Lower bound for the ratios of eigenvalues of Schrödinger with nonpositive single‐barrier potentials

Horváth and Kiss (Proc. Amer. Math. Soc., 2005) proved the upper bound estimate for Dirichlet eigenvalue ratios of the Schrödinger problem ?y^′′ + q(x)y = λy with nonnegative and single‐well potential q. In this paper, we prove that if q(x) is a nonpositive, continuous, and single‐barrier potential, then for λ_n > λ_m≥ ? 2q^?, where . In particular, if q(x) satisfies the additional condition , then λ₁ > 0 and for n > m ≥ 1. For this result, we develop a new approach to study the monotonicity of the modified Prüfer angle function.     

Quantitative Uncertainty Principles for the Canonical Fourier—Bessel Transform

Acta Mathematica Sinica, English Series - The aim of this paper is to establish an extension of quantitative uncertainty principles and an algorithm for signal recovery about the essential supports...     

Exact observability and controllability for a nonsimple elastic rod

In this paper, we prove exact observability, boundary and internal exact controllability results for a nonsimple elastic rod. By the spectral analysis method, it is shown that the considered problem has a sequence of eigenfunctions, which forms a Riesz basis for the state space. The exact observability is proved by application of a modified version of Ingham’s Theorem. Then, via the Hilbert Uniqueness method, we prove a boundary controllability result with only one controller and an internal controllability result.     

Unprecedented stereoselective synthesis of 3-methylisoxazolidine-5-aryl-1,2,4-oxadiazoles via 1,3-dipolar cycloaddition and study of their in vitro antioxidant activity

The stereoselective 1,3-dipolar cycloaddition between allyl cyanide and a menthone-derived nitrone led to the desired isoxazolidine in good yield. Once the nitrile group transformed to an amidoxime group, the cyclocondensation of various aldehydes with chiral amidoxime led to unprecedented enantiopure 3-methylisoxazolidine-5-aryl-1,2,4-oxadiazoles. The menthone chiral auxiliary was then removed with acid hydrolysis. The new compounds were also screened for their in vitro antioxidant activity using DPPH and FRAP assays. Some of the compounds showed promising antioxidant activity.