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News

# AVA Solar Completes $104 Million Equity Financing September 01, 2008 by Jeff Shepard AVA Solar, an advanced thin-film photovoltaic (PV) module manufacturer located in Fort Collins, Colorado, announced that it has completed its second institutional equity round of financing. This$104 million funding was led by DCM and included new investors Technology Partners, GLG Partners and Bohemian Companies, LLC as well as prior investors, including Invus, LP. The funds will be used to complete AVA Solar’s first production facility in Longmont, Colorado, which will have the capacity to produce 200MW of PV modules annually.

"I am very pleased to have closed this financing with such a prestigious group of investors," said Pascal Noronha, CEO of AVA Solar. "This funding will allow us to move quickly to establish our first manufacturing facility to produce cost-effective PV modules."

Tom Blaisdell, a general partner at DCM, said "In the last few years, we’ve seen many companies with plans to commercialize products for the solar industry. Our commitment to AVA Solar is founded both on the commercial readiness of the product and their management and engineering team’s demonstrated ability to deliver."

Ira Ehrenpreis, a general partner at Technology Partners, added that "AVA’s approach to manufacturing technology is uniquely positioned to meet the market’s need for large-scale installations at a cost that enables grid parity."


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# [Pw_forum] fermi level

Eyvaz Isaev eyvaz_isaev at yahoo.com
Thu Aug 14 20:35:46 CEST 2008

Dear Vega,

> How about the case for semi-conductors like TiO2? When
> calculations for TiO2 was achieved, I can't find any > information about Fermi energy in my output file.

Let me resend you my reply to almost the same question of Padmaja Patnaik. Please see below.

Bests,
Eyvaz.

--- On Thu, 7/8/08, Eyvaz Isaev <eyvaz_isaev at yahoo.com> wrote:
From: Eyvaz Isaev <eyvaz_isaev at yahoo.com>
Subject: Re: [Pw_forum] doubt in occupations:2
Date: Thursday, 7 August, 2008, 4:52 AM

I have repeated your calculations and do not see any problem with them.

First of all I have to note that THERE IS NO FERMI LEVEL FOR SEMICONDUCTORS AND INSULATORS. Why, please read a textbook, say Kittel, Ashcroft&Mermin, etc.
For them we can consider only the top of the valence bands and the bottom of the conductivity bands.

As the Fermi level is defined as \int_{e_min}^{e_F}n(E)dE=N, where e_F is the Fermi level, n(E) is the DOS and N is the number of valence electrons, it does not matter where you put the Fermi level in the band gap. Because the DOS for E>E_{top valence level} (but E<E_{bottom of conductivity levels}), i.e. inside the band gap, is zero.

So, even with the calculated Fermi level for semiconductor or insulator one can put it in the middle of the band gap (or everywhere you want!).

But the best choice is the top level of the valence bands which is well defined.

1. Scf calculations do not give the Fermi level, because you use fixed occupations used by default for semiconductors (insulators). I.e. if nothing specified as occupations, your system is considered as a semiconductor
(insulator). In this case the number of bands treated is exactly half of the number of valence electrons, i.e. 4 in your case.

So, you do it correct, but the Fermi level should not be expected in this case.

2. As concerns the second remark, now it should be clear that you have to shift all your bands with respect to the highest filled band, 9.2464eV (the 4th band
at the Gamma point) in your particular case.
In this case there are no states at "the Fermi level".

a) as you use norm-conserving pseudopotentials there is no need to take ecutrho >4*ecutwf, even skip this keyword.
b) Use more k-points for the DOS calculations.

Bests,
Eyvaz.

> Subject: Re: [Pw_forum] doubt in occupations:2
> To: eyvaz_isaev at yahoo.com
> Date: Tuesday, August 5, 2008, 6:16 PM
> Dear Sir
> Hope you remember our communication regarding the
> occupations and Fermi energy.
> I am here sending you the input and output file for > DOS plotting of SiC. Please have a look. The scf
> calculations doesn't give Fermi energy. Am i doing
> anything wrong again?
> Secondly, when plot the DOS, usually we take the Fermi
> energy as zero of the energy axis. So when i make > Fermi energy as zero(by subtracting the Fermi energy
> from dos.out file, from the energy values of the
> plot) then the valence band maximum is not at zero > but is less than zero. Why is this happening?

> Attaching here both scf and dos files.
>
>  Regards
>
> Research Scholar
> Dept of Physics
> IIT Bombay
> Mumbai, India

-------------------------------------------------------------------
Prof. Eyvaz Isaev,
Theoretical Physics Department, Moscow State Institute of Steel & Alloys, Russia,
Department of Physics, Chemistry, and Biology (IFM), Linkoping University, Sweden
Condensed Matter Theory Group, Uppsala University, Sweden
Eyvaz.Isaev at fysik.uu.se, isaev at ifm.liu.se, eyvaz_isaev at yahoo.com

--- On Thu, 8/14/08, vega <vegalew at hotmail.com> wrote:

> From: vega <vegalew at hotmail.com>
> Subject: Re: [Pw_forum] fermi level
> To: "PWSCF Forum" <pw_forum at pwscf.org>
> Date: Thursday, August 14, 2008, 8:51 PM
> Dear all,
>
> > Fermi energy is written in output whenever it is
> calculated, that is in any scf calculations for metals.
>
> How about the case for semi-conductors like TiO2? When
> calculations for TiO2 was achieved, I can't find any > information about Fermi  energy in my output file.
>
> could anyone can help me.
>
> vega
>
> ==============================================================
> Vega Lew ( weijia liu)
> PH.D Candidate in Chemical Engineering
> State Key Laboratory of Materials-oriented Chemical
> Engineering
> College of Chemistry and Chemical Engineering
> Nanjing University of Technology, 210009, Nanjing, Jiangsu,
> China
>
>
>
> _______________________________________________
> Pw_forum mailing list
> Pw_forum at pwscf.org
> http://www.democritos.it/mailman/listinfo/pw_forum

format: plain


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/ Dynamics

[X++] DMF/DIXF Custom generate method R2/R3

With the Data Import/Export Framework => DIXF (old: Data Migration Framework => DMF) we have the possibillity to import all kinds of data.
But when it comes to writing custom logic for imports, there is a small but significant change between Dynamics AX 2012 R2 and R3.

First of all, lets have a look what we are talking about.
I took the standard entitiy DMFDlvModeEntity for this example. The requirement was to import translations to the field Description also.

New Field in staging table

To accomplish this, I added a new Field called TxtNL to the staging table DMFDlvModeEntity to be able import my translated text for the language NL from my source file.

Generate and getReturnFields Methods

Next, the method to generate the translation record has to be written.
We need to customize two methods:

• generateTranslation (the new method)
• getReturnFields (standard method, needs to be overriden if not exists)
Generate method

Those methods has to be prefixed with generate and will do the actual job.
And here comes the mentioned difference between R2 and R3:
In R2 we didn't need to specify attributes in this methods and had to create a field group on the table with the same name as the method.
In R3 we just need to specify the attributes in the method and the Framework will automatically recognize the method!
The method has always to be in the same schema:

• Returning a container
• Having parameter bool _stagingToTarget = true (to control if the code should run only when pushing the data from staging to target or also/just when importing to staging)
• R3: Having Attribute DMFTargetTransformationAttribute(true)[1]
• R3: Having Attribute DMFTargetTransformationDescAttribute("@labelID")[1:1]
• R3: Having Attribute DMFTargetTransformationSequenceAttribute(Sequence)[1:2]
• R3: Having Attribute DMFTargetTransFieldListAttribute([ListOfFields])[1:3]
getReturnFields method

This method needs to be overridden if not exists and is responsible to map the method to the fields later in the mapping setup. If the generate method does not update any fields in the target table, an empty case statement can be added.

Customizing the target mapping

Under /Data import export framework/Area page/Setup/Target entities the button Modify target mapping will show the form to allow to modify the mapping as needed.

The Framework will automatically recognize our generation method by the attributes and will show it in the mapping:

If we want to change the order or the mapping itself, we can customize it at Mapping details:

Hope this helps!

Footnotes:

1. Attribute DMFTargetTransFieldListAttribute([ListOfFields]): The list of fields to be used insted of creating the field group on the table. ↩ ↩ ↩ ↩

Robin

Software Engineer / Application Developer

[X++] DMF/DIXF Custom generate method R2/R3


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## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$(z+3)(z^2+7)$
Using factoring by grouping, the factored form of the given expression, $z^3+3z^2+7z+21 ,$ is \begin{array}{l} (z^3+3z^2)+(7z+21) \\\\= z^2(z+3)+7(z+3) \\\\= (z+3)(z^2+7) .\end{array}


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# Math Help - l'Hopital's rule

1. ## l'Hopital's rule

Find the limit by using the l' Hopital's rule!
\lim_{x\to0}\frac{\cos\!\left(x\right)-1}{2x^2}

So far i got to
\lim_{x\to0}\frac{\-sin\!\left(x}{4x}

What do you do next

2. Originally Posted by Emmeyh15@hotmail.com
Find the limit by using the l' Hopital's rule!
\lim_{x\to0}\frac{\cos\!\left(x\right)-1}{2x^2}

So far i got to
\lim_{x\to 0}\frac{\-sin\!\left(x}{4x}

What do you do next

$\lim_{x\to0}\frac{\cos\!\left(x\right)-1}{2x^2}=\lim_{x\to 0}\frac{-\sin(x)}{4x}=\text{ use L'Hopital again } = \lim_{x\to0}\frac{-\cos(x)}{4} = -\frac{1}{4}$

Tell your teacher that it is Bernoulli's rule actually.

