The present paper introduces a method of basis transformation of vector fields that is specifically applicable to polynomials space and differential equations with certain polynomials solutions such as Hermite, Laguerre and Legendre polynomials. The method based on separated transformation of vector space basis by a set of operators that are equivalent to the formal basis transformation and connected to it by linear combination with projection operators. Applying the Forbenius covariants yields a general method that incorporates the Rodrigues formula as a special case in polynomial space. Using the Lie algebra modules, specifically , on polynomial space results in isomorphic algebras whose Cartan sub-algebras are Hermite, Laguerre and Legendre differential operators. Commutation relations of these algebras and Baker-Campbell-Hausdorff formula gives new formulas for special polynomials

While the language model using the stop sign as an independent token has been widely used to decide when the model should stop, it may lead to the growth of vocabulary dimensions and further problems. Similarly, present research on game algorithms usually estimate stopping point related problems based on the evaluation of the winning rate. However, information redundancy may also exist in such models, thus increasing the training difficulty. Above two types of tasks (and similar autoregressive tasks) show a common problem of stopping point prediction. In this paper, we describe a design of separated model, trying to separate the complexity of stopping point prediction from the main task model, so that the information used for estimating stopping point can be reduced. On this basis, in order to verify the rationality of using separated model, we propose a model-free test method. It judges the separability of transformed data based on point difference and sequence difference metrics. In this way, it can predict the credibility of the separated model inference.

The goal of newborn screening is to improve health outcomes by identifying and treating affected newborns. This manuscript provides an overview of a data tool to facilitate the longitudinal collection of health information on newborns diagnosed with a condition through NBS. The Newborn Screening Translational Research Network (NBSTRN) developed the Longitudinal Pediatric Data Resource (LPDR) to capture, store, analyze, visualize, and share genomic and phenotypic data over the lifespan of NBS identified newborns to facilitate understanding of genetic disease, and to assess the impact of early identification and treatment. NBSTRN developed a consensus-based process using clinical care experts to create, maintain, and evolve question and answer sets organized into common data elements (CDEs). The LPDR contains 24,172 core and disease-specific CDEs for 118 rare genetic diseases, and the CDEs are being made available through the NIH CDE Repository. The number of CDEs for each condition average of 2,200 with a range from 69 to 7,944. The LPDR is used by state NBS programs, clinical researchers, and community-based organizations. Case level, de-identified data sets are available for secondary research and data mining. The development of the LPDR for longitudinal data gathering, sharing, and analysis supports research and facilitates the translation of new discoveries into clinical practice.