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 Frequency of Debris Disks around Solar-Type Stars: First Results from a Spitzer MIPS SurveyWe have searched for infrared excesses around a well-defined sample of69 FGK main-sequence field stars. These stars were selected withoutregard to their age, metallicity, or any previous detection of IRexcess; they have a median age of ~4 Gyr. We have detected 70 μmexcesses around seven stars at the 3 σ confidence level. Thisextra emission is produced by cool material (<100 K) located beyond10 AU, well outside the habitable zones'' of these systems andconsistent with the presence of Kuiper Belt analogs with ~100 times moreemitting surface area than in our own planetary system. Only one star,HD 69830, shows excess emission at 24 μm, corresponding to dust withtemperatures >~300 K located inside of 1 AU. While debris disks withLdust/L*>=10-3 are rare around oldFGK stars, we find that the disk frequency increases from 2%+/-2% forLdust/L*>=10-4 to 12%+/-5% forLdust/L*>=10-5. This trend in thedisk luminosity distribution is consistent with the estimated dust inour solar system being within an order of magnitude greater or less thanthe typical level around similar nearby stars. Although there is nocorrelation of IR excess with metallicity or spectral type, there is aweak correlation with stellar age, with stars younger than a gigayearmore likely to have excess emission. Detection Limits from the McDonald Observatory Planet Search ProgramBased on the long-term radial velocity surveys carried out with theMcDonald Observatory 2.7 m Harlan J. Smith Telescope from 1988 to thepresent, we derive upper limits to long-period giant planet companionsfor 31 nearby stars. Data from three phases of the McDonald Observatory2.7 m planet-search program have been merged together, and for 17objects data from the pioneering Canada-France-Hawaii Telescope radialvelocity program have also been included in the companion-limitsdetermination. For those 17 objects, the baseline of observations is inexcess of 23 yr, enabling the detection or exclusion of giant planets inorbits beyond 8 AU. We also consider the possibility of eccentric orbitsin our computations. At an orbital separation of 5.2 AU, we can excludeon average planets of Msini>~(2.0+/-1.1)MJ (e=0) andMsini>~(4.0+/-2.8)MJ (e=0.6) for 25 of the 31 stars inthis survey. However, we are not yet able to rule out true Jupiters,''i.e., planets of Msini~1MJ in 5.2 AU orbits. These limits areof interest for the Space Interferometry Mission, Terrestrial PlanetFinder, and Darwin missions, which will search for terrestrial planetsorbiting nearby stars, many of which are included in this work. Dwarfs in the Local RegionWe present lithium, carbon, and oxygen abundance data for a sample ofnearby dwarfs-a total of 216 stars-including samples within 15 pc of theSun, as well as a sample of local close giant planet (CGP) hosts (55stars) and comparison stars. The spectroscopic data for this work have aresolution of R~60,000, a signal-to-noise ratio >150, and spectralcoverage from 475 to 685 nm. We have redetermined parameters and derivedadditional abundances (Z>10) for the CGP host and comparison samples.From our abundances for elements with Z>6 we determine the meanabundance of all elements in the CGP hosts to range from 0.1 to 0.2 dexhigher than nonhosts. However, when relative abundances ([x/Fe]) areconsidered we detect no differences in the samples. We find nodifference in the lithium contents of the hosts versus the nonhosts. Theplanet hosts appear to be the metal-rich extension of local regionabundances, and overall trends in the abundances are dominated byGalactic chemical evolution. A consideration of the kinematics of thesample shows that the planet hosts are spread through velocity space;they are not exclusively stars of the thin disk. A Comparative Study on Lithium Abundances in Solar-Type Stars With and Without PlanetsWe have investigated the abundance anomalies of lithium for stars withplanets in the temperature range of 5600-5900 K reported by Israelianand coworkers, as compared to 20 normal stars in the same temperatureand metallicity ranges. Our result indicates a higher probability oflithium depletion for stars with planets in the main-sequence stage. Itseems that stellar photospheric abundances of lithium in stars withplanets may be somewhat affected by the presence of planets. Twopossible mechanisms are considered to account for the lower Liabundances of stars with planets. One is related to the rotation-inducedmixing due to the conservation of angular momentum by the protoplanetarydisk, and the other is a shear instability triggered by planetmigration. These results provide new information on stellar evolutionand the lithium evolution of the Galaxy. Lithium Abundances of F-, G-, and K-Type Stars: Profile-Fitting Analysis of the Li I 6708 DoubletAn extensive profile-fitting analysis was performed for the Li(+Fe)6707-6708Å feature of nearby 160 F-K dwarfs/subgiants (including27 planet-host stars) in the Galactic disk ( 7000 K ≳Teff ≳ 5000 K, -1 ≲ [Fe/H] ≲ +0.4), in orderto establish the photospheric lithium abundances of these stars. Thenon-LTE effect (though quantitatively insignificant) was taken intoaccount based on our statistical equilibrium calculations, which werecarried out on an adequate grid of models. Our results confirmed most ofthe interesting observational characteristics revealed by recentlypublished studies, such as the bimodal distribution of the Li abundancesfor stars at Teff ≳ 6000 K, the satisfactory agreementof the upper envelope of the A(Li) vs. [Fe/H] distribution with thetheoretical models, the existence of a positive correlation betweenA(Li) and the stellar mass, and the tendency of lower lithium abundancesof planet-host stars (as compared to stars without planets) at thenarrow transition'' region of 5900 K ≳ Teff ≳5800 K. The solar Li abundance derived from this analysis is 0.92 (H =12.00), which is by 0.24dex lower than the widely referenced standardvalue of 1.16. Spectroscopic Study on the Atmospheric Parameters of Nearby F--K Dwarfs and SubgiantsBased on a collection of high-dispersion spectra obtained at OkayamaAstrophysical Observatory, the atmospheric parameters (Teff,log g, vt, and [Fe/H]) of 160 mid-F through early-K starswere extensively determined by the spectroscopic method using theequivalent widths of Fe I and Fe II lines along with the numericaltechnique of Takeda et al. (2002, PASJ, 54, 451). The results arecomprehensively discussed and compared with the parameter values derivedby different approaches (e.g., photometric colors, theoreticalevolutionary tracks, Hipparcos parallaxes, etc.) as well as with thepublished values found in various literature. It has been confirmed thatour purely spectroscopic approach yields fairly reliable and consistentresults. Spectroscopic Properties of Cool Stars (SPOCS). I. 1040 F, G, and K Dwarfs from Keck, Lick, and AAT Planet Search ProgramsWe present a uniform catalog of stellar properties for 1040 nearby F, G,and K stars that have been observed by the Keck, Lick, and AAT planetsearch programs. Fitting observed echelle spectra with synthetic spectrayielded effective temperature, surface gravity, metallicity, projectedrotational velocity, and abundances of the elements Na, Si, Ti, Fe, andNi, for every star in the catalog. Combining V-band photometry andHipparcos parallaxes with a bolometric correction based on thespectroscopic results yielded stellar luminosity, radius, and mass.Interpolating Yonsei-Yale isochrones to the luminosity, effectivetemperature, metallicity, and α-element enhancement of each staryielded a theoretical mass, radius, gravity, and age range for moststars in the catalog. Automated tools provide uniform results and makeanalysis of such a large sample practical. Our analysis method differsfrom traditional abundance analyses in that we fit the observed spectrumdirectly, rather than trying to match equivalent widths, and wedetermine effective temperature and surface gravity from the spectrumitself, rather than adopting values based on measured photometry orparallax. As part of our analysis, we determined a new relationshipbetween macroturbulence and effective temperature on the main sequence.Detailed error analysis revealed small systematic offsets with respectto the Sun and spurious abundance trends as a function of effectivetemperature that would be inobvious in smaller samples. We attempted toremove these errors by applying empirical corrections, achieving aprecision per spectrum of 44 K in effective temperature, 0.03 dex inmetallicity, 0.06 dex in the logarithm of gravity, and 0.5 kms-1 in projected rotational velocity. Comparisons withprevious studies show only small discrepancies. Our spectroscopicallydetermined masses have a median fractional precision of 15%, but theyare systematically 10% higher than masses obtained by interpolatingisochrones. Our spectroscopic radii have a median fractional precisionof 3%. Our ages from isochrones have a precision that variesdramatically with location in the Hertzsprung-Russell diagram. We planto extend the catalog by applying our automated analysis technique toother large stellar samples. Single-Visit Photometric and Obscurational CompletenessWe report a method that uses completeness'' to estimate the number ofextrasolar planets discovered by an observing program with adirect-imaging instrument. We develop a completeness function forEarth-like planets on habitable'' orbits for an instrument with acentral field obscuration, uniform sensitivity in an annular detectionzone, and limiting sensitivity that is expressed as a deltamagnitude'' with respect to the star, determined by systematic effects(given adequate exposure time). We demonstrate our method of estimationby applying it to our understanding of the coronagraphic version of theTerrestrial Planet Finder (TPF-C) mission as of 2004 October. Weestablish an initial relationship between the size, quality, andstability of the instrument's optics and its ability to meet missionscience requirements. We provide options for increasing the fidelity andversatility of the models on which our method is based, and we discusshow the method could be extended to model the TPF-C mission as a wholeto verify that its design can meet the science requirements. The Planet-Metallicity CorrelationWe have recently carried out spectral synthesis modeling to determineTeff, logg, vsini, and [Fe/H] for 1040 FGK-type stars on theKeck, Lick, and Anglo-Australian Telescope planet search programs. Thisis the first time that a single, uniform spectroscopic analysis has beenmade for every star on a large Doppler planet search survey. We identifya subset of 850 stars that have Doppler observations sufficient todetect uniformly all planets with radial velocity semiamplitudes K>30m s-1 and orbital periods shorter than 4 yr. From this subsetof stars, we determine that fewer than 3% of stars with-0.5<[Fe/H]<0.0 have Doppler-detected planets. Above solarmetallicity, there is a smooth and rapid rise in the fraction of starswith planets. At [Fe/H]>+0.3 dex, 25% of observed stars have detectedgas giant planets. A power-law fit to these data relates the formationprobability for gas giant planets to the square of the number of metalatoms. High stellar metallicity also appears to be correlated with thepresence of multiple-planet systems and with the total detected planetmass. This data set was examined to better understand the origin of highmetallicity in stars with planets. None of the expected fossilsignatures of