3. Originally Posted by courteous
$\lim_{x\to0}\frac{\cos\!\left(x\right)-1}{2x^2}=\lim_{x\to 0}\frac{-\sin(x)}{4x}=\text{ use L'Hopital again } = \lim_{x\to0}\frac{-\cos(x)}{4} = -\frac{1}{4}$

Tell your teacher that it is Bernoulli's rule actually.
But is you call it Bernoulli's rule most (if not all) people won't know what your talking about.

Here's an interesting site.

Calculus I Notes, Section 4-5


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WIKISKY.ORG
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# NGC 4976

Contents

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### Related articles

 Peculiarities and populations in elliptical galaxies. III. Dating the last star formation eventUsing 6 colours and 4 Lick line-indices we derive two-component modelsof the populations of ellipticals, involving a "primary" and a"juvenile" population. The first component is defined by the regressionsof indices against the central velocity dispersion found in Papers I andII for the {Nop} sample of non-peculiar objects. The second one isapproximated by an SSP, and the modeling derives its age A, metallicityZ and fractional V-luminosity q_V, the fractional mass qMbeing found therefrom. The model is designed for "blueish" peculiargalaxies, i.e. the {Pec} sample and NGC 2865 family in the terminologyof Paper I. The morphological peculiarities and the population anomalyare then believed to involve the same event, i.e. a merger plusstarburst. It is possible to improve the models in a few cases byintroducing diffuse dust (as suggested by far IR data), and/or by takinginto account the fact that Lick- and colour indices do not relate toidentical galaxy volumes. In most of the cases, the mass ratio of youngstars qM seems too small for the product of a recent majormerger: the events under consideration might be minor mergers bringing"the final touch" to the build-up of the structure of the E-type object.The same modeling has been successfully applied to blueish galaxies ofthe {Nop} sample, without morphological peculiarities however, tosupport the occurence of a distinct perturbing event. A few reddishobjects of the {Pec} sample (NGC 3923 family) and of the {Nop} sampleare also modeled, in terms of an excess of high metallicity stars, ordiffuse dust, or both, but the results are inconclusive. Peculiarities and populations in elliptical galaxies. II. Visual-near IR colours as population indicesAs a complement to the data collected and discussed in Paper I of thisseries, 2MASS near-IR images have been used, in connection withavailable V light aperture photometry, to derive the colours V-J, V-K,J-H and J-K within the effective aperture A_e: nearly the same completesample of 110 E-type galaxies is treated. In Paper I these wereclassified, based on morphological criteria, into the peculiar'' (orPec) and normal'' (or Nop) subsamples. For the Nop subsample, thederived colour indices are tightly related to the galaxy masses, asmeasured by the central velocity dispersion σ0,although with rather small slopes as regards J-H and J-K. For the Pecsubsample, the V-J and V-K colours behave as UBV and line-indices: partof the objects show blue residuals from the appropriatecolour-σ0 regression, which is evidence of a youngerpopulation mixed with the normal'' one traced by the Nop regressions;the other shows no deviations from the Nop subsample. The distinctionamong Pec objects between the YP family (NGC 2865 type), and the NP one(NGC 3923 type), is statistically supported, and generally confirmed inspecific cases.Based in part on observations collected at the Observatoire deHaute-Provence.Table 4 is only available in electonic form at the CDS via anonymous ftpto cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/429/819 Variability of Active Galactic Nuclei in the UVNot Available Deepsky challenges.Not Available Peculiarities and populations in elliptical galaxies. I. An old question revisitedMorphological peculiarities, as defined from isophote asymmetries andnumber of detected shells, jets or similar features, have been estimatedin a sample of 117 E classified galaxies, and qualified by an ad hocΣ2 index. The overall frequency of peculiar'' objects(Pec subsample) is 32.5%. It decreases with the cosmic density of theenvironment, being minimal for the Virgo cluster, the densestenvironment in the sampled volume. This environmental effect is strongerfor galaxies with relatively large Σ2.The Pec subsample objects are compared with normal'' objects (Nopsubsample) as regards their basic properties. Firstly, theysystematically deviate from the Fundamental Plane and the Faber-Jacksonrelation derived for the Nop subsample, being too bright for their mass.Secondly, the dust content of galaxies, as estimated from IRAS fluxes,are similar in both subsamples. Third, the same is true of the frequencyof Kinematically Distinct cores (KDC), suggesting that KDC andmorphological peculiarities do not result from the same events in thehistory of E-galaxies.Using the Nop sample alone, we obtain very tight reference relationsbetween stellar population indicators (U-B, B-V, B-R, V-I,Mg2, Hβ, , Mgb) and the central velocitydispersion σ0. The discussion of the residuals of theserelations allows us to classify the Pec galaxies in two families i.e.the YP or NGC 2865 family, and the NP or NGC 3923 one. Galaxies in thefirst group show consistent evidence for a younger stellar populationmixed with the old one, in agreement with classical results (Schweizeret al. \cite{Schweizer1990}; Schweizer & Seitzer\cite{Schweizer1992}). The second group, however, has normal, orreddish, populations. It is remarkable that a fraction (circa 40%) ofmorphologically perturbed objects do not display any signature of ayoung population, either because the event responsible for thepecularity is too ancient, or because it did not produce significantstar formation (or eventually that the young sub-population has highmetallicity).A preliminary attempt is made to interpret the populations of Pecobjects by combining a young Single Stellar Population with a Nopgalaxy, with only limited success, perhaps largely due to uncertaintiesin the SSP indices used.Based in part on observations collected at the Observatoire deHaute-Provence.Figures \ref{fig1}-\ref{fig3} are only available in electronic form athttp://www.edpsciences.orgTable 10 is only available in electronic form at the CDS via anonymousftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/423/833 Line-of-Sight Reddening Predictions: Zero Points, Accuracies, the Interstellar Medium, and the Stellar Populations of Elliptical GalaxiesRevised (B-V)0-Mg2 data, which are used to testreddening predictions, are presented for 402 elliptical galaxies. Thesereddening predictions can tell us both what the intrinsic errors are inthis relationship among gE galaxy stellar populations as well as detailsof nearby structure in the interstellar medium (ISM) of our Galaxy, andof the intrinsic errors in reddening predictions. Using least-squaresfits, the explicit 1 σ errors in reddenings predicted by theBurstein-Heiles (BH) method and the Schlegel and coworkers (IR) methodare calculated, as well as the 1 σ observational error in the(B-V)0-Mg2 for gE galaxies. It is found that indirections with E(B-V)<0.100 mag (where most of these galaxies lie),1 σ errors in the IR reddening predictions are 0.006-0.009 mag inE(B-V), those for BH reddening predictions are 0.011 mag, and the 1σ agreement between the two reddening predictions is 0.007 mag.The IR predictions have an accuracy of 0.010-0.011 mag in directionswith E(B-V)>=0.100 mag, significantly better than those of the BHpredictions (0.024-0.025). Both methods yield good evidence thatgas-to-dust variations that vary by a factor of 3, both high and low,exist along many lines of sight in our Galaxy. Both methods also predictmany directions with E(B-V)<0.015 mag, despite the difference in zeropoint that each has assumed. The ~0.02 higher reddening zero point inE(B-V) previously determined by Schlegel and coworkers is confirmed,primarily at the Galactic poles. Independent evidence of reddening atthe north Galactic pole (NGP) is reviewed, with the conclusion thatdirections still exist at the NGP that have E(B-V)<<0.01. Twolines of evidence suggest that IR reddenings are overpredicted indirections with high gas-to-dust ratios. As high gas-to-dust directionsin the ISM also include the Galactic poles, this overprediction is thelikely cause of the E(B-V)~0.02 mag larger IR reddening zero pointrelative to that of BH. On the calibration of the COBE/IRAS dust emission reddening mapsIn this work we study the spectral properties (3600-6800 Å) of thenuclear region of early-type galaxies at low (|b|<25deg),intermediate (including surroundings of the Magellanic Clouds) and high(South Polar Cap) Galactic latitudes. We determine the E(B-V) reddeningvalues of the galaxies by matching their continuum distribution withrespect to those of reddening-free spectral galaxy templates withsimilar stellar populations. We also compare the spectroscopic reddeningvalue of each galaxy with that derived from 100 mu m dust emission(E(B-V)FIR) in its line of sight, and we find that there isagreement up to E(B-V)=0.25. Beyond this limit E(B-V)FIRvalues are higher. Taking into account the data up to E(B-V) ~ 0.7, wederive a calibration factor of 0.016 between the spectroscopic E(B-V)values and Schlegel et al.'s (\cite{Schlegel1998}) opacities. Bycombining this result with an AK extinction map built withinten degrees of the Galactic centre using Bulge giants as probes (Dutraet al. \cite{Dutra2003}), we extended the calibration of dust emissionreddening maps to low Galactic latitudes down to |b|=4deg andE(B-V)= 1.6 (AV ~ 5). According to this new calibration, amultiplicative factor of ~0.75 must be applied to the COBE/IRAS dustemission reddening maps.Based on observations made at Complejo Astronómico El Leoncito,which is operated under agreement between the Consejo Nacional deInvestigaciones Científicas y Técnicas de laRepública Argentina and the National Universities of La Pata,Córdoba and San Juan, Argentina. A new catalogue of ISM content of normal galaxiesWe have compiled a catalogue of the gas content for a sample of 1916galaxies, considered to be a fair representation of normality''. Thedefinition of a normal'' galaxy adopted in this work implies that wehave purposely excluded from the catalogue galaxies having distortedmorphology (such as interaction bridges, tails or lopsidedness) and/orany signature of peculiar kinematics (such as polar rings,counterrotating disks or other decoupled components). In contrast, wehave included systems hosting active galactic nuclei (AGN) in thecatalogue. This catalogue revises previous compendia on the ISM contentof galaxies published by \citet{bregman} and \citet{casoli}, andcompiles data available in the literature from several small samples ofgalaxies. Masses for warm dust, atomic and molecular gas, as well asX-ray luminosities have been converted to a uniform distance scale takenfrom the Catalogue of Principal Galaxies (PGC). We have used twodifferent normalization factors to explore the variation of the gascontent along the Hubble sequence: the blue luminosity (LB)and the square of linear diameter (D225). Ourcatalogue significantly improves the statistics of previous referencecatalogues and can be used in future studies to define a template ISMcontent for normal'' galaxies along the Hubble sequence. The cataloguecan be accessed on-line and is also available at the Centre desDonnées Stellaires (CDS).The catalogue is available in electronic form athttp://dipastro.pd.astro.it/galletta/ismcat and at the CDS via