arXiv subjects
Publications and source records attributed to Vuong Bui.
We prove an old conjecture of McMullen, Schneider and Shephard that every polytope with the generating property is strongly monotypic. The other direction is already known, which implies that strong monotypy and the generating property for polytopes are the same notion. A criterion for monotypic and strongly monotypic polytopes is also given.
A topological version of the famous Hedetniemi conjecture says: The mapping index of the Cartesian product of two $\mathbb Z/2$-spaces is equal to the minimum of their $\mathbb Z/2$-indexes. The main purpose of this article is to study the topological version of the Hedetniemi conjecture for $G$-spaces. Indeed, we show that the topological Hedetniemi conjecture cannot be valid for general pairs of $G$-spaces. More precisely, we show that this conjecture can possibly survive if the group $G$ is either a cyclic $p$-group or a generalized quaternion group whose size is a power of 2.
The following game in a similar formulation to Petri nets and chip-firing games is studied: Given a finite collection of baskets, each has an infinite number of balls of the same value. Initially, a ball from some basket is chosen to put on the table. Subsequently, in each step a ball from the table is chosen to be replaced by some $2$ balls from some baskets. Which baskets to take depend only on the ball to be replaced and they are decided in advance. Given some $n$, the object of the game is to find the maximum possible sum of values $g(n)$ for a table of $n$ balls. In this article, the sequence $g(n)/n$ for $n=1,2,\dots$ will be shown to converge to a growth rate $λ$. Furthermore, this value $λ$ is also the rate of a structure called pseudo-loop and the solution of a rather simple linear program. The structure and the linear program are closely related, e.g. a solution of the linear program gives a pseudo-loop with the rate $λ$ in linear time of the number of baskets, and vice versa with the pseudo-loop giving a solution to the dual linear program. A method to test in quadratic time whether a given $λ_0$ is smaller than $λ$ is provided to approximate $λ$. When the values of the balls are all rational, we can compute the precise value of $λ$ in cubic time, using the quadratic time rate test algorithm and the binary search with a special condition to stop. Four proofs of the limit $λ$ are given: one just uses the relation between the baskets, one uses pseudo-loops, one uses the linear program and one uses Fekete's lemma (the latest proof assumes a condition on the rule of replacements).
We give an effective bound of the joint spectral radius $ρ(Σ)$ for a finite set $Σ$ of nonnegative matrices: For every $n$, \[ \sqrt[n]{\left(\frac{V}{UD}\right)^{D} \max_C \max_{i,j\in C} \max_{A_1,\dots,A_n\inΣ}(A_1\dots A_n)_{i,j}} \le ρ(Σ) \le \sqrt[n]{D \max_C \max_{i,j\in C} \max_{A_1,\dots,A_n\inΣ}(A_1\dots A_n)_{i,j}}, \] where $D\times D$ is the dimension of the matrices, $U,V$ are respectively the largest entry and the smallest entry over all the positive entries of the matrices in $Σ$, and $C$ is taken over all strongly connected components in the dependency graph. The dependency graph is a directed graph where the vertices are the dimensions and there is an edge from $i$ to $j$ if and only if $A_{i,j}\ne 0$ for some matrix $A\inΣ$. Furthermore, a bound on the norm is also given: If $ρ(Σ)>0$ then there exist a nonnegative integer $r$ and two positive numbers $α,β$ so that for every $n$, \[ αn^r{ρ(Σ)}^n \le \max_{A_1,\dots,A_n\inΣ} \|A_1\dots A_n\| \le βn^r{ρ(Σ)}^n. \] Corollaries of the approach include a simple proof for the joint spectral theorem for finite sets of nonnegative matrices and the convergence rate of some sequences. The method in use is mostly based on Fekete's lemma, for both submultiplicative and supermultiplicative sequences.
We characterize all the strongly monotypic polytopes. Hadwiger's conjecture for this class of polytopes is deduced from the characterization.
For a bilinear map $*:\mathbb R^d\times \mathbb R^d\to \mathbb R^d$ of nonnegative coefficients and a vector $s\in \mathbb R^d$ of positive entries, among an exponentially number of ways combining $n$ instances of $s$ using $n-1$ applications of $*$ for a given $n$, we are interested in the largest entry over all the resulting vectors. An asymptotic behavior is that the $n$-th root of this largest entry converges to a growth rate $λ$ when $n$ tends to infinity. In this paper, we prove the existence of this limit by a special structure called linear pattern. We also pose a question on the possibility of a relation between the structure and whether $λ$ is algebraic.
We give an improvement of the Carathéodory theorem for strong convexity (ball convexity) in $\mathbb R^n$, reducing the Carathéodory number to $n$ in several cases; and show that the Carathéodory number cannot be smaller than $n$ for an arbitrary gauge body $K$. We also give an improved topological criterion for one convex body to be a Minkowski summand of another.