In this paper, we present a counterexample of the existence of holomorphic interpolation map in the unit ball of the complex euclidean space of dimension two.

We establish an existence result for a positive solution to the Schrödinger-type p-Laplacian semipositone problem: -𝚫_pu+V(x)|u|^p-2u = λf(u) in Ω, u = 0 on 𝜕Ω, where 𝚫_p = div(|∇_u|^p-2∇_u), p ≥ 2, Ω is a bounded domain in ℝ^N, N > 2, λ ∈ ℝ is a positive parameter. We assume V ∈ L^∞(Ω) and f ∈ C([0, ∞), ℝ) with f(0) < 0. In particular, when f(s) is p-sublinear at infinity, we establish the existence of positive solutions for sufficiently large λ. The proofs primarily are rely on the sub and supersolution method. Furthermore, we extend the existence result to p-Laplacian semipositone problems with mixed boundary conditions.

We concern the definability of a strict Cohen subring for a complete finitely ramified ℤ-valued field. We show that a strict Cohen subring is type-definable in the one-sorted language {+, ·, 0, 1, Div} of valued fields when the residue field has a finite imperfection degree. If the residue field has an infinite imperfection degree, we show that the set of elements contained in a strict Cohen subring is the countable union of type-definable sets.

Power options are derivatives whose payoffs involve a power of the underlying asset price. They are important instruments with broad potential applications in financial markets, including equity-indexed annuities. However, much of the existing literature assumes constant volatility, which tends to deviate from real financial market behavior. To address this limitation, we price power options under a hybrid volatility structure that combines local and stochastic volatility.

This paper investigates the well-posedness of linear elliptic equations, focusing on the divergence-free transformation introduced in the author's recent work [J. Math. Anal. Appl. 548 (2025), 129425]. By comparing this approach with classical bilinear form methods, we demonstrate that while standard techniques encounter limitations in handling zero-order coefficients c ∈ L¹(U), the divergence-free transformation successfully establishes well-posedness in this setting. Furthermore, utilizing the Riesz-Thorin interpolation theorem between the cases c ∈ L¹(U) and $c{\in}L^{\frac{2d}{d+2}}(U)$, we establish the existence and uniqueness of weak solutions under the assumption c ∈ L^s(U) for $s{\in}[1,\,{\frac{2d}{d+2}}]$.