All Questions
8 questions
1
vote
2
answers
204
views
Only trivial solutions to system of linear diophantine equations possibly related to hamiltonian cycles in graphs
This might be related to counting hamiltonian cycles.
@Peter Taylor gave negative result about the one dimensional case, but we believe his attack is
not directly applicable to this question.
Given ...
- 25.4k
1
vote
1
answer
100
views
$\ell^1$-bound on graph laplacian with weight
Consider the $\mathbb Z^2$ lattice, we then define for $u=(u_{ij})_{i,j \in \mathbb Z}$ the discrete Laplacian
$$(\Delta u)_{i,j}=u_{i+1,j}+u_{i-1,j}+ u_{i,j+1}+u_{i,j-1}$$
and the weight which pushes ...
- 536
5
votes
2
answers
1k
views
Cardinality of certain subsets in vector spaces over finite fields
Assume that you have an $n$-dimensional vector space over a finite field (therefore the number of elements in the vector space is finite) and $F$ is a subset of this vector space which contains $m$ ...
- 228
13
votes
1
answer
468
views
Near-linear mappings from $\mathbb F_p$ to $\mathbb R$
$\newcommand{\F}{{\mathbb F}}$
$\newcommand{\R}{{\mathbb R}}$
$\renewcommand{\phi}{\varphi}$
Let $p\ge 5$ be a prime.
If the functions $\phi_1,\phi_2,\phi_3\colon\F_p\to\R$ satisfy $\phi_1(x)+\...
- 23k
5
votes
1
answer
394
views
Disjoint union of affine subspaces contains a larger affine subspace
I'd like to say that a large structured subset of the $n$-dimensional Boolean cube $\{0,1\}^n$ contains a non-trivial affine subspace. To be more specific, I want to prove/disprove that for some ...
- 211
1
vote
1
answer
433
views
Approximation of sets
Is the following true? For every $\varepsilon>0$ there is a finite
subset $W$ of $\mathbb{N}\times \mathbb{N}\times \mathbb{N}$, such that
$$|p_1(W)\cap p_2(W)\cap \{p_1(x)+p_2(x):x\in W\}\cap \{...
- 13
-1
votes
1
answer
200
views
Collecting terms of a linear expression with nested sums and combinatorics in coefficients
I need to collect the $\Pr(\cdot)$ terms of the following expression:
$\sum_{m=3}^{n}\frac{g_{m}\left( \cdot \right) }{\left( \sqrt{\theta \left(
1-\theta \right) }\right) ^{m}}\left[ \sum_{j=2}^{m-1}...
3
votes
1
answer
419
views
Maximal size of an almost-disjoint linearly independent family in $K^{\mathbb{N}}$
Let $K$ be a field, say infinite, and denote by $L$ the $K$-vector space $K^{\mathbb{N}}$. What is the maximal cardinality of a $K$-linearly independent subset $X$ of $L$ such that any two distinct ...
- 76