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# Posts Tagged galaxy m87

## Recent Postings from galaxy m87

### Optical Proper Motion Measurements of the M87 Jet: New Results from the Hubble Space Telescope

We report new results from an HST archival program to study proper motions in the optical jet of the nearby radio galaxy M87. Using over 13 years of archival imaging, we reach accuracies below 0.1c in measuring the apparent velocities of individual knots in the jet. We confirm previous findings of speeds up to 4.5c in the inner 6" of the jet, and report new speeds for optical components in the outer part of the jet. We find evidence of significant motion transverse to the jet axis on the order of 0.6c in the inner jet features, and superluminal velocities parallel and transverse to the jet in the outer knot components, with an apparent ordering of velocity vectors possibly consistent with a helical jet pattern. Previous results suggested a global deceleration over the length of the jet in the form of decreasing maximum speeds of knot components from HST-1 outward, but our results suggest that superluminal speeds persist out to knot C, with large differentials in very nearby features all along the jet. We find significant apparent accelerations in directions parallel and transverse to the jet axis, along with evidence for stationary features in knots D, E, and I. These results are expected to place important constraints on detailed models of kpc-scale relativistic jets.

### Monitoring the Very-Long-Term Variability of X-ray Sources in the Giant Elliptical Galaxy M87

We report on our search for very-long-term variability (weeks to years) in X-ray binaries (XRBs) in the giant elliptical galaxy M87. We have used archival Chandra imaging observations to characterise the long-term variability of 8 of the brightest members of the XRB population in M87. The peak brightness of some of the sources exceeded the ultra luminous X-ray source (ULX) threshold luminosity of ~ 10^{39} erg/s, and one source could exhibit dips or eclipses. We show that for one source, if it has similar modulation amplitude as in SS433, then period recoverability analysis on the current data would detect periodic modulations, but only for a narrow range of periods less than 120 days. We conclude that a dedicated monitoring campaign, with appropriately defined sampling, is essential if we are to investigate properly the nature of the long-term modulations such as those seen in Galactic sources.

### Monitoring the Very-Long-Term Variability of X-ray Sources in the Giant Elliptical Galaxy M87 [Replacement]

We report on our search for very-long-term variability (weeks to years) in X-ray binaries (XRBs) in the giant elliptical galaxy M87. We have used archival Chandra imaging observations to characterise the long-term variability of 8 of the brightest members of the XRB population in M87. The peak brightness of some of the sources exceeded the ultra luminous X-ray source (ULX) threshold luminosity of ~ 10^{39} erg/s, and one source could exhibit dips or eclipses. We show that for one source, if it has similar modulation amplitude as in SS433, then period recoverability analysis on the current data would detect periodic modulations, but only for a narrow range of periods less than 120 days. We conclude that a dedicated monitoring campaign, with appropriately defined sampling, is essential if we are to investigate properly the nature of the long-term modulations such as those seen in Galactic sources.

### The LOFAR View of Cosmic Magnetism

The origin of magnetic fields in the Universe is an open problem in astrophysics and fundamental physics. Polarization observations with the forthcoming large radio telescopes will open a new era in the observation of magnetic fields and should help to understand their origin. At low frequencies, LOFAR (10-240 MHz) will allow us to map the structure of weak magnetic fields in the outer regions and halos of galaxies, in galaxy clusters and in the Milky Way via their synchrotron emission. Even weaker magnetic fields can be measured at low frequencies with help of Faraday rotation measures. A detailed view of the magnetic fields in the local Milky Way will be derived by Faraday rotation measures from pulsars. First promising images with LOFAR have been obtained for the Crab pulsar-wind nebula, the spiral galaxy M51, the radio galaxy M87 and the galaxy clusters A2255 and A2256. With help of the polarimetric technique of "Rotation Measure Synthesis", diffuse polarized emission has been detected from a magnetic bubble in the local Milky Way. Polarized emission and rotation measures were measured for more than 20 pulsars so far.

### The LOFAR View of Cosmic Magnetism [Replacement]

The origin of magnetic fields in the Universe is an open problem in astrophysics and fundamental physics. Polarization observations with the forthcoming large radio telescopes will open a new era in the observation of magnetic fields and should help to understand their origin. At low frequencies, LOFAR (10-240 MHz) will allow us to map the structure of weak magnetic fields in the outer regions and halos of galaxies, in galaxy clusters and in the Milky Way via their synchrotron emission. Even weaker magnetic fields can be measured at low frequencies with help of Faraday rotation measures. A detailed view of the magnetic fields in the local Milky Way will be derived by Faraday rotation measures from pulsars. First promising images with LOFAR have been obtained for the Crab pulsar-wind nebula, the spiral galaxy M51, the radio galaxy M87 and the galaxy clusters A2255 and A2256. With help of the polarimetric technique of "Rotation Measure Synthesis", diffuse polarized emission has been detected from a magnetic bubble in the local Milky Way. Polarized emission and rotation measures were measured for more than 20 pulsars so far.

### The LOFAR View of Cosmic Magnetism [Replacement]

The origin of magnetic fields in the Universe is an open problem in astrophysics and fundamental physics. Polarization observations with the forthcoming large radio telescopes will open a new era in the observation of magnetic fields and should help to understand their origin. At low frequencies, LOFAR (10-240 MHz) will allow us to map the structure of weak magnetic fields in the outer regions and halos of galaxies, in galaxy clusters and in the Milky Way via their synchrotron emission. Even weaker magnetic fields can be measured at low frequencies with help of Faraday rotation measures. A detailed view of the magnetic fields in the local Milky Way will be derived by Faraday rotation measures from pulsars. First promising images with LOFAR have been obtained for the Crab pulsar-wind nebula, the spiral galaxy M51, the radio galaxy M87 and the galaxy clusters A2255 and A2256. With help of the polarimetric technique of "Rotation Measure Synthesis", diffuse polarized emission has been detected from a magnetic bubble in the local Milky Way. Polarized emission and rotation measures were measured for more than 20 pulsars so far.

### Transmission Line Analogy for Relativistic Poynting-Flux Jets

Radio emission, polarization, and Faraday rotation maps of the radio jet of the galaxy 3C 303 have shown that one knot of this jet carries a {\it galactic}-scale electric current and that it is magnetically dominated. We develop the theory of magnetically dominated or Poynting-flux jets by making an analogy of a Poynting jet with a transmission line or waveguide carrying a net current and having a potential drop across it (from the jet’s axis to its radius) and a definite impedance which we derive. Time-dependent but not necessarily small perturbations of a Poynting-flux jet are described by the "telegrapher’s equations." These predict the propagation speed of disturbances and the effective wave impedance for forward and backward propagating wave components. A localized disturbance of a Poynting jet gives rise to localized dissipation in the jet which may explain the enhanced synchrotron radiation in the knots of the 3C 303 jet, and also in the apparently stationary knot HST-1 in the jet near the nucleus of the nearby galaxy M87. For a relativistic Poynting jet on parsec scales, the reflected voltage wave from an inductive termination or load can lead to a backward propagating wave which breaks down the magnetic insulation of the jet giving $|{\bf E}| /|{\bf B}|\geq 1$. At the threshold for breakdown, $|{\bf E}|/|{\bf B}|=1$, positive and negative particles are directly accelerated in the ${\bf E \times B}$ direction which is approximately along the jet axis. Acceleration can occur up to Lorentz factors $\sim 10^7$. This particle acceleration mechanism is distinct from that in shock waves and that in magnetic field reconnection.

### Transmission Line Analogy for Relativistic Poynting-Flux Jets [Replacement]

Radio emission, polarization, and Faraday rotation maps of the radio jet of the galaxy 3C 303 have shown that one knot of this jet carries a {\it galactic}-scale electric current and that it is magnetically dominated. We develop the theory of magnetically dominated or Poynting-flux jets by making an analogy of a Poynting jet with a transmission line or waveguide carrying a net current and having a potential drop across it (from the jet’s axis to its radius) and a definite impedance which we derive. Time-dependent but not necessarily small perturbations of a Poynting-flux jet are described by the "telegrapher’s equations." These predict the propagation speed of disturbances and the effective wave impedance for forward and backward propagating wave components. A localized disturbance of a Poynting jet gives rise to localized dissipation in the jet which may explain the enhanced synchrotron radiation in the knots of the 3C 303 jet, and also in the apparently stationary knot HST-1 in the jet near the nucleus of the nearby galaxy M87. For a relativistic Poynting jet on parsec scales, the reflected voltage wave from an inductive termination or load can lead to a backward propagating wave which breaks down the magnetic insulation of the jet giving $|{\bf E}| /|{\bf B}|\geq 1$. At the threshold for breakdown, $|{\bf E}|/|{\bf B}|=1$, positive and negative particles are directly accelerated in the ${\bf E \times B}$ direction which is approximately along the jet axis. Acceleration can occur up to Lorentz factors $\sim 10^7$. This particle acceleration mechanism is distinct from that in shock waves and that in magnetic field reconnection.

