Sections of Convex Bodies in John’s and Minimal Surface Area Position

We prove several estimates for the volume, the mean width, and the value of the Wills functional of sections of convex bodies in John’s position, as well as for their polar bodies. These estimates extend some well-known results for convex bodies in John’s position to the case of lower-dimensional sections, which had mainly been studied for the cube and the regular simplex. Some estimates for centrally symmetric convex bodies in minimal surface area position are also obtained.

A molecular framework for control of oriented cell division in the Arabidopsis embryo

SummaryPremitotic control of cell division orientation is critical for plant development, as cell walls prevent extensive cell remodelling or migration. Whilst many divisions are proliferative and add cells to existing tissues, some divisions are formative, and generate new tissue layers or growth axes. Such formative divisions are often asymmetric in nature, producing daughters with different fates. We have previously shown that in the Arabidopsis thaliana embryo, developmental asymmetry is correlated with geometric asymmetry, creating daughter cells of unequal volume. Such divisions are generated by division planes that deviate from a default “minimal surface area” rule. Inhibition of auxin response leads to reversal to this default, yet the mechanisms underlying division plane choice in the embryo have been unclear. Here we show that auxin-dependent division plane control involves alterations in cell geometry, but not in cell polarity or nuclear position. Through transcriptome profiling, we find that auxin regulates genes controlling cell wall and cytoskeleton properties. We confirm the involvement of microtubule (MT)-binding proteins in embryo division control. Topology of both MT and Actin cytoskeleton depend on auxin response, and genetically controlled MT or Actin depolymerization in embryos leads to disruption of asymmetric divisions, including reversion to the default. Our work shows how auxin-dependent control of MT- and Actin cytoskeleton properties interacts with cell geometry to generate asymmetric divisions during the earliest steps in plant development.

Quantum holographic entanglement entropy to all orders in 1/N expansion

We study holographic entanglement entropy in four-dimensional quantum gravity with negative cosmological constant. By using the replica trick and evaluating path integrals in the minisuperspace approximation, in conjunction with the Wheeler–DeWitt equation, we compute quantum corrections to the holographic entanglement entropy for a circular entangling surface on the boundary three-sphere. Similarly to our previous work on the sphere partition function, the path integrals are dominated by a replica version of asymptotically anti-de Sitter conic geometries at saddle points. As expected from a general conformal field theory argument, the final result is minus the free energy on the three-sphere, which agrees with the logarithm of the Airy partition function for the Aharony–Bergman–Jafferis–Maldacena theory that sums up all perturbative $1/N$ corrections despite the absence of supersymmetries. The all-order holographic entanglement entropy cleanly splits into two parts, (1) the $1/N$-corrected Ryu–Takayanagi minimal surface area and (2) the bulk entanglement entropy across the minimal surface, as suggested in the earlier literature. It is explicitly shown that the former comes from the localized conical singularity of the replica geometries and the latter from the replication of the bulk volume.

Origin and structure of Ubisch bodies in Pinus sylvestris

In <em>Pinus sylvestris</em> Ubisch bodies are produced repeatedly, and each crop is formed at a distinct phase in the secretory cycles of tapetal cells. While each production has a Ubisch body wall similar to the then current state of the exine with regard to thickening and ornamentation, the survivers of previous productions do not change. Examples of all the structurally different Ubisch body wall forms can be seen when terminally, at the time of pollen shedding, the relict Ubisch bodies become spatially concentrated on the minimal surface area of the senescent cells of the tapetum. In angiosperms after one or a few periods of initiation Ubisch bodies may remain in association with the surface of tapetal cells where the Ubisch body wall undergoes changes like those of the maturing pollen exine. We conclude that as a consequence of Ubisch body detachment from the plasma membrane of tapetal cells there is in <em>Pinus sylvestris</em> no updated information for modification of the wall and the Ubisch body wall remains static.

Comparing the M-position with some classical positions of convex bodies

The purpose of this paper is to compare some classical positions of convex bodies. We provide exact quantitative results which show that the minimal surface area position and the minimal mean width position are not necessarily M-positions. We also construct examples of unconditional convex bodies of minimal surface area that exhibit the worst possible behavior with respect to their mean width or their minimal hyperplane projection.

1999 Pure or Applied Inorganic Chemistry Award Lecture Curves in chemistry: supramolecular materials taking shape

A large part of the development of mathematics was driven by the desire to understand why objects in nature are found in a myriad of shapes and sizes and why certain forms are preferred over others. The area of mathematics called the calculus of variations has been utilized to handle the optimal forms in geometry and nature. It has been used to comprehend morphogenesis and the similarity, yet variety, of forms in the natural world. The mathematical underpinnings of natural form are beginning to influence the inorganic materials world, especially the unusual morphologies that arise in the field of self-assembly. These morphologies often exhibit curved shapes resembling those of minimal surfaces rather than familiar Platonic, polyhedral crystal habits. The curved shapes of mesostructured inorganic materials synthesized by supramolecular templating are striking examples of minimal surface area and energy principles at work in their growth and form.Key words: mesoporous materials, morphosynthesis, silica, supramolecular.