The smallest $I_5$-free and triangle-free binary matroids
We determine the smallest simple triangle-free binary matroids that have no five-element independent flat. This solves a special case of a conjecture of Nelson and Norin.
math.CO↗
arXiv subjects
Peter Nelson
Publications and source records attributed to Peter Nelson.
Brownian motion in attenuated or renormalized inverse-square Poisson potential
We consider the parabolic Anderson problem with random potentials having inverse-square singularities around the points of a standard Poisson point process in $\mathbb{R}^d$, $d \geq 3$. The potentials we consider are obtained via superposition of translations over the points of the Poisson point process of a kernel $\mathfrak{K}$ behaving as $\mathfrak{K}(x) \approx θ|x|^{-2}$ near the origin, where $θ\in (0,(d-2)^2/16]$. In order to make sense of the corresponding path integrals, we require the potential to be either attenuated (meaning that $\mathfrak{K}$ is integrable at infinity) or, when $d=3$, renormalized, as introduced by Chen and Kulik in [8]. Our main results include existence and large-time asymptotics of non-negative solutions via Feynman-Kac representation. In particular, we settle for the renormalized potential in $d=3$ the problem with critical parameter $θ= 1/16$, left open by Chen and Rosinski in [arXiv:1103.5717].
math.PR↗
The structure of $I_4$-free and triangle-free binary matroids
A simple binary matroid is called $I_4$-free if none of its rank-4 flats are independent sets. These objects can be equivalently defined as the sets $E$ of points in $PG(n-1,2)$ for which $|E \cap F|$ is not a basis of $F$ for any four-dimensional flat $F$. We prove a decomposition theorem that exactly determines the structure of all $I_4$-free and triangle-free matroids. In particular, our theorem implies that the $I_4$-free and triangle-free matroids have critical number at most $2$.
math.CO↗
The Rise and Fall of the King: The Correlation between FO Aquarii's Low States and the White Dwarf's Spindown
The intermediate polar FO Aquarii (FO Aqr) experienced its first-reported low-accretion states in 2016, 2017, and 2018, and using newly available photographic plates, we identify pre-discovery low states in 1965, 1966, and 1974. The primary focus of our analysis, however, is an extensive set of time-series photometry obtained between 2002 and 2018, with particularly intensive coverage of the 2016-2018 low states. After computing an updated spin ephemeris for the white dwarf (WD), we show that its spin period began to increase in 2014 after having spent 27 years decreasing; no other intermediate polar has experienced a sign change of its period derivative, but FO Aqr has now done so twice. Our central finding is that the recent low states all occurred shortly after the WD began to spin down, even though no low states were reported in the preceding quarter-century, when it was spinning up. Additionally, the system's mode of accretion is extremely sensitive to the mass-transfer rate, with accretion being almost exclusively disk-fed when FO Aqr is brighter than V~14 and substantially stream-fed when it is not. Even in the low states, a grazing eclipse remains detectable, confirming the presence of a disk-like structure (but not necessarily a Keplerian accretion disk). We relate these various observations to theoretical predictions that during the low state, the system's accretion disk dissipates into a non-Keplerian ring of diamagnetic blobs. Finally, a new XMM-Newton observation from a high state in 2017 reveals an anomalously soft X-ray spectrum and diminished X-ray luminosity compared to pre-2016 observations.
astro-ph.SR↗
Survey of Period Variations of Superhumps in SU UMa-Type Dwarf Novae. X: The Tenth Year (2017)
Continuing the project described by Kato et al. (2009, PASJ, 61, S395, arXiv/0905.1757), we collected times of superhump maxima for 102 SU UMa-type dwarf novae observed mainly during the 2017 season and characterized these objects. WZ Sge-type stars identified in this study are PT And, ASASSN-17ei, ASASSN-17el, ASASSN-17es, ASASSN-17fn, ASASSN-17fz, ASASSN-17hw, ASASSN-17kd, ASASSN-17la, PNV J20205397+2508145 and TCP J00332502-3518565. We obtained new mass ratios for 7 objects using growing superhumps (stage A). ASASSN-17gf is an EI Psc-type object below the period minimum. CRTS J080941.3+171528 and DDE 51 are objects in the period gap and both showed long-lasting phase of stage A superhumps. We also summarized the recent advances in understanding of SU UMa-type and WZ Sge-type dwarf novae.