### Probing the central black hole in M87 with gamma-rays

Recent high-sensitivity observation of the nearby radio galaxy M87 have provided important insights into the central engine that drives the large-scale outflows seen in radio, optical and X-rays. This review summarizes the observational status achieved in the high energy (HE;<100 GeV) and very high energy (VHE; >100 GeV) gamma-ray domains, and discusses the theoretical progress in understanding the physical origin of this emission and its relation to the activity of the central black hole.

### MAGIC observations of the giant radio galaxy M87 in a low-emission state between 2005 and 2007

We present the results of a long M87 monitoring campaign in very high energy $\gamma$-rays with the MAGIC-I Cherenkov telescope. We aim to model the persistent non-thermal jet emission by monitoring and characterizing the very high energy $\gamma$-ray emission of M87 during a low state. A total of 150\,h of data were taken between 2005 and 2007 with the single MAGIC-I telescope, out of which 128.6\,h survived the data quality selection. We also collected data in the X-ray and \textit{Fermi}–LAT bands from the literature (partially contemporaneous). No flaring activity was found during the campaign. The source was found to be in a persistent low-emission state, which was at a confidence level of $7\sigma$. We present the spectrum between 100\,GeV and 2\,TeV, which is consistent with a simple power law with a photon index $\Gamma=2.21\pm0.21$ and a flux normalization at 300\,GeV of $(7.7\pm1.3) \times 10^{-8}$ TeV$^{-1}$ s$^{-1}$ m$^{-2}$. The extrapolation of the MAGIC spectrum into the GeV energy range matches the previously published \textit{Fermi}–LAT spectrum well, covering a combined energy range of four orders of magnitude with the same spectral index. We model the broad band energy spectrum with a spine layer model, which can satisfactorily describe our data.

### MAGIC observations of the giant radio galaxy M87 in a low-emission state between 2005 and 2007 [Replacement]

We present the results of a long M87 monitoring campaign in very high energy $\gamma$-rays with the MAGIC-I Cherenkov telescope. We aim to model the persistent non-thermal jet emission by monitoring and characterizing the very high energy $\gamma$-ray emission of M87 during a low state. A total of 150\,h of data were taken between 2005 and 2007 with the single MAGIC-I telescope, out of which 128.6\,h survived the data quality selection. We also collected data in the X-ray and \textit{Fermi}–LAT bands from the literature (partially contemporaneous). No flaring activity was found during the campaign. The source was found to be in a persistent low-emission state, which was at a confidence level of $7\sigma$. We present the spectrum between 100\,GeV and 2\,TeV, which is consistent with a simple power law with a photon index $\Gamma=2.21\pm0.21$ and a flux normalization at 300\,GeV of $(7.7\pm1.3) \times 10^{-8}$ TeV$^{-1}$ s$^{-1}$ m$^{-2}$. The extrapolation of the MAGIC spectrum into the GeV energy range matches the previously published \textit{Fermi}–LAT spectrum well, covering a combined energy range of four orders of magnitude with the same spectral index. We model the broad band energy spectrum with a spine layer model, which can satisfactorily describe our data.

### A cosmic ray cocoon along the X-ray jet of M87?

Relativistic jets propagating through an ambient medium must produce some observational effects along their side boundaries because of interactions across the large velocity gradient. One possible effect of such an interaction would be a sheared magnetic field structure at the jet boundaries, leading to a characteristic radio polarization pattern. As proposed by Ostrowski, another effect can come from the generation of a high energy cosmic ray component at the boundary, producing dynamic effects on the medium surrounding the jet and forming a cocoon dominated by cosmic rays with a decreased thermal gas emissivity. We selected this process for our first attempt to look for the effects of this type of interaction. We analyzed the Chandra X-ray data for the radio galaxy M87 in order to verify if the expected regions of diminished emissivity may be present near the spectacular X-ray jet in this source. The detailed analysis of the data, merged from 42 separate observations, shows signatures of lower emissivity surrounding the jet. In particular we detect an intensity dip along the part of the jet, which would be approximately 150 pc x 2 kpc in size, if situated along the jet which is inclined toward us. Due to a highly non-uniform X-ray background in the central region we are not able to claim the discovery of a cosmic ray cocoon around the M87 jet: we only have demonstrated that the data show morphological structures which could be accounted for if a cosmic ray cocoon exists.

### Hundred Thousand Degree Gas in the Virgo Cluster of Galaxies

The physical relationship between low-excitation gas filaments at ~10^4 K, seen in optical line emission, and diffuse X-ray emitting coronal gas at ~10^7 K in the centers of many galaxy clusters is not understood. It is unclear whether the ~10^4 K filaments have cooled and condensed from the ambient hot (~10^7 K) medium or have some other origin such as the infall of cold gas in a merger, or the disturbance of an internal cool reservoir of gas by nuclear activity. Observations of gas at intermediate temperatures (~10^5-10^6 K) can potentially reveal whether the central massive galaxies are gaining cool gas through condensation or losing it through conductive evaporation and hence identify plausible scenarios for transport processes in galaxy cluster gas. Here we present spectroscopic detection of ~10^5 K gas spatially associated with the H-alpha filaments in a central cluster galaxy, M87 in the Virgo Cluster. The measured emission-line fluxes from triply ionized carbon (CIV 1549 A) and singly ionized helium (HeII 1640 A) are consistent with a model in which thermal conduction determines the interaction between hot and cold phases.

### Relativistic Poynting-Flux Jets as Transmission Lines

Recent radio emission, polarization, and Faraday rotation maps of the radio jet of the galaxy 3C 303 have shown that one knot of this jet has a {\it galactic}-scale electric current of $\sim 3\times 10^{18}$ Amp\`ere flowing along the jet axis (Kronberg et al. 2011). We develop the theory of relativistic Poynting-flux jets which are modeled as a transmission line carrying a DC current $I_0$, having a potential drop $V_0$, and a definite impedance ${\cal Z}_0 =90(u_z/c)\Omega$, where $u_z$ is the bulk velocity of the jet plasma. The electromagnetic energy flow in the jet is ${\cal Z}_0 I_0^2$. The observed current in 3C 303 can be used to calculate the electromagnetic energy flow in this magnetically dominated jet. Time-dependent but not necessarily small perturbations of a Poynting-flux jet – possibly triggered by a gas cloud penetrating the jet – are described by “telegrapher’s equations,” which predict the propagation speed of disturbances and the effective wave impedance ${\cal Z}$. The disturbance of a Poynting jet by the cloud gives rise to localized dissipation in the jet which may explain the enhanced synchrotron radiation in the knots of the 3C 303 jet, and in the apparently stationary knot HST-1 in the jet from the nucleus of the galaxy M87 (Biretta et al. 1999).

### VERITAS Observations of day-scale flaring of M87 in 2010 April

VERITAS has been monitoring the very-high-energy (VHE; >100GeV) gamma-ray activity of the radio galaxy M87 since 2007. During 2008, flaring activity on a timescale of a few days was observed with a peak flux of (0.70 +- 0.16) X 10^{-11} cm^{-2} s^{-1} at energies above 350GeV. In 2010 April, VERITAS detected a flare from M87 with peak flux of (2.71 +- 0.68) X 10^{-11} cm^{-2} s^{-1} for E>350GeV. The source was observed for six consecutive nights during the flare, resulting in a total of 21 hr of good quality data. The most rapid flux variation occurred on the trailing edge of the flare with an exponential flux decay time of 0.90^{+0.22}_{-0.15} days. The shortest detected exponential rise time is three times as long, at 2.87^{+1.65}_{-0.99} days. The quality of the data sample is such that spectral analysis can be performed for three periods: rising flux, peak flux, and falling flux. The spectra obtained are consistent with power-law forms. The spectral index at the peak of the flare is equal to 2.19 +- 0.07. There is some indication that the spectrum is softer in the falling phase of the flare than the peak phase, with a confidence level corresponding to 3.6 standard deviations. We discuss the implications of these results for the acceleration and cooling rates of VHE electrons in M87 and the constraints they provide on the physical size of the emitting region.

