3
votes
0answers
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
1answer
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
0answers
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
0answers
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
3answers
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
0answers
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
2answers
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
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
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
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
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
0answers
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
2answers
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
0answers
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
1answer
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
1answer
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
3answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
2answers
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 ...

15 30 50 per page