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 Radial velocity measurements of B stars in the Scorpius-Centaurus associationWe derive single-epoch radial velocities for a sample of 56 B-type starsmembers of the subgroups Upper Scorpius, Upper Centaurus Lupus and LowerCentaurus Crux of the nearby Sco-Cen OB association. The radial velocitymeasurements were obtained by means of high-resolution echelle spectravia analysis of individual lines. The internal accuracy obtained in themeasurements is estimated to be typically 2-3 km s-1, butdepends on the projected rotational velocity of the target. Radialvelocity measurements taken for 2-3 epochs for the targets HD 120307, HD142990 and HD 139365 are variable and confirm that they arespectroscopic binaries, as previously identified in the literature.Spectral lines from two stellar components are resolved in the observedspectra of target stars HD 133242, HD 133955 and HD 143018, identifyingthem as spectroscopic binaries. Formation scenarios for the young stellar associations between galactic longitudes l = 280degr - 360degrWe investigate the spatial distribution, the space velocities and agedistribution of the pre-main sequence (PMS) stars belonging toOphiuchus, Lupus and Chamaeleon star-forming regions (SFRs), and of theyoung early-type star members of the Scorpius-Centaurus OB association.These young stellar associations extend over the galactic longituderange from 280degr to 360degr , and are at a distance interval ofaround 100 and 200 pc. This study is based on a compilation ofdistances, proper motions and radial velocities from the literature forthe kinematic properties, and of basic stellar data for the constructionof Hertzsprung-Russel diagrams. Although there was no well-known OBassociation in Chamaeleon, the distances and the proper motions of agroup of 21 B- and A-type stars, taken from the Hipparcos Catalogue,lead us to propose that they form a young association. We show that theyoung early-type stars of the OB associations and the PMS stars of theSFRs follow a similar spatial distribution, i.e., there is no separationbetween the low and the high-mass young stars. We find no difference inthe kinematics nor in the ages of these two populations studied.Considering not only the stars selected by kinematic criteria but thewhole sample of young early-type stars, the scattering of their propermotions is similar to that of the PMS stars and all the young starsexhibit a common direction of motion. The space velocities of theHipparcos PMS stars of each SFR are compatible with the mean values ofthe OB associations. The PMS stars in each SFR span a wide range of ages(from 1 to 20 Myr). The ages of the OB subgroups are 8-10 Myr for UpperScorpius (US), and 16-20 Myr for Upper Centaurus Lupus (UCL) and forLower Centaurus Crux (LCC). Thus, our results do not confirm that UCL isolder than the LCC association. Based on these results and theuncertainties associated with the age determination, we cannot say thatthere is indeed a difference in the age of the two populations. Weanalyze the different scenarios for the triggering of large-scalestar-formation that have been proposed up to now, and argue that mostprobably we are observing a spiral arm that passes close to the Sun. Thealignment of young stars and molecular clouds and the average velocityof the stars in the opposite direction to the Galactic rotation agreewith the expected behavior of star formation in nearby spiral arms.Tables 1 to 4 are only available in electronic form at the CDS viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/404/913 The mass ratio distribution of B-type visual binaries in the Sco OB2 associationA sample of 115 B-type stars in the Sco OB2 association is examined forexistence of visual companions in the J and K_s bands, using the ADONISnear-infrared adaptive optics system and coronograph. Practically allthe components in the separation range 0farcs3 -6farcs4 (45-900 AU) andmagnitudes down to K = 16 were detected. The K and J - K photometry ofthe primaries and differential photometry and astrometry of the 96secondaries are presented. Ten secondaries are new physical components,as inferred from the photometric and statistical criteria, while therest of the newly detected objects are faint background stars. After asmall correction for detection incompleteness and a conversion of thefluxes into masses, an unbiased distribution of the components massratio q was derived. The power law f(q)~ q-0.5 fits theobservations well, whereas a q-1.8 distribution, whichcorresponds to a random pairing of stars, is rejected. The companionstar fraction is 0.20+/-0.04 per decade of separation which iscomparable to the highest measured binary fraction among low-mass PMSstars and ~ 1.6 times higher than the binary fraction of low-mass dwarfsin the solar neighborhood and in open clusters in the same separationrange. Based on observations collected at the European SouthernObservatory, La Silla, Chile (ESO programme 65.H-0179). Tables 1, 3 andthe full version of Table 2 are only available in electronic form at theCDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/382/92 Astrometric radial velocities. III. Hipparcos measurements of nearby star clusters and associationsRadial motions of stars in nearby moving clusters are determined fromaccurate proper motions and trigonometric parallaxes, without any use ofspectroscopy. Assuming that cluster members share the same velocityvector (apart from a random dispersion), we apply a maximum-likelihoodmethod on astrometric data from Hipparcos to compute radial and spacevelocities (and their dispersions) in the Ursa Major, Hyades, ComaBerenices, Pleiades, and Praesepe clusters, and for theScorpius-Centaurus, alpha Persei, and HIP 98321'' associations. Theradial motion of the Hyades cluster is determined to within 0.4 kms-1 (standard error), and that of its individual stars towithin 0.6 km s-1. For other clusters, Hipparcos data yieldastrometric radial velocities with typical accuracies of a few kms-1. A comparison of these astrometric values withspectroscopic radial velocities in the literature shows a good generalagreement and, in the case of the best-determined Hyades cluster, alsopermits searches for subtle astrophysical differences, such as evidencefor enhanced convective blueshifts of F-dwarf spectra, and decreasedgravitational redshifts in giants. Similar comparisons for the ScorpiusOB2 complex indicate some expansion of its associations, albeit slowerthan expected from their ages. As a by-product from the radial-velocitysolutions, kinematically improved parallaxes for individual stars areobtained, enabling Hertzsprung-Russell diagrams with unprecedentedaccuracy in luminosity. For the Hyades (parallax accuracy 0.3 mas), itsmain sequence resembles a thin line, possibly with wiggles in it.Although this main sequence has underpopulated regions at certaincolours (previously suggested to be Böhm-Vitense gaps''), suchare not visible for other clusters, and are probably spurious. Futurespace astrometry missions carry a great potential for absoluteradial-velocity determinations, insensitive to the complexities ofstellar spectra. Based on observations by the ESA Hipparcos satellite.Extended versions of Tables \ref{tab1} and \ref{tab2} are available inelectronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr(130.79.125.8) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/381/446 A Second Catalog of Orbiting Astronomical Observatory 2 Filter Photometry: Ultraviolet Photometry of 614 StarsUltraviolet photometry from the Wisconsin Experiment Package on theOrbiting Astronomical Observatory 2 (OAO 2) is presented for 614 stars.Previously unpublished magnitudes from 12 filter bandpasses withwavelengths ranging from 1330 to 4250 Å have been placed on thewhite dwarf model atmosphere absolute flux scale. The fluxes wereconverted to magnitudes using V=0 for F(V)=3.46x10^-9 ergs cm^-2 s^-1Å^-1, or m_lambda=-2.5logF_lambda-21.15. This second catalogeffectively doubles the amount of OAO 2 photometry available in theliterature and includes many objects too bright to be observed withmodern space observatories. A HIPPARCOS Census of the Nearby OB AssociationsA comprehensive census of the stellar content of the OB associationswithin 1 kpc from the Sun is presented, based on Hipparcos positions,proper motions, and parallaxes. It is a key part of a long-term projectto study the formation, structure, and evolution of nearby young stellargroups and related star-forming regions. OB associations are unboundmoving groups,'' which can be detected kinematically because of theirsmall internal velocity dispersion. The nearby associations have a largeextent on the sky, which traditionally has limited astrometricmembership determination to bright stars (V<~6 mag), with spectraltypes earlier than ~B5. The Hipparcos measurements allow a majorimprovement in this situation. Moving groups are identified in theHipparcos Catalog by combining de Bruijne's refurbished convergent pointmethod with the Spaghetti method'' of Hoogerwerf & Aguilar.Astrometric members are listed for 12 young stellar groups, out to adistance of ~650 pc. These are the three subgroups Upper Scorpius, UpperCentaurus Lupus, and Lower Centaurus Crux of Sco OB2, as well as VelOB2, Tr 10, Col 121, Per OB2, alpha Persei (Per OB3), Cas-Tau, Lac OB1,Cep OB2, and a new group in Cepheus, designated as Cep OB6. Theselection procedure corrects the list of previously known astrometricand photometric B- and A-type members in these groups and identifiesmany new members, including a significant number of F stars, as well asevolved stars, e.g., the Wolf-Rayet stars gamma^2 Vel (WR 11) in Vel OB2and EZ CMa (WR 6) in Col 121, and the classical Cepheid delta Cep in CepOB6. Membership probabilities are given for all selected stars. MonteCarlo simulations are used to estimate the expected number of interloperfield stars. In the nearest associations, notably in Sco OB2, thelater-type members include T Tauri objects and other stars in the finalpre-main-sequence phase. This provides a firm link between the classicalhigh-mass stellar content and ongoing low-mass star formation. Detailedstudies of these 12 groups, and their relation to the surroundinginterstellar medium, will be presented elsewhere. Astrometric evidencefor moving groups in the fields of R CrA, CMa OB1, Mon OB1, Ori OB1, CamOB1, Cep OB3, Cep OB4, Cyg OB4, Cyg OB7, and Sct OB2, is inconclusive.OB associations do exist in many of these regions, but they are eitherat distances beyond ~500 pc where the Hipparcos parallaxes are oflimited use, or they have unfavorable kinematics, so that the groupproper motion does not distinguish it from the field stars in theGalactic disk. The mean distances of the well-established groups aresystematically smaller than the pre-Hipparcos photometric estimates.While part of this may be caused by the improved membership lists, arecalibration of the upper main sequence in the Hertzsprung-Russelldiagram may be called for. The mean motions display a systematicpattern, which is discussed in relation to the Gould Belt. Six of the 12detected moving groups do not appear in the classical list of nearby OBassociations. This is sometimes caused by the absence of O stars, but inother cases a previously known open cluster turns out to be (part of) anextended OB association. The number of unbound young stellar groups inthe solar neighborhood may be significantly larger than thoughtpreviously. Five-colour photometry of OB-stars in the Southern HemisphereObservations of OB-stars, made in 1959 and 1960 at the Leiden SouthernStation near Hartebeespoortdam, South Africa, with the VBLUW photometerattached to the 90 cm light-collector, are given in this paper. They arecompared with photometry obtained by \cite[Graham (1968),]{gra68}\cite[Walraven & Walraven (1977),]{wal77} \cite[Lub & Pel(1977)]{lub77} and \cite[Van Genderen et al. (1984).]{gen84} Formulaefor the transformation of the present observations to those of\cite[Walraven & Walraven (1977)]{wal77} and \cite[Lub & Pel(1977)]{lub77} are given. Table 4 is only available in electronic format the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) orvia http://cdsweb.u-strasbg.fr/Abstract.html The Southern Vilnius Photometric System. IV - The E region standard starsThis paper is the fourth in a series on the extension of the Vilniusphotometric system to the Southern Hemisphere. Observations were made of60 stars in the Harvard Standard E regions to increase a set of standardstars. Kinematical Structure of the Local Interstellar Medium: The Galactic Center HemisphereA survey of interstellar Na I D1 and D2 absorption features found in thespectra of 31 early-type stars in the first and fourth Galacticquadrants reveals a new picture of the large-scale kinematics ofinterstellar gas in the vicinity of the Sun. While absorption featuresfound in the spectra of stars in the fourth Galactic quadrant show thegas expanding from stars in the Scorpio-Centaurus OB association, starsin the first Galactic quadrant, in the range of distances covered bythis survey (74--312 pc from the Sun), show interstellar gas fallingtoward the Galactic plane with a bulk motion whose component parallel tothat plane is directed toward the Scorpio-Centaurus association. Thispicture differs completely from generally accepted models in which thenearby interstellar gas flows away from the Scorpio-Centaurusassociation toward the Galactic anticenter. Evidence for the presence ofan expanding shell-like feature that is reaching the solar system ispresented. High S/N Echelle spectroscopy in young stellar groups. II. Rotational velocities of early-type stars in SCO OB2.We investigate the rotational velocities of early-type stars in the ScoOB2 association. We measure v.sin(i) for 156 established and probablemembers of the association. The measurements are performed with threedifferent techniques, which are in increasing order of expectedv.sin(i): 1) converting the widths of spectral lines directly tov.sin(i), 2) comparing artificially broadened spectra of low v.sin(i)stars to the target spectrum, 3) comparing the HeI λ4026 lineprofile to theoretical models. The sample is extended with literaturedata for 47 established members of Sco OB2. Analysis of the v.sin(i)distributions shows that there are no significant differences betweenthe subgroups of Sco OB2. We find that members of the binary populationof Sco OB2 on the whole rotate more slowly than the single stars. Inaddition, we find that the B7-B9 single star members rotatesignificantly faster than their B0-B6 counterparts. We test varioushypotheses for the distribution of v.sin(i) in the association. Theresults show that we cannot clearly exclude any form of randomdistribution of the direction and/or magnitude of the intrinsicrotational velocity vector. We also investigate the effects of rotationon colours in the Walraven photometric system. We show that positions ofB7-B9 single dwarfs above the main sequence are a consequence ofrotation. This establishes the influence of rotation on the Walravencolours, due primarily to surface gravity effects. The ROSAT all-sky survey catalogue of optically bright OB-type stars.For the detailed statistical analysis of the X-ray emission of hot starswe selected all stars of spectral type O and B listed in the Yale BrightStar Catalogue and searched for them in the ROSAT All-Sky Survey. Inthis paper we describe the selection and preparation of the data andpresent a compilation of the derived X-ray data for a complete sample ofbright OB stars. Vitesses radiales. Catalogue WEB: Wilson Evans Batten. Subtittle: Radial velocities: The Wilson-Evans-Batten catalogue.We give a common version of the two catalogues of Mean Radial Velocitiesby Wilson (1963) and Evans (1978) to which we have added the catalogueof spectroscopic binary systems (Batten et al. 1989). For each star,when possible, we give: 1) an acronym to enter SIMBAD (Set ofIdentifications Measurements and Bibliography for Astronomical Data) ofthe CDS (Centre de Donnees Astronomiques de Strasbourg). 