### The ongoing assembly of a central cluster galaxy: Phase-space substructures in the halo of M87

The halos of galaxies preserve unique records of their formation histories. We carry out the first combined observational and theoretical study of phase-space halo substructure in an early-type galaxy: M87, the central galaxy in the Virgo cluster. We analyze an unprecedented wide-field, high-precision photometric and spectroscopic data set for 488 globular clusters (GCs), which includes new, large-radius Subaru/Suprime-Cam and Keck/DEIMOS observations. We find signatures of two substructures in position-velocity phase-space. One is a small, cold stream associated with a known stellar filament in the outer halo; the other is a large shell-like pattern in the inner halo that implies a massive, hitherto unrecognized accretion event. We perform extensive statistical tests and independent metallicity analyses to verify the presence and characterize the properties of these features, and to provide more general methodologies for future extragalactic studies of phase-space substructure. The cold outer stream is consistent with a dwarf galaxy accretion event, while for the inner shell there is tension between a low progenitor mass implied by the cold velocity dispersion, and a high mass from the large number of GCs, which might be resolved by a ~0.5 L* E/S0 progenitor. We also carry out proof-of-principle numerical simulations of the accretion of smaller galaxies in an M87-like gravitational potential. These produce analogous features to the observed substructures, which should have observable lifetimes of ~1 Gyr. The shell and stream GCs together support a scenario where the extended stellar envelope of M87 has been built up by a steady rain of material that continues until the present day. This phase-space method demonstrates unique potential for detailed tests of galaxy formation beyond the Local Group.

### Dissipation Efficiency of Reconfinement Shocks in Relativistic Jets

We calculate the dissipation efficiency of relativistic reconfinement shocks. Building on previous work (Nalewajko & Sikora 2009), we consider different distributions of the external pressure. The average dissipation efficiency epsilon_diss is a function of the product of two parameters – the jet Lorentz factor Gamma_j and the reconfinement angle Theta_r, which is related to the opening angle Theta_j and the external pressure index eta. The spatial distribution of the dissipation rate strongly depends on eta. We discuss the significance of these results for the properties of relativistic jets in gamma-ray bursts and active galactic nuclei and propose that reconfinement shocks may explain a very high dissipation efficiency of the former and a moderate dissipation efficiency of the latter. Finally, we estimate the dissipation efficiency of the reconfinement shock associated with the quasi-stationary knot HST-1 in the jet of radio galaxy M87 and show that it is roughly consistent with the observational constraints.

### Multiwavelength study of TeV Blazar Mrk421 during giant flare and observations of TeV AGNs with HAGAR

The radiation mechanism of very high energy $\gamma$-ray emission from blazars and crucial parameters like magnetic field, and size of the emitting region are not well understood yet. To understand the above mentioned properties of blazars, we observed five nearby TeV $\gamma$-ray emitting blazars (Mrk421, Mrk501, 1ES2344+514, 1ES1218+304 and 3C454.3) and one radio galaxy (M87) using the High Altitude GAmma Ray (HAGAR) telescope. HAGAR is an array of seven telescopes located at Hanle, India to detect Cherenkov light caused by extensive air showers initiated by $\gamma$-rays. Mrk421 was observed to undergo one of its brightest flaring episodes on 2010 February 17, and detected by various experiments in X-rays and $\gamma$- rays. HAGAR observations of this source during 2010 February 13 – 19, in the energies above 250 GeV show an enhancement in the flux level, with a flux of 6-7 Crab units being detected on 2010 February 17. We present the spectral energy distribution of the source during this flaring episode. In addition to this, the analysis procedure to extract $\gamma$-ray signal from HAGAR data is discussed and preliminary results on all the AGNs are presented.

### MAGIC observations of the giant radio galaxy M87 in a low emission state between 2005 and 2007

We present the results of a long M87 monitoring campaign in very high energy $\gamma$-rays with the MAGIC-I Cherenkov telescope. A total of 150 hours of data was gathered between 2005 and 2007. No flaring activity was found during that time. Nevertheless, we have found an apparently steady and weak signal at the level of $7\sigma$. We present the spectrum between 100 GeV and 2 TeV, which is consistent with a simple power law with a spectral index $-2.21\pm0.21$ and a flux normalization (at 300 GeV) of $5.4\pm1.1 \times 10^{-8} \frac{1}{\mathrm{TeV s m}^{2}}$. It complements well with the previously published Fermi spectrum, covering an energy range of four orders of magnitude without apparent change in the spectral index.

### Search for Nuclei Sources in the Ultra-High Energy Cosmic Ray Data

We present a new method to search for heavy nuclei sources, on top of background, in the Ultra-High Energy Cosmic Ray data. We apply it to the 69 events with energies E>55 EeV published by the Pierre Auger Collaboration. We find a set of events for which the method reconstructs the source near the Virgo galaxy cluster. The probability to have a comparable set of events in some background is ~ 0.7 %. The reconstructed source is located at ~ 8.5 degrees from the active galaxy M87. The probability to reconstruct the source at less than 10 degrees from M87 for data already containing a comparable set of events is ~ 0.4 %. This may be a hint at the Virgo galaxy cluster as an ultra-high energy heavy nuclei source. We discuss the capability of current and near future experiments to test this possibility. Such a scenario gives a self-consistent description of the Auger anisotropy and composition data at the highest energies.

### The Black-Hole Mass in M87 from Gemini/NIFS Adaptive Optics Observations

We present the stellar kinematics in the central 2″ of the luminous elliptical galaxy M87 (NGC 4486), using laser adaptive optics to feed the Gemini telescope integral-field spectrograph, NIFS. The velocity dispersion rises to 480 km/s at 0.2″. We combine these data with extensive stellar kinematics out to large radii to derive a black-hole mass equal to (6.6+-0.4)x10^9 Msun, using orbit-based axisymmetric models and including only the NIFS data in the central region. Including previously-reported ground-based data in the central region drops the uncertainty to 0.25×10^9 Msun with no change in the best-fit mass; however, we rely on the values derived from the NIFS-only data in the central region in order to limit systematic differences. The best-fit model shows a significant increase in the tangential velocity anisotropy of stars orbiting in the central region with decreasing radius; similar to that seen in the centers of other core galaxies. The black-hole mass is insensitive to the inclusion of a dark halo in the models — the high angular-resolution provided by the adaptive optics breaks the degeneracy between black-hole mass and stellar mass-to-light ratio. The present black-hole mass is in excellent agreement with the Gebhardt & Thomas value, implying that the dark halo must be included when the kinematic influence of the black hole is poorly resolved. This degeneracy implies that the black-hole masses of luminous core galaxies, where this effect is important, may need to be re-evaluated. The present value exceeds the prediction of the black hole-dispersion and black hole-luminosity relations, both of which predict about 1×10^9 Msun for M87, by close to twice the intrinsic scatter in the relations. The high-end of the black hole correlations may be poorly determined at present.

### Effect of a dark matter halo on the determination of black hole masses

Stellar dynamical modeling is a powerful method to determine the mass of black holes in quiescent galaxies. However, in previous work the presence of a dark matter halo has been ignored in the modeling. Gebhardt & Thomas (2009) showed that accounting for a dark matter halo increased the black-hole mass of the massive galaxy M87 by a factor of 2. We used a sample of 12 galaxies to investigate the effect of accounting for a dark matter halo in the dynamical modeling in more detail, and also updated the masses using improved modeling. The sample of galaxies possess HST and ground based observations of stellar kinematics. Their black-hole masses have been presented before, but without including a dark matter halo in the models. Without a dark halo, we find a mean increase in the estimated mass of 1.5 for the whole sample compared to previous results. We attribute this change to using a more complete orbit library. When we include a dark matter halo, along with the updated models, we find an additional increase in black-hole mass by a factor of 1.2 in the mean, much less than for M87. We attribute the smaller discrepancy in black-hole mass to using data that better resolves the black hole’s sphere of influence. We redetermined the M-sigma and M-L relationships using our updated black-hole masses and find a slight increase in both normalization and intrinsic scatter.

### Effect of a dark matter halo on the determination of black hole masses [Replacement]

Stellar dynamical modeling is a powerful method to determine the mass of black holes in quiescent galaxies. However, in previous work the presence of a dark matter halo has been ignored in the modeling. Gebhardt & Thomas (2009) showed that accounting for a dark matter halo increased the black hole mass of the massive galaxy M87 by a factor of two. We used a sample of 12 galaxies to investigate the effect of accounting for a dark matter halo in the dynamical modeling in more detail, and also updated the masses using improved modeling. The sample of galaxies possesses Hubble Space Telescope and ground based observations of stellar kinematics. Their black hole masses have been presented before, but without including a dark matter halo in the models. Without a dark halo, we find a mean increase in the estimated mass of 1.5 for the whole sample compared to previous results. We attribute this change to using a more complete orbit library. When we include a dark matter halo, along with the updated models, we find an additional increase in black hole mass by a factor of 1.2 in the mean, much less than for M87. We attribute the smaller discrepancy in black hole mass to using data that better resolves the black hole’s sphere of influence. We redetermined the M-sigma and M-L relationships using our updated black hole masses and found a slight increase in both normalization and intrinsic scatter.

