arXiv subjects

Paul Smith

Publications and source records attributed to Paul Smith.

At least 37 records · Page 2Linked to original sources

arXiv2020

On the Estimation of Entropy in the FastICA Algorithm

The fastICA method is a popular dimension reduction technique used to reveal patterns in data. Here we show both theoretically and in practice that the approximations used in fastICA can result in patterns not being successfully recognised. We demonstrate this problem using a two-dimensional example where a clear structure is immediately visible to the naked eye, but where the projection chosen by fastICA fails to reveal this structure. This implies that care is needed when applying fastICA. We discuss how the problem arises and how it is intrinsically connected to the approximations that form the basis of the computational efficiency of fastICA.

stat.ML↗

arXiv2020

SN 2014ab: An Aspherical Type IIn Supernova with Low Polarization

We present photometry, spectra, and spectropolarimetry of supernova (SN) 2014ab, obtained through $\sim 200$ days after peak brightness. SN 2014ab was a luminous Type IIn SN ($M_V < -19.14$ mag) discovered after peak brightness near the nucleus of its host galaxy, VV 306c. Prediscovery upper limits constrain the time of explosion to within 200 days prior to discovery. While SN 2014ab declined by $\sim 1$ mag over the course of our observations, the observed spectrum remained remarkably unchanged. Spectra exhibit an asymmetric emission-line profile with a consistently stronger blueshifted component, suggesting the presence of dust or a lack of symmetry between the far side and near side of the SN. The Pa$β$ emission line shows a profile very similar to that of H$α$, implying that this stronger blueshifted component is caused either through obscuration by large dust grains, occultation by optically thick material, or a lack of symmetry between the far side and near side of the interaction region. Despite these asymmetric line profiles, our spectropolarimetric data show that SN 2014ab has little detected polarization after accounting for the interstellar polarization. This suggests that we are seeing emission from a photosphere that has only small deviation from circular symmetry face-on. We are likely seeing a SN IIn with nearly circular symmetry in the plane normal to our line of sight, but with either large-grain dust or significant asymmetry in the density of circumstellar material or SN ejecta along our line of sight. We suggest that SN 2014ab and SN 2010jl (as well as other SNe IIn) may be similar events viewed from different directions.

astro-ph.SR↗

arXiv2020

What Makes Ly$α$ Nebulae Glow? Mapping the Polarization of LABd05

"Ly$α$ nebulae" are giant ($\sim$100 kpc), glowing gas clouds in the distant universe. The origin of their extended Ly$α$ emission remains a mystery. Some models posit that Ly$α$ emission is produced when the cloud is photoionized by UV emission from embedded or nearby sources, while others suggest that the Ly$α$ photons originate from an embedded galaxy or AGN and are then resonantly scattered by the cloud. At least in the latter scenario, the observed Ly$α$ emission will be polarized. To test these possibilities, we are conducting imaging polarimetric observations of seven Ly$α$ nebulae. Here we present our results for LABd05, a cloud at $z$ = 2.656 with an obscured, embedded AGN to the northeast of the peak of Ly$α$ emission. We detect significant polarization. The highest polarization fractions $P$ are $\sim$10-20% at $\sim$20-40 kpc southeast of the Ly$α$ peak, away from the AGN. The lowest $P$, including upper-limits, are $\sim$5% and lie between the Ly$α$ peak and AGN. In other words, the polarization map is lopsided, with $P$ increasing from the Ly$α$ peak to the southeast. The measured polarization angles $θ$ are oriented northeast, roughly perpendicular to the $P$ gradient. This unique polarization pattern suggests that 1) the spatially-offset AGN is photoionizing nearby gas and 2) escaping Ly$α$ photons are scattered by the nebula at larger radii and into our sightline, producing tangentially-oriented, radially-increasing polarization away from the photoionized region. Finally we conclude that the interplay between the gas density and ionization profiles produces the observed central peak in the Ly$α$ emission. This also implies that the structure of LABd05 is more complex than assumed by current theoretical spherical or cylindrical models.

