be a free Lie algebra (over $\mathbb{Q}$) on generators $x_1, x_2, \ldots, x_n$, and let $V_k$ be the subspace spanned by the $k$-fold brackets. Let $U_1 = \mathrm{span}\{ x_i | i< ...

**2**

votes

**0**answers

76 views

### Galois group for 0-dimensional motives

$\newcommand{\M}{\mathcal{M}_0}$$\newcommand{\Q}{\mathbb{Q}}$
It is my understanding that in dimension 0, the theory of motives should just be Galois theory for fields. I am hoping to find a reference ...

**4**

votes

**0**answers

214 views

### If 2-manifolds are homeomorphic and smooth, are they diffeomorphic? [on hold]

Perhaps this question has already been asked on Mathoverflow. I mean this question in a global sense. A friend mentioned it to me today, and I started thinking about it. I'm not sure how to prove it. ...

**-1**

votes

**0**answers

40 views

### A fredholm index associated with two vector fields generating a 2 dimensional foliation

Let $M$ be a compact manifold and $X,Y$ be two independent vector fields on $M$ with $[X,Y]=0$. Let $\mathcal{F}$ be the 2 dimensional foliation associated with the distribution ...

**-7**

votes

**0**answers

89 views

### Does anyone want to see the critical figure for n= 7? [on hold]

I watched Ronald Lewis Graham's youtube blurb for the "happy ending problem". it's about 5 minutes long. I was able to supply him with the positions of the points for the case: n=7. it looks like a ...

**4**

votes

**2**answers

157 views

### Relation between Turing degrees and functions computable with them

Suppose $A<_T B$ ($A$ is a set computable from $B$ but not vice versa). Is it always the case that there exists a $B$-computable function which eventually outgrows all $A$-computable functions?
Of ...