Complete reducibility: variations on a theme of Serre

In this note, we unify and extend various concepts in the area of G-complete reducibility, where G is a reductive algebraic group. By results of Serre and Bate–Martin–Röhrle, the usual notion of G-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of G. We show that other variations of this notion, such as relative complete reducibility and $$\sigma $$ σ -complete reducibility, can also be viewed as special cases of this building-theoretic definition, and hence a number of results from these areas are special cases of more general properties.

Formation and Evolution of Nanoscale Calcium Phosphate Precursors Under Biomimetic Conditions

Simulated body fluids that mimic human blood plasma are widespread media for in-vitro studies in an extensive array of research fields, from biomineralization to surface and corrosion sciences. We show that these solutions undergo dynamic nanoscopic conformational rearrangements on the timescale of minutes to hours, even though they are commonly considered stable or metastable. In particular, we find and characterize nanoscale inhomogeneities made of calcium phosphate (CaP) aggregates that emerge from homogeneous SBF within a few hours and evolve into prenucleation species (PNS) that act as precursors in CaP crystallization processes. These ionic clusters consist of about 2 nm large spherical building units that can aggregate into supra-structures with sizes of over 200 nm. We show that the residence times of phosphate ions in the PNS depend critically on the total PNS surface. These findings are particularly relevant for understanding non-classical crystallization phenomena, in which PNS are assumed to act as building blocks for the final crystal structure.<br>

Formation and Evolution of Nanoscale Calcium Phosphate Precursors Under Biomimetic Conditions

Simulated body fluids that mimic human blood plasma are widespread media for in-vitro studies in an extensive array of research fields, from biomineralization to surface and corrosion sciences. We show that these solutions undergo dynamic nanoscopic conformational rearrangements on the timescale of minutes to hours, even though they are commonly considered stable or metastable. In particular, we find and characterize nanoscale inhomogeneities made of calcium phosphate (CaP) aggregates that emerge from homogeneous SBF within a few hours and evolve into prenucleation species (PNS) that act as precursors in CaP crystallization processes. These ionic clusters consist of about 2 nm large spherical building units that can aggregate into supra-structures with sizes of over 200 nm. We show that the residence times of phosphate ions in the PNS depend critically on the total PNS surface. These findings are particularly relevant for understanding non-classical crystallization phenomena, in which PNS are assumed to act as building blocks for the final crystal structure.<br>

Layered assemblers for scalable parallel integration

Many complex natural and artificial systems are composed of large numbers of elementary building blocks, such as organisms made of many biological cells or processors made of many electronic transistors. This modular substrate is essential to the evolution of biological and technological complexity, but has been difficult to replicate for mechanical systems. This study seeks to answer if layered assembly can engender exponential gains in the speed and efficacy of block or cell-based manufacturing processes. A key challenge is how to deterministically assemble large numbers of small building blocks in a scalable manner. Here, we describe two new layered assembly principles that allow assembly faster than linear time, integrating n modules in O( n 2/3 ) and O( n 1/3 ) time: one process uses a novel opto-capillary effect to selectively deposit entire layers of building blocks at a time, and a second process jets building block rows in rapid succession. We demonstrate the fabrication of multi-component structures out of up to 20 000 millimetre scale spherical building blocks in 3 h. While these building blocks and structures are still simple, we suggest that scalable layered assembly approaches, combined with a growing repertoire of standardized passive and active building blocks could help bridge the meso-scale assembly gap, and open the door to the fabrication of increasingly complex, adaptive and recyclable systems.

A rank rigidity result for CAT(0) spaces with one-dimensional Tits boundaries

We prove the following rank rigidity result for proper {\operatorname{CAT}(0)} spaces with one-dimensional Tits boundaries: Let Γ be a group acting properly discontinuously, cocompactly, and by isometries on such a space X. If the Tits diameter of {\partial X} equals π and Γ does not act minimally on {\partial X}, then {\partial X} is a spherical building or a spherical join. If X is also geodesically complete, then X is a Euclidean building, higher rank symmetric space, or a nontrivial product. Much of the proof, which involves finding a Tits-closed convex building-like subset of {\partial X}, does not require the Tits diameter to be π, and we give an alternate condition that guarantees rigidity when this hypothesis is removed, which is that a certain invariant of the group action be even.

Generalized Non-Crossing Partitions and Buildings

For any finite Coxeter group $W$ of rank $n$ we show that the order complex of the lattice of non-crossing partitions $\mathrm{NC}(W)$ embeds as a chamber subcomplex into a spherical building of type $A_{n-1}$. We use this to give a new proof of the fact that the non-crossing partition lattice in type $A_n$ is supersolvable for all $n$. Moreover, we show that in case $B_n$, this is only the case if $n<4$. We also obtain a lower bound on the radius of the Hurwitz graph $H(W)$ in all types and re-prove that in type $A_n$ the radius is $\binom{n}{2}$. A Corrigendum for this paper was added on May 17, 2018.

Galois Involutions

This chapter focuses on the fixed points of a strictly semi-linear automorphism of order 2 of a spherical building which satisfies the conditions laid out in Hypothesis 30.1. It begins with the fhe definition of a spherical building satisfying the Moufang condition and a Galois involution of Δ‎, described as an automorphism of Δ‎ of order 2 that is strictly semi-linear. It can be recalled that Δ‎ can have a non-type-preserving semi-linear automorphism only if its Coxeter diagram is simply laced. The chapter assumes that the building Δ‎ being discussed is as in 30.1 and that τ‎ is a Galois involution of Δ‎. It also considers the notation stating that the polar region of a root α‎ of Δ‎ is the unique residue of Δ‎ containing the arctic region of α‎.

Strictly Semi-linear Automorphisms

This chapter considers the action of a strictly semi-linear automorphism fixing a root on the corresponding root group. It begins with the hypothesis whereby Δ‎ is a Moufang spherical building and Π‎ is the Coxeter diagram of Δ‎; here the chapter fixes an apartment Σ‎ of Δ‎ and a root α‎ of Σ‎. The discussion then turns to a number of assumptions about an isomorphism of Moufang sets, anisotropic quadratic space, and root group sequence, followed by a lemma where E is an octonion division algebra with center F and norm N and D is a quaternion subalgebra of E. The chapter concludes with three versions of what is really one result about fixed points of non-linear automorphisms of the Moufang sets.

Linear Automorphisms

This chapter considers the notion of a linear automorphism of an arbitrary spherical building satisfying the Moufang property. It begins with the notation whereby Ω‎ = (U₊, U₁, ..., Uₙ) is the root group sequence and x₁, ... , xₙ the isomorphisms obtained by applying the recipe in [60, 16.x] for x = 1, 2, 3, ... or 9 to a parameter system Λ‎ of the suitable type (and for suitable n) and Δ‎ is the corresponding Moufang n-gon. The chapter proceeds by looking at cases where Λ‎ is a proper anisotropic pseudo-quadratic space defined over an involutory set or a quadratic space of type E⁶, E₇ or E₈. It also describes a notation dealing with the Moufang spherical building with Coxeter diagram Λ‎, an apartment of Δ‎, and a chamber of Σ‎.