### The far-infrared view of M87 as seen by the Herschel Space Observatory

The origin of the far-infrared emission from the nearby radio galaxy M87 remains a matter of debate. Some studies find evidence of a far-infrared excess due to thermal dust emission, whereas others propose that the far-infrared emission can be explained by synchrotron emission without the need for an additional dust emission component. We observed M87 with PACS and SPIRE as part of the Herschel Virgo Cluster Survey (HeViCS). We compare the new Herschel data with a synchrotron model based on infrared, submm and radio data to investigate the origin of the far-infrared emission. We find that both the integrated SED and the Herschel surface brightness maps are adequately explained by synchrotron emission. At odds with previous claims, we find no evidence of a diffuse dust component in M87.

### Launching proton-dominated jets from accreting Kerr black holes: the case of M87 [Replacement]

A general relativistic model for the formation and acceleration of low mass-loaded jets from systems containing accreting black holes is presented. The model is based on previous numerical results and theoretical studies in the Newtonian regime, but modified to include the effects of space-time curvature in the vicinity of the event horizon of a spinning black hole. It is argued that the boundary layer between the Keplerian accretion disk and the event horizon is best suited for the formation and acceleration of the accretion-powered jets in active galactic nuclei and micro-quasars. The model is based on matching the solutions of three different regions: i- a weakly magnetized Keplerian accretion disk in the outer part, where the transport of angular momentum is mediated through the magentorotational instability, ii- a strongly magnetized, advection-dominated and turbulent-free boundary layer (BL) between the outer cold accretion disk and the event horizon, where the plasma rotates sub-Keplerian and iii- a transition zone (TZ) between the BL and the overlying corona, where the electrons and protons are thermally uncoupled, highly dissipative and rotate super-Keplerian. Our model predicts the known correlation between the Lorentz-factor and the spin parameter of the BH. It also shows that the effective surface of the BL, through which the baryons flow into the TZ, shrinks with increasing the spin parameter, implying therefore that low mass-loaded jets most likely originate from around Kerr black holes. When applying our model to the jet in the elliptical galaxy M87, we find a spin parameter in the range 0.99 – 0.998, a transition radius of about 30 gravitational radii and a fraction of 0.05 – 0.1 of the mass accretion rate goes into the TZ, where the plasma speeds up its outward-oriented motion to reach a Lorentz factor of 2.5 – 5.0 at the transition radius.

### The Herschel Virgo Cluster Survey: VI. The far-infrared view of M87

The origin of the far-infrared emission from the nearby radio galaxy M87 remains a matter of debate. Some studies find evidence of a far-infrared excess due to thermal dust emission, whereas others propose that the far-infrared emission can be explained by synchrotron emission without the need for an additional dust emission component. We present Herschel PACS and SPIRE observations of M87, taken as part of the science demonstration phase observations of the Herschel Virgo Cluster Survey. We compare these data with a synchrotron model based on mid-infrared, far-infrared, submm and radio data from the literature to investigate the origin of the far-infrared emission. Both the integrated SED and the Herschel surface brightness maps are adequately explained by synchrotron emission. At odds with previous claims, we find no evidence of a diffuse dust component in M87, which is not unexpected in the harsh X-ray environment of this radio galaxy sitting at the core of the Virgo Cluster.

### NIKA: A millimeter-wave kinetic inductance camera [Replacement]

Current generation millimeter wavelength detectors suffer from scaling limits imposed by complex cryogenic readout electronics. To circumvent this it is imperative to investigate technologies that intrinsically incorporate strong multiplexing. One possible solution is the kinetic inductance detector (KID). In order to assess the potential of this nascent technology, a prototype instrument optimized for the 2 mm atmospheric window was constructed. Known as the N\’eel IRAM KIDs Array (NIKA), it was recently tested at the Institute for Millimetric Radio Astronomy (IRAM) 30-meter telescope at Pico Veleta, Spain. The measurement resulted in the imaging of a number of sources, including planets, quasars, and galaxies. The images for Mars, radio star MWC349, quasar 3C345, and galaxy M87 are presented. From these results, the optical NEP was calculated to be around $1 \times 10^{-15}$ W$/$Hz$^{1/2}$. A factor of 10 improvement is expected to be readily feasible by improvements in the detector materials and reduction of performance-degrading spurious radiation.

### NIKA: A Millimeter-Wave Kinetic Inductance Camera

Current generation millimeter wavelength detectors suffer from scaling limits imposed by complex cryogenic readout electronics. To circumvent this it is imperative to investigate technologies that intrinsically incorporate strong multiplexing. One possible solution is the Kinetic-Inductance Detector (KID). In order to assess the potential of this nascent technology, a prototype instrument optimized for the 2 mm atmospheric window was constructed. Known as the N\’eel IRAM KIDs Array (NIKA), it was recently tested at the Institute for Millimetric Radio Astronomy (IRAM) 30-meter telescope at Pico Veleta, Spain. The measurement resulted in the imaging of a number of sources, including planets, quasars, and galaxies. The images for Mars, radio star MWC349, quasar 3C345, and galaxy M87 are presented. From these results, the optical NEP was calculated to be around $1 \times 10^{-15}$ W$/$Hz$^{1/2}$. A factor of 10 improvement is expected to be readily feasible by improvements in the detector materials and reduction of performance-degrading spurious radiation.

### Optical Colors of Intracluster Light in the Virgo Cluster Core [Replacement]

We continue our deep optical imaging survey of the Virgo cluster using the CWRU Burrell Schmidt telescope by presenting B-band surface photometry of the core of the Virgo cluster in order to study the cluster’s intracluster light (ICL). We find ICL features down to mu_b ~ 29 mag sq. arcsec, confirming the results of Mihos et al. (2005), who saw a vast web of low-surface brightness streams, arcs, plumes, and diffuse light in the Virgo cluster core using V-band imaging. By combining these two data sets, we are able to measure the optical colors of many of the cluster’s low-surface brightness features. While much of our imaging area is contaminated by galactic cirrus, the cluster core near the cD galaxy, M87, is unobscured. We trace the color profile of M87 out to over 2000 arcsec, and find a blueing trend with radius, continuing out to the largest radii. Moreover, we have measured the colors of several ICL features which extend beyond M87′s outermost reaches and find that they have similar colors to the M87′s halo itself, B-V ~ 0.8. The common colors of these features suggests that the extended outer envelopes of cD galaxies, such as M87, may be formed from similar streams, created by tidal interactions within the cluster, that have since dissolved into a smooth background in the cluster potential.

### Optical Colors of Intracluster Light in the Virgo Cluster Core

We continue our deep optical imaging survey of the Virgo cluster using the CWRU Burrell Schmidt telescope by presenting B-band surface photometry of the core of the Virgo cluster in order to study the cluster’s intracluster light (ICL). We find ICL features down to mu_b ~ 29 mag sq. arcsec, confirming the results of Mihos et al. (2005), who saw a vast web of low-surface brightness streams, arcs, plumes, and diffuse light in the Virgo cluster core using V-band imaging. By combining these two data sets, we are able to measure the optical colors of many of the cluster’s low-surface brightness features. While much of our imaging area is contaminated by galactic cirrus, the cluster core near the cD galaxy, M87, is unobscured. We trace the color profile of M87 out to over 2000 arcsec, and find a blueing trend with radius, continuing out to the largest radii. Moreover, we have measured the colors of several ICL features which extend beyond M87′s outermost reaches and find that they have similar colors to the M87′s halo itself, B-V ~ 0.8. The common colors of these features suggests that the extended outer envelopes of cD galaxies, such as M87, may be formed from similar streams, created by tidal interactions within the cluster, that have since dissolved into a smooth background in the cluster potential.

### Metal transport by gas sloshing in M87

We present the results of an XMM-Newton mosaic covering the central ~200 kpc of the nearby Virgo cluster. We focus on a strong surface brightness discontinuity in the outskirts of the brightest cluster galaxy, M87. Using both XMM-Newton and Suzaku, we derive accurate temperature and metallicity profiles across this feature and show that it is a cold front probably due to sloshing of the Virgo ICM. It is also associated with a discontinuity in the chemical composition. The gas in the inner, bright region of the front is ~40% more abundant in Fe than the gas outside the front, suggesting the important role of sloshing in transporting metals through the ICM. For the first time, we provide a quantitative estimate of the mass of Fe transported by a cold front. This amounts to ~6% of the total Fe mass within the radial range affected by sloshing, significantly more than the amount of metals transported by the AGN in the same cluster core. The very low Fe abundance of only ~0.2 solar immediately outside the cold front at a radius of 90 kpc suggests we are witnessing first-hand the transport of higher metallicity gas into a pristine region, whose abundance is typical of the cluster outskirts. The Mg/Fe and O/Fe abundance ratios remain approximately constant over the entire radial range between the centre of M87 and the faint side of the cold front, which requires the presence of a centrally peaked distribution not only for Fe but also for core-collapse type supernova products. This peak may stem from the star formation triggered as the BCG assembled during the protocluster phase.

