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r/3 are multiplicative inverses.
In 1940, an algebraist named Graham Higman made a daring conjecture in his doctoral thesis: The worst of this cancellation weirdness, he proposed, will only happen if the group that is used to construct the group algebra contains elements for which some power equals 1, as with
r in the example above. In all other group algebras, he posited, while elements with just a single term, like 7
a or 8
r + 2 or 3
r â 5
s can never have multiplicative inverses. Since elements with multiplicative inverses are called units, Higmanâs hypothesis came to be known as the unit conjecture.
Over the next few decades, Irving Kaplansky, one of the leading mathematicians of the 20th century, popularized this conjecture along with two other group algebra conjectures called the zero divisor and idempotent conjectures; the three came to be known as the Kaplansky conjectures. Collectively, the three conjectures posit that group algebras are not too radically
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Latest coronavirus news as of 5pm on 13 April
There have been six reports of rare blood clots among more than 6.8 million people in the US who received the Johnson & Johnson vaccine
US health authorities, including the Food and Drug Administration (FDA) and the Centers for Disease Control and Prevention (CDC), recommended a pause in the use of the Johnson & Johnson covid-19 vaccine on 13 April as a precautionary measure, following reports of rare blood clots in six people who had received the vaccine. More than 6.8 million doses of the single-shot vaccine had been administered across the US as of 13 April. Among these, there were six reports of a rare blood clotting condition called cerebral venous sinus thrombosis (CVST), which affects blood vessels in the brain, all of which were among women aged 18 to 48. A special meeting of the CDC’s Advisory Committee on Immunization Practices will review the reports on 14 April,
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