2) the numberHIC of the HIPPARCOS catalogue (Turon 1992). 3) the CCDM number(Catalogue des Composantes des etoiles Doubles et Multiples) byDommanget & Nys (1994). For the cluster stars, a precise study hasbeen done, on the identificator numbers. Numerous remarks point out theproblems we have had to deal with. Simultaneous photometric and spectroscopic monitoring of rapid variations of the Be star η Centauri.Results of the simultaneous monitoring of line-profile and light andcolour variations of the bright southern Be star ηCen in May-June1992 are reported. The new data were analyzed along with numerousearlier b and uvby observations of the star, secured between 1987 and1991. The brightness of ηCen varies on three distinct time scalesbut only rapid variations on a time scale of hours to days are analyzedhere. They can be described as a superposition of a periodic componentwith period 0.6424241d, which is secularly stable over the wholesix-year interval covered by our data, and systematic but obviouslyaperiodic cycle-to-cycle disturbances. The 0.6424-d light curve isslightly non-sinusoidal (light minima occur 0.64^P^ after light maxima)and has an average peak-to-peak amplitude of 0.05mag in y, b and v andabout 0.10mag in u. The u-b and 1 indices also vary with the sameperiod, with full amplitudes of 0.04mag and 0.03mag, respectively, whilethe variations of b-y and .1mag - if present - are within 0.01mag only.During light maximum, the star is bluest in u-b. The radial velocity ofSi III 455.26nm as measured in the outer wings of the profile exhibits anearly sinusoidal variation with the 0.6424-d period and definitivelyexcludes the triple-wave photometric period of 1.927d consideredearlier. For variations of other line-profile parameters of the Si III455.26 nm line, the most significant periods differ by up to +/-5% fromthe 0.6424-d period. More extended spectroscopic material will be neededto confirm our preliminary conclusion that this disparity is ofstochastic nature only and that the entire line-profile variability maybe reconciled with the photometric period. The exactly simultaneousmonitoring shows that radial velocity, first moment, equivalent width,and line depth of the Si III 455.26nm line increase when the star isgetting brighter and bluer. Very pronounced higher-order line-profilevariations of the Si III line were also detected. They have thecharacter of apparent absorption or emission bumps, which move acrossthe line profile from blue to red. Most of them disappear close to theline center. The analysis of their radial velocities leads to asuper-period, which is close to the photometric one. Occurrence andstrength of the bumps also seem to follow the same period. The light,colour, radial-velocity and equivalent-width variations can only be dueto the low-order line profile variations or duplicity. The observedradial-velocity amplitude seems to be larger than expected for therotational modulation and nonradial pulsation models. Theequivalent-width variation cannot be understood unless a phase-dependentmicroturbulent velocity is assumed in both models. Available data cannotalso exclude a model of a contact binary with 0.6-day orbital period. The local distribution of NA I interstellar gasWe present high-resolution absorption measurements (lambda/Delta lambdaapproximately 75,000) of the interstellar Na I D lines at 5890 A toward80 southern hemisphere early-type stars located in the localinterstellar medium (LISM). Combining these results with other sodiummeasurements taken from the literature, we produce galactic maps of thedistribution of neutral sodium column density for a total of 293 starsgenerally lying within approximately 250 pc of the Sun. These mapsreveal the approximate shape of the mid-plane contours of the rarefiedregion of interstellar space termed the Local Bubble. Its shape is seenas highly asymmetric, with a radius ranging from 30 to 300 pc, and withan average radius of 60 pc. Similar plots of the Galactic mid-planedistribution of sources emitting extreme ultraviolet radiation show thatthey also trace out similar contours of the Local Bubble derived from NaI absorption measurements. We conclude that the Local Bubble absorptioninterface can be represented by a hydrogen column density,NuETA = 2 x 1019 cm-2, which explainsboth the local distribution of Na I absorption and the observed galacticdistribution of extreme ultraviolet sources. The derived mid-planecontours of the Bubble generally reproduce the large-scale featurescarved out in the interstellar medium by several nearby galactic shellstructures. High-resolution CA II observations of the local interstellar mediumHigh-resolution absorption measurements of the interstellar Ca II K lineobserved toward 46 early-type stars in the local ISM (LISM) arepresented. Ca II was detected in 36 of the 46 stars with 82 individualcloud components identified. Ca II was detected to most of the starscloser than 50 pc, except in the region of the Galactic quadrant l =180-270 degrees which also contains the empty line of sight to B CMa at220 pc. The mean local standard of rest velocity of the 82 Ca IIcomponents implies that the LISM clouds are associated with the motionof the solar neighborhood and not the sun. If the present data arecombined with other nearby Ca II component velocities taken from theliterature, then a cloud centered approximately at l = 90 deg, b = -40deg moving coherently with the local interstellar wind vector issupported at a significance level of 99 percent. The Ca II data havebeen combined with Na I data for the same stars to produce a N(NaI)/N(Ca II) ratio for each identified absorption feature. This ratioplotted against the local standard of rest velocities of the cloudsshows that the Routly-Spitzer effect exists down to +/- 10 km/s, whichsupports grain desorption/destruction models that are efficient atreturning calcium to the gas phase at these low velocities. ICCD speckle observations of binary stars. VIII - Measurements during 1989-1991 from the Cerro Tololo 4 M telescopeOne-thousand eighty-eight observations of 947 binary star systems,observed by means of speckle interferometry with the 4 m telescope onCerro Tololo, are presented. These measurements, made during the period1989-1991, comprise the second installment of results stemming from theexpansion of our speckle program to the southern hemisphere. A homogeneous catalog of new UBV and H-beta photometry of B- and A-type stars in and around the Scorpius-Centaurus OB associationB- and A-type stars in and near the Sco-Cen OB association areinvestigated with UBV and H-beta photometry to acquire data relevant tothe luminosity function of Sco-Cen. The measurements generally consistof two 10-s integrations of U, B, V, (W, N) filters, and theobservations are corrected iteratively for atmospheric extinction andinstrumental response. The data presented give the mean V magnitude,mean B-V, mean U-B, and the estimated uncertainties for these values.The catalog provides a homogeneous catalog of data for a large fieldwith stellar objects delineating membership to the association Sco-Cenand that affect the luminosity function of the aggregate. High resolution observations of interstellar NA I and CA II towards the Scorpio-Centaurus associationHigh-resolution observations (3.6 km/s FWHM) of interstellar Na I and CaII toward 28 stars in the direction of the Sco-Cen association arepresented. The observed velocity structure, velocity dispersions, columndensities and Na I/Ca II ratios are discussed. Of the 58 identifiedvelocity components (toward 23 stars), 46 (or 79 percent) were found tohave negative LSR velocities. This is in agreement with previousresults, and indicates a general outflow from the Sco-Cen association.By fitting an expanding shell model to the observed velocities anexpansion velocity of 7 km/s was obtained. If the further assumption ismade that only one component per star arises in such a shell, thebest-fitting expansion velocity was found to be 9 km/s. In the lattercase, the remaining components were variously identified (depending ontheir velocities and Na I/Ca II ratios). Secondary standards for H-beta photometry in the Southern Hemisphere (second series).Not Available ICCD speckle observations of binary stars. V - Measurements during 1988-1989 from the Kitt Peak and the Cerro Tololo 4 M telescopesAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1990AJ.....99..965M&db_key=AST Walraven photometry of nearby southern OB associationsHomogeneous Walraven (VBLUW) photometry is presented for 5260 stars inthe regions of five nearby southern OB associations: Scorpio Centaurus(Sco OB2), Orion OB1, Canis Major OB1, Monoceros OB1, and Scutum OB2.Derived V and (B - V) in the Johnson system are included. Catalogue of i and w/w crit values for rotating early type starsNot Available Secondary standards for H-beta photometry in the E regions.Not Available UBV (RI)c standard stars in the E- and F-regions and in the Magellanic Clouds - a revised catalogue.Not Available The distance and structure of the Coalsack. I - Photometric dataThe results of UBV and H-beta photometry for 284 stars of spectral typesB to F located in the area of the dark cloud Coalsack are presented.Most of the present stars are brighter than 10.0 mag and are classifiedin the MK system of Houk and Cowley (1975). For each star, the bestvalues for the extinction and distance modulus are provided, withtypical estimated errors of + or - 0.08 mag for the extinction and + or- 0.32 mag for the distance modulus. Possible misclassifications andspectral peculiarities are noted for several of the stars. Physical parameters of stars in the Scorpio-Centaurus OB associationWalraven photometry is presented of established and probable members ofthe Scorpio-Centaurus OB association. For each star, effectivetemperature and surface gravity are derived using Kurucz (1979)atmosphere models. From the Straizys and Kuriliene (1981) tables,absolute magnitudes are calculated. Distance moduli and visualextinctions are determined for all stars. From a comparison of theHR-diagrams of the stars in each subgroup with theoretical isochrones,the ages of the three subgroups are derived. The distances to the threesubgroups are shown to be different; there is a general trend (alsowithin each subgroup) for the distances to be larger at higher galacticlongitudes. The visual extinction in the youngest subgroupUpper-Scorpius, is well correlated with the IRAS 100-micron map. Thedistance toward the Ophiuchus dark clouds is found to be 125 pc, basedon the photometric distances to the stars. Most of the early-type starsin Upper-Scorpius are located at the far side of the dark clouds. Secondary standards for the Stromgren UVBY systemObservations of 158 E region stars have been made in the Stromgrensystem, using the 46-cm reflector at Cape Town. They are mostly brighterthan eighth magnitude and are intended for use as secondary standardsfor the four-color system. The E region relative zero points are definedwith a precision of + or - 0.001 mag, and the internal standard errorsof the colors life between + or - 0.001 and + or - 0.002 mag. A photometric survey of the bright southern Be starsRepeated UBV photometric measurements were made of the 86 bright Bestars south of declination -20 deg, and a network of comparison starswas set up. From a statistical study of the differential photometry itwas found that short- or intermediate-term variability seems to beoccurring in about half of the Be stars, and to be more evident in thestars of earlier spectral type. It was also possible to identify 11individual short- or intermediate-term variables. Four of these (all ofearly B spectral type) appear to exhibit significant variability on atime-scale of a day or less. More intensive observations of one of thesestars, 28 Omega CMA, indicate short-term variations consistent with thepublished spectroscopic period of 1.37 day. The distance of the Pleiades cluster and the calibration of photometric luminosities for early-type starsAn examination of the lower main-sequence (mode-A) stars in the Pleiadescluster suggests an Fe/H abundance ratio between 0.0 and 0.1 dex with aresulting modulus of 5.65 + or - 0.1 mag, and fundamental defects in thecalculation of Balona and Shobbrook (1984), with an adopted modulus of5.50 mag, are discussed. It is suggested that the ZAMS of Balona andShobbrook, and of Mermilliod (1981), are too bright due to theirassumption that the color-luminosity arrays of such clusters as thePleiades represent isochrones, leading to uncertainties in the ZAMS,particularly with respect to slope. Several recently publishedphotometric luminosity calibrations for early-type stars may beincorrect due to their failing to recognize the probable presence of atleast two evolutionary modes and the apparent absence of ZAMS stars nearthe sun. A catalog of ultraviolet interstellar extinction excesses for 1415 starsUltraviolet interstellar extinction excesses are presented for 1415stars with spectral types B7 and earlier. The excesses with respect to Vare derived from Astronomical Netherlands Satellite (ANS) 5-channel UVphotometry at central wavelengths of approximately 1550, 1800, 2500, and3300 A. A measure of the excess extinction in the 2200-A extinction bumpis also given. The data are valuable for investigating the systematicsof peculiar interstellar extinction and for studying the character of UVinterstellar extinction in the general direction of stars for which theextinction-curve shape is unknown.