anonymousftp to\ cdsarc.u-strasbg.fr (130.79.128.5) or via\http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/405/5 An Infrared Space Observatory Atlas of Bright Spiral GalaxiesIn this first paper in a series we present an atlas of infrared imagesand photometry from 1.2 to 180 μm for a sample of bright spiralgalaxies. The atlas galaxies are an optically selected,magnitude-limited sample of 77 spiral and S0 galaxies chosen from theRevised Shapley-Ames Catalog (RSA). The sample is a representativesample of spiral galaxies and includes Seyfert galaxies, LINERs,interacting galaxies, and peculiar galaxies. Using the Infrared SpaceObservatory (ISO), we have obtained 12 μm images and photometry at60, 100, and 180 μm for the galaxies. In addition to its imagingcapabilities, ISO provides substantially better angular resolution thanis available in the IRAS survey, and this permits discrimination betweeninfrared activity in the central regions and global infrared emission inthe disks of these galaxies. These ISO data have been supplemented withJHK imaging using ground-based telescopes. The atlas includes 2 and 12μm images. Following an analysis of the properties of the galaxies,we have compared the mid-infrared and far-infrared ISO photometry withIRAS photometry. The systematic differences we find between the IRASFaint Source Catalog and ISO measurements are directly related to thespatial extent of the ISO fluxes, and we discuss the reliability of IRASFaint Source Catalog total flux densities and flux ratios for nearbygalaxies. In our analysis of the 12 μm morphological features we findthat most but not all galaxies have bright nuclear emission. We find 12μm structures such as rings, spiral arm fragments, knotted spiralarms, and bright sources in the disks that are sometimes brighter thanthe nuclei at mid-infrared wavelengths. These features, which arepresumably associated with extranuclear star formation, are common inthe disks of Sb and later galaxies but are relatively unimportant inS0-Sab galaxies. Based on observations with the Infrared SpaceObservatory (ISO), an ESA project with instruments funded by ESA MemberStates (especially the PI countries: France, Germany, Netherlands, andUnited Kingdom) and with the participation of ISAS and NASA. A catalogue and analysis of X-ray luminosities of early-type galaxiesWe present a catalogue of X-ray luminosities for 401 early-typegalaxies, of which 136 are based on newly analysed ROSAT PSPC pointedobservations. The remaining luminosities are taken from the literatureand converted to a common energy band, spectral model and distancescale. Using this sample we fit the LX:LB relationfor early-type galaxies and find a best-fit slope for the catalogue of~2.2. We demonstrate the influence of group-dominant galaxies on the fitand present evidence that the relation is not well modelled by a singlepower-law fit. We also derive estimates of the contribution to galaxyX-ray luminosities from discrete-sources and conclude that they provideLdscr/LB~=29.5ergs-1LBsolar-1. Wecompare this result with luminosities from our catalogue. Lastly, weexamine the influence of environment on galaxy X-ray luminosity and onthe form of the LX:LB relation. We conclude thatalthough environment undoubtedly affects the X-ray properties ofindividual galaxies, particularly those in the centres of groups andclusters, it does not change the nature of whole populations. A synthesis of data from fundamental plane and surface brightness fluctuation surveysWe perform a series of comparisons between distance-independentphotometric and spectroscopic properties used in the surface brightnessfluctuation (SBF) and fundamental plane (FP) methods of early-typegalaxy distance estimation. The data are taken from two recent surveys:the SBF Survey of Galaxy Distances and the Streaming Motions of AbellClusters (SMAC) FP survey. We derive a relation between(V-I)0 colour and Mg2 index using nearly 200galaxies and discuss implications for Galactic extinction estimates andearly-type galaxy stellar populations. We find that the reddenings fromSchlegel et al. for galaxies with E(B-V)>~0.2mag appear to beoverestimated by 5-10 per cent, but we do not find significant evidencefor large-scale dipole errors in the extinction map. In comparison withstellar population models having solar elemental abundance ratios, thegalaxies in our sample are generally too blue at a given Mg2;we ascribe this to the well-known enhancement of the α-elements inluminous early-type galaxies. We confirm a tight relation betweenstellar velocity dispersion σ and the SBF `fluctuation count'parameter N, which is a luminosity-weighted measure of the total numberof stars in a galaxy. The correlation between N and σ is eventighter than that between Mg2 and σ. Finally, we deriveFP photometric parameters for 280 galaxies from the SBF survey data set.Comparisons with external sources allow us to estimate the errors onthese parameters and derive the correction necessary to bring them on tothe SMAC system. The data are used in a forthcoming paper, whichcompares the distances derived from the FP and SBF methods. A Test for Large-Scale Systematic Errors in Maps of Galactic ReddeningAccurate maps of Galactic reddening are important for a number ofapplications, such as mapping the peculiar velocity field in the nearbyuniverse. Of particular concern are systematic errors which vary slowlyas a function of position on the sky, as these would induce spuriousbulk flow. We have compared the reddenings of Burstein & Heiles (BH)and those of Schlegel, Finkbeiner, & Davis (SFD) to independentestimates of the reddening, for Galactic latitudes |b|>10^deg. Ourprimary source of Galactic reddening estimates comes from comparing thedifference between the observed B-V colors of early-type galaxies, andthe predicted B-V color determined from the B-V-Mg_2 relation. We havefitted a dipole to the residuals in order to look for large-scalesystematic deviations. There is marginal evidence for a dipolar residualin the comparison between the SFD maps and the observed early-typegalaxy reddenings. If this is due to an error in the SFD maps, then itcan be corrected with a small (13%) multiplicative dipole term. Weargue, however, that this difference is more likely to be due to a small(0.01 mag) systematic error in the measured B-V colors of the early-typegalaxies. This interpretation is supported by a smaller, independentdata set (globular cluster and RR Lyrae stars), which yields a resultinconsistent with the early-type galaxy residual dipole. BH reddeningsare found to have no significant systematic residuals, apart from theknown problem in the region 230^deg X-ray luminosities for a magnitude-limited sample of early-type galaxies from the ROSAT All-Sky SurveyFor a magnitude-limited optical sample (B_T <= 13.5 mag) ofearly-type galaxies, we have derived X-ray luminosities from the ROSATAll-Sky Survey. The results are 101 detections and 192 useful upperlimits in the range from 10^36 to 10^44 erg s^-1. For most of thegalaxies no X-ray data have been available until now. On the basis ofthis sample with its full sky coverage, we find no galaxy with anunusually low flux from discrete emitters. Below log (L_B) ~ 9.2L_⊗ the X-ray emission is compatible with being entirely due todiscrete sources. Above log (L_B) ~ 11.2 L_osolar no galaxy with onlydiscrete emission is found. We further confirm earlier findings that L_xis strongly correlated with L_B. Over the entire data range the slope isfound to be 2.23 (+/- 0.12). We also find a luminosity dependence ofthis correlation. Below log L_x = 40.5 erg s^-1 it is consistent with aslope of 1, as expected from discrete emission. Above this value theslope is close to 2, as expected from gaseous emission. Comparing thedistribution of X-ray luminosities with the models of Ciotti et al.leads to the conclusion that the vast majority of early-type galaxiesare in the wind or outflow phase. Some of the galaxies may have alreadyexperienced the transition to the inflow phase. They show X-rayluminosities in excess of the value predicted by cooling flow modelswith the largest plausible standard supernova rates. A possibleexplanation for these super X-ray-luminous galaxies is suggested by thesmooth transition in the L_x--L_B plane from galaxies to clusters ofgalaxies. Gas connected to the group environment might cause the X-rayoverluminosity. A catalogue of Mg_2 indices of galaxies and globular clustersWe present a catalogue of published absorption-line Mg_2 indices ofgalaxies and globular clusters. The catalogue is maintained up-to-datein the HYPERCAT database. The measurements are listed together with thereferences to the articles where the data were published. A codeddescription of the observations is provided. The catalogue gathers 3541measurements for 1491 objects (galaxies or globular clusters) from 55datasets. Compiled raw data for 1060 galaxies are zero-point correctedand transformed to a homogeneous system. Tables 1, 3, and 4 areavailable in electronic form only at the CDS, Strasbourg, via anonymousftp 130.79.128.5. Table 2 is available both in text and electronic form. Total magnitude, radius, colour indices, colour gradients and photometric type of galaxiesWe present a catalogue of aperture photometry of galaxies, in UBVRI,assembled from three different origins: (i) an update of the catalogueof Buta et al. (1995) (ii) published photometric profiles and (iii)aperture photometry performed on CCD images. We explored different setsof growth curves to fit these data: (i) The Sersic law, (ii) The net ofgrowth curves used for the preparation of the RC3 and (iii) A linearinterpolation between the de Vaucouleurs (r(1/4) ) and exponential laws.Finally we adopted the latter solution. Fitting these growth curves, wederive (1) the total magnitude, (2) the effective radius, (3) the colourindices and (4) gradients and (5) the photometric type of 5169 galaxies.The photometric type is defined to statistically match the revisedmorphologic type and parametrizes the shape of the growth curve. It iscoded from -9, for very concentrated galaxies, to +10, for diffusegalaxies. Based in part on observations collected at the Haute-ProvenceObservatory. A catalogue of spatially resolved kinematics of galaxies: BibliographyWe present a catalogue of galaxies for which spatially resolved data ontheir internal kinematics have been published; there is no a priorirestriction regarding their morphological type. The catalogue lists thereferences to the articles where the data are published, as well as acoded description of these data: observed emission or absorption lines,velocity or velocity dispersion, radial profile or 2D field, positionangle. Tables 1, 2, and 3 are proposed in electronic form only, and areavailable from the CDS, via anonymous ftp to cdsarc.u-strasbg.fr (to130.79.128.5) or via http://cdsweb.u-strasbg.fr/Abstract.html Galactic BulgesWe discuss the present observational and theoretical understanding ofthe stellar populations of bulges and their implications for galaxyformation and evolution. The place of bulges as key to the HubbleSequence remains secure, but some old paradigms are giving way to newones as observations develop. Detailed studies of Local Group bulges andhaloes provide a basis on which we consider higher redshift data. Wepresent the evidence for and against the currently common preconceptionsthat bulges are old, above solar metallicity in the mean, and simplyscaled-down versions of ellipticals. We conclude life is not so simple:Bulges are diverse and heterogeneous, and although their properties varysystematically, sometimes they are reminiscent of disks, sometimes ofellipticals. The