astro-ph.GA↗

arXiv2019

Repeated out of Sample Fusion in the Estimation of Small Tail Probabilities

Often, it is required to estimate the probability that a quantity such as toxicity level, plutonium, temperature, rainfall, damage, wind speed, wave size, earthquake magnitude, risk, etc., exceeds an unsafe high threshold. The probability in question is then very small. To estimate such a probability, information is needed about large values of the quantity of interest. However, in many cases, the data only contain values below or even far below the designated threshold, let alone exceedingly large values. It is shown that by repeated fusion of the data with externally generated random data, more information about small tail probabilities is obtained with the aid of certain new statistical functions. This provides relatively short, yet reliable interval estimates based on moderately large samples. A comparison of the approach with a method from extreme values theory (Peaks over Threshold, or POT), using both artificial and real data, points to the merit of repeated out of sample fusion.

stat.ME↗

arXiv2018

Nucleation and growth in two dimensions

We consider a dynamical process on a graph $G$, in which vertices are infected (randomly) at a rate which depends on the number of their neighbours that are already infected. This model includes bootstrap percolation and first-passage percolation as its extreme points. We give a precise description of the evolution of this process on the graph $\mathbb{Z}^2$, significantly sharpening results of Dehghanpour and Schonmann. In particular, we determine the typical infection time up to a constant factor for almost all natural values of the parameters, and in a large range we obtain a stronger, sharp threshold.

math.PR↗

arXiv2017

SN2012ab: A Peculiar Type IIn Supernova with Aspherical Circumstellar Material

We present photometry, spectra, and spectropolarimetry of supernova (SN) 2012ab, mostly obtained over the course of $\sim 300$ days after discovery. SN 2012ab was a Type IIn (SN IIn) event discovered near the nucleus of spiral galaxy 2MASXJ12224762+0536247. While its light curve resembles that of SN 1998S, its spectral evolution does not. We see indications of CSM interaction in the strong intermediate-width emission features, the high luminosity (peak at absolute magnitude $M=-19.5$), and the lack of broad absorption features in the spectrum. The H$α$ emission undergoes a peculiar transition. At early times it shows a broad blue emission wing out to $-14{,}000$ km $\mathrm{s^{-1}}$ and a truncated red wing. Then at late times ($>$ 100$\,$days) it shows a truncated blue wing and a very broad red emission wing out to roughly $+20{,}000$ km $\mathrm{s^{-1}}$. This late-time broad red wing probably arises in the reverse shock. Spectra also show an asymmetric intermediate-width H$α$ component with stronger emission on the red side at late times. The evolution of the asymmetric profiles requires a density structure in the distant CSM that is highly aspherical. Our spectropolarimetric data also suggest asphericity with a strong continuum polarization of $\sim 1-3$% and depolarization in the H$α$ line, indicating asphericity in the CSM at a level comparable to that in other SNe IIn. We estimate a mass-loss rate of $\dot{M} = 0.050\, {\rm M}_{\odot}\,\mathrm{yr^{-1}}$ for $v_{\rm pre} = 100$$\,$km$\,$$\mathrm{s^{-1}}$ extending back at least 75$\,$yr prior to the SN. The strong departure from axisymmetry in the CSM of SN 2012ab may suggest that the progenitor was an eccentric binary system undergoing eruptive mass loss.

astro-ph.SR↗

arXiv2017

The threshold for jigsaw percolation on random graphs

Jigsaw percolation is a model for the process of solving puzzles within a social network, which was recently proposed by Brummitt, Chatterjee, Dey and Sivakoff. In the model there are two graphs on a single vertex set (the `people' graph and the `puzzle' graph), and vertices merge to form components if they are joined by an edge of each graph. These components then merge to form larger components if again there is an edge of each graph joining them, and so on. Percolation is said to occur if the process terminates with a single component containing every vertex. In this note we determine the threshold for percolation up to a constant factor, in the case where both graphs are Erdős--Rényi random graphs.

