text:
The MSR Mass and the \mathcal{O}(\Lambda_{\mathrm{QCD}}) Renormalon Sum Rule

# The MSR Mass and the O(ΛQCD) Renormalon Sum Rule

André H. Hoang Erwin Schrödinger International Institute for Mathematical Physics,
University of Vienna, Boltzmanngasse 9, A-1090 Wien, AustriaDepartamento de Física Fundamental e IUFFyM,
Universidad de Salamanca, E-37008 Salamanca, SpainInstituto de Física Teórica UAM-CSIC,
Ambar Jain Erwin Schrödinger International Institute for Mathematical Physics,
University of Vienna, Boltzmanngasse 9, A-1090 Wien, AustriaDepartamento de Física Fundamental e IUFFyM,
Universidad de Salamanca, E-37008 Salamanca, SpainInstituto de Física Teórica UAM-CSIC,
Christopher Lepenik Erwin Schrödinger International Institute for Mathematical Physics,
University of Vienna, Boltzmanngasse 9, A-1090 Wien, AustriaDepartamento de Física Fundamental e IUFFyM,
Universidad de Salamanca, E-37008 Salamanca, SpainInstituto de Física Teórica UAM-CSIC,
Vicent Mateu Erwin Schrödinger International Institute for Mathematical Physics,
University of Vienna, Boltzmanngasse 9, A-1090 Wien, AustriaDepartamento de Física Fundamental e IUFFyM,
Universidad de Salamanca, E-37008 Salamanca, SpainInstituto de Física Teórica UAM-CSIC,
Moritz Preisser Erwin Schrödinger International Institute for Mathematical Physics,
University of Vienna, Boltzmanngasse 9, A-1090 Wien, AustriaDepartamento de Física Fundamental e IUFFyM,
Universidad de Salamanca, E-37008 Salamanca, SpainInstituto de Física Teórica UAM-CSIC,
Ignazio Scimemi Erwin Schrödinger International Institute for Mathematical Physics,
University of Vienna, Boltzmanngasse 9, A-1090 Wien, AustriaDepartamento de Física Fundamental e IUFFyM,
Universidad de Salamanca, E-37008 Salamanca, SpainInstituto de Física Teórica UAM-CSIC,
Iain W. Stewart Erwin Schrödinger International Institute for Mathematical Physics,
University of Vienna, Boltzmanngasse 9, A-1090 Wien, AustriaDepartamento de Física Fundamental e IUFFyM,
Universidad de Salamanca, E-37008 Salamanca, SpainInstituto de Física Teórica UAM-CSIC,
###### Abstract

We provide a detailed description and analysis of a low-scale short-distance mass scheme, called the MSR mass, that is useful for high-precision top quark mass determinations, but can be applied for any heavy quark . In contrast to earlier low-scale short-distance mass schemes, the MSR scheme has a direct connection to the well known mass commonly used for high-energy applications, and is determined by heavy quark on-shell self-energy Feynman diagrams. Indeed, the MSR mass scheme can be viewed as the simplest extension of the mass concept to renormalization scales . The MSR mass depends on a scale that can be chosen freely, and its renormalization group evolution has a linear dependence on , which is known as R-evolution. Using R-evolution for the MSR mass we provide details of the derivation of an analytic expression for the normalization of the renormalon asymptotic behavior of the pole mass in perturbation theory. This is referred to as the renormalon sum rule, and can be applied to any perturbative series. The relations of the MSR mass scheme to other low-scale short-distance masses are analyzed as well.

\preprint

UWThPh-2017-6

MIT-CTP 4896

IFT-UAM/CSIC-17-034

Indian Institute of Science Education and Research Bhopal,
Bhopal Bypass Road, Bhopal 462066, IndiaUniversity of Vienna, Faculty of Physics,
E-28040 Madrid, SpainCenter for Theoretical Physics, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA

## 1 Introduction

Achieving higher precision in theoretical predictions in the framework of quantum chromo dynamics (QCD) is one of the main goals in high-energy physics and an essential ingredient in the indirect search for physics beyond the Standard Model. In this endeavor accurate determinations of the masses of the heavy charm, bottom and top quarks play an important role since they enter the description of many observables that are employed in consistency tests of the Standard Model and in the exploration of models of new physics. Because quark masses are formally-defined renormalized quantities and not physical observables, the quantities from which the heavy quark masses are extracted need to be computed in perturbative QCD to high order. Among the most precise recent high-order analyses to determine the heavy quark masses are QCD sum rules and the analysis of quarkonium energies for the charm and bottom quark masses Dehnadi:2011gc (); Bodenstein:2011ma (); Bodenstein:2011fv (); Hoang:2012us (); Chakraborty:2014aca (); Colquhoun:2014ica (); Beneke:2014pta (); Ayala:2014yxa (); Dehnadi:2015fra (); Erler:2016atg () and the top pair production threshold cross section at a future lepton collider for the top quark mass Hoang:2000yr (); Hoang:2013uda (); Beneke:2015kwa (). Over time all of these analyses have been continuously updated and improved by computations of new QCD corrections, and more are being designed and studied currently to also allow for more precise determinations of the top quark mass from available LHC data Czakon:2013goa (); Khachatryan:2016mqs (); Aaboud:2016pbd (); Czakon:2016ckf (); Alioli:2013mxa (); Frixione:2014ala (); Chatrchyan:2013boa (); Kharchilava:1999yj ().

In all the analyses of Refs. Dehnadi:2011gc (); Bodenstein:2011ma (); Bodenstein:2011fv (); Hoang:2012us (); Chakraborty:2014aca (); Colquhoun:2014ica (); Beneke:2014pta (); Ayala:2014yxa (); Dehnadi:2015fra (); Erler:2016atg (); Hoang:2000yr (); Hoang:2013uda (); Beneke:2015kwa () the use of short-distance mass schemes was essential to achieve a well-converging perturbative expansion and a precision in the mass determination well below the hadronization scale  MeV. The heavy quark pole mass , which is the perturbation theory equivalent of the rest mass of an on-shell quark, on the other hand, leads to a substantially worse perturbative behavior due to its linear infrared-sensitivity, also known as the renormalon problem Bigi:1994em (); Beneke:1994sw (), and was therefore not adopted as a relevant mass scheme for analyses where a precision better than could be achieved. Nevertheless, the pole mass still served as an important intermediate mass scheme during computations because it determines the partonic (but unphysical) poles of heavy quark Green functions. Typical short-distance quark mass schemes which have been employed were the renormalization-scale dependent mass and so-called low-scale short-distance masses such as the kinetic mass Czarnecki:1997sz (), the potential-subtracted (PS) mass Beneke:1998rk (), the 1S mass Hoang:1998ng (); Hoang:1998hm (); Hoang:1999ye (), the renormalon-subtracted (RS) mass Pineda:2001zq () or the jet mass Jain:2008gb (); Fleming:2007qr (). The basic difference between the mass to the low-scale short-distance mass schemes is that the perturbative coefficients of its relation to the pole mass scale linearly with the heavy quark mass, , while for the low-scale short-distance mass schemes the corresponding series scales linearly with a scale . This feature enables the low-scale short-distance quark mass schemes to be used for predictions of quantities where the heavy quark dynamics is non-relativistic in nature and fluctuations at the scale of are integrated out. This is because radiative corrections to the mass in such quantities involve physical scales much smaller than . One very prominent example in the context of top quark physics is the non-relativistic heavy quarkonium dynamics inherent to the top-antitop pair production cross section at threshold at a future lepton collider Hoang:2000yr (); Hoang:2013uda (); Beneke:2015kwa (), where the most important dynamical scale is the inverse Bohr radius  GeV . On the other hand, the mass is a good scheme choice for quantities that involve energies much larger than , such as for high-energy total cross sections, or when the massive quark causes virtual and off-shell effects. This is because in such cases the heavy quark mass yields corrections that either scale with positive or negative powers of such that QCD corrections associated with the mass have a scaling that is linear in as well. The difference between the mass and the low-scale short-distance masses is most important for the case of the top quark because in this case the difference between and the dynamical low-energy scales can be very large numerically.

For the top quark mass there are excellent prospects for very precise measurements in low-scale short-distance schemes such as the PS mass or the 1S mass from the top-antitop threshold inclusive cross section at a future lepton collider Hoang:2000yr (); Hoang:2013uda (); Beneke:2015kwa (). Current studies indicate that a precision well below  MeV can be achieved accounting for theoretical as well as experimental uncertainties Seidel:2013sqa (); Horiguchi:2013wra (); Vos:2016til (). Currently, the most precise measurements of the top quark mass come from reconstruction analyses at the LHC Khachatryan:2015hba (); Aaboud:2016igd () and the Tevatron Tevatron:2014cka () and have uncertainties at the level of  MeV or larger. Moreover, the mass is obtained from multivariate fits involving multipurpose Monte Carlo (MC) event generators and thus represents a determination of the top quark mass parameter contained in the particular MC event generator. Recently, a first high-precision analysis on how the MC top quark mass parameter can be related to a field theoretically well-defined short-distance top quark mass was provided in Refs. Butenschoen:2016lpz (); Hoang:2017kmk () and general considerations on the relation were discussed in Ref. Hoang:2008xm (); Hoang:2014oea (). For the analysis, hadron level predictions for the 2-jettiness distribution Stewart:2010tn () for electron-positron collisions and QCD corrections together with the resummation of large logarithms at next-to-next-to leading order Fleming:2007qr (); Fleming:2007tv (); Hoang:2007vb () were employed. Since the 2-jettiness distribution is closely related to the invariant mass distribution of a single reconstructed top quark, the relevant dynamical scales inherent to the problem are governed by the width of the mass distribution which amounts to only about  GeV in the peak region of the distribution where the sensitivity to the top mass is the highest. Interestingly, as was shown in Ref. Butenschoen:2016lpz (), the dynamical scales increase continuously considering the 2-jettiness distribution further away from the peak. In the analysis of Butenschoen:2016lpz () the MSR mass scheme was employed which depends on a scale and for which the dependence on is described by a renormalization group flow such that can be continuously adapted according to which part of the distribution is predicted. Other applications of the MSR mass using a flavor number dependent evolution in to account for the mass effects of lighter quarks were given in Ref. Hoang:2017btd (); Mateu:2017hlz (). In contrast to the -dependent mass , which evolves only logarithmically in , the MSR mass has logarithmic as well as linear dependence on .

The MSR mass scheme was succinctly introduced in Ref. Hoang:2008yj () and discussed conceptually in Ref. Hoang:2014oea (), but a detailed discussion has so far not been provided. A key purpose of this paper is to provide sufficient details such that phenomenological MSR mass analyses, such as the results of Ref. Butenschoen:2016lpz (), can be easily related to other common short-distance mass schemes that are being used in the literature.

The definition of the MSR mass given by the perturbative series for the MSR-pole mass difference is obtained directly from the -pole mass relation and is therefore the only low-scale short-distance mass suggested in the literature that is derived directly from on-shell heavy quark self-energy diagrams just like the mass.111The name ‘MSR mass’ arises from a combination of the letters ‘MS’ standing for the close relation to the mass and the letter ‘R’ standing for R-evolution. The MSR mass thus automatically inherits the clean and good infrared properties of the mass. Furthermore, by construction, the MSR mass matches to the mass for and is known to the same order as the series of without any further effort, which is currently from the results of Refs. Tarrach:1980up (); Gray:1990yh (); Melnikov:2000qh (); Chetyrkin:1999ys (); Chetyrkin:1999qi (); Marquard:2007uj (); Marquard:2015qpa (); Marquard:2016dcn (). As already argued in Refs. Hoang:2008yj (); Hoang:2008xm (), the MSR mass can therefore be considered as the natural modification of the “running” mass scheme concept for renormalization scales below , where the logarithmic evolution of the regular mass is known to be unphysical.

