Small and medium-sized enterprises (SMEs) are vital drivers of China’s economy. As in any other country, SMEs in Chinaare exceptionally exposed to the devastating effects of the COVID-19 outbreak. The aim of the paper is to assess the impact of the pandemic on SMEs in China and study the effectiveness of the government’s support for SMEs through the crisis. The methodologies applied by the authors included the historical and logical method, the method of the rising from the abstract to the concrete, synthesis, comparative factor analysis, grouping and graphical methods, as well as a systematic and statistical approach. The authors investigate the main policies and initiatives launched in support of smaller businesses and implemented by the People’s Bank of China, the Ministry of Finance, the National Development and Reform Commission, the Ministry of Industry and Information Technology, as well as by the two national regulatory authorities –– China Banking and Insurance Regulatory Commission and China Securities Regulatory Commission. In this paper the authors analyze the direct and indirect support available to SMEs through financial institutions. The study leads to the conclusions that the state support for SMEs has been effective and helped to avoid a sharp decline in production. However, the spread of the disease in other countries may threaten the recovery of the Chinese economy.

By definition, reciprocal matrices are tridiagonal n-by-n matrices A with constant main diagonal and such that ai , i +1 ai +1, i = 1 for i = 1, . . ., n − 1. We establish some properties of the numerical range generating curves C(A) (also called Kippenhahn curves) of such matrices, in particular concerning the location of their elliptical components. For n ≤ 6, in particular, we describe completely the cases when C(A) consist entirely of ellipses. As a corollary, we also provide a complete description of higher rank numerical ranges when these criteria are met.

Studies on the effects of school openings or closures during the Covid-19 pandemic seem to reach contrasting conclusions even in similar contexts. We aim at clarifying this controversy. A mathematical analysis of compartmental models with subpopulations has been conducted, starting from the SIR model, and progressively adding features modeling outbreaks or upsurge of variants, lockdowns, and vaccinations. We find that in all cases, the in-school transmission rates only affect the overall course of the pandemic above a certain context dependent threshold. We provide rigorous proofs and computations of the thresdhold through linearization. We then confirm our theoretical findings through simulations and the review of data-driven studies that exhibit an often unnoticed phase transition. Specific implications are: awareness about the threshold could inform choice of data collection, analysis and release, such as in-school transmission rates, and clarify the reason for divergent conclusions in similar studies; schools may remain open at any stage of the Covid-19 pandemic, including variants upsurge, given suitable containment rules; these rules would be extremely strict and hardly sustainable if only adults are vaccinated, making a compelling argument for vaccinating children whenever possible.

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We aim at clarifying the controversy about the effects of school openings or closures on the course of the Covid-19 pandemic.The mathematical analysis of compartmental models with subpopulations shows that the in-school contact rates affects the overall course of the pandemic only above a certain threshold that separates an influence phase from a non-influence one. The threshold, that we calculate via linear approximation in several cases, seems to appear in all contexts, including outbreaks or new strains upsurge, lockdowns, and vaccination campaigns excluding children, albeit with different values. Our theoretical findings are then confirmed by several data driven studies that have previously identified the phase transition in specific cases.Specific outcomes of this study are:• opposite conclusions reached by studies of the same or similar situations might depend on, possibly small, differences in modeling or in parameter estimation from the very noisy Covid-19 data, that result in identifying different phases;