Consecutive-term vectors associated with second-order linear recurrences have been studied in several settings, including generalized Fibonacci vectors, Fibonacci/Lucas vectors, Horadam vectors, and Pell–Pell/Lucas vectors. In this paper, we give a unified geometric formulation for length-\(m\) vectors generated by the recurrence \(U_{n}=pU_{n-1}+qU_{n-2}\) under the hypothesis of two distinct real characteristic roots. Using the intrinsic two-dimensional structure of the solution space, we derive Binet-type vector formulas and show that every exterior product of two consecutive-term vectors is a scalar multiple of one fixed bivector. This exterior-algebra formulation explains the planar confinement observed in three dimensions and yields closed-form inner products in arbitrary dimension, including mixed products involving the normalized Lucas-type companion sequence. The Pell specialization recovers the consistent Pell/Pell–Lucas vector identities and makes their rank-two origin explicit.
In this paper, we prove some inequalities of the operator norm and the numerical radius for Hilbert spaces operators. Among other inequalities, it is shown that if \(A, B \in \mathcal{B}(\mathcal{H})\) and \(\mathrm{Re}(A) > 0\), then \(\displaystyle \omega(BA) \leq \left(\frac{\omega(B)+\|B\|}{2}+D_{B}\right)\omega(A) \leq \frac{7}{2}\omega(A)\omega(B),\) where \(D_{B}=\inf_{\lambda\in\mathbb{C}}\|B-\lambda I\|\). Moreover, we prove that if \(A, B \in \mathcal{B}(\mathcal{H})\) and \(A > 0\), then \(\displaystyle \omega(AB+BA) \leq \frac{3\|A\|-\min_{\lambda\in\sigma(A)}\lambda}{2}\,\omega(B) < \frac{3}{2}\|A\|\omega(B).\)
In this paper we consider a class of maps (whose convex hull has a nice selection property) with another general class of maps and we present coincidence results between these two classes.
Vertex and edge labellings have played a significant role in the study of graphs since the foundation of the theory of graph labellings in the 1960s. This paper is focused on the specific case of vertex labellings called graceful labellings. It is a companion to a paper published in this journal in 2024, in which we characterized “kites” formed by the cycles known to be graceful. It was left there as a possible further research direction to characterize the remaining kites formed by the non-graceful cycles. The present paper completes this task. The kites formed by the non-graceful cycles are characterized among all simple graphs via Sheppard’s labelling sequences, and via labelling relations and graph chessboards. While the labelling relations are rather closely related to the labelling sequences, the graph chessboards provide a nice visualization of the gracefulness of the graphs they represent.
In this paper we find the necessary and sufficient conditionsfor the geometrical dimension of the real nerve of the modulispace of Riemann surfaces of odd genusg to be maximal.Furthermore, we prove some properties of Riemann surfaceswith extremal configuration of symmetries which lead to theconclusion that certain homology groups of the real nerveNgof the moduli space of Riemann surfaces of odd genusgare nontrivial.
