Almost commuting unitary matrices related to time reversal

The behavior of fermionic systems depends on the geometry of the system and the symmetry class of the Hamiltonian and observables. Almost commuting matrices arise from band-projected position observables in such systems. One expects the mathematical behavior of almost commuting Hermitian matrices to depend on two factors. One factor will be the approximate polynomial relations satisfied by the matrices. The other factor is what algebra the matrices are in, either the matrices over A for A the real numbers, A the complex numbers or A the algebra of quaternions. There are potential obstructions keeping k-tuples of almost commuting operators from being close to a commuting k-tuple. We consider two-dimensional geometries and so this obstruction lives in KO_{-2}(A). This obstruction corresponds to either the Chern number or spin Chern number in physics. We show that if this obstruction is the trivial element in K-theory then the approximation by commuting matrices is possible.Comment: 33 pages, 2 figures. In version 2 some formulas have been corrected and some proofs have been rewritten to improve the expositio

Sensibilidade de cultivares de soja (Glycine max) aos herbicidas diclosulam e flumetsulam Soybean (Glycine max) cultivars sensibility to the herbicides diclosulam and flumetsulam

O objetivo deste trabalho foi verificar, no campo, no município de Londrina-PR, a resposta de 12 cultivares de soja em relação aos herbicidas diclosulam e flumetsulam aplicados e incorporados em Latossolo Roxo distrófico (80,84 % de argila, pH 5,4 e matéria orgânica 3,07 %). As cultivares foram: 'FT-Guaira', 'Embrapa-48', 'Ocepar-14', 'BR-16', 'Embrapa-4', 'Ocepar-13', 'BR-36', 'FT10-Princesa', 'FT-2000', 'FTAbyara', 'BR-37' e 'RS7- Jacuí'. O diclosulam foi utilizado nas doses de 35 g/ha (normal) e 70 g/ha (dobrada); e o flumetsulam nas doses de 120 g/ha (normal) e 240 g/ha (dobrada); também foi utilizada a mistura de diclosulam + flumetsulam (20 + 100 g/ha); e imazaquin à 150 g/ha. Nas doses normais utilizadas dos dois herbicidas não foi possível distinguir nenhuma cultivar suscetível. Em dose dobrada do diclosulam os rendimentos das cultivares 'FT-Guaira', 'Embrapa 4' e 'BR-37 ' foram reduzidos em 20,9 %, 11,8 % e 11,0 % respectivamente, em relação à testemunha. O flumetsulam à 240 g/ha e sua mistura com diclosulam também reduziu o rendimento destas cultivares, mais o da 'Ocepar-14'. A maioria das cultivares teve o estande reduzido, nas doses dobradas dos dois herbicidas, mas também sem significância. Somente o estande da cultivar 'Embrapa-48' não foi afetado, ao contrário da 'RS-7 Jacuí'. A altura de plantas foi menor em relação à testemunha no tratamento com flumetsulam à 240 g/ha. Na maioria das cultivares o peso da biomassa seca de nódulos foi menor nas doses dobradas, e em algumas também o peso por unidade de nódulos. A biomassa seca da parte aérea e da raiz não foram reduzidas.<br>The objective of the present paper was to check the behaviour of 12 different soybean cultivars in relation to the tolerance to the herbicides flumetsulam and diclosulam applied on a soil classified as Latossolo Roxo distrófico (80,87% of clay, pH of 5,4 and 3,07 of organic matter) in Londrina - PR. The cultivars were: 'FTGuaira', 'Embrapa-48', 'Ocepar-14', 'BR-16', 'Embrapa-4', 'Ocepar-13', 'BR-36', 'FT 10- Princesa', 'FT-2000', 'FT-Abyara', 'BR-37' and 'RS 7-Jacuí'. Diclosulam was used in the rates of 35 g/ha (normal rate) and 70 g/ha (double rate) and flumetsulam in the rates of 120 g and 240 g/ha (normal and double rate respectively). It was also used a mix of diclosulam and flumetsulam (20 + 100 g/ha) and imazaquin at 150 g/ha. At the normal rates of both herbicides it was not poss to find any difference in suscetibily of the cultivars. At double rates of diclosulam the yields of 'FT-Guaira', 'Embrapa-4'and 'BR-37' were reduced in 20,9%, 11,8% and 11,0%, respectively, in relation to the check. Flumetsulam at 240 g/ha and its mix with diclosulam also reduced the yields of these cultivars, plus "Ocepar-14'. The major of the cultivars showed a reduced stand at the double rates of the both herbicides, but without significance. Only the stand of 'Embrapa- 48' was not affected, contrary to 'RS 7-Jacuí'. The plant height was reduced by flumetsulam at 240 g/ha. In all cultivars the nodule dry biomass was reduced in double rates, and in some of them the weight per unit of nodule. The dry biomass of aerial port and roots werw nor reduced