User talk:LokiClock/Sandbox

Snippets

{{Disambiguation needed}}

{{anchor|Measure spaces}}

A tuple|triple $\left(X,\Sigma ,\mu \right)$ is called a {{visible anchor|measure space}}.

transitive functional dependent

{{contradict-other-multiple|Permutation|Lehmer code|Factorial number system|date=March 2013}}

Old Norse morphology

Strong nouns

Fem. A declension

plain ā^3.3.2 - gjǫf -ing, -ung (in plural — i-decl. in sg.)

-j- (Gothic jō cog.s sibja-sif (this stem in pl. only), mawi-mær, and þiwi-þýr) fit hęl Frigg ey

-r = Gothic feminines in -is = stem(brúðr,bruþs,*brudhiz) assum. iyā stems i-umlauted Dat.&Acc. sg., as well for most roots ęrmr from armr heiðr heiðar heiði heiði kýr and other contracted roots -uðr/-unnr, -unn (e.g. Iðunn), -dís

-v- (assum. wā) ǫr(var) stǫð(var) bǫð(var) gǫt(var)

Strong verbs

Morpho tables

Note the reversal of mood and tense in the column hierarchy.

Inflection (-suff) & umlaut (ę←a→ǫ) in a Norse strong verb
		Inf	-a	Imp	∅	Pr P	-andi
		Pa P
		N	-it	M	-inn	F	-in
		ek	þú	þat	vér	þér	þau
Ind	Pr T	ę	ę-r		ǫ-um	-ið	-a
	Pa T	∅	-t	∅	ǫ-um	ǫ-uð	ǫ-u
Subj	Pr T	-a	-ir	-i	-im	-ið	-i
	Pa T	ę-a	ę-ir	ę-i	ę-im	ę-ið	ę-i

-ra verbs

Sweet gives Pa T S sló, P slógu, Pa P slęginn for slá.

Unsourced past tense forms very likely, given the declension's name and the fact that the first form of a strong verb used to specify the conjugation in C-V entries is most often the past 1st-person sg. (e.g. bera)

Reflexives of contracted -ra can be constructed from part B of the entry for slá.

Table of inflections

Image - Morpho stem map

	Infl.	Root/Infl.
	-um	ę	-r
Noun	dat pl
Adj	dat pl, masc dat sg
Verb	Ind 1st plur

Articles needing diagrams

• Cauchy–Riemann equations - orthogonality
• Plate trick - cone between shoulder's (identity) and hand's rotation as a homotopy

List of opposite categories

See: Presheaf (category theory), Opposite category, Dagger category, Spectrum of a ring#Functoriality, Chu space, Categorical algebra#Dual, Pontryagin duality#Categorical considerations, Category of relations, Grassmanian Algebras w/ Superpoints

Piecing things together

Monoidal category:

1. R-Mod, the category of modules over a commutative ring R, is a monoidal category with the tensor product of modules ⊗R serving as the monoidal product and the ring R (thought of as a module over itself) serving as the unit. As special cases one has:

• K-Vect, the category of vector spaces over a field K, with the one-dimensional vector space K serving as the unit.
• Ab, the category of abelian groups, with the group of integers Z serving as the unit.

Generator (category theory):

• In the category of abelian groups, the group of integers $\mathbf {Z}$ is a generator: If f and g are different, then there is an element $x\in X$, such that f(x)≠g(x). Hence the map $\mathbf {Z} \rightarrow X,n\mapsto n\cdot x$ suffices.

Group scheme:

"There are cases of intermediate abstraction, such as commutative algebraic groups over a field, where Cartier duality gives an antiequivalence with commutative affine formal groups." "Dieudonne and Cartier constructed an antiequivalence of categories between finite commutative group schemes over k of order a power of "p" and modules over D with finite W(k)-length."