Real Vector Space and Related NotionsThis study was supported in part by JSPS KAKENHI Grant Numbers 17K00182 and 20K19863.

Summary. In this paper, we discuss the properties that hold in finite dimensional vector spaces and related spaces. In the Mizar language [1], [2], variables are strictly typed, and their type conversion requires a complicated process. Our purpose is to formalize that some properties of finite dimensional vector spaces are preserved in type transformations, and to contain the complexity of type transformations into this paper. Specifically, we show that properties such as algebraic structure, subsets, finite sequences and their sums, linear combination, linear independence, and affine independence are preserved in type conversions among TOP-REAL(n), REAL-NS(n), and n-VectSp over F Real. We referred to [4], [9], and [8] in the formalization.

Molecular spaces and the dimension paradox

In this study, the dimension or dimensionality paradox is defined and discussed in a dedicated context. This paradox appears when discrete vector representations of the elements of a molecular set are constructed employing several descriptor parameters, adopting specific values for each molecule. The dimension paradox consists in that when constructing discrete N-dimensional molecular vectors, the primal structure of the attached molecular set, chosen as a collection of different objects, cannot be well-defined if the number of descriptors N and the number of molecules M do not bear a convenient relation like: N ≥ M $N\ge M$ . This has implications for the linear independence of the vectors connected with each molecule.

Linear Independence of T-Spline Blending Functions of Degree One for Isogeometric Analysis

Linear independence of the blending functions is a necessary requirement for T-spline in isogeometric analysis. The main work in this paper focuses on the analysis about T-splines of degree one, we demonstrate that all the blending functions of such T-spline of degree one are linearly independent. The advantage owned by one degree T-spline is that it can avoid the problem of judging whether the model is analysis-suitable or not, especially for occasions that need a quick response from the analysis results. This may provide a new way of using T-spline for a CAD and CAE integrating scenario, since one degree T-spline still guarantees the topology flexibility and is compatible with the spline-based modeling system. In addition, we compare the numerical approximations of isogeometric analysis and finite element analysis, and the experiment indicates that isogeometric analysis using T-spline of degree one can reach a comparable result with classical method.

ELLIPTIC CURVES AND -ADIC ELLIPTIC TRANSCENDENCE

We prove a necessary and sufficient condition for isogenous elliptic curves based on the algebraic dependence of p-adic elliptic functions. As a consequence, we give a short proof of the p-adic analogue of Schneider’s theorem on the linear independence of p-adic elliptic logarithms of algebraic points on two nonisogenous elliptic curves defined over the field of algebraic numbers.