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## Electronically Available Publications and Pre-Prints

#### 2016

High Resolution Linear Polarimetric Imaging for the Event Horizon Telescope by Chael, Andrew A.; Johnson, Michael D.; Narayan, Ramesh; Doeleman, Sheperd S.; Wardle, John F. C.; Bouman, Katherine L., Astrophys. J. in press (2016)

Abstract: Images of the linear polarization of synchrotron radiation around Active Galactic Nuclei (AGN) identify their projected magnetic field lines and provide key data for understanding the physics of accretion and outflow from supermassive black holes. The highest resolution polarimetric images of AGN are produced with Very Long Baseline Interferometry (VLBI). Because VLBI incompletely samples the Fourier transform of the source image, any image reconstruction that fills in unmeasured spatial frequencies will not be unique and reconstruction algorithms are required. In this paper, we explore extensions of the Maximum Entropy Method (MEM) to linear polarimetric VLBI imaging. In contrast to previous work, our polarimetric MEM algorithm combines a Stokes I imager that uses only bispectrum measurements that are immune to atmospheric phase corruption with a joint Stokes Q and U imager that operates on robust polarimetric ratios. We demonstrate the effectiveness of our technique on 7- and 3-mm wavelength quasar observations from the VLBA and simulated 1.3-mm Event Horizon Telescope observations of Sgr A* and M87. Consistent with past studies, we find that polarimetric MEM can produce superior resolution compared to the standard CLEAN algorithm when imaging smooth and compact source distributions. As an imaging framework, MEM is highly adaptable, allowing a range of constraints on polarization structure. Polarimetric MEM is thus an attractive choice for image reconstruction with the EHT.

#### 2015

"X-Shaped Radio Galaxies and the Nanohertz Gravitational Waves Background", by David Roberts, L. Saripalli, & R. Subrahmanyan in Astronomy in Focus, Vol. I, in press (2015).

Abstract: Coalescence of super massive black holes (SMBH’s) in galaxy mergers is potentially the dominant contributor to the low frequency gravitational wave background (GWB). It was proposed by Merritt & Ekers (2002) that X-shaped radio galaxies are signposts of such coalescences, and that their abundance might be used to predict the magnitude of the gravitational wave background. In Roberts et al. (2015) we present radio images of all 52 X-shaped radio source candidates out of the sample of 100 selected by Cheung (2007) for which archival VLA data were available. These images indicate that at most 21% of the candidates might be genuine X-shaped radio sources that were formed by a restarting of beams in a new direction following a major merger. This suggests that fewer than 1.3% of extended radio sources appear to be candidates for genuine axis reorientations (“spin flips”), much smaller than the 7% suggested by Leahy & Parma (1992). Thus the associated gravitational wave background may be substantially smaller than previous estimates. These results can be used to normalize detailed calculations of the SMBH coalescence rate and the GWB. Subject headings: galaxies: active — gravitational waves — radio continuum: galaxies

#### 2013

The Structure and Linear Polarization of the Kiloparsec-scale Jet of the Quasar 3C 345, by David Roberts, John Wardle & Valerie Marchenko (Brandeis University) in Astron. J. 145, 49 (2013)

Abstract: Deep Very Large Array imaging of the quasar 3C 345 at 4.86 and 8.44 GHz has been used to study the structure and linear polarization of its radio jet on scales ranging from 2 to 30 kpc. There is a 7-8 Jy unresolved core with spectral index α ~= –0.24 (I νvpropνα). The jet (typical intensity 15 mJy beam–1) consists of a 2.''5 straight section containing two knots, and two additional non-co-linear knots at the end. The jet's total projected length is about 27 kpc. The spectral index of the jet varies over –1.1 <~ α <~ –0.5. The jet diverges with a semi-opening angle of about 9°, and is nearly constant in integrated brightness over its length. A faint feature northeast of the core does not appear to be a true counter-jet, but rather an extended lobe of this FR-II radio source seen in projection. The absence of a counter-jet is sufficient to place modest constraints on the speed of the jet on these scales, requiring β >~ 0.5. Despite the indication of jet precession in the total intensity structure, the polarization images suggest instead a jet re-directed at least twice by collisions with the external medium. Surprisingly, the electric vector position angles in the main body of the jet are neither longitudinal nor transverse, but make an angle of about 55° with the jet axis in the middle while along the edges the vectors are transverse, suggesting a helical magnetic field. There is no significant Faraday rotation in the source, so that is not the cause of the twist. The fractional polarization in the jet averages 25% and is higher at the edges. In a companion paper, Roberts & Wardle show that differential Doppler boosting in a diverging relativistic velocity field can explain the electric vector pattern in the jet.

#### 2012

Discovery of a Kiloparsec-scale X-Ray/Radio Jet in the z = 4.72 Quasar GB 1428+4217, by C. Cheung (Naval Research Laboratory), L. Starwarz (ISAS, Japan), A. Siemiginowska (Harvard-Smithsonian Center for Astrophysics), D. Gobeille (University of South Florida), J. Wardle (Brandeis University), D. Harris (Harvard-Smithsonian Center for Astrophysics) & D. Schwartz (Harvard-Smithsonian Center for Astrophysics) in Astrophys. J. Letters, 756, 20 (2012)

Abstract: We report the discovery of a one-sided 3farcs6 (24 kpc, projected) long jet in the high-redshift, z = 4.72, quasar GB 1428+4217 in new Chandra X-ray and Very Large Array (VLA) radio observations. This is the highest redshift kiloparsec-scale X-ray/radio jet known. Analysis of archival very long baseline interferometry 2.3 and 8.6 GHz data reveal a faint one-sided jet extending out to ~200 pc and aligned to within ~30° of the Chandra/VLA emission. The 3farcs6 distant knot is not detected in an archival Hubble Space Telescope image, and its broadband spectral energy distribution is consistent with an origin from inverse Compton scattering of cosmic microwave background photons for the X-rays. Assuming also equipartition between the radiating particles and magnetic field, the implied jet Lorentz factor is ≈5. This is similar to the other two known z ~ 4 kpc scale X-ray jet cases and smaller than typically inferred in lower-redshift cases. Although there are still but a few such very high redshift quasar X-ray jets known, for an inverse Compton origin, the present data suggest that they are less relativistic on large scales than their lower-redshift counterparts.

Evidence for Highly Relativistic Velocities in the Kiloparsec-scale Jet of the Quasar 3C 345, by David Roberts & John Wardle (Brandeis University) in Astrophys. J. Letters 759, 35 (2012)

Abstract: In this paper we use radio polarimetric observations of the jet of the nearby bright quasar 3C 345 to estimate the fluid velocity on kiloparsec scales. The jet is highly polarized, and surprisingly, the electric vector position angles in the jet are "twisted" with respect to the jet axis. Simple models of magnetized jets are investigated in order to study various possible origins of the electric vector distribution. In a cylindrically symmetric transparent jet a helical magnetic field will appear either transverse or longitudinal due to partial cancellations of Stokes parameters between the front and back of the jet. Synchrotron opacity can break the symmetry, but it leads to fractional polarization less than that observed and to strong frequency dependence that is not seen. Modeling shows that differential Doppler boosting in a diverging jet can break the symmetry, allowing a helical magnetic field to produce a twisted electric vector pattern. Constraints on the jet inclination, magnetic field properties, intrinsic opening angle, and fluid velocities are obtained and show that highly relativistic speeds (β >~ 0.95) are required. This is consistent with the observed jet opening angle, with the absence of a counter-jet, with the polarization of the knots at the end of the jet, and with some inverse-Compton models for the X-ray emission from the 3C 345 jet. This model can also apply on parsec scales and may help explain those sources where the electric vector position angles in the jet are neither parallel nor transverse to the jet axis.

