In this paper, we investigate a novel fuzzy extension of the classical Jacobsthal number sequence, called fuzzy Jacobsthal numbers. By embedding the Jacobsthal recurrence relation into the framework of fuzzy set theory, each term of the sequence is represented as an interval-valued quantity parameterized by a fuzziness coefficient α ∈ [0,1]. This parameter regulates the uncertainty width of each term, such that the classical Jacobsthal sequence is recovered when α = 1, whereas maximal uncertainty occurs for α = 0. A series of new theoretical results are established. First, the well-definedness and structural consistency of the fuzzy intervals are rigorously proven. Second, the fuzzy Jacobsthal numbers are represented through matrix and interval companion matrix formulations, enabling algebraic manipulation within the framework of interval arithmetic. Third, the spectral characteristics of the associated interval matrices are analyzed to determine asymptotic behavior and stability conditions. The proposed model not only generalizes the Jacobsthal sequence into the fuzzy domain but also reveals a clear connection between fuzziness propagation and exponential growth dynamics. These results provide a systematic foundation for the development of fuzzy recurrence relations and open avenues for uncertainty quantification in discrete dynamical systems.
Let Ψ d ( x ) denote the primitive minimal polynomial over ℚ of cos(2π/ d ). We place the factorizations of T n (x) − c , c ∈ {0,±1/2, ±1}, in the uniform form T n (x) − c = α n,c ∏ d∈D(n,c) ψ d (x) ed , where the index set, multiplicities, and normalization are explicit. The five individual cases have substantial precedents: the zero fiber was treated by Rayes–Trevisan–Wang; the fibers at ±1 by Wolfram, with related square-divisibility formulas of Doi; and the fibers at ±1/2 follow, after rescaling, from Grubb's composition factorizations, with special irreducible cases also implicit in Kéri. Our principal contribution is their synthesis in a single primitive Ψ d normalization and an independent proof obtained by determining the exact orders of the roots of unity underlying each fiber. The divisor sets immediately give the splitting fields and abelian Galois groups, the number of irreducible factors, complete irreducibility criteria, and composition recursions; we record these as consequences of the uniform factorization. We also give a self-contained derivation, in the present normalization and for arbitrary degree, of the discriminant of T n ( x ) – c , relating it to earlier general critical-value formulas and normalized Chebyshev formulas. Finally, we express the power sums of the multiset T n −1 (c) as polynomials in c . For real fibers this recovers a known full-circle trigonometric power-sum identity; polynomial continuation extends the formulation to all c ∈ ℂ, including multiple fibers. It follows in particular that, for c ∈ [−1, 1], the normalized counting measure on T n −1 (c) has the same moments as the arcsine measure in every degree below n .
In the present article, we consider two families of integer sequences, the (k,r)-Lorenz-Fibonacci sequences of the first and second kind, whose characteristic polynomial is p_k,r(x)=x^k-x^k-1-⋯- x^r+x^r-1+⋯ +x+1, where k≥ 2r+1, and k≥ 3. These families include the well-known k-bonacci sequence, when r=0. These sequences arise naturally in the study of the dynamics of the Lorenz attractor. We introduce two families of substitutions (in an alphabet of k symbols) so that they are associated with each of the families of integer sequences and share the same polynomial. We study the main combinatorial properties of these substitutions.
We study the canonical normalization map behind Zeckendorf addition: a finite word over {0,1, 2} is replaced by the unique admissible Zeckendorf word with the same Fibonacci value. We first present this map as the canonical normalization induced by a section of the Fibonacci congruence on finite-support digit-count sequences and give a terminating local rewrite presentation whose terminal irreducible word is independent of the rewrite strategy. The main results give exact least-significant-digit-first (LSDF) noncausality invariants for this canonical map. For every output coordinate j, the LSDF anticipation horizon is infinite, even after restricting the domain to genuine digitwise sums of two admissible Zeckendorf inputs: A{ 0,1,2 }(& Nopf;)(j) = AAdd2(j) = infinity. The map has no continuous extension at the natural alternating one-sided limit tails. For LSDF finite words, the online commitment deficit is maximal: B(n) = n; equivalently, the prefix-destruction index - the largest number of initial normalized-output digits that may be invalidated by extending an input - satisfies Delta(n) = n. The same maximal prefix erasure holds on the genuine addition image, Delta Add2(n) = n, and the input-scale reduced residual complexity satisfies Rin(n) >= n. Non-subsequentiality - that is, impossibility of realization by a deterministic one-pass finite transducer with bounded terminal output - is then a corollary of these sharper quantitative obstructions. These statements are specific to the canonical greedy Zeckendorf normalizer in the LSDF one-sided model and are not claims of a new general normalization theory for Fibonacci or confluent numeration systems. We contrast them with the classical most-significant-digit-first Berstel adder. The existence and ten-state presentation of that adder are classical. We clarify its output-delay quotient: the ten displayed complete-output residuals are distinct as a raw letter-to-letter kernel, but standard subsequential output-delay normalization gives a six-state quotient. We then study the operation of iterating the Berstel recoding until the greedy Zeckendorf normal form is reached. Binary cleanup has exact worst-case depth & LeftFloor; L/2 & RightFloor; on trimmed binary words of length , and on genuine addition inputs the sharp depth is maxw is an element of Add2MSD(n)tau(w) = [n/2] ; the maximum is already attained at significant length, without leading-zero padding. A standard local-propagation obstruction is recorded separately in an appendix.
In this paper, we introduce and study new matrix-domain sequence spaces generated by Perrin numbers. We construct an infinite lower triangular matrix & Pscr; = (& Pscr;nk) and define the associated sequence spaces & ell;& Pscr;(& Pscr;), & ell;infinity(& Pscr;), c(& Pscr;), and c0(& Pscr;). We investigate several algebraic, topological, and geometric properties of these spaces. We further determine Schauder bases and study various inclusion relations among the Perrin sequence spaces. Moreover, we examine certain geometric properties including the approximation property, the Kadec-Klee property, the Hahn-Banach extension property, and rotundity. The alpha-, beta-, and gamma-duals of the newly introduced spaces are also computed. In addition, we characterize several classes of infinite matrix transformations involving Perrin sequence spaces and obtain necessary and sufficient conditions for matrix mappings of the forms (X(& Pscr;), Y) and (X(& Pscr;),Y(& Pscr;)), where X,Y is an element of {& ell;p, & ell;infinity, c, c0}.