math.PR↗

arXiv2016

Mapping the Polarization of the Radio-Loud Ly$α$ Nebula B3 J2330+3927

Lya nebulae, or "Lya blobs", are extended (up to ~100 kpc), bright (L[Lya] > 10^43 erg/s) clouds of Lya emitting gas that tend to lie in overdense regions at z ~ 2--5. The origin of the Lya emission remains unknown, but recent theoretical work suggests that measuring the polarization might discriminate among powering mechanisms. Here we present the first narrowband, imaging polarimetry of a radio-loud Lya nebula, B3 J2330+3927 at z=3.09, with an embedded active galactic nucleus (AGN). The AGN lies near the blob's Lya emission peak and its radio lobes align roughly with the blob's major axis. With the SPOL polarimeter on the 6.5m MMT telescope, we map the total (Lya + continuum) polarization in a grid of circular apertures of radius 0.6" (4.4kpc), detecting a significant (>2sigma) polarization fraction P in nine apertures and achieving strong upper-limits (as low as 2%) elsewhere. P increases from <2% at ~5kpc from the blob center to ~17% at ~15-25kpc. The detections are distributed asymmetrically, roughly along the nebula's major axis. The polarization angles theta are mostly perpendicular to this axis. Comparing the Lya flux to that of the continuum, and conservatively assuming that the continuum is highly polarized (20-100%) and aligned with the total polarization, we place lower limits on the polarization of the Lya emission P(Lya) ranging from no significant polarization at ~5 kpc from the blob center to ~ 3--17% at 10--25kpc. Like the total polarization, the Lya polarization detections occur more often along the blob's major axis.

astro-ph.GA↗

arXiv2016

Monotone cellular automata in a random environment

In this paper we study in complete generality the family of two-state, deterministic, monotone, local, homogeneous cellular automata in $\mathbb{Z}^d$ with random initial configurations. Formally, we are given a set $\mathcal{U}=\{X_1,\dots,X_m\}$ of finite subsets of $\mathbb{Z}^d\setminus\{\mathbf{0}\}$, and an initial set $A_0\subset\mathbb{Z}^d$ of `infected' sites, which we take to be random according to the product measure with density $p$. At time $t\in\mathbb{N}$, the set of infected sites $A_t$ is the union of $A_{t-1}$ and the set of all $x\in\mathbb{Z}^d$ such that $x+X\in A_{t-1}$ for some $X\in\mathcal{U}$. Our model may alternatively be thought of as bootstrap percolation on $\mathbb{Z}^d$ with arbitrary update rules, and for this reason we call it $\mathcal{U}$-bootstrap percolation. In two dimensions, we give a classification of $\mathcal{U}$-bootstrap percolation models into three classes -- supercritical, critical and subcritical -- and we prove results about the phase transitions of all models belonging to the first two of these classes. More precisely, we show that the critical probability for percolation on $(\mathbb{Z}/n\mathbb{Z})^2$ is $(\log n)^{-Θ(1)}$ for all models in the critical class, and that it is $n^{-Θ(1)}$ for all models in the supercritical class. The results in this paper are the first of any kind on bootstrap percolation considered in this level of generality, and in particular they are the first that make no assumptions of symmetry. It is the hope of the authors that this work will initiate a new, unified theory of bootstrap percolation on $\mathbb{Z}^d$.

math.PR↗

arXiv2016

The sharp threshold for the Duarte model

The class of critical bootstrap percolation models in two dimensions was recently introduced by Bollobás, Smith and Uzzell, and the critical threshold for percolation was determined up to a constant factor for all such models by the authors of this paper. Here we develop and refine the techniques introduced in that paper in order to determine a sharp threshold for the Duarte model. This resolves a question of Mountford from 1995, and is the first result of its type for a model with drift.

