Archives for Pure Math
Invariant Subspaces — Definition & Examples
Invariant Subspaces — Definition & Examples Invariant Subspaces An invariant subspace is a subspace that a given linear transformation cannot “move you out of.” Definition Let V be a vector…
Partial-Sum Probabilities vs. Bell Correlations
Partial-Sum Probabilities vs. Bell Correlations — Illustrated Partial-Sum Probabilities ⇄ Successive Measurements in Bell Experiments Figure: Left — running-partial-sum Markov transitions; Right — Bell measurement settings and entangled state. (Generated…
Randomy generated numbers – Probability of Running Sums
Draws are independent and uniformly chosen from the set \(\{1,2,\dots,k\}\) (with unlimited repetition). Let \(p(n)\) denote the probability that at some time the running sum equals exactly \(n\). 1. Recurrence…
Uncountability of (0,1): Cantor’s Diagonal Argument
The Reals in \((0,1)\) Are Not Countably Infinite See also - Cardinality of the Rationals - Positive and Negative included Claim The interval \((0,1)\subset \mathbb{R}\) is uncountable. Equivalently, there is…
Cardinality of the Rationals (Including Negative Rationals)
Also read 'Cardinality of the Reals' Do Negative Rationals Change the Cardinality? Short answer: No. Adding negative rationals keeps the set countably infinite, the same cardinality as the integers \(\mathbb{Z}\).…
Godel’s Consistency of Axiom of Choice Paper
Gödel's Landmark Paper The Consistency of the Axiom of Choice and the Generalized Continuum-Hypothesis with the Axioms of Set Theory Background Set Theory and ZFC: The Zermelo-Fraenkel axioms with the…
Construct a conformal equivalence f between the “angle” {z ∈ C | z 6= 0, 0 < arg(z) < π/3} and the unit disk D ⊂ C
Constructing a Conformal Equivalence We aim to construct a conformal equivalence f between the "angle" A = { z ∈ ℂ | z ≠ 0, 0 < arg(z) < π/3…
Nowhere Differentiable Functions and Integration of such functions
Nowhere Differentiable Functions Nowhere differentiable functions are functions that are continuous everywhere but do not have a well-defined derivative at any point. They exhibit erratic behavior, and although they can…
Analytic Within and On the Unit Circle
Analytic Within and On the Unit Circle When discussing whether a function is analytic within or on the unit circle, we are referring to complex analysis, which deals with functions…
Curves Over Finite Fields
Curves Over Finite Fields Curves over finite fields are algebraic curves defined by equations where the coefficients come from a finite field. A finite field is a field with a…