#### 2011

Structure and Magnetic Fields in the Precessing Jet System SS 433. III. Evolution of the Intrinsic Brightness of the Jets from a Deep Multi-epoch Very Large Array Campaign, by Michael Bell, David Roberts & John Wardle (Brandeis University) in Astrophys. J. 736, 118 (2011)

Abstract: We present a sequence of five deep observations of SS 433 made over the summer of 2007 using the Very Large Array in the A configuration at 5 and 8 GHz. In this paper, we study the brightness profiles of the jets and their time evolution. We also examine the spectral index distribution in the source. We find (as previously reported from the analysis of a single earlier image) that the profiles of the east and west jets are remarkably similar if projection and Doppler beaming are taken into account. The sequence of five images allows us to disentangle the evolution of brightness of individual pieces of jet from the variations of jet power originating at the core. We find that the brightness of each piece of the jet fades as an exponential function of age (or distance from the core), e^{-\tau /\tau ^{\prime }}, where τ is the age at emission and τ' = 55.9 ± 1.7 days. This evolutionary model describes both the east and west jets equally well. There is also significant variation (by a factor of at least five) in jet power with birth epoch, with the east and west jets varying in synchrony. The lack of deceleration between the scale of the optical Balmer line emission (1015 cm) and that of the radio emission (1017 cm) requires that the jet material is much denser than its surroundings. We find that the density ratio must exceed 300:1.

#### 2010

Structure and Magnetic Fields in the Precessing Jet System SS 433. II. Intrinsic Brightness of the Jets, by David Roberts, John Wardle, Michael Bell, Matthew Mallory, Valerie Marchenko & Phoebe Sanderbeck (Brandeis University) in Astrophys. J. 719, 1918 (2010)

Abstract: Deep Very Large Array imaging of the binary X-ray source SS 433, also classified as a microquasar, has been used to study the intrinsic brightness distribution and evolution of its radio jets. The intrinsic brightness of the jets as a function of age at emission of the jet material τ is recovered by removal of the Doppler boosting and projection effects. We find that, intrinsically, the two jets are remarkably similar when compared for equal τ, and that they are best described by Doppler boosting of the form D 2+α, as expected for continuous jets. The intrinsic brightnesses of the jets as functions of age behave in complex ways. In the age range 60 days < τ < 150 days, the jet decays are best represented by exponential functions of τ, but linear or power-law functions are not statistically excluded. This is followed by a region out to τ ~= 250 days during which the intrinsic brightness is essentially constant. At later times, the jet decay can be fit roughly as exponential or power-law functions of τ.

#### 2009

Full Polarization Spectra of 3C 279, by D. Homan (Dennison University), M. Lister (Purdue University), H. Aller (University of Michigan), M. Aller (University of Michigan), J. Wardle (Brandeis) in Astrophys. J. 696, 328 (2009)

Abstract: We report the results of parsec-scale, multifrequency Very Long Baseline Array observations of the core region of 3C 279 in Stokes I, linear polarization, and circular polarization. These full polarization spectra are modeled by radiative transfer simulations to constrain the magnetic field and particle properties of the parsec-scale jet in 3C 279. We find that the polarization properties of the core region, including the amount of linear polarization, the amount and sign of Faraday rotation, and the amount and sign of circular polarization can be explained by a consistent physical picture. The base of the jet, component D, is modeled as an inhomogeneous Blandford-Königl style conical jet dominated by a vector-ordered poloidal magnetic field along the jet axis, and we estimate its net magnetic flux. This poloidal field is responsible for the linear and circular polarization from this inhomogeneous component. Farther down the jet, the magnetic field in two homogeneous features is dominated by local shocks and a smaller fraction of vector-ordered poloidal field remains along the jet axis. This remaining poloidal field provides internal Faraday rotation which drives Faraday conversion of linear polarization into circular polarization from these components. In this picture, we find the jet to be kinetically dominated by protons with the radiating particles being dominated by electrons at an approximate fraction of gsim75%, still allowing the potential for a significant admixture of positrons. Based on the amounts of Faraday conversion deduced for the homogeneous components, we find a plausible range for the lower cutoff in the relativistic particle energy spectrum to be 5 lsim γ l lsim 35. The physical picture described here is not unique if the observed Faraday rotation and depolarization occur in screens external to the jet; however, we find the joint explanation of linear and circular polarization observations from a single set of magnetic fields and particle properties internal to the jet to be compelling evidence for this picture.

#### 2008

Structure and Magnetic Fields in the Precessing Jet System SS 433. I. Multifrequency Imaging from 1998, by David Roberts, John Wardle, Scott Lipnick, Phillip Selesnik & Simon Slutsky (Brandeis University) in Astrophys. J. 676, 584 (2008)

Abstract: The Very Large Array has been used at five frequencies to study the structure and linear polarization of SS 433 on scales as small as ~0.1''~=500 AU. Each jet consists of a sharp, curving ridgeline at the leading edge, plus significant trailing off-jet emission, showing that they are enveloped by diffuse relativistic plasma. No kinematic model with constant jet speed fits our images on all scales, but they are consistent with variations in jet speed of around +/-10% around the optical value. Our images show continuous jets with bright components occurring simultaneously in the two jets roughly every 35 days. When corrected for projection effects and Doppler boosting, the intensities of the two jets are intrinsically very similar. Fractional linear polarization up to 20% is present along the ridgelines, while the core is essentially unpolarized. The rotation measures are consistent with a foreground screen with RM~=+110 rad m-2 plus a larger, asymmetrical contribution close to the core. The apparent magnetic fields in the jets are roughly aligned with the ridgelines in most but not all of each jet. The jet is more highly polarized between the components than in the components themselves, suggesting that the mechanism that creates them compresses a tangled part of the magnetic field into a partially ordered transverse layer. The off-jet emission is remarkably highly polarized, with m~=50% in places, suggesting large-scale order of the magnetic field surrounding the jets. This polarized signal may confuse the determination of magnetic field orientations in the jets themselves. However, the images are consistent with a jet magnetic field that is everywhere parallel to the helices.

#### 2006

Deep Chandra and Multicolor HST Follow-up of the Jets in Two Powerful Radio Quasars, by R. Sambruna (NASA GSFC), M. Gliozzi (George Mason), D. Donato (George Mason), L. Maraschi (Osservatorio di Brera), F. Tavecchio (Osservatorio di Brera), C. Cheung (MIT), M. Urry (Yale) & J. Wardle (Brandeis) in Astrophys. J. 641, 717 (2006)

Abstract: We present deep (70-80 ks) Chandra and multicolor HST ACS images of two jets hosted by the powerful quasars 1136-135 and 1150+497, together with new radio observations. The sources have an FR II morphology and were selected from our previous X-ray and optical jet survey for detailed follow-up aimed at obtaining better constraints on the jet multiwavelength morphology and X-ray and optical spectra of individual knots and to test emission models to derive physical parameters more accurately. All the X-ray and optical knots detected in our previous short exposures are confirmed, together with a few new faint features. The overlaid maps and the emissivity profiles along the jet show good correspondence between emission regions at the various wavelengths; a few show offsets between the knot peaks of <1". In 1150+497 the X-ray, optical, and radio profiles decrease in similar ways with distance from the core up to ~7", after which the radio emission increases more than does the X-ray one. No X-ray spectral variations are observed in 1150+497. In 1136-135 an interesting behavior is observed, whereby, downstream of the most prominent knot at ~6.5" from the core, the X-ray emission fades, while the radio emission brightens. The X-ray spectrum also varies, with the X-ray photon index flattening from ΓX~2 in the inner part to ΓX~1.7 to the end of the jet. We interpret the jet behavior in 1136-135 in a scenario in which the relativistic flow suffers systematic deceleration along the jet, and we briefly discuss the major consequences of this scenario. The latter is discussed in more detail in our companion paper (Tavecchio et al.).

