Deuterium Toward WD1634-573: Results from the Far Ultraviolet Spectroscopic Explorer (FUSE) Mission

We use Far Ultraviolet Spectrocopic Explorer (FUSE) observations to study interstellar absorption along the line of sight to the white dwarf WD1634-573 (d=37.1+/-2.6 pc). Combining our measurement of D I with a measurement of H I from Extreme Ultraviolet Explorer data, we find a D/H ratio toward WD1634-573 of D/H=(1.6+/-0.5)e-5. In contrast, multiplying our measurements of D I/O I=0.035+/-0.006 and D I/N I=0.27+/-0.05 with published mean Galactic ISM gas phase O/H and N/H ratios yields D/H(O)=(1.2+/-0.2)e-5 and D/H(N)=(2.0+/-0.4)e-5, respectively. Note that all uncertainties quoted above are 2 sigma. The inconsistency between D/H(O) and D/H(N) suggests that either the O I/H I and/or the N I/H I ratio toward WD1634-573 must be different from the previously measured average ISM O/H and N/H values. The computation of D/H(N) from D I/N I is more suspect, since the relative N and H ionization states could conceivably vary within the LISM, while the O and H ionization states will be more tightly coupled by charge exchange.Comment: 23 pages, 5 figures; AASTEX v5.0 plus EPSF extensions in mkfig.sty; accepted by ApJ Supplemen

Functional identities of one variable

Let $A$ be a centrally closed prime algebra over a characteristic 0 field $k$, and let $q:A\to A$ be the trace of a $d$-linear map (i.e., $q(x)=M(x,...,x)$ where $M:A^d\to A$ is a $d$-linear map). If $[q(x),x]=0$ for every $x\in A$, then $q$ is of the form $q(x) =\sum_{i=0}^{d} \mu_i(x)x^i$ where each $\mu_i$ is the trace of a $(d-i)$-linear map from $A$ into $k$. For infinite dimensional algebras and algebras of dimension $>d^2$ this was proved by Lee, Lin, Wang, and Wong in 1997. In this paper we cover the remaining case where the dimension is $\le d^2$. Using this result we are able to handle general functional identities of one variable on $A$; more specifically, we describe the traces of $d$-linear maps $q_i:A\to A$ that satisfy $\sum_{i=0}^m x^i q_i(x)x^{m-i}\in k$ for every $x\in A$.Comment: 10 page