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It is known that the class of p-vector spaces $(0 < p \leq 1)$ ( 0 < p ≤ 1 ) is an important generalization of the usual norm spaces with rich topological and geometrical structure, but most tools and general principles with nature in nonlinearity have not been developed yet. The goal of this paper is to develop some useful tools in nonlinear analysis by applying the best approximation approach for the classes of 1-set contractive set-valued mappings in p-vector spaces. In particular, we first develop general fixed point theorems of compact (single-valued) continuous mappings for closed p-convex subsets, which also provide an answer to Schauder’s conjecture of 1930s in the affirmative way under the setting of topological vector spaces for $0 < p \leq 1$ 0 < p ≤ 1 . Then one best approximation result for upper semicontinuous and 1-set contractive set-valued mappings is established, which is used as a useful tool to establish fixed points of nonself set-valued mappings with either inward or outward set conditions and related various boundary conditions under the framework of locally p-convex spaces for $0 < p \leq 1$ 0 < p ≤ 1 . In addition, based on the framework for the study of nonlinear analysis obtained for set-valued mappings with closed p-convex values in this paper, we conclude that development of nonlinear analysis and related tools for singe-valued mappings in locally p-convex spaces for $0 < p \leq 1$ 0 < p ≤ 1 seems even more important, and can be done by the approach established in this paper.

The goal of this study is to establish a general framework for predicting the so-called critical "Turning Period" in an infectious disease epidemic such as the COVID-19 outbreak in China early this year. This framework enabled a timely prediction of the turning period when applied to Wuhan COVID-19 epidemic and informed the relevant authority for taking appropriate and timely actions to control the epidemic. It is expected to provide insightful information on turning period for the world's current battle against the COVID-19 pandemic.The underlying mathematical model in our framework is the individual Susceptible-Exposed-Infective-Removed (iSEIR) model, which is a set of differential equations extending the classic SEIR model. We used the observed daily cases of COVID-19 in Wuhan from February 6 to 10, 2020 as the input to the iSEIR model and were able to generate the trajectory of COVID-19 cases dynamics for the following days at midnight of February 10 based on the updated model, from which we predicted that the turning period of CIVID-19 outbreak in Wuhan would arrive within one week after February 14. This prediction turned to be timely and accurate, providing adequate time for the government, hospitals, essential industry sectors and services to meet peak demands and to prepare aftermath planning.Our study also supports the observed effectiveness on flatting the epidemic curve by decisively imposing the "Lockdown and Isolation Control Program" in Wuhan since January 23, 2020. The Wuhan experience provides an exemplary lesson for the whole world to learn in combating COVID-19.

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The purpose of this paper is to discuss the general risk assessment under the Hologram framework for the enterprise based on big data language; and to illustrate the Hologram as a new tool for establishing a mechanism to evaluate SMEs growth and change in financial technology dynamically (here we mainly focus on SMEs as they are one of the very important classes for enterprises with less information available from financial accounting report and associated assets. Indeed, the approach discussed here is applicable to general enterprises). The key idea of our new approach is to introduce and use the “Hologram” (similar to, “holographic portrait” used in portrait holography), a platform for data fusion dynamically, as a tool and mechanism to describe the dynamic evolution of SMEs based on their business dynamic behavior. Through processing structured and/or unstructured data in terms of “related-party” information sets which analyze (1) “investment” and (2) “management” information provided by SMEs’ business behavior, and extracting “Risk Genes” from complex financial network structures in the business ecosystem, we can establish a “good” or “bad” rating for SMEs by using data fusion dynamically and financial technology. This method to assess SMEs is a new approach to evaluating SMEs’ development dynamically based on the network structure information of enterprise and business behavior. The framework introduced in this paper for the dynamic mechanism of SMEs’ development and evolution allows us to assess the risk of any SMEs (in particular to evaluate SMEs’ loan applications) even not available for critical data required in traditional finance analysis including information such as financial accounting and associated assets, etc. This new “Hologram” approach for SMEs assessment is a pioneering innovation that incorporates big data and financial technology for inclusive financial services in practical application. Ultimately, the Hologram approach offers a new theoretical solution for the long-standing problem of credit risk assessment for SMEs and individuals in practice. Since the information embedded in SMEs’ business behavior reveals the competition and cooperation mechanism that drives its stochastic resonance (SR) behavior which is associated with successful SMEs development, the two concepts of SAI and URR under the Hologram approach to risk assessment that identifies if an SME is “good” are based on the network generated from an SMEs’ related-party information in terms of “investment” and “management” dynamically, along with other available information such as related investment capital and risk control. Significantly, the Hologram approach to risk assessment for SMEs does not require critical data of traditional financial account and related assets, etc. which heavily depend on financial accounting and associated assets used by financial risk analysis in practice. Using big data and FinTech Hologram method discussed in this paper utilizes the related-party information (in term of investment and management) of each SME which exists in an embedded business network to overcome the situation for SMEs which always have not or have not enough in providing accounting and associated asset information in the practice. By the feature of each Hologram for a given SME, one always has the related-party information in terms of either investment, or management dynamically, which is indeed also an explanation for the reason why the new approach proposed only comes true only until the era of big data’s occurring by using ideas from financial technology today. Furthermore, this paper explores the implementation of the “Holo Credit Loan”, a pure credit loan without any collateral and guarantee launched in 2016, as practical applications of the Hologram approach. We illustrate the framework of SMEs risk assessment under the Holograms new theoretical basis for solving the long-standing problem of credit risk assessment for SMEs (and individuals). Moreover, this paper’ conclusion will address the performance of the “Holo Credit Loan”.