Deceleration from Entrainment in the Jet of the Quasar 1136-135?, by F. Tavecchio (Osservatorio di Brera), L. Maraschi (Osservatorio di Brera), R. Sambruna (NASA GSFC), M. Gliozzi (George Mason), D.Donato (George Mason), C. Cheung (MIT), J. Wardle (Brandeis) & M. Urry (Yale) ) in Astrophys. J. 641, 732 (2006)

Abstract:  By modeling the multiwavelength emission of successive regions in the jet of the quasar PKS 1136-135, we find indications that the jet suffers deceleration near its end on a (deprojected) scale of ~400 kpc. We adopt a continuous flow approximation, and we discuss the possibility that the inferred deceleration from a Lorentz factor of Γ=6.5 to 2.5 is induced by entrainment of external gas. Some consequences of this scenario are discussed.

#### 2005

Discovery of Optical Emission in the Hot Spots of Three 3CR Quasars: High-Energy Particle Acceleration in Powerful Radio Hot Spots, by C. Cheung, J. Wardle & T. Chen (Brandeis University) in Astrophys. J.  628, 104 (2005)

Abstract: Archival Hubble Space Telescope WFPC2 images were used to search for optical emission associated with the radio jets of a number of luminous quasars. From this search, we report new optical hot spot detections in the well-known blazar 3C 454.3 and the lobe-dominated quasars 3C 275.1 and 3C 336. We also find optical emission in the vicinity of the hot spot in 3C 208 but believe this is a chance alignment. Optical emission from the arcsecond-scale jet in 3C 454.3 is also detected. Multi-frequency archival radio data from the VLA and MERLIN are analyzed, and the synchrotron spectra of these high-power hot spots are presented. We estimate that their break frequencies are in the range of 1010-1011 Hz, with large uncertainties due to the wide gap in frequency coverage between the radio and optical bands. We also calculate their equipartition magnetic fields and find that the anti-correlation between break frequency and magnetic field found by Brunetti et al. for lower power hot spots extends to these high-power hot spots. This supports their model of hot spots based on shock acceleration and synchrotron losses.

Concurrent 43 and 86 GHz Very Long Baseline Polarimetry of 3C 273 , by Joanne Attridge (MIT Haystack Observatory), John Wardle (Brandeis University) & Daniel Homan (Dennison University) in Astrophys. J. Letters 633, 85 (2005)

Abstract: We present submilliarcsecond resolution total intensity and linear polarization VLBI images of 3C 273, using concurrent 43 and 86 GHz data taken with the Very Long Baseline Array in 2002 May. The structure seen in the innermost jet suggest that we have fortuitously caught the jet in the act of changing direction. The polarization images confirm that the core is unpolarized (fractional polarization <=1%) at 86 GHz, but also show well ordered magnetic fields (m~15%) in the inner jet, at a projected distance of 2.3 pc from the core. In this strongly polarized region, the rotation measure changes across the jet by ~4.2×104 rad m-2 over an angular width of about 0.3 mas. If the lack of polarization in the core is also attributed to a Faraday screen, then a rotation measure dispersion >~5.2×104 rad m-2 must be present in or in front of that region. These are among the highest rotation measures reported so far in the nucleus of any active galaxy or quasar, and must occur outside (but probably close to) the radio emitting region. The transverse rotation measure gradient is in the same sense as that observed by Asada and coworkers and by Zavala and Taylor at greater core distances. The magnitude of the transverse gradient decreases rapidly with distance down the jet, and appears to be variable.

#### 2004

Jets from Sub-Parsec to Kiloparsec Scales: A Physical Connection, by F. Tavecchio, L. Maraschi (Brera), R.M. Sambruna (GMU), C.M. Urry (Yale), C.C. Cheung (Brandeis), J.K. Gambill (GMU), R. Scarpa (ESO). 2004, ApJ, accepted

Abstract: The Chandra discovery of bright X-ray emission from kpc-scale jets allows insight into the physical parameters of the jet flow at large scale. At the opposite extreme, extensive studies of the inner relativistic jets in Blazars with multiwavelength observations, yield comparable information on sub-parsec scales. In the framework of simple radiation models for the emission regions we compare the physical parameters of jets on these two very different scales in the only two well studied Blazars for which large-scale emission has been resolved by Chandra. Notably, we find that the relativistic Doppler factors and powers derived independently at the two scales are consistent, suggesting that the jet does not suffer severe deceleration or dissipation. Moreover the internal equipartition pressures in the inner jet and in the external X-ray bright knots scale inversely with the jet cross section as expected in the simple picture of a freely expanding jet in equipartition.

High Levels of Circularly Polarized Emission from the Radio Jet in NGC 1275 (3C84), by D.C. Homan (NRAO, Denison) and J.F.C. Wardle (Brandeis). 2004, ApJL, accepted

Abstract: We present multi-frequency, high resolution VLBA circular polarization images of the radio source 3C 84 in the center of NGC 1275. Our images reveal a complex distribution of circular polarization in the inner parsec of the radio jet, with local levels exceeding 3% polarization, the highest yet detected with VLBI techniques. The circular polarization changes sign along the jet, making 3C 84 also the first radio jet to show both signs of circular polarization simultaneously. The spectrum and changing sign of the circular polarization indicate that it is unlikely to be purely intrinsic to the emitted synchrotron radiation. The Faraday conversion process makes a significant and perhaps dominant contribution to the circular polarization, and the observed spectrum suggests the conversion process is near saturation. The sign change in the circular polarization along the jet may result from this saturation or may be due to a change in magnetic field order after an apparent bend in the jet. From the small spatial scales probed here, ~ 0.15 pc, and the comparably high levels of circular polarization inferred for the intra-day variable source PKS 1519-273, we suggest a connection between small spatial scales and efficient production of circular polarization.

A Survey of Extended Radio Jets with Chandra and HST, by R.M. Sambruna, J.K. Gambill (George Mason Univ.), L. Maraschi, F. Tavecchio, R. Cerutti (Oss Brera), C. C. Cheung (Brandeis Univ.), C. M. Urry (Yale Univ.), G. Chartas (Penn State Univ.), 2004, ApJ, accepted

Parsec-Scale Blazar Monitoring: The Data, by Roopesh Ojha, Daniel C. Homan, David H. Roberts, John F. C. Wardle, Margo F. Aller, Hugh D. Aller, and Philip A. Hughes. 2004, ApJS, 150, 187:237

Abstract: We present the images and modeling data obtained from a dual frequency, six-epoch, VLBA polarization experiment monitoring a sample of 12 blazars. The observations were made at 15 and 22 GHz at evenly spaced, bimonthly intervals over 1996. The advent of the VLBA makes possible a data set with reliable calibration as well as regular and frequent temporal sampling. Detection of circular polarization, proper motion studies, and flux and polarization variability in the sample are some of the topics that such a data set makes available for robust investigation.

#### 2003

Radio Identification of the X-ray Jet in the z=4.3 Quasar GB 1508+5714, by C.C. Cheung (Brandeis). 2003, ApJL, accepted

Abstract: The recent discovery of an X-ray jet in the z=4.3 quasar GB~1508+5714 by Yuan et al. (astro-ph/0309318) and Siemiginowska et al. (astro-ph/0310241) prompted a search for its radio counterpart. Here, we report the successful discovery of faint radio emission from the jet at 1.4 GHz using archival VLA data. The X-ray emission is best interpreted as inverse Compton (IC) emission off the CMB as discussed by the previous investigators. In this scenario, its high X-ray to radio monochromatic luminosity ratio, compared to previously detected IC/CMB X-ray jets at lower redshift, is a natural consequence of its high redshift.