astro-ph.SR↗
The structure of binary matroids with no induced claw or Fano plane restriction
An 'induced restriction' of a simple binary matroid $M$ is a restriction $M|F$, where $F$ is a flat of $M$. We consider the class $\mathcal{M}$ of all simple binary matroids $M$ containing neither a free matroid on three elements (which we call a 'claw'), nor a Fano plane as an induced restriction. We give an exact structure theorem for this class; two of its consequences are that the matroids in $\mathcal{M}$ have unbounded critical number, while the matroids in $\mathcal{M}$ not containing the clique $M(K_5)$ as an induced restriction have critical number at most $2$.
math.CO↗
The matroid secretary problem for minor-closed classes and random matroids
We prove that for every proper minor-closed class $M$ of matroids representable over a prime field, there exists a constant-competitive matroid secretary algorithm for the matroids in $M$. This result relies on the extremely powerful matroid minor structure theory being developed by Geelen, Gerards and Whittle. We also note that for asymptotically almost all matroids, the matroid secretary algorithm that selects a random basis, ignoring weights, is $(2+o(1))$-competitive. In fact, assuming the conjecture that almost all matroids are paving, there is a $(1+o(1))$-competitive algorithm for almost all matroids.
math.CO↗
CS Indi: SU UMa-Type Dwarf Nova with Long Precursor Outburst
We observed the 2018 November outburst of CS Ind and confirmed that it was a genuine superoutburst with a very long [0.12471(1) d in average] superhump period. The superoutburst was preceded by a long precursor, which was recorded first time in SU UMa-type dwarf novae. We interpret that the combination of a sufficient amount of mass in the disk before the ignition of the outburst and the slow development of tidal instability near the borderline of the 3:1 resonance caused a cooling front to start before the full development of tidal instability. This finding provides another support to the recent interpretation of slow development of the tidal instability causing various phenomena similar to WZ Sge-type dwarf novae in SU UMa-type dwarf novae with very long orbital periods.
astro-ph.SR↗
TESS Hunt for Young and Maturing Exoplanets (THYME): A planet in the 45 Myr Tucana-Horologium association
Young exoplanets are snapshots of the planetary evolution process. Planets that orbit stars in young associations are particularly important because the age of the planetary system is well constrained. We present the discovery of a transiting planet larger than Neptune but smaller than Saturn in the 45 Myr Tucana-Horologium young moving group. The host star is a visual binary, and our follow-up observations demonstrate that the planet orbits the G6V primary component, DS Tuc A (HD 222259A, TIC 410214986). We first identified transits using photometry from the Transiting Exoplanet Survey Satellite (TESS; alerted as TOI 200.01). We validated the planet and improved the stellar parameters using a suite of new and archival data, including spectra from SOAR/Goodman, SALT/HRS and LCO/NRES; transit photometry from Spitzer; and deep adaptive optics imaging from Gemini/GPI. No additional stellar or planetary signals are seen in the data. We measured the planetary parameters by simultaneously modeling the photometry with a transit model and a Gaussian process to account for stellar variability. We determined that the planetary radius is $5.70\pm0.17$ Earth radii and that the orbital period is 8.1 days. The inclination angles of the host star's spin axis, the planet's orbital axis, and the visual binary's orbital axis are aligned within 15 degrees to within the uncertainties of the relevant data. DS Tuc Ab is bright enough (V=8.5) for detailed characterization using radial velocities and transmission spectroscopy.
