On the characterization of 2×2ρ-contraction matrices

AbstractWe give an explicit description of all ρ-contractive (in Nagy–Foiaş sense) 2×2 matrices. This description leads to the formulas for ρ-numerical radii when the eigenvalues of such matrices either have equal absolute values or equal (modπ) arguments. We also discuss (natural) generalizations to the case of decomposable operators with at most two-dimensional blocks covering, in particular, the quadratic operators

ϱ-contraction and 2 × 2 matrix

AbstractIn this paper, the following is proved. When |a|⩽1 and |b|⩽1, A = acOb is a g9-contraction if and only if |c|2 + |a − |2 ⩽ infζ∈D{ρ + (1 −ρ)aζ}{ρ + (1 − ρ)bζ} − aζ|ζ|2ρζ2 where D is the open unit disc. The result is then extended to quadratic operators. Several special cases of the result are analysed in detail

Varietal Variation of Earliness in a Narrow Sense and Its Significance for Adaptation in Barley

Heading time of is a complex character, which is controlled by several internal factors, namely, vernalization requirement, photoperiodic response and earliness in a narrow sense (ENS). The ecological significance of vernalization requirement and photoperiodic response has been clarified. However, the significance of ENS is not explained well. In this study, we examined the varietal variation of ENS, its geographical distribution and significance of ENS for regional adaptability in barley. The ENS was estimated from the number of days from sowing to flag-leaf emergence of sufficiently vernalized plants grown under a 24-h. photoperiod at 20℃ condition. The variation of ENS was determined from the number of leaves and leaf emergence intervals under the controlled condition. The ENS of 1,017 barley varieties ranged from 12 to 28 days. ENS was shortest in varieties from low-latitudes, while it was longer in varieties from high-latitudes and very low-latitudes. Multiple regression analysis for heading time(Y) on the ENS(X1) and the photoperiodic response(X2) was carried out using various data obtained from different experiment sites and sowing times. The multiple correlation was highly significant in all cases. It was clear that the ENS played an important role for determining heading time in the spring-sown conditions, while the photoperiodic response did in the fall-sown condition.本研究ではオオムギの出穂早晩性に関与する内的要因のひとつである純粋早晩性の品種変異とその地理的差異について検討するとともに、播種期と試験地を異にした場合の出穂期に対する内的要因の影響を解析し、純粋早晩性の適応的意義について検討した。春化後高（20℃）24時間日長条件下で栽培したときの止葉展開まで日数を純粋早晩性とする。世界各地のオオムギ1,017品種の純粋早晩性は12日から28日まで変異した。純粋早晩性の変異を地域別にみると、高緯度地域に分布する品種群は晩生であり、低緯度地域の品種群は早生であった。中緯度地域および極低緯度地域（エチオピア）の品種群では純粋早晩性の変異が大きく、この地域には高緯度地域よりも晩生の品種が多く存在した。重回帰分析の結果、倉敷における出穂期の早晩は秋播栽培条件では主として光周反応性によって支配されるが、春播栽培条件では主として純粋早晩性によって支配され、遅播きするほど純粋早晩性の影響が強いことが示された。春播栽培地域である北見においても出穂期は純粋早晩性の影響を強く受けていた。幼穂形成期における日長が十分に長い栽培条件では光周反応性は出穂期の調整要因として作用せず、純粋早晩性が出穂期を支配するものと考えられた