The XMM-Newton View of the X-ray Halo and Jet of NGC 6251, by R.M. Sambruna (1), M. Gliozzi (1), D. Donato (1), F. Tavecchio (2), C.C. Cheung (3), R. Mushotzky (4) ((1) GMU, (2) OAB Merate, (3) Brandeis University, (4) NASA GSFC). 2003, Astronomy and Astrophysics, accepted

Abstract: We present an XMM observation of the radio jet and diffuse halo of the nearby radio galaxy NGC6251. The EPIC spectrum of the galaxy's halo is best-fitted by a thermal model with temperature kT~1.6 keV and subsolar abundances. Interestingly, an additional hard X-ray component is required to fit the EPIC spectra of the halo above 3 keV, and is independently confirmed by an archival Chandra observation. However, its physical origin is not clear. Contribution from a population of undetected Low Mass X-ray Binaries seems unlikely. Instead, the hard X-ray component could be due to inverse Compton scattering of the CMB photons off relativistic electrons scattered throughout the halo of the galaxy, or non-thermal bremsstrahlung emission. The IC/CMB interpretation, together with limits on the diffuse radio emission, implies a very weak magnetic field, while a non-thermal bremsstrahlung origin implies the presence of a large number of very energetic electrons. We also detect X-ray emission from the outer (~3.5') jet, confirming previous ROSAT findings. Both the EPIC and ACIS spectra of the jet are best-fitted by a power law with photon index \~1.2. A thermal model is formally ruled out by the data. Assuming an origin of the X-rays from the jet via IC/CMB, as suggested by energetic arguments, and assuming equipartition implies a large Doppler factor (delta~10). Alternatively, weaker beaming is possible for magnetic fields several orders of magnitude lower than the equipartition field.

Near-Infrared Observations of BL Lacertae Host Galaxies, by C. C. Cheung (1,4), C. Megan Urry (2,4), Riccardo Scarpa (3,4), and Mauro Giavalisco (4) (1) Brandeis Univ. (2) Yale Univ. (3) European Southern Observatory (4) STScI. 2003, ApJ, accepted

Abstract: Multi-band near-infrared images of twelve BL Lacertae objects were obtained with the 2.5m telescope at the Las Campanas Observatory in order to determine the properties of their underlying host galaxies. Resolved emission was clearly detected in eight of the lowest redshift targets (up to z~0.3), and was modeled with a de Vaucouleurs r^{1/4} surface brightness law. We find that the morphologies match the elliptical galaxy profiles well, and that the BL Lac objects reside in large and luminous, but otherwise normal hosts -- consistent with previous studies done predominantly at optical wavelengths. The median absolute K-band magnitude of the galaxies in this study is -26.2, the average half-light radius is 4.2 +/- 2.3 kpc, and their average integrated R-K color is 2.7 +/- 0.3 mag. These are well within the range of values measured previously in the H-band by Kotilainen et al. and Scarpa et al. in a comparable number of targets. Taking their data together with our results, we find a best-fit K-band Kormendy relation of \mu_{e} = 4.3 log_{10} (r_{e}/kpc) + 14.2 mag arcsec^{-2}, virtually identical to that obtained for normal ellipticals. Finally, the near-infrared colors determined for five galaxies (average J-K = 0.8 +/- 0.3 mag) are the first such measurements for BL Lac hosts, and match those expected from old stellar populations at the BL Lac redshifts.

Theoretical Models for Producing Circularly Polarized Radiation in Extragalactic Radio Sources, by John F. C. Wardle (1) and Daniel C. Homan (2), (1) Brandeis Univ. (2) National Radio Astronomy Obs. 2003, To appear in the proceedings of "Circular Polarization of Relativistic Jet Sources", eds R. P. Fender and J.-P. Macquart, in Astrophysics and Space Science.

Abstract: We discuss the production of circular polarization in compact radio sources both by the intrinsic mechanism and by Faraday conversion. We pay particular attention to the magnetic field structure, considering partially ordered fields and Laing sheets, and distinguishing between uniform and unidirectional fields. (The latter can be constrained by flux conservation arguments.) In most cases, Faraday conversion is the more important mechanism. Conversion opera tes on Stokes U, which can be generated by internal Faraday rotation, or by magnetic field fluctuations, which can therefore produce circular polarization even in a pure pair plasma. We also show that the spectrum of circular polarization in an inhomogeneous jet can be quite different from that in a uniform source, being flat or even inverted.

Chandra Observations of Nuclear X-ray Emission from a Sample of Radio Sources, by J. K. Gambill (1), R. M. Sambruna (1), G. Chartas (2), C. C. Cheung (3), L. Maraschi (4), F. Tavecchio (4), C. M. Urry (5), J. E. Pesce (1); (1) George Mason University, (2) Penn State University, (3) Brandeis University, (4) Osservatorio Astronomico di Brera, (5) Yale University; accepted for publication in Astronomy and Astrophysics

Abstract: We present the X-ray properties of a sample of 17 radio sources observed with the Chandra X-ray Observatory as part of a project aimed at studying the X-ray emission from their radio jets. In this paper, we concentrate on the X-ray properties of the unresolved cores. The sample includes 16 quasars (11 core-dominated and 5 lobe-dominated) in the redshift range z=0.30--1.96, and one low-power radio-galaxy at z=0.064. No diffuse X-ray emission is present around the cores of the quasars, except for the nearby low-power galaxy that has diffuse emission on a scale and with a luminosity consistent with other FRIs. No high-amplitude, short-term variability is detected within the relatively short Chandra exposures. However, 1510-089 shows low-amplitude flux changes with a timescale of $\sim$25 minutes. The X-ray spectra of the quasar cores are generally well described by a single power law model with Galactic absorption. However, in six quasars we find soft X-ray excess emission below 1.6 keV. Interestingly, we detect an Fe K-shell emission line, consistent with fluorescent Kalpha emission from cold Iron, in one lobe- and two core-dominated sources. The average X-ray photon index for the quasars in the sample is Gamma=1.66 and dispersion, sigma=0.23. The average spectral slope for our sample is flatter than the slope found for radio-quiet quasars and for radio-loud AGNs with larger jet orientations; this indicates that beaming affects the X-ray emission from the cores in our sample of quasars.

#### 2002

New Detections of Optical Emission from Kiloparsec-scale Quasar Jets, by C. C. Cheung, J. F. C. Wardle, Tingdong Chen, and S. P. Hariton (Brandeis University). to appear in proceedings of: "The Physics of Relativistic Jets in the Chandra and XMM Era,'' eds. G. Brunetti, D.E. Harris, R.M. Sambruna, and G. Setti, submitted to New Astronomy Review.

Abstract: We report initial results from the detection of optical emission in the arcsecond-scale radio jets of two quasars utilizing images from the Hubble Space Telescope archive. The optical emission has a very knotty appearance and is consistent with synchrotron emission from highly relativistic electrons in the jet. Combining these observations with those of previously reported features in other quasars, an emerging trend appears to be that their radio-to-optical spectral indices are steeper than those of similar features in jets of lower power radio sources.

Chandra Observations of Three SDSS Quasars at z~6, by D. A. Schwartz (Harvard-Smithsonian Center for Astrophysics), C. C. Cheung, J. F. C. Wardle (Brandeis University). to appear in proceedings of: "Active Galactic Nuclei: from Central Engine to Host Galaxy,'' Meudon, July 2002 meeting; ASP Conference Series, Eds.: S. Collin, F. Combes and I. Shlosman

Abstract: Chandra snapshot observations of the three most distant quasars then known, at redshifts 5.82, 5.99, and 6.28, gave signficant detections even in the short, 6 -- 8 ks, observations. The X-ray to optical luminosity ratios indicate that quasars will be detectable in X-rays if they exist at even larger redshifts. The present observations hint at two exciting discoveries. An extended X-ray source 23 arcsec from SDSS1306+0356 may be a jet emitting inverse Compton radiation from the Cosmic Microwave Background. SDSS 1030+0524 does not appear to be a point source, and may be a gravitationally lensed system, or contain a small scale X-ray jet.

