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This paper presents a necessary and sufficient condition for a topological vector group to be locally compact. We also introduce several sufficient conditions that ensure the local compactness of topological vector groups. Furthermore, we establish a sufficient condition for a topological vector group to be first countable.
In the presentwork,we are interested in the linear operators of the form $$$S= T(a_+)R(a_-)$$$, where $$$a_-$$$ and $$$a_+$$$ are the annihilation and creation operators, respectively defined in irreducible representation of a deformed oscillator algebra and $$$T$$$, $$$R$$$ are analytic functions. We characterize all real sequences $$$(x_k)_{k\geq0}$$$ and functions $$$T$$$ for which the matrix elements associated to the operator $$$S$$$ are expressed in terms of polynomial sets on the discrete variable $$$x_k$$$ and we show when the considered polynomial sets are $$$d$$$-orthogonal. The analytic function $$$R$$$, in most specific cases is expressed in terms of exponential or $$$q$$$-exponential functions. As a consequence, several known results are recovered and extended, including those related to the Heisenberg-Weyl algebra, and $$$q$$$-deformed oscillator algebras. Explicit realizations are given in terms ofMeixner and Charlier-type $$$d$$$-orthogonal polynomials, together with their $$$q$$$-analogues.
We introduce a hierarchical clustering framework for weighted projective spaces $$$ℙ_{\parallel}$$$ built on Finsler geometry. From an optimization-based Finsler norm that quotients out the weighted scaling action, we construct a scaling-invariant distance $$$d_F([z], [w])$$$ and a rational analogue $$$d_{F,ℚ}([z], [w])$$$ for points of $$$ℙ_{\parallel}(ℚ)$$$. The norm carries a shape parameter $$$p:$$$ the case $$$p=2$$$ is Riemannian and admits a closed-form distance, while $$$p\neq 2$$$ is genuinely Finsler, and the metric and clustering guarantees below hold for every $$$p\in[1,\infty)$$$. Whereas earlier work measured proximity in these spaces through non-metric dissimilarities, we prove that $$$d_F$$$ satisfies the triangle inequality and is therefore a genuine metric ; this is what equips the induced clustering with its theoretical guarantees, including monotone dendrograms and Gromov–Hausdorff stability under perturbation of the data. The metric respects the intrinsic scaling symmetry and weighted topology of $$$ℙ_{\parallel}$$$, avoiding the distortions of a flat-space embedding. We develop the framework’s arithmetic applications—clustering rational points in the moduli space of genus two curves and analyzing rational functions in arithmetic dynamics—and indicate prospective extensions to quantum state spaces, where the weights $$${\parallel}$$$ model anisotropic noise. More broadly, the construction offers a rigorous metric foundation for graded neural networks and related machine-learning techniques on graded algebraic varieties.
2026
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