Since the MSR mass is designed to be employed for scales , it can be useful – for applications where a clean treatment of virtual massive-flavor effects is important – to integrate out the virtual effects of the massive quark from the MSR mass definition. We therefore introduce two types of MSR masses, one where the virtual effects of the massive quark are integrated out, called the natural MSR mass, and one where these effects are not integrated out, called the practical MSR mass. The difference between these two versions of the MSR mass is quite small and very well behaved for all values in the perturbative region, and the practical definition should be perfectly fine for most phenomenological applications. But the natural definition has conceptual advantages as its evolution for scales does not include the virtual effects of the massive quark , which is conceptually cleaner since these belong physically to the scale .

We note that the R-evolution concept of a running heavy quark mass scheme for scales elaborated in Ref. Hoang:2008yj () has already been suggested a long time ago in Refs. Voloshin:1992wg (); Bigi:1997fj (). The R-evolution equation we discuss for the MSR mass was already quoted explicitly for the renormalization group evolution of the kinetic mass Czarnecki:1997sz () at in these references, but the conceptual implications of R-evolution and its connection to the renormalon problem in the perturbative relations between short-distance masses and the pole mass were first studied systematically in Ref. Hoang:2008yj (). The second main purpose of this paper is to give further details on R-evolution and also to discuss its relation to the Borel transformation focusing mainly on the case of the MSR mass. We note that the concept of R-evolution is quite general and can in principle be applied to any short-distance mass which depends on a variable infrared cutoff scale (such as the PS and the RS masses) or to cutoff-dependent QCD matrix elements with arbitrary dimensions. In fact, R-evolution has already been examined and applied in a number of other applications which include the factorization-scale dependence in the context of the operator product expansion Hoang:2009yr (), the scale dependence of the non-perturbative soft radiation matrix element in high-precision determinations of the strong coupling from event-shape distributions Abbate:2010xh (); Abbate:2012jh (); Hoang:2014wka (); Hoang:2015hka (), even accounting for the finite mass effects of light quarks Gritschacher:2013tza (); Pietrulewicz:2014qza () and hadrons Mateu:2012nk (); Hoang:2014wka ().

The basic feature of the R-evolution concept is that for the difference of MSR masses at two scales, , its linear dependence on the renormalization scale provides, completely within perturbation theory, a resummation of the terms in the asymptotic series associated to the pole-mass renormalon ambiguity to all orders. The R-evolution then resums the factorially growing terms in a systematic way that is -renormalon free and, at the same time also sums all large logarithms that arise if and are widely separated. This cannot be achieved by more common purely logarithmic renormalization group equations, but is fully compatible with a Wilsonian renormalization group setup. We note that the summations carried out by the R-evolution was achieved prior to Ref. Hoang:2008yj () for the RS mass in Bali:2003jq () (see also Ref. Campanario:2005np ()). Their method (and the RS mass) is based on using an approximate expression for the Borel transform function. The summation for a difference of RS masses (for scales and ) is obtained by computing the inverse Borel integral over the difference of the two respective Borel functions. This method and R-evolution lead to consistent results, but the R-evolution does not rely on the knowledge of the Borel functions.

The essential and probably most interesting conceptual feature of the perturbative series of the R-evolution equations is that it provides a systematic reordering of the terms in the asymptotic series associated to the renormalon ambiguity in leading, subleading, subsubleading, etc. contributions. So using the analytic solution of the R-evolution equations allows one to derive analytically (i.e. without any numerical procedure or modeling) the Borel-transform of a given perturbative series from the perspective that it carries an renormalon ambiguity. As a result one can rigorously derive an analytic expression for the normalization of the non-analytic terms in the Borel transform that are characteristic for the renormalon. The analytic result for this normalization factor was already given and discussed in Ref. Hoang:2008yj (), but no details on the derivation were provided. We take the opportunity to show the details of the derivation here. We call the analytic result for the normalization of the renormalon ambiguity the sum rule, because it can be quickly applied to any given perturbative series. To demonstrate the use and the high sensitivity of the renormalon sum rule we apply it also to a number of other cases, pointing out subtleties in its application to avoid inconsistencies and misinterpretations of the results.

We note that also other methods to determine the normalization factor have been used. In Ref. Pineda:2001zq () it was determined from a computation of the residue of the Borel transform of the series following a proposal in Ref. Lee:1996yk (). This approach, which we call Borel method can also be carried out analytically and provides the correct result, but has been observed to converge very slowly. We can identify the reason for this analytically from the solutions for the R-evolution equations, and we also discuss the connection of this method to our sum rule based on explicit analytic expressions. In Ref. Bali:2013pla () the normalization factor was computed taking the ratio of the -th term of the series to the asymptotic behavior. This ratio method converges very fast and provides results very similar to the sum rule. Recently, the ratio method was applied in Ref. Beneke:2016cbu (), accounting for the corrections to the pole- mass relation Marquard:2015qpa (); Marquard:2016dcn (). We show that our sum rule provides results that are in full agreement with the ones obtained in Ref. Beneke:2016cbu () and also leads to very similar uncertainties.

The paper is organized as follows: In Sec. 2 we provide the definition of the natural and practical MSR masses, and , based on the perturbative series of the -pole mass relation , and we also analyze the difference between these two MSR masses. This section provides the conventions we use for the coefficients of perturbative series, but it can otherwise be skipped by the reader not interested in the MSR masses. In Sec. 3 we present the R-evolution equations which describe the scale dependence of the MSR masses and we also show explicitly how the solutions of the R-evolution equations sum large logarithms together with the high-order asymptotic series terms related to the renormalon. We in particular show for the top quark mass under which conditions the use of the R-evolution equations and its resummation is essential and superior to renormalon-free fixed-order perturbation theory, which does not sum any large logarithms. To our knowledge, such an analysis has not been provided in the literature before. We also point out that the solution of the R-evolution equations is intrinsically related to carrying out an inverse Borel transform over differences of functions in the Borel plane such that the singularities related to the renormalon cancel. In Sec. 4 we present the analytic derivation of the renormalon sum rule and demonstrate its utility by a detailed analysis concerning the normalization of the renormalon ambiguity in the series for the difference of the pole mass and the MSR masses. The derivation of the sum rule allows to derive a new alternative expression for the high-order asymptotic behavior of a series that contains an renormalon which we discuss as well. To demonstrate the high sensitivity of the sum rule and to explain its consistent (and inconsistent) application we discuss its strong flavor number dependence and apply it to the massive quark vacuum polarization function, the series for the PS mass-pole mass difference, the QCD -function, and the hadronic R-ratio. This section can be bypassed by the reader not interested in applications of the sum rule, but we note that Sec. 4.5.3 discusses implications for the PS mass that are relevant for Sec. 5 and may be important for high-precision top quark mass determinations. Some subtle issues in the relation of the MSR masses to the PS, 1S and masses are discussed in Sec. 5. Finally, we conclude in Sec. 6. The paper also contains two appendices. In App. A we specify our convention for the QCD -function coefficients and present a number of expressions and formulae for coefficients, quantities and matching relations that arise in the discussion of R-evolution, the renormalon and on various mass definitions throughout this paper. In App. B we provide details on the relation of the Borel method and our sum rule method to determine the normalization of the renormalon ambiguity of the pole mass. Finally, in App. C we quote the coefficients that define the PS and the 1S masses for the convenience of the reader and also show how the MSR masses can be obtained from a given value of the 1S mass in the non-relativistic and -expansion counting scheme Hoang:1998hm (); Hoang:1998ng ().

## 2 MSR Mass Setup

### 2.1 Basic Idea of the MSR Mass

The mass serves as the standard short-distance mass scheme for many high-energy applications with physical scales of the order or larger than the mass of the quark . It relies on the subtraction of the divergences in the common scheme in the on-shell self-energy corrections calculated in dimensional regularization. Despite the fact that it is an unphysical (i.e. theoretically designed) mass definition, it is infrared-safe and gauge invariant to all orders Tarrach:1980up (); Kronfeld:1998di () and its series relation to the pole mass thus serves as the cleanest way to precisely quantify the renormalon ambiguity of the pole mass. The relation of to the pole mass in the approximation that the masses of all quarks lighter than are zero reads

 mpoleQ−¯¯¯¯¯mQ=¯¯¯¯¯mQ∞∑n=1a¯¯¯¯¯¯¯MSn(nℓ,nh)(α(nℓ+1)s(¯¯¯¯¯mQ)4π)n, (1)

with

 a¯¯¯¯¯¯¯MS1(nℓ,nh) =163, (2) a¯¯¯¯¯¯¯MS2(nℓ,nh) =213.437+1.65707nh−16.6619nℓ, a¯¯¯¯¯¯¯MS3(nℓ,nh) =12075.+118.986nh+4.10115n2h−1707.35nℓ+1.42358nhnℓ+41.7722n2ℓ, a¯¯¯¯¯¯¯MS4(nℓ,nh) =(911588.±417.)+(1781.61±30.72)nh−(60.1637±0.6912)n2h −(231.201±0.102)nhnℓ−(190683.±10.)nℓ+9.25995n2hnℓ +6.35819n3h+4.40363nhn2ℓ+11105.n2ℓ−173.604n3ℓ,

where stands for the strong coupling that renormalization-group (RG) evolves with active flavors, see Eq. (67). The coefficients at are known analytically from Refs. Tarrach:1980up (); Gray:1990yh (); Chetyrkin:1999ys (); Chetyrkin:1999qi (); Melnikov:2000qh (); Marquard:2007uj (). The coefficient was determined numerically in Refs. Marquard:2015qpa (); Marquard:2016dcn (), and the quoted numerical uncertainties have been taken from Ref. Marquard:2016dcn (). Using the method of Ref. Kataev:2015gvt () the uncertainties of the -dependent terms may be further reduced. Using renormalon calculus Bigi:1994em (); Beneke:1994sw (); Beneke:1998ui () one can show that the high-order asymptotic behavior series of Eq. (1) has an ambiguity of order , which depends on the number of massless quarks (indicated by the superscript) but is independent of the actual value of .

A coherent treatment of the mass effects of lighter quarks is beyond the scope of this paper, and we therefore use the approximation that all flavors lighter than are massless. These mass corrections come from the insertion of massive virtual quark loops in the self-energy Feynman diagrams and start at . At this order and at the mass corrections from the virtual massive quark loops have been calculated analytically for all mass values in Ref. Gray:1990yh () and Bekavac:2007tk (), respectively. The dominant linear mass corrections at were determined in Ref. Hoang:2000fm (). At and the mass corrections are not yet known, but the corrections in the limit of large virtual quark masses are encoded in the ultraheavy flavor threshold matching relations of the RG-evolution at scales above  Chetyrkin:1997un ().

