Rm Sh Research Articles

Shocks in the stacked Sunyaev-Zel’dovich profiles of clusters II: Measurements from SPT-SZ + Planck Compton-y map

ABSTRACT We search for the signature of cosmological shocks in stacked gas pressure profiles of galaxy clusters using data from the South Pole Telescope (SPT). Specifically, we stack the latest Compton-y maps from the 2500 deg2 SPT-SZ survey on the locations of clusters identified in that same data set. The sample contains 516 clusters with mean mass $\langle M_{\rm 200m}\rangle = 10^{14.9} \, {\rm M}_\odot$ and redshift 〈z〉 = 0.55. We analyse in parallel a set of zoom-in hydrodynamical simulations from the three hundred project. The SPT-SZ data show two features: (i) a pressure deficit at R/R200m = 1.08 ± 0.09, measured at 3.1σ significance and not observed in the simulations, and; (ii) a sharp decrease in pressure at R/R200m = 4.58 ± 1.24 at 2.0σ significance. The pressure deficit is qualitatively consistent with a shock-induced thermal non-equilibrium between electrons and ions, and the second feature is consistent with accretion shocks seen in previous studies. We split the cluster sample by redshift and mass, and find both features exist in all cases. There are also no significant differences in features along and across the cluster major axis, whose orientation roughly points towards filamentary structure. As a consistency test, we also analyse clusters from the Planck and Atacama Cosmology Telescope Polarimeter surveys and find quantitatively similar features in the pressure profiles. Finally, we compare the accretion shock radius ($R_{\rm sh,\, acc}$) with existing measurements of the splashback radius (Rsp) for SPT-SZ and constrain the lower limit of the ratio, $R_{\rm sh,\, acc}/R_{\rm sp}\gt 2.16 \pm 0.59$.

Spin Hall effect in a thin Pt film

A density-functional-theory based relativistic scattering formalism is used to study charge transport through thin Pt films with room temperature lattice disorder. A Fuchs-Sondheimer specularity coefficient $p \sim 0.5$ is needed to describe the suppression of the charge current at the surface even in the absence of surface roughness. The charge current drives a spin Hall current perpendicular to the surface. Analysing the latter with a model that is universally used to interpret the spin Hall effect in thin films and layered materials, we are unable to recover values of the spin-flip diffusion length $l_{\rm sf}$ and spin Hall angle $\Theta_{\rm sH}$ that we obtain for bulk Pt using the same approximations. We trace this to the boundary conditions used and develop a generalized model that takes surface effects into account. A reduced value of $\Theta_{\rm sH}$ at the surface is then found to describe the first-principles transport results extremely well. The in-plane spin Hall effect is substantially enhanced at the surface.

Spectroscopic Studies of 30 Short-period Cataclysmic Variable Stars and Remarks on the Evolution and Population of Similar Objects

Abstract We present spectroscopy and orbital periods P orb for 30 apparently nonmagnetic cataclysmic binaries with periods below ∼3 hr, nearly all of which are dwarf novae, mostly of the SU Ursae Majoris subclass. We then turn to the evidence supporting the prediction that short-period dwarf novae evolve toward longer periods after passing through a minimum period—the “period bounce” phenomenon. Plotting data from the literature reveals that for superhump period excess ϵ = (P sh − P orb)/P orb below ∼0.015, the period appears to increase with decreasing ϵ, agreeing at least qualitatively with the predicted behavior. Next, motivated by the long (decadal) outburst intervals of the WZ Sagittae subclass of short-period dwarf novae, we ask whether there could be a sizable population of “lurkers”—systems that resemble dwarf novae at minimum light, but which do not outburst over accessible timescales (or at all) and therefore do not draw attention to themselves. By examining the outburst history of the Sloan Digital Sky Survey sample of CVs, which were selected by color and not by outburst, we find that a large majority of the color-selected dwarf-nova-like objects have been observed to outburst, and we conclude that “lurkers,” if they exist, are a relatively minor part of the CV population.