accretion are observed in stars with planets relative tothe general sample: (1) metallicity does not appear to increase as themass of the convective envelopes decreases, (2) subgiants with planetsdo not show dilution of metallicity, (3) no abundance variations for Na,Si, Ti, or Ni are found as a function of condensation temperature, and(4) no correlations between metallicity and orbital period oreccentricity could be identified. We conclude that stars with extrasolarplanets do not have an accretion signature that distinguishes them fromother stars; more likely, they are simply born in higher metallicitymolecular clouds.Based on observations obtained at Lick and Keck Observatories, operatedby the University of California, and the Anglo-Australian Observatories. Stars within 15 Parsecs: Abundances for a Northern SampleWe present an abundance analysis for stars within 15 pc of the Sunlocated north of -30° declination. We have limited our abundancesample to absolute magnitudes brighter than +7.5 and have eliminatedseveral A stars in the local vicinity. Our final analysis list numbers114 stars. Unlike Allende Prieto et al. in their consideration of a verysimilar sample, we have enforced strict spectroscopic criteria in thedetermination of atmospheric parameters. Nevertheless, our results arevery similar to theirs. We determine the mean metallicity of the localregion to be <[Fe/H]>=-0.07 using all stars and -0.04 when interlopersfrom the thick disk are eliminated. Abundance trends in kinematical groups of the Milky Way's diskWe have compiled a large catalogue of metallicities and abundance ratiosfrom the literature in order to investigate abundance trends of severalalpha and iron peak elements in the thin disk and the thick disk of theGalaxy. The catalogue includes 743 stars with abundances of Fe, O, Mg,Ca, Ti, Si, Na, Ni and Al in the metallicity range -1.30 < [Fe/H]< +0.50. We have checked that systematic differences betweenabundances measured in the different studies were lower than randomerrors before combining them. Accurate distances and proper motions fromHipparcos and radial velocities from several sources have been retreivedfor 639 stars and their velocities (U, V, W) and galactic orbits havebeen computed. Ages of 322 stars have been estimated with a Bayesianmethod of isochrone fitting. Two samples kinematically representative ofthe thin and thick disks have been selected, taking into account theHercules stream which is intermediate in kinematics, but with a probabledynamical origin. Our results show that the two disks are chemicallywell separated, they overlap greatly in metallicity and both showparallel decreasing alpha elements with increasing metallicity, in theinterval -0.80 < [Fe/H] < -0.30. The Mg enhancement with respectto Fe of the thick disk is measured to be 0.14 dex. An even largerenhancement is observed for Al. The thick disk is clearly older than thethin disk with tentative evidence of an AMR over 2-3 Gyr and a hiatus instar formation before the formation of the thin disk. We do not observea vertical gradient in the metallicity of the thick disk. The Herculesstream has properties similar to that of the thin disk, with a widerrange of metallicity. Metal-rich stars assigned to the thick disk andsuper-metal-rich stars assigned to the thin disk appear as outliers inall their properties. CHARM2: An updated Catalog of High Angular Resolution MeasurementsWe present an update of the Catalog of High Angular ResolutionMeasurements (CHARM, Richichi & Percheron \cite{CHARM}, A&A,386, 492), which includes results available until July 2004. CHARM2 is acompilation of direct measurements by high angular resolution methods,as well as indirect estimates of stellar diameters. Its main goal is toprovide a reference list of sources which can be used for calibrationand verification observations with long-baseline optical and near-IRinterferometers. Single and binary stars are included, as are complexobjects from circumstellar shells to extragalactic sources. The presentupdate provides an increase of almost a factor of two over the previousedition. Additionally, it includes several corrections and improvements,as well as a cross-check with the valuable public release observationsof the ESO Very Large Telescope Interferometer (VLTI). A total of 8231entries for 3238 unique sources are now present in CHARM2. Thisrepresents an increase of a factor of 3.4 and 2.0, respectively, overthe contents of the previous version of CHARM.The catalog is only available in electronic form at the CDS viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/431/773 Isochrone ages for field dwarfs: method and application to the age-metallicity relation A new method is presented to compute age estimates from theoreticalisochrones using temperature, luminosity and metallicity data forindividual stars. Based on Bayesian probability theory, this methodavoids the systematic biases affecting simpler strategies and providesreliable estimates of the age probability distribution function forlate-type dwarfs. Basic assumptions concerning the a priori parameterdistribution suitable for the solar neighbourhood are combined with thelikelihood assigned to the observed data to yield the complete posteriorage probability. This method is especially relevant for G dwarfs in the3-15 Gyr range of ages, crucial to the study of the chemical anddynamical history of the Galaxy. In many cases, it yields markedlydifferent results from the traditional approach of reading the derivedage from the isochrone nearest to the data point. We show that thestrongest process affecting the traditional approach is that of stronglyfavouring computed ages near the end-of-main-sequence lifetime. TheBayesian method compensates for this potential bias and generallyassigns much higher probabilities to lower main-sequence ages, comparedwith short-lived evolved stages. This has a strong influence on anyapplication to galactic studies, especially given the presentuncertainties on the absolute temperature scale of the stellar evolutionmodels. In particular, the known mismatch between the model predictionsand the observations for moderately metal-poor dwarfs (-1 < [Fe/H]< -0.3) has a dramatic effect on the traditional age determination.We apply our method to the classic sample of Edvardsson et al., whoderived the age-metallicity relation (AMR) of 189 field dwarfs withprecisely determined abundances. We show how much of the observedscatter in the AMR is caused by the interplay between the systematicbiases affecting the traditional age determination, the colour mismatchwith the evolution models and the presence of undetected binaries. Usingnew parallax, temperature and metallicity data, our age determinationfor the same sample indicates that the intrinsic dispersion in the AMRis at most 0.15 dex and probably lower. In particular, we show that old,metal-rich objects ([Fe/H]~ 0.0 dex, age > 5 Gyr) and young,metal-poor objects ([Fe/H] < -0.5 dex, age < 6 Gyr) in manyobserved AMR plots are artefacts caused by too simple a treatment of theage determination. The incompatibility of those AMR plots with awell-mixed interstellar medium may therefore only be apparent.Incidentally, our results tend to restore confidence in the method ofage determination from the chromospheric activity for field dwarfs. Determination of fundamental characteristics for stars of the F, G, and K spectral types. The surface gravities and metallicity parameters.Not Available The Atmospheric Lithium Abundances of Solar AnaloguesWe have analyzed the lithium abundance in the atmospheres of 20 starsthat are solar analogues based on high-resolution echelle spectra usingmodel atmospheres in a non-LTE approach. In terms of their lithiumabundances, the stars (which are located in a narrow range oftemperatures of 5650 5900 K) can be divided into two groups: one withlow lithium abundances, as in the solar atmosphere, and one with lithiumabundances that are higher than the solar value by about 1 dex (with theaccuracy of the lithium abundances being 0.15 dex). Chemical Composition of 15 Photometric Analogues of the SunThe results of a spectroscopic analysis of 15 stars that are photometricanalogues of the Sun are reported. The effective temperatures andsurface gravities in the stellar atmospheres are derived from publishedphotometric indices and the HIPPARCOS parallaxes. The abundances of 33elements ranging from lithium to europium are analyzed based onhigh-dispersion spectra taken with the new Coudé echellespectrometer of the Terskol Observatory in the northern Caucasus. Themain parameters of most of the stars agree with the data of an [Fe/H]catalog published in 2001. Our study of the chemical compositions of thesample stars indicates that photometric analogues of the Sun can bedivided into three groups according to their elemental abundances: sixstars have solar chemical composition, four have abundance excesses, andfive have some abundance deficiencies. The sample contains twometal-deficient subgiants (HD 133002 and HD 225239). Our resultsdemonstrate that photometric similarity is not a sufficient criterion toconsider a star as solar analogue. When several criteria, includingchemical composition, are simultaneously taken into account, only fourstars from the sample can be considered true solar analogues: HD 10307,HD 34411, HD 146233 (18 Sco), and HD 186427 (16 Cyg B). These resultsconfirm the previously published suggestion that 18 Sco is the mostprobable twin of the Sun: essentially all the parameters of the twostars coincide within the errors. Chromospheric Ca II Emission in Nearby F, G, K, and M StarsWe present chromospheric Ca II H and K activity measurements, rotationperiods, and ages for ~1200 F, G, K, and M type main-sequence stars from~18,000 archival spectra taken at Keck and Lick Observatories as a partof the California and Carnegie Planet Search Project. We have calibratedour chromospheric S-values against the Mount Wilson chromosphericactivity data. From these measurements we have calculated medianactivity levels and derived R'HK, stellar ages,and rotation periods from general parameterizations for 1228 stars,~1000 of which have no previously published S-values. We also presentprecise time series of activity measurements for these stars.Based on observations obtained at Lick Observatory, which is operated bythe University of California, and on observations obtained at the W. M.Keck Observatory, which is operated jointly by the University ofCalifornia and the California Institute of Technology. The KeckObservatory was made possible by the generous financial support of theW. M. Keck Foundation. The Indo-US Library of Coudé Feed Stellar SpectraWe have obtained spectra for 1273 stars using the 0.9 m coudéfeed telescope at Kitt Peak National Observatory. This telescope feedsthe coudé spectrograph of the 2.1 m telescope. The spectra havebeen obtained with the no. 5 camera of the coudé spectrograph anda Loral 3K×1K CCD. Two gratings have been used to provide spectralcoverage from 3460 to 9464 Å, at a resolution of ~1 Å FWHMand at an original dispersion of 0.44 Å pixel-1. For885 stars we have complete spectra over the entire 3460 to 9464 Åwavelength region (neglecting small gaps of less than 50 Å), andpartial spectral coverage for the remaining stars. The 1273 stars havebeen selected to provide broad coverage of the atmospheric parametersTeff, logg, and [Fe/H], as well as spectral type. The goal ofthe project is to provide a comprehensive library of stellar spectra foruse in the automated classification of stellar and galaxy spectra and