### Results from the observation of extragalactic objects with the MAGIC telescope

In the last couple of years the Magic air Cherenkov telescope has made significant contributions to very high energy $\gamma$-ray astronomy. These include the detection of the galactic binary system LSI +61 303 and the observation of pulsed emission from the Crab pulsar. Extragalactic objects like the famous FSRQ 3C 279 and the LBL S5 0716+714 have both been detected during optical high states, and the radio galaxy M87 could be observed during an unexpected strong $\gamma$-ray outburst. Given its low energy trigger threshold (~50 GeV) and fast repositioning time of less than 30s the Magic air Cherenkov telescope is particularly well suited for the observation of fast transient objects like AGN or GRBs. So far no GRB could be detected with Magic, however. In this paper we present selected highlights from recent MAGIC observations of extragalactic objects.

### Pinpointing the TeV gamma-ray emission region in M87 using TeV and 43 GHz radio monitoring

The TeV radio galaxy M87 is the first radio galaxy detected in the TeV regime. The structure of its jet, which is not pointing towards our line of sight, is spatially resolved in X-ray (by Chandra), optical and radio observations. In 2008, the three main Atmospheric Cherenkov Telescope observatories VERITAS, MAGIC and H.E.S.S. coordinated their observations in a joint campaign from January to May with a total observation time of approx. 120 hours. In February, strong and rapid day-scale TeV flares were detected. VLBA monitoring observations during the same period showed that the 43 GHz radio flux density of the unresolved core began to rise at the time of the TeV flares and eventually reached levels above any previously seen with VLBI. New jet components appeared during the flare. The localization accuracy of the TeV instruments of many arcseconds, even for strong sources, is inadequate to constrain the origin of the emission in the inner jets of AGNs. For M87, with a 6 billion solar mass black hole and a distance of 16.7 Mpc, the VLBA resolution instead corresponds to 30 by 60 Schwarzschild radii. This is starting to resolve the jet collimation region. The temporal coincidence of the TeV and radio flares indicates that they are related and provides the first direct evidence that the TeV radiation from this source is produced within a few tens of RS of the radio core, thought to be coincident to within the VLBA resolution with the black hole.

### Fermi Large Area Telescope Gamma-Ray Detection of the Radio Galaxy M87

We report the Fermi-LAT discovery of high-energy (MeV/GeV) gamma-ray emission positionally consistent with the center of the radio galaxy M87, at a source significance of over 10 sigma in ten-months of all-sky survey data. Following the detections of Cen A and Per A, this makes M87 the third radio galaxy seen with the LAT. The faint point-like gamma-ray source has a >100 MeV flux of 2.45 (+/- 0.63) x 10^-8 ph cm^-2 s^-1 (photon index = 2.26 +/- 0.13) with no significant variability detected within the LAT observation. This flux is comparable with the previous EGRET upper limit (< 2.18 x 10^-8 ph cm^-2 s^-1, 2 sigma), thus there is no evidence for a significant MeV/GeV flare on decade timescales. Contemporaneous Chandra and VLBA data indicate low activity in the unresolved X-ray and radio core relative to previous observations, suggesting M87 is in a quiescent overall level over the first year of Fermi-LAT observations. The LAT gamma-ray spectrum is modeled as synchrotron self-Compton (SSC) emission from the electron population producing the radio-to-X-ray emission in the core. The resultant SSC spectrum extrapolates smoothly from the LAT band to the historical-minimum TeV emission. Alternative models for the core and possible contributions from the kiloparsec-scale jet in M87 are considered, and can not be excluded.

### Radio Imaging of the Very-High-Energy Gamma-Ray Emission Region in the Central Engine of a Radio Galaxy

The accretion of matter onto a massive black hole is believed to feed the relativistic plasma jets found in many active galactic nuclei (AGN). Although some AGN accelerate particles to energies exceeding 10^12 electron Volts (eV) and are bright sources of very-high-energy (VHE) gamma-ray emission, it is not yet known where the VHE emission originates. Here we report on radio and VHE observations of the radio galaxy M87, revealing a period of extremely strong VHE gamma-ray flares accompanied by a strong increase of the radio flux from its nucleus. These results imply that charged particles are accelerated to very high energies in the immediate vicinity of the black hole.

### Fast TeV variability from misaligned minijets in the jet of M87

The jet of the radio galaxy M87 is misaligned, resulting in a Doppler factor delta~1 for emission of plasma moving parallel to the jet. This makes the observed fast TeV flares on timescales of t_v~5R_g/c harder to understand as emission from the jet. In previous work, we have proposed a jets-in-a-jet model for the ultra-fast TeV flares with t_v<<R_g/c seen in Mrk 501 and PKS 2155-304. Here, we show that about half of the minijets beam their emission outside the jet cone. Minijets emitting off the jet axis result in rapidly evolving TeV (and maybe lower energy) flares that can be observed in nearby radio galaxies. The TeV flaring from M87 fits well into this picture if M87 is a misaligned blazar.

### Fast TeV variability from misaligned minijets in the jet of M87 [Replacement]

The jet of the radio galaxy M87 is misaligned, resulting in a Doppler factor delta~1 for emission of plasma moving parallel to the jet. This makes the observed fast TeV flares on timescales of t_v~5R_g/c harder to understand as emission from the jet. In previous work, we have proposed a jets-in-a-jet model for the ultra-fast TeV flares with t_v<<R_g/c seen in Mrk 501 and PKS 2155-304. Here, we show that about half of the minijets beam their emission outside the jet cone. Minijets emitting off the jet axis result in rapidly evolving TeV (and maybe lower energy) flares that can be observed in nearby radio galaxies. The TeV flaring from M87 fits well into this picture, if M87 is a misaligned blazar.

### Scientific Highlights from Observations of Active Galactic Nuclei with the MAGIC Telescope

Since 2004, the MAGIC gamma-ray telescope has newly discovered 6 TeV blazars. The total set of 13 MAGIC-detected active galactic nuclei includes well-studied objects at other wavelengths like Markarian 501 and the giant radio galaxy M87, but also the distant the flat-spectrum radio quasar 3C 279, and the newly discovered TeV gamma-ray emitter S5 0716+71. In addition, also long-term and multi-wavelength studies on well-known TeV blazars and systematic searches for new TeV blazars have been carried out. Here we report selected highlights from recent MAGIC observations of extragalactic TeV gamma-ray sources, emphasizing the new physics insights MAGIC was able to contribute.

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# Thread: Analytic continuation

1. ## Analytic continuation

This seems to be a hard problem for me. Can someone help?
Suppose $\displaystyle F(z)=\sum_{n=1}^{\infty} d(n)z^n, for\mid{z}\mid <1$ $\displaystyle d(n)$ is the number of divisors of n. The radius of convergence of this series is 1. Show that $\displaystyle \sum_{n=1}^{\infty} d(n)z^n=\sum_{n=1}^{\infty}\frac{z^n}{1-z^n}$. Then use this to show that if $\displaystyle z=r$ with $\displaystyle 0<r<1$, then $\displaystyle \mid{F(r)}\mid\geq c\frac{1}{1-r}log(1/(1-r))$ as $\displaystyle r\rightarrow 1$.
Smilarly, if $\displaystyle \theta=2\pi p/q$ $\displaystyle p,q$ are positive integers and $\displaystyle z=re^{i\theta}$, then $\displaystyle \mid{F(re^{i\theta})}\mid\geq c_{p/q}\frac{1}{1-r}log(1/(1-r))$ as $\displaystyle r\rightarrow 1$. Conclude that there can not be any regular point on the unit disc.

2. I can perhaps start it:

$\displaystyle \frac{q^n}{1-q^n}=\sum_{k=1}^{\infty} \left(q^{n}\right)^k$

So that

$\displaystyle \sum_{n=1}^{\infty}\frac{q^n}{1-q^n}=\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}\left(q ^n\right)^k$

$\displaystyle =q+q^2+q^3+q^4+\cdots$
$\displaystyle +q^2+q^4+q^6+q^8+\cdots$
$\displaystyle +q^3+q^6+q^9+q^{12}+\cdots$
$\displaystyle +q^4+q^8+q^{12}+q^{16}+\cdots$
$\displaystyle +q^5+q^{10}+q^{15}+q^{20}+\cdots$

$\displaystyle +\vdots$

Grouping like-powers, it becomes obvious that the number of terms for each power is the divisor function $\displaystyle d(nk)$ so that that sum can be written as:

$\displaystyle \sum_{n=1}^{\infty}\frac{q^n}{1-q^n}=\sum_{n=1}^{\infty}d(n)q^n$

The rest I assume may be approached from the general theory of Lacunary series and the Hadamard Gap Theorem although I wouldn't know how.

