27 Matching Annotations
  1. Apr 2023
  2. Oct 2021
    1. Alicia Boole Stott

      Alicia was the only Boole sister to inherit the mathematical career of her parents, although her mother Mary Everest Boole had brought up all of her five children from an early age 'to acquaint them with the flow of geometry' by projecting shapes onto paper, hanging pendulums etc. She was first exposed to geometric models by her brother-in-law Charles Howard Hinton when she was 17, and developed the ability to visualise in a fourth dimension. She found that there were exactly six regular polytopes in four dimensions and that they are bounded by 5, 16 or 600 tetrahedra, 8 cubes, 24 octahedra or 120 dodecahedra.

  3. Mar 2021
    1. In computer science, a tree is a widely used abstract data type that simulates a hierarchical tree structure

      a tree (data structure) is the computer science analogue/dual to tree structure in mathematics

  4. Feb 2021
    1. In fact, the Product comonad is just the dual of the Writer monad and effectively the same as the Reader monad (both discussed below)
  5. Jul 2020
    1. In logic, functions or relations A and B are considered dual if A(¬x) = ¬B(x), where ¬ is logical negation. The basic duality of this type is the duality of the ∃ and ∀ quantifiers in classical logic. These are dual because ∃x.¬P(x) and ¬∀x.P(x) are equivalent for all predicates P in classical logic
    2. the ∧ and ∨ operators are dual in this sense, because (¬x ∧ ¬y) and ¬(x ∨ y) are equivalent. This means that for every theorem of classical logic there is an equivalent dual theorem. De Morgan's laws are examples
  6. May 2020
    1. Related concepts in other fields are: In natural language, the coordinating conjunction "and". In programming languages, the short-circuit and control structure. In set theory, intersection. In predicate logic, universal quantification.

      Strictly speaking, are these examples of dualities (https://en.wikipedia.org/wiki/Duality_(mathematics))? Or can I only, at strongest, say they are analogous (a looser coonection)?

    1. Mathematically speaking, necessity and sufficiency are dual to one another. For any statements S and N, the assertion that "N is necessary for S" is equivalent to the assertion that "S is sufficient for N".
    1. In mathematical contexts, duality has numerous meanings[1] although it is "a very pervasive and important concept in (modern) mathematics"[2] and "an important general theme that has manifestations in almost every area of mathematics".[3]
  7. Apr 2020
    1. the phrase up to is used to convey the idea that some objects in the same class — while distinct — may be considered to be equivalent under some condition or transformation
    2. "a and b are equivalent up to X" means that a and b are equivalent, if criterion X, such as rotation or permutation, is ignored
    1. If solutions that differ only by the symmetry operations of rotation and reflection of the board are counted as one, the puzzle has 12 solutions. These are called fundamental solutions; representatives of each are shown below
  8. Jul 2019
    1. In Hardy's words, "Exposition, criticism, appreciation, is work for second-rate minds. [...] It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done."

      similar to Nassim Taleb's "History is written by losers"

  9. Jan 2019
    1. A rooted binary tree is full if every vertex has either two children or no children.

      Example of Catalan Numbers use case.

    1. Catalan numbers notation and short explanation of it.

      Use - LookUP: Combinatorics (non crossing combinations) ex: ((())), ()(()), ()()(), (())(), (()())

      ref: https://www.geeksforgeeks.org/program-nth-catalan-number/

  10. Sep 2016
  11. Sep 2013
    1. A computable Dedekind cut is a computable function which when provided with a rational number as input returns or ,

      This definition of computable Dedekind cut is wrong. The correct definition is that the lower and the upper cut be computably enumerable.