ingalaxy population synthesis. In this paper we discuss thecharacteristics of the spectral library, viz., details of theobservations, data reduction procedures, and selection of stars. We alsopresent a few illustrations of the quality and information available inthe spectra. The first version of the complete spectral library is nowpublicly available from the National Optical Astronomy Observatory(NOAO) via ftp and http. The Rise of the s-Process in the GalaxyFrom newly obtained high-resolution, high signal-to-noise ratio spectrathe abundances of the elements La and Eu have been determined over thestellar metallicity range -3<[Fe/H]<+0.3 in 159 giant and dwarfstars. Lanthanum is predominantly made by the s-process in the solarsystem, while Eu owes most of its solar system abundance to ther-process. The changing ratio of these elements in stars over a widemetallicity range traces the changing contributions of these twoprocesses to the Galactic abundance mix. Large s-process abundances canbe the result of mass transfer from very evolved stars, so to identifythese cases we also report carbon abundances in our metal-poor stars.Results indicate that the s-process may be active as early as[Fe/H]=-2.6, although we also find that some stars as metal-rich as[Fe/H]=-1 show no strong indication of s-process enrichment. There is asignificant spread in the level of s-process enrichment even at solarmetallicity. Synthetic Lick Indices and Detection of α-enhanced Stars. II. F, G, and K Stars in the -1.0 < [Fe/H] < +0.50 RangeWe present an analysis of 402 F, G, and K solar neighborhood stars, withaccurate estimates of [Fe/H] in the range -1.0 to +0.5 dex, aimed at thedetection of α-enhanced stars and at the investigation of theirkinematical properties. The analysis is based on the comparison of 571sets of spectral indices in the Lick/IDS system, coming from fourdifferent observational data sets, with synthetic indices computed withsolar-scaled abundances and with α-element enhancement. We useselected combinations of indices to single out α-enhanced starswithout requiring previous knowledge of their main atmosphericparameters. By applying this approach to the total data set, we obtain alist of 60 bona fide α-enhanced stars and of 146 stars withsolar-scaled abundances. The properties of the detected α-enhancedand solar-scaled abundance stars with respect to their [Fe/H] values andkinematics are presented. A clear kinematic distinction betweensolar-scaled and α-enhanced stars was found, although a one-to-onecorrespondence to thin disk'' and thick disk'' components cannot besupported with the present data. Stellar Chemical Signatures and Hierarchical Galaxy FormationTo compare the chemistries of stars in the Milky Way dwarf spheroidal(dSph) satellite galaxies with stars in the Galaxy, we have compiled alarge sample of Galactic stellar abundances from the literature. Whenkinematic information is available, we have assigned the stars tostandard Galactic components through Bayesian classification based onGaussian velocity ellipsoids. As found in previous studies, the[α/Fe] ratios of most stars in the dSph galaxies are generallylower than similar metallicity Galactic stars in this extended sample.Our kinematically selected stars confirm this for the Galactic halo,thin-disk, and thick-disk components. There is marginal overlap in thelow [α/Fe] ratios between dSph stars and Galactic halo stars onextreme retrograde orbits (V<-420 km s-1), but this is notsupported by other element ratios. Other element ratios compared in thispaper include r- and s-process abundances, where we find a significantoffset in the [Y/Fe] ratios, which results in a large overabundance in[Ba/Y] in most dSph stars compared with Galactic stars. Thus, thechemical signatures of most of the dSph stars are distinct from thestars in each of the kinematic components of the Galaxy. This resultrules out continuous merging of low-mass galaxies similar to these dSphsatellites during the formation of the Galaxy. However, we do not ruleout very early merging of low-mass dwarf galaxies, since up to one-halfof the most metal-poor stars ([Fe/H]<=-1.8) have chemistries that arein fair agreement with Galactic halo stars. We also do not rule outmerging with higher mass galaxies, although we note that the LMC and theremnants of the Sgr dwarf galaxy are also chemically distinct from themajority of the Galactic halo stars. Formation of the Galaxy's thickdisk by heating of an old thin disk during a merger is also not ruledout; however, the Galaxy's thick disk itself cannot be comprised of theremnants from a low-mass (dSph) dwarf galaxy, nor of a high-mass dwarfgalaxy like the LMC or Sgr, because of differences in chemistry.The new and independent environments offered by the dSph galaxies alsoallow us to examine fundamental assumptions related to thenucleosynthesis of the elements. The metal-poor stars ([Fe/H]<=-1.8)in the dSph galaxies appear to have lower [Ca/Fe] and [Ti/Fe] than[Mg/Fe] ratios, unlike similar metallicity stars in the Galaxy.Predictions from the α-process (α-rich freeze-out) would beconsistent with this result if there have been a lack of hypernovae indSph galaxies. The α-process could also be responsible for thevery low Y abundances in the metal-poor stars in dSph's; since [La/Eu](and possibly [Ba/Eu]) are consistent with pure r-process results, thelow [Y/Eu] suggests a separate r-process site for this light(first-peak) r-process element. We also discuss SNe II rates and yieldsas other alternatives, however. In stars with higher metallicities([Fe/H]>=-1.8), contributions from the s-process are expected; [(Y,La, and Ba)/Eu] all rise as expected, and yet [Ba/Y] is still muchhigher in the dSph stars than similar metallicity Galactic stars. Thisresult is consistent with s-process contributions from lower metallicityAGB stars in dSph galaxies, and is in good agreement with the slowerchemical evolution expected in the low-mass dSph galaxies relative tothe Galaxy, such that the build-up of metals occurs over much longertimescales. Future investigations of nucleosynthetic constraints (aswell as galaxy formation and evolution) will require an examination ofmany stars within individual dwarf galaxies.Finally, the Na-Ni trend reported in 1997 by Nissen & Schuster isconfirmed in Galactic halo stars, but we discuss this in terms of thegeneral nucleosynthesis of neutron-rich elements. We do not confirm thatthe Na-Ni trend is related to the accretion of dSph galaxies in theGalactic halo. Empirically Constrained Color-Temperature Relations. II. uvbyA new grid of theoretical color indices for the Strömgren uvbyphotometric system has been derived from MARCS model atmospheres and SSGsynthetic spectra for cool dwarf and giant stars having-3.0<=[Fe/H]<=+0.5 and 3000<=Teff<=8000 K. Atwarmer temperatures (i.e., 8000-2.0. To overcome thisproblem, the theoretical indices at intermediate and high metallicitieshave been corrected using a set of color calibrations based on fieldstars having well-determined distances from Hipparcos, accurateTeff estimates from the infrared flux method, andspectroscopic [Fe/H] values. In contrast with Paper I, star clustersplayed only a minor role in this analysis in that they provided asupplementary constraint on the color corrections for cool dwarf starswith Teff<=5500 K. They were mainly used to test thecolor-Teff relations and, encouragingly, isochrones thatemploy the transformations derived in this study are able to reproducethe observed CMDs (involving u-v, v-b, and b-y colors) for a number ofopen and globular clusters (including M67, the Hyades, and 47 Tuc)rather well. Moreover, our interpretations of such data are verysimilar, if not identical, with those given in Paper I from aconsideration of BV(RI)C observations for the sameclusters-which provides a compelling argument in support of thecolor-Teff relations that are reported in both studies. Inthe present investigation, we have also analyzed the observedStrömgren photometry for the classic Population II subdwarfs,compared our final'' (b-y)-Teff relationship with thosederived empirically in a number of recent studies and examined in somedetail the dependence of the m1 index on [Fe/H].Based, in part, on observations made with the Nordic Optical Telescope,operated jointly on the island of La Palma by Denmark, Finland, Iceland,Norway, and Sweden, in the Spanish Observatorio del Roque de losMuchachos of the Instituto de Astrofisica de Canarias.Based, in part, on observations obtained with the Danish 1.54 mtelescope at the European Southern Observatory, La Silla, Chile. A new Böhm-Vitense gap in the temperature range 5560 to 5610 K in the main sequence hm-Vitense gap in the main sequenceHighly precise temperatures (σ = 10-15 K) have been determinedfrom line depth ratios for a set of 248 F-K field dwarfs of about solarmetallicity (-0.5 < [Fe/H] < +0.4), based on high resolution (R=42000), high S/N echelle spectra. A new gap has been discovered in thedistribution of stars on the Main Sequence in the temperature range 5560to 5610 K. This gap coincides with a jump in the microturbulent velocityVt and the well-known Li depression near 5600 K in fielddwarfs and open clusters. As the principal cause of the observeddiscontinuities in stellar properties we propose the penetration of theconvective zone into the inner layers of stars slightly less massivethan the Sun and related to it, a change in the temperature gradient.Based on spectra collected with the ELODIE spectrograph at the 1.93-mtelescope of the Observatoire de Haute-Provence (France).Full Table 1 is only available in electronic form athttp://www.edpsciences.org The [Zn/Fe] - [Fe/H] trend for disk and halo starsZn abundances, derived from a model atmosphere analysis of theλ6362.35 Å Zn I line, are presented for 44 thin disk, 10thick disk and 8 halo dwarf stars in the metallicity range -1.0 <[Fe/H] < +0.2. It is found that [Zn/Fe] in thin disk stars shows aslight increasing trend with decreasing metallicity reaching a value[Zn/Fe] ≃ +0.1 at [Fe/H] = -0.6. The thick disk stars in themetallicity range -0.9 < [Fe/H] < -0.6 have an average [Zn/Fe]≃ +0.15 dex, whereas five alpha-poor and Ni-poor halo stars in thesame metallicity range have [Zn/Fe] ≃ 0.0 dex. These resultsindicate that Zn is not an exact tracer of Fe as often assumed inabundance studies of damped Lyman-alpha systems (DLAs). A betterunderstanding of the nucleosynthesis of Zn is needed in order to obtainmore detailed information on the past history of star formation in DLAsfrom e.g. the observed sulphur/zinc ratio.Based on observations collected at the National AstronomicalObservatories, Xinglong, China and the European Southern Observatory, LaSilla, Chile (ESO No. 67.D-0106). Sodium abundances in nearby disk starsWe present sodium abundances for a sample of nearby stars. All resultshave been derived from NLTE statistical equilibrium calculations. Theinfluence of collisional interactions with electrons and hydrogen atomsis evaluated by comparison of the solar spectrum with very precise fitsto the Na I line cores. The NLTE effects are more pronounced inmetal-poor stars since the statistical equilibrium is dominated bycollisions of which at least the electronic component is substantiallyreduced. The resulting influence on the determination of sodiumabundances is in a direction opposite to that found previously for Mgand Al. The NLTE corrections are about -0.1 in thick-disk stars with[Fe/H] -0.6. Our [Na/Fe] abundance ratios are about solar forthick- and thin-disk stars. The increase in [Na/Fe] as a function of[Fe/H] for