Schwarzschild black holes have the largest size

We consider static black holes in Einstein gravity and study parameters characterizing the black hole size, namely the radii of the horizon $R_+$, photon sphere $R_{\rm ph}$ and black hole shadow $R_{\rm sh}$. We find a sequence of inequalities $\frac32 R_+\, \le\, R_{\rm ph}\, \le \,\frac{1}{\sqrt3} R_{\rm sh}\,\le\, 3 M$, where $M$ is the black hole mass. These are consistent with and beyond the previously known upper bounds in literature. The Schwarzschild black hole saturates all the inequalities, making it the biggest of all for given mass. The inequalities include an upper bound of entropy for any quantum system with given energy. We also point out that some black holes satisfying the dominant energy condition can trap photons to form a stable photon shield outside the event horizon, but the shadow hides it from an observer at infinity.

Calculating spin transport properties from first principles: Spin currents

Local charge and spin currents are evaluated from the solutions of fully relativistic quantum mechanical scattering calculations for systems that include temperature-induced lattice and spin disorder as well as intrinsic alloy disorder. This makes it possible to determine material-specific spin transport parameters at finite temperatures. Illustrations are given for a number of important materials and parameters at 300 K. The spin-flip diffusion length $l_{\rm sf}$ of Pt is determined from the exponential decay of a spin current injected into a long length of thermally disordered Pt; we find $l_{\rm sf}^{\rm Pt}= 5.3\pm0.4 \,$nm. For the ferromagnetic substitutional disordered alloy Permalloy (Py), we inject currents that are fully polarized parallel and antiparallel to the magnetization and calculate $l_{\rm sf}$ from the exponential decay of their difference; we find $l_{\rm sf}^{\rm Py}= 2.8 \pm 0.1 \,$nm. The transport polarization $\beta$ is found from the asymptotic polarization of a charge current in a long length of Py to be $\beta = 0.75 \pm 0.01$. The spin Hall angle $\Theta_{\rm sH}$ is determined from the transverse spin current induced by the passage of a longitudinal charge current in thermally disordered Pt; our best estimate is $\Theta_{\rm sH}^{\rm Pt}=4.5 \pm 1 \%$ corresponding to the experimental room temperature bulk resistivity $\rho =10.8 \mu \Omega \,$cm.

Ambient noise Love wave tomography of China

We first report on the Love wave tomography of China based on ambient noise cross-correlations. We used 3 years of continuous waveform data recorded by 206 broadband seismic stations on the Chinese Mainland and 36 neighboring global stations and obtained Love wave empirical Green’s functions from cross-correlations of the horizontal components. The Love wave group velocity dispersion measurements were used to construct dispersion maps of 8- to 40-s periods, which were then inverted to obtain a three-dimensional horizontally polarized S-wave (SH) velocity structure. The resolution was approximately 4° × 4° and 8° × 8° for eastern and western China, respectively, and extended to a depth of approximately 50 km. The SH model was generally consistent with a previously published vertically polarized S-wave (SV) model and showed large-scale features that were consistent with geological units, such as the major basins and changes in the crustal thickness across the north-south gravity lineament. The SH and SV models also showed substantial differences, which were used to examine the subsurface radial anisotropy. We define the radial anisotropy parameter as \begin{document}$\psi = 2\left( {{V_{\rm SH}} - {V_{\rm SV}}} \right)/\left( {{V_{\rm SH}} + {V_{\rm SV}}} \right)$\end{document} . At a shallow depth, we observed significant radial anisotropy under major basins, which may be related to thick sedimentary layers. At the mid to lower crust, most of the Chinese continent showed strong positive radial anisotropy (SH > SV). Central and southern Tibet showed strong positive anisotropy, whereas the radial anisotropy was relatively weak at the northern and eastern margins, which suggests a change in deformation style from the plateau interior to its margins. The North China craton showed prominent positive radial anisotropy, which may be related to decratonization and strong extension since the Mesozoic Era. Love waves are less well retrieved than Rayleigh waves from ambient noise cross-correlations. Increasing the duration of the cross-correlation data beyond 4 to 8 years may not aid in retrieving Love waves of longer periods, for which improved methods need to be explored.

