Finding Small Complete Subgraphs Efficiently

(I) We revisit the algorithmic problem of finding all triangles in a graph $G=(V,E)$ with $n$ vertices and $m$ edges. According to a result of Chiba and Nishizeki (1985), this task can be achieved by a combinatorial algorithm running in $O(m \alpha) = O(m^{3/2})$ time, where $\alpha= \alpha(G)$ is the graph arboricity. We provide a new very simple combinatorial algorithm for finding all triangles in a graph and show that is amenable to the same running time analysis. We derive these worst-case bounds from first principles and with very simple proofs that do not rely on classic results due to Nash-Williams from the 1960s. (II) We extend our arguments to the problem of finding all small complete subgraphs of a given fixed size. We show that the dependency on $m$ and $\alpha$ in the running time $O(\alpha^{\ell-2} \cdot m)$ of the algorithm of Chiba and Nishizeki for listing all copies of $K_\ell$, where $\ell \geq 3$, is asymptotically tight. (III) We give improved arboricity-sensitive running times for counting and/or detection of copies of $K_\ell$, for small $\ell \geq 4$. A key ingredient in our algorithms is, once again, the algorithm of Chiba and Nishizeki. Our new algorithms are faster than all previous algorithms in certain high-range arboricity intervals for every $\ell \geq 7$.Comment: 14 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:2105.0126

Dynamic Set Intersection

Consider the problem of maintaining a family $F$ of dynamic sets subject to insertions, deletions, and set-intersection reporting queries: given $S,S'\in F$, report every member of $S\cap S'$ in any order. We show that in the word RAM model, where $w$ is the word size, given a cap $d$ on the maximum size of any set, we can support set intersection queries in $O(\frac{d}{w/\log^2 w})$ expected time, and updates in $O(\log w)$ expected time. Using this algorithm we can list all $t$ triangles of a graph $G=(V,E)$ in $O(m+\frac{m\alpha}{w/\log^2 w} +t)$ expected time, where $m=|E|$ and $\alpha$ is the arboricity of $G$. This improves a 30-year old triangle enumeration algorithm of Chiba and Nishizeki running in $O(m \alpha)$ time. We provide an incremental data structure on $F$ that supports intersection {\em witness} queries, where we only need to find {\em one} $e\in S\cap S'$. Both queries and insertions take O\paren{\sqrt \frac{N}{w/\log^2 w}} expected time, where $N=\sum_{S\in F} |S|$. Finally, we provide time/space tradeoffs for the fully dynamic set intersection reporting problem. Using $M$ words of space, each update costs $O(\sqrt {M \log N})$ expected time, each reporting query costs $O(\frac{N\sqrt{\log N}}{\sqrt M}\sqrt{op+1})$ expected time where $op$ is the size of the output, and each witness query costs $O(\frac{N\sqrt{\log N}}{\sqrt M} + \log N)$ expected time.Comment: Accepted to WADS 201

Chemical Abundances of the Outer Halo Stars in the Milky Way

We present chemical abundances of 57 metal-poor stars that are likely constituents of the outer stellar halo in the Milky Way. Almost all of the sample stars have an orbit reaching a maximum vertical distance (Z_max) of >5 kpc above and below the Galactic plane. High-resolution, high signal-to-noise spectra for the sample stars obtained with Subaru/HDS are used to derive chemical abundances of Na, Mg, Ca, Ti, Cr, Mn, Fe, Ni, Zn, Y and Ba with an LTE abundance analysis code. The resulting abundance data are combined with those presented in literature that mostly targeted at smaller Z_max stars, and both data are used to investigate any systematic trends in detailed abundance patterns depending on their kinematics. It is shown that, in the metallicity range of -25 kpc are systematically lower (~0.1 dex) than those with smaller Z_max. This result of the lower [alpha/Fe] for the assumed outer halo stars is consistent with previous studies that found a signature of lower [alpha/Fe] ratios for stars with extreme kinematics. A distribution of the [Mg/Fe] ratios for the outer halo stars partly overlaps with that for stars belonging to the Milky Way dwarf satellites in the metallicity interval of -2<[Fe/H]<-1 and spans a range intermediate between the distributions for the inner halo stars and the stars belonging to the satellites. Our results confirm inhomogeneous nature of chemical abundances within the Milky Way stellar halo depending on kinematic properties of constituent stars as suggested by earlier studies. Possible implications for the formation of the Milky Way halo and its relevance to the suggested dual nature of the halo are discussed.Comment: 68 pages with 23 figures, Accepted for publication in PAS

