Vol 11, No 3 (2019)

The necessary and sufficient conditions under which a given family ℱ of subsets of finite set X coincides with the family B_X of all balls generated by some ultrametric d on X are found. It is shown that the representing tree of the ultrametric space (B_X, d_H) with the Hausdorff distance d_H can be obtained from the representing tree T_X of ultrametric space (X, d) by adding a leaf to every internal vertex of T_X.

In this paper, we develop the foundation of a new mathematical language, which we term “Soft Logic”. This language enables us to present an extension of the number 0 from a singular point to a continuous line. We create a distinction between −0 and +0 and generate a new type of numbers, which we call ‘Bridge Numbers’ (BN):

\({\boldsymbol{a}}\overline {\bf{0}} \bot {\boldsymbol{b}}\overline {\bf{1}} ,\)

where a, b are real numbers, “a” is the value on the \(\overline {\bf{0}} \) axis, and “b” is the value on the \(\overline {\bf{1}} \) axis. We proceed by defining arithmetic and algebraic operations on the Bridge Numbers, investigate their properties, and conclude by defining goals for further research. In the Attachment, we continue comparing our results with existing mathematical work on Infinitesimals, Dual numbers, and Nonstandard analysis. The research is a part of “Digital living 2030” project with Stanford University.

We investigate the spectral geometry and spectral action functionals associated to 1D Supersymmetry Algebras, using the classification of these superalgebras in terms of Adinkra graphs and the construction of associated dessin d’enfant and origami curves. The resulting spectral action functionals are computed in terms of the Selberg (super) trace formula and of a Poisson summation formula, respectively.