SL$ _n(\mathbb{Z}) $-normalizer of a principal congruence subgroup

<abstract><p>Let SL$ _n(\mathbb{Q}) $ be the set of matrices of order $ n $ over the rational numbers with determinant equal to 1. We study in this paper a subset $ \Lambda $ of SL$ _n(\mathbb{Q}) $, where a matrix $ B $ belongs to $ \Lambda $ if and only if the conjugate subgroup $ B\Gamma_q(n)B^{-1} $ of principal congruence subgroup $ \Gamma_q(n) $ of lever $ q $ is contained in modular group SL$ _n(\mathbb{Z}) $. The notion of least common denominator (LCD for convenience) of a rational matrix plays a key role in determining whether <italic>B</italic> belongs to $ \Lambda $. We show that LCD can be described by the prime decomposition of $ q $. Generally $ \Lambda $ is not a group, and not even a subsemigroup of SL$ _n(\mathbb{Q}) $. Nevertheless, for the case $ n = 2 $, we present two families of subgroups that are maximal in $ \Lambda $ in this paper.</p></abstract>

Structure Synthesis and Reconfiguration Analysis of Variable-DOF Single-Loop Mechanisms With Prismatic Joints Using Dual Quaternions

This paper deals with the structure synthesis and reconfiguration analysis of variable-DOF (variable degree-of-freedom) single-loop mechanisms with prismatic joints based on a unified tool - the dual quaternion. According to motion polynomials over dual quaternions, an algebraic method is presented to synthesize variable-DOF single-loop 5R2P mechanisms (R and P denote revolute and prismatic joints respectively), which are composed of the Bennett and RPRP mechanisms. Using this approach, variable-DOF single-loop RRPRPRR and RRPRRPR mechanisms are constructed by joints obtained from the factorization of motion polynomials. Then reconfiguration analysis of these variable-DOF single-loop mechanisms is performed in light of the kinematic mapping and the prime decomposition. The results show that the variable-DOF 5R2P mechanisms have a 1-DOF spatial 5P2P motion mode and a 2-DOF Bennett-RPRP motion mode. Furthermore, the variable-DOF 5R2P mechanisms have two transition configurations, from which the mechanisms can switch among their two motion modes.

Phenolic acid-degrading Paraburkholderia prime decomposition in forest soil

Plant-derived phenolic acids are catabolized by soil microorganisms whose activity may enhance the decomposition of soil organic carbon (SOC). We characterized whether phenolic acid-degrading bacteria enhance SOC mineralization in forest soils when primed with 13C-labeled p-hydroxybenzoic acid (pHB). We further tested whether pHB-induced priming could explain differences in SOC content among mono-specific tree plantations in a 70-year-old common garden experiment. pHB addition primed significant losses of SOC (3–13 µmols C g−1 dry wt soil over 7 days) compared to glucose, which reduced mineralization (-3 to -8 µmols C g−1 dry wt soil over 7 days). The principal degraders of pHB were Paraburkholderia and Caballeronia in all plantations regardless of tree species or soil type, with one predominant phylotype (RP11ASV) enriched 23-fold following peak pHB respiration. We isolated and confirmed the phenolic degrading activity of a strain matching this phylotype (RP11T), which encoded numerous oxidative enzymes, including secretion signal-bearing laccase, Dyp-type peroxidase and aryl-alcohol oxidase. Increased relative abundance of RP11ASV corresponded with higher pHB respiration and expression of pHB monooxygenase (pobA), which was inversely proportional to SOC content among plantations. pobA expression proved a responsive measure of priming activity. We found that stimulating phenolic-acid degrading bacteria can prime decomposition and that this activity, corresponding with differences in tree species, is a potential mechanism in SOC cycling in forests. Overall, this study highlights the ecology and function of Paraburkholderia whose associations with plant roots and capacity to degrade phenolics suggest a role for specialized bacteria in the priming effect.

On a conjecture of Chen and Yui: Resultants and discriminants

In [5], Chen and Yui conjectured that Gross–Zagier type formulas may also exist for Thompson series. In this work, we verify Chen and Yui’s conjecture for the cases for Thompson series $j_{p}(\tau )$ for $\Gamma _{0}(p)$ for p prime, and equivalently establish formulas for the prime decomposition of the resultants of two ring class polynomials associated to $j_{p}(\tau )$ and imaginary quadratic fields and the prime decomposition of the discriminant of a ring class polynomial associated to $j_{p}(\tau )$ and an imaginary quadratic field. Our method for tackling Chen and Yui’s conjecture on resultants can be used to give a different proof to a recent result of Yang and Yin. In addition, as an implication, we verify a conjecture recently raised by Yang, Yin, and Yu.

Primeness of Relative Annihilators in BCK-Algebra

Conditions that are necessary for the relative annihilator in lower B C K -semilattices to be a prime ideal are discussed. Given the minimal prime decomposition of an ideal A, a condition for any prime ideal to be one of the minimal prime factors of A is provided. Homomorphic image and pre-image of the minimal prime decomposition of an ideal are considered. Using a semi-prime closure operation “ c l ”, we show that every minimal prime factor of a c l -closed ideal A is also c l -closed.