3. Originally Posted by shawsend
I can perhaps start it:

$\displaystyle \frac{q^n}{1-q^n}=\sum_{k=1}^{\infty} \left(q^{n}\right)^k$

So that

$\displaystyle \sum_{n=1}^{\infty}\frac{q^n}{1-q^n}=\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}\left(q ^n\right)^k$

$\displaystyle =q+q^2+q^3+q^4+\cdots$
$\displaystyle +q^2+q^4+q^6+q^8+\cdots$
$\displaystyle +q^3+q^6+q^9+q^{12}+\cdots$
$\displaystyle +q^4+q^8+q^{12}+q^{16}+\cdots$
$\displaystyle +q^5+q^{10}+q^{15}+q^{20}+\cdots$

$\displaystyle +\vdots$

Grouping like-powers, it becomes obvious that the number of terms for each power is the divisor function $\displaystyle d(nk)$ so that that sum can be written as:

$\displaystyle \sum_{n=1}^{\infty}\frac{q^n}{1-q^n}=\sum_{n=1}^{\infty}d(n)q^n$

The rest I assume may be approached from the general theory of Lacunary series and the Hadamard Gap Theorem although I wouldn't know how.
Thanks shawsend. I don't think I have learned either of the theorems you mentioned yet.

4. Ok. Then I probably misunderstand the problem. Don't want to lead you down the wrong path.

I think I was confusing convergence of the series on the disc of convergence and the ability to analytically continue the series past the boundary. These two I think are not related. My understanding of the problem then is that there are no point on the boundary that the series can be analytically continued to a disc containing a point on the boundary. Or no?

5. You can get the result for z=r by an elementary calculation:

\displaystyle \begin{aligned}F(r) = \frac r{1-r} + \frac{r^2}{1-r^2} + \frac{r^3}{1-r^3} + \ldots &= \frac1{1-r}\left(r + \frac {r^2}{1+r} + \frac{r^3}{1+r+r^2} + \ldots\right)\\ &\geqslant \frac1{1-r}\left(r + \frac {r^2}2 + \frac{r^3}3 + \ldots\right)\\ &= \frac1{1-r}\ln(1/(1-r)).\end{aligned}

But the result for $\displaystyle z = re^{2\pi pi/q}$ doesn't look as though it will come out easily from that method.

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doFORC Help

#### Current version: rc6 Release date: 18.03.2020

doFORC is a portable (standalone) application working on various operating systems, is made using only free libraries, and it is made freely available to the scientific community.
Even if doFORC is mainly dedicated to FORC diagrams computation, it can smooth and approximate the derivatives of a general set of arbitrarily distributed two-dimensional points.

The main features of the doFORC tool are:

• runs naturally on Microsoft Windows operating systems, and via Wine compatibility layer on Linux-like and Macintosh operating systems
• the easy to use graphical user interface (GUI) allows users to import the input data, set the fitting parameters, to graphical represent (2D, 3D, and projection) both input and output data, and to export the graphs to image files
• the implemented algorithms and GUI are implemented in Fortran. The graphical interface is realized using the DISLIN library. The fact that both the algorithms and the interface are implemented in Fortran allows a direct interaction between them, without the need to use temporary working files saved on the computer.
• allows the choice of one of the four implemented nonparametric regression procedures: LOESS and three modified Shepard methods further modified for noisy data. The procedures allow great flexibility because no assumptions about the parametric form of the regression surface are needed. Thus the users can try different methods using their data and select one (or more) that is best suited to their needs.
• allows the use of different kernel functions
• input data may have various formats, including the PMC MicroMag format for which the drift correction can be performed
• allows the use of user weights associated with each data point, weights that indicates the precision of the information contained in the associated observation
• removes points that are closer than some tolerance (duplicate or nearby points) from the input data
• the input data can be cropped to ignore certain parts of the input points
• allows the use of a scale factor for input data to change the shape of the neighborhood, considering the points lying on an ellipse centered at the given point to be equidistant from the given point. This feature is useful when the variables have different scales.
• allows the standardization of the input data to change the shape of the neighborhood. This feature is useful when the variables have significantly different scales. The standardization is accomplished using Winsorized mean and standard deviation of each variable. Winsorized values are robust scale estimators in that extreme values of a variable are discarded (the smallest and largest 5% of the data) before estimating the data scaling.
• robust smoothing allows to minimize the influence of outlying data (outliers) allowing the user to choose different robustness weight functions
• output data can be provided at the user's choice in:
1. input points
2. a regular grid in the $\left( {h_{{\mathrm{applied}}} \ge h_{{\mathrm{reversal}}},\; h_{{\mathrm{reversal}}}} \right)$ half-plane
3. a regular grid in the $\left( {h_{{\rm{coercive}}} \ge 0,\; h_{{\rm{interaction}}}} \right)$ half-plane
4. a general rectangular regular grid
5. user points
• output consist in:
1. predicted smoothed values in the points from the input file, along with other information (residuals, smoothed residuals, histogram of the residuals, influences, confidence intervals for fit)
2. requested derivatives in the output points
3. requested statistics from (RSS, RSSm, RSE, DF1, DF2, DF3, $\delta _1$, $\delta _2$, $\rho$, AICC, AICC1, GCV)
• allows the visual analysis of the agreement between the input data and the smoothed values, and we note that this is a mandatory step, as it allows the identification of the underfitting and overfitting situations. If the two sets of data differ significantly or if the smoothed data follows too closely a noisy input, then the errors in estimating the derivatives may be quite large.
• performs statistical inference (deduces properties of data sets from a set of observations and hypotheses) provided that the error distribution satisfies some basic assumptions
• in order to perform the diagnostics and goodness of fit doFORC compute the residuals to characterize the difference between the actual observed value and the predicted value, generalized cross-validation GCV to measure the predictive performance of the model, two information criteria AICC and AICC1 to quantify the information that is lost by using an approximate model on the available data, and three degrees of freedom DF1, DF2, DF3 to compare different amounts of smoothing being performed by different smoothing methods.
• based on the above criteria doFORC can perform automatic smoothing parameter selection. Although the default method for selecting the smoothing parameter value is often satisfactory, it is often a good practice to examine how the fit varies with the smoothing parameter. In some cases, fits with different smoothing parameters might reveal important features of the data that cannot be discerned by looking at a fit with just a single "best" smoothing parameter.
• there are several ways in which user ca control the sequence of fitting parameters (number of neighbors $nn$) examined:
1. specifying a list of $nn$ values
• if no criterion is specified then a separate fit is provided for each $nn$ value
• if a criterion is specified then all values specified in $nn$ list are examined and the value that minimizes the specified criterion is selected
2. specifying a range $\left( {lower,upper} \right)$ of $nn$ values examined, for which the golden section search method is used to find a local minimum of the specified criterion in the given range
• provides a graphical illustration of the $nn$ nearest neighbors for each input point
• provides test problems that consist of sets of data obtained using various known functions, over which a known normal (Gaussian) noise and a certain percentage of outliers are added. These test problems allow users to see the limits of each method, to observe any numerical artifacts. The test problems can also be used to test, asses the accuracy, and validate other FORC type (or two dimensional smoothing) software tools that exist in the scientific literature.

