The modified Coulomb-Born approximation including the proton impact internuclear interaction (MCB-PI) is applied to study single ionization of helium by 75 keV proton impact. Fully differential cross-sections (FDCS) are calculated both in the scattering and perpendicular planes. The results are compared with experimental data and theoretical predictions from the three-body distorted wave (3DW) and the continuum distorted wave-eikonal initial state (CDW-EIS). It is shown that the features of the FDCS are better reproduced by the MCB-PI results. We also assess the influence of the PI interaction on the FDCS and we find that the PI interaction is particularly important in the perpendicular plane.

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In this paper, we are interested in the existence of Pareto solutions to vector polynomial optimization problems over a basic closed semi-algebraic set. By invoking some powerful tools from real semi-algebraic geometry, we first introduce the concept called tangency varieties; then we establish connections of the Palais-Smale condition, Cerami condition, M-tameness, and properness related to the considered problem, in which the condition of Mangasarian-Fromovitz constraint qualification at infinity plays an essential role in deriving these connections. According to the obtained connections, we provide some sufficient conditions for existence of Pareto solutions to the problem in consideration, and we also give some examples to illustrate our main findings.

<p style='text-indent:20px;'>In this paper, we first establish a surjectivity result for the sum of a maximal monotone mapping and a generalized pseudomontone mapping by using the generalized Yosida approximation technique. Then, we study a double obstacle quasilinear parabolic variational inequality problem <inline-formula><tex-math id="M1">\begin{document}$ {{\rm{(VIP)}}} $\end{document}</tex-math></inline-formula> by using the surjectivity result and penalty approximation method. In order to deal with the double obstacle constraints, we construct a quasilinear parabolic partial differential penalty equation, then we obtain the solvability of the quasilinear parabolic partial differential penalty equation. Moreover, we show that the set of the solutions to the penalty equation is bounded and every weak cluster point of this set is a solution of the problem <inline-formula><tex-math id="M2">\begin{document}$ {{\rm{(VIP)}}} $\end{document}</tex-math></inline-formula>. At last, as an application, we obtain numerical solutions of two double obstacle parabolic variational inequality problems by using the power penalty approximation method. Through the figures in the examples, we can intuitively see the different numerical solutions of the problems at different times.</p>

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