相关论文: It is undecidable whether the growth rate of a giv…
The following notion of growth rate can be seen as a generalization of joint spectral radius: Given a bilinear map $*:\mathbb R^d\times\mathbb R^d\to\mathbb R^d$ with nonnegative coefficients and a nonnegative vector $s\in\mathbb R^d$,…
For a given finite set $\Sigma$ of matrices with nonnegative integer entries we study the growth of $$ \max_t(\Sigma) = \max\{\|A_{1}... A_{t}\|: A_i \in \Sigma\}.$$ We show how to determine in polynomial time whether the growth with $t$ is…
We present a notion of bilinear stability, which is to numerical stability what bilinear complexity is to time complexity. In bilinear complexity, an algorithm for evaluating a bilinear operator $\beta : \mathbb{U} \times \mathbb{V} \to…
We give necessary and sufficient conditions for a sequence to be exactly realizable as the sequence of numbers of periodic points in a dynamical system. Using these conditions, we show that no non-constant polynomial is realizable, and give…
We study the rate of growth of $p(n,S,M)$, the number of partitions of $n$ whose parts all belong to $S$ and whose multiplicities all belong to $M$, where $S$ (resp. $M$) are given infinite sets of positive (resp. nonnegative) integers. We…
A good range of problems on trees can be described by the following general setting: Given a bilinear map $*:\mathbb R^d\times\mathbb R^d\to\mathbb R^d$ and a vector $s\in\mathbb R^d$, we need to estimate the largest possible absolute value…
We study the growth of polynomials on semialgebraic sets. For this purpose we associate a graded algebra to the set, and address all kinds of questions about finite generation. We show that for a certain class of sets, the algebra is…
We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By "accurate" we mean that the computed…
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 $n-1$…
In this paper we complete B\"{u}chi's proof that there is no decision algorithm for the solubility in integers of arbitrary systems of diagonal quadratic form equations, by proving the assertion that whenever $x_1^2, \cdots, x_5^2$ are five…
In 2008, Ben-Amram, Jones and Kristiansen showed that for a simple programming language - representing non-deterministic imperative programs with bounded loops, and arithmetics limited to addition and multiplication - it is possible to…
We consider a branching particle system where each particle moves as an independent Brownian motion and breeds at a rate proportional to its distance from the origin raised to the power $p$, for $p\in[0,2)$. The asymptotic behaviour of the…
This paper derives a somewhat surprising but interesting enough result on the stabilizability of discrete-time parameterized uncertain systems. Contrary to an intuition, it shows that the growth rate of a discrete-time stabilizable system…
We show that for any rational p \in [1,\infty) except p = 1, 2, unless P = NP, there is no polynomial-time algorithm for approximating the matrix p-norm to arbitrary relative precision. We also show that for any rational p\in [1,\infty)…
We study the computational complexity of two Boolean nonlinearity measures: the nonlinearity and the multiplicative complexity. We show that if one-way functions exist, no algorithm can compute the multiplicative complexity in time…
Monadic second order logic can be used to express many classical notions of sets of vertices of a graph as for instance: dominating sets, induced matchings, perfect codes, independent sets or irredundant sets. Bounds on the number of sets…
It is known that Hilbert's Tenth Problem over the Gaussian ring $\mathbb Z[i]=\{a+bi:\ a,b\in\mathbb Z\}$ is undecidable. In this paper we obtain the following further result: There is no algorithm to decide whether an arbitrarily given…
Consider the set of those binary words with no non-empty factors of the form $xxx^R$. Du, Mousavi, Schaeffer, and Shallit asked whether this set of words grows polynomially or exponentially with length. In this paper, we demonstrate the…
We consider the computational problem of finding short paths in the skeleton of the perfect matching polytope of a bipartite graph. We prove that unless $P=NP$, there is no polynomial-time algorithm that computes a path of constant length…
This paper derives two stabilizability theorems for a basic class of discrete-time nonlinear systems with multiple unknown parameters. First, we claim that a discrete-time multi-parameter system is stabilizable if its nonlinear growth rate…