Let \(D\subset \mathbb{R}^3\) be a bounded domain. \(q\in C(D)\) be a real-valued compactly supported potential, \(A(\beta, \alpha,k)\) be its scattering amplitude, \(k>0\) be fixed, without loss of generality we assume \(k=1\), \(\beta\) be the unit vector in the direction of scattered field, \(\alpha\) be the unit vector in the direction of the incident field. Assume that the boundary of \(D\) is a smooth surface \(S\). Assume that \(D\subset Q_a:=\{x: |x|\le a\}\), and \(a>0\) is the minimal number such that \(q(x)=0\) for \(|x|>a\). Formula is derived for \(a\) in terms of the scattering amplitude.

We develop and analyze an adaptive spacetime finite element method for nonlinear parabolic equations of \(p\)–Laplace type. The model problem is governed by a strongly nonlinear diffusion operator that may be degenerate or singular depending on the exponent \(p\), which typically leads to limited regularity of weak solutions. To address these challenges, we formulate the problem in a unified spacetime variational framework and discretize it using conforming finite element spaces defined on adaptive spacetime meshes. We prove the well-posedness of both the continuous problem and the spacetime discrete formulation, and establish a discrete energy stability estimate that is uniform with respect to the mesh size. Based on residuals in the spacetime domain, we construct a posteriori error estimators and prove their reliability and local efficiency. These results form the foundation for an adaptive spacetime refinement strategy, for which we prove global convergence and quasi-optimal convergence rates without assuming additional regularity of the exact solution. Numerical experiments confirm the theoretical findings and demonstrate that the adaptive spacetime finite element method significantly outperforms uniform refinement and classical time-stepping finite element approaches, particularly for problems exhibiting localized spatial and temporal singularities.

This article introduces what we term Hardy-Hilbert-Mulholland-type integral inequalities, which unify features of Hardy-Hilbert-type and Mulholland-type integral inequalities. These inequalities are parameterized by an adjustable parameter. The obtained constant factors are derived in singular form involving a logarithmic-tangent expression, and their optimality is discussed in detail. Several new secondary inequalities are also established. Complete proofs are provided, including detailed steps and references to intermediate results.

In this paper, we give the definitions of \(s\)-convex set and \(s\)-convex function on Heisenberg group. And some inequalities of Jensen’s type for this class of mappings are pointed out.

This article concerns the problem on the growth and the oscillation of some differential polynomials generated by solutions of the second order non-homogeneous linear differential equation \[\begin{equation*} f^{\prime \prime }+P\left( z\right) e^{a_{n}z^{n}}f^{\prime }+B\left( z\right) e^{b_{n}z^{n}}f=F\left( z\right) e^{a_{n}z^{n}}, \end{equation*}\] where \(a_{n}\), \(b_{n}\) are complex numbers, \(P\left( z\right)\) \(\left( \not\equiv 0\right)\) is a polynomial, \(B\left( z\right)\) \(\left( \not\equiv 0\right)\) and \(F\left( z\right)\) \(\left( \not\equiv 0\right)\) are entire functions with order less than \(n\). Because of the control of differential equation, we can obtain some estimates of their hyper-order and fixed points.