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 星座: 豺狼座 右阿森松: 14h51m38.40s 赤纬: -43Â°34'32.0" 视星: 4.32 距离: 125 天文距离 右阿森松适当运动: -27.1 赤纬适当运动: -26.2 B-T magnitude: 4.126 V-T magnitude: 4.289


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### Arkadiusz Ploski - One of the best experts on this subject based on the ideXlab platform.

• ##### a bound for the milnor number of plane curve singularities
Open Mathematics, 2014
Abstract:

Let f = 0 be a plane algebraic curve of degree d > 1 with an isolated singular point at 0 ∈ ℂ2. We show that the Milnor number μ0(f) is less than or equal to (d−1)2 − [d/2], unless f = 0 is a set of d concurrent lines passing through 0, and characterize the curves f = 0 for which μ0(f) = (d−1)2 − [d/2].

• ##### a bound for the milnor number of plane curve singularities
arXiv: Algebraic Geometry, 2013
Abstract:

Let $f=0$ be a plane algebraic curve of degree $d>1$ with an isolated singular point at the origin of the complex plane. We show that the Milnor number $\mu_0(f)$ is less than or equal to $(d-1)^2-\left[\frac{d}{2}\right]$, unless $f=0$ is a set of $d$ concurrent lines passing through 0. Then we characterize the curves $f=0$ for which $\mu_0(f)=(d-1)^2-\left[\frac{d}{2}\right]$.

### Y Polatoglu - One of the best experts on this subject based on the ideXlab platform.

• ##### bounded log harmonic functions with positive real part
Journal of Mathematical Analysis and Applications, 2013
Co-Authors: Y Polatoglu
Abstract:

Abstract Let H ( D ) be the linear space of all analytic functions defined on the open unit disc D = { z | | z | 1 } , and let B be the set of all functions w ( z ) ∈ H ( D ) such that | w ( z ) | 1 for all z ∈ D . A log-harmonic mapping is a solution of the non-linear elliptic partial differential equation f z ¯ ¯ = w ( z ) ( f ¯ / f ) f z , where w ( z ) is the second dilatation of f and w ( z ) ∈ B . In the present paper we investigate the set of all log-harmonic mappings R defined on the unit disc D which are of the form R = H ( z ) G ( z ) ¯ , where H ( z ) and G ( z ) are in H ( D ) , H ( 0 ) = G ( 0 ) = 1 , and R e ( R ) > 0 for all z ∈ D . The class of such functions is denoted by P L H .

### Arnaud Chéritat - One of the best experts on this subject based on the ideXlab platform.

• ##### Upper bound for the size of quadratic Siegel disks
Inventiones mathematicae, 2004
Co-Authors: Xavier Buff, Arnaud Chéritat
Abstract:

If α is an irrational number, we let { p _ n / q _ n }_ n ≥0, be the approximants given by its continued fraction expansion. The Bruno series B (α) is defined as $$B(\alpha)=\sum_{n\geq 0} \frac{\log q_{n+1}}{q_n}.$$ The quadratic polynomial P _α: z ↦ e ^2 i πα z + z ^2 has an indifferent fixed point at the origin. If P _α is linearizable, we let r (α) be the conformal radius of the Siegel disk and we set r (α)=0 otherwise. Yoccoz proved that if B (α)=∞, then r (α)=0 and P _α is not linearizable. In this article, we present a different proof and we show that there exists a constant C such that for all irrational number α with B (α)

• ##### On the Size of Quadratic Siegel Disks: Part I
arXiv: Dynamical Systems, 2003
Co-Authors: Xavier Buff, Arnaud Chéritat
Abstract:

If $\a$ is an irrational number, we let $\{p_n/q_n\}_{n\geq 0}$, be the approximants given by its continued fraction expansion. The Bruno series $B(\a)$ is defined as $$B(\a)=\sum_{n\geq 0} \frac{\log q_{n+1}}{q_n}.$$ The quadratic polynomial $P_\a:z\mapsto e^{2i\pi \a}z+z^2$ has an indifferent fixed point at the origin. If $P_\a$ is linearizable, we let $r(\a)$ be the conformal radius of the Siegel disk and we set $r(\a)=0$ otherwise. Yoccoz proved that if $B(\a)=\infty$, then $r(\a)=0$ and $P_\a$ is not linearizable. In this article, we present a different proof and we show that there exists a constant $C$ such that for all irrational number $\a$ with \$B(\a)

### Mei-chu Chang - One of the best experts on this subject based on the ideXlab platform.

• ##### A sum–product theorem in semi-simple commutative Banach algebras
Journal of Functional Analysis, 2004
Co-Authors: Mei-chu Chang
Abstract:

The following analogue of the Erdos–Szemeredi sum-product theorem is shown. Let A=f1,⋯,fN be a finite set of N arbitrary distinct functions on some set. Then either the sum set fi+fj or the product set fifj has at least N1+c elements, where c>0 is an absolute constant. We use Freiman's lemma and Balog–Szemeredi–Gowers Theorem on graphs and combinatorics. As a corollary, we obtain an Erdos–Szemeredi type theorem for semi-simple commutative Banach algebras R. Thus if A⊂R is a finite set, |A| large enough, then |A+A|+|A.A|>|A|1+c, where c>0 is an absolute constant. The result and method have various consequences, for instance decay estimates on the convolution powers of finite multiplication subgroups. Let H be a finite multiplicative subgroup of R (as above) and let N=|H|, v=1N∑x∈Hδx. Then, for all constant c, there is a k=k(c) such that maxz∈Rv(k)(z)

### Pietro Rigo - One of the best experts on this subject based on the ideXlab platform.

• ##### A uniform limit theorem for predictive distributions
Statistics & Probability Letters, 2002
Co-Authors: Patrizia Berti, Pietro Rigo
Abstract:

Abstract Let { F n } be a filtration, {Xn} an adapted sequence of real random variables, and {αn} a predictable sequence of non-negative random variables with α1>0. Set β n = ∑ i=1 n α i and define the random distribution functions F n (t)=(1/β n ) ∑ i=1 n α i I {X i ⩽t} and B n (t)=(1/β n ) ∑ i=1 n α i P(X i ⩽t| F i−1 ) . Under mild assumptions on {αn}, it is shown that sup t |F n (t)−B n (t)|→0 , a.s. on the set {Fn or B n converges uniformly } . Moreover, conditions are given under which Fn converges uniformly with probability 1.

format: plain


text:
Deepak Scored 45->99%ile with Bounce Back Crack Course. You can do it too!

# Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.

Question:

Find a G.P. for which sum of the first two terms is $-4$ and the fifth term is 4 times the third term.

Solution:

Let a be the first term and r be the common ratio of the G.P.

According to the given conditions,

$S_{2}=-4=\frac{a\left(1-r^{2}\right)}{1-r}$ $\ldots(1)$

$a_{5}=4 \times a_{3}$

$a r^{4}=4 a r^{2}$

$\Rightarrow r^{2}=4$

$\therefore r=\pm 2$

From (1), we obtain

$-4=\frac{a\left[1-(2)^{2}\right]}{1-2}$ for $r=2$

$\Rightarrow-4=\frac{a(1-4)}{-1}$

$\Rightarrow-4=a(3)$

$\Rightarrow a=\frac{-4}{3}$

Also, $-4=\frac{a\left[1-(-2)^{2}\right]}{1-(-2)}$ for $r=-2$

$\Rightarrow-4=\frac{a(1-4)}{1+2}$

$\Rightarrow-4=\frac{a(-3)}{3}$

$\Rightarrow a=4$

Thus, the required G.P. is

$\frac{-4}{3}, \frac{-8}{3}, \frac{-16}{3}, \ldots$ or $4,-8,16,-32, \ldots$


text:
# What are the following types of numbers called?

#### Chris11

Let n be a natural number, then s is the number of ways to write n as the sum of at least one natural number. What is the number s called?

#### undefined

MHF Hall of Honor
Let n be a natural number, then s is the number of ways to write n as the sum of at least one natural number. What is the number s called?
If the addends are not necessarily distinct and order is not considered, then we have Partition Function P. And here is the Wikipedia article.

#### Chris11

What about when order is considered?

#### undefined

MHF Hall of Honor

#### Chris11

Thanks for replying. I made the conjecture that the number of compositions is $$\displaystyle 2^{n-1}$$, but I haven't been able to prove it. I don't understand the proof on wikipedia much either... I shamefully don't know anything about combinatorical proof..

#### undefined

MHF Hall of Honor
Thanks for replying. I made the conjecture that the number of compositions is $$\displaystyle 2^{n-1}$$, but I haven't been able to prove it. I don't understand the proof on wikipedia much either... I shamefully don't know anything about combinatorical proof..
The proof is not too hard, let's try an example: n=3.

1+1+1 = (3)
1+1,1 = (2,1)
1,1+1 = (1,2)
1,1,1 = (1,1,1)

You can look at the "+" and "," symbols as 0 and 1.

00
01
10
11

Does it make more sense now?

#### Chris11

yeah, it makes alot more sense. Thanks.


text:
The Formula Factory

The formula factory is the central concept of LogicNG and is always required when working with LogicNG. A formula factory is an object consisting of two major components:

A factory, which creates formulas, and a container, which stores created formulas.

The container function is 'smart': A formula factory guarantees that syntactically equivalent formulas are created only once. This mechanism also extends to variants of the formula in terms of associativity and commutativity. Therefore, if the user creates formulas for

• A & B
• B & A
• (B & A)

all of them are represented by only one formula in memory. This approach is only possible, because formulas in LogicNG are immutable data structures. So once created, a formula can never be altered again.

In order to use the fact that formula factories avoid unnecessary formula creations, it is generally recommended to use only one formula factory for a certain task.

During this documentation, we consider the formulas f1, f2 and f3 and use the following notion of a formula factory:

FormulaFactory f = new FormulaFactory();

Creating Formulas with a Formula Factory

Formulas in LogicNG cannot be constructed directly but must be created by an instance of a formula factory. This factory ensures the following five invariants:

1. A constant (true or false) cannot be an operand to any other formula, i.e. constants are automatically removed.
2. The operand of a conjunction may not be another conjunction; the same applies to disjunctions. These cases are merged in one big conjunction/disjunction.
3. The operand of a negation may only be a binary operator, an n-ary operator or a pseudo-Boolean constraint. For other operands the respective simplifications are performed.
4. An n-ary operator has unique operands. Duplicate operands in a disjunction or conjunction are filtered.
5. Every positive literal is guaranteed to be an instance of class Variable.

In the default configuration there is also a sixth invariant, which however can be deactivated by a configuration flag (see below):

1. Inverse operands of an n-ary operator are simplified, this means f1 & ~f1 is parsed to $false, and f1 | ~f1 is parsed to $true.

Furthermore, some further simplifications are performed when parsing or creating formulas, such as A <=> A is equivalent to $true, or A <=> ~A is equivalent to $false.

While being rather easy to realize, these restrictions simplify reasoning about the structure of a formula and thus significantly reduce the number of corner cases algorithms have to face.

Together with the smart container function presented in the last section, the example can be extended. Not only are formulas A & B, B & A, or (B & A) represented by only one formula object in memory, but also variants like

• A & A & B
• B & A & $true • (B & A) & B & ($false | A) & ($true | C) There are two ways to create formulas using a formula factory: Firstly, one can parse a formula from a string: E.g. Formula created = f.parse("A & B");. Secondly, one can create a formula of a certain type with the methods for formula creation in the formula factory. An overview about how to create those formulas is here: Object Factory Method Syntax True f.verum() $true
False f.falsum() $false Variable f.variable("A") A Literal f.literal("A", false) ~A Negation f.not(f1) ~f1 Implication f.implication(f1, f2) f1 => f2 Equivalence f.equivalence(f1, f2) f1 <=> f2 Conjunction f.and(f1, f2, f3) f1 & f2 & f3 Disjunction f.or(f1, f2, f3) f1 | f2 | f3 The order of operands in the resulting formula does not follow an overall ordering but depends on the formulas created first. If a formula B & A was created, the order of the operands will be always B, A. Thus, when creating another formula A & B it will result in B & A. If another formula A & B & C is created, the operands occur in the order A, B, C, since A & B & C was the first created formula. It is also possible to write Pseudo-Boolean constraints, especially cardinality constraints like e.g. A + B + C <= 1: f.cc(CType.LE, 1, Arrays.asList(f.variable("A"), f.variable("B"), f.variable("C")))  which means from variables A, B, C can be at most one variable assigned to true. An example for a Pseudo-Boolean constraint is A + 2* ~B - 3*C = 2: List<Literal> lits = Arrays.asList(f.variable("A"), f.literal("B", false), f.variable("C")); List<Integer> coeffs = Arrays.asList(1, 2, -3); f.pbc(CType.EQ, 2, lits, coeffs);  Beside the mentioned factory methods there are many convenience methods to create formulas. Examples are: • constant(boolean value) which creates a $true or $false constant depending on the given Boolean value • cnf(Formula... clauses) creating a conjunctive normal form (CNF) for the given clauses • clause(Literal... literals) creating a clause for the given literals • variables(Collection<String> names) and variables(String... names) creating a set of variables from the given names (since 2.4.0) For information about properties of formulas check out the next chapter Formulas. When creating a formula factory, the factory does not contain any formulas except from the constants true and false, which are always generated when creating a formula factory. Another option for using a formula within a formula factory without creating it, is to import a formula from a different formula factory (see further below). Configuring the Formula Factory The user can configure some settings of the formula factory. Those are given over via the FormulaFactoryConfiguration (see FormulaFactoryConfig), with which the formula factory can be generated. If the formula factory is generated with no parameter, then the default formula factory configuration is chosen. The parameters that can be configured are: • The formula merge strategy: The formula merge strategy is concerned with combining formulas of different formula factories. A formula from a different factory can only be used in the current factory after importing it. In order to avoid bugs by using formulas from different factories without importing it, the default behaviour of the formula factory is to panic and throw an exception. This however can be changed to the import strategy - then a formula from a different factory is automatically imported. The respective configuration flags are PANIC, and IMPORT. • A flag, whether complementary operands shall be simplified. This is the flag whether complementary operands within a conjunction or disjunction should be simplified. For example, shall a formula like A & ~A, (A | ~A) & C, or (A & B) | ~(A & B) be simplified or not. If the flag is set to true, these formulas will be simplified during creation to $false, C, and \$true respectively. If the flag is set to false such trivial tautologies and contradictions are not simplified but kept.
• The string representation: This is the default string representation of formulas when using Java's standard toString() method. In LogicNG there are three default implementations:
• UTF8StringRepresentation for a UTF-8 representation with nice symbols for the operands and sub-script for indexed variables
• LatexStringRepresentation for a LaTeX notation of the formulas
• DefaultStringRepresentation, which is the default string representation and outputs the formulas in the LogicNG syntax which can be parsed again by the LogicNG parsers.

Consider the formula A & B | C. The UTF-8 representation gives the formula out as A ∧ B ∨ C and the Latex style representation as A \land B \lor C. The default representation is A & B | C.

• the name: An optional name for the formula factory (which can be useful when using more than one formula factory at the time). The default is the empty string "". Note that the name will appear in all auxiliary variables (constructed variables for CNF, PB and CC encodings) and will not be changed when imported to other factories.

The parameters of the default configuration is summarized here:

Default Configuration

When constructing a formula factory without configuration, the default configuration FormulaFactoryConfig is constructed. The settings are:

• name: ""
• formula merge strategy: PANIC
• string representation: DefaultStringRepresentation
• simplify complementary operands: true
• ccPrefix: @RESERVED_CC_, pbPrefix: @RESERVED_PB_, cnfPrefix: @RESERVED_CNF_

In order to create a formula factory with different parameters, one can use the FormulaFactoryConfig builder as follows:

FormulaFactory f = new FormulaFactory(
FormulaFactoryConfig.builder()
.formulaMergeStrategy(FormulaFactoryConfig.FormulaMergeStrategy.IMPORT)
.stringRepresentation(UTF8StringRepresentation::new)
.name("New factory")
.build());

Configuring Further Aspects of LogicNG in the Formula Factory

A formula factory can also hold configurations for different algorithms in LogicNG. These algorithms can be configured when using or by putting an initial configuration in the formula factory.

E.g. currently the following configurations can be put on the formula factory:

• CNF configuration (see CNFConfig), defining the encoding procedure used to generate a CNF of a formula, e.g. when f1.cnf() is called.
• Cardinality constraint configuration (see CCConfig), defining the encoding procedure used when a cardinality constraints is encoded, e.g. when f1.cc() is called.
• Pseudo-Boolean configuration (see PBConfig), defining the encoding procedure used when a Pseudo-Boolean constraint is encoded, e.g. when f1.pb() is called.
• Formula randomizer (see FormulaRandomizerConfig), a configuration for random formula generation, which can be a useful tool for testing.
• MUS (see MUSConfig), the configuration object for the MUS (minimal unsatisfiable set) generation.

Further, one can specify parameters for the different solvers:

Using f.putConfiguration(config), for some configuration config, one can put a new configuration into the configuration database. Existing configurations for this type will be overwritten. f.configurationFor(configType) (for some configuration type configType) returns the configuration of a specific configuration type.