The idea of the MSR mass is based on the fact that the ambiguity of the perturbative series on the RHS of Eq. (1) does not depend on the value , as already mentioned above. This is an exact mathematical statement within the context of the calculus for asymptotic series and means that we can replace the term by the arbitrary scale on the RHS of Eq. (1) and use the resulting perturbative series as the definition of the -dependent MSR mass scheme. It was pointed out in Ref. Hoang:2014oea () that, for a given value of , one can also interpret the MSR mass field theoretically as having a mass renormalization constant that contains the on-shell self-energy corrections of the pole mass only for scales larger than . In other words, the pole mass and the MSR mass at the scale differ by self-energy corrections from scales below : while the pole mass absorbs all self-energy corrections for quantum fluctuations up to scales , the MSR mass at the scale absorbs only self-energy corrections between and . Since the pole mass renormalon problem is related to the self-energy corrections from the scale , this explains why the MSR mass is a short-distance mass. In this illustrative context the mass absorbs no self-energy corrections up to the scale . Since the scale is variable, the MSR mass can serve as a short-distance mass definition for applications governed by different physical scales and thus can also interpolate between them. Since the MSR mass is expected to have applications primarily for , it is further suitable to change the scheme from dynamical flavors, which includes the UV effects of the quark , to a scheme with dynamical flavors. This can be achieved in two ways, either by simply rewriting in terms of , or by integrating out the virtual loop corrections of the quark . This results in two different ways to define the MSR mass, where we call the former the practical MSR mass and the latter the natural MSR mass, either one having advantages depending on the application.

We note that the notion of a scale-dependent short-distance mass which was first suggested in Refs. Voloshin:1992wg (); Bigi:1997fj () has also been adopted for the kinetic Czarnecki:1997sz (), the PS Beneke:1998rk (), RS Pineda:2001zq () and jet masses Jain:2008gb (); Fleming:2007tv (). However, none of these short-distance masses is defined directly from the on-shell self-energy diagrams of the massive quark such as the MSR mass. This has a number of advantages, for example when discussing heavy flavor symmetry properties in the pole- mass relation of different heavy quarks.

### 2.2 Natural MSR Mass

The natural MSR mass definition is obtained by integrating out the corrections from the heavy quark virtual loops in the self-energy diagrams of the massive quark , such that its relation to the pole mass reads

 mpoleQ−mMSRnQ(R)=R∞∑n=1a¯¯¯¯¯¯¯MSn(nℓ,0)(α(nℓ)s(R)4π)n, (3)

where the coefficients are given in Eq. (2). The natural MSR mass only accounts for gluonic and massless quark corrections, and has a non-trivial matching relation to the mass. The matching between the natural MSR mass and the mass can be derived from the relation [  ]

 mMSRnQ(¯¯¯¯¯mQ)−¯¯¯¯¯mQ=¯¯¯¯¯mQ∞∑k=1[a¯¯¯¯¯¯¯MSk(nℓ,1)(α(nℓ+1)s(¯¯¯¯¯mQ)4π)k−a¯¯¯¯¯¯¯MSk(nℓ,0)(α(nℓ)s(¯¯¯¯¯mQ)4π)k], (4)

and will be discussed in more detail in Sec. 5.3.

We note that, formally, the natural MSR mass (as well as the practical MSR mass discussed in the next subsection) agrees with the pole mass in the limit . However, taking this limit is ambiguous as it involves evolving through the Landau pole of the strong coupling and dealing with its non-perturbative definition for . This issue is a manifestation of the renormalon problem of the pole mass.

### 2.3 Practical MSR Mass

The practical MSR mass definition is directly related to the -pole perturbative series of Eq. (1). To obtain its defining series one rewrites as a series in in Eq. (1) using the matching relation given in Eq. (73) and then replaces by , obtaining

 mpoleQ−mMSRpQ(R)=R∞∑n=1aMSRpn(nℓ)(α(nℓ)s(R)4π)n, (5)

with

 aMSRp1(nℓ) =163, (6) aMSRp2(nℓ) =215.094−16.6619nℓ, aMSRp3(nℓ) =12185.−1705.93nℓ+41.7722n2ℓ, aMSRp4(nℓ) =(911932.±418.)−(190794.±10.)nℓ+11109.4n2ℓ−173.604n3ℓ.

The practical MSR mass still accounts for the virtual corrections from the massive quark Q with an evolving mass and has the convenient feature that it agrees with the mass at the scale of the mass to all orders in perturbation theory [  ]:

 mMSRpQ(mMSRpQ)=¯¯¯¯¯mQ(¯¯¯¯¯mQ). (7)

The formula for the difference of the natural and practical MSR masses at the same scale up to reads

 +((344.±31.)−(111.59±0.10)nℓ+4.40n2ℓ)(α(nℓ)s(R)4π)4+…]. (8)

In Fig. 1 the difference between the natural and the practical MSR top quark masses is shown for between and  GeV (here ).222Throughout this article we use and GeV. The numerical difference between these two masses is quite small. The natural MSR mass is larger than the practical MSR mass and the difference increases with reaching about  MeV at  GeV. The error bands reflect variations of the renormalization scale in between and , showing very good convergence, exhibiting a perturbative error of  MeV for  GeV and below  MeV for  GeV due to missing terms of and higher. This indicates that the different way how the natural and practical MSR masses treat the virtual massive quark effects does not reintroduce any infrared sensitivity, as is expected since the mass of the virtual quark provides an infrared cutoff. The numerical uncertainties in the correction are below the level of  MeV and negligible. Note that the difference between the natural and the practical MSR masses at the common scale starts at and that the uncertainty band from scale variation is an underestimate at this lowest order. However, the series results and error bands at show good behavior and convergence. In Ref. Butenschoen:2016lpz () the practical MSR mass was employed, but the numerical difference to the natural MSR mass is subdominant to the uncertainties obtained in the analysis there.

In the rest of the paper we will simply use the notation of the MSR mass with the definition when the difference between the natural and practical definitions and the value of are insignificant but we will specify explicitly our use of the practical or the natural MSR masses (or any other mass scheme) and the massless flavor number for any numerical analysis.

## 3 R-Evolution

The dependence of the MSR mass on the scale is described by the R-evolution equation Hoang:2008yj (), which is derived from the logarithmic derivative of the defining equations  (3) and (5) and using that the pole mass is independent:

 RddRmMSRQ(R)=−RγR(αs(R))=−R∞∑n=0γRn(αs(R)4π)n+1, (9)

where

 γR0 =a1, (10) γR1 =a2−2β0a1, γR2 =a3−4β0a2−2β1a1, γRn =an+1−2n−1∑j=0(n−j)βjan−j.

The overall minus sign on the RHS of Eq. (9) indicates that the MSR mass always decreases with . Note that this equation applies to all MSR schemes and we have therefore suppressed the superscript on the ’s. The crucial feature of the R-evolution equation is that it is free from the ambiguity contained in the series that relates the MSR mass to the pole mass because the ambiguity is -independent. This is directly related to the fact that for determining the R-evolution equation also the overall linear factor of on the RHS of Eqs. (3) and (5) has to be accounted for. Therefore the R-evolution equation does not only have a logarithmic dependence on , as common to usual renormalization group equations (RGEs), but also a linear one. Both of these issues are actually tied together conceptually. The numerical expressions for the coefficients for the natural and practical MSR masses are given explicitly in Eqs. (77) and (78). We implement renormalization scale variation in the R-evolution equation by simply expanding in Eq. (9) as a series in and by varying , typically in the range . In principle one may also consider varying the boundaries of integration, as it is common for usual RGEs, but only the former way of implementing scale variations in the R-evolution leads to variations of the scale solely in logarithms, which is the standard used for the usual logarithmic RGEs.

By solving the R-evolution equation one sums, at the same time and systematically, the asymptotic renormalon series as well as the large logarithmic terms in to all orders in a manner free from the renormalon:

 mMSRQ(R0)−mMSRQ(R1)=−∞∑n=0γRn∫R0R1dR(αs(R)4π)n+1. (11)

It is straightforward to solve the R-evolution equation numerically and it shows very good perturbative stability even for low values of very close to the Landau pole Hoang:2009yr () in the perturbative strong coupling. Details of how to solve the R-evolution equations analytically have already been given in Hoang:2008yj () and shall not be repeated here.

It is instructive to briefly discuss what the solution of the R-evolution achieves by considering the difference of the MSR mass, , in the context of fixed-order perturbation theory (FOPT), where it is well-known that the renormalon ambiguity contained in the series for and the series for only cancel if one expands in with a common renormalization scale . This is nicely illustrated in the /LL (leading log) approximation where the pole-MSR mass relation has the all order form

 [mpoleQ−mMSRQ(R)]β0/LL =a12β0R∞∑n=0(β0αs(R)2π)n+1n! (12) =a12β0R∞∑n=0(β0αs(μ)2π)n+1n!n∑k=01k!logkμR.

The series by itself is divergent and not summable, but

 [mMSRQ(R0) −mMSRQ(R1)]β0/LL= (13) =a12β0∞∑n=0(β0αs(μ)2π)n+1n!(R1n∑k=01k!logkμR1−R0n∑k=01k!logkμR0) =a12β0∞∑n=0(β0αs(R1)2π)n+1n!(R1−R0n∑k=01k!logkR1R0),

is easily seen to be convergent. In the context of FOPT, when the sum over is truncated, the unavoidable appearance of large logarithms for let’s say may degrade the convergence and cause sizable perturbative uncertainties. Due to the additional linear dependence on and , as shown in Eq. (13), these logarithms cannot be summed by common logarithmic renormalization group (RG) equations. The same type of logarithms also appear for example in the relation of any other low-scale short-distance mass to the mass and their effects can be significant particularly for the top quark. By solving the R-evolution equation one sums, at the same time and systematically, the asymptotic terms in the renormalon series as well as the large logarithmic terms in to all orders in a manner free from the renormalon. It is again instructive to see how this is achieved in the /LL approximation of Eq. (12), which explicitly shows the factorial growth of the perturbative series. When calculating the derivative to get the R-evolution equation, the whole series collapses exactly (i.e. without any truncation!) to

 [RddRmMSRQ(R)]β0/LL=−a1R(αs(R)4π), (14)

which is the one-loop version of Eq. (9). Moreover, the exact solution of the R-evolution equation at this order

 [mMSRQ(R0)−mMSRQ(R1)]β0/LL=−a1∫R0R1dR(αs(R)4π), (15)

can be easily seen to be exactly equal to the RHS of Eq. (13) which sums the renormalon series and the large logarithms at the same time into a convergent series.

Conceptually, the solution of the R-evolution equation is directly related to the Borel space integral over the Borel transform for the series for . Since this has not been shown in Hoang:2008yj () we briefly outline this calculation here at the /LL level. Starting from Eq. (15) one can shuffle the integration over into an integral over by using the QCD -function and the relation . Using the variable one can then rewrite the integral as [  ]

 [mMSRQ(R0)−mMSRQ(R1)]β0/LL =−a12β0ΛLLQCD∫t0t1dtte−t (16) =−a12β0ΛLLQCD[∫∞t1dtte−t−∫∞t0dtte−t],

where the two integrals in the last line are just the difference of the MSR masses at to the pole mass, and the pole mass ambiguity is encoded in the singularity at , which arises because ,

 [mMSRQ(Ri)−mpoleQ]β0/LL=a12β0ΛLLQCD∫∞tidtte−t. (17)

Upon changing variables to the Borel plane parameter and writing in terms of and in both integrals, this gives

 [mMSRQ(R0)−mMSRQ(R1)]β0/LL=∫∞0du[B(R0,μ,u)−B(R1,μ,u)]e−4πuβ0αs(μ). (18)

Here

 B(R,μ,u)=a12β0R(μR)2u1u−12, (19)

is the well-known Borel transform with respect to of the /LL series in Eq. (12). In Eq. (18) the singular and non-analytic contributions contained in the individual Borel functions cancel and the integral becomes ambiguity-free.