The Multi-Dimensional Structure of Radiative Shocks: Suppressed Thermal X-rays and Relativistic Ion Acceleration

Radiative shocks, behind which gas cools faster than the dynamical time, play a key role in many astrophysical transients, including classical novae and young supernovae interacting with circumstellar material. The dense layer behind high Mach number $\mathcal{M} \gg 1$ radiative shocks is susceptible to thin-shell instabilities, creating a "corrugated" shock interface. We present two and three-dimensional hydrodynamical simulations of optically-thin radiative shocks to study their thermal radiation and acceleration of non-thermal relativistic ions. We employ a moving-mesh code and a specialized numerical technique to eliminate artificial heat conduction across grid cells. The fraction of the shock's luminosity $L_{\rm tot}$ radiated at X-ray temperatures $kT_{\rm sh} \approx (3/16)\mu m_p v_{\rm sh}^{2}$ expected from a one-dimensional analysis is suppressed by a factor $L(>T_{\rm sh}/3)/L_{\rm tot} \approx 4.5/\mathcal{M}^{4/3}$ for $\mathcal{M} \approx 4-36$. This suppression results in part from weak shocks driven into under-pressured cold filaments by hot shocked gas, which sap thermal energy from the latter faster than it is radiated. Combining particle-in-cell simulation results for diffusive shock acceleration with the inclination angle distribution across the shock (relative to an upstream magnetic field in the shock plane$-$the expected geometry for transient outflows), we predict the efficiency and energy spectrum of ion acceleration. Though negligible acceleration is predicted for adiabatic shocks, the corrugated shock front enables local regions to satisfy the quasi-parallel magnetic field geometry required for efficient acceleration, resulting in an average acceleration efficiency of $\epsilon_{\rm nth} \sim 0.005-0.02$ for $\mathcal{M} \approx 12-36$, in agreement with modeling of the gamma-ray nova ASASSN-16ma.

The Spin Seebeck and Spin Peltier Reciprocal Relation

We derive the reciprocal relation between electric and thermal quantities of the spin Seebeck and the spin Peltier effects. For a Pt/yttrium-iron garnet bilayer device, the reciprocity assumes the remarkably simple form $T \Delta V_{e,y}/I_{q,x} = \Delta T_{\text{SP},x}/I_{e,y}$ , where $\Delta V_{e,y}$ is the voltage generated at the Pt metal in a spin Seebeck experiment under the heat current $I_{q,x}$ and $\Delta T_{\text{SP},x}$ is the temperature difference generated in a spin Peltier experiment with an electric current $I_{e,y}$ flowing in the Pt metal. The ratios are related to intrinsic coefficients: the spin Hall angle $\theta _{{\rm SH}}$ of Pt and the thermomagnetic power coefficient $\epsilon _M$ of yttrium-iron garnet.

Shocks in nova outflows – II. Synchrotron radio emission

The discovery of GeV gamma-rays from classical novae indicates that shocks and relativistic particle acceleration are energetically key in these events. Further evidence for shocks comes from thermal keV X-ray emission and an early peak in the radio light curve on a timescale of months with a brightness temperature which is too high to result from freely expanding photo-ionized gas. Paper I developed a one dimensional model for the thermal emission from nova shocks. This work concluded that the shock-powered radio peak cannot be thermal if line cooling operates in the post-shock gas at the rate determined by collisional ionization equilibrium. Here we extend this calculation to include non-thermal synchrotron emission. Applying our model to three classical novae, we constrain the amplification of the magnetic field $\epsilon_B$ and the efficiency $\epsilon_e$ of accelerating relativistic electrons of characteristic Lorentz factor $\gamma \sim 100$. If the shocks are radiative (low velocity $v_{\rm sh} \lesssim 1000$ km s$^{-1}$) and cover a large solid angle of the nova outflow, as likely characterize those producing gamma-rays, then values of $\epsilon_e \sim 0.01-0.1$ are required to achieve the peak radio brightness for $\epsilon_B = 10^{-2}$. Such high efficiencies exclude secondary pairs from pion decay as the source of the radio-emitting particles, instead favoring the direct acceleration of electrons at the shock. If the radio-emitting shocks are instead adiabatic (high velocity), as likely characterize those responsible for the thermal X-rays, then much higher brightness temperatures are possible, allowing the radio-emitting shocks to cover a smaller outflow solid angle.

