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In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables (,) = + +,where a, b, c are the coefficients.When the coefficients can be arbitrary complex numbers, most results are not specific to the case of two variables, so they are described in quadratic form.
2,1,3-Benzothiadiazole has been known since the 19th century. It is readily prepared in at least 85% yield from o-phenylenediamine by reaction with two equivalents of thionyl chloride in pyridine. The by-products are sulfur dioxide and HCl. [1] There are a number of alternative methods used to make this heterocycle and these have been reviewed.
In the absence of protic solvent, this dimer collapses to form 1,2-diols. In the presence of a proton source, however, the dimer may undergo either disproportionation to form a samarium alkoxide and carbonyl compound, or protonation to form a carbinol radical ii followed by a second reduction and protonation, yielding an alcohol. (5)
In mathematics, in number theory, Gauss composition law is a rule, invented by Carl Friedrich Gauss, for performing a binary operation on integral binary quadratic forms (IBQFs). Gauss presented this rule in his Disquisitiones Arithmeticae , [ 1 ] a textbook on number theory published in 1801, in Articles 234 - 244.
The compound can effect regioselective conjugate reductions of various carbonyl derivatives including unsaturated aldehydes, ketones, and esters.This reagent was assigned as the "Reagent of the year" in 1991 for its functional group tolerance, high overall efficiency, and mild reaction conditions in the reduction reactions.
Samarium(II) iodide is an inorganic compound with the formula SmI 2. When employed as a solution for organic synthesis, it is known as Kagan's reagent. SmI 2 is a green solid and forms a dark blue solution in THF. [1] It is a strong one-electron reducing agent that is used in organic synthesis.
The CBS reduction has proven to be an effective and powerful method to reduce a wide range of different types of ketones in both a stereoselective and chemoselective manner. Substrates include a large variety of aryl-aliphatic, di-aliphatic, di-aryl, α,β unsaturated enone and ynone systems, as well as ketones containing heteroatoms .
Over F 2, the Arf invariant is 0 if the quadratic form is equivalent to a direct sum of copies of the binary form , and it is 1 if the form is a direct sum of + + with a number of copies of . William Browder has called the Arf invariant the democratic invariant [ 3 ] because it is the value which is assumed most often by the quadratic form. [ 4 ]