## Main doFORC Interface

The main doFORC interface allows to:

1. Configure the parameters for computation.
2. Process the input data.
3. [ Optional ] Graphically represent the input and output data, as well as statistics.
• the selected data can be represented as:
• 2D plots of the selected $y$ data versus the selected $x$ data
• if several curves are represented simultaneously then each of them is drawn with different colors
• if only the residuals are selected as $y$ data then the histogram of the residuals can be represented using "Residuals histogram"
• 3D shaded surface of the $z$ triangulated data
• xy projection of the $z$ triangulated data onto the coordinate plane $\left(x, y \right)$
• the triangles are ploted with interpolated colors
• contours can be added using "contours"
• xy projection scattered of the $z$ surface onto the coordinate plane $\left(x, y \right)$
• each data point is plotted as a colored rectangle where the $x$ and $y$ coordinates determine the position of the rectangle and the $z$ coordinate deﬁnes the color
• the size of the rectangles can be set with "Marker size"
• only the points from the selected data are plotted, without the addition of any other points to fill any possible empty spaces or to create the visual sensation of continuity. Adding new points (by interpolation for example) could make it difficult to identify possible local numerical artifacts.
• if the imput data or the output data obtained with output_points = user_points option are represented as a 2D plot (with only one $y$ selected column) or as a xy scattered projection, then the nearest neighbors for each input point can be represented graphically using "Show neighbors"
4. [ Optional ] Export graphs as bmp, eps, gif, pdf, png, svg or tiff files in the working directory.
• both gif and png are exported with transparent background
• gif is saved using a 8-bit color palette
• all other formats are saved using a 24-bit color palette (true color)

## Configuration Interfaces

The parameters needed for data processing can be set by one of the two methods:

1. through a configuration file that is read in the configuration interface using "Browse config file"
• optionally after reading the parameter values can be changed by user using the interface
• an erroneous value in the configuration file is replaced with the corresponding default value (except the name of the input file)
2. by setting the parameters in the configuration interface

For both cases:

• optionally the new parameters can be saved in a new configuration file using "Save as"
• saving is not possible if errors are present
• the selected parameter values can be further used for data processing even if they are not saved in a configuration file
• mandatory the user must verify the correctness of the parameter values using "Check config data"
• the verification is also made by "Save as" and "Plot input data"
• any error or warning is signaled both in the interface and the command prompt
• if the interface is closed before all errors are solved, then data processing is not possible

For an existing configuration file:

• blank lines and comments beginning with an exclamation point (!) are ignored
• lines that do not comply with the 'keyword = value' format are ignored
• line terminator (the character or sequence of characters that marks the end of a line of text) can be CR (usually Macintosh files), LF (usually Unix files), or CRLF (usually Windows files). All lines in a given file must have the same terminator.
• the configuration files can also be created and/or modified using a plain text editor

The configuration file provided as an example contains extensive documentation in the form of comments.

## Lite Configuration Interface

• contains only the mandatory parameters
• the optional parameters are set to the default values
• lite configuration interface can be used to:
• set new parameters
• create a new configuration file
• input data are graphically represented as:
• a plot of the magnetization vs. the applied field $\left( z \;\mathrm{vs.}\; x \right)$
• a projection of the magnetization surface onto the coordinate planes $\left( z \;\mathrm{vs.}\; x \;\mathrm{and}\; y \right)$
• each data point is plotted as a colored rectangle where the $x$ and $y$ coordinates determine the position of the rectangle and the $z$ coordinate deﬁnes the color
• the size of the rectangles can be set with "Marker size"
• values entered by the user in the editable field are not read until the Enter key is pressed
• "Save as," "Check config data," and "Plot input data" read all the displayed fields
• "New config file" resets all the parameters to their default values

## Main Configuration Interface

• similar to Lite Configuration Interface, but additionally contains the optional parameters
• initially all parameters are set to their default values
• the displayed fields may change depending on the value selected from a list
• the values entered by the user in the editable fields are not read until the Enter key is pressed

## Plot details

The Plot details interface allows to:

• edit the axes titles
• set the position and the number of the ticks on the axes
• add text on the graphs
• the text characters are interpreted using LaTeX markup, allowing to include Greek letters, special characters (such as integral and summation symbols), superscripts, subscripts, to modify the text type and color

## Command Prompt

The command prompt contains both the messages displayed in GUI and others in addition.

The error or warning messages from the numerical subroutines are displayed only in the command prompt.

## Files and data saved by doFORC

doFORC save the processed data into new files whose names are obtained from the input_file by appending new strings before the extension. If files with the same name already exist, then they will be overwritten.

1. input_file__smoothed_input contains the smoothed values in the points from the input file, along with other information, according to the table below:
ColumnDescriptionRestriction
h_app applied field (column $x$) N/A
h_rev reversal field (column $y$) N/A
m magnetic moment (column $z$) N/A
diagL $\text{diag} \left( L \right) \equiv$ influence ihat ≥ 1 or istat ≥ 1
confi confidence intervals ihat = 2 or istat = 2
m_fit smoothed magnetic moment N/A
res residuals N/A
res_fit smoothed residuals smoothresidual = true
2. input_file__smoothed_output contains the requested derivatives in the output_points saved as columns $\left( x,y,z \right)$

3. input_file__smoothed_output_matrix_order_of_derivative_nn contains the requested derivatives in the output_points saved as matrices, one file for each value of order_of_derivative and/or nn

Restriction: only for output_points = ha_hr_regular_grid, hc_hu_regular_grid, or rectangular_grid

4. input_file__statistics contains the requested statistics according to ihat or istat

If nn_list = true and a CRITERION is specified, or if nn_range = true, then the statistics for all examined nn values are saved, and the selected value that minimizes the specified CRITERION is on the last line.

## Test Functions Generator

The pseudorandom number generator is initialized by the $seed$ number, so that each value of the seed generates other realization of the true values for the selected function:

• if $seed=0$ then the program generates a different pseudorandom sequence each time it runs, using the system clock
• if $seed\neq 0$ then each value of the $seed$ generates other sequence of random numbers, the same at every run

## Screenshots

"fill_x-steps_gt" resamples the individual FORCs by filling gaps greater than 'fill_x-steps_gt' on each individual FORC [curves given by the points with the $y$ ($h_{\mathrm{reversal}}$) coordinate], in the preprocessing step.

This feature is useful in the case of missing data or of large gaps in the input data.

A statistic regarding the values of the increment $dx$ on each individual curve and respectively of the increment $dy$ between curves is provided in the Command Prompt window.

"merge_x-steps_lt" resamples the individual FORCs by merging the points that are separated by steps less than 'merge_x-steps_lt' on each individual FORC [curves given by the points with the same $y$ ($h_{\mathrm{reversal}}$) coordinate], in the preprocessing step.

The merging procedure may return points that were not input points, and it can even have a "smoothing effect".

"nn_iFORC" sets the number of nearest neighbors used to smooth each individual FORC with respect to $h_{applied}$, in the preprocessing step.

"Display curve" can be used to plot only an individual curve or only a subset of the input curves.

"Curves to be processed" sets the curves (FORCs) that will be processed in the processing step.

.

2D plot of both the input data and the smoothed data.

The visual analysis of the agreement between the input data and the smoothed values is a mandatory step, as it allows the identiﬁcation of the underﬁtting and overﬁtting situations.

If the imput data or the output data obtained with output_points = user_points option are represented as a 2D plot with only one $y$ selected column, then the nearest neighbors for each input point can be represented graphically using "Show neighbors".

If in a 2D plot only the residuals are selected as $y$ data then the histogram of the residuals can be represented using "Residuals histogram".

3D shaded surface of the $z$ triangulated data

xy projection of the $z$ triangulated data onto the coordinate plane $\left(x, y \right)$

• the triangles are ploted with interpolated colors
• contours can be added using "contours", specifying one of the following parameters:
• values at which contours are drawn
• "nc = number_of_contours" number of equidistant contours drawn between zmin and zmax
• "step = spacing_of_contours" spacing of contour lines
• starting from zmin if step > 0
• starting from zmax if step < 0

xy projection scattered of the $z$ surface onto the coordinate plane $\left(x, y \right)$

• the size of the rectangles can be set with "Marker size"
• if the imput data or the output data obtained with output_points = user_points option are represented, then the nearest neighbors for each input point can be represented graphically using "Show neighbors"

Even if doFORC is mainly dedicated to FORC diagrams computation, it can smooth and approximate the derivatives of a general set of arbitrarily distributed two-dimensional points.

The Plot details interface allows to edit the axes titles, to set the position and the number of the ticks on the axes, and to add text on the graphs.

The text characters are interpreted using LaTeX markup, allowing to include Greek letters, special characters (such as integral and summation symbols), superscripts, subscripts, to modify the text type and color.


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## Thursday, March 10, 2011

### Image Segmentation using Active Contours, Level Sets

Here is a new image segmentation Python code which is a port of this Matlab file. It uses level sets and mean curvature motion, and is able to segment the sample image after few iterations. We had to write our own bwdist which is a wrapper for distance_transform_edt wrapper so bwdist acted the same way as Matlab bwdist(), some conversions were necessary to make this work. We also changed the initialization of \phi, making it simpler and closer to Dr. Li's code we shared before.

