On the multiple Borsuk numbers of sets

The Borsuk number of a set S of diameter d >0 in Euclidean n-space is the smallest value of m such that S can be partitioned into m sets of diameters less than d. Our aim is to generalize this notion in the following way: The k-fold Borsuk number of such a set S is the smallest value of m such that there is a k-fold cover of S with m sets of diameters less than d. In this paper we characterize the k-fold Borsuk numbers of sets in the Euclidean plane, give bounds for those of centrally symmetric sets, smooth bodies and convex bodies of constant width, and examine them for finite point sets in the Euclidean 3-space.Comment: 16 pages, 3 figure

Mixed Electronic and Ionic Conduction Properties of Lithium Lanthanum Titanate

With the continued increase in Li‐metal anode rate capability, there is an equally important need to develop high‐rate cathode architectures for solid‐state batteries. A proposed method of improving charge transport in the cathode is introducing a mixed electronic and ionic conductor (MEIC) which can eliminate the need for conductive additives that occlude electrolyte–electrode interfaces and lower the net additive required in the cathode. This study takes advantage of a reduced perovskite electrolyte, Li0.33La0.57TiO3 (LLTO), to act as a model MEIC. It is found that the ionic conductivity of reduced LLTO is comparable to oxidized LLTO (σbulk = 10−3–10−4 S cm−1, σGB = 10−5–10−6 S cm−1) and the electronic conductivity is 1 mS cm−1. The ionic transference numbers are 0.9995 and 0.0095 in the oxidized and reduced state, respectively. Furthermore, two methods for controlling the transference numbers are evaluated. It is found that the electronic conductivity cannot easily be controlled by changing O2 overpressures, but increasing the ionic conductivity can be achieved by increasing grain size. This work identifies a possible class of MEIC materials that may improve rate capabilities of cathodes in solid‐state architectures and motivate a deeper understanding of MEICs in the context of solid‐state batteries.Lithium lanthanum titanate can exhibit both high ionic conductivity and high electronic conductivity depending on its oxidation state. While electronic conductivity in solid electrolytes is typically undesirable, the ability to conduct both Li‐ions and electrons may pose useful for cathode composites. This work investigates the conduction properties of lithium lanthanum titanate and potential methods to control transference number.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/154428/1/adfm201909140_am.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/154428/2/adfm201909140-sup-0001-SuppMat.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/154428/3/adfm201909140.pd

Some recent approaches to smaller, faster devices for remote PET radiolabeling

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/90295/1/25802601204_ftp.pd