Single Peaks in Truncated Stieltjes Integral Equations
Overview
The paper studies curves defined by an integral equation. For a positive profile h that rises to one peak and then falls, a strictly decreasing relative slope xh'(x)/h(x) on its falling side forces a unique peak of each curve that exists, proving Viakhirev's uniqueness conjecture. The results cover every positive power of the integral kernel and determine the exact range of possible levels and how the peak moves. For the proposed normalized log-normal cutoffs, the peak location and height approach limits independent of the cutoff exponent with an explicit first-order correction. Smooth profiles with only one peak can nevertheless produce curves with arbitrarily many stationary points, where the derivative vanishes.
Original public questions
Viakhirev Conjecture 10: an integral self-consistency equation
Ivan Viakhirev: Structural properties of the implicit function defined by an integral self-consistency equation — arXiv 2606.04243v1: Conjecture 10 (uniqueness, with its additional existence hypotheses derived), the log-normal cutoff question in Section 6.3, and the unnumbered question on multiple critical points for unimodal profiles.
Original abstract (English)
We study maxima of curves defined by normalized, truncated Stieltjes integrals. For a positive profile h, strict decrease of its elasticity xh'(x)/h(x) on its declining side gives a unique nondegenerate maximum at every level for which the curve exists. This proves a uniqueness conjecture of Viakhirev and makes its additional existence hypotheses unnecessary. We treat every positive power of the kernel, on a finite interval or the positive half-line: all positive levels occur for powers at least one, whereas a uniquely maximized boundary integral gives the exact level threshold for smaller powers. We determine the movement and limiting positions of the peak. For log-normal densities, the normalized polynomial cutoffs proposed by Viakhirev have maximizing locations and heights converging at rate O(R^-1) to limits independent of the cutoff exponent. In contrast, smooth strictly unimodal profiles can produce arbitrarily many critical points.
Mathematical review
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