Generalized de Boor–Cox Formulas and Pyramids for Multi-Degree Spline Basis Functions

The conventional B-splines possess the de Boor–Cox formula, which relates to a pyramid algorithm. However, for multi-degree splines, a de Boor–Cox-type evaluation algorithm only exists in some special cases. This paper considers any multi-degree spline with arbitrary degree and continuity, and provides two generalized de Boor–Cox-type relations. One uses several lower degree polynomials to build a combination to evaluate basis functions, whose form is similar to using the de Boor–Cox formula several times. The other is a linear combination of two functions out of the recursive definition, which keeps the combination coefficient polynomials of degree 1, so it is more similar to the de Boor–Cox formula and can be illustrated by several pyramids with different heights. In the process of calculating the recursions, a recursive representation using the Bernstein basis is used and numerically analyzed

Integrated Anodes from Heteroatoms (N, S, and F) Co-Doping Antimony/Carbon Composite for Efficient Alkaline Ion (Li⁺/K⁺) Storage

Sb-based materials are widely used in alkali metal-ion batteries due to large specific capacity, low cost, and a suitable operating voltage platform. However, they suffer from tardy dynamic performance and structural instability, along with a vague storage mechanism for alkali metal ion (Li+/K+) confining their further popularization. Herein, a scalable strategy is proposed to make Sb nanoparticles encapsulated in the N, S, and F co-doping carbon skeleton (Sb@NSF-C) with a three-dimensional ordered hierarchical porous structure. The Sb@NSF-C composite shows remarkable electrochemical performances in lithium-ion batteries (LIBs) and potassium-ion batteries (PIBs) due to improved Li+/K+ diffusion kinetics, rapid ion transport path, and stable structure. The Li+/K+ storage mechanism is detailedly investigated for the Sb@NSF-C composite. This work may provide a feasible method to design electrode materials for LIBs and PIBs with excellent performances