Probing the Circular Polarization of Relativistic Jets on VLBI Scales, by Daniel C. Homan (1) and John F. C. Wardle (2) ((1) National Radio Astronomy Obs. (2) Brandeis Univ.), 2002. To appear in the proceedings of "Circular Polarization of Relativistic Jet Sources", eds R. P. Fender and J.-P. Macquart, in Astrophysics and Space Science.

Abstract: High resolution studies of circular polarization allow us see where it arises in a jet, study its local fractional level and spectrum, and compare these results to local measures of linear polarization and Faraday rotation. Here we not only review past results from Very Long Baseline Array (VLBA) circular polarization studies, but we also present preliminary new results on two quasars. In the core of PKS 0607-157, we find strong circular polarization at 8 GHz and much weaker levels at 15 GHz. Combined with the linear polarization data, we favor a simple model where the circular is produced by Faraday conversion driven by a small amount of Faraday rotation. In the core of 3C345, we find strong circular polarization at 15 GHz in a component with distinct linear polarization. This core component is optically thick at 8 GHz, where we detect no circular polarization. With opposite trends in frequency for PKS 0607-157 and 3C345, it seems clear that local conditions in a jet can have a strong effect on circular polarization and need to be taken into account when studying inhomogeneous objects with multi-frequency observations.

Detection of Optical Synchrotron Emission from the Radio Jet of 3C279, by C. C. Cheung, Brandeis University, 2002. Accepted for publication in The Astrophysical Journal Letters.

Abstract: We report the detection of optical and ultraviolet emission from the kiloparsec scale jet of the well-known quasar 3C~279. A bright knot, discovered in archival V and U band Hubble Space Telescope Faint Object Camera images, is coincident with a peak in the radio jet ~0.6 arcsec from the nucleus. The detection was also confirmed in Wide Field Planetary Camera-2 images. Archival Very Large Array and MERLIN radio data are also analyzed which help to show that the high-energy optical/UV continuum, and spectrum, are consistent with a synchrotron origin from the same population of relativistic electrons responsible for the radio emission.

PKS 1510-089: A Head-On View of a Relativistic Jet, by Daniel C. Homan (1), John F.C. Wardle (2), Chi C. Cheung (2), David H. Roberts (2), Joanne M. Attridge (3) ((1) NRAO, (2) Brandeis University, (3) MIT Haystack Observatory), 2002. Accepted for publication in The Astrophysical Journal.

Abstract: The gamma-ray blazar PKS 1510-089 has a highly superluminal milli-arcsecond jet at a position angle (PA) of -28 degrees and an arcsecond jet with an initial PA of 155 degrees. With a PA difference of 177 degrees between the arcsecond and milli-arcsecond jets, PKS 1510-089 is perhaps the most highly misaligned radio jet ever observed and serves as a graphic example of projection effects in a highly beamed relativistic jet. Here we present the results of observations designed to bridge the gap between the milli-arcsecond and arcsecond scales. We find that a previously detected counter-feature'' to the arcsecond jet is directly fed by the milli-arcsecond jet. This feature is located 0.3'' from the core, corresponding to a de-projected distance of 30 kiloparsecs. The feature appears to be dominated by shocked emission and has an almost perfectly ordered magnetic field along its outside edge. We conclude that it is most likely a shocked bend, viewed end-on, where the jet crosses our line of sight to form the southern arcsecond jet. While the bend appears to be nearly 180 degrees when viewed in projection, we estimate the intrinsic bending angle to be between 12 and 24 degrees. The cause of the bend is uncertain; however, we favor a scenario where the jet is bent after it departs the galaxy, either by ram pressure due to winds in the intracluster medium or simply by the density gradient in the transition to the intergalactic medium.

A Survey of Extended Radio Jets in AGN with Chandra and HST: First Results, by Rita M. Sambruna (George Mason Univ.), L. Maraschi, F. Tavecchio (Oss Brera), C. Megan Urry (Yale Univ.), C. C. Cheung (Brandeis Univ.), G. Chartas (Penn State Univ.), R. Scarpa (ESO), Jessica K. Gambill (George Mason Univ.) 2002. Accepted for publication in The Astrophysical Journal.

Abstract: We present the first results from an X-ray and optical survey of a sample of AGN radio jets with Chandra and HST. We focus here on the first six sources observed at X-rays, in four of which a bright X-ray jet was detected for the first time. In three out of four cases optical emission from the jet is also detected in our HST images. We compare the X-ray morphology with the radio as derived from improved processing of archival VLA data and we construct spectral energy distributions (SED) for the most conspicuous emission knots. In most cases the SEDs, together with the similarity of the X-ray and radio morphologies, favor an inverse Compton origin of the X-rays. The most likely origin of the seed photons is the Cosmic Microwave Background, implying the jets are still relativistic on kiloparsec scales. However, in the first knot of the PKS 1136-135 jet, X-rays are likely produced via the synchrotron process. In all four cases bulk Lorentz factors of a few are required. The radio maps of the two jets not detected by either Chandra or HST suggest that they are less beamed at large scales than the other four detected sources. Our results demonstrate that, at the sensitivity and resolution of Chandra, X-ray emission from extragalactic jets is common, yielding essential information on their physical properties.

#### 2001

Parsec-Scale Blazar Monitoring: Flux and Polarization Variability, by Daniel C. Homan, Roopesh Ojha, John F. C. Wardle, David H. Roberts, Margo F. Aller, Hugh D. Aller, Philip A. Hughes. 2001. Submitted to The Astrophysical Journal.

Abstract: We present analysis of the flux and polarization variability of parsec scale radio jets from a dual-frequency, six-epoch, VLBA polarization experiment monitoring 12 blazars. The observations were made at 15 and 22 GHz at bimonthly intervals over 1996. Here we analyze the flux, fractional polarization, and polarization position angle behavior of core regions and jet features, considering both the linear trends of these quantities with time and more rapid fluctuations about the linear trends. The dual frequency nature of the observations allows us to examine spectral evolution, to separate Faraday effects from changes in magnetic field order, and also to deduce empirical estimates for the uncertainties in measuring properties of VLBI jet features (see the Appendix). [abridged]

Detection of an X-ray jet in 3C 371 with Chandra, by Joseph E. Pesce, Rita M. Sambruna, F. Tavecchio, L. Maraschi, C. C. Cheung, C. Megan Urry, and R. Scarpa. 2001. Accepted by The Astrophysical Journal Letters. Download PDF. Figure 1 (in color) is available in GIF and PDF format.

Abstract: We report the detection at X-rays of the radio/optical jet of 3C 371, from a short (10 ks) Chandra exposure in March 2000. We also present a new MERLIN observation at 1.4 GHz together with a renalysis of the archival HST WFPC2 F555W image. Despite the limited signal-to-noise ratio of the Chandra data, the X-ray morphology is clearly different from that of the radio/optical emission, with the brightest X-ray knot at 1.7" from the nucleus and little X-ray emission from the brightest radio/optical knot at 3.1". We construct the spectral energy distributions for the two emission regions at 1.7" and 3.1". Both show that the X-ray flux is below the extrapolation from the radio-to-optical continuum, suggesting moderately beamed synchrotron from an electron population with decreasing high energy cut-off as a plausible emission mechanism.

Making Movies from Radio Astronomical Images with AIPS, by C. C. Cheung, D. C. Homan, J. F. C. Wardle, D. H. Roberts. 2001. NRAO AIPS Memo # 106.

Abstract: We present a detailed recipe for making movies from multi-epoch radio observations of astronomical sources. Images are interpolated linearly in time to create a smooth succession of frames so that a continuous movie can be compiled. Here, we outline the procedure, and draw attention to specific details necessary for making a successful movie. In particular, we discuss the issues pertaining specifically to making polarization movies. The procedure described here has been implemented into scripts in NRAO's AIPS package (Brandeis AIPS Movie Maker -- BAMM) that are available for public use (http://www.astro.brandeis.edu).