Code

import matplotlib.pyplot as plt
import numpy as np
import scipy.signal as signal
import scipy.ndimage as image
import time
from scipy import ndimage

def bwdist(a):
"""
this is an intermediary function, 'a' has only True, False vals,
so we convert them into 0, 1 values -- in reverse. True is 0,
False is 1, distance_transform_edt wants it that way.
"""
b = np.ones(a.shape)
b[a==True] = 0.
return ndimage.distance_transform_edt(b)

def gauss_kern():
""" Returns a normalized 2D gauss kernel array for convolutions """
h1 = 15
h2 = 15
x, y = np.mgrid[0:h2, 0:h1]
x = x-h2/2
y = y-h1/2
sigma = 1.5
g = np.exp( -( x**2 + y**2 ) / (2*sigma**2) );
return g / g.sum()

Img = Img[::-1]
g = gauss_kern()
Img_smooth = signal.convolve(Img,g,mode='same')
rows, cols = Img.shape

# initial function phi - level set is a square 4 pixels
# away from borders on each side, in 3D it looks like an empty
# box
c0=4
w=4
nrow, ncol=Img.shape
phi=c0*np.ones((nrow,ncol))
phi[w+1:-w-1, w+1:-w-1]=-c0

# edge-stopping function

dt=.4

# number of iterations after which we reinitialize the surface
num_reinit=10

phiOld=np.zeros((rows,cols))

# number of iterations after which we reinitialize the surface
iter=0

plt.ion()

while np.sum(np.sum(np.abs(phi-phiOld))) != 0:
# magnitude of gradient of phi
# normalized gradient of phi - eliminating singularities

# curvature is the divergence of normalized gradient of phi
tmp1 = g * K * absGradPhi
dPhiBydT =tmp1 + tmp2 + tmp3
phiOld=phi
# level set evolution equation
phi = phi + ( dt * dPhiBydT )
iter=iter+1
if np.mod(iter,num_reinit)==0:
# reinitialize the embedding function after num_reinit iterations
phi=np.sign(phi)
phi = (phi > 0) * (bwdist(phi <> 0))

if np.mod(iter,10)==0:
time.sleep(0.6)
plt.imshow(Img, cmap='gray')
plt.hold(True)
CS = plt.contour(phi,0, colors='r')
plt.draw()
plt.hold(False)


text:
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# In the formula above, a, b, c, and d are positive numbers.

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In the formula above, a, b, c, and d are positive numbers.  [#permalink]

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Updated on: 12 Sep 2013, 09:37
1
1
15
00:00

Difficulty:

85% (hard)

Question Stats:

54% (02:30) correct 46% (02:44) wrong based on 201 sessions

### HideShow timer Statistics

x = 8(a/c) + 16(b/c)

In the formula above, a, b, c, and d are positive numbers. If c = a + b and b > a, which of the following could be the value of x?

(A) 8
(B) 10
(C) 12
(D) 15
(E) 21

_________________

Originally posted by carcass on 12 Sep 2013, 09:32.
Last edited by Bunuel on 12 Sep 2013, 09:37, edited 1 time in total.
Moved to PS forum.
Math Expert
Joined: 02 Sep 2009
Posts: 58098
Re: In the formula above, a, b, c, and d are positive numbers.  [#permalink]

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12 Sep 2013, 09:44
3
7
carcass wrote:
x = 8(a/c) + 16(b/c)

In the formula above, a, b, c, and d are positive numbers. If c = a + b and b > a, which of the following could be the value of x?

(A) 8
(B) 10
(C) 12
(D) 15
(E) 21

$$\frac{8a}{c} + \frac{16b}{c}=\frac{8a+16b}{c}=\frac{8a+16b}{a+b}$$.

You can notice that it's a weighted average formula for $$a+b$$ items, where the weight of each of the items of $$a$$ is 8 and the weight of each of the items of $$b$$ is 16. Now, since b > a, then the average must be closer to 16, then to 8. Only D fits.

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##### General Discussion
Math Expert
Joined: 02 Sep 2009
Posts: 58098
Re: In the formula above, a, b, c, and d are positive numbers.  [#permalink]

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12 Sep 2013, 09:45
2
Bunuel wrote:
carcass wrote:
x = 8(a/c) + 16(b/c)

In the formula above, a, b, c, and d are positive numbers. If c = a + b and b > a, which of the following could be the value of x?

(A) 8
(B) 10
(C) 12
(D) 15
(E) 21

$$\frac{8a}{c} + \frac{16b}{c}=\frac{8a+16b}{c}=\frac{8a+16b}{a+b}$$.

You can notice that it's a weighted average formula for $$a+b$$ items, where the weight of each of the items of $$a$$ is 8 and the weight of each of the items of $$b$$ is 16. Now, since b > a, then the average must be closer to 16, then to 8. Only D fits.

Similar question to practice from OG:
Quote:
If x, y, and k are positive numbers such that (x/(x+y))(10) + (y/(x+y))(20) = k and if x < y, which of the following could be the value of k?

A. 10
B. 12
C. 15
D. 18
E. 30

Discussed here: if-x-y-and-k-are-positive-numbers-such-that-x-x-y-128231.html
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Re: In the formula above, a, b, c, and d are positive numbers.  [#permalink]

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08 Sep 2014, 03:50
1
1
carcass wrote:
x = 8(a/c) + 16(b/c)

In the formula above, a, b, c, and d are positive numbers. If c = a + b and b > a, which of the following could be the value of x?

(A) 8
(B) 10
(C) 12
(D) 15
(E) 21

x = 8a/c + 16b/c
x= 8(a+b)/c + 8b/c
now substituting c = a + b in the above equation
x = 8 + 8b/(a+b)
The ratio b/(a+b) is smaller than 1 thus answer choice E is eliminated... and as a < b ... thus the ratio b/(a+b) is larger than 1/2 or 0.5... thus
8*b/(a+b) is larger than 4... thus the only answer choice left is D
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In the formula above, a, b, c, and d are positive numbers.  [#permalink]

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13 Sep 2014, 01:09
Can anyone explain this in detail?
Math Expert
Joined: 02 Sep 2009
Posts: 58098
Re: In the formula above, a, b, c, and d are positive numbers.  [#permalink]

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13 Sep 2014, 07:18
vijayann wrote:
Can anyone explain this in detail?

Please specify what part in the solutions above is unclear.
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In the formula above, a, b, c, and d are positive numbers.  [#permalink]

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15 Sep 2015, 13:30
x = 8(a/c) + 16(b/c)

just by looking at the options .. we should be able to understand that 8 or its multiple i.e 16 should be divisible by C..why ? Because none of the options are in p/q form.

let us say c =4 which gives us a=1 and b=3

the equation thus becomes x=8(1/4) + 16(3/4) =2+12=14 --close enough
the other value c can take is 8
which gives us possible values for (a,b) as (1,7) (2,6) (3,5)

take 1,7

x=8(1/8)+16(7/8)=15

That's how I managed to do.
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Re: In the formula above, a, b, c, and d are positive numbers.  [#permalink]

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11 Jan 2016, 19:17
oh man..I did the long way..
8a+8b/a+b = 8
now, 8b/a+b =??
we can eliminate A right away..
tried several numbers..then got to:
a=1
b=7
a+b=8
8b=56.
56/8=7.
7+8=15.

D
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Re: In the formula above, a, b, c, and d are positive numbers.  [#permalink]

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27 May 2019, 14:45
CounterSniper wrote:
x = 8(a/c) + 16(b/c)

just by looking at the options .. we should be able to understand that 8 or its multiple i.e 16 should be divisible by C..why ? Because none of the options are in p/q form.

let us say c =4 which gives us a=1 and b=3

the equation thus becomes x=8(1/4) + 16(3/4) =2+12=14 --close enough
the other value c can take is 8
which gives us possible values for (a,b) as (1,7) (2,6) (3,5)

take 1,7

x=8(1/8)+16(7/8)=15

That's how I managed to do.

I did exactly the same way! Can someone clarify if this method is OK to use for future such questions? Feels like guess work and lucked out with this one.
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Re: In the formula above, a, b, c, and d are positive numbers.  [#permalink]

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28 May 2019, 18:06
carcass wrote:
x = 8(a/c) + 16(b/c)

In the formula above, a, b, c, and d are positive numbers. If c = a + b and b > a, which of the following could be the value of x?

(A) 8
(B) 10
(C) 12
(D) 15
(E) 21

Simplifying, we have:

x = (8a + 16b)/c

x = (8a + 16b)/(a + b)

We see that we have a weighted average, and since b is greater than a, then x must be closer to 16 and also between 8 and 16, so x could be 15.

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Re: In the formula above, a, b, c, and d are positive numbers.   [#permalink] 28 May 2019, 18:06
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text:
## The counting matrix of a simplicial complex

The counting matrix of a simplicial complex has determinant 1 and is isospectral to its inverse. The sum of the matrix entries of the inverse is the number of elements in the complex.

## A Dyadic Riemann hypothesis

When replacing the circle group with the dyadic group of integers, the Riemann zeta function becomes an explicit entire function for which all roots are on the imaginary axes. This is the Dyadic Riemann Hypothesis.