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2
votes
1answer
545 views

Probability generating function zero implies random variable is infinite

Let $V$ be a random variable supported on the nonnegative integers (including $\infty$) and $f(x) = \mathbf E x^V$ be the probability generating function. In our model $V$ is the number of visits to ...
11
votes
0answers
302 views

Generating random finite groups

I would like a method to efficiently generate a random finite group of a given order $n$. If there are $g(n)$ non-isomorphic groups of order $n$, ideally each group would occur with probability ...
6
votes
0answers
735 views

How many 2L-bit numbers are the product of two L-bit numbers?

If I multiply two integers $x, y $ in $[0,2^L)$, I get an integer in $[0,2^{2L})$. Clearly, this map from $[0,2^L) \times [0,2^L) \to [0,2^{2L})$ is not bijective. I am interested in the size of ...
5
votes
0answers
487 views

Optimal Gear Trains

Suppose you need to slow down a turning motor so that a gear turns at an angular velocity $\frac{a}{b}$ of that of the motor shaft, where $a$ and $b$ are natural numbers. For example, this set of ...
5
votes
0answers
752 views

On “The Average Height of Planted Plane Trees” by Knuth, de Bruijn and Rice (1972)

I am trying to derive the classic paper in the title only by elementary means (no generating functions, no complex analysis, no Fourier analysis) although with much less precision. In short, I "only" ...
4
votes
0answers
218 views

Domination in Nice Lattices

Let an integer vector be nice when it has only two nonzero components, which sum to zero. So (0, 0, 3, 0, -3) and (-1, 0, 1, 0, 0) are examples of nice vectors in $n=5$ dimensions. Call a lattice ...
3
votes
0answers
97 views

“Standard” notation for symmetric functions?

Here is what I encountered in the paper "The Optimal Lattice Quantizer in Three dimensions" by Barnes and Sloane. Here is the setup: Let $\Lambda$ be a lattice in $\mathbb{R}^3$. Around each ...
3
votes
0answers
124 views

Citation for subset complement result

Let $S=\lbrace s_1,\ldots,s_n \rbrace \subset \lbrace1,\ldots,2n\rbrace$. Consider two operations on $S$: the complement $C(S)=\lbrace 1,\ldots,2n \rbrace \setminus S$ and a reflection* ...
3
votes
0answers
119 views

Are there existing resources on modular-esque recurrence relations?

Does anyone know where I would be able to get information on analyzing a class of polynomial recurrence relations of a form like this? $\begin{align*} f_{n,k}(x) & ...
3
votes
0answers
134 views

A fast way to test whether a partial function can be extended to a chirotope of rank 3 ?

Hello, What is a fast way to test whether a partial function can be extended to a chirotope of rank 3 ? That is: I have a domain $E = \{1,...,n\}$ and a partial function $f: E^3 \to \{-1, 0, 1\}$ ...
3
votes
0answers
724 views

Method for variable substitution in multiple summation

I want to ask: is there any general method for variable substitution in multiple summation? For example in the following equation a new variable $\lambda=n+m-2\mu$ is introduced to transform the LHS ...
2
votes
0answers
39 views

A weaker version of Randell Isotopy Theorem

I am studying a problem in hyperplane arrangement theory related to the homotopy type of the complement manifold of a certain class of hyperplane arrangements. In a well celebrated paper Richard ...
2
votes
0answers
399 views

Guessing game with guess cost

This is a question about Problem 328 on the website Project Euler. A description of the problem is provided in the previous link. I was wondering if there has been any research done on this question. ...
2
votes
0answers
164 views

Number of breakpoints in parametric maximum flow problems

The parametric maximum flow problem can be formulated as $$f(\lambda) = \min_{x\in\{0,1\}^n} \left( \sum_{i}(a_i + b_i\lambda)x_i + \sum_{i,j}c_{ij}x_ix_j \right), $$ where all $c_{ij}<0$ (so that ...
1
vote
0answers
55 views

Presentation of the Rybnikov matroid

In this well celebrated work Gregory Rybnikov exhibit an example of two arrangements with the same underlying matroid, but with fundamental groups which are not isomorphic. This is a key ...
1
vote
0answers
27 views

Estimates for derivatives of a positive discrete harmonic function

There is the following estimation (Duffin, Discrete potential theory, Theorem 5): Let $f$ be a discrete harmonic function in a sphere of radius $R$ with the center $p$, all in $\mathbb Z^3$. Then, if ...
1
vote
0answers
29 views

Simplifying closed form for Meta Operator

I was consider the set of linear operators: $$O_{a,k} = \frac{f(ax^k) - f(x)}{ax^k - x} $$' Particularly I am looking for the closed forms of the eigenfunctions of this operator, that is the ...
1
vote
0answers
68 views

Cross-correlation of two functions which are not fixed

I am trying to cross-correlate two functions, but one of which is changing each 'step' of the cross-correlation. I want to cross-correlate T(f) and ...
1
vote
0answers
210 views

Trigonometric semialgebraic conditions for two floors to be unequal [edited]

EDIT: Since posting my original question I have simplified the problem to finding a sufficient condition for $\lfloor \frac{a}{\phi}\rfloor \neq \lfloor \frac{b}{\phi}\rfloor$ which is trigonometric ...
1
vote
0answers
169 views

recursion formula for odd holonomic function

suppose we have a map $f:\mathbb{Z}\longrightarrow\mathbb{C}[t^{\pm}]$ with property that $f(i)=-f(-i)$. The algebra $\mathcal{T}=\mathbb{C}[t^{\pm}][L^{\pm},M^{\pm}]/[LM=tML]$ acts on $f$ by ...
1
vote
0answers
427 views

Zeroes of a tricky function.

I am attempting to show that there does not exist an N past which every open unit interval (k, k+1) -where k is an integer- contains a zero of the following function: $h(x)=\sum_{n=2}^{[\sqrt(x)]} ...
0
votes
0answers
36 views

3D matching modification

Consider all instances of the $3D$ matching problem where all edges that intersect-intersect in "exactly" one vertex (1-edge intersection). Consider all instances of the $3D$ matching problem where ...
0
votes
0answers
19 views

On the stability analysis of a discrete difference system with multiplicative noise

If we assume that \begin{equation*} \rho \{\phi \otimes \phi+\psi \otimes \psi\}<1 \end{equation*} where $\rho$ denotes the spectral radius, then can we verify the following inequality holds ...
0
votes
0answers
142 views

Lattice basis reductions and finding minimal values

While reading several articles about lattice basis reduction I am left with a few questions. For one, I came across this piece of text Let $\alpha$ and $\beta \in \mathbb{R}$. Also let $X>0$ and ...
0
votes
0answers
131 views

A good upper bound on the size of k-biclique in random bipartite graphs.

Let $G = (X \cup Y, E)$ be a random bipartite graph, where $X$ and $Y$ are the set of vertices and $E$ the set of edges. I want to find an upper bound of the largest biclique in which exactly $k$ ...
0
votes
0answers
85 views

An upperbound related to inductively reducing a set by adding the two least elements

I hope this is not too trivial to be asked here, but here it goes anyway. This is just out of curiosity (regarding a problem with graphs but I have "reduced" it to the problem below:) Let ...
0
votes
0answers
205 views

Partial Recurrence Equation

Hey people I have the following equation, which I don't manage to solve. The background of the problem is the clustergrowth of two chemical species, resulting in a final relation I'd like to solve: ...