metal-rich stars found by Edvardsson et al. (\cite{EAG93}) isconfirmed. Our results suggest that sodium yields increase with themetallicity, and quite large amounts of sodium may be produced by AGBstars. We find that [Na/Fe] ratios, together with either [Mg/Fe] ratio,kinematic data or stellar evolutionary ages, make possible theindividual discrimination between thin- and thick-disk membership.Based on observations collected at the Germany-Spanish AstronomicalCenter, Calar Alto, Spain.Tables \ref{table2} and \ref{table3} are only available in electronicform at http://www.edpsciences.org On the determination of oxygen abundances in chromospherically active starsWe discuss oxygen abundances derived from [O I] λ6300s and the OI triplet in stars spanning a wide range in chromospheric activitylevel, and show that these two indicators yield increasingly discrepantresults with higher chromospheric/coronal activity measures. While theforbidden and permitted lines give fairly consistent results forsolar-type disk dwarfs, spuriously high O I triplet abundances areobserved in young Hyades and Pleiades stars, as well as in individualcomponents of RS CVn binaries (up to 1.8 dex). The distinct behaviour ofthe [O I]-based abundances which consistently remain near-solar suggeststhat this phenomenon mostly results from large departures from LTEaffecting the O I triplet at high activity level that are currentlyunaccounted for, but also possibly from a failure to adequately modelthe atmospheres of K-type stars. These results suggest that some cautionshould be exercised when interpreting oxygen abundances in activebinaries or young open cluster stars.Based on observations collected at the European Southern Observatory,Chile (Proposals 64.L-0249 and 071.D-0260).Table \ref{tab_data} is only available in electronic form athttp://www.edpsciences.org Chemical enrichment and star formation in the Milky Way disk. III. Chemodynamical constraintsIn this paper, we investigate some chemokinematical properties of theMilky Way disk, by using a sample composed by 424 late-type dwarfs. Weshow that the velocity dispersion of a stellar group correlates with theage of this group, according to a law proportional to t0.26,where t is the age of the stellar group. The temporal evolution of thevertex deviation is considered in detail. It is shown that the vertexdeviation does not seem to depend strongly on the age of the stellargroup. Previous studies in the literature seem to not have found it dueto the use of statistical ages for stellar groups, rather thanindividual ages. The possibility to use the orbital parameters of a starto derive information about its birthplace is investigated, and we showthat the mean galactocentric radius is likely to be the most reliablestellar birthplace indicator. However, this information cannot bepresently used to derive radial evolutionary constraints, due to anintrinsic bias present in all samples constructed from nearby stars. Anextensive discussion of the secular and stochastic heating mechanismscommonly invoked to explain the age-velocity dispersion relation ispresented. We suggest that the age-velocity dispersion relation couldreflect the gradual decrease in the turbulent velocity dispersion fromwhich disk stars form, a suggestion originally made by Tinsley &Larson (\cite{tinsley}, ApJ, 221, 554) and supported by several morerecent disk evolution calculations. A test to distinguish between thetwo types of models using high-redshift galaxies is proposed.Full Table 1 is 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/423/517 S4N: A spectroscopic survey of stars in the solar neighborhood. The Nearest 15 pcWe report the results of a high-resolution spectroscopic survey of allthe stars more luminous than M_V = 6.5 mag within 14.5 pc from the Sun.The Hipparcos catalog's completeness limits guarantee that our survey iscomprehensive and free from some of the selection effects in othersamples of nearby stars. The resulting spectroscopic database, which wehave made publicly available, includes spectra for 118 stars obtainedwith a resolving power of R ≃ 50 000, continuous spectral coveragebetween  362-921 nm, and typical signal-to-noise ratios in therange 150-600. We derive stellar parameters and perform a preliminaryabundance and kinematic analysis of the F-G-K stars in the sample. Theinferred metallicity ([Fe/H]) distribution is centered at about -0.1dex, and shows a standard deviation of 0.2 dex. A comparison with largersamples of Hipparcos stars, some of which have been part of previousabundance studies, suggests that our limited sample is representative ofa larger volume of the local thin disk. We identify a number ofmetal-rich K-type stars which appear to be very old, confirming theclaims for the existence of such stars in the solar neighborhood. Withatmospheric effective temperatures and gravities derived independentlyof the spectra, we find that our classical LTE model-atmosphere analysisof metal-rich (and mainly K-type) stars provides discrepant abundancesfrom neutral and ionized lines of several metals. This ionizationimbalance could be a sign of departures from LTE or inhomogeneousstructure, which are ignored in the interpretation of the spectra.Alternatively, but seemingly unlikely, the mismatch could be explainedby systematic errors in the scale of effective temperatures. Based ontransitions of majority species, we discuss abundances of 16 chemicalelements. In agreement with earlier studies we find that the abundanceratios to iron of Si, Sc, Ti, Co, and Zn become smaller as the ironabundance increases until approaching the solar values, but the trendsreverse for higher iron abundances. At any given metallicity, stars witha low galactic rotational velocity tend to have high abundances of Mg,Si, Ca, Sc, Ti, Co, Zn, and Eu, but low abundances of Ba, Ce, and Nd.The Sun appears deficient by roughly 0.1 dex in O, Si, Ca, Sc, Ti, Y,Ce, Nd, and Eu, compared to its immediate neighbors with similar ironabundances.Based on observations made with the 2.7 m telescope at the McDonaldObservatory of the University of Texas at Austin (Texas), and the 1.52 mtelescope at the European Southern Observatory (La Silla, Chile) underthe agreement with the CNPq/Observatorio Nacional (Brazil).Tables 3-5 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/420/183 The Geneva-Copenhagen survey of the Solar neighbourhood. Ages, metallicities, and kinematic properties of 14 000 F and G dwarfsWe present and discuss new determinations of metallicity, rotation, age,kinematics, and Galactic orbits for a complete, magnitude-limited, andkinematically unbiased sample of 16 682 nearby F and G dwarf stars. Our63 000 new, accurate radial-velocity observations for nearly 13 500stars allow identification of most of the binary stars in the sampleand, together with published uvbyβ photometry, Hipparcosparallaxes, Tycho-2 proper motions, and a few earlier radial velocities,complete the kinematic information for 14 139 stars. These high-qualityvelocity data are supplemented by effective temperatures andmetallicities newly derived from recent and/or revised calibrations. Theremaining stars either lack Hipparcos data or have fast rotation. Amajor effort has been devoted to the determination of new isochrone agesfor all stars for which this is possible. Particular attention has beengiven to a realistic treatment of statistical biases and errorestimates, as standard techniques tend to underestimate these effectsand introduce spurious features in the age distributions. Our ages agreewell with those by Edvardsson et al. (\cite{edv93}), despite severalastrophysical and computational improvements since then. We demonstrate,however, how strong observational and theoretical biases cause thedistribution of the observed ages to be very different from that of thetrue age distribution of the sample. Among the many basic relations ofthe Galactic disk that can be reinvestigated from the data presentedhere, we revisit the metallicity distribution of the G dwarfs and theage-metallicity, age-velocity, and metallicity-velocity relations of theSolar neighbourhood. Our first results confirm the lack of metal-poor Gdwarfs relative to closed-box model predictions (the G dwarfproblem''), the existence of radial metallicity gradients in the disk,the small change in mean metallicity of the thin disk since itsformation and the substantial scatter in metallicity at all ages, andthe continuing kinematic heating of the thin disk with an efficiencyconsistent with that expected for a combination of spiral arms and giantmolecular clouds. Distinct features in the distribution of the Vcomponent of the space motion are extended in age and metallicity,corresponding to the effects of stochastic spiral waves rather thanclassical moving groups, and may complicate the identification ofthick-disk stars from kinematic criteria. More advanced analyses of thisrich material will require careful simulations of the selection criteriafor the sample and the distribution of observational errors.Based on observations made with the Danish 1.5-m telescope at ESO, LaSilla, Chile, and with the Swiss 1-m telescope at Observatoire deHaute-Provence, France.Complete Tables 1 and 2 are only available in electronic form at the CDSvia anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/418/989 On the correlation of elemental abundances with kinematics among galactic disk starsWe have performed the detailed analysis of 174 high-resolution spectraof FGK dwarfs obtained with the ELODIE echelle spectrograph at theObservatoire de Haute-Provence. Abundances of Fe, Si and Ni have beendetermined from equivalent widths under LTE approximation, whereasabundances of Mg have been determined under NLTE approximation usingequivalent widths of 4 lines and profiles of 5 lines. Spatial velocitieswith an accuracy better than 1 km s-1, as well as orbits,have been computed for all stars. They have been used to define 2subsamples kinematically representative of the thin disk and the thickdisk in order to highlight their respective properties. A transitionoccurs at [Fe/H] =-0.3. Stars more metal-rich than this value have aflat distribution with Zmax;<1 kpc and σW<20 km s-1, and a narrow distribution of [α/Fe].There exist stars in this metallicity regime which cannot belong to thethin disk because of their excentric orbits, neither to the thick diskbecause of their low scale height. Several thin disk stars areidentified down to [Fe/H] =-0.80. Their Mg enrichment is lower thanthick disk stars with the same metallicity. We confirm from a largersample the results of Feltzing et al. (\cite{felt03}) and Bensby et al.(\cite{ben03}) showing a decrease of [α/Fe] with [Fe/H] in thethick disk interpreted as the signature of the SNIa which haveprogressively enriched the ISM with iron. However our data suggest thatthe star formation in the thick disk stopped when the enrichment was[Fe/H] =-0.30, [Mg/Fe] =+0.20, [Si/Fe] =+0.17. A vertical gradient in[α/Fe] may exist in the thick disk but should be confirmed with alarger sample. Finally we have identified 2 new candidates of the HR1614moving group.Based on spectra collected with the ELODIE spectrograph at the 1.93-mtelescope of the Observatoire de Haute Provence (France).Tables 3 and 8 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/418/551
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 Constellation: Auriga Right ascension: 05h19m08.50s Declination: +40Â°05'57.0" Apparent magnitude: 4.71 Distance: 12.645 parsecs Proper motion RA: 0 Proper motion Dec: 0 B-T magnitude: 5.438 V-T magnitude: 4.761

