Abstract The fractal characteristics of the Catalan sequence and its reciprocal version have been systematically investigated. In particular, the box dimension and the Hausdorff dimension of the reciprocal sequence are rigorously calculated. It has been proved that both the box and the Hausdorff dimensions of the reciprocal sequence are equal to zero, which reflects the ultra-fast convergence rates of the sequence toward zero. Furthermore, the convergence of the series formed by the reciprocal Catalan sequence is verified based on the fractal dimension criterion and the double monotonicity of the sequence. Numerical experiments have been conducted to validate the theoretical results. The results obtained provide a theoretical framework for the fractal analysis of classical combinatorial sequences and demonstrate the intrinsic relationship between the fractal dimension of point sequences and their convergence rates.
Abstract Let T , S ∈ L ( H ) $T,S\in \mathcal{L}\left(H\right)$ be commuting operators on a Hilbert space H , where T is w -hyponormal with ker T ⊆ ker T * and S is normal. Consider the elementary operator ψ T , S defined on L ( H ) $\mathcal{L}\left(H\right)$ by ψ T , S ( X ) = TXT * − SXS *, for all X ∈ L ( H ) $X\in \mathcal{L}\left(H\right)$ , and let C p ${\mathcal{C}}_{p}$ denote the Schatten p -class. It is shown that: 1. For all X ∈ C p $X\in {\mathcal{C}}_{p}$ , TXT * = SXS * implies T * XT = S * XS , i.e., ker ( ψ T , S ∣ C p ) ⊆ ker ψ T * , S * ∣ C p $\mathrm{ker}\left({\psi }_{T,S}\mid {\mathcal{C}}_{p}\right)\subseteq \mathrm{ker}\left({\psi }_{{T}^{{\ast}},{S}^{{\ast}}}\mid {\mathcal{C}}_{p}\right)$ ; and 2. For all X , Y ∈ L ( H ) $X,Y\in \mathcal{L}\left(H\right)$ with Y ∈ ker ( ψ T , S ∣ C p ) $Y\in \mathrm{ker}\left({\psi }_{T,S}\mid {\mathcal{C}}_{p}\right)$ , we have T X T * − S X S * + Y p ≥ ‖ Y ‖ p , $${{\Vert}TX{T}^{{\ast}}-SX{S}^{{\ast}}+Y{\Vert}}_{p}\ge {\Vert}Y{{\Vert}}_{p},$$ i.e., the range of ψ T , S ∣ C p ${\psi }_{T,S}\mid {\mathcal{C}}_{p}$ is orthogonal to its kernel if and only if ker T ∩ ker S = {0}. These results extend to the operator Ψ( X ) = AXB − CXD , where A , B * are w -hyponormal with ker A ⊆ ker A * and ker B * ⊆ ker B , and C , D are normal commuting with A and B respectively. This generalizes classical results by Turnšek, the Putnam-Fuglede property, and Weiss’s theorem.
Abstract Let k be a perfect field. Assuming that the Mordell–Weil groups of certain elliptic curves are Z / 2 Z $\mathbb{Z}/2\mathbb{Z}$ , we find solutions of the equation x 4 − y 4 = D in all cubic extensions of k . This significantly extends the result of Tho in [“On the difference of two fourth powers,” Proc. Edinb. Math. Soc., II. Ser. , vol. 67, no. 1, pp. 142–150, 2024]. As an application, for prime p ≡ 3 (mod 8), we completely solve the equation x 4 − y 4 = p n in all cyclic cubic number fields.
Abstract In this paper, we introduce a new family of κ -Hermite polynomials as a natural κ -extension of the classical Hermite polynomials within the framework of Kaniadakis calculus. The construction is based on a κ -modified generating function obtained by replacing the classical exponential function with the κ -exponential function together with the corresponding κ -factorial structure. This approach allows us to derive explicit representations of the κ -Hermite polynomials and the associated κ -Hermite numbers and to establish several fundamental properties that generalize those of the classical Hermite family. In particular, recurrence relations, κ -derivative formulas, and operational identities in the κ -calculus framework are obtained. Furthermore, a two-variable extension H n , κ x , y ${H}_{n,\kappa }\left(x,y\right)$ is introduced and its main structural properties are investigated. Mixed-type κ -Hermite–Bernoulli, κ -Hermite–Euler, and κ -Hermite–Stirling polynomials are also constructed, and explicit connection formulas are established. It is further shown that, for 0 < | κ | < 1, no nontrivial nonnegative scalar weight with finite moments through order four yields orthogonality in the natural κ -integral setting. Graphical comparisons for the first three degrees exhibiting nontrivial κ -dependence, namely n = 3, 4, and 5, illustrate how the polynomials depart from their classical counterparts as | κ | increases. It is shown that all formulae and constructions that admit classical counterparts consistently reduce to those counterparts in the limit κ → 0, confirming the mathematical consistency of the proposed κ -extension. The present work complements recent studies on κ -generalized polynomial families, such as κ -Bernoulli, κ -Euler, and κ -Stirling polynomials, and provides a systematic κ -extension of the Hermite polynomials.
Abstract In this note, we extend and refine the sufficient part of the Feller weak law of large numbers by establishing mean convergence for weighted sums and the “maximum” of weighted sums of adapted sequences of measurable operators under regularly varying normalization. The corresponding convergence in measure results are obtained as immediate consequences of these stronger mean convergence results. The results are new even in the classical setting of random variables.