Parsec-Scale Circular Polarization Observations of 40 Blazars, by D. C. Homan, J. M. Attridge, and J. F. C. Wardle. 2001. Accepted by The Astrophysical Journal.

Abstract: We present circular polarization results from a 5 GHz survey of the parsec-scale polarization properties of 40 AGN made with the VLBA. We find 11 circular polarization detections at the 3 sigma level or higher. This nearly quadruples the number of sources detected in circular polarization at VLBI resolution. We find no correlation between fractional linear and circular polarization across our sample. A likely explanation is external Faraday depolarization in the cores of AGN which reduces linear polarization but leaves circular polarization unchanged. In comparing ours and other recent results to observations made ~ 20 years ago, we find that, in five of six cases, sources have the same sign of circular polarization today as they did 20 years ago. This suggests the presence of a long term property of the jets, such as the polarity of a net magnetic flux, which is stable on time-scales much longer than those of individual outbursts.

#### 2000

Parsec-Scale Blazar Monitoring: Proper Motions, by D. C. Homan, R. Ojha, J. F. C. Wardle, D. H. Roberts, M. F. Aller, H. D. Aller, and P. A. Hughes. Appeared in The Astrophysical Journal 2001, vol 549, p. 840.

Abstract: We present proper motions obtained from a dual frequency, six-epoch, VLBA polarization experiment monitoring a sample of 12 blazars. The observations were made at 15 GHz and 22 GHz at bi-monthly intervals over 1996. Ten of the eleven sources for which proper motion could be reliably determined are superluminal. Only J2005+77 has no superluminal components. Three sources (OJ287, J1224+21, and J1512-09) show motion faster than 10h^{-1} c, requiring $\gamma_{pattern}$ of at least 10h^{-1} (H_0 = 100h km/s/Mpc). We compare our results to those in the literature and find motions outside the previously observed range for four sources. While some jet components exhibit significant non-radial motion, most motion is radial. In at least two sources there are components moving radially at significantly different structural position angles. In five of six sources (3C120, J1224+21, 3C273, 3C279, J1512-09, and J1927+73) that have multiple components with measurable proper motion, the innermost component is significantly slower than the others, suggesting that acceleration occurs in the jet. In the motions of individual components we observe at least one decelerating motion and two bending'' accelerations which tend to align their motions with larger scale structure. We also discuss in detail our procedures for obtaining robust kinematical results from multi-frequency VLBI data spanning several epochs.

The Nature of Jets: Evidence from Circular Polarization Observations, by John F. C. Wardle, Daniel C. Homan. To appear in "Particles and Fields in Radio Galaxies", Eds. Robert A. Laing and Katherine M. Blundell, ASP Conference Series

Abstract: We review recent observations of circularly polarized radiation from AGN made with the VLBA and with the ATCA. We also discuss briefly the detections of the Galactic sources Sag A* and SS433. The origin of the circular polarization is still an open question in most cases, and we discuss four possible mechanisms. Detectable circular polarization is a common property of quasars, but not of radio galaxies, and is always associated with the compact core. There is growing evidence that the sign of the circular polarization stays the same over at least two or three decades in time, suggesting it is a fundamental property of the jet.

Direct Distance Measurements to Superluminal Radio Sources, by D. C. Homan and J. F. C. Wardle. Appeared in The Astrophysical Journal (2000) vol 535, pg. 575.

Abstract: We present a new technique for directly measuring the distances to superluminal radio sources. By comparing the observed proper motions of components in a parsec scale radio jet to their measured Doppler factors, we can deduce the distance to the radio source independent of the standard rungs in the cosmological distance ladder. This technique requires that the jet angle to the line of sight and the ratio of pattern to flow velocities are sufficiently constrained. We evaluate a number of possibilities for constraining these parameters and demonstrate the technique on a well defined component in the parsec scale jet of the quasar 3C279 (z = 0.536). We find an angular size distance to 3C279 of greater than 1.8 (+0.5,-0.3) n^{1/8} Gpc, where n is the ratio of the energy density in the magnetic field to the energy density in the radiating particles in that jet component. For an Einstein-de Sitter Universe, this measurement would constrain the Hubble constant to be H < 65 n^{-1/8} km/s/Mpc at the two sigma level. Similar measurements on higher redshift sources may help discriminate between cosmological models.

#### 1999 and before

Detection and Measurement of Parsec-Scale Circular Polarization in Four AGN, by D. C. Homan and J. F. C. Wardle. Appeared in The Astronomical Journal (1999) vol 118, pg 1942. Download PDF

Abstract: We present five epochs of 15 GHz VLBA observations of 13 AGN. These observations were specially calibrated to detect parsec scale circular polarization and our calibration techniques are discussed and analyzed in detail. We obtained reliable detections of parsec scale circular polarization in the radio jets of 4 AGN: 3C84, PKS 0528+134, 3C273, and 3C279. For each of these objects our detections are at the level of ~ 0.3-1% local fractional circular polarization. Our observations are consistent across multiple epochs (and different calibration techniques) in the sign and magnitude of the circular polarization observed. 3C273 and 3C279 both undergo core outbursts during our observations and changes in the circular polarization of both sources are correlated with these outbursts. In general, we observe the circular polarization to be nearly coincident with the strong VLBI cores of these objects; however, in 3C84 the circular polarization is located a full milli-arcsecond south of the source peak, and in the 1996.73 epoch of 3C273 the circular polarization is predominately associated with the newly emerging jet component. Our observations support the theoretical conclusion that emission of circular polarization is a sensitive function of opacity, being strongest when the optical depth is near unity. Circular polarization may be produced as an intrinsic component of synchrotron radiation or by the Faraday conversion of linear to circular polarization. Our single frequency observations do not easily distinguish between these possible mechanisms, but independent of mechanism, the remarkable consistency across epoch of the sign of the observed circular polarization suggests the existence of a long term, stable, uni-directional magnetic field. Single dish observations of 3C273 and 3C279 at 8 GHz by Hodge and Aller suggest that this stability may persist for decades in our frame of observation.

Radio Jet-Ambient Medium Interactions on Parsec Scales in the Blazar 1055+018, by J. M. Attridge, D. H. Roberts, and J. F. C. Wardle. Appeared in Astrophys.J. (1999) vol 518, pg. 87.

Abstract: As part of our study of the magnetic fields of AGN we have recently observed a large sample of blazars with the Very Long Baseline Array. Here we report the discovery of a striking two-component jet in the source 1055+018, consisting of an inner spine with a transverse magnetic field, and a fragmentary but distinct boundary layer with a longitudinal magnetic field. The polarization distribution in the spine strongly supports shocked-jet models while that in the boundary layer suggests interaction with the surrounding medium. This behavior suggests a new way to understand the differing polarization properties of strong- and weak-lined blazars.