Codiffusion Sources and Barriers for the Assembly of Back-Contact Back-Junction Solar Cells

In this study, several diffusion sources are investigated, aiming at codiffusion for the fabrication of back-contact back-junction (BC-BJ) silicon solar cell. As a gaseous diffusion source, a POCl3-diffusion process is investigated, and for solid diffusion sources, phosphorus- and boron-doped silicate glass (PSG and BSG) deposited by the means of plasma-enhanced chemical vapor deposition are considered. The n+-doped areas diffused from a solid PSG layer allow for a precise adjustment of the sheet resistance $(R_{{\rm sh}})$ in the range of 40–400 Ω/sq, along with a dark saturation current density $(J_{o})$ of 55 fA/cm $^{2}$ . Subsequently, boron diffusion from solid BSG layer leads to p+-doped areas with high doping levels ( $R_{\rm sh}=$ 50 Ω/sq). However, gaseous POCl3 diffusion in combination with solid boron diffusion from the BSG layer can only be successfully performed if the BSG layer is protected with an SiO x layer. Furthermore, by adjusting the gas flows during POCl3 diffusion, n+-doped areas with $R_{{\rm sh}}$ in the range of 150– 300 Ω/sq are achieved. The corresponding surfaces feature $J_{o}$ values of 30 fA/cm $^{2}$ . The result of this study is a flexible codiffusion setup allowing for the efficient integration in advanced process chains of BC-BJ solar cells which results in the cell efficiencies well above above $\eta =$ 20%.

Numerical Analysis of Radiative Recombination and Reabsorption in GaAs/Si Tandem

We demonstrate a numerical analysis of the device impact of photon reabsorption on single-junction GaAs and tandem GaAs/Si solar cells. A self-consistent optical–electrical model that considers nonideal losses within the devices is developed. For single-junction devices, we find that the impact of photon recycling on the voltage increases monotonically with the injection level. For record-level GaAs solar cells, the voltage boost is 33 mV under open-circuit conditions and 13 mV at the maximum power point. For tandem GaAs/Si solar cells, photon reabsorption moderates the sensitivity of tandem efficiency to both obvious parameters like absorber thickness and implicit parameters like shunt resistance ( $R_{{\rm sh}}$ ) and bulk lifetime. Considering luminescent coupling results in a GaAs top cell that is 9.5% thicker than without luminescent coupling. The tandem device is 50% more sensitive to $R_{{\rm sh}}$ changes in the GaAs cell than $R_{{\rm sh}}$ changes in the Si cell. The impact of the GaAs top-cell bulk lifetime on tandem efficiency is reduced by 61% if photon reabsorption is not considered. This integrated optoelectronic device model allows one quantification of the implicit effects of photon recycling and luminescent coupling on device parameters for GaAs/Si tandem, providing a valuable tool for high-performance device optimization.

Intelligent Islanding and Seamless Reconnection Technique for Microgrid With UPQC

A new proposal for the placement, integration, and control of unified power quality conditioner (UPQC) in distributed generation (DG)-based grid connected/autonomous microgrid/microgeneration ( $\mu $ G) system has been presented here. The DG converters (with storage) and the shunt part of the UPQC Active Power Filter (APF $_{\rm sh}$ ) is placed at the Point of Common Coupling (PCC). The series part of the UPQC (APF $_{\rm se}$ ) is connected before the PCC and in series with the grid. The dc link can also be integrated with the storage system. An intelligent islanding detection and reconnection technique (IR) are introduced in the UPQC as a secondary control. Hence, it is termed as ${\rm UPQC}_{{\mu {\rm G}{-}{\rm IR}}}$ . The advantages of the proposed ${\rm UPQC}_{{\mu {\rm G}{-}{\rm IR}}}$ over the normal UPQC are to compensate voltage interruption in addition to voltage sag/swell, harmonic, and reactive power compensation in the interconnected mode. During the interconnected and islanded mode, DG converter with storage will supply the active power only and the shunt part of the UPQC will compensate the reactive and harmonic power of the load. It also offers the DG converter to remain connected during the voltage disturbance including phase jump.