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# Propositional Logic CNF

gefragt 2017-08-06 13:25:56 +0200

How can I compute this in a faster way?

d<->a<->b

I want to know if I can directly apply any formulas here to reduce it to CNF. Right now it takes me much time to compute it.

bearbeiten retag schließen löschen

## 3 Antworten

geantwortet 2017-08-07 09:55:39 +0200

The first answer of contradictioned is great. Thanks for answering that! However, there was a small typo. The first two connectives in the the right most formula should be switched. So it should be $(\neg x \wedge \neg y) \vee (x \wedge y)$. (which has been fixed now)

For the given example $d \leftrightarrow a \leftrightarrow b$ , I want to point out that connectives are left associative (slide 6 in the VRS-Chapter on propositional logic).

Your formula is hence equivalent to $(d \leftrightarrow a) \leftrightarrow b$. If you look at the inner subformula, it can be written as $(\neg d \vee a) \wedge(d \vee \neg a)$ or $(d \wedge a) \vee (\neg d \wedge \neg a)$. Thus, you could transform the formula like this:

• $d \leftrightarrow a \leftrightarrow b =$
• $(d \leftrightarrow a) \leftrightarrow b =$
• $(\neg (d \leftrightarrow a) \vee b) \wedge ((d \leftrightarrow a) \vee \neg b) =$
• $(\neg (d \leftrightarrow a) \vee b) \wedge (((\neg d \vee a) \wedge (d \vee \neg a)) \vee \neg b) =$
• $(\neg (d \leftrightarrow a) \vee b) \wedge (\neg d \vee a \vee \neg b) \wedge (d \vee \neg a \vee \neg b) =$
• $(\neg ((d \wedge a) \vee (\neg d \wedge \neg a)) \vee b) \wedge (\neg d \vee a \vee \neg b) \wedge (d \vee \neg a \vee \neg b) =$
• $(((\neg d \vee \neg a) \wedge (d \vee a)) \vee b) \wedge (\neg d \vee a \vee \neg b) \wedge (d \vee \neg a \vee \neg b) =$
• $(\neg d \vee \neg a \vee b) \wedge (d \vee a \vee b) \wedge (\neg d \vee a \vee \neg b) \wedge (d \vee \neg a \vee \neg b)$
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## Kommentare

1

Thanks for spotting :)

( 2017-08-07 10:42:52 +0200 )bearbeiten
1

Thanks for the explanation!

( 2017-08-07 11:44:28 +0200 )bearbeiten

if you think, the answer is right, you can click the checkmark below the voting tool next to the answer. This marks the answer as correct for the other users.

( 2017-08-07 11:46:21 +0200 )bearbeiten

geantwortet 2017-08-09 10:49:10 +0200

The answers so far are great and explain to really compute the CNF. As you were asking actually for a faster way to do that, here are two:

Fast Way #1:

If you just need the result, then you may remember the parity function: Given n variables x1,...,xn the parity function is the boolean function associated with the formula x1⊕...⊕xn and it is true if and only if an odd number of the variables is true. Next, observe that (a⊕b) is equivalent to ¬(a↔b), so that (d↔a)↔b is equivalent to (d⊕a)⊕b. Hence, the formula is true if and only if an odd number of variables is true, i.e., either exactly one or all three.

Hence, its DNF is obtained by listing these assignments:

a∧b∧d ∨ a∧¬b∧¬d ∨ ¬a∧b∧¬d ∨ ¬a∧¬b∧d,

and the DNF of its negation is obtained by listing the other assignments (where none or exactly two variables are true):

a∧b∧¬d ∨ a∧¬b∧d ∨ ¬a∧b∧d ∨ ¬a∧¬b∧¬d

The CNF of (d↔a)↔b can now be obtained by negating the latter, which clearly yields

(¬a∨¬b∨d)  ∧ (¬a∨b∨¬d) ∧ (a∨¬b∨¬d) ∧ (a∨b∨d)

Fast Way #2:

BDDs can also compute CNFs and DNFs by simply enumerating all paths from the root node to the 1 or 0 leaves, respectively. Hence, you may first transform the formula into a BDD, i.e., if-then-else form which is then a=>(d↔b)|¬(d↔b) by making case distinctions on a, and then a=>(b=>d|¬d)|(b=>¬d|d) by a further case distinction on b. From here you can read the DNF a∧b∧d ∨ a∧¬b∧¬d ∨ ¬a∧b∧¬d ∨ ¬a∧¬b∧d and proceed as above.

mehr

geantwortet 2017-08-07 09:44:16 +0200

If you are free to choose a procedure to do that, I would probably (up to four variables) just write down a truth-table of the formula and read the CNF from that. That is in my opinion the least error prone way.

If you are asked to transform it by applying formulas, then why not use iteratively

$$x \leftrightarrow y \equiv (\neg x \lor y) \land (x \lor \neg y) \equiv (\neg x \land \neg y) \lor (x \land y)$$

until you have all other operators eliminated and then bring it into CNF? Again, using many simple steps--in my opinion--reduces the amount of mistakes. However, you have to practice these small steps ;)

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## Statistik

Gefragt: 2017-08-06 13:25:56 +0200

Gesehen: 662 mal

Letztes Update: Aug 09 '17

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# Imagifractalous! 4: Fractal Binary Trees II

Now that the paper Nick and I wrote on binary trees was accepted for Bridges 2017 (yay!), I’d like to say a little more about what we discovered.  I’ll presume you’ve already read the first Imagifractalous! post on binary trees (see Day077 for a refresher if you need it).

Recall that in that post, I discussed creating binary trees with branching ratios which were 1 or larger.  Below are three examples of binary trees, with branching ratios less that 1, equal to 1, and larger than 1, respectively.

It was Nick’s insight to consider the following question:  how are trees with branching ratio r related to those with branching ratio 1/r?  He had done a lot of exploring with graphics in Python, and observed that there was definitely some relationship.

Let’s look at an example.  The red tree is a binary tree with branching ratio r less than one, and the gray tree has a branching ratio which is the reciprocal r.  Both are drawn to the same depth.

Of course you notice what’s happening — the leaves of the trees are overlapping!  This was happening so frequently, it just couldn’t be coincidence.  Here is another example.

Notice how three copies of the trees with branching ratio less than one are covering some of the leaves of a tree with the reciprocal ratio.

Now if you’ve ever created your own binary trees, you’ll likely have noticed that I left out a particularly important piece of information:  the size of the trunks of the trees.  You can imagine that if the sizes of the trunks of the r trees and the 1/r trees were not precisely related, you wouldn’t have the nice overlap.

Here is a figure taken from our paper which explains just how to find the correct relationship between the trunk sizes.  It illustrates the main idea which we used to rigorously prove just about everything we observed about these reciprocal trees.

Let’s take a look at what’s happening.  The thick, black tree has a branching ratio of 5/8, and a branching angle of 25°.  The thick, black path going from O to P is created by following the sequence of instructions RRRLL (and so the tree is rendered to a depth of 5).

Now make a symmetric path (thick, gray, dashed) starting at P and going to O.  If we start at P with the same trunk length we started with at O, and follow the exact same instructions, we have to end up back at O.

The trick is to now look at this gray path backwards, starting from O.  The branches now get larger each time, by a factor of 8/5 (since they were getting smaller by a factor of 5/8 when going in the opposite direction).  The size of the trunk, you can readily see, is the length of the last branch drawn in following the black path from O to P.  This must be (5/8)5 times the length of the trunk, since the tree is of depth 5.

The sequence of instructions needed to follow this gray path is RRLLL.  It turns out this is easy to predict from the geometry.  Recall that beginning at P, we followed the instructions RRRLL along the gray path to get to O.  When we reverse this path and go from O to P, we follow the instructions in reverse — except that in going in the reverse direction, what was previously a left turn becomes a right turn, and vice versa.

So all we need to do to get the reverse instructions is to reverse the string RRRLL to get LLRRR, and then change the L‘s to R‘s and the R‘s to L‘s, yielding RRLLL.

There’s one important detail to address:  the fact that the black tree with branching ratio 5/8 is rotated by 25° to make everything work out.  Again, this is easy to see from the geometry of the figure.  Look at the thick gray path for a moment.  Since following the instructions RRLLL means that in total, you make one more left turn than you do right turns, the last branch of the path must be oriented 25° to the left of your starting orientation (which was vertical).  This tells you precisely how much you need to rotate the black tree to make the two paths have the same starting and ending points.

Of course one example does not make a proof — but in fact all the important ideas are contained in this one illustration.  It is not difficult to make the argument more general, and we have successfully accomplished that (though this blog is not the place for it!).

If you look carefully at the diagram, you’ll count that there are exactly 10 leaves in common with these two trees with reciprocal branching ratios.  There is some nice combinatorics going on here, which is again easy to explain from the geometry.

You can see that these common leaves (illustrated with small, black dots) are at the ends of gray branches which are oriented 25° from the vertical.  Recall that this specific angle came from the fact that there was one more L than there were R‘s in the string RRLLL.

Now if you have a sequence of 5 instructions, the only way to have exactly one more L than R‘s is to have precisely three L‘s (and hence two R‘s).  And the number of ways to have three L‘s in a string of length 5 is just

$\displaystyle{5\choose3}=10.$

Again, these observations are easy to generalize and prove rigorously.

And where does this take us?

On the right are 12 copies of a tree with a braching ratio of r less than one and a branching angle of 30°, and on the left are 12 copies of a tree with a reciprocal branching ratio of 1/r, also with a branching angle of 30°.  All are drawn to depth 4, and the trunks are appropriately scaled as previously discussed.

These sets of trees produce exactly the same leaves!  We called this the Dual Tree Theorem, which was the culmination of all these observations.  Here is an illustration with both sets of trees on top of each other.

As intriguing as this discovery was, it was only the beginning of a much broader and deeper exploration into the fractal world of binary trees.  I’ll continue a discussion of our adventures in the next installment of Imagifractalous!


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# Function Rules for Input-Output Tables

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Practice Function Rules for Input-Output Tables
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Writing Function Rules

Have you ever eaten a churro? Take a look at this dilemma.

