On a generalization of distance sets

A subset $X$ in the $d$-dimensional Euclidean space is called a $k$-distance set if there are exactly $k$ distinct distances between two distinct points in $X$ and a subset $X$ is called a locally $k$-distance set if for any point $x$ in $X$, there are at most $k$ distinct distances between $x$ and other points in $X$. Delsarte, Goethals, and Seidel gave the Fisher type upper bound for the cardinalities of $k$-distance sets on a sphere in 1977. In the same way, we are able to give the same bound for locally $k$-distance sets on a sphere. In the first part of this paper, we prove that if $X$ is a locally $k$-distance set attaining the Fisher type upper bound, then determining a weight function $w$, $(X,w)$ is a tight weighted spherical $2k$-design. This result implies that locally $k$-distance sets attaining the Fisher type upper bound are $k$-distance sets. In the second part, we give a new absolute bound for the cardinalities of $k$-distance sets on a sphere. This upper bound is useful for $k$-distance sets for which the linear programming bound is not applicable. In the third part, we discuss about locally two-distance sets in Euclidean spaces. We give an upper bound for the cardinalities of locally two-distance sets in Euclidean spaces. Moreover, we prove that the existence of a spherical two-distance set in $(d-1)$-space which attains the Fisher type upper bound is equivalent to the existence of a locally two-distance set but not a two-distance set in $d$-space with more than $d(d+1)/2$ points. We also classify optimal (largest possible) locally two-distance sets for dimensions less than eight. In addition, we determine the maximum cardinalities of locally two-distance sets on a sphere for dimensions less than forty.Comment: 17 pages, 1 figur

Modified Perfluorocarbon Tracer Method for Measuring Effective Multizone Air Exchange Rates

A modified procedure was developed for the measurement of the effective air exchange rate, which represents the relationship between the pollutants emitted from indoor sources and the residents’ level of exposure, by placing the dosers of tracer gas at locations that resemble indoor emission sources. To measure the 24-h-average effective air exchange rates in future surveys based on this procedure, a low-cost, easy-to-use perfluorocarbon tracer (PFT) doser with a stable dosing rate was developed by using double glass vials, a needle, a polyethylene-sintered filter, and a diffusion tube. Carbon molecular sieve cartridges and carbon disulfide (CS2) were used for passive sampling and extraction of the tracer gas, respectively. Recovery efficiencies, sampling rates, and lower detection limits for 24-h sampling of hexafluorobenzene, octafluorotoluene, and perfluoroallylbenzene were 40% ± 3%, 72% ± 5%, and 84% ± 6%; 10.5 ± 1.1, 14.4 ± 1.4, and 12.2 ± 0.49 mL min−1; and 0.20, 0.17, and 0.26 μg m−3, respectively