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<p>In this paper, we introduce a notion of a probabilistic measure which takes values in <i>t</i>-scaled hyperbolic numbers for <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="6_2025_1394_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">\(t\in \mathbb {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>, with a system of axioms generalizing directly Kolmogorov’s axioms. i.e., we establish a suitable measure theory in the set <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="6_2025_1394_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {D}_{t}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">D</mi> <mi>t</mi> </msub> </math></EquationSource> </InlineEquation> of all <i>t</i>-scaled hyperbolic numbers for arbitrarily fixed <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="6_2025_1394_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">\(t\in \mathbb {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Probabilities with Values in Scaled Hyperbolic Numbers

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年6月15日

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  • Daniel Alpay,
  • Ilwoo Cho

摘要

In this paper, we introduce a notion of a probabilistic measure which takes values in t-scaled hyperbolic numbers for \(t\in \mathbb {R}\) t ∈ R , with a system of axioms generalizing directly Kolmogorov’s axioms. i.e., we establish a suitable measure theory in the set \(\mathbb {D}_{t}\) D t of all t-scaled hyperbolic numbers for arbitrarily fixed \(t\in \mathbb {R}\) t ∈ R .