To illustrate the impact of using R-evolution compared to using FOPT we show in Fig. 2 the difference of natural MSR masses for in fixed-order perturbation theory (FOPT) and with R-evolution. The curves in Fig. 2 show for  GeV in FOPT for the common renormalization scale between and at 1 loop (cyan), 2 loop (green), 3 loop (blue) and 4 loops (red). We see a good convergence for around , but a deterioration of the series when gets closer to either or . For the series even gets out of bounds and breaks down completely. If one uses scale variation as an estimate of the remaining perturbative error, one therefore obtains a significant dependence on the choice of the lower bound of the variation, and one has no other choice than to abandon in an ad hoc manner scales closer to to estimate the scale variation error. The curves in Fig. 2 show for  GeV from numerically solving the R-evolution equation as a function of the renormalization scale parameter between and . The color coding for the order of the R-evolution equation used for the evaluation is the same as for Fig. 2. As explained below Eq. (9), the parameter is the renormalization scaling parameter in the R-evolution equation which determines by how much the scale in differs from the scale . Thus a variation between and means that in the solution of the R-evolution equations scales between and are covered at each value of along the evolution, which in this case includes scales between and  GeV. Comparing the curves in Fig. 2 and 2 we see that the renormalization scale variation in the R-evolved results is much smaller than the one of FOPT. For the FOPT result with scale variation between – which we pick by hand – and we obtain  GeV at (1, 2, 3, 4) loops. Using R-evolution with variation between and we obtain  GeV which is fully compatible with the FOPT result, but shows more stability and smaller errors. It is also quite instructive to see that using R-evolution the 3-loop result is significantly closer to the 4-loop result than the corresponding 3-loop FOPT result. The results show that for employing R-evolution to calculate MSR mass differences is clearly superior to FO perturbation theory.

To compare to a situation where the scales and are of similar size we have also shown in Figs. 2 and 2 the results for in FOPT and from R-evolution for  GeV. Here the results from both approaches are completely equivalent showing that the logarithm is not large and the summation of the renormalon contributions from higher orders only constitutes very small effects. Furthermore using renormalization scales close to or in FOPT is not problematic. Numerically, using FOPT with scale variations between and we obtain  GeV at loops, while using R-evolution with variations between and we obtain  GeV. We find that FOPT and R-evolution give equivalent results even for  GeV, and that the use of R-evolution is essential for . Overall we see that, if and are of similar size, FO perturbation theory and R-evolution lead to equivalent results, but that it is in general safer to use R-evolution. So the situation is very similar to the one we encounter when considering the relation of the strong coupling for two different renormalization scales.

We note that the possibility to sum the renormalon-type logarithms displayed in Eq. (13) by considering the Borel integral over the difference of Borel transforms as shown in Eq. (18) was pointed out already in Ref. Bali:2003jq () prior to Ref. Hoang:2008yj (). However, this exact equivalence [ via a transformation of variables as given below Eq. (17) ] of R-evolution and the method using the integration over Borel transform differences can only be analytically shown at the /LL approximation. Beyond that, both approaches sum up the same type of logarithms but differ in subleading terms. Numerically, both approaches converge to the same result and have comparable order-by-order convergence. From a practical point of view, however, the concept of R-evolution may be considered more general. This is because R-evolution can be applied directly to any series having the form of (3) or (5) while using the Borel integration method requires that the corresponding Borel transforms are known or constructed beforehand. For general series, such as for the difference of MSR masses as discussed above, this is not possible without making additional approximations. In practice, the approach of Ref. Bali:2003jq () to sum the renormalon-type logarithms has therefore only been applied for series (referred to as RS-schemes) which were explicitly derived from a given expression for the Borel transform.

## 4 Analytic Borel Transform and Renormalon Sum Rule

Using the solution of the R-evolution equation it is possible to derive, analytically and rigorously, an expression for the Borel transform of the MSR-pole mass relation. This Borel transform is designed to focus on the singular contributions that quantify the renormalon of the pole mass. This result was already quoted in the letter Hoang:2008yj () where, however, no details on the derivation could be given due to lack of space. In the following we provide these details on how to obtain the analytic result for the normalization of the singular terms. The analytic results for the normalization can be applied to other perturbative series as a probe of renormalon ambiguities, and we therefore call it the renormalon sum rule. This sum rule was first given in Ref. Hoang:2008yj (), and is very sensitive to even subtle effects if corrections are known. We apply the sum rule to obtain an updated determination of the size of the pole mass ambiguity, accounting for the results of Refs. Marquard:2015qpa (); Marquard:2016dcn () which became available recently but were unknown when Ref. Hoang:2008yj () appeared. To demonstrate the sum rule’s capabilities to probe renormalon ambiguities in perturbative series and to clarify subtleties in how to use it properly, we also apply it to a few other cases. Interestingly, the analytic manipulations arising in the derivation of the sum rule lead to an alternative expression for the high-order asymptotic behavior of a series that contains an renormalon. This expression differs from the well known asymptotic formula which is known since a long time from Beneke:1994rs (), and we therefore discuss it as well.

### 4.1 Derivation

The analytic derivation for the Borel transform of the MSR-pole mass relation starts from its expression related to the solution of the R-evolution equation given in Eq. (9) which was already derived in Ref. Hoang:2008yj ().

 mMSRQ(R)−mpoleQ =−∫R0d¯RγR(αs(¯R)) (20) =−ΛQCD∫∞tRdtγR(t)^b(t)e−G(t) =ΛQCD∞∑k=0eiπ(^b1+k)Sk∫∞tRdtt−1−k−^b1e−t =ΛQCD∞∑k=0eiπ(^b1+k)SkΓ(−^b1−k,tR),

where in the second line we changed variable to and used the identity (72) to scale out , and in the third line we employed the coefficients given in Eq. (81). The expression in Eq. (20) gives an all-order representation of the original series that is more useful for analyzing renormalon issues than Eqs. (3) and (5). This is because using the R-evolution equation of Eq. (9) (which is linear in ) and its solution, provides, through the sum in , a reordering of the original series in leading and subleading series of terms from the perspective of their numerical importance in the asymptotic high order behavior related to the renormalon. This allows to derive rigorously a representation of the Borel transform [ given in Eq. (4.1) ] reflecting efficiently the hierarchy of leading and subleading terms with respect to the renormalon, which is the information that is not contained in the original series. That such a separation is possible in a systematic way may not be obvious, but it is achieved by the R-evolution equation. We stress that the result of Eq. (4.1) should not be considered as the exact expression for the Borel transform because it does not encode information on possible poles (or non-analytic cuts) other than at . We note that these poles and the associated renormalons can be studied by considering solutions of R-evolution equations involving powers of different from the linear dependence shown in Eq. (9), see Ambar:thesis ().

We note that the expression in the last line of Eq. (20), which involves the incomplete gamma function , also arises in the analytic solution of the mass difference (11),

 mMSRQ(R0)−mMSRQ(R1)=ΛQCD∞∑k=0eiπ(^b1+k)Sk[Γ(−^b1−k,t0)−Γ(−^b1−k,t1)]. (21)

Here the cut in the gamma functions for cancels in the difference for each in the sum, and the result on the RHS is real. We mention that the first term () in the sum over provides the summation of the leading terms in the approximation shown in Eqs. (13) and (15). In Eq. (20) the cut still remains and arises from the integration of the Landau pole in the strong coupling located at in the integral in the next-to-last line. The resulting imaginary part in the numerical expression corresponds to the imaginary part that arises in the inverse Borel integral for , see Eq. (18), and simply reflects the ambiguity of the pole mass. From the point of view of the analytic solution of Eq. (20) based on a perturbative expansion, the imaginary part is well-defined and analytically unique.

To proceed we asymptotically expand the incomplete gamma function in inverse powers of (i.e. powers of )

 ΛQCDeiπ(^b1+k)Γ(−^b1−k,t) =−R[eG(t)e−t(−t)−^b1]∞∑m=0Γ(1+^b1+k+m)Γ(1+^b1+k)(−t)−1−k−m =−R∞∑ℓ=0gℓ∞∑m=0Γ(1+^b1+k+m)Γ(1+^b1+k)(−t)−1−ℓ−k−m, (22)

where the coefficients are given in Eq. (79), and coincide with the coefficients defined in Ref. Beneke:1994rs (). We stress that the equality in Eq. (4.1) is the asymptotic expansion and is not an identity, so that the imaginary part due to the cut in the incomplete gamma function does not arise on the RHS. Inserting Eq. (4.1) in Eq. (20) gives

 mMSRQ(R)−mpoleQ =−R∞∑k=0Sk∞∑ℓ=0gℓ∞∑m=0Γ(1+^b1+k+m)Γ(1+^b1+k)(−t)−1−ℓ−k−m. (23)

We then perform the Borel transform with respect to powers of according to the rule giving

 Bαs(R)[ mMSRQ(R)−mpoleQ](u)= (24) =−2R∞∑ℓ=0gℓ∞∑k=0Sk∞∑m=0Γ(1+^b1+k+m)Γ(1+^b1+k)Γ(1+k+ℓ+m)(2u)ℓ+k+m =−2R∞∑ℓ=0gℓ∞∑k=0Sk(2u)ℓ+kΓ(1+k+ℓ)2F1(1,1+^b1+k,1+k+ℓ,2u).

Using identities for the hypergeometric function we can rewrite

 (2u)ℓ+kΓ(1+k+ℓ) 2F1(1,1+^b1+k,1+k+ℓ,2u)=Γ(1+^b1−ℓ)Γ(1+^b1+k)(1−2u)−1−^b1+ℓ (25) −1(1+^b1−ℓ)Γ(k+ℓ)2F1(1+^b1−ℓ,1−k−ℓ,2+^b1−ℓ


text:
Factorial

What is Factorial?

In Mathematics, factorial is a simple thing. Factorials are just products. An exclamation mark indicates the factorial. Factorial is a multiplication operation of natural numbers with all the natural numbers that are less than it. In this article, let’s discuss the factorial definition, formula and examples.

Also, Check: Factorial Calculator

Factorial Notation

The multiplication of all positive integers says “n”, that will be smaller than or equivalent to n is known as the factorial. The factorial of a positive integer is represented by the symbol “n!”. The formula to find the factorial of a number is

n! = n × (n-1) × (n-2) × (n-3) × ….× 3 × 2 × 1

For an integer n≥ 1, the factorial representation in terms of pi product notation is

$n! = \prod_{i=1}^{n}i$

From the above formulas, the recurrence relation for the factorial of a number is defined as the product of factorial number and factorial of that number minus 1. It is given by

n! = n. (n-1) !

For example, the factorial of 10 is written as

10! = 10. 9 !

10! = 10 (9 × 8 × 7 × 6 × 5× 4 × 3 × 2 × 1)

10! = 10 ( 362,880 )

10! = 3,628,800

Therefore, the value of 10 factorial is 3,628,800

The factorial operation is encountered in many areas of Mathematics such as algebra, permutation and combination, and mathematical analysis. Its primary use is to count “n” possible distinct objects.

Factorial Examples

Factorial of 5

We know that the factorial formula is

n! = n × (n-1)×(n-2)×(n-3)× ….× 3×2×1

So the factorial of 5 is

5! = 5× (5-1)× (5-2)× (5-3)× 1

5! = 5 ×4 ×3 ×2 ×1

5! = 120

Therefore, the factorial of 5 is 120

Factorial of 0 is 1

The factorial of 0 is 1

i.e., 0 ! = 1

According to the convention of empty product, the result of multiplying no factors is a nullary product. It means that the convention is equal to the multiplicative identity.

