{"ID":3053256,"CreatedAt":"2026-06-04T04:41:36.695875263Z","UpdatedAt":"2026-06-05T21:30:13.696557441Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.04243","arxiv_id":"2606.04243","title":"Structural properties of the implicit function defined by an integral self-consistency equation","abstract":"We study the integral equation $\\int_0^m ηρ(η)/(C-η)\\,dη= 1$ with $C\u003em$, where $ρ$ is a $C^1$ probability density on $[0,M]$ vanishing polynomially at $η=M$. Setting $\\mathcal{I}^+(m) := \\lim_{C \\downarrow m}\\int_0^m ηρ(η)/(C-η)\\,dη$ and $Ω:= \\{m \\in (0,M) : \\mathcal{I}^+(m) \u003e 1\\}$, the equation determines $C$ implicitly as a function of $m$ on $Ω$, and our object of study is the dimensionless ratio $β(m) := C(m)/m$. Writing $h(η) := ηρ(η)$, our main theorem establishes openness of $Ω$, $C^1$-smoothness of $β$, a sign formula identifying $β'(m)$ with a positively-weighted integral of $dh/d\\lnη$, transfer of monotonicity from $h$ to $β$, and existence of an interior critical point of $β$ when $h$ is unimodal and two technical hypotheses hold. Numerically, $β$ has a single critical point in seven log-concave test densities (mostly Beta-type), in support of a separate uniqueness conjecture. A bimodal density that violates both unimodality and log-concavity exhibits three critical points; this shows that dropping the two hypotheses jointly admits multiple critical points, but does not separate their roles.","short_abstract":"We study the integral equation $\\int_0^m ηρ(η)/(C-η)\\,dη= 1$ with $C\u003em$, where $ρ$ is a $C^1$ probability density on $[0,M]$ vanishing polynomially at $η=M$. Setting $\\mathcal{I}^+(m) := \\lim_{C \\downarrow m}\\int_0^m ηρ(η)/(C-η)\\,dη$ and $Ω:= \\{m \\in (0,M) : \\mathcal{I}^+(m) \u003e 1\\}$, the equation determines $C$ implicit...","url_abs":"https://arxiv.org/abs/2606.04243","url_pdf":"https://arxiv.org/pdf/2606.04243v1","authors":"[\"Ivan Viakhirev\"]","published":"2026-06-02T21:45:00Z","proceeding":"cs.SI","tasks":"[\"cs.SI\",\"math.PR\"]","methods":"[]","has_code":false}