You are going to order a bunch of churros from a vender for your family. Each one costs $1.50. How much will they cost if you buy 6 or 8 or 10? What’s the function rule? This Concept will show you how to write a function rule for a situation like this one. ### Guidance Do you remember how to identify a function, a relation, a range and a domain? A function is a relation in which each member of the domain is paired with exactly one member of the range. In other words, a number in the domain cannot have two values for the range. When we look at the values in the domain and the range, we can figure out if the relation is a function or not. A set of ordered pairs is a relation. The values in the domain and range help us to understand a relation. The domain is made up of the values in first column or the $x$ coordinate in the relation. The range is made up of the second column or the $y$ value of the relation. One of the great things about functions is that they can be applied to all kinds of situations. Just remember that in order for a relation to be a function, that the values of the domain need to be assigned to only one value of the range. One way of thinking about functions is through the use of function tables. A function table is an input/output table where the input is the domain and the output is the range. There are function rules that go with function tables. Do you know what a function rule is? A function rule can be written in words or in the form of an equation. The function rule tells you what operation or operations to perform with the input to get the output. We can also use input/output tables to help us to write function rules. When we look at the input and decipher what happened to it to create the output, then we can write a function rule based on our discoveries. This is like being a detective! You will have to use what you have learned and look for clues. Take a look. $x$ $f(x)$ 0 5 3 8 6 11 9 14 The first thing to notice is that the words input and output have been replaced by the $x$ and the $f(x)$ . This means that we are using function notation to say that $x$ is the input and that the output is a function of $x$ . That is what the $f(x)$ means. That is exactly what you need to do. In looking at this pattern, you can see that the $x$ value is increased in each step of the table. Each $x$ value is increased by 5. This means that we can write the following rule for our function. $f(x) = x + 5$ This is our answer. Write down an example of function notation and that you need to look for a pattern when figuring out function rules. Put this information in your notebook. Write a function rule for each example. #### Example A $x$ $f(x)$ 9 8 11 10 15 14 17 16 Solution: $x-1$ #### Example B $x$ $f(x)$ 3 7 9 19 10 21 12 25 Solution: $2x+1$ #### Example C $x$ $f(x)$ 9 $-27$ 11 $-33$ 15 $-45$ 16 $-48$ Solution: $-3x$ Now let's go back to the dilemma at the beginning of the Concept. First, we can take the given information to write the rule. $p(c)=1.50c$ where $p$ is total price and $c$ is the number of churros. Next, we can we can substitute different values into the function rule to figure out the cost for 6, 8 or 10 churros. $& p(c) =1.50c && p(c)=1.50c && p(c) =1.50c\\& p(6) =1.50 \cdot 6 && p(8)=1.50 \cdot 8 && p(10)=1.50 \cdot 10\\& p(6) =9 && p(8) =12 && p(10)=15\\& \ 9 \text{ for } 6 \text{ churrors} && \ 12 \text{ for } 8 \text{ churros} && \ 15 \text{ for } 10 \text{ churros}$ Based on the number of churros, we can figure out the differences in cost. This is our answer. ### Guided Practice Here is one for you to try on your own. Use function notation to write a function rule for the given table. $x$ $f(x)$ 12 6 9 4.5 7 3.5 4 2 Solution Let's start by looking for a pattern. Do you notice one? Each value of the input has been divided in half. We can write this function rule in two ways. $f(x)&= \frac{1}{2}x \\or \ f(x)&= \frac{x}{2}$ Both of these will work as a correct answer. ### Video Review ### Explore More Directions: Write function rules. 1. Write a function rule for the following data. $x$ $f(x)$ 9 27 11 33 15 45 16 48 1. Write a function rule for the following data: $x$ $f(x)$ $-2$ -6 0 $-4$ 2 $-2$ 4 0 1. Write a function rule for the following data: $x$ $f(x)$ 0 0 1 2 4 8 5 10 1. Write a function rule for the following table. $x$ $f(x)$ 1 0 2 2 4 6 8 14 1. Write a function rule for the following table. $x$ $f(x)$ 2 1 4 2 8 4 10 5 18 9 1. Write a function rule for each table. $x$ $f(x)$ 6 2 9 3 15 5 21 7 30 10 1. Write a function rule for each table. $x$ $f(x)$ 2 3 9 10 15 16 21 22 30 31 1. Write a function rule for each table. $x$ $f(x)$ 3 $-6$ 9 $-18$ 15 $-30$ 20 $-40$ 24 $-48$ Directions: Solve the following problem. 1. Sandwich cost$3.45 each. Write a function rule for the cost, $c$ , for a number of sandwiches, $s$ .
2. Now, find the cost of 3 sandwiches.
3. Find the cost of 6 sandwiches.
4. Find the cost of 9 sandwiches.
5. Find the cost of 2 sandwiches.
6. Find the cost of 8 sandwiches.
7. Find the cost of a dozen sandwiches.

### Vocabulary Language: English

domain

domain

The domain of a function is the set of $x$-values for which the function is defined.
Function

Function

A function is a relation where there is only one output for every input. In other words, for every value of $x$, there is only one value for $y$.
Function Rule

Function Rule

A function rule describes how to convert an input value ($x$) into an output value ($y$) for a given function. An example of a function rule is $f(x) = x^2 + 3$.
input

input

The input of a function is the value on which the function is performed (commonly the $x$ value).
Output

Output

The output of a function is the result of the operations performed on the independent variable (commonly $x$). The output values are commonly the values of $y$ or $f(x)$.
Range

Range

The range of a function is the set of $y$ values for which the function is defined.
Relation

Relation

A relation is any set of ordered pairs $(x, y)$. A relation can have more than one output for a given input.


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# Proof for copula determined by correlation matrix

How can I proof that the copula of an elliptical distribution $$El(\mu, \sigma^2, g_n)$$ is fully determined by the generator function $$g_n$$ and the correlation matrix extracted from $$\sigma^2$$.

• Best of my knowledge, the elliptical copula does not determine by the generator function. A good book to read about Elliptical copula and how to generate it is: An introduction to copula by Nelsen 2006. – Maryam Nov 23 '18 at 8:37
• what about: $X\backsim El(\mu,\sigma^{2},g_{n})$ -> $X-\mu\backsim El(0,\sigma^{2},g_{n})$ This is a monotonic tranformation, therefore $X-\mu$ has the same copula therefore it doesn't depend on $\mu$? – Nickpick Nov 23 '18 at 10:28
• Could you please add the book or source for your question? Or could you please expand your question? – Maryam Nov 25 '18 at 5:12


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# Elegant representations of graphs in R^3

If I have a graph of a reasonable size (e.g. ~100 nodes, ~40 edges coming out of each node) and I want to represent it in R^3 (i.e. map each node to a point in R^3 and draw a straight line between any two nodes which are connected in the original graph) in a way which would make it easy to understand its structure, what do you think would make a good drawing criterion?

I know this question is ill-posed; it's not objective. The idea behind it is easier to understand with an extreme case. Suppose you have a connected graph in which each node connects to two and only two other nodes, except for two nodes which only connect to one other node. It's not difficult to see that this graph, when drawn in R^3, can be drawn as a straight line (with nodes sprinkled over the line). Nevertheless, it is possible to draw it in a way which makes it almost impossible to see its very simple structure, e.g. by "twisting" it as much as possible around some fixed point in R^3. So, for this simple case, it's clear that a simple 3D representation is that of a straight line. However, it is not clear what this simplicity property is in the general case.

So, the question is: how would you define this simplicity property?

I'm happy with any kind of answer, be it a definition of "simplicity" computable for graphs, or a greedy approximated algorithm which transforms graphs and that converges to "simpler" 3D representations.

Thanks!

EDITED

In the mean time I've put force-based graph drawing ideas into practice and wrote a computer program to simulate how imposing an electrical repulsive force between nodes (Coulomb's Law) and a spring-like behaviour on edges (Hooke's law) would turn out. I've posted the video on youtube. The video starts by randomly generating a graph of 100 nodes each with approximately 1-2 outgoing edges, placing the nodes randomly in 3D space. Then all the forces I mentioned are put into place and the system is left to move around subject to those forces. In the beginning, the graph is a mess and it's very difficult to see the structure. Closer to the end, it is clear that the graph is almost linear. I've also experience with larger-sized graphs but sometimes the geometry of the graph is just a mess and no matter how you plot it, you won't be able to visualise anything. And here is an even more extreme example with 500 nodes.

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One standard algorithm constructs such a "representation of the graph in $\mathbb{R}^3$" using the first 3 eigenvectors of a weighting of the graph's adjacency matrix. With this representation, the distance between vertices A and B is an approximate measure of the "number" of "short" random walks between A and B. For details see, eg., pnas.org/content/102/21/7426 by Coifman-Lafon-Lee-Maggioni-Nadler-Warner-Zucker. –  macbeth Feb 15 '11 at 3:37
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## 6 Answers

You might start by exploring the various tools that are available. For example, Mathematica's GraphPlot3D[] does a nice job with small graphs. Here is a 5-node graph:

And here is a somewhat dense 50-node graph:

Both were produced without giving Mathematica any advice. Like most such tools, it has options for applying different methods, e.g., SpringEmbedding.