Example: Changing a configuration of the formula factory

Consider the cardinality constraint

A + B + C <= 1

Using the default cardinality constraint configuration, the cardinality constraint is encoded to:

(~A | ~B) & (~A | ~C) & (~B | ~C)

However, if you'd like to encode the cardinality constraint differently, you may choose to set a different algorithm for the encoding of cardinality constraints in the formula factory. For example, changing the formula factory's default algorithm for encoding at-most-one constraints with the LADDER encoding, one could configure the formula factory in the following way:

CCConfig config = CCConfig.builder().amoEncoding(CCConfig.AMO_ENCODER.LADDER).build();
f.putConfiguration(config);

The resulting cardinality constraint encoding is then, for the auxiliary variables x1 and x2:

(~A | x1) & (~B | x2) & (~x1 | x2) & (~B | ~x1) & (~C | ~x1)

More Methods on Formula Factories

Some more useful methods on formula factories are:

• Remove all formulas from the factories cache with f.clear(): Use this method if you want to hold on to the "settings" of the formula factory, but get rid of all formulas created so far
• Import a formula from another formula factory into this factory and return it: f.importFormula(f1). If the current factory of the formula is already this formula factory (f), the same instance will be returned
• Check whether a variable is a generated variable with f.isGeneratedVariable(v) for some variable v. A variable is considered to be a generated variable (or auxiliary variable) if it was generated during a CNF, PG or cardinality constraints encoding
• Get some statistics about the formula factory: f.statistics() returns a FormulaFactoryStatistics which includes some relevant information about the factory, for example:
• The number of formulas the formula factory holds in total
• The number of formulas the formula factory holds for each formula type

Caching Formulas in Practice and the Extended Formula Factory

While caching formulas in the formula factory in most cases greatly improves heap space usage and performance, there are some cases where this behaviour could be undesirable. An example is an application which holds a large base rule set and there are many requests for other large formulas, which are checked against this base rule set. If the application runs in a server environment and uses one formula factory, this formula factory would potentially grow with every request. Usually this does not happen too often in practice, since in our experience, also the base rule set is something which is not static and is created with each request. However, for the described scenario, the ever-caching formula factory would be a suboptimal approach.

Therefore, there is an ExtendedFormulaFactory which extends the normal formula factory by the ability of saving and loading the state of the factory, therefore temporarily adding formulas to the factory and then removing them again. Thus, in the described scenario, one could add the base rule set and save the formula factory state. Then we add some large other formulas, perform our computations, and when we are done, we load the saved state. By these operations, all internal data structures of the formula factory are shrunk to their length as stored in the saved state and therefore the formulas added after the state save are removed. Also, for internal consistency reasons, the caches of all formulas are cleared.

The following code example shows the usage of this feature

ExtendedFormulaFactory f = new ExtendedFormulaFactory();

// read and work with a large base formula

FormulaFactoryState state = f.save(); // (1)!
f.parse("..."); // (2)!


text:
Sat 25 Jan 2020 11:30 - 12:00 at Orleans - Programming features Chair(s): Giuseppe Castagna

In practice, FFIs are implemented by translation to a target language and by conversions at the level of the target. We wish to establish type soundness in such a setting, where there are two languages making foreign calls to one another. In particular, we want a notion of \emph{convertibility}, that a type $\tau_A$ from language $A$ is convertible to a type $\tau_B$ from language $B$, which we will write $\tau_A \sim \tau_B$, such that conversions between these types maintain type soundness (dynamically or statically) of the overall system.

In this paper, we leverage the idea that sound FFIs are a generalization of sound gradual typing, except unlike gradual typing, FFIs mediate between languages \emph{without} a common underlying term language. We start with two languages and define type soundness in a manner inspired by the practical implementation strategy: that the languages will be translated to a common target. We do this using a realizability model, that is, by setting up a logical relation indexed by source types but inhabited by target terms that behave as dictated by the source types. The conversions $\tau_A \sim \tau_B$ that \emph{should} be allowed, are the ones implemented by target-level translations that convert terms that semantically behave like $\tau_A$ to terms that semantically behave like $\tau_B$ (and vice versa). This means that the converted term must either produce a runtime cast failure or produce some term or value that behaves as $\tau_B$. With the notion of $\tau_A \sim \tau_B$ in hand, we can type check a foreign function call, and based on the semantics of
$\tau_A \sim \tau_B$, we can prove that typing the foreign function call is sound.

#### Sat 25 JanDisplayed time zone: Saskatchewan, Central America change

 10:30 - 12:30 Programming featuresWGT at Orleans Chair(s): Giuseppe Castagna CNRS - Université de Paris, France 10:3030mTalk Gradual Algebraic Data TypesWGTMichael Greenberg Pomona College, Stefan Malewski University of Santiago de Chile, Éric Tanter University of Chile Pre-print File Attached 11:0030mTalk Gradual Typing for Extensibility by RowsWGTTaro Sekiyama National Institute of Informatics, Atsushi Igarashi Kyoto University, Japan Pre-print 11:3030mTalk Foreign Function Typing: Semantic Type Soundness for FFIsWGTDaniel Patterson Northeastern University, Amal Ahmed Northeastern University, USA Pre-print 12:0030mTalk Space-Efficient Monotonic ReferencesWGTDeyaaeldeen Almahallawi Indiana University, Jeremy G. Siek Indiana University, USA Pre-print


text:
Question

# Find the sum of the following series to n terms:$$1.2.4+2.3.7+3.4.10+...$$

Solution

## Let $$T_n$$ be the terms of the given series. Then$$T_n (n^{th}\ term\ of\ 1,2,3...)\times (n^{th}\ term\ of\ 2,3,4...)\times (n^{th}\ term\ of\ 2,3,4...)$$$$=[1+(n-1)\times 1].[2+(n-1)\times 1] [4+(n-1)\times 3]$$$$=[1+n-1][2+n-1][4+3n-3]$$$$=n (n+1)(3n+1)$$$$=3n^3+n^2 +3n^2+n$$$$=3n^3+4n^2 +n$$$$T_n =3n^3+4n^2+n$$Let $$S_n$$ denoted sum of $$n$$ terms of the given series $$S_n =\displaystyle \sum _{n=1}^{n} T_n =\displaystyle \sum _{n=1}^{n}n^2 +\displaystyle \sum _{n=1}^{n}n$$$$=\dfrac{3}{1} \left [\dfrac {n(n+1)}{2}\right]^2 +4\left [\dfrac {n(n+1) (2n+1)}{6}\right]+\dfrac {n(n+1)}{2}$$$$=\dfrac {3}{4}[n(n+1)]^2+\dfrac {2n (n+1) (2n+1)}{3}+\dfrac {n(n+1)}{2}$$$$=\dfrac {n}{12}(n+1)[9n^2 +9n+16n+8+6]$$$$S_n =\dfrac {n}{12}(n+1)[9n^2+25n+14]$$Mathematics

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torus - Maple Help

plottools

 torus
 generate 3-D plot object for a torus

 Calling Sequence torus([x, y, z], r, R, options)

Parameters

 [x, y, z] - (optional) center of the torus; default=[0,0,0] r - (optional) radius of the meridian; default=1 R - (optional) distance from center of meridian to center of the torus; default=2 options - (optional) equations of the form option=value. For a complete list, see plot3d/options.

Description

 • The torus command creates a three-dimensional plot data object, which when displayed is a torus centered at [x,y,z], whose meridian has radius r and distance from the center of a meridian to the center of the torus is R.
 • The plot data object produced by the torus command can be used in a PLOT3D data structure, or displayed using the plots[display] command.
 • Remaining arguments are interpreted as options, which are specified as equations of the form option = value. For more information, see plottools and plot3d/options.

Examples

 > $\mathrm{with}\left(\mathrm{plottools}\right):$
 > $\mathrm{with}\left(\mathrm{plots}\right):$
 > $\mathrm{display}\left(\mathrm{torus}\left(\left[1,1,1\right],1,2\right),\mathrm{scaling}=\mathrm{constrained},\mathrm{lightmodel}=\mathrm{light1},\mathrm{shading}=\mathrm{zgrayscale}\right)$


text:
# What am I doing wrong in this integration problem?

• January 18th 2009, 09:04 PM
Dana_Scully
What am I doing wrong in this integration problem?
Hi, I've been working on this integration problem, but when I upload it to our homework program the computer keeps saying that it's wrong. It's:

$\int \frac{z^5}{\sqrt[3]{9+z^6}} du$

So what I did was:

$u = 9 + z^6$

$\frac{du}{6} = z^5 dz$

$\frac{1}{6}\int u^{\frac{-1}{3}} du$

$\frac{-1}{18 \sqrt[3]{(9+z^6)^4}}$

Do you think it's a syntax problem, or am I making an obvious mistake?
• January 18th 2009, 09:10 PM
Bilbo Baggins
Quote:

Originally Posted by Dana_Scully
Hi, I've been working on this integration problem, but when I upload it to our homework program the computer keeps saying that it's wrong. It's:

$\int \frac{z^5}{\sqrt[3]{9+z^6}} du$

So what I did was:

$u = 9 + z^6$

$\frac{du}{6} = z^5 dz$

$\frac{1}{6}\int u^{\frac{-1}{3}} du$

$\frac{-1}{18 \sqrt[3]{(9+z^6)^4}}$

Do you think it's a syntax problem, or am I making an obvious mistake?

You made a mistake. Your substitution is good, but you went astray. After your substitution, simply treat the entity as a polynomial. Raise the exponent by one and divide by the new exponent. Then, resubstitute the original expression back in. Lastly, the sure-fire proof is to differentiate your answer and see that you get the original expression.
• January 18th 2009, 09:11 PM
Prove It
Quote:

Originally Posted by Dana_Scully
Hi, I've been working on this integration problem, but when I upload it to our homework program the computer keeps saying that it's wrong. It's:

$\int \frac{z^5}{\sqrt[3]{9+z^6}} du$

So what I did was:

$u = 9 + z^6$

$\frac{du}{6} = z^5 dz$

$\frac{1}{6}\int u^{\frac{-1}{3}} du$

$\frac{-1}{18 \sqrt[3]{(9+z^6)^4}}$

Do you think it's a syntax problem, or am I making an obvious mistake?