astro-ph.EP↗
Fe alloy slurry and a compacting cumulate pile across Earth's inner-core boundary
Seismic observations show a reduced compressional-wave gradient at the base of the outer core relative to the preliminary reference Earth model and seismic wave asymmetry between the east-west hemispheres at the top of the inner core. Here, we propose a model for the inner core boundary (ICB), where a slurry layer forms through fractional crystallization of an Fe alloy at the base of the outer core (F layer) above a compacting cumulate pile at the top of the inner core (F' layer). Using recent mineral physics data, we show that fractional crystallization of an Fe alloy (e.g., Fe-Si-O) with light element partitioning can explain the observed reduced velocity gradient in the F layer, in cases with a solid fraction of ~15(5)% in liquid with a compositional gradient due to preferential light element partitioning into liquid. The compacting cumulate pile in the F' layer may exhibit lateral variations in thickness between the east-west hemispheres due to lateral variations of large-scale heat flow in the outer core, which may explain the east-west asymmetry observed in the seismic velocity. Our interpretations suggest that the inner core with solid Fe alloy has a high shear viscosity of ~10^23 Pa s.
physics.geo-ph↗
A Multi-Year Search For Transits Of Proxima Centauri. II: No Evidence For Transit Events With Periods Between 1-30 Days
Using a global network of small telescopes, we have obtained light curves of Proxima Centauri at 329 observation epochs from 2006 - 2017. The planet Proxima b discovered by Anglada-Escude et al. (2016) with an orbital period of 11.186 d has an a priori transit probability of ~1.5%; if it transits, the predicted transit depth is about 5 millimagnitudes. In Blank et al. (2018), we analyzed 96 of our light curves that overlapped with predicted transit ephemerides from previously published tentative transit detections, and found no evidence in our data that would corroborate claims of transits with a period of 11.186 d. Here we broaden our analysis, using 262 high-quality light curves from our data set to search for any periodic transit-like events over a range of periods from 1 - 30 d. We also inject a series of simulated planet transits and find that our data are sufficiently sensitive to have detected transits of 5 millimagnitude depth, with recoverability ranging from ~100% for an orbital period of 1 d to ~20% for an orbital period of 20 d for the parameter spaces tested. Specifically at the 11.186 d period and 5 millimagnitude transit depth, we rule out transits in our data with high confidence. We are able to rule out virtually all transits of other planets at periods shorter than 5 d and depths greater than 3 millimagnitudes; however, we cannot confidently rule out transits at the period of Proxima b due to incomplete orbital phase coverage and a lack of sensitivity to transits shallower than 4 millimagnitudes.
astro-ph.EP↗
The structure of claw-free binary matroids
A simple binary matroid is called claw-free if none of its rank-3 flats are independent sets. These objects can be equivalently defined as the sets $E$ of points in $\mathrm{PG}(n-1,2)$ for which $|E \cap P|$ is not a basis of $P$ for any plane $P$, or as the subsets $X$ of $\mathbb{F}_2^n$ containing no linearly independent triple $x,y,z$ for which $x+y,y+z,x+z,x+y+z \notin X$. We prove a decomposition theorem that exactly determines the structure of all claw-free matroids. The theorem states that claw-free matroids either belong to one of three particular basic classes of claw-free matroids, or can be constructed from these basic classes using a certain 'join' operation.
math.CO↗
Matroids with no $U_{2,n}$-minor and many hyperplanes
We construct, for every $r \ge 3$ and every prime power $q > 10$, a rank-$r$ matroid with no $U_{2,q+2}$-minor, having more hyperplanes than the rank-$r$ projective geometry over $\mathrm{GF}(q)$.
