Root Sublattices and Torus-Restriction of Prehomogeneous Vector Spaces Associated with Dynkin Quivers

For a Dynkin quiver Γ with r vertices, a subset S of the vertices of Γ, and an r-tuple d = (d⁽¹⁾, d⁽²⁾,…, d^(r)) of positive integers, we define a “torus-restricted” representation (G_S, R _d (Γ)) in natural way. Here we put G_S = G₁ × G₂ × … ×G_r, where each G_i is either SL(d⁽ⁱ⁾) or GL(d⁽ⁱ⁾) according to S containing i or not. In this paper, for a prescribed torus-restriction S, we give a necessary and sufficient condition on d that R _d (Γ) has only finitely many G_S-orbits. This can be paraphrased as a condition whether or not d is contained in a certain lattice spanned by positive roots of Γ. We also discuss the prehomogeneity of (G_S, R _d (Γ)).     

A Generalization of Extended Affine Lie Algebras

We introduce a new class of possibly infinite dimensional Lie algebras and study their structural properties. Examples of this new class of Lie algebras are finite dimensional simple Lie algebras containing a nonzero split torus, affine and extended affine Lie algebras. Our results generalize well-known properties of these examples.     

Roots of Irreducible Polynomials in Tame Henselian Extension Fields

A class of irreducible polynomials 𝒫 over a valued field (F, v) is introduced, which is the set of all monic irreducible polynomials over F when (F, v) is maximally complete. A “best-possible” criterion is given for when the existence of an approximate root in a tamely ramified Henselian extension K of F of a polynomial f in 𝒫 guarantees the existence of an exact root of f in K.     

Root Systems for Asymmetric Geometric Representations of Coxeter Groups

Results are obtained concerning root systems for asymmetric geometric representations of Coxeter groups. These representations were independently introduced by Vinberg and Eriksson, and generalize the standard geometric representation of a Coxeter group in such a way as to include certain restrictions of all Kac–Moody Weyl groups. In particular, a characterization of when a nontrivial multiple of a root may also be a root is given in the general context. Characterizations of when the number of such multiples of a root is finite and when the number of positive roots sent to negative roots by a group element is finite are also given. These characterizations are stated in terms of combinatorial conditions on a graph closely related to the Coxeter graph for the group. Other finiteness results for the symmetric case which are connected to the Tits cone and to a natural partial order on positive roots are extended to this asymmetric setting.     

精确自动识别莫尔条纹方向角的算法

由于常规莫尔条纹计算方法难以满足速度和精度的要求,本文提出一种快速准确求取莫尔条纹方向角的算法。该算法首先根据直线方向与莫尔条纹方向越接近则均方误差值越小的原理,初步求得莫尔方向角α,之后采用Mean-Shift算法在方向角α附近进一步精确求取莫尔条纹方向角。实验结果表明：采用误差评判指标测试对比度满足5.4%时,提出的算法所获得的莫尔条纹方向角计算精度达到29＇,同时计算速度相对频域处理方法大幅提高,计算时间为15 ms,满足算法处理速度对实时性的要求。     

Effect of subsoil compaction on rooting behavior and yields of wheat

Soil compactions are widely dispersed in the world but tend to be the most prevalent, where heavy machinery is used in agriculture. The increasing use of heavy machinery is the primary cause of soil and subsoil compaction. The impact of subsoil compaction on root growth and yield of wheat (Triticum aestivum L) were evaluated during 2006 and 2007. Sub-soil compactions were created by three normal loads, i.e. 4.40, 6.40 and 8.40 kg and four number of passes of tractor, i.e. 1, 6, 11 and 16. The field was divided into 39 plots including a control plot, i.e. no passes of the tractor. The size of each plot was 400 square meter. A factorial randomized block design was followed in laying out the experiment and care was taken that all the 13 treatments and their replications are included in field experiments. It was observed that for all the compaction treatments in the field experiment on the wheat crop, 51–61% of wheat roots were confined in 0–15 cm, 17–20% in 15–30 cm and the rest 22–28% is below 30 cm soil layer. Sub-soil compaction reduced the wheat crop yields to a maximum of 23%. A statistical model is developed to predict crop yield considering the root length density of the crop. Average root diameters increased with the increase of the sub-soil compaction level. In sub-soil zone, average root diameter decreased with the increase of sub-soil compaction level.     

Generalized B(m,n), C(n),D(m,n)-graded Lie superalgebras arising from fermionic-bosonic representations

We construct fermionic-bosonic representations for a class of generalized B(m, n), C(n), D(m, n)-graded Lie superalgebras coordinatized by quantum tori with nontrivial central extensions.     

Modified Jarratt method with sixth-order convergence

In this paper, we present a variant of Jarratt method with order of convergence six for solving non-linear equations. Per iteration the method requires two evaluations of the function and two of its first derivatives. The new multistep iteration scheme, based on the new method, is developed and numerical tests verifying the theory are also given.