<p style='text-indent:20px;'>This paper presents a new mathematical signal transform that is especially suitable for decoding information related to non-rigid signal displacements. We provide a measure theoretic framework to extend the existing Cumulative Distribution Transform [<xref ref-type="bibr" rid="b29">29</xref>] to arbitrary (signed) signals on <inline-formula><tex-math id="M1">\begin{document}$ \overline {\mathbb{R}} $\end{document}</tex-math></inline-formula>. We present both forward (analysis) and inverse (synthesis) formulas for the transform, and describe several of its properties including translation, scaling, convexity, linear separability and others. Finally, we describe a metric in transform space, and demonstrate the application of the transform in classifying (detecting) signals under random displacements.</p>

Let Ω be a bounded open interval, and let p > 1 and q ∈ (0, p − 1). Let m ∈ L p (Ω) and 0 ≤ c ∈ L ∞ (Ω). We study the existence of strictly positive solutions for elliptic problems of the form −(|u | p−2 u ) + c(x)u p−1 = m(x)u q in Ω, u = 0 on ∂Ω. We mention that our results are new even in the case c ≡ 0.2010 Mathematics subject classification: primary 34B15; secondary 34B18, 35J25, 35J61.

Let Ω be a bounded open interval, let p > and > , and let m : Ω → ℝ be a function that may change sign in Ω. In this article we study the existence and nonexistence of positive solutions for one-dimensional singular problems of the form −(|u ὔ | p− u ὔ ) ὔ = m(x)u − in Ω, u = on ∂Ω. As a consequence we also derive existence results for other related nonlinearities.

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