Related papers: Hyperquadratic continued fractions and automatic s…
Motivated by some recent developments in abstract theories of quadratic forms, we start to develop in this work an expansion of Linear Algebra to multivalued structures (a multialgebraic structure is essentially an algebraic structure but…
The connection between continued fractions and orthogonality which is familiar for $J$-fractions and $T$-fractions is extended to what we call $R$-fractions of type I and II. These continued fractions are associated with recurrence…
We are concerned with power series in 1/T over a finite field of 3 elements $\F_3$. In a previous article, Alain Lasjaunias investigated the existence of particular power series of elements algebraic over $\F_3[T]$, having all partial…
We give characterizations of unital uniform topological algebras and saturated locally multiplicatively convex algebras by means of multiplicative linear functionals. Some automatic continuity theorems in advertibly complete uniform…
The study of combinatorial properties of mathematical objects is a very important research field and continued fractions have been deeply studied in this sense. However, multidimensional continued fractions, which are a generalization…
We show several properties related to the structure of the family of classes of two-dimensional periodic continued fractions. This approach to the study of the family of classes of nonequivalent two dimexsional periodic continued fractions…
Finite automata with weights in the max-plus semiring are considered. The main result is: it is decidable in an effective way whether a series that is recognized by a finitely ambiguous max-plus automaton is unambiguous, or is sequential. A…
We present a form convergence theorem for sequences of sectorial forms and their associated semigroups in a complex Hilbert space. Roughly speaking, the approximating forms $a_n$ are all `bounded below' by the limiting form $a$, but in…
We put forward several general conjectures concerning the algebraicity or transcendence of continued fractions and Stieltjes continued fractions defined by the Thue-Morse and period-doubling sequences in characteristic $2$. We present our…
In a previous paper, we studied certain sequences of simultaneous rational approximations in ${\bf R}^2$ which present some analogy with the continued fractions. We got results around the Littlewood conjecture by using such approximations.…
Let $f(x)$ be a square free quartic polynomial defined over a quadratic field $K$ such that its leading coefficient is a square. If the continued fraction expansion of $\displaystyle \sqrt{f(x)}$ is periodic, then its period $n$ lies in the…
We introduced a new continued fraction expansions in our previous paper. For these expansions, we show formulae of probability about incomplete quotients. Furthermore, we prove the existence of invariant measures with respect to the…
In this paper we investigate the continuity of $\mathbb{Q}_+$-homogeneous superadditive correspondences defined on the cones with finite basis in real topological vector spaces. It is also shown that every lower semicontinuous and…
We consider continuous structures which are obtained from finite dimensional Hilbert spaces over $\mathbb{C}$ by adding some unitary operators. Quantum automata and circuits are naturally interpretable in such structures. We consider…
We build, for real quadratic fields, infinitely many periodic continuous fractions uniformly bounded, with a seemingly better bound than the known ones. We do that using continuous fraction expansions with the same shape as those of real…
Our main result is that any real cubic algebraic number has a continued fraction expansion with polynomial coefficients. Some generalizations are mentioned.
Abstract numeration systems encode natural numbers using radix ordered words of an infinite regular language and linear recurrence sequences play a key role in their valuation. Sequence automata, which are deterministic finite automata with…
A sequence is difference algebraic (or D-algebraic) if finitely many shifts of its general term satisfy a polynomial relationship; that is, they are the coordinates of a generic point on an affine hypersurface. The corresponding equations…
We show the finiteness of perfect powers in orbits of polynomial dynamical systems over an algebraic number field. We also obtain similar results for perfect powers represented by ratios of consecutive elements in orbits. Assuming the…
In this short note we present a set of interesting and useful properties of a one-parameter family of sequences including factorial and subfactorial, and their relations to the Gamma function and the incomplete Gamma function.