Real coordinate space - Wikipedia

Real coordinate space - Wikipedia: "it is the prototypical real vector space and is a frequently used representation of Euclidean n-space."



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Euclidean space - Wikipedia

Euclidean space - Wikipedia: "A Euclidean space is not technically a vector space but rather an affine space,"



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Unit sphere - Wikipedia

Unit sphere - Wikipedia: " forms[edit source]
If V is a linear space with a real quadratic form F:V → R, then { p ∈ V : F(p) = 1 } may be called the unit sphere[1][2] or unit quasi-sphere of V."



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Orthogonal group - Wikipedia

Orthogonal group - Wikipedia: "The term "orthogonal group" may also refer to a generalization of the above case: the group of invertible linear operators that preserve a non-degenerate symmetric bilinear form or quadratic form[1] on a vector space over a field. In particular, when the bilinear form is the scalar product on the vector space F n of dimension n over a field F, with quadratic form the sum of squares, then the corresponding orthogonal group, denoted O(n, F ), "



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Symmetry group - Wikipedia

Symmetry group - Wikipedia: "The proper symmetry group is then a subgroup of the special orthogonal group SO(n), and is therefore also called rotation group of the figure."



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Symmetry group - Wikipedia

Symmetry group - Wikipedia: "The proper symmetry group of an object is equal to its full symmetry group if and only if the object is chiral (and thus there are no orientation-reversing isometries under which it is invariant)."



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Symmetry group - Wikipedia

Symmetry group - Wikipedia: "The proper symmetry group of an object is equal to its full symmetry group if and only if the object is chiral (and thus there are no orientation-reversing isometries under which it is invariant)."



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Permutation group - Wikipedia

Permutation group - Wikipedia: "The group of all permutations of a set M is the symmetric group of M, often written as Sym(M)."



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Group action - Wikipedia

Group action - Wikipedia: " Actions of groups on vector spaces are called representations of the group."



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Introduction

Introduction: "The Lie algebra is a vector space, but it has additional structure:"



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Categories

Categories: "Group representations are really only a special case of category representations."



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Symmetry (physics) - Wikipedia

Symmetry (physics) - Wikipedia: "Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group)."



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Introduction to Category Theory/Categories - Wikiversity

Introduction to Category Theory/Categories - Wikiversity: "Monoids in general have one feature that free categories can lack, in that different compositions of arrows can result in the same monoid element. This can happen with categories too, but not with free ones."



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Introduction to Category Theory/Sets and Functions - Wikiversity

Introduction to Category Theory/Sets and Functions - Wikiversity: "For every set X and every two functions {\displaystyle f_{1},f_{2}:X\to Y} the following holds: {\displaystyle g\circ f_{1}=g\circ f_{2}} if and only if {\displaystyle f_{1}=f_{2}} . In this case, the function g is called monic."



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Category (mathematics) - Wikipedia

Category (mathematics) - Wikipedia: "The category Cat consists of all small categories, with functors between them as morphisms.

"



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Category

Read $centerdot as “after” and you'll be able to avoid treating a categorical arrow as a function.
ob (C) and hom (C) are complementary and conjugate.

Monoid as a single object category - Mathematics Stack Exchange

Monoid as a single object category - Mathematics Stack Exchange: "But now if we just observe that the operations on MM are exactlyexactly the same as the operations on Morph(C)Morph(C), we may regard the category CC as the monoid MM. "



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Category (mathematics) - Wikipedia

Category (mathematics) - Wikipedia: "a small category with a single object x. (Here, x is any fixed set."



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word choice - "As following" vs "as follows" - English Language Learners Stack Exchange

word choice - "As following" vs "as follows" - English Language Learners Stack Exchange: "as follows is always singular"



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Category (mathematics) - Wikipedia

Category (mathematics) - Wikipedia: "Any monoid can be understood as a special sort of category (with a single object whose self-morphisms are represented by the elements of the monoid), and so can any preorder.

"



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SpaceWeather.com -- News and information about meteor showers, solar flares, auroras, and near-Earth asteroids

SpaceWeather.com -- News and information about meteor showers, solar flares, auroras, and near-Earth asteroids: "no other spacecraft could match--before or since,"



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Categories

Categories: "spinors (fancy for 4-tuples of complex numbers),"



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Categories

Categories: "the fundamental group is really a functor from the category Top to the category Group."



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Topological group - Wikipedia

Topological group - Wikipedia:



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Group action - Wikipedia

Group action - Wikipedia: "When there is a natural correspondence between the set of group elements and the set of space transformations, a group can be interpreted as acting on the space in a canonical way."



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Topological group - Wikipedia

Topological group - Wikipedia: "The real numbers form a topological group under addition"



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Categories

Categories: "Diff - smooth manifolds as objects, smooth maps as morphisms
"



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space-time-wave

For any conjugate pair, there's some constant (could they be a smooth function) which relates one to the other. Waves show up representing some number which relates some space and time aspects of that media.

Mathematical structure - Wikipedia

Mathematical structure - Wikipedia: "Examples are homomorphisms, which preserve algebraic structures; homeomorphisms, which preserve topological structures; and diffeomorphisms, which preserve differential structures."



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Join and meet - Wikipedia

Join and meet - Wikipedia: "A partially ordered set that is both a join-semilattice and a meet-semilattice is a lattice"



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Ubiquitous octonions | plus.maths.org

Ubiquitous octonions | plus.maths.org: "bounce back and forth between visualising things and scribbling formulas on paper, and neither one by itself is good enough"



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Vector space - Wikipedia

Vector space - Wikipedia: "multiplied ("scaled") by numbers, called scalars."



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Form - Wikipedia

Form - Wikipedia: "Differential form, a concept from differential topology that combines multilinear forms and smooth functions"



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Homogeneous polynomial - Wikipedia

Homogeneous polynomial - Wikipedia: "A form is also a function defined on a vector space,"



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Ring (mathematics) - Wikipedia

Ring (mathematics) - Wikipedia: "A commutative ring such that every nonzero element has a multiplicative inverse is called a field."



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Division algebra - Wikipedia

Division algebra - Wikipedia: "constructing a division algebra of three dimensions"



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Group action - Wikipedia

Group action - Wikipedia: "Actions of groups on vector spaces are called representations of the group."



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