extant observational data are, however, limited. Newand surveys will rectify this, and we discuss the questions those datawill address. Detectability of High-Redshift Elliptical Galaxies in the Hubble Deep FieldRelatively few intensively star-forming galaxies at redshifts ofz>2.5 have been found in the Hubble Deep Field (HDF). This has beeninterpreted to imply a low space density of elliptical galaxies at highz, possibly due to a late ( z<2.5 ) epoch of formation or to dustobscuration of the ellipticals that are forming at z~3 . I use HubbleSpace Telescope UV (~2300 A) images of 25 local early-type galaxies toinvestigate a third option, that ellipticals formed at z>4.5 and werefading passively by 24.5 and have faded passivelyto their present brightnesses at UV wavelengths, they would generally bebelow the HDF detection limits in any of its bands at z>2.5 .Quiescent z~3 ellipticals, if they exist, should turn up in sufficientlydeep IR images. An Atlas of Hubble Space Telescope Ultraviolet Images of Nearby GalaxiesWe present an atlas of UV (~2300 A) images, obtained with the HubbleSpace Telescope (HST) Faint Object Camera, of the central 22" x 22" of110 galaxies. The observed galaxies are an unbiased selectionconstituting about one-half of a complete sample of all large (D >6') and nearby (V < 2000 km s^-1^) galaxies. This is the firstextensive UV imaging survey of normal galaxies. The data are useful forstudying star formation, low-level nuclear activity, and UV emission byevolved stellar populations in galaxies. At the HST resolution (~0.05"),the images display an assortment of morphologies and UV brightnesses.These include bright nuclear point sources, compact young star clustersscattered in the field or arranged in circumnuclear rings, centrallypeaked diffuse light distributions, and galaxies with weak or undetectedUV emission. We measure the integrated ~2300 A flux in each image,classify the UV morphology, and examine trends between these parametersand the optical properties of the galaxies. The fundamental plane of early-type galaxies: stellar populations and mass-to-light ratio.We analyse the residuals to the fundamental plane (FP) of ellipticalgalaxies as a function of stellar-population indicators; these are basedon the line-strength parameter Mg_2_ and on UBVRI broad-band colors, andare partly derived from new observations. The effect of the stellarpopulations accounts for approximately half the observed variation ofthe mass-to-light ratio responsible for the FP tilt. The residual tiltcan be explained by the contribution of two additional effects: thedependence of the rotational support, and possibly that of the spatialstructure, on the luminosity. We conclude to a constancy of thedynamical-to-stellar mass ratio. This probably extends to globularclusters as well, but the dominant factor would be here the luminositydependence of the structure rather than that of the stellar population.This result also implies a constancy of the fraction of dark matter overall the scalelength covered by stellar systems. Our compilation ofinternal stellar kinematics of galaxies is appended. A Catalog of Stellar Velocity Dispersions. II. 1994 UpdateA catalog of central velocity dispersion measurements is presented,current through 1993 September. The catalog includes 2474 measurementsof 1563 galaxies. A standard set of 86 galaxies is defined, consistingof galaxies with at least three reliable, concordant measurements. It issuggested that future studies observe some of these standard galaxies sothat different studies can be normalized to a consistent system. Allmeasurements are reduced to a normalized system using these standards. Integrated photoelectric magnitudes and color indices of bright galaxies in the Johnson UBV systemThe photoelectric total magnitudes and color indices published in theThird Reference Catalogue of Bright Galaxies (RC3) are based on ananalysis of approximately equals 26,000 B, 25,000 B-V, and 17,000 U-Bmultiaperture measurements available up to mid 1987 from nearly 350sources. This paper provides the full details of the analysis andestimates of internal and external errors in the parameters. Thederivation of the parameters is based on techniques described by theVaucouleurs & Corwin (1977) whereby photoelectric multiaperture dataare fitted by mean Hubble-type-dependent curves which describe theintegral of the B-band flux and the typical B-V and U-B integrated colorgradients. A sophisticated analysis of the residuals of thesemeasurements from the curves was made to allow for the random andsystematic errors that effect such data. The result is a homogeneous setof total magnitudes BTA total colors(B-V)T and (U-B)T, and effective colors(B-V)e and (U-B)e for more than 3000 brightgalaxies in RC3. Photometrically distinct nuclei in elliptical and early-type disks galaxies.Not Available Kinematics of 12 elliptical galaxiesLong-slit spectroscopic data for a sample of 12 elliptical galaxies havebeen analyzed to derive velocity dispersion and rotational velocityprofiles. Minor axis profiles are also given for two galaxies, NGC 2768and IC 1459. These data clearly demonstrate the large range ofrotational properties known to exist in luminous elliptical galaxiesMB less than or equal to -20 for Hzero = 50km/sec/Mpc. The spectroscopic data typically have quite high S/N and canbe used to derive velocity dispersion and rotation velocity profiles outto radii r approximately re. The mean logarithmic gradient ofthe velocity dispersion profiles is small; sigma decreases by only 20%for an order of magnitude change in radius from approximately 0.1 to 1 re. The ellipticals in this sample are also found to have avery small range in their logarithmic gradients. Even though rotation isenergetically unimportant, the rotation curves in ellipticals aretypically flat at radii larger than 0.2re, and often at radiibeyond 0.1re. A spectrum offset from the nucleus of the E6/S0galaxy NGC 2768 is used to characterize the form of the velocitydistribution in this quite flattened galaxy (it decreases rapidly withz). Observations of the gas (from O II) show that there exists a polargas disk in the inner regions of NGC 2768. UBVRI photoelectric photometry of bright southern early-type galaxiesUBVRI multi-aperture photometry of 207 bright southern galaxies and of72 objects of an additional list is presented. These observations weremade for obtaining the magnitude scale zero-point as an accuratedetermination of the sky background for a two-dimensional photometryprogram concerning these galaxies. We have also inferred the asymptoticmagnitudes, color indices and effective apertures of these objects. Ourresults are in good agreement with those of others authors. The dependence of the cool matter content on galaxy morphology in galaxies of types E/S0, S0, and SAUsing the material assembled in earlier papers, we examine the manner inwhich the interstellar matter content varies along the Hubble sequencefrom S0 galaxies to Sa galaxies selected from the RSA2 compilation. Forthis we make use of a new and more detailed classification which isdescribed here as applied to these early disk/spiral galaxies. Theprominence of the disk in S0's and the visibility of features (H IIregions) in the Sa's serve as the basis for the subtypes. Three S0categories: subtle, intermediate, and pronounced, and four Sadescriptors: very early, early, intermediate, and late are assigned tothe galaxies. It is found that the total amount of hydrogen (H I + H2)is a function of subtype, being low in the S0's and rising smoothly fromthe early Sa's to the later Sa's. The average surface density ofhydrogen exceeds 3 solar masses/pc-squared only in the latest subtypesof the Sa's. We conclude that the prominence of the disk of a galaxyclosely follows the amount of cool gas which the disk contains. General study of group membership. II - Determination of nearby groupsWe present a whole sky catalog of nearby groups of galaxies taken fromthe Lyon-Meudon Extragalactic Database. From the 78,000 objects in thedatabase, we extracted a sample of 6392 galaxies, complete up to thelimiting apparent magnitude B0 = 14.0. Moreover, in order to considersolely the galaxies of the local universe, all the selected galaxieshave a known recession velocity smaller than 5500 km/s. Two methods wereused in group construction: a Huchra-Geller (1982) derived percolationmethod and a Tully (1980) derived hierarchical method. Each method gaveus one catalog. These were then compared and synthesized to obtain asingle catalog containing the most reliable groups. There are 485 groupsof a least three members in the final catalog. Low-luminosity early-type galaxies. I - Photometry and morphologyNew multiaperture photoelectric photometry in U, B, V, R, and I for 50southern low-luminosity early-type galaxies (LLEs) is presented.Asymptotic magnitude and mean surface brightness within the effectiveaperture are derived from fits to r exp 1/4 growth curves, and colorsare reduced at an effective radius for 154 galaxies. Morphological andstructural analysis of the LLE sample, based on the ESO-LV image database, shows that the average flattening of these galaxies is high. Thesample can be divided into four main categories. There is no correlationbetween the morphological type for t of not greater than -3 and any ofthe structural and geometrical properties investigated. Only a fewgalaxies show a boxy isophote shape, while a greater fraction showsdisky isophotes. Groups of galaxies within 80 Mpc. II - The catalogue of groups and group membersThis paper gives a catalog of the groups and associations obtained bymeans of a revised hierarchical algorithm applied to a sample of 4143galaxies with diameters larger than 100 arcsec and redshifts smallerthan 6000 km/s. The 264 groups of galaxies obtained in this way (andwhich contain at least three sample galaxies) are listed, with the looseassociations surrounding them and the individual members of eachaggregate as well; moreover, the location of every entity among 13regions corresponding roughly to superclusters is specified. Finally,1729 galaxies belong to the groups, and 466 to the associations, i.e.,the total fraction of galaxies within the various aggregates amounts to53 percent.
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This paper tackles the problem of designing state observers for flexible link mechanisms: An investigation is made on the possibility of employing observers making use of suitable piecewise-linear truncated dynamics models. A general and novel approach is proposed, which provides an objective way of synthesizing observers preventing the instability that may arise from using reduced-order linearized models. The approach leads to the identification of the regions of the domain of the state variables where the linear approximations of the nonlinear model can be considered acceptable. To this purpose, first of all, the stability of the equilibrium points of the closed-loop system is assessed by applying the eigenvalue analysis to appropriate piecewise-linear models. Admittedly, the dynamics of such a closed-loop system is affected by the perturbation of the poles caused by spillover and by the discrepancies between the linearized models of the plant and the one of the observer. Additionally, when nodal elastic displacements and velocities are not bounded in the infinitesimal neighborhoods of the equilibrium points, the difference between the nonlinear model and the locally linearized one is expressed in terms of unstructured uncertainty and stability is assessed through $H∞$ robust analysis. The method is demonstrated by applying it to a closed-chain flexible link mechanism.

