Distance k-Sectors Exist

The bisector of two nonempty sets P and Q in a metric space is the set of all points with equal distance to P and to Q. A distance k-sector of P and Q, where k is an integer, is a (k-1)-tuple (C_1, C_2, ..., C_{k-1}) such that C_i is the bisector of C_{i-1} and C_{i+1} for every i = 1, 2, ..., k-1, where C_0 = P and C_k = Q. This notion, for the case where P and Q are points in Euclidean plane, was introduced by Asano, Matousek, and Tokuyama, motivated by a question of Murata in VLSI design. They established the existence and uniqueness of the distance trisector in this special case. We prove the existence of a distance k-sector for all k and for every two disjoint, nonempty, closed sets P and Q in Euclidean spaces of any (finite) dimension, or more generally, in proper geodesic spaces (uniqueness remains open). The core of the proof is a new notion of k-gradation for P and Q, whose existence (even in an arbitrary metric space) is proved using the Knaster-Tarski fixed point theorem, by a method introduced by Reem and Reich for a slightly different purpose.Comment: 10 pages, 5 figure

Fabrication and characterization of an L3 nanocavity designed by an iterative machine-learning method

Optical nanocavities formed by defects in a two-dimensional photonic crystal (PC) slab can simultaneously realize a very small modal volume and an ultrahigh quality factor (Q). Therefore, such nanocavities are expected to be useful for the enhancement of light-matter interaction and slowdown of light in devices. In the past, it was difficult to design a PC hole pattern that makes sufficient use of the high degree of structural freedom of this type of optical nanocavity, but very recently, an iterative optimization method based on machine learning was proposed that efficiently explores a wide parameter space. Here, we fabricate and characterize an L3 nanocavity that was designed by using this method and has a theoretical Q value of 29 x 10(6) and a modal volume of 0.7 cubic wavelength in the material. The highest unloaded Q value of the fabricated cavities is 4.3 x 10(6); this value significantly exceeds those reported previously for an L3 cavity, i.e., approximate to 2.1 x 10(6). The experimental result shows that the iterative optimization method based on machine learning is effective in improving cavity Q values

Zone Diagrams in Euclidean Spaces and in Other Normed Spaces

Zone diagram is a variation on the classical concept of a Voronoi diagram. Given n sites in a metric space that compete for territory, the zone diagram is an equilibrium state in the competition. Formally it is defined as a fixed point of a certain "dominance" map. Asano, Matousek, and Tokuyama proved the existence and uniqueness of a zone diagram for point sites in Euclidean plane, and Reem and Reich showed existence for two arbitrary sites in an arbitrary metric space. We establish existence and uniqueness for n disjoint compact sites in a Euclidean space of arbitrary (finite) dimension, and more generally, in a finite-dimensional normed space with a smooth and rotund norm. The proof is considerably simpler than that of Asano et al. We also provide an example of non-uniqueness for a norm that is rotund but not smooth. Finally, we prove existence and uniqueness for two point sites in the plane with a smooth (but not necessarily rotund) norm.Comment: Title page + 16 pages, 20 figure

In situ hole doping of wide-gap semiconductors by dual-target simultaneous laser ablation: GaN and SiC epitaxial films

Apparatus for dual-target simultaneous laser ablation deposition and in situdoping techniques have been developed to achieve p-type doping during epitaxialgrowth of wide-band-gap semiconductors. The apparatus has two target holders with a target-rotation mechanism and a rotation-axis adjusting mechanism to obtain homogeneously dopedfilms. Mg-doped GaNfilms have been fabricated on 6H–SiC(0001) and Si(111) substrates in NH₃ ambient by simultaneous ablation of GaN and Mg-metal targets using two lasers. Junctions of the films with n-type substrates show a diode curve characteristic of p-n junctions, but not for junction with p-Si, indicating hole doping without further procedures. In situ p-type doping to SiC was also achieved by using SiC and Al₄C₃ targets

Partial construction of an arrangement of lines and its application to optimal partitioning of bichromatic point set

金沢大学This paper presents an efficient algorithm for construction at-most-k levels of an arrangement of n lines in the plane in ime O(nk + nlogn), which is optimal since Ω(nk) line segments are included there. The algorithm can sweep the at-most-k levels of the arrangement using O(n) space. Although Everett recently gave an algorithm for constructing the at-most-k levels with the same time complexity independently, our algorithm is superior with respect to the space complexity as a sweep algorithm. Then, we apply the algorithm to a bipartitioning problem of a bichromatic point set

Role of the C-terminal extension stacked on the re-face of the isoalloxazine ring moiety of the flavin adenine dinucleotide prosthetic group in ferredoxin-NADP+ oxidoreductase from Bacillus subtilis

Ferredoxin-NADP+ oxidoreductase [EC 1.18.1.2] from Bacillus subtilis (BsFNR) is homologous to the bacterial NADPH-thioredoxin reductase, but possesses a unique C-terminal extension that covers the re-face of the isoalloxazine ring moiety of the flavin adenine dinucleotide (FAD) prosthetic group. In this report, we utilize BsFNR mutants depleted of their C-terminal residues to examine the importance of the C-terminal extension in reactions with NADPH and ferredoxin (Fd) from B. subtilis by spectroscopic and steady-state reaction analyses. The depletions of residues Y313 to K332 (whole C-terminal extension region) and S325 to K332 (His324 intact) resulted in significant increases in the catalytic efficiency with NADPH in diaphorase assay with ferricyanide, whereas Km values for ferricyanide were increased. In the cytochrome c reduction assay in the presence of B. subtilis ferredoxin, the S325-K332 depleted mutant displayed a significant decrease in the turnover rate with an Fd concentration range of 1-10 μM. The Y313-K332 depleted mutant demonstrated an increase in the rate of the direct reduction of horse heart cytochrome c in the absence of Fd. These data indicated that depletion of the C-terminal extension plays an important role in the reaction of BsFNR with ferredoxin. © 2014 Elsevier Masson SAS. All rights reserved

Build-up functionalization of anti-EGFR × anti-CD3 bispecific diabodies by integrating high-affinity mutants and functional molecular formats

Designing non-natural antibody formats is a practical method for developing highly functional next-generation antibody drugs, particularly for improving the therapeutic efficacy of cancer treatments. One approach is constructing bispecific antibodies (bsAbs). We previously reported a functional humanized bispecific diabody (bsDb) that targeted epidermal growth factor receptor and CD3 (hEx3-Db). We enhanced its cytotoxicity by constructing an Fc fusion protein and rearranging order of the V domain. In this study, we created an additional functional bsAb, by integrating the molecular formats of bsAb and high-affinity mutants previously isolated by phage display in the form of Fv. Introducing the high-affinity mutations into bsDbs successfully increased their affinities and enhanced their cytotoxicity in vitro and in vivo. However, there were some limitations to affinity maturation of bsDb by integrating high-affinity Fv mutants, particularly in Fc-fused bsDb with intrinsic high affinity, because of their bivalency. The tetramers fractionated from the bsDb mutant exhibited the highest in vitro growth inhibition among the small bsAbs and was comparable to the in vivo anti-tumor effects of Fc-fused bsDbs. This molecule shows cost-efficient bacterial production and high therapeutic potential