Study of Field-Induced Magnetic Order in Singlet-Ground-State Magnet CsFeCl3_3

The field-induced magnetic order in the singlet-ground-state system CsFeCl$_3$ has been studied by measuring magnetization and neutron diffraction. The field dependence of intensity for the neutron magnetic reflection has clearly demonstrated that the field-induced ordered phase is described by the order parameter $$. A condensate growth of magnons is investigated through the temperature dependence of $M_z$ and $M_{\perp}$, and this ordering is discussed in the context of a magnon Bose-Einstein condensation. Development of the coherent state and the static correlation length has been observed in the incommensurate phase in the field region of $5 H_{\rm c}$, a satellite peak was found in coexistence with the commensurate peak at the phase boundary around 10 T, which indicates that the tilt of the c-axis would be less than $\sim 0.5^{\circ}$ in the whole experiments.Comment: 5 pages, 5 figure

Thoron in the environment and its related issues

Since radon is internationally noted as the second cause of lung cancer, many countries are trying about to solve the problem worldwide. In addition, a new evidence of lung cancer risk has been recently found out with a low level below 200 Bq m–3. Thus the action level will have to be set lower than before. Importance of radon exposure has been further recognized and accurate radon concentrations will be required. Recently thoron has also been recognized from the viewpoint of accurate radon measurements. The present paper describes specification of the NIRS radon and thoron chambers, passive measurement technique of radon and thoron and thoron interference on radon measurements from both experimental studies and field experiences on epidemiological study area.Thoron in the environment and its related issues Shinji Tokonami National Institute of Radiological Sciences, 4-9-1 Anagawa, Inage-ku, Chiba 263-8555, Japan E-mail : [email protected] Institute of Radiological Sciences, 4-9-1 Anagawa, Inage-ku, Chiba 263-8555, Japa

Chemical Abundances of the Milky Way Thick Disk and Stellar Halo I.: Implications of [alpha/Fe] for Star Formation Histories in Their Progenitors

We present the abundance analysis of 97 nearby metal-poor (-3.3<[Fe/H]<-0.5) stars having kinematics characteristics of the Milky Way (MW) thick disk, inner, and outer stellar halos. The high-resolution, high-signal-to-noise optical spectra for the sample stars have been obtained with the High Dispersion Spectrograph mounted on the Subaru Telescope. Abundances of Fe, Mg, Si, Ca and Ti have been derived using a one-dimensional LTE abundance analysis code with Kurucz NEWODF model atmospheres. By assigning membership of the sample stars to the thick disk, inner or outer halo components based on their orbital parameters, we examine abundance ratios as a function of [Fe/H] and kinematics for the three subsamples in wide metallicity and orbital parameter ranges. We show that, in the metallicity range of -1.5<[Fe/H]<= -0.5, the thick disk stars show constantly high mean [Mg/Fe] and [Si/Fe] ratios with small scatter. In contrast, the inner, and the outer halo stars show lower mean values of these abundance ratios with larger scatter. The [Mg/Fe], [Si/Fe] and [Ca/Fe] for the inner and the outer halo stars also show weak decreasing trends with [Fe/H] in the range [Fe/H]$>-2$. These results favor the scenarios that the MW thick disk formed through rapid chemical enrichment primarily through Type II supernovae of massive stars, while the stellar halo has formed at least in part via accretion of progenitor stellar systems having been chemically enriched with different timescales.Comment: Accepted for publication in Ap