Factorials of Numbers 1 to 10 Table

The list of factorial values from 1 to 10 are:

 Factorial of a Number Expansion Value 1! 1 1 2! 2×1 2 3! 3 ×2× 1 6 4! 4 ×3× 2 ×1 24 5! 5× 4× 3× 2× 1 120 6! 6× 5 ×4 ×3× 2× 1 720 7! 7× 6× 5 ×4× 3× 2 ×1 5,040 8! 8× 7× 6× 5× 4× 3 ×2× 1 40,320 9! 9 ×8× 7× 6× 5× 4 ×3× 2 ×1 362,880 10! 10× 9×8× 7× 6× 5×4× 3× 2× 1 3,628,800

Visit BYJU’S – The Learning App for more information on factorial of numbers and explore Maths-related videos to learn with ease.

 Related Links Multiplication and Division Integers Whole Numbers Number Theory

format: plain


text:
# Unable to activate an Nmosfet gate with raspberry pi

So I've been trying to control a motor using a MOSFET transistor, where the gate is controlled by a raspberry pi.

My circuit is wired as follow:

The diodes represent the raspberry pi ground and gpio pin 17

The problem is that when the pin #17 is set to high, the motor should stop, or at least drastically slow down, but nothing happens. Help would be greatly appreciated, thanks.

The transistor model is the IRF730

Edit :

Ok, first sorry for the mess of my first post; second here is the transistor data sheet: http://www.vishay.com/docs/91047/91047.pdf

and the proper schematic I've done to the best of my knowledge:

simulate this circuit – Schematic created using CircuitLab

• Please draw a proper schematic. You have resistors floating out in space and connections that are hidden underneath the transistor body. You should also provide a link to the transistor data sheet. – Elliot Alderson Jan 11 at 22:11
• The reason we ask for a schematic is that it shows the schema of the circuit and the function of each pin. What you have provided is a wiring diagram and without having to read the datasheet for the device we have no idea which pin is which. There is a CircuitLab button on the editor toolbar and it is very easy to use. You can use the LAMP symbol to represent the motor. – Transistor Jan 11 at 22:28
• Assuming that's really the way you have it connected, you've connected the Raspberry-Pi GPIO pin and ground to the FET's gate. You need to connect the R-Pi's ground to your circuit ground. I can't remember if the R-Pi's GPIO is 5V or 3.3V -- if it's 5V that FET will be marginal; if it's 3.3V it probably isn't enough. You want a logic-level FET. – TimWescott Jan 11 at 22:29
• @Transistor: true, but the OP seems to be a rank enough beginner that they may not be able to do a schematic. If they can, though, they should. Witness questions like this one: electronics.stackexchange.com/questions/86028/…. – TimWescott Jan 11 at 22:31

The R'Pi is a 3.3V MCU so has insufficient voltage output on the GPIO to use the IRF730 you have.

To make any progress you need to be able to understand the components you are using.

1. Your MCU. R'Pi output is only 3.3V. You should be able to measure this using a multimeter. Set the GPIO pin high and measure the output voltage, it should be 3.3V. Set the GPIO low and the output in should read zero.
2. Your N-channel FET, the IRF730. This is an unusual device to select, but you should be able to make it work with some minor modifications to your drive.

For your FET you critically need to understand the gate voltage used to turn it on.
This is specified in the datasheet and you should put effort into understanding the datasheet if you expect to experiment further.

From the datasheet:

The above table shows that to get just 250uA of current to flow from Drain to Source you need at least 2V and perhaps as much as 4V on the gate.
So this device is likely not to work at all with a 3.3V MCU GPIO output.

The chart above is typical characteristics and shows that with a V(GS) voltage of 4.5V you typically would not be able to pass much more than 200mA from Drain to Source.
In any design you should ensure you have enough drive to ensure the device is driven at a voltage suitable for your load current.
In your case with a small motor (as shown in your diagram) you'd expect current flows in the 50mA to perhaps as much as 700mA under load.
Since you have a 6V battery to drive your motor, you do have enough voltage to drive the gate, you just need to interface this to your R'Pi.

A schematic that would work for you is shown below:

simulate this circuit – Schematic created using CircuitLab

The extra transistor interfaces the 3V3 output level of the R'Pi to the 6V level of your motor battery. A high on the GPIO pin cuts off the transistor and R1 pulls the gate of the FET to 6V to turn it on. This is more than enough to drive your motor.

I said the IRF730 was a quite strange device selection. This is because it's a very high voltage device with an avalanche body diode. Most schematics you see driving DC (brush) motors include a diode to prevent a high voltage kick when you turn it off. D1 would 'catch' this energy. However because the IRF730 includes an avalanche body diode you can allow this back EMF to occur, it is clamped at the avalanche voltage which is guaranteed to be above 400V. So in this particular circuit you could leave D1 out.

I leave it to you to be able to translate the schematic into a breadboard layout.

Update: The choice of FETs used tends to start religious wars with each EE having their own favorite devices to use.
There are many 'logic level' FETs produced, but sticking with your choice of a TO-220 case which is easy to work with you could try the IRF3709. This has very low RDS(on) even at 3.3V gate drive.

My personal favorites are the Alpha&Omega SOT-23 range for general purpose N and P channel FET I/O. Consider devices such as the AO3400A which is easily driven from 3V3 logic and with very low (<50 mOhms) RDS(on). The very low RDS(on) results in extremely low device dissipation in PWM applications.

• All right, thanks for all of this, but you said the IRF730 was an odd device choice, due to the low voltages used in the circuit, thus what type of transistor should I be looking for? – matkiller333 Jan 12 at 6:19
• @matkiller333 I updated the answer with a couple of device selections. – Jack Creasey Jan 12 at 7:13

From the fritizing diagram that you posted, it looks like this is your circuit, with the motor modeled as a resistor (really squinting here to see what G, D, and S are on the generic NMOS part that you placed...please follow Elliot's comments for future posts.)

simulate this circuit – Schematic created using CircuitLab

A couple issues:

1. The GPIO pin on your Raspberry Pi and ground are shorted at the gate pin of the NMOS?? This alone will prevent the NMOS from switching, and will destroy your GPIO driver, since you have a zero-impedance path between the GPIO 3V3 and Ground rails.

2. It looks like you have the drain and source pins switched. Your N-MOSFET is currently set up as a high-side driver, but it needs to be in low-side configuration, since the $$\ V_{GS} > V_{thresh} \$$ condition must be met for it to conduct.

Here's a general schematic for what you should have, but this isn't too much help without part numbers and design requirements.

simulate this circuit

• Unfortunately with the IRF730 device selected the R'Pi is unlikely to be able to turn it on enough to make the motor turn. The VGS(th) is 2-4V. The IRF530 shown here has the same VGS(th) and is therefore just as unlikely to work. – Jack Creasey Jan 12 at 4:14
• @JackCreasey yes, you’re right. I didn’t mean to suggest using the IRF4530, it’s just the default schematic part :P I wanted to just introduce the general practice of driving loads with a MOSFET. Depending on the motor and expected load, I was suggesting that if a MOSFET could be found that can be driven from 3V3, this is how you would connect it. – Rahul Iyer Jan 12 at 5:20


text:
# Test series for convergence

1. May 21, 2012

### sharks

1. The problem statement, all variables and given/known data
$$(a)\;\sum^{\infty}_{n=1}\frac{n-5}{n^2}\;(solved)$$
$$(b)\;\sum^{\infty}_{r=1}\frac{2r}{1+r^2}\;(solved)$$
$$(c)\;\sum^{\infty}_{n=1}\frac{\cos^4 nx}{n^2}\;(solved)$$
$$(d)\;\sum^{\infty}_{n=1}\frac{3^r+4^r}{4^r+5^r}$$
$$(e)\;\sum^{\infty}_{r=1}\frac{r^r}{r!}\;(solved)$$
$$(f)\;\sum^{\infty}_{n=1}\frac{1.2.3...n}{4.7.10... (3n+1)}\;(solved)$$

2. Relevant equations
nth-term test, ratio test, comparison test, limit comparison test, geometric series, etc.

3. The attempt at a solution
$$(a)\;\sum^{\infty}_{n=1}\frac{n-5}{n^2}=\sum^{\infty}_{n=1}\frac{1}{n}-\sum^{\infty}_{n=1}\frac{5}{n^2}$$This is contradictory, as according to the p-series test, the first series diverges but the second series converges. I also tried to factorize out $n$ or $n^2$ out of the numerator and denominator, but got limit = 0, which is inconclusive.

$$(b)\;\lim_{r\to \infty}\frac{2}{1/r+r}=0$$I factorized $r$ out of the numerator and denominator, then took the limit but got 0, which is inconclusive. I also tried comparison test, since $v_n=2/n \ge u_n$ and the series $v_n$ diverges, the test is inconclusive.

$$(c)\;0\le \frac{\cos^4 nx}{n^2}\le 1/n^2$$
I used the squeeze theorem on the sequence, but got 0, which is inconclusive.

$$(d)\;\lim_{n\to \infty}\frac{1+(3/4)^r}{1+(5/4)^r}=1/∞=0$$
Since the limit = 0, it is inconclusive.

$$(e)\;\lim_{r\to \infty}\left( \frac{r+1}{r} \right)^r$$
I used the ratio test. Then got stuck. Took logs and got the limit = $e^∞$?

$$(f)\;\lim_{n\to \infty}\frac{(4+4/n)}{(3+4/n)}=4/3$$
I used the ratio test. L>1, so the series diverges. But the answer is convergent series.

Last edited: May 22, 2012
2. May 21, 2012

### gopher_p

(a), (b) Which tests apply? Which have you tried?

(c) How is the squeeze theorem inconclusive?

(d)-(e) These are sequences. The tests for convergence/divergence (which are for series) do not apply.

Edit: Wow sorry. I really need to look at the whole problem.

Last edited: May 21, 2012
3. May 21, 2012

### gopher_p

(c) Looks like you have a pretty good setup for using one of the tests (not the test for divergence).

(d) I'd split the ratio into two pieces and examine them individually.

(e) This is a very common limit. I guarantee you've seen it at least two or three times in your calculus career.

(f) I'd check for an algebra mistake. You've got the right idea, though.

4. May 21, 2012

### sharks

For (c): Use the comparison test. Since the original series is less than $1/n^2$ and the latter converges, therefore the original series converges.
$$(d)\;\sum^{\infty}_{n=1}\frac{3^r+4^r}{4^r+5^r}= \sum^{\infty}_{n=1}\frac{3^r}{4^r+5^r}+ \sum^{ \infty}_{n=1}\frac{4^r}{4^r+5^r}$$The limits of both sequences give: 1/(∞+∞) + 1/∞ = 0 and the result is inconclusive.

For (f): Using the ratio test, here is what i end up with: $$\frac{u_{n+1}}{u_n}=\frac{4n+4}{3n+4}$$I get: L = 4/3

5. May 21, 2012

### Bohrok

Comparison test is very good for most of these series

For a), after splitting the series up like that and showing that part of the series diverges, the original series diverges too.

For b), try starting with 1+r2 < 2r2

For e), what's the limit of the sequence?

f) is a little more complicated than how you worked it out, since all those numbers are multiplied together and you're not just considering n and (3n+1). Still thinking about that one...

