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16 páginas.- 6 figuras.- 1 tabla.- referencias.- Supplementary information The online version contains supplementary material available at https://doi. org/10.1007/s11104-023-06029-z, Purpose Incorporating biosolids into the soil improves plant yield compared with surface application, but it can result in the increased uptake of trace elements. However, there is a lack of knowledge about how different types of biosolids applications affect soil quality. We aimed to determine the effect of the type and rate of biosolids application on soil quality and the mobility of contaminants. Methods Soil quality was determined by soil fertility (inorganic N, exchangeable P, Mg, Ca, K), exchangeable trace and non-essential elements (Al, Mn, Zn, Cu and Cd) and biological activity (dehydrogenase activity). We measured the properties of soil pore water, bulk soil and rhizosphere in a pot and a rhizobox experiment, with increasing concentration of biosolids (equiv. 16 t ha− 1, 48 t ha− 1 and 145 t ha− 1 dry weight), applied on the surface, incorporated to 25 cm, or incorporated into a patch. Results and discussion The incorporation of biosolids into the soil increased the exchangeable Zn, Cu, Cr, Ni and Cd, compared with surface application. The surface application of biosolids increased the inorganic N in the soil compared with biosolids incorporation (680 mg kg− 1 vs. 380 mg kg− 1), and decreased soil pH by 1.1 units. This aligned with solubilisation of Al (43 mg kg− 1 vs. 6 mg kg− 1) and Mn (43 mg kg− 1 vs. 33 mg kg− 1) and explains the decreased microbial activity in the soil compared with the unamended soil. Incorporating biosolids in the soil increased the biological activity, likely due to biosolids-borne microbes. The root systems significantly increased microbial activity, pH, and the concentration of NH4+, NO3−, and exchangeable P, S, Mg, Na, Zn, Cu and Ni, and significantly decreased exchangeable concentration of Mn and Fe., his study was funded by the Centre for Integrated Biowaste Research which was supported by ESR Strategic Science Investment Funding from the Ministry of Business, Innovation and Employment, New Zealand (C03X1701). We acknowledge the staff in the Lincoln University greenhouse facilities and in the laboratories of the Soil Department, where all the experimental work took place. We thank Christchurch City Council for providing the biosolids. During her postdoc position in Lincoln University (New Zealand), MJGG held a fellowship granted by Fundación Ramón Areces (Spain). EM thanks the financial support for her work at Lincoln University from Program Salvador de Madariaga of the Spanish MECD.

Real physical systems with reflective and rotational symmetries such as viruses, fullerenes and quasicrystals have recently been modeled successfully in terms of three-dimensional (affine) Coxeter groups. Motivated by this progress, we explore here the benefits of performing the relevant computations in a Geometric Algebra framework, which is particularly suited to describing reflections. Starting from the Coxeter generators of the reflections, we describe how the relevant chiral (rotational), full (Coxeter) and binary polyhedral groups can be easily generated and treated in a unified way in a versor formalism. In particular, this yields a simple construction of the binary polyhedral groups as discrete spinor groups. These in turn are known to generate Lie and Coxeter groups in dimension four, notably the exceptional groups D_4, F_4 and H_4. A Clifford algebra approach thus reveals an unexpected connection between Coxeter groups of ranks 3 and 4. We discuss how to extend these considerations and computations to the Conformal Geometric Algebra setup, in particular for the non-crystallographic groups, and construct root systems and quasicrystalline point arrays. We finally show how a Clifford versor framework sheds light on the geometry of the Coxeter element and the Coxeter plane for the examples of the two-dimensional non-crystallographic Coxeter groups I_2(n) and the three-dimensional groups A_3, B_3, as well as the icosahedral group H_3., Comment: Best Paper Award at AGACSE 2012, La Rochelle, France; 18 pages, 4 figures, 2 tables; accepted in Advances in Applied Clifford Algebras

Quaternionic representations of Coxeter (reflection) groups of ranks 3 and 4, as well as those of E_8, have been used extensively in the literature. The present paper analyses such Coxeter groups in the Clifford Geometric Algebra framework, which affords a simple way of performing reflections and rotations whilst exposing more clearly the underlying geometry. The Clifford approach shows that the quaternionic representations in fact have very simple geometric interpretations. The representations of the groups A_1 x A_1 x A_1, A_3, B_3 and H_3 of rank 3 in terms of pure quaternions are shown to be simply the Hodge dualised root vectors, which determine the reflection planes of the Coxeter groups. Two successive reflections result in a rotation, described by the geometric product of the two reflection vectors, giving a Clifford spinor. The spinors for the rank-3 groups A_1 x A_1 x A_1, A_3, B_3 and H_3 yield a new simple construction of binary polyhedral groups. These in turn generate the groups A_1 x A_1 x A_1 x A_1, D_4, F_4 and H_4 of rank 4, and their widely used quaternionic representations are shown to be spinors in disguise. Therefore, the Clifford geometric product in fact induces the rank-4 groups from the rank-3 groups. In particular, the groups D_4, F_4 and H_4 are exceptional structures, which our study sheds new light on., Comment: 20 pages, 1 figure; to appear in Advances in Applied Clifford Algebras