The chloride derivative [ Th ( C 5 H 5 ) 3 Cl ] is prepared by heating thorium tetrachloride with limiting K ( C 5 H 5 ) used ( other univalent metal cyclopentadienyls can also be used ).
42.
(i ) by substitution in the phenyl ring to any extent with alkyl, alkoxy, alkylenedioxy, haloalkyl or halide substituents, whether or not further substituted in the phenyl ring by one or more other univalent substituents;
43.
By Hurwitz's theorem, since each " g " " n " is univalent and normalized, i . e . fixes 0 and has derivative 1 there, their limit is also univalent.
44.
By Hurwitz's theorem, since each " g " " n " is univalent and normalized, i . e . fixes 0 and has derivative 1 there, their limit is also univalent.
45.
The book produced by participants in the IAS program was titled " Homotopy type theory : Univalent foundations of mathematics "; although this could refer to either usage, since the book only discusses HoTT as a mathematical foundation.
46.
An account of Voevodsky's construction of a univalent model of the Martin-L�f type theory with values in Kan simplicial sets can be found in a paper by Chris Kapulkin, Peter LeFanu Lumsdaine and Vladimir Voevodsky.
47.
He also proved, using an idea of A . K . Bousfield, that this universal fibration was univalent : the associated fibration of pairwise homotopy equivalences between the fibers is equivalent to the paths-space fibration of the base.
48.
Based on this work other models with non-trivial identity types were studied, including homotopy type theory which has been proposed as a foundation for mathematics in Vladimir Voevodsky's research program " Univalent Foundations of Mathematics ".
49.
Where " c " ( H + ) denotes the amount-of-substance concentration of hydrogen ion H + and " y " 1 denotes the activity coefficient of a typical uni-univalent electrolyte in the solution.
50.
It is claimed by the creators of the univalent foundations that the univalent formalization of sets in the Martin-L�f type theory is the best available today environment for formal reasoning about all aspects of set-theoretical mathematics, both constructive and classical.