THROUGH THE RING OF FIRE:γ-RAY VARIABILITY IN BLAZARS BY A MOVING PLASMOID PASSING A LOCAL SOURCE OF SEED PHOTONS

Blazars exhibit flares across the electromagnetic spectrum. Many $\gamma$-ray flares are highly correlated with flares detected at optical wavelengths; however, a small subset appears to occur in isolation, with little or no variability detected at longer wavelengths. These "orphan" $\gamma$-ray flares challenge current models of blazar variability, most of which are unable to reproduce this type of behavior. We present numerical calculations of the time-variable emission of a blazar based on a proposal by Marscher et al. (2010) to explain such events. In this model, a plasmoid ("blob") propagates relativistically along the spine of a blazar jet and passes through a synchrotron-emitting ring of electrons representing a shocked portion of the jet sheath. This ring supplies a source of seed photons that are inverse-Compton scattered by the electrons in the moving blob. The model includes the effects of radiative cooling, a spatially varying magnetic field, and acceleration of the blob's bulk velocity. Synthetic light curves produced by our model are compared to the observed light curves from an orphan flare that was coincident with the passage of a superluminal knot through the inner jet of the blazar PKS 1510$-$089. In addition, we present Very Long Baseline Array polarimetric observations that point to the existence of a jet sheath in PKS 1510$-$089, thus providing further observational support for the plausibility of our model. An estimate of the bolometric luminosity of the sheath within PKS 1510$-$089 is made, yielding $L_{\rm sh} \sim 3 \times 10^{45} ~ \rm erg ~ \rm s^{-1}$. This indicates that the sheath within PKS 1510$-$089 is potentially a very important source of seed photons.

Correlation Between Enhanced $T_{\rm c}$ , Orthorhombicity and the Volume of the Unit Cell in Argon Preheated $( \hbox{Y}_{1 - x}\hbox{Nd}_{x})\hbox{SrBaCu}_{3}\hbox{O}_{6 + z}$

We report on the preparation, X-ray diffraction with Rietveld refinement, ac magnetic susceptibility measurements, and effect of heat treatments in $(\hbox{Y}_{1 - {\rm x}} \hbox{Nd}x)\hbox{SrBaCu}_{3}\hbox{O}_{6+{\rm z}}$ .( $x = 0$ , 0.2, 0.4, 0.5, 0.6, 0.8, 1). Each sample was subject to two types of heat treatment: oxygen annealing [O] and argon annealing followed by oxygen annealing [AO]. For each $x$ , the [AO] heat treatment increases the orthorhombicity $\varepsilon = (b - a)/(b + a)$ (for $0 \leq x ), the distance $d[\hbox{Cu}(1)-(\hbox{Sr/Ba})]$ for $x and $T_{\rm c}$ (for $x > 0.2$ ). The latter increases by 9.8 to 77.2 K for $x = 1[\hbox{AO}]$ . When $x$ decreases from 1 to 0, $T_{\rm c}$ increases with $\varepsilon$ , with a transition from tetragonal to orthorhombic structure in the samples [O]. Remarkable correlations were observed between $T_{\rm c}(x)$ and the volume of the unit cell $V(x)$ ; and between $\delta T_{\rm c}(x) = T_{\rm c}[\hbox{AO}] - T_{\rm c}[\hbox{O}]$ and $ \delta\varepsilon(x)$ . A combination of several factors such as decrease in $d[\hbox{Cu}(1)-(\hbox{Sr/Ba})]$ ; increase in cationic and chain oxygen ordering; the number $p_{\rm sh}(x)$ of holes by $\hbox{Cu}(2)-\hbox{O}_{2}$ superconducting planes and in-phase purity for the [AO] samples may account for the observed data.

