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]
[knowledge base]
The MEMBERSHIP RELATION and its underlying definitions are presented in this white paper (knowledge base).
https://doi.org/10.31219/osf.io/r9p2m
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 221, 2021 [IR]
[knowledge base]
MODULES over RINGS and its underlying definitions are presented in this white paper (knowledge base).
https://doi.org/10.31219/osf.io/xjsc7
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 220, 2021 [IQ]
[knowledge base]
VECTOR SPACE OVER A FIELD and its underlying definitions are presented in this white paper (knowledge base).
https://doi.org/10.31219/osf.io/g6ux5
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 219, 2021 [IP]
[knowledge base]
FIELD, three propositions, and their underlying definitions are presented in this white paper (knowledge base).
https://doi.org/10.31219/osf.io/wmjst
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 217, 2021 [IN]
[knowledge base]
RING, commutative ring, almost a ring, semiring, zero ring, zero property, zero divisors, domain, integral domain, and their underlying definitions are presented in this white paper (knowledge base).
https://doi.org/10.31219/osf.io/bzugr
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 216, 2021 [IM]
[knowledge base]
CYCLIC GROUP and its underlying definitions are presented in this white paper (knowledge base).
https://doi.org/10.31219/osf.io/nvkab
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 215, 2021 [IL]
[knowledge base]
STRONG INDUCTION and its underlying definitions are presented in this white paper (knowledge base).
https://doi.org/10.31219/osf.io/jysg5
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 213, 2021 [IJ]
[knowledge base]
SUBSTITUTIONS, SUBSTITUTABILITY, and their underlying definitions are presented in this white paper (knowledge base).
https://doi.org/10.31219/osf.io/k6w7u
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 212, 2021 [II]
We present a discussion on the definition of the positional argument and the continuum hypothesis.
https://doi.org/10.31219/osf.io/tvg64
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 211, 2021 [IH]
We prove Cantor-Schröder-Bernstein theorem using the diagonal argument.
https://doi.org/10.31219/osf.io/2qkpx
Open Mathematics Collaboration
Op. J. Math. Phys.
Volume 3, Article 210, 2021 [IG]