math.PR↗

arXiv2016

Asymmetries in SN 2014J Near Maximum Light Revealed Through Spectropolarimetry

We present spectropolarimetric observations of the nearby Type Ia SN 2014J in M82 over six epochs: +0, +7, +23, +51, +77, +109, and +111 days with respect to B-band maximum. The strong continuum polarization, which is constant with time, shows a wavelength dependence unlike that produced by linear dichroism in Milky Way dust. The observed polarization may be due entirely to interstellar dust or include a circumstellar scattering component. We find that the polarization angle aligns with the magnetic field of the host galaxy, arguing for an interstellar origin. Additionally, we confirm a peak in polarization at short wavelengths that would imply $R_V < 2 $ along the light of sight, in agreement with earlier polarization measurements. For illustrative purposes, we include a two component fit to the continuum polarization of our +51 day epoch that combines a circumstellar scattering component with interstellar dust where scattering can account for over half of the polarization at $4000$ Å. Upon removal of the interstellar polarization signal, SN 2014J exhibits very low levels of continuum polarization. Asymmetries in the distribution of elements within the ejecta are visible through moderate levels of time-variable polarization in accordance with the Si II 6355 Åabsorption line. At maximum light, the line polarization reaches $\sim0.6$% and decreases to $\sim0.4\%$ one week later. This feature also forms a loop on the $q_{RSP}$-$u_{RSP}$ plane illustrating that the ion does not have an axisymmetric distribution. The observed polarization properties suggest the explosion geometry of SN 2014J is generally spheroidal with a clumpy distribution of silicon.

astro-ph.HE↗

arXiv2015

The time of bootstrap percolation in two dimensions

We study the distribution of the percolation time $T$ of two-neighbour bootstrap percolation on $[n]^2$ with initial set $A\sim\mathrm{Bin}([n]^2,p)$. We determine $T$ with high probability up to a constant factor for all $p$ above the critical probability for percolation, and to within a $1+o(1)$ factor for a large range of $p$.

math.PR↗

arXiv2015

Enigmas from the Sloan Digital Sky Survey DR7 Kleinman White Dwarf Catalog

We report results from a continuation of our searches for high field magnetic white dwarfs paired in a detached binary with non degenerate companions. We made use of the Sloan Digital Sky Survey DR7 catalog of Kleinman et al. (2013) with 19,712 spectroscopically-identified white dwarfs. These include 1,735 white dwarf plus M dwarf detached pairs (almost 10\% of the Kleinman at al.'s list). No new pairs were found, although we did recover the polar (AM~Herculis system) ST\,LMi in a low state of accretion. With the larger sample the original situation reported ten years ago remains intact now at a much higher level of statistical significance: in the selected SDSS sample, high field magnetic white dwarfs are not found in an apparently-detached pairing with an M dwarf, unless they are a magnetic CV in a low state of accretion. This finding strengthens the case that the fields in the isolated high field magnetic white dwarfs are generated by stellar mergers but also raises questions on the nature of the progenitors of the magnetic CVs.

astro-ph.SR↗

arXiv2014

On the unconditional uniqueness of solutions to the infinite radial Chern-Simons-Schrödinger hierarchy

In this article we establish the unconditional uniqueness of solutions to an Infinite Radial Chern-Simons-Schrödinger (IRCSS) hierarchy in two spatial dimensions. The IRCSS hierarchy is a system of infinitely many coupled PDEs that describes the limiting Chern-Simons-Schrödinger dynamics of infinitely many interacting anyons. The anyons are two dimensional objects which interact through a self-generated field. Due to the interactions with the self-generated field, the IRCSS hierarchy is a system of nonlinear PDEs, which distinguishes it from the linear infinite hierarchies studied previously. Factorized solutions of the IRCSS hierarchy are determined by solutions of the Chern-Simons-Schrödinger system. Our result therefore implies the unconditional uniqueness of solutions to the radial Chern-Simons-Schrödinger system as well.