Electron-positron Jets Associated with the Quasar 3C279, by J. F. C. Wardle, D. C. Homan, R. Ojha, and D. H. Roberts. Appeared in Nature (1998) vol 395, pg. 457. Download PDF

Abstract: A long outstanding problem in extragalactic astrophysics is the composition of the relativistic jets of plasma streaming from the nuclei of quasars and active galaxies -- whether it is predominantly a "normal" (electron - proton) plasma, or a "pair" (electron - positron) plasma. Distinguishing between these possibilities is crucial for understanding the physical processes occuring close to the putative supermassive black holes in galactic nuclei that create the jets. Here we present a new approach to the problem of jet composition, by detecting circularly polarized radio emission from the archtypal quasar 3C279, using the VLBA. The circular polarization is produced by Faraday conversion, which requires that the energy distribution of the radiating particles extends to very low energies, indicating that electron - positron pairs are a major component of the jet plasma. Similar detections in three other radio sources, suggest that in general, extragalactic radio jets are composed primarily of an electron - positron plasma.


text: Theorem 2says thatitisnecessary for u(x,y)and v(x,y)toobey the Cauchy–Riemann equations in order for f(x+iy) = u(x+iy)+v(x+iy) to be diﬀerentiable. Theorem 4.14. Q.E.D. If a function f is analytic at all points interior to and on a simple closed contour C (i.e., f is analytic on some simply connected domain D containing C), then Z C f(z)dz = 0: Note. Problems of the Cayley-Hamilton Theorem. Lagranges mean value theorem is defined for one function but this is defined for two functions. We prove the Cauchy-Schwarz inequality in the n-dimensional vector space R^n. This document is highly rated by Mathematics students and has been viewed 195 times. 1.7.. Let (x n) be a sequence of positive real numbers. Browse other questions tagged complex-analysis cauchy-integral-formula or ask your own question. Introduction to Engineering Mathematics. It is a very simple proof and only assumes Rolle’s Theorem. To check whether given set is compact set. This is perhaps the most important theorem in the area of complex analysis. Although not the original proof, it is perhaps the most widely known; it is certainly the author’s favorite. when condition (5) holds for all $x _ {0} \in S$. If jGjis even, consider the set of pairs fg;g 1g, where g 6= g 1. Since the integrand in Eq. If pdivides jGj, then Ghas Ended on Jun 3, 2020. The publication first elaborates on evolution equations, Lax-Mizohata theorem, and Cauchy problems in Gevrey class. If a function f is analytic on a simply connected domain D and C is a simple closed contour lying in D then Vishal Soni. Continuous on . Karumudi Umamaheswara Rao. Solutions to practice problems for the nal Holomorphicity, Cauchy-Riemann equations, and Cauchy-Goursat theorem 1. Unique solution of Cauchy problem in a neighbourhood of given set. Basic to advanced level. English General Aptitude. The following theorem says that, provided the ﬁrst order partial derivatives of u and v are continuous, the converse is also true — if u(x,y) and v(x,y) obey the Cauchy–Riemann equations then If we assume that f0 is continuous (and therefore the partial derivatives of u … when internal efforts are bounded, and for fixed normal n (at point M), the linear mapping n ↦ t (M; n) is continuous, then t(M;n) is a linear function of n, so that there exists a second order spatial tensor called Cauchy stress σ such that The history of this theorem begins in the 1300's with the Indian Mathematician Parameshvara , and is eventually based on the academic work of Mathematicians Michel Rolle in 1691 and Augustin Louis Cauchy in 1823. Irina V. Melnikova, Regularized solutions to Cauchy problems well posed in the extended sense, Integral Transforms and Special Functions, 10.1080/10652460500438003, 17, 2-3, (185 … The formal statement of this theorem together with an illustration of the theorem will follow. Then by Fermat’s theorem, the derivative at this point is equal to zero: $f’\left( c \right) = 0.$ Physical interpretation. 2 LECTURE 7: CAUCHY’S THEOREM Figure 2 Example 4. For z0 2 Cand r > 0 the curve °(z0;r) given by the function °(t) = z0+reit; t 2 [0;2…) is a prototype of a simple closed curve (which is the circle around z0 with radius r). 3. Theorem 7.4.If Dis a simply connected domain, f 2A(D) and is any loop in D;then Z f(z)dz= 0: Proof: The proof follows immediately from the fact that each closed curve in Dcan be shrunk to a point. Our calculation in the example at the beginning of the section gives Res(f,a) = 1. From introductory exercise problems to linear algebra exam problems from various universities. Show that the sequence (x n) de ned below satis es the Cauchy criterion. (In particular, does not blow up at 0.) Before treating Cauchy’s theorem, let’s prove the special case p = 2. Featured on Meta New Feature: Table Support Let Gbe a nite group and let pbe a prime number. Here is a set of practice problems to accompany the Stokes' Theorem section of the Surface Integrals chapter of the notes for Paul Dawkins Calculus III course at Lamar University. Suppose that ${u}_{k}$ is the solution, prove that: ... Theorem of Cauchy-Lipschitz reverse? Doubt about Cauchy-Lipshitz theorem use. 0. The Cauchy-Goursat Theorem Cauchy-Goursat Theorem. A generalization of Cauchy’s theorem is the following residue theorem: Corollary 1.5 (The residue theorem) Consider the following Cauchy problems. f(z) = (z −a)−1 and D = {|z −a| < 1}. Cauchy’s Theorem The theorem states that if f(z) is analytic everywhere within a simply-connected region then: I C f(z)dz = 0 for every simple closed path C lying in the region. Cauchy stated his theorem for permutation groups (i.e., subgroups of S n), not abstract nite groups, since the concept of an abstract nite group was not yet available [1], [2]. 1 Cauchy’s Theorem Here we present a simple proof of Cauchy’s theorem that makes use of the cyclic permutation action of Z=nZ on n-tuples. Theorem 1 (Cauchy). Two solutions are given. Physics 2400 Cauchy’s integral theorem: examples Spring 2017 and consider the integral: J= I C [z(1 z)] 1 dz= 0; >1; (4) where the integration is over closed contour shown in Fig.1. 1. Dec 19, 2020 - Contour Integral, Cauchy’s Theorem, Cauchy’s Integral Formula - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev is made by best teachers of Mathematics. 33 CAUCHY INTEGRAL FORMULA October 27, 2006 We have shown that | R C f(z)dz| < 2π for all , so that R C f(z)dz = 0. In this session problems of cauchy residue theorem will be discussed. In mathematicskowalswski Cauchy—Kowalevski theorem also written as the Cauchy—Kovalevskaya theorem is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems. Similar Classes. Cauchy problems are usually studied when the carrier of the initial data is a non-characteristic surface, i.e. Problems on Cauchy Residue Theorem. Then, ( ) = 0 ∫ for all closed curves in . Solution: Call the given function f(z). is mildly well posed (i.e., for each x ∈ X there exists a unique mild solution) if and only if the resolvent of A is a Laplace transform; and this in turn is the same as saying that A generates a C 0-semigroup.Well-posedness in a weaker sense will lead to generators of integrated semigroups (Section 3.2). By Cauchy’s theorem, the value does not depend on D. Example. We can use this to prove the Cauchy integral formula. Share. A holomorphic function has a primitive if the integral on any triangle in the domain is zero. ləm] (mathematics) The problem of determining the solution of a system of partial differential equation of order m from the prescribed values of the solution and of its derivatives of order less than m on a given surface. Then according to Cauchy’s Mean Value Theorem there exists a point c in the open interval a < c < b such that: The conditions (1) and (2) are exactly same as the first two conditions of Lagranges Mean Value Theorem for the functions individually. Proof. Cauchy’s integral theorem An easy consequence of Theorem 7.3. is the following, familiarly known as Cauchy’s integral theorem. Then there is a neighbourhood of 0 in W on which the quasilinear Cauchy problem. Oct 30, 2020 • 2h 33m . The other formula can be derived by using the Cauchy-Riemann equations or by the fact that in the proof of the Cauchy-Riemann theorem we also have that: (10) \begin{align} \quad f'(z) = \frac{\partial v}{\partial y} -i\frac{\partial u}{\partial y} \end{align} Practice Problems 3 : Cauchy criterion, Subsequence 1. The condensed formulation of a Cauchy problem (as phrased by J. Hadamard) in an infinite-dimensional topological vector space.While it seems to have arisen between the two World Wars (F. Browder in , Foreword), it was apparently introduced as such by E. Hille in 1952, , Sec. 1 Analytic functions and power series The subject of complex analysis and analytic function theory was founded by Augustin Cauchy THEOREM Suppose f is analytic everywhere inside and on a simple closed positive contour C. If z 0 is any point interior to C, then f(z 0) = 1 2πi Z C f(z) z− z One uses the discriminant of a quadratic equation. 3M watch mins. Cauchy Mean Value Theorem Let f(x) and g(x) be continuous on [a;b] and di … Cauchy Theorem. THE CAUCHY MEAN VALUE THEOREM JAMES KEESLING In this post we give a proof of the Cauchy Mean Value Theorem. (A second extension of Cauchy’s theorem) Suppose that is a simply connected region containing the point 0. Watch Now. (a)Show that there is a holomorphic function on = fzjjzj>2gwhose derivative is z (z 1)(z2 + 1): Hint. 5 The Cauchy-Taylor theorem and analytic continuation 63 6 Laurent’s theorem and the residue theorem 76 7 Maximum principles and harmonic functions 85 2. 1. Suppose is a function which is. The Cauchy–Kovalevskaya theorem occupies an important position in the theory of Cauchy problems; it runs as follows. Rolle’s theorem has a clear physical meaning. 1. (a) x 1 = 1 and x n+1 = 1 + 1 xn for all n 1 (b) x 1 = 1 and x n+1 = 1 2+x2 n for all n 1: (c) x 1 = 1 and x n+1 = 1 6 (x2 n + 8) for all n 1: 2. Discussions focus on fundamental proposition, proof of theorem 4, Gevrey property in t of solutions, basic facts on pseudo-differential, and proof of theorem 3. Theorem 5. Analytic on −{ 0} 2. Suppose that a body moves along a straight line, and after a certain period of time returns to the starting point. 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Tuesday, July 28, 2020 | History

