Normal form methods allow one to compute quasi-invariants of a Hamiltonian system, which are referred to as proper elements. The computation of the proper elements turns out to be useful to associate dynamical properties that lead to identify families of space debris, as it was done in the past for families of asteroids. In particular, through proper elements we are able to group fragments generated by the same break-up event and we possibly associate them to a parent body. A qualitative analysis of the results is given by the computation of the Pearson correlation coefficient and the probability of the Kolmogorov-Smirnov statistical test.

We show that every finitely generated algebra that is a finitely generated module over a finitely generated commutative subalgebra is an automaton algebra in the sense of Ufnarovskii.

We display a gallery of Lorenz-like attractors that emerge in a class ofthree-dimensional maps. We review the theory of Lorenz-like attractors for diffeomorphisms(as opposed to flows), define various types of such attractors, and find sufficientconditions for three-dimensional Henon-like maps to possess pseudohyperbolic Lorenz-likeattractors. The numerically obtained scenarios of the creation and destruction of theseattractors are also presented.

We investigate the possibility of extending Chrobak normal form to the probabilisticcase. While in the nondeterministic case a unary automaton can be simulated by anautomaton in Chrobak normal form without increasing the number of the states in thecycles, we show that in the probabilistic case the simulation is not possible by keepingthe same number of ergodic states. This negative result is proved by considering thenatural extension to the probabilistic case of Chrobak normal form, obtained by replacingnondeterministic choices with probabilistic choices. We then propose a different kind ofnormal form, namely, cyclic normal form, which does not suffer from the same problem: weprove that each unary probabilistic automaton can be simulated by a probabilisticautomaton in cyclic normal form, with at most the same number of ergodic states. In thenondeterministic case there are trivial simulations between Chrobak normal form and cyclicnormal form, preserving the total number of states in the automata and in theircycles.

Non-degenerate monoids of skew type are considered. This is a class of monoids S defined by n generators and quadratic relations of certain type, which includes the class of monoids yielding set-theoretic solutions of the quantum Yang–Baxter equation, also called binomial monoids (or monoids of I-type with square-free defining relations). It is shown that under any degree-lexicographic order on the associated free monoid FMn. of rank n the set of normal forms of elements of S is a regular language in FMn. As one of the key ingredients of the proof, it is shown that an identity of the form xN yN = yN xN holds in S. The latter is derived via an investigation of the structure of S viewed as a semigroup of matrices over a field. It also follows that the semigroup algebra K[S] is a finite module over a finitely generated commutative subalgebra of the form K[A] for a submonoid A of S.

An $n$-dimensional quantum torus is a twisted group algebra of the group ${{\mathbb{Z}}^{n}}.$ It is called rational if all invertible commutators are roots of unity. In the present note we describe a normal form for rational $n$-dimensional quantum tori over any field. Moreover, we show that for $n\,=\,2$ the natural exact sequence describing the automorphism group of the quantum torus splits over any field.

This paper presents the role of vector relative degree in theformulation of stationarity conditions of optimal control problemsfor affine control systems. After translating the dynamics into anormal form, we study the Hamiltonian structure. Stationarityconditions are rewritten with a limited number of variables. Theapproach is demonstrated on two and three inputs systems, then, weprove a formal result in the general case. A mechanical systemexample serves as illustration.

We investigate a locally full HNN extension of an inverse semigroup. A normal form theorem is obtained and applied to the word problem. We construct a tree and show that a maximal subgroup of a locally full HNN extension acts on the tree without inversion. Bass-Serre theory is employed to obtain a group presentation of the maximal subgroup as a fundamental group of a certain graph of groups associated with the D-structure of the original semigroup.