User:Tokenzero/sandbox

User:Jorge_Stolfi/DoW Wall Street Journal: Wikipedia loosing Volunteers

Lemma. For any two elements a and b (st. ab=ba), of order i and j, there is an element of order lcm(i,j). Proof: We can assume $i\bot j$, since otherwise we can replace a with an element of order $i/{\text{lcd}}(i,j)$, namely $a^{{\text{lcd}}(i,j)}$. Consider ord(ab)=k. k divides ij. a^k=(b^{-1})^k where b^{-1} has also order j. ord(a^k) divides i = ord(b^k) = divides j, so a^k=b^k=1. So i divides k and j divides k.

• Degeneracy (graph theory) (e.g. O(d)-time adjacency structure).
• De_Bruijn_graph, Line graph
• non-trivial "densest subgraph" algorithms, see non-dynamic refs in Near-Optimal Fully Dynamic Densest Subgraph
• Wikipedia_talk:WikiProject_Mathematics/Typography
• max-width
• PageRank

• fokonolona (maybe also Balinese 'seka')

• Modern approaches to early Islamic history – The New Cambridge History of Islam

Arabic

Arabic letters usage in Literary Arabic

Trans- literation	Value in Literary Arabic (IPA)	Closest English equivalent in pronunciation	Contextual forms			Isolated form
			Final	Medial	Initial
ā / ʾ (also â )	various, including /aː/	car, father	ـا		ا
b	/b/	barn	ـب	ـبـ	بـ	ب
t	/t/	stick	ـت	ـتـ	تـ	ت
th (also ṯ )	/θ/	think, thumb	ـث	ـثـ	ثـ	ث
j (also ǧ )	/d͡ʒ/	Jim, gym	ـج	ـجـ	جـ	ج
ḥ (also ḩ )	/ħ/	no equivalent ("guttural" h, may be approximated as heart)	ـح	ـحـ	حـ	ح
kh (also ḫ, ḵ, ẖ )	/x/	loch; German Bach	ـخ	ـخـ	خـ	خ
d	/d/	dear	ـد		د
dh (also ḏ )	/ð/	that, them, though	ـذ		ذ
r	/r/	rap	ـر		ر
z	/z/	zebra	ـز		ز
s	/s/	sin	ـس	ـسـ	سـ	س
sh (also š )	/ʃ/	shin	ـش	ـشـ	شـ	ش
ṣ (also ş )	/sˤ/	swarm, bass	ـص	ـصـ	صـ	ص
ḍ (also ḑ )	/dˤ/	dwarf	ـض	ـضـ	ضـ	ض
ṭ (also ţ )	/tˤ/	star	ـط	ـطـ	طـ	ط
ẓ (also z̧ )	/ðˤ/	buzzword	ـظ	ـظـ	ظـ	ظ
ʿ	/ʕ/	no equivalent ("guttural" voiced h; similar to ḥāʾ above)	ـع	ـعـ	عـ	ع
gh (also ġ, ḡ )	/ɣ/	Spanish higo	ـغ	ـغـ	غـ	غ
f	/f/	far	ـف	ـفـ	فـ	ف
q (also q )	/q/	scar	ـق	ـقـ	قـ	ق
k	/k/	key, cap	ـك	ـكـ	كـ	ك
l	/l/	lamp	ـل	ـلـ	لـ	ل
m	/m/	me	ـم	ـمـ	مـ	م
n	/n/	nun	ـن	ـنـ	نـ	ن
h	/h/	hat	ـه	ـهـ	هـ	ه
w / ū	/w/, /uː/	we	ـو		و
y / ī	/j/, /iː/	yacht	ـي	ـيـ	يـ	ي
ʾ	/ʔ/	uh–oh (aka "glottal stop")	ء (used mainly in medial and final position, which is an unlinked letter)
			ـأ		أ
			ـإ		إ
			ـؤ		ؤ
			ـئ	ـئـ	ئـ	ئ
ʾā	/ʔaː/		ـآ		آ