math.AP↗

arXiv2014

Subcritical $\mathcal{U}$-bootstrap percolation models have non-trivial phase transitions

We prove that there exist natural generalizations of the classical bootstrap percolation model on $\mathbb{Z}^2$ that have non-trivial critical probabilities, and moreover we characterize all homogeneous, local, monotone models with this property. Van Enter (in the case $d=r=2$) and Schonmann (for all $d \geq r \geq 2$) proved that $r$-neighbour bootstrap percolation models have trivial critical probabilities on $\mathbb{Z}^d$ for every choice of the parameters $d \geq r \geq 2$: that is, an initial set of density $p$ almost surely percolates $\mathbb{Z}^d$ for every $p>0$. These results effectively ended the study of bootstrap percolation on infinite lattices. Recently Bollobás, Smith and Uzzell introduced a broad class of percolation models called $\mathcal{U}$-bootstrap percolation, which includes $r$-neighbour bootstrap percolation as a special case. They divided two-dimensional $\mathcal{U}$-bootstrap percolation models into three classes -- subcritical, critical and supercritical -- and they proved that, like classical 2-neighbour bootstrap percolation, critical and supercritical $\mathcal{U}$-bootstrap percolation models have trivial critical probabilities on $\mathbb{Z}^2$. They left open the question as to what happens in the case of subcritical families. In this paper we answer that question: we show that every subcritical $\mathcal{U}$-bootstrap percolation model has a non-trivial critical probability on $\mathbb{Z}^2$. This is new except for a certain `degenerate' subclass of symmetric models that can be coupled from below with oriented site percolation. Our results re-open the study of critical probabilities in bootstrap percolation on infinite lattices, and they allow one to ask many questions of subcritical bootstrap percolation models that are typically asked of site or bond percolation.

math.PR↗

arXiv2014

The sharp threshold for maximum-size sum-free subsets in even-order abelian groups

We study sum-free sets in sparse random subsets of even order abelian groups. In particular, we determine the sharp threshold for the following property: the largest such set is contained in some maximum-size sum-free subset of the group. This theorem extends recent work of Balogh, Morris and Samotij, who resolved the case G = Z_{2n}, and who obtained a weaker threshold (up to a constant factor) in general.

math.CO↗

arXiv2014

An unconstrained Lagrangian formulation and conservation laws for the Schrödinger map system

We consider energy-critical Schrödinger maps from R^2 into the sphere and hyperbolic plane. Viewing such maps with respect to orthonormal frames on the pullback bundle provides a gauge field formulation of the evolution. We show that this gauge field system is the set of Euler-Lagrange equations corresponding to an action that includes a Chern-Simons term. We also introduce the stress-energy tensor and derive conservation laws. In conclusion we offer comparisons between Schrödinger maps and the closely related Chern-Simons-Schrödinger system.

math.AP↗

arXiv2014

Global wellposedness of the equivariant Chern-Simons-Schrödinger equation

In this article we consider the initial value problem for the m-equivariant Chern-Simons-Schrödinger model in two spatial dimensions with real-valued coupling parameter g. This is a covariant NLS type problem that is L^2-critical. We prove that at the critical regularity, for any integer-valued equivariance index m, the initial value problem in the defocusing case (g < 1) is globally wellposed and the solution scatters. The problem is focusing when g >= 1, and in this case we prove that for nonnegative integer-valued equivariance indices m there exist constants c = c_{m, g} such that, at the critical regularity, the initial value problem is globally wellposed and the solution scatters when the L^2 initial data phi_0 is m-equivariant and has L^2-norm less than the square root of c_{m, g}. We also show that c_{m, g}^{1/2} is equal to the minimum L^2 norm of a nontrivial m-equivariant standing wave solution. In the self-dual g = 1 case, we have the exact numerical values c_{m, 1} = 8*pi*(m + 1).

math.AP↗