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# Derivation of 2$\pi$ in the period of a simple pendulum

Most of us know that the period of a simple pendulum is given by

$$2 \pi \sqrt{\dfrac{\ell}{g}}.$$

But how did the $$2\pi$$ term get into that argument. From dimensional analysis, we can find the period but not the constant.

Is there a calculus argument as some limit is taken? Is it based on an energy equation? Surely there is a way to derive that $$2 \pi$$.

• Hint: The motion of a simple pendulum is a type of sinusoidal type motion. – John Omielan Sep 16 at 4:24
• Na that's an easy cop out. There must be a better argument. – KennyB Sep 16 at 5:00
• @HansLundmark As evidenced by the range of different types of answers and by OP's comments to those answers, this question lacks way too many details to be properly answerable. I have no idea what OP actually is asking. For example, at first it just sounds like a question of how to derive the period of a pendulum, but apparently OP is also confused about how $2\pi$ comes into the period of $\sin(\omega t)$ – and one answer doesn't discuss the derivation at all but OP seems to be quite happy about it. – JiK Sep 16 at 14:02
• @HansLundmark I thought it was an interesting question. I'm lecturing dimensional analysis and I derived the period for the students. One student (who studies physics) asked how we can solve for the constant without knowing anything about the problem. I told him that you need to look at the DE and the behavior of the pendulum to know its 2pi periodic. While this was a satisfying answer to the student im not happy with it. Dimensional analysis is powerful because you don't need insight into the problem, only the dimensions and variables. Similarly, can we find the constant in the same manner. – KennyB Sep 18 at 9:34
• @KennyB: I think that if you had added that information to the question, it would have been received much more positively! Context matters. – Hans Lundmark Sep 18 at 9:49

The pendulum movement equation is given by \begin{align*} \frac{\mathrm{d}^{2}\theta}{\mathrm{d}t^{2}} + \frac{g}{\ell}\sin(\theta) = 0 \end{align*}

For small values of $$\theta$$, we can make the approximation $$\sin(\theta) \approx \theta$$, from whence we obtain the equation \begin{align*} \ddot{\theta} + \frac{g}{\ell}\theta = 0 \end{align*}

whose associated characteristic equation is \begin{align*} x^{2} + \frac{g}{\ell} = 0 \Longrightarrow x = \pm i\sqrt{\frac{g}{\ell}} \end{align*}

Thus the solutions are $$\theta(t) = c_{1}\sin\left(\displaystyle t\sqrt{\frac{g}{\ell}}\right) + c_{2}\cos\left(\displaystyle t\sqrt{\frac{g}{\ell}}\right)$$. Finally, for $$T = 2\pi\sqrt{\displaystyle\frac{\ell}{g}}$$, one has \begin{align*} \theta\left(t + 2\pi\sqrt{\frac{\ell}{g}}\right) = c_{1}\sin\left(t\sqrt{\frac{g}{\ell}} + 2\pi\right) + c_{2}\cos\left(t\sqrt{\frac{g}{\ell}} + 2\pi\right) = \theta(t) \end{align*} Therefore we conclude that $$T$$ is the period indeed.

• I prefer this reasoning to the others presented. However, it's more like a good guess to make that substitution. I was hoping there might be a derivation of sorts. Maybe such a derivation doesn't exist? – KennyB Sep 16 at 5:03
• @KennyB This is a standard method for solving a differential equation. No guessing is involved. – Jyrki Lahtonen Sep 16 at 5:11
• No no, what I mean is, the reasoning to substitute that form of T. It's not explicit if you don't know the behavior of the pendulum. – KennyB Sep 16 at 5:25
• @KennyB it's explicit if you know that both sine and cosine have a periodicity of $2\pi$ and then look for time displacements that give you a displacement of $2\pi$. I have a simple demonstration for this but it is too big for this comment. – bracco23 Sep 16 at 14:27

The conserved energy is $$E = \tfrac12 m l^2 \dot \theta^2 + \tfrac12 mgl \theta^2$$ so that the system moves on ellipses in the $$(\theta,\dot\theta)$$ plane.

The period is the time required to go once around an ellipse, which is twice the time it takes to go half a lap, say between $$(\theta,\dot\theta)=(-\theta_0,0)$$ and $$(\theta_0,0)$$ with $$\dot\theta>0$$, where $$\theta_0$$ is the amplitude of the oscillation, determined by the equation $$E=0+ \tfrac12 mgl \theta_0^2$$, that is, $$\theta_0 = \sqrt{\frac{2E}{mgl}} .$$ This gives $$T = 2 \int_{0}^{T/2} dt = 2 \int_{-\theta_0}^{\theta_0} \frac{d\theta}{d\theta/dt} = 2 \int_{-\theta_0}^{\theta_0} \frac{d\theta}{\sqrt{\frac{E- \tfrac12 mgl \theta^2}{\tfrac12 ml^2}}} = 2 \int_{-\theta_0}^{\theta_0} \frac{d\theta}{\sqrt{\tfrac{g}{l}} \sqrt{\theta_0^2 - \theta^2}} .$$ Now let $$\theta = \theta_0 \sin u$$ to get $$T = 2 \sqrt{\tfrac{l}{g}} \int_{-\pi/2}^{\pi/2} \frac{1}{\sqrt{\theta_0^2 (1-\sin^2 u)}} \, \theta_0 \cos u \, du = 2 \sqrt{\tfrac{l}{g}} \int_{-\pi/2}^{\pi/2} du = 2 \sqrt{\tfrac{l}{g}} \cdot \pi ,$$ as desired.