6. May 21, 2012

### gopher_p

There are finitely many (seven by my count) tests:

The nth-term test rarely gives you anything conclusive, but it's usually easy to compute. In the problems you run across in calculus classes, it almost never "works". You can't just give up when it's inconclusive, because it almost always will be.

The integral test and the alternating series test usually don't apply due to the conditions you need in order to use them. When they do apply, they give you definitive answers and you're done.

Ratio/root are algorithmic; once you decide to try, you just do it. Either you get something conclusive or you don't.

So really you only have two tests where any decisions need to be made; comparison and limit comparison. Protip: unless the direct comparison is obvious (like it was in c above) use the limit comparison. It's way easier.

So at the risk of sounding mean, do the flipping tests. There's no pressure time wise right now. Just do them. Figure out which ones work in certain cases and which ones don't. That kind of intuition only comes with practice. So, again, just do it.

P.S. When all of your efforts seem to be in vain, make sure you haven't got a telescoping series sitting in front of you.

Last edited: May 21, 2012
7. May 21, 2012

### vela

Staff Emeritus
It often helps to look at how the terms behave far out in the sequence. For example, for (a), when n gets large, the terms look like
$$\frac{n-5}{n^2} \approx \frac{n}{n^2} = \frac{1}{n}$$ This suggests the series will diverge. To prove it, try using the limit comparison test with 1/n.

8. May 21, 2012

### Bohrok

Depending on whether you think the sum in f) converges or diverges, you can set up an inequality with 4·7·10···(3n+1) and 3·6·9···(3n) or 6·9·12···(3n+3).

9. May 21, 2012

### sharks

I'll be sure to keep that in mind.
I used the ratio test and i am quite sure that the limit L = 4/3
Thanks for the help, vela. I tried with comparison test but the result was inconclusive, then i tried with the limit comparison test and got the answer = 1. Since $0≤L≤∞$ therefore either both series converge or both diverge. Since 1/n diverges, therefore the original series diverges. I used the same method to solve (b).

Last edited: May 21, 2012
10. May 21, 2012

### vela

Staff Emeritus
Show us how you got that ratio. I got something different and found the series converges.

11. May 21, 2012

### sharks

For part (f): using the ratio test
$$u_n=\frac{1.2.3...n}{4.7.10... (3n+1)} \\u_{n+1}=\frac{2.3.4...(n+1)}{7.10.13... (3n+4)} \\\frac{u_{n+1}}{u_n}=\frac{2.3.4...(n+1)}{7.10.13... (3n+4)}\times \frac{4.7.10... (3n+1)}{1.2.3...n}=\frac{4(n+1)}{(3n+4)}=\frac{4n+4}{3n+4} \\\lim_{n \to \infty}\frac{4n+4}{3n+4}=4/3$$

12. May 21, 2012

### vela

Staff Emeritus
Your expression for un+1 isn't correct. You shouldn't be dropping the first factor in the numerator and denominator.

13. May 21, 2012

### sharks

$$u_n=\frac{1.2.3...n}{4.7.10... (3n+1)} \\u_{n+1}=\frac{1.2.3.4...(n+1)}{4.7.10.13... (3n+4)} \\\frac{u_{n+1}}{u_n}=\frac{1.2.3.4...(n+1)}{4.7.10.13... (3n+4)}\times \frac{4.7.10... (3n+1)}{1.2.3...n}=\frac{(n+1)}{(3n+4)} \\\lim_{n \to \infty}\frac{n+1}{3n+4}=\frac{1}{3}$$Since L < 1, the original series converges.

14. May 22, 2012

### sharks

$$(e)\;\sum^{\infty}_{r=1}\frac{r^r}{r!}$$Using the ratio test:
$$u_n=\frac{r^r}{r!} \\u_{n+1}=\frac{(r+1)^{r+1}}{(r+1)!}=\frac{(r+1)^r.(r+1)}{r!(r+1)} \\\frac{u_{n+1}}{u_n}=\frac{(r+1)^r.(r+1)}{r!(r+1)}\times \frac{r!}{r^r}=\frac{(r+1)^r}{r^r}=\left( \frac{r+1}{r} \right)^r \\\lim_{n\to \infty} \frac{u_{n+1}}{u_n}=\lim_{n\to \infty}\left( \frac{r+1}{r} \right)^r \\y=\left( \frac{r+1}{r} \right)^r \\\ln y=\ln \left( \frac{r+1}{r} \right)^r=r\ln \left( \frac{r+1}{r} \right)=\frac{\ln \left( \frac{r+1}{r} \right)}{1/r}=-r^3/(r+1) \\y=e^{-r^3/(r+1)} \\\lim_{n\to \infty}e^{-r^3/(r+1)}=\lim_{n\to \infty}e^{-\infty}=-∞$$Since L=-∞, meaning L<1, therefore the original series converges. Is that correct?

Last edited: May 22, 2012
15. May 22, 2012

### the_kid

I don't think so. I had trouble spotting your error, but lim$_{r\rightarrow\infty}$=$\infty$, so the series should diverge.

16. May 22, 2012

### vela

Staff Emeritus
You're fine up to here. Simplify the argument of the log to get
$$\lim_{r \to \infty} \frac{\log \big(1+\frac{1}{r}\big)}{1/r}$$ and try evaluating it again using the Hospital rule.

That limit can't possibly be right. The function you're taking the limit of is positive. How can you end up with a negative limit?

17. May 22, 2012

### sharks

$$\\\ln y=\ln \left( \frac{r+1}{r} \right)^r=r\ln \left( \frac{r+1}{r} \right)=\frac{\ln \left( \frac{r+1}{r} \right)}{1/r}=\frac{\ln \left( 1+\frac{1}{r} \right)}{1/r}$$Now, using L'Hopital's rule:$$\lim_{r\to \infty}\ln y=\frac{1/ \left( 1+\frac{1}{r} \right)}{-1/r^2}=\frac{-r^2}{ \left( 1+\frac{1}{r} \right)} \\\ln \lim_{r\to \infty}y=\frac{-r^2}{ \left( 1+\frac{1}{r} \right)} \\\lim_{r\to \infty}y=e^{\frac{-r^2}{ \left( 1+\frac{1}{r} \right)}}=e^{-∞}$$Is that correct? I had to analyze it on graph to confirm but i think that as $r→∞, e^{-∞}=0$? Since L < 1, the original series converges? But the answer is divergent series.

18. May 22, 2012

### vela

Staff Emeritus
You need to use the chain rule when you differentiate the logarithm.

Again, the negative result should have set off alarms that you made a mistake. ln(1+1/r) > 0 and 1/r>0. How can the limit diverge to -∞?

19. May 22, 2012

### sharks

Using L'Hopital's rule:$$\lim_{r\to \infty}\ln y=\frac{1/ \left( 1+\frac{1}{r} \right).(-1/r^2)}{-1/r^2}=\frac{1}{ \left( 1+\frac{1}{r} \right)} \\\ln \lim_{r\to \infty}y=\frac{1}{ \left( 1+\frac{1}{r} \right)}=\frac{r}{r+1}=1+\frac{1}{r} \\\lim_{r\to \infty}y=e^{1+\frac{1}{r}}=e^1$$Since L >1, the original series diverges.

20. May 22, 2012

### sharks

I have no idea what to do next.


text:
# Étale morphism over unirational/uniruled variety

Suppose we have an étale morphism between smooth quasi-projective (complex) varieties $$X \rightarrow Y$$ and assume that $$Y$$ is unirational. I am wondering whether we can somehow deduce that $$X$$ is unirational, or rationally connected, or uniruled, or has negative Kodaira dimension.
And what if $$Y$$ is just uniruled but non-necessarily unirational?

• An étale cover of a rational curve is a disjoint union of rational curves... – Francesco Polizzi May 27 '19 at 14:11
• @FrancescoPolizzi What you write is correct for an etale morphism that is also finite, e.g., proper. However, the OP only indicates that the morphism is etale, not etale and finite. – Jason Starr May 27 '19 at 14:17
• @JasonStarr: I guessed that finite was implict, but of course you are right... – Francesco Polizzi May 27 '19 at 14:18
• Without properness assumptions you cannot deduce anything. For instance take any projective variety $X$ of dimension $n$; a general projection to $Y=\mathbb{P^n}$ is finite. Just take out of $X$ the ramification locus, you get an étale map to $\mathbb{P}^n$. – abx May 27 '19 at 15:34
• "assuming all the other hypotheses": I am lost with which hypotheses you want to make. Anyway there is no reasonable statement of the type you ask for. – abx May 28 '19 at 16:13

In the case where $$Y$$ is unirational and projective, there exists no non-trivial étale cover of $$Y$$. In fact, the fundamental group of a complex, projective, smooth unirational variety is trivial, see


text:
# Kerr black hole

Kerr solution$^{*}$ is the solution of Einstein's equations in vacuum that describes a rotating black hole (or the metric outside of a rotating axially symmetric body) . In the Boyer-Lindquist coordinates$^{**}$ it takes the form \begin{align}\label{Kerr} &&ds^2=\bigg(1-\frac{2\mu r}{\rho^2}\bigg)dt^2 +\frac{4\mu a \,r\;\sin^{2}\theta}{\rho^2} \;dt\,d\varphi -\frac{\rho^2}{\Delta}\;dr^2-\rho^2\, d\theta^2 +\qquad\nonumber\\ &&-\bigg( r^2+a^2+\frac{2\mu r\,a^2 \,\sin^{2}\theta}{\rho^2} \bigg) \sin^2 \theta\;d\varphi^2;\\ \label{Kerr-RhoDelta} &&\text{where}\quad \rho^2=r^2+a^2 \cos^2 \theta,\qquad \Delta=r^2-2\mu r+a^2. \end{align} Here $\mu$ is the black hole's mass, $J$ its angular momentum, $a=J/\mu$; $t$ and $\varphi$ are time and usual azimuth angle, while $r$ and $\theta$ are some coordinates that become the other two coordinates of the spherical coordinate system at $r\to\infty$.

$^{*}$ R.P. Kerr, Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11 (5), 237 (1963).

$^{**}$ R.H. Boyer, R.W. Lindquist. Maximal Analytic Extension of the Kerr Metric. J. Math. Phys 8, 265–281 (1967).

## General axially symmetric metric

A number of properties of the Kerr solution can be understood qualitatively without use of its specific form. In this problem we consider the axially symmetric metric of quite general kind $$\label{AxiSimmMetric} ds^2=A dt^2-B(d\varphi-\omega dt)^{2}- C\,dr^2-D\,d\theta^{2},$$ where functions $A,B,C,D,\omega$ depend only on $r$ and $\theta$.

### Problem 1: preliminary algebra

Find the components of metric tensor $g_{\mu\nu}$ and its inverse $g^{\mu\nu}$.

### Problem 2: integrals of motion

Write down the integrals of motion corresponding to Killing vectors $\boldsymbol{\xi}_{t}=\partial_t$ and $\boldsymbol{\xi}_{\varphi}=\partial_\varphi$.

### Problem 3: Zero Angular Momentum Observer/particle

Find the coordinate angular velocity $\Omega=\tfrac{d\varphi}{dt}$ of a particle with zero angular momentum $u_{\mu}(\partial_{\varphi})^{\mu}=0$.

### Problem 4: some more simple algebra

Calculate $A,B,C,D,\omega$ for the Kerr metric.

## Limiting cases

### Problem 5: Schwarzshild limit

Show that in the limit $a\to 0$ the Kerr metric turns into Schwarzschild with $r_{g}=2\mu$.

