We demonstrate that while the intersection of a finite number of open sets in $\mathbb R$ remains open in $\mathbb R$, the same cannot be universally said for intersections involving an infinite number of open sets. An illustrative example is presented to highlight this distinction.
https://doi.org/10.31219/osf.io/4kjdq
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 5, Article 279, 2023 [LB]
We introduce a set of abstract objects, the so-called ultranumbers, in order to generalize different classes of numbers. The idea is to find equations with nonexistent solutions in the original set, and then use them to extend the numerical system.
https://doi.org/10.31219/osf.io/jrqax
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 5, Article 278, 2023 [LA]
We present nonzero numbers whose square vanishes for pedagogical purposes.
https://doi.org/10.31219/osf.io/ur7me
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 5, Article 277, 2023 [KZ]
[knowledge base]
LATTICES, ORDER and their underlying definitions and theorems are presented in this white paper (knowledge base).
https://doi.org/10.31219/osf.io/fzpc4
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 4, Article 276, 2022 [KX]
We propose the equivalence between one Wolfram model and the qubit permutation semigroup.
https://doi.org/10.31219/osf.io/qkdxv
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 4, Article 270, 2022 [KP]
Based on the idea that time is computation, we discuss one interpretation regarding the inner structure of time that explains quantum superposition.
https://doi.org/10.31219/osf.io/chmqy
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 4, Article 269, 2022 [KO]
We prove a theorem in calculus LJ and in LJT (LJ with stoup) as well. We show that while there are many proofs to one single theorem in LJ, there is exactly one proof in LJT.
https://doi.org/10.31219/osf.io/z6h4q
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 4, Article 268, 2022 [KN]
We prove some theorems in topology using the fewest number of symbols at each step. Our purpose is pedagogical.
https://doi.org/10.31219/osf.io/wn24y
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 4, Article 267, 2022 [KM]
This is an introductory collection of theorems in topology.
https://doi.org/10.31219/osf.io/zm56w
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 4, Article 260, 2022 [KE]
This is a personal collection of definitions and results from first-order logic.
https://doi.org/10.31219/osf.io/8wck9
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 252, 2021 [JX]
An introduction to the language of category theory is presented in a very minimalistic fashion, using the fewest number of symbols as possible. In this white paper, the focus is on the main concepts underlying categories.
https://doi.org/10.31219/osf.io/k9fbc
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 247, 2021 [JR]
[microresearch]
We calculate the Lyapunov exponents of the complex stretch factor $f^-(z)=(1-z^2)^{-1/2}$ from the Lorentz transformations and of its reciprocal $f^+(z)=(1-z^2)^{+1/2}$ as well.
https://doi.org/10.31219/osf.io/jsncm
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 239, 2021 [JJ]
We conjecture that quantum vacuum operates its discrete dynamics in a superposition of a class of iterating functions such that each physical system operates within a distinct function.
https://doi.org/10.31219/osf.io/8f4yg
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 238, 2021 [JI]
[white paper: pedagogical]
For pedagogical purposes, we define a simple language to show that two different structures with the same number of elements in their universes are not isomorphic.
https://doi.org/10.31219/osf.io/ytcru
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 234, 2021 [JE]
[white paper: pedagogical]
We present a pedagogical proof that the function of an isomorphism between two structures is an equivalence relation.
https://doi.org/10.31219/osf.io/7uwnh
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 233, 2021 [JD]
[white paper: pedagogical]
We show using induction on complexity that all terms of a language with one constant, one binary function, and one 4-ary function have an odd number of symbols.
https://doi.org/10.31219/osf.io/ue32a
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 231, 2021 [JB]
[white paper: pedagogical]
We prove that if A is a set consisting of n elements, then A has 2^n subsets.
https://doi.org/10.31219/osf.io/mptx7
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 230, 2021 [JA]
[knowledge base]
LINEAR ALGEBRA and its underlying definitions are presented in this white paper (knowledge base).
https://doi.org/10.31219/osf.io/8emp4
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 225, 2021 [IV]
[knowledge base]
LINEAR TRANSFORMATION and its underlying definitions are presented in this white paper [knowledge base (http://omkb.org)].
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 224, 2021 [IU]
[knowledge base]
CAYLEY’s THEOREM, the SYMMETRIC GROUP THEOREM, and their underlying definitions are presented in this white paper (knowledge base = http://omkb.org).
https://doi.org/10.31219/osf.io/63pmy
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 223, 2021 [IT]
[pedagogical]
We translate the proof of the theorem stated in the title, accomplished by Prover9, into a human readable form.
https://doi.org/10.31219/osf.io/wsbne
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 222, 2021 [IS]
Open Journal of Mathematics and Physics 