I think you'll find that $\int{u^{-\frac{1}{3}}\,du} = \frac{3}{2}u^{\frac{2}{3}}$.
• January 18th 2009, 09:15 PM
Aryth
The mistake is somewhat obvious, we'll start at this part:

$\frac 16 \int u^{-\frac 13} ~du$

Now, the integral of $u^n$ is:

$\frac{u^{n+1}}{n+1}$

So, that gives us:

$\frac 16 \left(\frac{u^{\frac{2}{3}}}{\frac{2}{3}}\right)$

Now we simplify:

$\frac 16 \left(\frac{3u^{\frac{2}{3}}}{2}\right)$

$= \frac{u^{\frac{2}{3}}}{4}$

$\frac{\sqrt[3]{(9 + z^6)^2}}{4}$

And there you go.
• January 18th 2009, 09:22 PM
Dana_Scully
Ah, thank you all so much. There goes me confusing derivation and integration again...not sure what's wrong with my head.


text:
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Inverted World [Christopher Priest]. A science-fiction novel, but of a piece with a writer like M. John Harrison — there’s a kind of disconnect and a focus on the conceptual world building rather than the nitty-gritty you get with Iain M. Banks. To avoid spoilers, I’ll just say it’s set in a city which moves through the world, always trying to be at a place called optimum. The city is on rails — it constantly builds fresh tracks ahead of it and winches itself forward a tenth of a mile per day. The city is run by a guild system of track layers, traction experts, bridge builders, surveyors, and the like. The protagonist, Helward Mann, takes an oath and joins a guild as an apprentice. The book follows his progress as he learns, and we learn, more about the strange world through which the city moves. Recommended if you like heady, somewhat retro, post-apocalyptic conceptual fiction.

Luka and the Fire of Life [Salman Rushdie]. A re-read for me, this didn’t hold up as well the second time around. I much prefer Haroun and the Sea of Stories, which I can read over and over again.

Boxers and Saints [Gene Luen Yang]. A great two-part graphic novel about the Boxer Rebellion in China. Chances are you don’t know much about this history. You won’t necessarily get a history lesson from this book, but you will want to learn more about it.

The Adventures of Augie March [Saul Bellow]. After leaving Chicago I have decided to read more books set in Chicago so that I can miss it more. I had read this book before but it was a rushed job. This time I let myself longer a bit more over Bellow’s language. It’s epic and scope and gave me a view of Chicago and the Great Depression that I hadn’t had before. Indeed, given our current economic woes, it was an interesting comparison to see the similarities (the rich are still pretty rich, and if you can get employed by them, you may do ok) and the dissimilarities.

The Idea Factory: Bell Labs and the Great Age of American Innovation [John Gertner]. A history of Bell Labs and a must-read for researchers who work on anything related to computing, communications, or applied physics and chemistry. It’s not all rah-rah, and while Gertner takes the “profiles of the personalities” approaches to writing about the place, I am sure there will be things in there that would surprise even the die-hard Shannonistas who may read this blog…

# SPCOM 2014: some more talks (and a plenary)

I did catch Greg Wornell’s plenary at SPCOM, which was called When Bits Absolutely, Positively, Have to be There as Soon as Possible, a riff on this FedEx commercial, which is older than I am. The talk was on link-aware PHY-layer design– basically looking at how ARQ enables incremental redundancy, and how to do a sort of layered superposition + incremental redundancy scheme in the sequential setting as well as a “multi-path” setting where blocks can arrive out of order. This was really digging into the signal issues in a way that a lot of non-communication engineering information theorists may get squeamish about. The nice thing is that I think the engineering problem is approachable without knowing a lot of heavy-duty math, but still requires some careful analysis.

Communication and Compression Via Sparse Linear Regression
Ramji Venkataramanan
This was on building codewords and codebooks out of a lower-complexity code dictionary $A \in \mathbb{R}^{n \times ML}$ where each codeword is a superposition of $L$ columns, one each from groups of size $M$. Thus encoding is $A \beta$ where $\beta$ is a sparse vector. I saw a talk by Barron and Joseph from a previous ISIT about this, but the framework extends to rate distortion (achieving the rate distortion function), and channel coding. The main point is to lower the complexity of the code at the expense of the gap to optimal rate — encoding and decoding are polynomial time but the rate gap for rate-distortion goes to zero as $1/\log n$. Ramji gave a really nice and clear talk on this — I hope he puts the slides up!

An Optimal Varentropy Bound for Log-Concave Distributions
Mokshay’s talk was also really clear and excellent. For a distribution $f(X)$ on $\mathbb{R}^n$, we can define $\tilde{h}(X) = - \log f(X)$. The entropy is the expectation of this random variable, and the varentropy is the variance. Their main result is a upper bound on the varentropu of log concave distributions $f(X)$. To wit, $\mathrm{Var}(\tilde{h}) \le n$. This bound doesn’t depend on the distribution and is sharp if $f$ is a product of exponentials. They then use this to prove a universal bound on the deviation of $\tilde{h}$ from its expectation, which gives a AEP that doesn’t really assume anything about the joint distribution of the variables except for log-concavity. There was more in the talk, but I eagerly await the paper.

Event-triggered Sampling and Reconstruction of Sparse Real-valued Trigonometric Polynomials
Neeraj Sharma; Thippur V. Sreenivas
This was on non-uniform sampling where the sampler tries to detect level crossings of the analog signal and samples at that point — the rate may not be uniform enough to use existing nonuniform sampling techniques. They come up with a method for reconstructing signals which are real-valued trigonometric polynomials with a few nonzero coefficients (e.g. sparse) and it seems to work pretty decently in experiments.

Removing Sampling Bias in Networked Stochastic Approximation
Vivek Borkar; Raaz Dwivedi
In networked stochastic approximation, the intermittent communication between nodes may mean that the system tracks a different ODE than the one we want. By modifying the method to account for “local clocks” on each edge, we can correct for this, but we end up with new conditions on the step size to make things work. I am pretty excited about this paper, but as usual, my notes were not quite up to getting the juicy bits. That’s what paper reading is for.

On Asymmetric Insertion and Deletion Errors
Ankur A. Kulkarni
The insertion/deletion channel model is notoriously hard. Ankur proposed a new model where $0$‘s are “indestructible” — they cannot be inserted or deleted. This asymmetric model leads to new asymptotic bounds on the capacity. I don’t really work on this channel model so I can’t get the finer points of the results, but once nice takeaway was that asymptotically, each indestructible $0$ in the codeword lets us correct around $1/2$ a deletion more.

# A teaser for ITAVision 2015

As part of ITAVision 2015 we are soliciting individuals and groups to submit videos documenting their love of information theory and/or its applications. During ISIT we put together a little example with our volunteers (it sounded better in rehearsal than at the banquet, alas). The song was Entropy is Awesome based on this, obviously. If you want to sing along, here is the Karaoke version:

The lyrics (so far) are:

Entropy is awesome!
Entropy is sum minus p log p
Entropy is awesome!
When you work on I.T.

Blockwise error vanishes as n gets bigger
Maximize I X Y
Polarize forever
Let’s party forever

I.I.D.
I get you, you get me
Communicating at capacity

Entropy is awesome…

This iteration of the lyrics is due to a number of contributors — truly a group effort. If you want to help flesh out the rest of the song, please feel free to email me and we’ll get a group effort going.

More details on the contest will be forthcoming!

# SPCOM 2014: some talks

Relevance Singular Vector Machine for Low-rank Matrix Sensing
Martin Sundin; Saikat Chatterjee; Magnus Jansson; Cristian Rojas
This talk was on designing Bayesian priors for sparse-PCA problems — the key is to find a prior which induces a low-rank structure on the matrix. The model was something like $y = A \mathrm{vec}(X) + n$ where $X$ is a low-rank matrix and $n$ is noise. The previous state of the art is by Babacan et al., a paper which I obviously haven’t read, but the method they propose here (which involved some heavy algebra/matrix factorizations) appears to be competitive in several regimes. Probably more of interest to those working on Bayesian methods…

Non-Convex Sparse Estimation for Signal Processing
David Wipf
More Bayesian methods! Although David (who I met at ICML) was not trying to say that the priors are particularly “correct,” but rather that the penalty functions that they induce on the problems he is studying actually make sense. More of an algorithmist’s approach, you might say. He set up the problem a bit more generally, to minimize problems of the form
$\min_{X_i} \sum_{i} \alpha_i \mathrm{rank}[X_i] \ \ \ \ \ \ \ Y = \sum_{i} A_i(X_i)$
where $A_i$ are some operators. He made the case that convex relaxations of many of these problems, while analytically beautiful, have restrictions which are not satisfied in practice, and indeed they often have poor performance. His approach is via Empirical Bayes, but this leads to non-convex problems. What he can show is that the algorithm he proposes is competitive with any method that tries to separate the error from the “low-rank” constraint, and that the new optimization is “smoother.” I’m sure more details are in his various papers, for those who are interested.

PCA-HDR: A Robust PCA Based Solution to HDR Imaging
My apologies for taking fewer notes on this one, but I don’t know much about HDR imaging, so this was mostly me learning about HDR image processing. There are several different ways of doing HDR, from multiple exposures to flash/no-flash, and so on. The idea is that artifacts introduced by the camera can be modeled using the robust PCA framework and that denoting in HDR imaging may be better using robust PCA. I think that looking at some of the approaches David mentioned may be good in this domain, since it seems unlikely to me that these images will satisfy the conditions necessary for convex relaxations to work…

On Communication Requirements for Secure Computation
Vinod M Prabhakaran
Vinod showed some information theoretic approaches to understanding how much communication is needed for secure computation protocols like remote oblivious transfer: Xavier has $\{X_0, X_1\}$, Yvonne has $Y \in \{0,1\}$ and Zelda wants $Z = X_Y$, but nobody should be able to infer each other’s values. Feige, Killian, and Naor have a protocol for this, which Vinod and Co. can show is communication-optimal. There were several ingredients here, including cut-set bounds, distribution switching, data processing inequalities, and special bounds for 3-party protocols. More details in his CRYPTO paper (and others).

Artificial Noise Revisited: When Eve Has More Antennas Than Alice
Shuiyin Liu; Yi Hong; Emanuele Viterbo
In a MIMO wiretap setting, if the receiver has more antennas than the transmitter, then the transmitter can send noise in the nullspace of the channel matrix of the direct channel — as long as the eavesdropper has fewer antennas than the transmitter then secure transmission is possible. In this paper they show that positive secrecy capacity is possible even when the eavesdropper has more antennas, but as the number of eavesdropper antennas grows, the achievable rate goes to $0$. Perhaps a little bit of a surprise here!

# SPCOM 2014: tutorials

I just attended SPCOM 2014 at the Indian Institute of Science in Bangalore — many thanks to the organizers for the invitation! SPCOM 2014 happens every two years and is a mix of invited and submitted papers (much like Allerton). This year they mixed the invited talks with the regular talks which I thought was a great idea — since invited papers were not for specific sessions, it makes a lot more sense to do it that way, plus it avoids a sort of “two-tier” system.