You can do the same thing for the full pendulum equation, without the small-angle approximation $$\sin \theta \approx \theta$$, where the energy is $$E = \tfrac12 m l^2 \dot \theta^2 - mgl \cos\theta ,$$ but then you get a non-elementary integral (a so-called complete elliptic integral) which depends on the amplitude $$\theta_0$$ (or, equivalently, on the energy $$E$$), so that the period is some amplitude-dependent constant (greater than $$2\pi$$) times $$\sqrt{\tfrac{l}{g}}$$.

• thanks for this answer. I think it's too advanced for the students I'm lecturing, but it's definitely useful and gives some insight into the problem. This also answers my question in the way that the 2pi term is only for the simple pendulum. This is perhaps why dimensional analysis just gives you a constant because it's independent of the small angle approximation. – KennyB Sep 18 at 9:43

The Lagrangian of a simple pendulum is $$L=\frac{1}{2}ml^2\dot{\varphi}^2+mgl\cos(\varphi)$$ Using harmonic approximation: $$L=\frac{1}{2} ml^2 \dot{\varphi}^2+mgl\left(1-\frac{\varphi^2}{2}\right)$$ Leaving the constant out: $$L=\frac{1}{2}ml^2 \dot{\varphi}^2-\frac{1}{2}mgl\varphi^2$$ Which will have the following equation of motion: $$\frac{\partial L}{\partial \varphi}-\frac{\mathrm{d}}{\mathrm{d}\varphi} \frac{\partial L}{\partial \dot{\varphi}}=0$$ I.e. $$-mgl\varphi +ml^2 \ddot{\varphi}=0$$ $$\ddot{\varphi}=\frac{g}{l}\varphi$$ Which is a simple harmonic oscillator, with frequency $$\sqrt{\frac{g}{l}}$$, so the period time will be $$2\pi \sqrt{\frac{l}{g}}$$.

• Unfortunately, you're using the fact that the period is 2pi/omega where omega is the angular frequency. So you have really derived the 2pi – KennyB Sep 16 at 5:06
• @KennyB The fact that the period is $2\pi/\omega$ comes from the fact that the solution is the linear combination of $\sin(\omega t)$ and $\cos(\omega t)$, and from the basic properties of the trigonometric functions. – Botond Sep 16 at 5:45

The expression $$\sqrt{\frac{g}{l}}$$ represents the frequency in radians per second. Divide by $$2\pi$$ radians per cycle to get $$\frac{1}{2\pi}\sqrt{\frac{g}{l}}$$ cycles per second.

Now take the reciprocal to get $$2\pi\sqrt{\frac{l}{g}}$$ seconds per cycle which, by definition, is the period.

• This is also one of the better reasonings for the 2pi term as compared to some of the other answers. – KennyB Sep 16 at 5:07
• Why is it radians/sec and not cycles/sec? Both radians and cycles has dimension $1$, so I can't really see the reason. – Botond Sep 16 at 5:47
• The dimension of $\omega$ is $T^{-1}$. The same goes for cycles/sec, in my opinion. – Botond Sep 16 at 7:07
• Yes, since I've not seen every physics book :) – Botond Sep 16 at 15:20
• @JohnDouma In what sense are cycles not unitless? What is the unit? – JiK Sep 16 at 18:38


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# How do the Euclidean and extended Euclidean algorithms work?

One can use the extended Euclidean algorithm to calculate the modular multiplicative inverse of a number, as it will be in the form $a x + b y = 1$, and if you take mod $b$ of both sides you get the inverse of $a$ in mod $b$. However, why does the Euclidean algorithm work? Specifically, why is the last non-zero remainder gcd(a,b)? And how come you can just substitute everything back and it magically give you gcd(a,b) in terms of integers? Thanks so much.

• What have you read about the algorithm? What don't you understand? – marty cohen Nov 15 '16 at 1:12
• @martycohen I read how to use it to find the inverse, but I have no idea why it works and how the substitution process with the remainders can get it in terms of a and b. – ConfusedMathStudent Nov 15 '16 at 2:46

## 1 Answer

That is because it relies on the following easy lemma:

The set of common divisors of $a$ and $b$ is the same as the set of common divisors of $b$ and $b\bmod a$. Hence $\gcd(a,b)=\gcd(b, b\bmod a)$.

The extended Euclidean algorithm relies on a more precise analysis of the successive divisions: if we denote by $r_0=a$, $r_1=b$, $r_2=r_0\bmod r_1$, … , $r_{n+1}=r_n\bmod r_{n-1}$, … , the $n$-th division equality: $$r_{n-1}=q_nr_n+r_{n+1}$$ can be rewritten as $$r_{n+1}=r_{n-1}-q_nr_n.$$ An easy induction shows that, not only the last non-zero remainder satisfies a Bézout's identity, but that all intermediate remainders satisfy such an identity: $$r_n=u_n a+v_nb.$$ The sequences of coefficients $(u_n)$ and $(v_n)$ are initialised as $$u_0=1,~v_0=0,\qquad u_1=0,~v_1=0,$$ and they satisfy the same recurrence relation as the remainders: $$u_{n+1}=u_{n-1}-q_nu_n, \qquad v_{n+1}=v_{n-1}-q_nv_n.$$


text:
# Is Hahn-Kolmogorov theorem a direct result of Carathéodory's extension theorem?

Both theorems assume the same condition and conclusion, except that

Since a field of subsets is also a ring of subsets, is Hahn-Kolmogorov theorem a direct result of Carathéodory's extension theorem?

Thanks!

-


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# differential equations – Finding a local normal form regarding distribution rank properties

I am working in geometry control field, fall last week on this exercice and I can’t figure it out. I have a distribution $$mathscr{D}$$ with $$rank(mathscr{D})=m+1$$ in $$mathbb{R}^n$$ with $$nleq 2m+1$$. I know that there exists an involutive sub-distribution $$mathscr{L}subsetmathscr{D}$$ with rank $$m$$. I also know that the growth vector is $$(m+1,n)$$. (So I guess $$mathscr{D}+(mathscr{D},mathscr{D})=Tmathbb{R}^n$$, am I right?).

What I need is to find a local normal form $$varphi$$ of $$mathscr{D}$$ so that $$varphi_*mathscr{D}=span{f’_1,dots,f’_n}$$ assuming $$mathscr{D}=span{f_1,dots,f_n}$$.

So far, I guess I can use the fact that $$mathscr{F}$$ is involutive to write it $$mathscr{F}=span{frac{partial}{partial x^1},dots,frac{partial}{partial x^m}}$$ and then add a vector to build $$mathscr{D}$$ but I don’t know how. I guess Fröbenius can provide some help but can’t find how. Someone has an idea ?


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# URIs

A request can only be sent if the request method & URI are defined. To represent URIs, sttp comes with a Uri case class, which captures all the parts of an address.

To specify the request method and URI, use one of the methods on the request definition corresponding to the name of the desired HTTP method: .post, .get, .put etc. All of them accept a single parameter, the URI to which the request should be sent (these methods only modify the request definition; they don’t send the requests).

The Uri class is immutable, and can be constructed by hand, but in many cases the URI interpolator will be easier to use.

## URI interpolator

Using the URI interpolator it’s possible to conveniently create Uri instances, for example:

import sttp.client3._
import sttp.model._

val user = "Mary Smith"
val filter = "programming languages"

val endpoint: Uri = uri"http://example.com/$user/skills?filter=$filter"

assert(endpoint.toString ==
"http://example.com/Mary%20Smith/skills?filter=programming+languages")

Note the uri prefix before the string and the standard Scala string-embedding syntax ($user, $filter).

Any values embedded in the URI will be URL-encoded, taking into account the context (e.g., the whitespace in user will be %-encoded as %20D, while the whitespace in filter will be query-encoded as +). On the other hand, parts of the URI given as literal strings (not embedded values), are assumed to be URL-encoded and thus will be decoded when creating a Uri instance.

All components of the URI can be embedded from values: scheme, username/password, host, port, path, query and fragment. The embedded values won’t be further parsed, except the : in the host part, which is commonly used to pass in both the host and port:

import sttp.client3._

// the embedded / is escaped
println(uri"http://example.org/${"a/b"}") // http://example.org/a%2Fb // the embedded / is not escaped println(uri"http://example.org/${"a"}/${"b"}") // http://example.org/a/b // the embedded : is not escaped println(uri"http://${"example.org:8080"}")
// http://example.org:8080

Both the Uri class, and the interpolator can be used stand-alone, without using the rest of sttp. Conversions are available both from and to java.net.URI; Uri.toString returns the URI as a String.

## Optional values

The URI interpolator supports optional values for hosts (subdomains), query parameters and the fragment. If the value is None, the appropriate URI component will be removed. For example:

val v1 = None
val v2 = Some("v2")

println(uri"http://example.com?p1=$v1&p2=v2") // http://example.com?p2=v2 println(uri"http://$v1.$v2.example.com") // http://v2.example.com println(uri"http://example.com#$v1")
// http://example.com

## Maps and sequences

Maps, sequences of tuples and sequences of values can be embedded in the query part. They will be expanded into query parameters. Maps and sequences of tuples can also contain optional values, for which mappings will be removed if None.