Uncertainty Analysis in Photovoltaic Cell Parameter Estimation

This paper describes a theoretical approach to evaluate the uncertainty on the series and shunt resistances estimated by the seven-parameter (double diode) model of a photovoltaic (PV) cell using data commonly provided by panel manufacturers, measured environmental parameters, and semiempirical equations. After a brief survey on the state of the art and the treatment of the double-diode model, the procedure proposed by the authors, to estimate the unknown parameters, is illustrated. The theoretical expression of the uncertainty, which affects the estimation of the series and shunt resistances (namely, <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$R_{s}$</tex></formula> and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$R_{\rm sh}$</tex></formula> ) of a PV cell, is then derived. A statistical analysis performed by means of a Monte Carlo simulation is in agreement with the theoretical expression of the uncertainty.

SDSS J162520.29+120308.7 – a new SU Ursae Majoris star in the period gap

We report results of an extensive world-wide observing campaign devoted to the recently discovered dwarf nova SDSS J162520.29+120308.7 (SDSS J1625). The data were obtained during the July 2010 eruption of the star and in August and September 2010 when the object was in quiescence. During the July 2010 superoutburst SDSS J1625 clearly displayed superhumps with a mean period of $P_{\rm sh}=0.095942(17)$ days ($138.16 \pm 0.02$ min) and a maximum amplitude reaching almost 0.4 mag. The superhump period was not stable, decreasing very rapidly at a rate of $\dot P = -1.63(14)\cdot 10^{-3}$ at the beginning of the superoutburst and increasing at a rate of $\dot P = 2.81(20)\cdot 10^{-4}$ in the middle phase. At the end of the superoutburst it stabilized around the value of $P_{\rm sh}=0.09531(5)$ day. During the first twelve hours of the superoutburst a low-amplitude double wave modulation was observed whose properties are almost identical to early superhumps observed in WZ Sge stars. The period of early superhumps, the period of modulations observed temporarily in quiescence and the period derived from radial velocity variations are the same within measurement errors, allowing us to estimate the most probable orbital period of the binary to be $P_{\rm orb}=0.09111(15)$ days ($131.20 \pm 0.22$ min). This value clearly indicates that SDSS J1625 is another dwarf nova in the period gap. Knowledge of the orbital and superhump periods allows us to estimate the mass ratio of the system to be $q\approx 0.25$. This high value poses serious problems both for the thermal and tidal instability (TTI) model describing the behaviour of dwarf novae and for some models explaining the origin of early superhumps.

SPECIAL RELATIVISTIC MAGNETOHYDRODYNAMIC SIMULATION OF A TWO-COMPONENT OUTFLOW POWERED BY MAGNETIC EXPLOSION ON COMPACT STARS

The nonlinear dynamics of outflows driven by magnetic explosion on the surface of a compact star is investigated through special relativistic magnetohydrodynamic simulations. We adopt, as the initial equilibrium state, a spherical stellar object embedded in hydrostatic plasma which has a density $\rho(r) \propto r^{- \alpha}$ and is threaded by a dipole magnetic field. The injection of magnetic energy at the surface of compact star breaks the equilibrium and triggers a two-component outflow. At the early evolutionary stage, the magnetic pressure increases rapidly around the stellar surface, initiating a magnetically driven outflow. A strong forward shock driven outflow is then excited. The expansion velocity of the magnetically driven outflow is characterized by the Alfv\'en velocity on the stellar surface, and follows a simple scaling relation $v_{\rm mag} \propto {v_{\rm A}}^{1/2}$. When the initial density profile declines steeply with radius, the strong shock is accelerated self-similarly to relativistic velocity ahead of the magnetically driven component. We find that it evolves according to a self-similar relation $\Gamma_{\rm sh} \propto r_{\rm sh}$, where $\Gamma_{\rm sh}$ is the Lorentz factor of the plasma measured at the shock surface $r_{\rm sh}$. Purely hydrodynamic process would be responsible for the acceleration mechanism of the shock driven outflow. Our two-component outflow model, which is the natural outcome of the magnetic explosion, can provide a better understanding of the magnetic active phenomena on various magnetized compact stars.

