Statistically p-Upward Quasi-Cauchy Sequences and Cone-Valued Continuity
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917275525709824 |
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| author | Açıkgöz. |
| author_facet | Açıkgöz. |
| contents | We introduce statistically $p$-upward quasi-Cauchy sequences, defined by the condition $\lim_{n\to\infty}\frac{1}{n}|\{k\leq n: x_k - x_{k+p}\geq\varepsilon\}|=0$ for every $\varepsilon>0$, and develop the corresponding notions of compactness and continuity. We prove that a subset of $\mathbb{R}$ is statistically $p$-upward compact if and only if it is bounded below, characterizing lower boundedness sequentially. Statistically $p$-upward continuity is shown to imply uniform continuity on below bounded sets. The function space $\mathrm{SUC}_p(E)$ is a closed convex cone that fails to be a vector subspace -- distinguishing it from all previously studied sequential continuity spaces. We establish that every non-decreasing uniformly continuous function belongs to $\mathrm{SUC}_p(E)$, use Weyl's equidistribution theorem to show $\sin x\notin\mathrm{SUC}_p(\mathbb{R})$, prove a step-parameter hierarchy, and show that $\mathrm{SUC}_p(E)\cap C_b(E)$ is nowhere dense in $C_b(E)$. As an application, we develop a one-sided error control theory for function approximation, illustrated by Bernstein operators on a pharmacokinetic model. The inclusion relations among the continuity types studied and open problems are provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14125 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Statistically p-Upward Quasi-Cauchy Sequences and Cone-Valued Continuity Açıkgöz. General Topology 40A05, 40A35, 40G15, 26A15 We introduce statistically $p$-upward quasi-Cauchy sequences, defined by the condition $\lim_{n\to\infty}\frac{1}{n}|\{k\leq n: x_k - x_{k+p}\geq\varepsilon\}|=0$ for every $\varepsilon>0$, and develop the corresponding notions of compactness and continuity. We prove that a subset of $\mathbb{R}$ is statistically $p$-upward compact if and only if it is bounded below, characterizing lower boundedness sequentially. Statistically $p$-upward continuity is shown to imply uniform continuity on below bounded sets. The function space $\mathrm{SUC}_p(E)$ is a closed convex cone that fails to be a vector subspace -- distinguishing it from all previously studied sequential continuity spaces. We establish that every non-decreasing uniformly continuous function belongs to $\mathrm{SUC}_p(E)$, use Weyl's equidistribution theorem to show $\sin x\notin\mathrm{SUC}_p(\mathbb{R})$, prove a step-parameter hierarchy, and show that $\mathrm{SUC}_p(E)\cap C_b(E)$ is nowhere dense in $C_b(E)$. As an application, we develop a one-sided error control theory for function approximation, illustrated by Bernstein operators on a pharmacokinetic model. The inclusion relations among the continuity types studied and open problems are provided. |
| title | Statistically p-Upward Quasi-Cauchy Sequences and Cone-Valued Continuity |
| topic | General Topology 40A05, 40A35, 40G15, 26A15 |
| url | https://arxiv.org/abs/2602.14125 |