Edit. As requested by Surikator, here are some random, more-sparse, 50-node graphs:

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This is interesting. I would be very interested to see what Mathematica draws if the 50 node graph you have there has only a couple of edges coming out of each node. (See my video link in the edited question.) Very dense graphs (many edges) will always be hard to visualise but some sparser graphs (fewer edges) are those which can gain most from a clever displacement of the nodes. Unfortunately, although it's great to see graphs drawn by Mathematica, we have no idea about the algorithm used to draw them. –  Surikator Feb 15 '11 at 23:20
@Surikator: Here are some quotes from the documentation: "Possible settings for Method include "SpiralEmbedding", "RandomEmbedding", "HighDimensionalEmbedding", "RadialDrawing", "SpringEmbedding" and "SpringElectricalEmbedding". With Method->Automatic, GraphPlot3D normally uses the "SpringElectricalEmbedding" method, though it uses the "RadialDrawing" method if the graph is a tree." My images were drawn by the default SpringElectricalEmbedding, which balances two forces, one attractive one repulsive, by minimizing an energy. It is defined precisely in the documentation. –  Joseph O'Rourke Feb 15 '11 at 23:55
@Joseph Nice! So, they're also using force-based graph drawing, possibly similarly to the wikipedia link I posted in the comment to Jim and to what I actually implemented and illustrated in the video. Good to know. Thanks! –  Surikator Feb 16 '11 at 0:05
I think your Wikipedia link is broken: en.wikipedia.org/wiki/… –  Joseph O'Rourke Feb 16 '11 at 0:20
@Joseph Yep, I've just corrected it. Thanks! The graphs are interesting to look at. Some are still quite intricate so it would be difficult to tell if they could be presented in a better way. Still, from the function name "SpringElectricalEmbedding", it's much clearer what's going on behind the scenes even if not in detail. –  Surikator Feb 16 '11 at 1:38
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There is a vast literature on Graph Drawing (mostly in two dimensions, but a lot of ideas can be extended to three). The first paper in the subject (almost exactly fifty years ago) was W. Tutte's "How to draw a graph", where the idea was that if you fixed some vertices to be the vertices of a convex polygon, you can make the other vertices satisfy the condition that every vertex is the barycenter of its neighbors. Most of that paper is devoted to a completely incomprehensible argument to show that such drawings are actual embeddings (so no pair of edges crosses) -- this has been reproved (or proved, depending on how cynical you are) by a number of people. That issue does not really arise in three dimensions, however. The Tutte embeddings (which have been generalized in various ways) are discrete harmonic maps.

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I'll mention this since no-one else has. Maple did it in the deprecated network package and now does it in the plots package as graphplot3d : Make the adjacency matrix of the graph. Then the graphplot3d command will do the following: find three eigenvectors (maybe of unit length) by default for eigenvalues 2,3,4 although you can choose. This give 3 vectors in $\mathbb{R}^n$ It will view them as $n$ triples in $\mathbb{R}^3$ and use these as the vertex locations. Sometimes it works nicely showing off symmetries, other times not so well. Some documentation

I am not praising Maple here rather the method. Below are 3 plots of random sparse 50 vertex graphs. I'd like thicker edges and cute balls at vertices. Also I had to fool around a bit to not have a number at each vertex.

The springs method is nice, and you can probably nudge and shake or pull and release to see what happens. At least in Maple the system of differential equations can start to really bog down for a big graph. I like that the representation by this eigenvalue method is completely determined by the graph. The results can be quite nice, when it works. I tried the Hoffman Singleton Graph (50 vertices, regular of degree 7) and none of the eigenvector choices I made came out with a nice graph.

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An idea that occurred to me, though I don't know how computationally feasible it is, is to insist all edges are of length one [Edit: this can't be done in general.], put a repelling charge on each vertex, and let the graph float to an equilibrium position. This would be somewhat analogous to the idea of minimizing "knot energy" in the case of embeddings of $S^1$ in $\mathbb R^3$.

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Insisting each edge has length one can be tricky. After you have drawn a lot of nodes, it will become difficult to choose the position of the next node in such a way that the distance to all the other nodes it connects to is one. I mean, how would you determine the initial position of the nodes before repelling forces come into place? Or am I missing something? –  Surikator Feb 15 '11 at 2:46
You can't require all edges to have the same length for an embedding into $\mathbb{R}^3$. There's no way to embed $K_5$, for example. –  Chris Eagle Feb 15 '11 at 2:53
I have also been pointed to en.wikipedia.org/wiki/… which follows the repelling charge and equilibrium positions you mention. –  Surikator Feb 15 '11 at 2:53
@Surikator: thanks for the link. I knew it was an obvious idea... –  Jim Conant Feb 15 '11 at 13:05
@Jim I've put these ideas into a program that simulates graph behaviour subject to repelling charge forces and spring-like behaviour. See my edit above. Note that if we only impose repelling charge forces, the graph will explode (literally). That's why you were trying to keep the edges at constant unit length. Although that is not possible, a nice approximation to that is to choose some random lengths and say that the lengths should not vary much from those initial lengths -- that can be achieved by giving them spring-like behaviour, i.e. a force preventing the edges from becoming too long. –  Surikator Feb 15 '11 at 23:26
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complexity(graph,f) := $\displaystyle\sum_{\langle g,r\rangle \in \; \operatorname{graph}^2} \begin{cases} 0 & \text{if }g=r \\\\ \operatorname{log}\left(\frac{d(f(g),f(r))}{d(g,r)}\right)^2 & \operatorname{if }g\neq r\end{cases}$

although I suppose this is more quality of embedding than simplicity ...

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Where does this definition come from? –  Igor Rivin Feb 15 '11 at 2:34
What is f and what is the definition of d (which I suppose is a distance)? –  Surikator Feb 15 '11 at 2:49
This definition comes from my having seen it somewhere (I forget where) as a measure for map projections. f is the embedding, d is the distance between either vertices or points in R^3, depending on what its inputs are. –  Ricky Demer Feb 15 '11 at 4:35
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A truncated icosahedron can be embedded in a plane, but is more symmetrical and easier to understand when embedded in a sphere. This goes for all the Platonic figures and their truncated, rectified, etc. derivatives.

That doesn't really answer the question "what do you think would make a good drawing criterion?" Ok, having a symmetrical embedding in R^3 makes a good drawing criterion.

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### variety-jones's blog

By variety-jones, 4 months ago,

cfstress.com

CF Stress is a highly customizable, no-code tool to stress test Codeforces problems.

You control all the parameters of the generator without writing a single line of code. Feeling lazy and want to pin down the error to an array of length 5? You can do that by adjusting the flags passed to the generator. Feeling adventurous and want to test your logic on test cases of length 1000? There's an option for that too. There's also a builtin diff checker for when you want to compare the output of large test cases. You can also seamlessly copy the input/output to your local machine.

Just plug in your submission ID and wait for the ticket to be processed.

It also has support for non-unique answers and interactive problems. The testcases are generated using testlib, the framework that's used to create the original test set.

Note: It only supports C++ and Python submissions for now. (For Java, I could not figure out how to make the name of the file the same as the class name).

A Sample Result

If this proves to be useful, I'll add more features, such as user accounts to view tickets in one place, emailing the user on successful completion of the ticket, better concurrency support to handle multiple tickets, revamped UI, etc.

If you have any feedback/suggestions, or if you'd like to collaborate, please let me know in the comments (or drop me an email).

• +126

 » 4 months ago, # |   +8 Website is down rn.
•  » » 4 months ago, # ^ |   0 Fixed.
•  » » » 4 months ago, # ^ |   +8 Down again :D
•  » » » » 4 months ago, # ^ |   0 Fixed again.
 » 4 months ago, # |   +11 It's not working for recent contests, like if I put 1631 E it does nothing
 » 4 months ago, # |   +13 It seems like it would be super cool to have this in general, but the amount of work it would require to accurately create generators for all problems on CF is insane and not really feasible to expand in general without a colossal amount of work being put into it. From my testing, it only works on a handful of rounds. For instance: https://codeforces.com/contest/1627/problem/E doesn't work in the generator at all, and this was the first test I did.
•  » » 4 months ago, # ^ |   +5 Yeah, fair enough. Although I should've mentioned in the blog that I don't really intend to support older problems, I've already created CF Debug (currently offline) for that which tells you the smallest testcase from the existing test-set on which your submission fails. However, it's not effective for problems with multiple test cases, the recent norm on Codeforces.It only supports contests that happened after it was launched (i,e 22nd January, 2022 onwards). The plan was (and is) to update the generator repo a day or two after the contest ends. Considering the frequency of CF contests, I don't think it'd require more than 1 hour commitment per contest, as I already have templates prepared for array, trees, graphs, etc. Creating a good generator that covers corner cases is tricky, but creating a naive one that is correct should not take much time, I believe. It's a stress test after all.Depending upon the usefulness, I also plan to make at least the generator repo open-source. In that case, all we need is just 1 participant contributing a generator per problem, and the benefits can be reaped by many.
 » 4 months ago, # |   +4 Update: Added Contest 768 and made some minor improvements to the site: Querying for Java submissions should now correctly display an error, rather than displaying the last stored test case. The status page should now display the last ticket number. If a counter example could not be found, the result would be displayed instantly upon completion, (previously it was only displayed once a valid ticket was processed). Upon failures, help message to increase the constraints are displayed.
 » 3 months ago, # |   +2 Update: Made some improvements to the site and added new features: The time limit for a single stress test has been reduced to 1 minute per submission (Earlier it was 3 minutes, which caused unnecessary delays if you accidentally enter an AC submission). A new feature of getting additional sample test cases for every problem has been added. The sample test case display page should now only display the Input and Expected Output (and not "Your output" and "Diff"). It also has new help messages for getting alternate counter examples.
 » 3 months ago, # |   +1 Why website is down :(
•  » » 3 months ago, # ^ |   +1 Will need to investigate, but fixed for now.
•  » » » 3 months ago, # ^ |   0 website down now.
•  » » » » 3 months ago, # ^ |   +1 Really? Seems to be working for me. (And the logs do not indicate any failure). Can you describe the exact issue that you faced?
 » 3 months ago, # | ← Rev. 2 →   0 Its working...
 » 3 months ago, # |   0 Tried changing parameters but did not get a test case on CF stress [1539D] which fails 146434582 Got wrong ans on test 8 for this.
•  » » 3 months ago, # ^ |   0 Changing $x$ to $x + 1$ is not what I meant by tuning the parameters. You need to be a bit more creative.This problem only contains a single array, and you've already deduced that choosing small $n$ and small $a[i]$ as flags produces no counter example. You could try choosing large $n$ and large $a[i]$, but you already have a failing test case on CF with these constraints. Plus you wont' be able to dry run it, even if you get the test case.Now, if you set the flags as large $n$ and small $a[i]$, you would still not be able to dry run it, even if you get a failing test case.So that leaves us with picking small $n$ and large $a[i]$ as flags. And sure enough, it gives a very simple counter example, as evident from Ticket 408. This produces a testcase of type 2 c b d a where $a < b < c < d$ with answer $2*a + c + (d - a)$
•  » » » 3 months ago, # ^ | ← Rev. 2 →   0 Thanks! I found a very small error and got AC which I wouldnt have found by myself imo.
 » 5 weeks ago, # |   0 Thanks a lot for creating this awesome website .
 » 2 weeks ago, # |   0 Update: I received some comments that the site was not able to retrieve submissions from Codeforces.The issue has now been resolved, thanks to woshiluo's PR for cf-tool.
 » 6 days ago, # |   0 Does this not work without logging in? And how to reset password if you don't remember?
•  » » 6 days ago, # ^ |   0 Yes, I've made some recent changes. I'm migrating the site to a freemium model.Currently, there's no way to reset a password. I'll add this feature soon.If you can DM me the email that you used for registering, I'll update the backend, and you can register again with the same email.