CCD Photometry of a Newly Confirmed SU UMa-Type Dwarf Nova, NSV 4838

Abstract We present time-resolved CCD photometry of a dwarf nova, NSV 4838 (UMa 8, SDSS J102320.27$+$440509.8), during 2005 June and 2007 February outbursts. The two light curves showed superhumps with mean periods of 0.0699(1)d for the 2005 outburst and 0.069824(83)d for the 2007 outburst. Using its known orbital period of 0.0678 d, we estimated the mass ratio of the system to be $q$$=$ 0.13 based on an empirical relation. Although the majority of SU UMa-type dwarf novae having similar superhump periods show negative period derivatives, we found that the superhump period increased at $\dot{P}/P_{\rm sh}$$=$$+$7($+$3, $-$4) $\times$ 10$^{-5}$ during the 2007 superoutburst. We also investigated long-term light curves of NSV 4838, from which we derived 340 d as a supercycle of this system.

A Few Results on Archimedean Sets

In a 1990 paper by R. Mabry, it is shown that for any constant $a\in \left(0,1\right)$ there exist sets $A$ on the real line with the property that for any bounded interval $I$, $\displaystyle\frac{\mu(A\bigcap I)}{\mu(I)}=a$, where $\mu$ is any Banach measure. \noindent Many of the constructed sets are Archimedean sets, which are sets that satisfy $A+t=A$ for densely many $t\in {\mathbb{R}}$. In that paper it is shown that if $A$ is an arbitrary Archimedean set, then for a fixed, $\mu$, $\displaystyle\frac{\mu(A\bigcap I)}{\mu(I)}$ is constant. (This constant is called the $\mu$-shade of $A$ and is denoted ${\rm sh}_{\mu}A$.) A problem is then proposed: For any Archimedean set $A$, any fixed Banach measure $\mu$, and any number $b$ between $0$ and ${\rm sh}_{\mu}A$, does there exist a subset $B$ of $A$ such that $\displaystyle\frac{\mu(B\bigcap I)}{\mu(I)}=b$ for any bounded interval $I$? In this paper, we partially answer this question. We also derive a lower bound formula for the $\mu$-shade of the difference set of an arbitrary Archimedean set. Finally, we generalize an intersection result from Mabry's original paper.

Photometric Observation and Numerical Simulation of Early Superhumps in BC Ursae Majoris during the 2003 Superoutburst

Abstract We carried out time-resolved photometric observations of an SU UMa-type dwarf nova, BC UMa, during its superoutburst in 2003 February. The period of early superhumps is 0.06258(13)d and the amplitude is 0.04 mag. After the early superhump phase, common superhumps with an amplitude of 0.3 mag and a period of 0.064466(16) d developed. The change rate of the superhump periods ($\dot{P}_{\rm SH}/P_{\rm SH}$) ($P_{\rm SH}$: superhump period) was positive throughout the superoutburst. The superhump period excess ($\varepsilon \equiv P_{\rm SH}P_{\rm orb}^{-1}-1$) is 3% and we derived a mass ratio of 0.13. We have modeled early superhump light curves including irradiation effects of the accretion disk and secondary star. The observed early superhumps can be reproduced when two-armed spirals appear on the accretion disk. The long-term data taken from AAVSO, VSOLJ, and VSNET indicate that BC UMa shows normal outbursts and two types of superoutbursts (short-faint and long-bright). BC UMa is the first example to show such two types of superoutbursts. The supercycle of BC UMa is between 600 d and 1000d. This suggests that BC UMa is an intermediate type of dwarf novae between WZ Sge and SU UMa.