For example:

val ps = Map("p1" -> "v1", "p2" -> "v2")

println(uri"http://example.com?$ps&p3=p4") // http://example.com?p1=v1&p2=v2&p3=p4  Sequences in the host part will be expanded to a subdomain sequence, and sequences in the path will be expanded to path components: val params = List("a", "b", "c")  println(uri"http://example.com/$params")
// http://example.com/a/b/c

If a string containing the protocol is embedded at the very beginning, it will not be escaped, allowing to embed entire addresses as prefixes, e.g.: uri"$endpoint/login", where val endpoint = "http://example.com/api". This is useful when a base URI is stored in a value, and can then be used as a base for constructing more specific URIs. ## Relative URIs The Uri class can represent both relative and absolute URIs. Hence, in terms of rfc3986, it is in fact a URI reference. Relative URIs can be created using the interpolator, same as absolute ones, e.g.: println(uri"/api/$params")
// /api/a/b/c

When sending requests using relative URIs, the ResolveRelativeUrisBackend backend wrapper might be useful to resolve them.

## All features combined

A fully-featured example:

import sttp.client3._
val secure = true
val scheme = if (secure) "https" else "http"
val subdomains = List("sub1", "sub2")
val vx = Some("y z")
val paramMap = Map("a" -> 1, "b" -> 2)
val jumpTo = Some("section2")

println(uri"$scheme://$subdomains.example.com?x=$vx&$paramMap#\$jumpTo")
// https://sub1.sub2.example.com?x=y+z&a=1&b=2#section2


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Talks
Spring 2016

# Approximately Counting Graph Homomorphisms

Monday, March 28th, 2016 9:30 am10:15 am

A homomorphism from a graph G to a graph H is a map from the vertices of G to the vertices of H that maps every edge of G to an edge of H. Graph homomorphisms capture many interesting combinatorial structures such as independent sets and proper colourings. This talk addresses the the classification program of counting complexity'' as it applies to the problem of approximately counting graph homomorphisms.


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# Math Help - Question regarding delta-epsion proof

1. ## Question regarding delta-epsion proof

Why is this sign, that I highlighted in yellow, changes to the equal sign? Sevens just cancel, why would something else have to happen? Also, we are proving that this expression we started with in our proof is less than $\epsilon$ as we show in the subsequent statement and not equal.

2. ## Re: Question regarding delta-epsion proof

The overall inequality you need still holds.

They are just saying that $7(\epsilon/7)=\epsilon$ which as you note is obvious.

The < sign on the line above still causes $|(7x-1)-27|<\epsilon$

3. ## Re: Question regarding delta-epsion proof

Oh, I think I got it. So they are just explicitly noting that we have this sign <, but the actual quantity after this sign is going to equal to $\epsilon$ hence this equality sign.


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# Thennarasu: Thennarasu's stability statistics In metan: Multi Environment Trials Analysis

## Description

Performs a stability analysis based on Thennarasu (1995) statistics.

## Usage

 1 Thennarasu(.data, env, gen, resp, verbose = TRUE)

## Arguments

 .data The dataset containing the columns related to Environments, Genotypes, replication/block and response variable(s). env The name of the column that contains the levels of the environments. gen The name of the column that contains the levels of the genotypes. resp The response variable(s). To analyze multiple variables in a single procedure use, for example, resp = c(var1, var2, var3). verbose Logical argument. If verbose = FALSE the code will run silently.

## Value

An object of class Thennarasu, which is a list containing the results for each variable used in the argument resp. For each variable, a tibble with the columns GEN, N1, N2, N3 and N4 is returned.

## Author(s)

Tiago Olivoto tiagoolivoto@gmail.com

## References

Thennarasu, K. 1995. On certain nonparametric procedures for studying genotype x environment interactions and yield stability. Ph.D. thesis. P.J. School, IARI, New Delhi, India.

## Examples

 1 2 3 library(metan) out <- Thennarasu(data_ge, ENV, GEN, GY) print(out)

metan documentation built on Nov. 10, 2021, 9:11 a.m.


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# Going behind Scala's recursion

There are times when we might want to call a function which calls itself repeatedly n number of times. In my career programming with Java so far, I have not written any single recursive functions, the main reason being that writing recursion was hard with Java. Java is best suited for an iterative approach. Sometimes a recursive code is a bit hard to read and follow. It is not that self intuitive as with an iteration.

Behind the scenes, during each function call the local variables used within the function are placed on something called a call stack. The stack is unfortunately and fortunately limited in size and resources. If our recursive logic spirals out of control, we could very quickly fill up such a stack, deplete all your resources and bring our entire application down. We certainly want to avoid this from happening, so we write iterative logic which has a certain definite exit point, thus preventing us from any stack overflow errors.

A functional language like Scala handles recursion much more elegantly and invites us to favour recursion instead of iteration.

Broadly speaking, there are two kinds of recursion in Scala, the head recursion and the tail recursion. The main difference between the two is in the way Scala handles the function evaluation with the function call. In head recursion, the function makes a recursive call, the body of the function is evaluated using the result of the recursive call. In case of tail recursion, the function body is evaluated first, the result of the evaluation is used to make the recursive call. If there was no concept called tail recursion in Scala, we would have been still left to deal with stack overflows like we experience in Java, but tail recursion does some magic behind the scenes for us. Not so clear? Let's look at some practical examples:

1def headRecursive(intList: List[Int]): Int = {
2  if (intList.length == 0) 0
4}
5
6val sumOfInts = headRecursive(List(1,2,3,4,5,6,7,8,9)) // prints 45

Other than me defining the method name in the code snippet above to be called headRecursive, we can say that it is a head recursive function. If you take a deeper look into the method again, in the else block, to evaluate the line:

1"intList.head + headRecursive(intList.tail)",

a call to the headRecursive(intList.tail) has to return. What happens effectively is that the recursive call is first made, then the head of the List is added to the result of the recursive call, the value is then finally returned.

So the runtime needs to know what was the value of the head element in the List, the result of the recursive call and it has to somehow store them into something called a call stack. So effectively, there is one call stack per recursive call. There is now an obvious reason to worry about stack overflows in case of a relatively huge List.

Let's now understand how tail recursion helps us solve this problem with the same example tweaked a little:

1@@scala.annotation.tailrec
2def tailRecursive(intList: List[Int], accumulator: Int): Int = {
3  if (intList.length == 0) accumulator
4  else tailRecursive(intList.tail, accumulator + intList.head)
5}
6
7val sumOfInts = tailRecursive(List(1,2,3,4,5,6,7,8,9), 0) // prints 45

The difference now is pretty straight forward, no not just in the method name, but the last call in the method is actually calling itself without any additional calculation which expects the method to first return the value and then perform the calculation. In the tail recursive case above, the sum is first evaluated (in the accumulator + intList.head) and then the recursive call is made.

This way, there is no need for the runtime to store the local variables during each recursive call in a call stack. The runtime can re-enter the same call stack again and again thereby eliminating completely a potential stack overflow error. Was that clear enough?


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# swirl Lesson 5: Missing Values