### Problem 6: Minkowski limit

Show that in the limit $\mu\to 0$ the Kerr metric describes Minkowski space with the spatial part in coordinates that are related to Cartesian as \begin{align*} &x=\sqrt{r^2+a^2}\;\sin\theta\cos\varphi, \nonumber\\ &y=\sqrt{r^2+a^2}\;\sin\theta\sin\varphi,\\ &z=r\;\cos\theta\nonumber,\\ &\text{where}\quad r\in[0,\infty),\quad \theta\in[0,\pi],\quad \varphi\in[0,2\pi).\nonumber \end{align*} Find equations of surfaces $r=const$ and $\theta=const$ in coordinates $(x,y,z)$. What is the surface $r=0$?

### Problem 7: weak field rotation effect

Write the Kerr metric in the limit $a/r \to 0$ up to linear terms.

## Horizons and singularity

Event horizon is a closed null surface. A null surface is a surface with null normal vector $n^\mu$: $n^{\mu}n_{\mu}=0.$ This same notation means that $n^\mu$ belongs to the considered surface (which is not to be wondered at, as a null vector is always orthogonal to self). It can be shown further, that a null surface can be divided into a set of null geodesics. Thus the light cone touches it in each point: the future light cone turns out to be on one side of the surface and the past cone on the other side. This means that world lines of particles, directed in the future, can only cross the null surface in one direction, and the latter works as a one-way valve, -- "event horizon"

### Problem 8: on null surfaces

Show that if a surface is defined by equation $f(r)=0$, and on it $g^{rr}=0$, it is a null surface.

### Problem 9: null surfaces in Kerr metric

Find the surfaces $g^{rr}=0$ for the Kerr metric. Are they spheres?

### Problem 10: horizon area

Calculate surface areas of the outer and inner horizons.

### Problem 11: black holes and naked singularities

What values of $a$ lead to existence of horizons?

On calculating curvature invariants, one can see they are regular on the horizons and diverge only at $\rho^2 \to 0$. Thus only the latter surface is a genuine singularity.

### Problem 12: $r=0$ is not a point.

Derive the internal metric of the surface $r=0$ in Kerr solution.

### Problem 13: circular singularity

Show that the set of points $\rho=0$ is a circle. How it it situated relative to the inner horizon?

## Stationary limit

Stationary limit is a surface that delimits areas in which particles can be stationary and those in which they cannot. An infinite redshift surface is a surface such that a phonon emitted on it escapes to infinity with frequency tending to zero (and thus its redshift tends to infinity). The event horizon of the Schwarzschild solution is both a stationary limit and an infinite redshift surface (see the problems on blackness of Schwarzshild black hole). In the general case the two do not necessarily have to coincide.

### Problem 14: geometry of the stationary limit surfaces in Kerr

Find the equations of surfaces $g_{tt}=0$ for the Kerr metric. How are they situated relative to the horizons? Are they spheres?

### Problem 15: natural angular velocities

Calculate the coordinate angular velocity of a massless particle moving along $\varphi$ in the general axially symmetric metric (\ref{AxiSimmMetric}). There should be two solutions, corresponding to light traveling in two opposite directions. Show that both solutions have the same sign on the surface $g_{tt}=0$. What does it mean? Show that on the horizon $g^{rr}=0$ the two solutions merge into one. Which one?

### Problem 16: angular velocities for massive particles and rigidity of horizon's rotation

What values of angular velocity can be realized for a massive particle? In what region angular velocity cannot be zero? What can it be equal to near the horizon?

### Problem 17: redshift

A stationary source radiates light of frequency $\omega_{em}$. What frequency will a stationary detector register? What happens if the source is close to the surface $g_{tt}=0$? What happens if the detector is close to this surface?

## Ergosphere and the Penrose process

Ergosphere is the area between the outer stationary limit and the outer horizon. As it lies before the horizon, a particle can enter it and escape back to infinity, but $g_{tt}<0$ there. This leads to the possibility of a particle's energy in ergosphere to be also negative, which leads in turn to interesting effects.

All we need to know of the Kerr solution in this problem is that it \emph{has an ergosphere}, i.e. the outer horizon lies beyond the outer static limit, and that on the external side of the horizon all the parameters $A,B,C,D,\omega$ are positive (you can check!). Otherwise, it is enough to consider the axially symmetric metric of general form.

### Problem 18: bounds on particle's energy

Let a massive particle move along the azimuth angle $\varphi$, with fixed $r$ and $\theta$. Express the first integral of motion $u_t$ through the second one $u_{\varphi}$ (tip: use the normalizing condition $u^\mu u_{\mu}=1$).

${}^{*}$ Note: relations ((7)) and ((8)) do not hold, as they were derived in assumption that $g_{00}>0$.

### Problem 19: negative energy

Under what condition a particle can have $u_{t}<0$? In what area can it be fulfilled? Can such a particle escape to infinity?

### Problem 20: unambiguity of negativeness

What is the meaning of negative energy? Why in this case (and in GR in general) energy is not defined up to an additive constant?

### Problem 21: profit!

Let a particle $A$ fall into the ergosphere, decay into two particles $B$ and $C$ there, and particle $C$ escape to infinity. Suppose $C$'s energy turns out to be greater than $A$'s. Find the bounds on energy and angular momentum carried by the particle $B$.

## Integrals of motion

### Problem 22: massless particles on circular orbits

Find the integrals of motion for a massless particle moving along the azimuth angle $\varphi$ (i.e. $dr=d\theta=0$). What signs of energy $E$ and angular momentum $L$ are possible for particles in the exterior region and in ergosphere?

### Problem 23: massive particles on circular orbits

Calculate the same integrals for massive particles. Derive the condition for negativity of energy in terms of its angular velocity $d\varphi/dt$. In what region can it be fulfilled? Show that it is equivalent to the condition on angular momentum found in the problem on negative energy.

### Problem 24: general case

Derive the integrals of motion for particles with arbitrary $4$-velocity $u^{\mu}$. What is the allowed interval of angular velocities $\Omega=d\varphi/dt$? Show that for any particle $(E-\tilde{\Omega} L )>0$ for any $\tilde{\Omega}\in(\Omega_{-},\Omega_{+})$.

## The laws of mechanics of black holes

If a Killing vector is null on some null hypersurface $\Sigma$, $\Sigma$ is called a Killing horizon.

### Problem 25: Killing horizons

Show that vector $K=\partial_{t}+\Omega_{H}\partial_{\varphi}$ is a Killing vector for the Kerr solution, and it is null on the outer horizon $r=r_{+}$. Here $\Omega_{H}=\omega\big|_{r=r_+}$ is the angular velocity of the horizon.

### Problem 26: surface gravity

Let us define the surface gravity for the Kerr black hole as the limit $\kappa=\lim\limits_{r\to r_{+}} \frac{\sqrt{a^{\mu}a_{\mu}}}{u^0}$ for a particle near the horizon with $4$-velocity $\boldsymbol{u}=u^{t}(\partial_{t}+\omega\partial_{\varphi})$. In the particular case of Schwarzschild metric this definition reduces to the one given in the corresponding problem. Calculate $\kappa$ for particles with zero angular momentum in the Kerr metric. What is it for the critical black hole, with $a=\mu$?

### Problem 27: horizon's area evolution

Find the change of (outer) horizon area of a black hole when a particle with energy $E$ and angular momentum $L$ falls into it. Show that it is always positive.

### Problem 28: irreducible mass

Let us define the irreducible mass $M_{irr}$ of Kerr black hole as the mass of Schwarzschild black hole with the same horizon area. Find $M_{irr}(\mu,J)$ and $\mu(M_{irr},J)$. Which part of the total mass of a black hole can be extracted from it with the help of Penrose process?

### Problem 29: extremal limit

Show that an underextremal Kerr black hole (with $a<\mu$) cannot be turned into the extremal one in any continuous accretion process.

This problem's results can be presented in the form that provides far-reaching analogy with the laws of thermodynamics.

0: Surface gravity $\kappa$ is constant on the horizon of a stationary black hole. The zeroth law of thermodynamics: a system in thermodynamic equilibrium has constant temperature $T$.

1: The relation $\delta\mu=\frac{\kappa}{8\pi}\delta A_{+} +\Omega_{H}\delta J$ gives an analogy of the first law of thermodynamics, energy conservation.

2: Horizon area $A_+$ is nondecreasing. This analogy with the second law of thermodynamics hints at a correspondence between the horizon area and entropy.

3: There exists no such continuous process, which can lead as a result to zero surface gravity. This is an analogy to the third law of thermodynamics: absolute zero is unreachable.

## Particles' motion in the equatorial plane

The following questions refer to a particle's motion in the equatorial plane $\theta=\pi/2$ of the Kerr metric.

### Problem 30: preparatory algebra

Put down explicit expressions for the metric components and parameters $A,B,C,D,\omega$

### Problem 31: zero energy particles

What is the angular velocity of a particle with zero energy?

### Problem 32: geodesics and effective potential

Use the normalizing conditions for the $4$-velocity $u^{\mu}u_{\mu}=\epsilon^2$ and two conservation laws to derive geodesic equations for particles, determine the effective potential for radial motion.

### Problem 33: principal null geodesics

Integrate the equations of motion for null geodesics with $L=aE$, investigate the asymptotes close to the horizons, limits $a\to 0$ and $a\to \mu$.

### Problem 34: innermost stable circular orbits, massless case

Find the minimal radii of circular geodesics for massless particles, the corresponding values of integrals of motion and angular velocities. Show that of the three solutions one lies beyond the horizon, one describes motion in positive direction and one in negative direction. Explore the limiting cases of Schwarzschild $a\to0$ and extreme Kerr $a\to\mu$.

### Problem 35: circular orbits for massive particles

Find $L^2$ and $E^2$ as functions of radii for circular geodesics of the massive particles.

### Problem 36: innermost stable circular orbits, massive case

Derive equation for the minimal radius of a stable circular orbit; find the energy and angular momentum of a particle on it, the minimal radius in the limiting cases $a/\mu\to 0,1$.


text:
# What is the amount of hydrochloric acid present in its aqueous solution having $$P^H = 2$$ in $$gL^{-1}$$? (molar mass of Hydrochloric acid - 36.5g $$mol^{-1}$$)

Free Practice With Testbook Mock Tests

## Options:

1. 3.65

2. 7.30

3. 0.365

4. 0.730

### Correct Answer: Option 3 (Solution Below)

This question was previously asked in

AP Police Constable Prelims Previous paper 1 (Held on : 6 January 2019)

## Solution:

The correct answer is option 3 i.e., 0.365.

• Given-  $$P^H = 2$$ in $$gL^{-1}$$
• Given - Molar mass of Hydrochloric acid = 36.5 g.
• Please note that HCl is a very strong acid. It would completely dissociate in the aqueous solution. It would give H3O and Chlorine.
• It means that H and Cl are dissociated in the aqueous solution.
• We know that pH= - log H+ (pH is the negative log of H+ concentration).
• It can be written as -2 = log (H+)
• Antilog (-2) = H+ = .01 which means there are .01 moles of hydrogen ions in 1 L of the solution.
• Now to find the amount of Hydrochloric acid-
• Amount of HCl in gm / 36.5 gm (molar mass) = 0.01 moles of HCl
• Amount of HCl in g = 0.365


text:
Module categories of finite Hopf algebroids, and self-duality - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Transactions of the American Mathematical Society Année : 2017

## Module categories of finite Hopf algebroids, and self-duality

Peter Schauenburg

Connectez-vous pour contacter l'auteur

#### Résumé

We characterize the module categories of suitably finite Hopf algebroids (more precisely, $X_R$-bialgebras in the sense of Takeuchi (1977) that are Hopf and finite in the sense of a work by the author (2000)) as those $k$-linear abelian monoidal categories that are module categories of some algebra, and admit dual objects for "sufficiently many" of their objects. Then we proceed to show that in many situations the Hopf algebroid can be chosen to be self-dual, in a sense to be made precise. This generalizes a result of Pfeiffer for pivotal fusion categories and the weak Hopf algebras associated to them.