3 edition of Blocked path. found in the catalog.

Blocked path.

David Shrigley

# Blocked path.

## by David Shrigley

Written in English

Subjects:
• Shrigley, David.

• Edition Notes

Artist"s book.

The Physical Object
Pagination1 v :
ID Numbers
Open LibraryOL16025488M
ISBN 108798706047
OCLC/WorldCa473910742

- Obstacles do not block the path, they are the path. – Zen Proverb. - Obstacles do not block the path, they are the path. – Zen Proverb. - Obstacles do not block the path, they are the path. – Zen Proverb.. Article.   Bad Religion is an institution. The legendary California punk outfit formed 40 years ago, when its founding members were all in high school. There have been a .

The barriers that block some workers from advancement, such as inadequate parental-leave policies, federal limits on imported brainpower, and overt racial discrimination, are holding us .   Michael Cohen has been spotted on the streets of New York City just a day after revealing bombshell details of his upcoming book about his .

This quest is given to you by Tarkleigh after entering The Ledge.. Tarkleigh asks you to find a way to the forest around the Prison for the betterment of the encampment. He suggests the Emperor's requires you to travel through the The Prison, and into the The Prisoner's Gate, where the quest will be updated upon witnessing the closing of the road. 19 hours ago  How Black women cleared a path for Harris to be the VP pick — A federal judge blocked the Trump administration on Monday from enforcing a .

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Trump niece's book blocked by New York judge but lawyer files appeal This article is more than 1 month old. President’s brother, Robert Trump, is trying to stop publication of niece’s. The Blocked Path When discussing undocumented immigrants, the ques-tion often arises, “Why don’t they just get in line and enter the U.S.

legally?” The answer is that for most of the women inter-viewed for this report — and for the vast majority of undocumented immigrants in the U.S. — there simply is no gateway to obtain legal status. “There was a smile dancing on his lips, although it was a wary smile, for the world is a bigger place than a little graveyard on a hill; and there would be dangers in it and mysteries, new friends to make, old friends Blocked path.

book rediscover, mistakes to be made and many paths to be walked before he would, finally, return to the graveyard or ride with the Lady on the broad back of her great grey. Block Path Mania is one of many topology puzzle game. To solve the puzzle draw a path to fill out entire board.

You are given the starting square-Blue must draw a continuous loop that covers the whole grid. Red squares are obstacles. You should first identify which square must be last. From there, it’s a 5/5(5). The Occupational Safety and Health Administration (OSHA) announced Jthat two companies in the New York City area have been cited and fined for safety violations involving blocked Author: Roy Maurer.

Michael Cohen released the foreword of his upcoming book on Thursday, teasing what he claims is a behind-the-scenes Blocked path. book of his acts as President Donald Trump's fixer -- from stiffing.

Synonyms for blocked at with free online thesaurus, antonyms, and definitions. Find descriptive alternatives for blocked. Wilco’s Jeff Tweedy will publish his new book, How to Write One Song, October 13th via Dutton.

The book, per a statement, will find Tweedy delving into the nature of his creative process. This is the most common way to block any website or IP address from a computer.

Anyone can block any website (including the ) on your PC using system host file. A Windows user (Windows XP, Windows Vista, Windows 7, Windows 8, Windows and Windows 10) can find the hosts file by navigating to the following path. Blocking means that the Actor components will be prevented from interpenetration in the engine, and any collision between two primitive shapes will be resolved.

The Control Blocks is an area in Act This area has a waypoint and is connected to The Ravaged entrance can always be found in the south west of The Ravaged Square at the very bottom of the map. Make a Library Books Quilt Block for NaNoWriMo.

November is National Novel Writing Month (NaNoWriMo for short). This year, more thanwriters have pledged to write an entire novel in just 30 days. Quilting Tips Quilting Tutorials Quilting Projects Sewing Projects Quilting Fabric Scrappy Quilts Mini Quilts Rag Quilt Pattern Blocks.

Sacred Pathways reveals nine distinct spiritual temperaments--and their strengths, weaknesses, and tendencies--to help you improve your spiritual life and deepen your personal walk with God.

It's time to strip away the frustration of a one-size-fits-all spirituality and discover a path of worship that frees you to be s:   1. If there are no variables being conditioned on, a path is blocked if and only if two arrowheads on the path collide at some variable on the path.

$$L \rightarrow A \rightarrow Y$$ is open. $$A \rightarrow Y \leftarrow L$$ is blocked at $$Y$$ 2. Any path that contains a noncollider that has been conditioned on is blocked.

1 day ago  Jill Biden's path from reluctant politico to possible FLOTUS. Jill Biden is known for being impeccably dressed, always offering up a good book recommendation, writing small notes or sending.

Michael Cohen's memoir about President Donald Trump will be released Sept. 8 by Skyhorse Publishing, which confirmed the news Thursday to The Associated Press. The book. 20 hours ago  How Black women cleared a path for Harris to be the VP pick.

Kat Stafford, Associated Press. Updated am EDT, Wednesday, Aug block a solid piece of hard material; an obstruction: block the street Not to be confused with: bloc – coalition of groups with the same purpose: a bloc of votes block (blŏk) n.

A solid piece of a hard substance, such as wood, having one or more flat sides. Such a piece used as a construction member or as a support. Such a piece upon. Using the 'block' command generates a block that is saved in the file you are working in only. You can view a list of all blocks in a drawing by selecting 'insert' under the insert tab (far left).

Then expand the window that drops down and scroll through. The last created block will be at the bottom of this list. Tech support scams are an industry-wide issue where scammers trick you into paying for unnecessary technical support services.

You can help protect yourself from scammers by verifying that the contact is a Microsoft Agent or Microsoft Employee and that the phone number is an official Microsoft global customer service number.

The Drunkard's Path block is known by multiple names and lay out variations. Names include Road to California, Rocky Road to Dublin, Vine of Friendship, Falling Timber, Polka Dots, Fool's Puzzle and Snowball.

No matter what the name or variation, Drunkard's Path is tons of fun with the AccuQuilt GO! or GO! Big and easy with the "no pin" method!I found a blocked footpath when doing the "Round Preston Walk" in The paths were in the correct places according to both the guide book and the Preston OS Map.

This one was in the Fulwood area; barbed wire had been placed across between 2 fence posts with a stile into the next field several feet behind the fence posts.Just add websites to your block list on and we’ll do the rest, we’ll even throw in some really funny images if you try and go on the sites while they’re blocked 💩 ⏰ Stay Focused Extension You can now remove all distracting and harmful websites from your life by blocking /5(K).