I arrived early enough to catch the tutorials on the first day. There was a 3 hour session in the morning and another on the in afternoon. For the morning I decided to expand my horizons by attending Manoj Gopalkrishnan‘s tutorial on the physics of computation. Manoj focused on the question of how much energy it takes to erase or copy a bit of information. He started with some historical context via von Neumann, Szilard, and Landauer to build a correspondence between familiar information theoretic concepts and their physical counterparts. So in this correspondence, relative entropy is the same as free energy. He then turned to look at what one might call “finite time” thermodynamics. Suppose that you have to apply a control that operates in finite time in order to change a bit. One way to look at this is through controlling the transition probabilities in a two-state Markov chain representing the value of the bit you want to fix. You want to drive the resting state (with stationary distribution $(1/2,1/2)$ to something like $(\epsilon, 1 - \epsilon)$ within time $T$. At this level I more or less understood what was going on, but since my physics background is pretty poor, I think I missed out on how the physical intuition/constraints impact what control strategies you can choose.

Prasad Santhanam gave the other tutorial, which was a bit more solid ground for me. This was not quite a tutorial on large-alphabet probability estimation, but more directly on universal compression and redundancy calculations. The basic setup is that you have a family of distributions $\mathcal{P}$ and you don’t know which distribution $p \in \mathcal{P}$ will generate your data. Based on the data sample you want to do something: estimate some property of the distribution, compress the sample to a size close to its entropy, etc. A class can be weakly or strongly compressible, or insurable (which means being able to estimate quantiles), and so on. These problems turn out to be a bit different from each other depending on some topological features of the class. One interesting thing to consider for the machine learners out there this stopping time that you need in some analyses. As you are going along, observing the data and doing your task (estimation, compression, etc) can you tell from the data that you are doing well? This has major implications for whether or not an online algorithm can even work the way we want it to, and is something Prasad calls “data-driven compressible.”

I’ll try to write another post or two about the talks I saw as well!

# ISIT 2014: a few more talks

Annina Bracher (ETH Zurich, Switzerland); Amos Lapidoth (ETHZ, Switzerland)
The title pretty much describes it — there are two receivers which are both looking out for a particular message. This is the identification problem, in which the receiver only cares about a particular message (but we don’t know which one) and we have to design a code such that they can detect the message. The number of messages is $2^{2^{nC}}$ where $C$ is the Shannon capacity of the DMC. In the broadcast setting we run into the problem that the errors for the two receivers are entangled. However, their message sets are disjoint. The way out is to look at the average error for each (averaged over the other user’s message). The main result is that the rates only depend on the conditional marginals, and they have a strong converse.

Efficient compression of monotone and m-modal distributions
Jayadev Acharya (University of California, San Diego, USA); Ashkan Jafarpour (University of California, San Diego, USA); Alon Orlitsky (University of California, San Diego, USA); Ananda Theertha Suresh (University of California, San Diego, USA)
A monotone distribution is a distribution on $\mathbb{N}$ such that the probabilities are non-increasing. The redundancy for this class is infinite, alas, so they restrict the support to size $k$ (where $k$ can be large). They propose a two-step compression scheme in which the first step is to approximate the true distribution with a piecewise constant step distribution, and then use a compression scheme for step distributions.

Writing on a Dirty Paper in the Presence of Jamming
Amitalok J Budkuley (Indian Institute of Technology, Bombay, India); Bikash K Dey (Indian Institute of Technology Bombay, India); Vinod M Prabhakaran (Tata Institute of Fundamental Research, India)
Ahh, jamming. A topic near and dear to my heart. This paper takes a game-theoretic approach to jamming in a DPC setup: “the capacity of the channel in the presence of the jammer is the unique Nash equilibrium utility of the zero sum communication game between the user and the jammer.” This is a mutual information game, and they show that i.i.d. Gaussian jamming and dirty paper coding are a Nash equilibrium. I looked at an AVC version of this problem in my thesis, and the structure is quite a bit different, so this was an interesting different take on the same problem — how can we use the state information to render adversarial interference as harmless as noise?

Stable Grassmann Manifold Embedding via Gaussian Random Matrices
Hailong Shi (Tsinghua University & Department of Electronic Engineering, P.R. China); Hao Zhang (TsinghuaUniversity, P.R. China); Gang Li (Tsinghua University, P.R. China); Xiqin Wang (Tsinghua University, P.R. China)
This was in the session I was chairing. The idea is that you are given a subspace (e.g., a point on the Grassman manifold) and you want to see what happens when you randomly project this into a lower-dimensional subspace using an i.i.d. Gaussian matrix a la Johnson-Lindenstrauss. The JL Lemma says that projections are length-preserving. Are they also volume-preserving? It turns out that they are (no surprise). The main tools are measure concentration results together with a union bound over a covering set.

Is “Shannon capacity of noisy computing” zero?
Pulkit Grover (Carnegie Mellon University, USA)
Yes. I think. Maybe? Pulkit set up a physical model for computation and used a cut-set argument to show that the total energy expenditure is high. I started looking at the paper in the proceedings and realized that it’s significantly different than the talk though, so I’m not sure I really understood the argument. I should read the paper more carefully. You should too, probably.

# ISIT 2014: two more plenaries

As I wrote before, I took pretty woeful notes during ISIT this year, so I don’t have much to write about. Andrea Goldsmith’s plenary was about how we always say IT/Comm is dead, but she thinks we should be more sanguine about it. She presented a glimpse of some recent work with Stefano Rini on a unified approach for providing achievable results for single-hop networks using a graph to represent superposition coding and binning operations among the auxiliary variables. If it is actually as easy to use as advertised, it might save over the 23+ rate inequalities defining some achievable rate regions. The moral of the story is that it’s sometimes better to clean up our existing results a bit. I think the El-Gamal and Kim book did a great job of this for basic multiterminal IT, for example.

Vijay Kumar’s plenary was on codes for distributed storage and repair-bandwidth tradeoffs, focusing on extensions of the model. There was a lot of discussion of other code constructions, and how asking for certain properties (such as “locality”) can cost you something in the tradeoff. This is important when you can’t repair a code from arbitrary nodes in the network/data center — because there’s an underlying network which supplies the data for repair, codes should probably respect that network. At least that was the moral I took from this talk. Since I don’t work on coding, some things were a little over my head, but I thought he did an excellent job of keeping it accessible with nice concrete examples.


text:
# RocketClubs/PracticeTesting

C#
Fetching latest commit…
Cannot retrieve the latest commit at this time.

# PracticeTesting

## Quick Usage

The easiest way to use the PracticeTesting library is to just load up an existing XML question pool and create a test from it. Then you ask each question and record the answer. In a console app, the main test function would be similar to this:

var questionPoolFileName = @"c:\the\full\path\to\your\question\pool.xml";
var practiceTest = questionPool.CreatePracticeTest();

foreach(var practiceTestQuestion in practiceTest.Questions)
{
// ask the question and get the answer, AskAndGetAnswer() is your function

}

If you are using this in a web application, the practice test is likely destroyed before the answers are provided. For this reason, QuestionPool.CreatePracticeTest() has an overload which takes an enumerable of question ids, which are generated from the PracticeTestQuestion.QuestionId property.

[HttpPost]
{
// get the test and apply the answers
var questionPool = GetQuestionPool();
var practiceTest = pool.CreatePracticeTest(questionIds);

for (int qIndex = 0, numberOfQuestions = answers.Count; qIndex < numberOfQuestions; qIndex += 1)
{
if (answerId == string.Empty) continue; // they skipped this question

var question = test.Questions.ElementAt(questionIndex);
}

}

## The XML Format

<?xml version="1.0" encoding="utf-8"?>
<QuestionPool xmlns="http://www.rocketclubs.com/study/practicetesting/" randomizedTestOrder="false">
<Name>The Example Test</Name>
<Version>2012</Version>
<Organization>Rocket Clubs</Organization>
<OrganizationWebsite>http://www.rocketclubs.com/</OrganizationWebsite>
<Link url="http://www.original.com/test/source" text="Question Pool Source" toolTip="Links are for posting relevant content such as where the original question pool source" />
<Link url="http://www.original.com/diagrams/source" text="Diagrams Source" toolTip="Make sure to give credit to the original source for your diagrams as well" />
<PassingScore>26</PassingScore>
<Section name="G1A" numberOnTest="1">
<Description>Many question pools are divided into sections and take a set number of questions from each section.</Description>
<Question number="01">
<Prompt>What is the question number?</Prompt>
<Answer choice="A" correct="false">It's a number starting from 1 and going to int.MaxValue</Answer>
<Answer choice="C" correct="true">It's a string that is combined with the section name to make a unique identifier for this question</Answer>
</Question>
<Question number="02">
<Prompt>How does the uri in a diagram work?</Prompt>
<Diagram title="My Diagram" uri="my-diagram.png" source="http://www.original.com/diagrams/source" />
<Answer choice="A" correct="false">It is a file in at the end of the source attribute</Answer>
<Answer choice="B" correct="true">It's and identifier to help the programmer decide where it is</Answer>

• The XML namespace is http://www.rocketclubs.com/study/practicetesting/


text:
## Models and Applications

### Learning Outcomes

• Write a linear equation to express the relationship between unknown quantities.
• Write a linear equation that models two different cell phone packages.
• Use a linear model to answer questions.
• Set up a linear equation involving distance, rate, and time.
• Find the dimensions of a rectangle given the area.
• Find the dimensions of a box given information about its side lengths.

Josh is hoping to get an A in his college algebra class. He has scores of 75, 82, 95, 91, and 94 on his first five tests. Only the final exam remains, and the maximum of points that can be earned is 100. Is it possible for Josh to end the course with an A? A simple linear equation will give Josh his answer.

College students taking an exam. Credit: Kevin Dooley

Many real-world applications can be modeled by linear equations. For example, a cell phone package may include a monthly service fee plus a charge per minute of talk-time; it costs a widget manufacturer a certain amount to produce x widgets per month plus monthly operating charges; a car rental company charges a daily fee plus an amount per mile driven. These are examples of applications we come across every day that are modeled by linear equations. In this section, we will set up and use linear equations to solve such problems.

## Using Formulas to Solve Problems

Many applications are solved using known formulas. The problem is stated, a formula is identified, the known quantities are substituted into the formula, the equation is solved for the unknown, and the problem’s question is answered. Typically, these problems involve two equations representing two trips, two investments, two areas, and so on. Examples of formulas include the area of a rectangular region, $A=LW$; the perimeter of a rectangle, $P=2L+2W$; and the volume of a rectangular solid, $V=LWH$. When there are two unknowns, we find a way to write one in terms of the other because we can solve for only one variable at a time.

### Example: Solving an Application Using a Formula

It takes Andrew 30 minutes to drive to work in the morning. He drives home using the same route, but it takes 10 minutes longer, and he averages 10 mi/h less than in the morning. How far does Andrew drive to work?

