Comparing q-ary relations on a set $𝒪$ of elementary objects is one of the most fundamental problems of classification and combinatorial data analysis. In this paper the specific comparison task that involves classification tree structures (binary or not) is considered in this context. Two mathematical representations are proposed. One is defined in terms of a weighted binary relation; the second uses a 4-ary relation. The most classical approaches to tree comparison are discussed in...

We prove that the cutwidth of the -dimensional mesh of -ary trees is of order $\Theta \left(d^{(r-1)n+1}\right)$, which improves and generalizes previous results.

Coalgebras for endofunctors $𝒞\to 𝒞$ can be used to model classes of object-oriented languages. However, binary methods do not fit directly into this approach. This paper proposes an extension of the coalgebraic framework, namely the use of ${𝒞}^{op}\times 𝒞\to 𝒞$. This extension allows the incorporation of binary methods into coalgebraic class specifications. The paper also discusses how to define bisimulation and invariants for coalgebras of extended polynomial functors and proves many...

We consider a finite-dimensional control system $\left(\Sigma \right)\dot{x}\left(t\right)=f(x\left(t\right),u\left(t\right))$, such that there exists a feedback stabilizer that renders $\dot{x}=f(x,k\left(x\right))$ globally asymptotically stable. Moreover, for with an output map and $1\le p\le q\le \infty$, we assume that there exists a ${𝒦}_{\infty }$-function such that $\parallel H\left(x_u\right){\parallel }_q\le {\alpha (\parallel u\parallel }_p)$, where is the maximal solution of ${\left(\Sigma \right)}_k\dot{x}\left(t\right)=f(x\left(t\right),k\left(x\left(t\right)\right)+u\left(t\right))$, corresponding to and to the initial condition . Then, the gain function $G_{(H,p,q)}$ of given by 14.5cm $$G_{(H,p,q)}\left(X\right)\stackrel{\mathrm{def}}{=}\underset{{\parallel u\parallel }_p=X}{sup}{\parallel H\left(x_u\right)\parallel }_q,$$ is well-defined. We call profile of for any ${𝒦}_{\infty }$-function which is of the same order of magnitude...