On globally solving nonconvex trust region subproblem via projected gradient method

The trust region subproblem (TRS) is to minimize a possibly nonconvex quadratic function over a Euclidean ball. There are typically two cases for (TRS), the so-called ``easy case'' and ``hard case''. Even in the ``easy case'', the sequence generated by the classical projected gradient method (PG) may converge to a saddle point at a sublinear local rate, when the initial point is arbitrarily selected from a nonzero measure feasible set. To our surprise, when applying (PG) to solve a cheap and possibly nonconvex reformulation of (TRS), the generated sequence initialized with {\it any} feasible point almost always converges to its global minimizer. The local convergence rate is at least linear for the ``easy case'', without assuming that we have possessed the information that the ``easy case'' holds. We also consider how to use (PG) to globally solve equality-constrained (TRS).Comment: 19 pages, 3 figure

A Study on the Impact of Urban Digitalization on the Urban-rural Income Gap

The empirical research topic for this paper is a panel dataset of 31 provinces and urban areas from my country from 2011 to 2020. On the one hand, it gauges the level of regional digital economic development. On the other side, we’ll talk about the structural impact of the level of digitalization on the urban-rural income difference and further debate whether the digital economy helps close or widen this gap. The findings show that the degree of digitization has a significant impact on reducing the income gap between urban and rural areas, while an increase in the Internet coverage index helps do so. However, the overall impact makes the digital economy unfavorable to reducing the income gap between urban and rural areas

A Parallel Local Reconnection Approach for Tetrahedral Mesh Improvement

AbstractA multi-threaded parallel local reconnection algorithm is proposed for tetrahedral meshes. It defines a feature point within the region involved in each operation, and sorts the features points along a Hilbert curve. The decomposition of this Hilbert curve results in a load-balanced distribution of local operations. Meanwhile, the regions of concurrently executed local operations are separated far away, such that the possibility of interference is reduced to a very low level. Finally, a parallel mesh improver is developed by combining the proposed algorithm with a parallel mesh smoothing algorithm, and its effectiveness and efficiency is verified in various numerical experiments

Minimal flag triangulations of lower-dimensional manifolds

We prove the following results on flag triangulations of 2- and 3-manifolds. In dimension 2, we prove that the vertex-minimal flag triangulations of $\mathbb{R} P^2$ and $\mathbb{S}^1\times \mathbb{S}^1$ have 11 and 12 vertices, respectively. In general, we show that $8+3k$ (resp. $8+4k$) vertices suffice to obtain a flag triangulation of the connected sum of $k$ copies of $\mathbb{R} P^2$ (resp. $\mathbb{S}^1\times \mathbb{S}^1$). In dimension 3, we describe an algorithm based on the Lutz-Nevo theorem which provides supporting computational evidence for the following generalization of the Charney-Davis conjecture: for any flag 3-manifold, $\gamma_2:=f_1-5f_0+16\geq 16 \beta_1$, where $f_i$ is the number of $i$-dimensional faces and $\beta_1$ is the first Betti number over a field. The conjecture is tight in the sense that for any value of $\beta_1$, there exists a flag 3-manifold for which the equality holds.Comment: 6 figures, 3 tables, 19 pages. Final version with a few typos correcte

Use of urban residential community parks for stress management during the COVID-19 lockdown period in China

During the pandemic lockdown period, residents had to stay at home and increased stress and other mental health problems have been associated with the lockdown period. Since most public parks were closed, community parks within gated residential areas became the most important green space in Chinese cities, and the use of such space might help to reduce the residents’ stress levels. This study aimed to investigate to what extent urban residents in China used community parks, engaged in outdoor activity during the lockdown period (23 January–8 April 2020) and if the use of such spaces helped to reduce their stress levels. An online questionnaire survey (n = 1342) was carried out from 23 March to 23 April 2020. Ordinary Least Squares regression was used to analyse the association between community park use, outdoor activity, willingness to engage in outdoor activity, and stress level. All results have been further analysed by two-sampled t-test to explore the difference between young and old age groups. We found that the overall self-reported stress level of the respondents was relatively moderate during the lockdown period. Respondents had generally reduced their use of community parks and engagement in outdoor activity. There was no significant association between stress level and the use of community parks or the engagement in outdoor activities. However, we found that older people showed much lower stress levels, used community parks more frequently, and engaged in more outdoor activities than younger adults. The findings suggest that outdoor activities and spatial characteristics in urban China differ from Western studies and advance the need to integrate the stress management role of community parks with urban green space policy to optimise the use of community parks blended in with everyday life, particularly during the lockdown period

Numerical investigation of multi-nozzle ejector device with inclined nozzles for marine gas turbine

The high-temperature exhaust gases and the hot surfaces of the ejector device in marine gas turbines generate significant levels of infrared radiation. An appropriate nozzle structure can effectively lower the exhaust gas temperature and reduce the high-temperature radiation surface area, thereby minimizing external infrared radiation. In this study, a numerical simulation of the nozzle structure in the ejector device was conducted using computational fluid dynamics (CFD) methods. By investigating the orthogonal combinations of nozzle inclination angles and the number of nozzles, the temperature distribution and flow characteristics under different operating conditions were analysed. The results showed that as the nozzle inclination angle increased, the entrainment coefficient (Ce) and the temperature ratio at the inlet and outlet (Rt) initially improved but then worsened. Simultaneously, the pressure loss coefficient (Cpl) increased with the inclination angle. The CRITIC weight method was employed to objectively allocate weights to Rt, Ce, and Cpl, determining the optimal solution. The results indicated that Rt and Cpl had significant and similar weights. The optimal solution was found in Case 10 (α = 5°, x = 4), with corresponding evaluation indices of Ce=2.38, Cpl=11.45, and =0.68. This study\u27s findings are of great importance for enhancing the performance of marine gas turbines and reducing external infrared radiation