### Try It

On Saturday morning, it took Jennifer 3.6 hours to drive to her mother’s house for the weekend. On Sunday evening, due to heavy traffic, it took Jennifer 4 hours to return home. Her speed was 5 mi/h slower on Sunday than on Saturday. What was her speed on Sunday?

### Example: Solving a Perimeter Problem

The perimeter of a rectangular outdoor patio is $54$ ft. The length is $3$ ft. greater than the width. What are the dimensions of the patio?

### Try It

Find the dimensions of a rectangle given that the perimeter is $110$ cm. and the length is 1 cm. more than twice the width.

### Example: Solving an Area Problem

The perimeter of a tablet of graph paper is 48 in2. The length is $6$ in. more than the width. Find the area of the graph paper.

### Try It

A game room has a perimeter of 70 ft. The length is five more than twice the width. How many ft2 of new carpeting should be ordered?

### Example: Solving a Volume Problem

Find the dimensions of a shipping box given that the length is twice the width, the height is $8$ inches, and the volume is 1,600 in.3.

## Key Concepts

• A linear equation can be used to solve for an unknown in a number problem.
• Applications can be written as mathematical problems by identifying known quantities and assigning a variable to unknown quantities.
• There are many known formulas that can be used to solve applications. Distance problems, for example, are solved using the $d=rt$ formula.
• Many geometry problems are solved using the perimeter formula $P=2L+2W$, the area formula $A=LW$, or the volume formula $V=LWH$.

## Glossary

area
in square units, the area formula used in this section is used to find the area of any two-dimensional rectangular region: $A=LW$
perimeter
in linear units, the perimeter formula is used to find the linear measurement, or outside length and width, around a two-dimensional regular object; for a rectangle: $P=2L+2W$
volume
in cubic units, the volume measurement includes length, width, and depth: $V=LWH$


text:
We are inviting IDEALS users, both people looking for materials in IDEALS and those who want to deposit their work, to give us feedback on improving this service through an interview. Participants will receive a $20 VISA gift card. Please sign up via https://forms.illinois.edu/sec/4069811 # Browse Dept. of Computer Science by Title • (2013-08-22) This thesis presents an investigation of the Collection of Parts Model for object categorization. Multiclass categorization is performed using the Collections of Parts model. Results using Support Vector Machines, L1 ... application/pdf PDF (934kB) • (2017-07-11) Object localization algorithms aim at finding out what objects exist in an image and where each object is. Object localization is fundamental to many computer vision problems. Simply knowing what is in the image is not ... application/pdf PDF (93MB) • (2005) Modelling by example has arisen as a powerful paradigm for reducing the artistic skill required for computer graphics. Instead of relying on the user's own modelling skills, a system that models by example allows users to ... application/pdf PDF (7MB) • (2005-05) Modelling by example has arisen as a powerful paradigm for reducing the artistic skill required for computer graphics. Instead of relying on the user's own modelling skills, a system that models by example allows users to ... application/pdf PDF (79MB) • (2010-08-20) As the web evolves, increasing quantities of structured information is embedded in web pages in disparate formats. For example, a digital camera’s description may include its price and megapixels whereas a professor’s ... application/pdf PDF (567kB) • (1995) This thesis defines an extensible transaction model and shows that an extensible and customizable transaction system based on the model can be built using an object-oriented framework. application/pdf PDF (5MB) • (1992) File systems are essential components of general purpose operating systems and are highly visible to computer users. To satisfy the growing diversity of persistent storage needs presented by application and system programs, ... application/pdf PDF (7MB) • (1992) Protection is the mechanism employed by operating systems to control access to resources. Object encapsulation in object-based systems requires control of access to every object. The incremental definition of objects through ... application/pdf PDF (6MB) • (1991) This thesis describes an experiment to use object-oriented programming and design techniques to design and implement an operating system. This experiment uses object-oriented techniques to address problems of operating ... application/pdf PDF (8MB) • (1991) The large size and inherent complexity of modern software products have made it difficult to use strictly manual techniques for the management of software production. In this thesis, object-oriented techniques are used to ... application/pdf PDF (5MB) • (2011-04) application/pdf PDF (867kB) • (2015-12-09) 3D Tele-immersion (3DTI) technology allows full-body, multi-modal content delivery among geographically dispersed users. In 3DTI, user’s 3D model will be captured by multiple RGB-D (color plus depth) cameras surround- ing ... application/pdf PDF (996kB) • (2015) 3D video is a new form of video featuring omni-view and immersive properties that are beneficial to many applications like telediagnosis, remote physiotherapy and e-learning. In these applications, the data are always ... application/pdf PDF (345kB) • (2012-04-25) This paper addresses the problem of collection and delivery of a representative subset of pictures, in participatory camera networks, to maximize coverage when a significant portion of the pictures may be redundant or ... application/pdf PDF (504kB) • (2004-04) In this paper, we examine the problem of large-volume data dissemination via overlay networks. A natural way to maximize the throughput of an overlay multicast session is to split the traffic and feed them into multiple ... application/pdf PDF (953kB) • (2018-08-23) Graph cut algorithms are an important tool for solving optimization problems in a variety of areas in computer science. Of particular importance is the min$s$-$t\$ cut problem and an efficient (polynomial time) algorithm ...

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• (2005-12)
The images of an object may look very different under different illumination conditions or viewing directions. This thesis considers the problem of clustering images of various objects into disjoint subsets according to ...

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• (2005)
When the lighting and pose variation occurs simultaneously, we do not have a reasonably good clustering algorithm yet. A few candidate algorithms are tested and the result are given in this thesis. Finding an effective ...

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• (2006-11)
Robots must complete their tasks in spite of unreliable actuators and limited, noisy sensing. In this paper, we consider the information requirements of such tasks. What sensing and actuation abilities are needed to complete ...

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This paper describes a completely automated method for generating efficient and competitive interpreters from formal semantics expressed in Rewriting Logic. The semantics are compiled into OCaml code, which then acts as ...

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text:
## Wednesday, 8 October 2014

### Reality of three-point structure constants in CFT, unitary or not

The reality of three-point structure constants in unitary CFT is a crucial ingredient of the numerical bootstrap in more than two dimensions. But what are the precise meaning and the proof of this property? Surely not all operators can have real three-point structure constants, since rescaling an operator by a complex scalar destroys this property. And the proof is not necessarily obvious, because the definition of unitarity as the existence of a positive definite scalar product is not directly related to three-point structure constants.

Fortunately I have received some explanations on these questions from Slava Rychkov. So here is what I understood from his arguments on unitary CFT, plus speculations of my own on non-unitary CFT.

A general proof

First of all, some notations. Consider a QFT with a space of states $S$, together with a sesquilinear form (i.e. a scalar product) which we do not yet assume to be positive definite. Let us denote states as $|i\rangle \in S$ and the scalar product as $\langle i| j\rangle$.  The Hermitian conjugate of an operator $O$ is by definition the operator $O^\dagger$ such that
$\langle i| O^\dagger |k\rangle = \langle k | O |i\rangle^*\ ,$
where the star denotes complex conjugation. In particular the identity is self-conjugate i.e. Hermitian. Finally we denote the state-operator correspondence as $|i\rangle \mapsto O_i$, and the Hermitian conjugate of such an operator as $O_i^\dagger = O_{i^\dagger}$.

If in addition our QFT is a CFT, then three-point functions are determined up to scalar factors by conformal symmetry. These scalar factors are called three-point structure constants, and denoted
$C_{ijk} = \langle O_i O_j O_k \rangle\ .$
We want to know how these structure constants behave under Hermitian conjugation of the operators. To do this, we use the state-operator correspondence and rewrite
$\langle O_i^\dagger O_j O_k \rangle = \langle i|O_j|k\rangle\ .$
This identity is particularly obvious if we assume that there exists a vacuum state such that $|k\rangle = O_k|0\rangle$ and $\langle \cdots \rangle = \langle 0|\cdots |0\rangle$, but this assumption is not really necessary.
From the three equations displayed so far, and from the symmetry of $C_{ijk}$ under permutations, we deduce
$C^*_{ijk} = C_{i^\dagger j^\dagger k^\dagger}\ ,$
and in particular a structure constant involving three Hermitian operators must be real.

So, if we want real structure constants, it is enough to consider Hermitian operators. But is there a Hermitian basis of operators? The answer is yes: given an operator $O$, either $O^\dagger = \lambda O$ in which case a rescaling makes it Hermitian, or $O$ and $O^\dagger$ are linearly independent in which case they can be replaced with the Hermitian combinations $O+O^\dagger$ and $i(O-O^\dagger)$.

The result seems too strong! Where's the mistake?

At this point the reader may protest that the argument does not rely on unitarity at all, and we seem to have proved that three-point structure constants are real in any CFT. This is actually true, but not as strong as it appears. The point is that Hermitian operators need not have positive two-point functions, except in unitary CFTs. Given a Hermitian operator $O_i$, we indeed have
$\langle O_i O_i \rangle = \langle O_i^\dagger O_i \rangle = \langle i | i \rangle \ ,$
which is always real, and positive in a unitary CFT. In a non-unitary CFT, insisting on the fundamental assumption that two-point functions are positive (or even normalized to one) can make structure constants imaginary. So we can tentatively formulate the result as
In any CFT, there are Hermitian bases of operators, and the corresponding three-point structure constants are real. In a unitary CFT, moreover, the two-point functions are positive.

The case of Minimal Models

Two-dimensional CFT provides a wealth of fully solved CFTs, unitary or not. So this is a good place to look for (counter)examples. Let us consider the A-series Virasoro Minimal Models (reviewed here). The simplest non-unitary model is the model of the Lee-Yang singularity $M_{5,2}$. In this model there is only one primary field $\phi$ which is not the identity, and the structure constant $C_{\phi\phi\phi}$ is purely imaginary. This suggests that this field is anti-Hermitian. Considering the Hermitian field $i\phi$ instead, we would have real structure constants, but a negative two-point function $\langle (i\phi)(i\phi) \rangle$.

More generally, whenever three-point functions are not real in Minimal Models, this is because some primary fields are anti-Hermitian. These fields can be renormalized and made Hermitian, but their two-point functions then become negative. So the example of Minimal Models supports our general claims.

Interestingly, non-unitary Minimal Models always have primary fields with negative conformal dimensions. The presence of negative dimensions is a sufficient condition for non-unitarity, and I wonder under which assumptions it is also necessary. (It cannot be necessary in general, cf the $H_3^+$ model.)

Numerical bootstrap for non-unitary theories?

Adapting the numerical bootstrap to non-unitary theories may not be very difficult. We can work with Hermitian operators, in order to have real structure constants. In particular, we can assume all primary operators to be Hermitian. The question is whether the two-point functions are positive or negative. There, a binary choice has to be made for each primary operator. (The choice can become more complicated if there are several primary operators with the same conformal dimension.) The case of Minimal Models suggests that the space of non-unitary CFTs may not be too much larger than the space of unitary CFTs.