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2009, Ernesto Damiani, Ottavio D'Antona, Vincenzo Marra, From Combinatorics to Philosophy: The Legacy of G.-C. Rota, (page 113)#:
In order to set up such an algorithm, we need the notion of multiplicative inverse of an umbra. Two umbrae are said to be multiplicative inverse to each other when αγ ≡ u.
“umbra”, in Kielitoimiston sanakirja [Dictionary of Contemporary Finnish] (in Finnish) (online dictionary, continuously updated), Kotimaisten kielten keskuksen verkkojulkaisuja 35, Helsinki: Kotimaisten kielten tutkimuskeskus (Institute for the Languages of Finland), 2004–, retrieved 2023-07-04
“ vel Pater omnipotēns adigat mē fulmine ad umbrās, pallentīs umbrās Erebī noctemque profundam, .”
“ or the Father almighty hurl me with his thunderbolt to the shades, the pallid shades and boundless night of Erebus .” (That is, the Underworld, or land of the dead. The repetition is an example of epanalepsis.)
“umbra”, in Charlton T. Lewis and Charles Short (1879) A Latin Dictionary, Oxford: Clarendon Press
“umbra”, in Charlton T. Lewis (1891) An Elementary Latin Dictionary, New York: Harper & Brothers
umbra in Gaffiot, Félix (1934) Dictionnaire illustré latin-français, Hachette.
Carl Meißner, Henry William Auden (1894) Latin Phrase-Book, London: Macmillan and Co.
to exert oneself in the schools: desudare in scholae umbra or umbraculis
“umbra”, in Harry Thurston Peck, editor (1898), Harper's Dictionary of Classical Antiquities, New York: Harper & Brothers
^ De Vaan, Michiel (2008) Etymological Dictionary of Latin and the other Italic Languages (Leiden Indo-European Etymological Dictionary Series; 7), Leiden, Boston: Brill, →ISBN, page 639
^ Whitehead, The Sound of Indo-European: Phonetics, Phonemics, and Morphophonemics, p. 13