| Please choose a course, or type 0 to exit swirl.
1: R Programming
2: Take me to the swirl course repository!
Selection: 1
1: Basic Building Blocks      2: Workspace and Files
3: Sequences of Numbers       4: Vectors
5: Missing Values             6: Subsetting Vectors
7: Matrices and Data Frames   8: Logic
9: Functions                 10: lapply and sapply
11: vapply and tapply         12: Looking at Data
13: Simulation                14: Dates and Times
15: Base Graphics
Selection: 5
|                                                     |   0%
| Missing values play an important role in statistics and data
| analysis. Often, missing values must not be ignored, but
| rather they should be carefully studied to see if there's an
| underlying pattern or cause for their missingness.
...
|===                                                  |   5%
| In R, NA is used to represent any value that is 'not
| available' or 'missing' (in the statistical sense). In this
| lesson, we'll explore missing values further.
...
|=====                                                |  10%
| Any operation involving NA generally yields NA as the
| result. To illustrate, let's create a vector c(44, NA, 5,
| NA) and assign it to a variable x.
> x<-c(44,NA,5,NA)
| Perseverance, that's the answer.
|========                                             |  15%
| Now, let's multiply x by 3.
> 3*x
[1] 132  NA  15  NA
| Excellent job!
|===========                                          |  20%
| Notice that the elements of the resulting vector that
| correspond with the NA values in x are also NA.
...
|=============                                        |  25%
| To make things a little more interesting, lets create a
| vector containing 1000 draws from a standard normal
| distribution with y <- rnorm(1000).
> y<-rnorm(1000)
| Keep up the great work!
|================                                     |  30%
| Next, let's create a vector containing 1000 NAs with z <-
| rep(NA, 1000).
> z<-rep(NA,1000)
| You're the best!
|===================                                  |  35%
| Finally, let's select 100 elements at random from these 2000
| values (combining y and z) such that we don't know how many
| NAs we'll wind up with or what positions they'll occupy in
| our final vector -- my_data <- sample(c(y, z), 100).
> my_data <- sample(c(y, z), 100)
| You are really on a roll!
|=====================                                |  40%
| Let's first ask the question of where our NAs are located in
| our data. The is.na() function tells us whether each element
| of a vector is NA. Call is.na() on my_data and assign the
| result to my_na.
> my_na<- is.na(my_data)
| You are doing so well!
|========================                             |  45%
| Now, print my_na to see what you came up with.
> my_na
[1] FALSE FALSE FALSE FALSE  TRUE FALSE FALSE  TRUE  TRUE
[10]  TRUE  TRUE FALSE  TRUE  TRUE FALSE  TRUE FALSE FALSE
[19]  TRUE FALSE  TRUE  TRUE  TRUE FALSE  TRUE  TRUE  TRUE
[28]  TRUE  TRUE FALSE FALSE  TRUE  TRUE  TRUE FALSE  TRUE
[37] FALSE FALSE FALSE  TRUE FALSE  TRUE  TRUE FALSE FALSE
[46] FALSE FALSE  TRUE FALSE  TRUE FALSE FALSE FALSE FALSE
[55] FALSE  TRUE FALSE  TRUE  TRUE FALSE FALSE  TRUE FALSE
[64]  TRUE FALSE  TRUE FALSE  TRUE FALSE  TRUE  TRUE FALSE
[73] FALSE  TRUE  TRUE FALSE FALSE FALSE FALSE FALSE  TRUE
[82] FALSE FALSE FALSE FALSE  TRUE FALSE  TRUE  TRUE FALSE
[91] FALSE  TRUE  TRUE FALSE FALSE FALSE  TRUE  TRUE FALSE
[100]  TRUE
| You nailed it! Good job!
|==========================                           |  50%
| Everywhere you see a TRUE, you know the corresponding
| element of my_data is NA. Likewise, everywhere you see a
| FALSE, you know the corresponding element of my_data is one
| of our random draws from the standard normal distribution.
...
|=============================                        |  55%
| In our previous discussion of logical operators, we
| introduced the == operator as a method of testing for
| equality between two objects. So, you might think the
| expression my_data == NA yields the same results as is.na().
| Give it a try.
> my_data == NA
[1] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
[20] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
[39] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
[58] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
[77] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
[96] NA NA NA NA NA
| You got it!
|================================                     |  60%
| The reason you got a vector of all NAs is that NA is not
| really a value, but just a placeholder for a quantity that
| is not available. Therefore the logical expression is
| incomplete and R has no choice but to return a vector of the
| same length as my_data that contains all NAs.
...
|==================================                   |  65%
| Don't worry if that's a little confusing. The key takeaway
| is to be cautious when using logical expressions anytime NAs
| might creep in, since a single NA value can derail the
| entire thing.
...
|=====================================                |  70%
| So, back to the task at hand. Now that we have a vector,
| my_na, that has a TRUE for every NA and FALSE for every
| numeric value, we can compute the total number of NAs in our
| data.
...
|========================================             |  75%
| The trick is to recognize that underneath the surface, R
| represents TRUE as the number 1 and FALSE as the number 0.
| Therefore, if we take the sum of a bunch of TRUEs and
| FALSEs, we get the total number of TRUEs.
...
|==========================================           |  80%
| Let's give that a try here. Call the sum() function on my_na
| to count the total number of TRUEs in my_na, and thus the
| total number of NAs in my_data. Don't assign the result to a
| new variable.
> sum(my_na)
[1] 46
| Great job!
|=============================================        |  85%
| Pretty cool, huh? Finally, let's take a look at the data to
| convince ourselves that everything 'adds up'. Print my_data
| to the console.
> my_data
[1]  1.28833738 -1.71566792 -2.10643156  0.52646288
[5]          NA -0.43369979 -0.73588709          NA
[9]          NA          NA          NA  0.04198942
[13]          NA          NA -0.07881950          NA
[17] -0.95076395  2.31028848          NA -0.73152858
[21]          NA          NA          NA  1.05294332
[25]          NA          NA          NA          NA
[29]          NA -0.59012503 -0.38427819          NA
[33]          NA          NA -0.48309105          NA
[37]  0.32180094 -1.05520535  0.27521195          NA
[41] -1.62668329          NA          NA -2.18910058
[45]  1.74314263 -0.07327235  1.49444965          NA
[49]  1.05747803          NA  0.71787615 -1.11394933
[53] -0.08661972  0.87982505 -0.51738201          NA
[57]  0.40314933          NA          NA  0.17416071
[61] -0.46598802          NA  0.01918326          NA
[65] -0.47643745          NA  0.96464973          NA
[69]  1.07043774          NA          NA  0.55960535
[73]  0.41772515          NA          NA -0.41373806
[77]  0.80355818 -1.01706160 -1.36218341 -1.55671571
[81]          NA  1.42576312 -0.49953533 -0.52994970
[85]  0.62437351          NA -0.53641307          NA
[89]          NA -0.08968300  0.18669337          NA
[93]          NA  1.05551995  1.19631709 -0.47725033
[97]          NA          NA -1.49508777          NA
| You're the best!
|================================================     |  90%
| Now that we've got NAs down pat, let's look at a second type
| of missing value -- NaN, which stands for 'not a number'. To
| generate NaN, try dividing (using a forward slash) 0 by 0
| now.
> 0/0
[1] NaN
| Your dedication is inspiring!
|==================================================   |  95%
| Let's do one more, just for fun. In R, Inf stands for
| infinity. What happens if you subtract Inf from Inf?
> Inf-Inf
[1] NaN
| All that hard work is paying off!
|=====================================================| 100%


text:
# Maximal Number of Pairs of Orthogonal vectors in a set of $n$ vectors in $\mathbb{R}^3$

Suppose you are given a set of $n$ non-zero vectors in $\mathbb{R}^3$. What is the maximum number of pairs of them that are orthogonal? The current guess is $\le 2n$.

EDIT: I forgot to add that no two vectors should be colinear.

• Definitely it is superlinear. Take all vectors with integer coordinates in a large ball $x^2+y^2+z^2<R^2$. – Fedor Petrov Aug 11 '15 at 19:48
• It's possible to get $n^{3/2}$. Just take $(k,0,0)$, and $(0,m,0)$ for $k,m=1,...,\sqrt{n}$. – Oleg Eroshkin Aug 11 '15 at 20:27
• Oleg, if $k,m$ vary up to $n/2$ we get even as many as $n^2/4$ pairs, but this is probably not permitted:) – Fedor Petrov Aug 11 '15 at 20:43
• Still better, take $n/3$ nonzero points on each coordinate axis to get $(2/3)n^2$ pairs. Since there are no four points pairwise orthogonal the story is completed by Turan's theorem. Perhaps a more interesting question was intended though. – Sean Eberhard Aug 11 '15 at 21:54
• @OlegEroshkin: You're getting $n^3/9$ pairs from $n$ vectors? I don't think so. – Robert Israel Aug 11 '15 at 22:45

The maximum is $cn^{4/3}$ for some constant $c$.

We may as well assume all our points are on the unit sphere $S$. Let $P$ be some plane not containing the origin, which we might think of as being far away. For each point $x\in S$ in our collection let $p_x$ be the intersection of the line through $0$ and $x$ with $P$ and let $\ell_x$ be the intersection of the hyperplane orthogonal to $x$ with $P$. Unless $P$ was chosen by your enemies then all these things are well defined. Note that $p_x\in\ell_y$ iff $x$ and $y$ are orthogonal, so orthogonality in our original collection becomes incidence in our new collection, and by the Szemeredi-Trotter theorem there are at most $O(n^{4/3})$ incidences.

To see that there is a construction with this many orthogonal pairs, take some example which shows that Szemeredi-Trotter is tight, and such that there are about as many points as lines, and read the above paragraph backwards. If $P$ is far away then this example will consist of two collections of about $n/2$ points carefully clustered around some two orthogonal points of $S$.

I read this construction in the following paper of Erdos, Hickerson, and Pach, which also contains references to many other things. See Theorem 2(ii) for this problem.

http://www.renyi.hu/~p_erdos/1989-02.pdf

• When we read the paragraph backwards, are we to construct different sets of points $x$ and $y$ on $S$ for the points $p_x$ and lines $l_y$ on $P$? Or is there some way of ensuring that every point $p_x$ agrees with some line $l_x$ on the resulting point $x$? Either way, you would still get $\sim n^{4/3}$, but I couldn't understand what "read the paragraph backwards" meant. – Yoav Kallus Aug 13 '15 at 14:20
• @YoavKallus Construct different sets of points $x$ and $y$. As you say, it only affects the constant. – Sean Eberhard Aug 13 '15 at 15:22
• I wonder whether this correspondence works for the finite vector space ${\mathbb F}_q^r$. Assuming we have a system of $n$ points and $l$ lines in ${\mathbb F}_q^r$ that determines $I$ incidences, how many points we get in ${\rm PG}(r,q)$ and how many pairs of them will be orthogonal? – Seva Aug 13 '15 at 17:39
• @Seva I think most of it carries over, assuming "line" means $(r-1)$-dimensional hyperplane. Realize our copy of $\mathbb{F}_q^r$ as an affine hyperplane $P$ of $\mathbb{F}_q^{r+1}$. For each point in our collection we get a point in $\text{PG}(r,q)$ by projection, and for each "line" in our collection we get an $r$-dimensional hyperplane in $\mathbb{F}_q^{r+1}$ and hence a point in $\text{PG}(r,q)$ by duality. As before incidence corresponds to orthogonality. The number of points we get is between $\max(n,l)$ and $n+l$ and the number of orthogonal pairs is at least the number of incidences. – Sean Eberhard Aug 13 '15 at 17:58