math.CO↗
A Multi-Year Search For Transits Of Proxima Centauri. I: Light Curves Corresponding To Published Ephemerides
Proxima Centauri has become the subject of intense study since the radial-velocity discovery by Anglada-Escudé et al. 2016 of a planet orbiting this nearby M-dwarf every ~ 11.2 days. If Proxima Centauri b transits its host star, independent confirmation of its existence is possible, and its mass and radius can be measured in units of the stellar host mass and radius. To date, there have been three independent claims of possible transit-like event detections in light curve observations obtained by the MOST satellite (in 2014-15), the BSST telescope in Antarctica (in 2016), and the Las Campanas Observatory (in 2016). The claimed possible detections are tentative, due in part to the variability intrinsic to the host star, and in the case of the ground-based observations, also due to the limited duration of the light curve observations. Here, we present preliminary results from an extensive photometric monitoring campaign of Proxima Centauri, using telescopes around the globe and spanning from 2006 to 2017, comprising a total of 329 observations. Considering our data that coincide directly and/or phased with the previously published tentative transit detections, we are unable to independently verify those claims. We do, however, verify the previously reported ubiquitous and complex variability of the host star. We discuss possible interpretations of the data in light of the previous claims, and we discuss future analyses of these data that could more definitively verify or refute the presence of transits associated with the radial-velocity discovered planet.
astro-ph.EP↗
A Ramsey theorem for biased graphs
A $biased\ graph$ is a pair $(G,\mathcal{B})$, where $G$ is a graph and $\mathcal{B}$ is a collection of `balanced' circuits of $G$ such that no $Θ$-subgraph of $G$ contains precisely two balanced circuits. We prove a Ramsey-type theorem, showing that if $(G,\mathcal{B})$ is a biased graph which $G$ is a very large complete graph, then $G$ contains a large complete subgraph $H$ such that the set of balanced cycles within $H$ has one of three specific, highly symmetric structures, all of which can be described naturally via group-labellings.
math.CO↗
On the Morava $E$-theory of wreath products of symmetric groups
Strickland gave an interpretation of a quotient of the Morava $E$-theory of symmetric groups in terms of algebraic geometry. We identify an analogous quotient of the Morava $E$-theory of wreath products of symmetric groups with a tensor product of Strickland's quotients. This allows an algebro-geometric interpretation similar to Strickland's.
math.AT↗
Bounding $χ$ by a fraction of $Δ$ for graphs without large cliques
The greedy coloring algorithm shows that a graph of maximum degree at most $Δ$ has chromatic number at most $Δ+ 1$, and this is tight for cliques. Much attention has been devoted to improving this "greedy bound" for graphs without large cliques. Brooks famously proved that this bound can be improved by one if $Δ\geq 3$ and the graph contains no clique of size $Δ+ 1$. Reed's Conjecture states that the "greedy bound" can be improved by $k$ if the graph contains no clique of size $Δ+ 1 - 2k$. Johansson proved that the "greedy bound" can be improved by a factor of $Ω(\ln(Δ)^{-1})$ or $Ω\left(\frac{\ln(\ln(Δ))}{\ln(Δ)}\right)$ for graphs with no triangles or no cliques of any fixed size, respectively. Notably missing is a $\textit{linear}$ improvement on the "greedy bound" for graphs without large cliques. In this paper, we prove that for sufficiently large $Δ$, if $G$ is a graph with maximum degree at most $Δ$ and no clique of size $ω$, then $$χ(G) \leq 72Δ\sqrt{\frac{\ln(ω)}{\ln(Δ)}}.$$ This implies that for sufficiently large $Δ$, if $ω^{(72c)^2} \leq Δ$ then $χ(G) \leq Δ/c$. This bound actually holds for the list-chromatic and even the correspondence-chromatic number (also known as the DP-chromatic number). In fact, we prove what we call a "local version" of it, a result implying the existence of a coloring when the number of available colors for each vertex depends on local parameters, like the degree and the clique number of its neighborhood. Our result simultaneously implies the linear improvement over the "greedy bound" and the two aforementioned results of Johansson.
math.CO↗
Almost all matroids are non-representable
We prove that, as $n$ approaches infinity, the proportion of $n$-element matroids that are representable tends to zero.
math.CO↗