WEAK MONOTONICITY OF INTERVAL MATRICES ∗

Abstract. This article concerns weak monotonicity of interval matrices, with specific emphasis on its relationship with a certain class of proper splittings. Key words. Weak monotonicity; Moore-Penrose inverse; Interval matrix; range kernal regularity; proper splitting AMS subject classifications. 15A09, 15B48. 1. Introduction and Preliminaries. A real n × n matrix A is monotone if Ax ≥ 0 implies x ≥ 0, where by x ≥ 0 we mean that all the components of x are nonnegative. It can be easily shown that A is monotone if and only if A is nonsingular and A −1 ≥ 0, where for a matrix B, we denote B ≥ 0, if all the entries of B are nonnegative. Due to this fact, monotone matrices are also referred to as inverse positiv

Sivakumar, Weak monotonicity of matrices and sub classes of proper splitting

Abstract. This article concerns weak monotonicity of matrices, with specific emphasis on its relationship with a certain class of proper splittings. The matrix A ∈ Rm×n is weak monotone provided Ax ≥ 0 = ⇒ x ∈ Rn ++N(A), where N(A) is the nullspace of A. In particular, the following extension of wellknown characterizations forM-matricesisobtained. Suppose that int(Rm +)∩R(A) = φ. Then the statements (a) A is weak-monotone. (b) R m + ∩R(A) ⊆ ARn +. (c) There exists x 0 ≥ 0 such that Ax 0> 0. satisfy (a) ⇔ (b) ⇒ (c). Suppose further that A can be written as A = U −V, where A and U have the same range space and null space, U and V are nonnegative, VU † ≥ 0 (where U † denotes the Moore-Penrose inverse of U), and Ax ≥ 0, Ux ≥ 0 = ⇒ x ∈ Rn + + N(A). Then each of the above statements is equivalent to the statement (d) ρ(VU † ) < 1. Key words. Proper splittings; Weak monotonicity; Nonnegativity; Moore-Penrose inverse

Synthesis of some new 2-(benzylsulfonyl¬ethylsulfonylmethyl)¬oxazolines and thiazolines

1494-1501A new class of five-membered heterocycles oxazolines and thiazolines have been prepared by multistep, one-pot and microwave irradiation methodologies. The microwave method provides an excellent approach for the safe, rapid, inexpensive and simple synthesis of oxazolines and thiazolines

Desempenho de clones de seringueira da série IAC 300 selecionados para a região noroeste do Estado de São Paulo Performance of Hevea clones from IAC 300 series selected in the northwestern region of São Paulo State, Brazil

É apresentado o desempenho de 19 novos clones de seringueira [Hevea brasiliensis (Willd. exAdr. de Juss.) Muell. Arg.], resultantes de hibridações conduzidas no Instituto Agronômico e avaliados em experimento de pequena escala, tendo o clone RRIM 600 como testemunha. O experimento em campo obedeceu ao delineamento em blocos ao acaso, com três repetições. Com relação à produção de borracha seca, o clone IAC 40 apresentou a maior média (62,22 g/árvore/sangria) nos três anos de avaliação, seguido pelo IAC 301 (57,67 g/árvore/sangria) e pelo IAC 300 (50,61 g/árvore/sangria), com produções 154%, 138% e 123% superiores em relação ao RRIM 600 (41,04 g/árvore/sangria). Todos os clones selecionados foram vigorosos, com perímetro do caule na abertura do painel variando de 37,81 cm (IAC 317) a 50,90 cm (IAC 315). A porcentagem de plantas aptas a sangria variou de 20,0% (IAC 317) a 100% (IAC 315). Todos os clones apresentaram baixas incidências de quebra pelo vento e de secamento do painel. Não foi detectada nenhuma doença foliar em caráter epidêmico. Dos clones estudados, 15 apresentaram alta resistência à antracnose do painel, e foram superiores ao RRIM 600; os outros cinco apresentaram resistência moderada semelhante ao RRIM 600.<br>The present paper shows the performance of 19 rubber tree [Hevea brasiliensis (Willd. ex Adr. de Juss.) Muell. Arg.] originated from Hevea breeding programme conducted at the Instituto Agronômico de Campinas (IAC) and evaluated in a small scale trial in the northwestern of São Paulo State, Brazil. The old popular clone RRIM 600 of Malaysian origin was used as control. The trial was laid out in a randomized block design with three replications. Concerned to yielding the clone IAC 40 recorded highest yield (62.22 g/tree/tap) over three years of tapping which was followed by IAC 301 (57.67 g/tree/tap) and IAC 300 (50.61 g/tree/tap) yielding 154%, 138% and 123% superior to the control clone and RRIM 600 whereas recorded 41.04 tree/tap. All selected clones were of vigorous growing, with the girth at opening ranging from 37.81 cm for IAC 317 to 50.90 cm for IAC 315 and tappability ranging from 20.0% for IAC 317 to 100% for IAC 315. All the clones experienced low to very low incidence of wind damage. All selected clones showed low to very low incidence of dry trees. Fifteen clones showed high resistance and five presented moderate resistance to anthrachnose panel canker disease similarly to RRIM 600