相关论文: Growth of bilinear maps III: Decidability
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 $n-1$…
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 show that there exists no algorithm that decides for any bilinear system $(B,v)$ if the growth rate of $(B,v)$ is $1$. This answers a question of Bui who showed that if the coefficients are positive the growth rate is computable (i.e.,…
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…
We prove lower bounds of order $n\log n$ for both the problem to multiply polynomials of degree $n$, and to divide polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers. These lower…
Overlap-free words are words over the binary alphabet $A=\{a, b\}$ that do not contain factors of the form $xvxvx$, where $x \in A$ and $v \in A^*$. We analyze the asymptotic growth of the number $u_n$ of overlap-free words of length $n$ as…
The exponential growth rate of non polynomially growing subgroups of $GL_d$ is conjectured to admit a uniform lower bound. This is known for non-amenable subgroups, while for amenable subgroups it is known to imply the Lehmer conjecture…
Generalizing some popular sequences like Catalan's number, Schr\"oder's number, etc, we consider the sequence $s_n$ with $s_0=1$ and for $n\ge 1$, \begin{multline*} s_n=\sum_{x_1+\dots+x_{\ell_1}=n-1} \kappa_1 s_{x_1}\dots s_{x_{\ell_1}} +…
This paper introduces a new combinatorial framework for modeling the growth of binary trees through a discrete evolution process that incorporates a growing rule and an extinction rule. Building upon the theory of increasingly labeled…
For a given natural number $n$, the second part of Hilbert's 16th Problem asks whether there exists a finite upper bound for the maximum number of limit cycles that planar polynomial vector fields of degree $n$ can have. This maximum number…
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…
The second part of the Hilbert's sixteenth problem consists in determining the upper bound $\mathcal{H}(n)$ for the number of limit cycles that planar polynomial vector fields of degree $n$ can have. For $n\geq2$, it is still unknown…
For a pair of coupled rectangular random matrices we consider the squared singular values of their product, which form a determinantal point process. We show that the limiting mean distribution of these squared singular values is described…
We investigate joint spectral characteristics of a family of matrices $\mathcal F $, associated with products in the semigroup generated by $\mathcal F$. In the literature, extremal measures such as the well-known joint spectral radius and…
An order-$s$ Davenport-Schinzel sequence over an $n$-letter alphabet is one avoiding immediate repetitions and alternating subsequences with length $s+2$. The main problem is to determine the maximum length of such a sequence, as a function…
Although known lower bounds for the growth rate $\lambda$ of polyominoes, or Klarner's constant, are already close to the empirically estimated value $4.06$, almost no conceptual progress on upper bounds has occurred since the seminal work…
Motivated by the classical Hilbert's Sixteenth Problem, we extend some main developments obtained for Hilbert's number in the polynomial setting to the piecewise polynomial context. Specifically, we study the growth of the maximum number of…
Motivated by complexity questions in integer programming, this paper aims to contribute to the understanding of combinatorial properties of integer matrices of row rank $r$ and with bounded subdeterminants. In particular, we study the…
The joint spectral radius of a bounded set of d times d real or complex matrices is defined to be the maximum exponential rate of growth of products of matrices drawn from that set. Under quite mild conditions such a set of matrices admits…
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…