We present a symbolic-numeric technique to find the closest multivariate polynomial system to a given one which has roots with prescribed multiplicity structure. Our method generalizes the “Weierstrass iteration”, defined by Ruatta, to the case when the input system is not exact, i.e. when it is near to a system with multiple roots, but itself might not have multiple roots. First, using interpo...

Tits has defined Kac–Moody groups for all root systems, over all commutative rings with unit. A central concept is the idea of a prenilpotent pair of (real) roots. In particular, writing down his group presentation explicitly would require knowing all the Weyl-group orbits of such pairs. We show that for the hyperbolic root system E10 there are so many orbits that any attempt at direct enumerat...

Nutrient availability is one of the limiting factors for plant growth and development, and the nutrients on Earth are unevenly distributed. To overcome this, plants adapt by modifying their morphology and physiology, especially of the root system. Seabuckthorn is a small tree growing in temperate regions of Europe and Asia. It performs symbiotic N2 fixation and has high adaptability to environm...

one of the main causes of endodontic treatment failure is the clinician’s inability to localize all the root canals. due to the complex anatomy of the root canal system, missed canals are not uncommon. there are several strategies to decrease the possibility of missed root canals starting with good pre-operative radiographies. in order to overcome the limitations of conventional radiographies, ...

We present a novel method for deriving a structural model of a plant root system from 3D Magnetic Resonance Imaging (MRI) data of soil grown plants and use it for plant root system analysis. The structural model allows calculation of physiologically relevant parameters. Roughly speaking, MRI images show local water content of the investigated sample. The small, local amounts of water in roots r...

To better assess the efficacy of mechanical preparation of root canals, transparent specimens of 250 extracted maxillary canines were investigated for canal configuration, thickness and curvature of the root canal, condition of any accessory canals, and location of the apical foramen. Fewer than 40% of the specimens showed accessory canals that were mechanically impossible to clean. The majorit...

New evidence confirms earlier postulates that root signals to shoots, including abscisic acid, nitrate flux, and cytokinins, modify whole plant growth processes including leaf expansion, stomatal behavior, and biosynthesis of photosynthetic enzymes. Root signals are thought to reflect soil water, nutrient, and mechanical attributes, as sensed by roots. Meristematic activities in root tips initi...

Given a root system R, the vector system R̃ is obtained by taking a representative v in each antipodal pair {v,−v}. The matroid M(R) is formed by all independent subsets of R̃. The automorphism group of a matroid is the group of permutations preserving its independent subsets. We prove that the automorphism groups of all irreducible root systems matroids M(R) are uniquely determined by their inde...

The G2 Root system is the only root system in which the angle of pi / 6 appears between two roots. This is made necessary in nature by the demand for circular objects which can be divided by six or twelve, according to Vedic literature concerning the 28 Nakshastra or astrological houses. In addition, G2 provides the key linkage between the Sedenions, which curiously contain properties related t...