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# zbMATH — the first resource for mathematics

Complexity of an SIR epidemic dynamics model with impulsive vaccination control. (English) Zbl 1065.92050
Summary: This paper considered an infected class – removed class (IR) epidemic model with impulsive vaccination, which may inherently oscillate. We studied the impulsive control and got conditions by which the epidemics elimination solution is globally asymptotically stable and conditions for the boundedness of the system. On the other hand, if the epidemic turns out to be endemic, we studied numerically the influence of impulsive vaccination on the periodic oscillations of the system without impulsions and found phenomena of chaos in this case.

##### MSC:
 92D30 Epidemiology 34A37 Ordinary differential equations with impulses 93C15 Control/observation systems governed by ordinary differential equations 34C60 Qualitative investigation and simulation of ordinary differential equation models 37N25 Dynamical systems in biology
Full Text:
##### References:
 [1] Chen, L.; Chen, J., Nonlinear biological dynamic systems, (1993), Science Beijing, [in Chinese] [2] Liu, W.; Levin, S.A.; Lwasa, Y., Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. math. biol., 23, 187-204, (1986) · Zbl 0582.92023 [3] Liu, W.M.; Hethcote, H.W.; Levin, S.A., Dynamical behavior of epidemiological models with nonlinear incidence rates, J. math. biol., 25, 359-380, (1987) · Zbl 0621.92014 [4] Wang, H., Existence of Hopf bifurcation periodic solution to SIRS epidemiological models with nonlinear incidence rates, J anhui agric univ no. 2, 29, 199-202, (2002), [in Chinese] [5] Shulgin, B.; Stone, L.; Agur, Z., Pulse vaccination strategy in the SIR epidemic model, Bull math biol, 60, 1-26, (1998) · Zbl 0941.92026 [6] Zhen, J., Research on impulsive ecological and epidemical model (the doctor degree’s article), (2001), Xi’an Jiaotong Univ, [in Chinese] [7] d’Onofrio, A., Stability properties of pulse vaccination strategy in SEIR epidemic model, Math. bios., 179, 1, 57-72, (2002) · Zbl 0991.92025 [8] Stone, L.; Shulgin, B.; Agur, Z., Theoretical examination of the pulse vaccination policy in the sir epidemic model, Math. comput. model, 31, 4-5, 207-215, (2000) · Zbl 1043.92527 [9] Lu, Z.; Chi, X.; Chen, L., The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission, Math comput model, 36, 9-10, 1039-1057, (2002) · Zbl 1023.92026 [10] Liu, X.; Chen, L., Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator, Chaos, solitions fractals, 16, 311-320, (2003) · Zbl 1085.34529 [11] Tang, S.; Chen, L., Chaos in functional response host cparasitoid ecosystem models, Chaos, solitons fractals, 13, 4, 875-884, (2002) · Zbl 1022.92042 [12] Liu, B.; Zhang, Y.; Chen, L., Dynamic complexities of a Holling I predator-prey model concerning periodic biological and chemical control, Chaos, solitons fractals, 22, 1, 123-134, (2004) · Zbl 1058.92047 [13] Zhang, S.; Chen, L., Chaos in three species food chain system with impulsive perturbations, Chaos, solitons fractals, 24, 1, 73-83, (2005) · Zbl 1066.92060 [14] Zhang, S.; Dong, L.; Chen, L., The study of predator-prey system with defensive ability of prey and impulsive perturbations on the predator, Chaos, solitons fractals, 23, 2, 631-643, (2005) · Zbl 1081.34041 [15] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002 [16] Parker, T.S.; Chua, L.O., Practical numerical algorithms for chaotic systems[M], (1989), Springer New York · Zbl 0692.58001 [17] Davies, B., Exploring chaos, theory and experiment, (1999), Perseus Books Reading, MA · Zbl 0959.37001 [18] Wolf, A.; Swift, J.B.; Swinney, L.; Vastano, A., Determining Lyapunov exponents from a time series, Physica D, 16, 285-317, (1985) · Zbl 0585.58037 [19] May, R.M., Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos, Science, 186, 645-647, (1974) [20] May, R.M.; Oster, G.F., Bifurcations and dynamic complexity in simple ecological models, Am natur, 110, 573-599, (1976) [21] Grebogi, C.; Ott, E.; Yorke, Ja., Crises, sudden changes in chaotic attractors and chaotic transients, Physica D, 7, 181-200, (1983)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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# Spring Oscillation in fluid

1. Feb 19, 2016

### SirR3D

So there is this ball held on a spring. It's radius R = 0.015m , and density ρ=7800kg/m^3.
It's oscillation period in air is 1.256 seconds and in the fluid it changes to 1.57 seconds.
Find the viscosity of the fluid considering that the drag force obeys Stokes law.

I first found the spring constant k to be k = 2.76.
But how do I find the drag force without knowing the terminal velocity? Or is there a way of solving this otherwise.

Darn I posted this in the wrong section. May someone please move it to the coursework section?
Sorry for the inconvenience

Last edited: Feb 19, 2016
2. Feb 19, 2016

### drvrm

well you have found k of the spring meaning thereby that you have used a relation between time period and k; this relation must have come from some dynamical equation involving forces ......proceed in the same manner using viscous drag then you can go ahead....

3. Feb 19, 2016

### SirR3D

As I said, in the manner I'm trying to solve it I have too many unknowns. I get 6*pi*η*R*vterminal = mg - ρVg - kx
But I do not know η and vterminal
Is there any way to solve this without needing to find vterminal

4. Feb 19, 2016

### drvrm

the above expression i think is not correct as your body is oscillating-not moving with a terminal velocity only downward- the above does not represent the physical situation- try drawing the force diagram and write equation of motion

actually one can not go your way as oscillator velocity is not a terminal one as in stokes law- but the forces operating on the body must be written out and viscous drag which opposes the motion may be proportional to speed of the body so its a case of damped oscillator witha damping force due to viscous drag- consult a text book on it -damped oscillations and this may help

5. Feb 19, 2016

### Staff: Mentor

You're supposed to assume that the terminal velocity drag force formula applies to the ball at all velocities. This is the approximation that they expect you to make.

Chet

6. Feb 20, 2016

### drvrm

Well i think he should express the drag force as dependent upon the velocity of the body -that gives him a damping factor and that is increasing the time period the coefficient of velocity dependence can give him coefficient of viscosity if he can use the stokes law in expressing the viscous drag force but the new time period must be calculated
i agree that your formula is something in error; i feel the forces operating are -ky- b(dy/dt) where b is the proportionality constant for drag dependence ; then naturally your solution for oscillator equation which you should write will be modified and naturally new time period will appear there fore you move as the physics goes.

7. Feb 20, 2016

### Staff: Mentor

I didn't say how to solve the problem. I only said that the drag force should be calculated using the Stokes equation, even though the ball is not at terminal velocity during any part of the oscillation. But now that you're asking, I will show specifically how to solve it:

If x represents the upward displacement of the ball relative to the buoyant equilibrium position, then the force balance on the oscillating ball is:

$$\frac{4}{3}\pi\rho R^3\frac{d^2x}{dt^2}=-kx-6\pi\mu R\frac{dx}{dt}$$where $\mu$ is the fluid viscosity and $\rho$ is the density of the ball. This second order homogeneous ODE with constant coefficients can easily be solved for the period of the oscillation.

Last edited: Feb 20, 2016


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[2] However, null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty). For example, when considered as a subset of the real numbers, with its usual ordering, represented by the real number line, every real number is both an upper and lower bound for the empty set. ∅ Darling (2004) explains that the empty set is not nothing, but rather "the set of all triangles with four sides, the set of all numbers that are bigger than nine but smaller than eight, and the set of all opening moves in chess that involve a king. In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets.Connectedness is one of the principal topological properties that are used to distinguish topological spaces.. A subset of a topological space X is a connected set if it is a connected space when viewed as a subspace of X. Many possible properties of sets are vacuously true for the empty set. The closure of the empty set is empty. Earliest Uses of Symbols of Set Theory and Logic. The prime spectrum of any commutative ring with the Zariski topology is a compact space important in algebraic geometry . Moreover, the empty set is compact by the fact that every finite set is compact. WikiIsNotaDictionary See also: MultiSet, SetOfAllSets CategoryMath CategoryNull If X is empty, this is the (unique) empty function. 0 { {\displaystyle \varnothing } In some textbooks and popularizations, the empty set is referred to as the "null set". This is the smallest T 1 topology on any infinite set. { ) JavaScript is disabled. In fact, the Cantor set contains uncountably many points. • The empty set ∅ is compact, since we can always just take the empty sub-cover. ∅ { ∅ The reason for this is that zero is the identity element for addition. Lemma 3: If every rectangle is compact, then every closed and bounded subset of is compact. The sum of a compact convex set and a closed convex set is closed. Moreover, the empty set is a compact set by the fact that every finite set is compact. The following famous theorem, proved by Dieudonné in 1966, gives a sufficient condition for the difference of two closed convex subsets to be closed. This is known as "preservation of nullary unions." Then X is a compact topological space. ∅ { α There is, namely the cover consisting of the empty set (but for this you need that the empty set is open, which Hurkyl just explained.) The following lists document of some of the most notable properties related to the empty set. The empty set is not the same thing as nothing; rather, it is a set with nothing inside it and a set is always something. Exercise 1: If (X, ) is a topological space and , then (A, ) is also a topological space. 1 This empty topological space is the unique initial object in the category of topological spaces with continuous maps. { School Harvard University; Course Title MATH 55b; Uploaded By unclaimed-conjecture. Thus, we have Set Theory, Logic, Probability, Statistics, Study revealing the secret behind a key cellular process refutes biology textbooks, Irreversible hotter and drier climate over inner East Asia, Study of threatened desert tortoises offers new conservation strategy, Compact Sets of Metric Spaces Which Are Also Open. {\displaystyle -\infty \!\,,}