#### Domaines

Mathématiques [math]

### Dates et versions

hal-01447298 , version 1 (26-01-2017)

### Identifiants

• HAL Id : hal-01447298 , version 1
• DOI :

### Citer

Peter Schauenburg. Module categories of finite Hopf algebroids, and self-duality. Transactions of the American Mathematical Society, 2017, 369 (2), pp.1127 - 1146. ⟨10.1090/tran6687⟩. ⟨hal-01447298⟩

### Exporter

BibTeX TEI Dublin Core DC Terms EndNote Datacite

### Collections

76 Consultations
0 Téléchargements


text:
### What is the Intraday Momentum Index?

The Intraday Momentum Index, or IMI, is a technical indicator that combines aspects of candlestick analysis with the relative strength index (RSI). The intraday indicator was developed by Tushar Chande to provide investors with a way to find optimal days to buy and sell.

### Understanding the Intraday Momentum Index (IMI)

Investors use technical indicators to estimate when a stock should be bought or sold. Technical analysis, which uses technical indicators, examines the relationship between stock price and volume over varied periods of time. Indicators, such as the relative strength index and Bollinger Bands, seek to generate buy and sell signals without examining a stock’s fundamentals. As such, they are generally considered more useful for short-term traders than long-term investors.

The Intraday Momentum Index looks at the relationship between a stock’s open and close price over the course of the day, rather than how the open/close price varies between days. It combines some features of the relative strength index, namely the relationship between up closes and down closes and whether there is an indication that a stock is overbought or oversold, with candlestick charts. Candlestick charts for a given day contain a “real body” highlighting the gap between the open and close price, and price points above the high and low called upper and lower shadows.

Technical analysts can use the IMI to anticipate when a stock is overbought:

\begin{aligned} &\text{IMI} = \left ( \frac{ \sum_{d=1}^{n} \text{Gains} }{ \sum_{d=1}^{n} \text{Gains} + \sum_{d=1}^{n} \text{Losses} } \right ) \times 100 \\ &\textbf{where:} \\ &\text{Gains} = \text{Closing price} - \text{Opening price, on up days -- i.e. close} > \text{open} \\ &\text{Losses} = \text{Opening price} - \text{Closing price, on down days -- i.e. open} < \text{close} \\ &d = \text{Days} \\ &n = \text{Number of days (14 is typical)} \\ \end{aligned}

The IMI is calculated as the sum of gains on up days divided by the sum of gains on up days plus the sum of losses on down days. This is then multiplied by 100. If the resulting number is greater than 70 then the stock is considered overbought, while a figure less than 30 means that a stock is oversold. The investor will look at the IMI over a period of days, with 14 days being the most common time frame to look at.

### Example of the Intraday Momentum Index

Let's take a look at the Intraday Momentum Index applied to the SPDR S&P 500 ETF (SPY):

The chart above shows how oversold or overbought IMI readings can generate buy and sell trade signals on a popular index. While these signals aren't always accurate, they may provide a greater degree of accuracy than simply using the RSI. Many traders combine these insights with other forms of technical analysis to maximize their chances of a successful trade. For example, they may look for oversold conditions and a breakout from a chart pattern before entering into a long position.


text:
# Definition and delimitation of regression model

An embarrassingly simple question -- but it seems it has not been asked on Cross Validated before:

1. What is the definition of a regression model?

Also a support question,

1. What is not a regression model?

With regards to the latter, I am interested in tricky examples where the answer is not immediately obvious, e.g. ARIMA or GARCH.

I would say that "regression model" is a kind of meta-concept, in the sense that you will not find a definition of "regression model", but more concrete concepts such as "linear regression", "non-linear regression", "robust regression" and so on. This in the same way as in mathemathics we usually do not define "number", but "natural number", "integers", "real number", "p-adic number" and so on, and if somebody will want to include the quaternions among numbers so be it! it doesn't really matter, what matters is what definitions is used by the book/paper you are reading at the moment.

Definitions are tools, and essentialism, that is discussing what is the essence of ..., what a word really means, are seldom worthwhile.

So, what distinguishes a "regression model" from other kinds of statistical models? Mostly, that there is a response variable, which you want to model as influenced by (or determined by) some set of predictor variables. We are not interested in influence the other direction, and we are not interested in relationships among the predictor variables. Mostly, we take the predictor variables as given, and treat them as constants in the model, not as random variables.

The relationship mentioned above can be linear or nonlinear, specified in a parametric or nonparametric way, and so on.

To delineate from other models we better have a look at some other words often taken to denote something different for "regression models", like "errors in variables", when we accept the possibility of measurement errors in the predictor variables. That could well be included in my description of "regression model" above, but is often taken as an alternative model.

Also, what is meant might vary among fields, see What is the difference between conditioning on regressors vs. treating them as fixed?

To repeat: what matters is the definition used by the authors you are reading now, and not some metaphysics about what it "really is".

• I agree with the essence of your answer. My question was motivated by having encountered statements about regression models that left me wondering what the statement really applies to (and what it does not apply to). Of course, now you could say, "use your best judgement and check the details carefully", but sometimes I might wish to reject the hypothesized statement right away saying that it is not true in general (perhaps true only in a very specific case). Then I need a definition to refer to. There are of course more such situations where having a precise definition is useful. – Richard Hardy Mar 16 '16 at 13:12
• Then you shouls ask specific questions about those uses you have encountered, with references. – kjetil b halvorsen Mar 16 '16 at 13:18
• I don't intend to be picky, but think about it: someone asks you what you are doing, you say "I am analyzing/forecasting/testing [something] using regression models." -- "What is a regression model?" -- (Silence). Or a situation in an introductory econometrics class: "Professor, what is a regression model?" -- (No answer). I think these are very natural questions, so it would be nice to have an answer. – Richard Hardy Mar 16 '16 at 19:24
• Yes, it would be nice to have an answer, but I am not sure there is one canonical answer all can agree about. I got a very different idea of regression from a statistical book such as Seber: "Linear Regression Analysis" as from a text in econometrics. But some ideas all can agree about. I guess it is really a family of models. Then we can ask what is the common core of all this models. – kjetil b halvorsen Mar 16 '16 at 20:44
• Perhaps you will be interested in a related question of mine: Definition of a simple linear regression model. – Richard Hardy Feb 13 at 7:28

In regression case we have some random variables $Y$ and $X_1,\dots,X_k$. The variables have some unknown distribution and complicated covariance structure. We simplify this problem to focusing solely on conditional distribution, or more precisely on conditional expectation of $Y$ given the other variables. We simplify it to

$$\mu = E(y|x_1,\dots,x_k) = f(x_1,\dots,x_k)$$

Where $f$ is a function of predictors that can take different forms (linear, non-linear) depending on particular regression model and $\mu$ is a mean of some distribution when thinking of regression models in terms of generalized linear models. In GLM's $\mu$ can be location of Poisson, Binomial, Gamma etc. distributions. With $L_1$ regularized regression it is a location of Laplace distribution, for robust model minimizing Huber loss so called Huber density is used. In case of quartile regression we focus on other feature of distribution, we estimate $\mu$ that is distribution's quartile rather then expected value.

So instead of looking on full joint distribution, we focus on conditional distribution of $Y$. This simplification is a key feature of regression models.

• Thanks. Intuition doesn't hurt, although I am looking for a more formal definition that I could throw at someone who asked me, So what is a regression model anyway? and then tried to pick on details. – Richard Hardy May 6 '16 at 16:47
• @RichardHardy I think that this is the key feature of regression models that is shared by all of them. – Tim May 6 '16 at 16:49
• I think this answer is a correct and useful approach, but it needs to be generalized so it can apply to situations commonly thought of as "regression" (including GLMs, multiplicative errors, regression with transformed responses, quantile regression, and so on). More broadly, a regression model specifies one or more properties of the entire distribution of the response $y$ in terms of the values of the regressors (within specific ranges, random or fixed). In particular, it can go far beyond merely specifying the expectation or assuming an additive error. – whuber May 6 '16 at 18:54

Some thoughts based on the literature:

F. Hayashi in Chapter 1 of his classic graduate textbook "Econometrics" (2000) states that the following assumptions comprise the classical linear regression model:

1. Linearity
2. Strict exogeneity
3. No multicollinearity
4. Spherical error variance
5. "Fixed" regressors

Wooldridge in Chapter 2 of his classic introductory econometrics textbook "Introductory Econometrics: A Modern Approach" (2012) states that the following equation defines the simple linear regression model:

$$y=\beta_0+\beta_1 x+u.$$

Greene in Chapter 2 of his popular econometrics textbook "Econometric Analysis" (2011) states

The classical linear regression model consists of a set of assumptions about how a data set will be produced by an underlying “data-generating process.”

and subsequently gives a list of assumptions similar to that of Hayashi's.

Regarding the OP's interest in the GARCH model, Bollerslev "Generalized autoregressive conditional heterosedasticity" (1986) includes a phrase "the GARCH regression model" in the title of section 5 and also in the first sentence of that section. So the father of the GARCH model did not mind calling GARCH a regression model.

• Your three references are all restricted to the linear regression model, but your question is wider than that. (Thus, using this as argument in your answer to another post, which I presume spawned the interest in this issue, is I think not completely valid.) If you'd say that latent variable models are not regression models, then using the immediate connection with measurement errors, regression models with measurement errors would no longer be regression models. Seems odd to me. Wiki just says that a reg model relates indep vars to dep in the sense that $Y\approx f(X, \beta)$. – hejseb Sep 23 '15 at 18:23
• True, my examples are for linear regression models; that is what I was able to find in reliable sources such as these textbooks that are widely used and have become classic. I do not trust Wikipedia that much for statistical and econometric questions. Anyway, even in Wikipedia there is a chapter "Underlying assumptions" that is similar to what I have cited from the textbooks. Regarding the other post, could you post the relevant part of your comment there so that I could respond there? In this post I did not say anything about latent variable models, but it's good to hear you opinion. – Richard Hardy Sep 23 '15 at 18:35
• Why point 3, "no multicollinearity"? I have never seen that used as an assumption in the proof of some result! – kjetil b halvorsen Mar 16 '16 at 13:00
• @kjetilbhalvorsen, please do not hold me responsible over what is written in a textbook which I am not the author of. But thanks for the comment, of course, and even more for the answer! – Richard Hardy Mar 16 '16 at 13:07


text:
# Choose a “middle” point from a set

I read a post which talks about pretty much the same problem. But here I simplify the problem hoping that a concrete proof can be offered.

There is a set $A$ which contains some discrete points (one-dimensional), like $\{1, 3, 37, 59\}$. I want to pick one point from $A$ which minimizes the sum of distances between this point and others.

There may be lot of posts out there, and my problem is just the one-dimensional version of those. I know how to prove it if $A$ is not discrete, but I fail when $A$ is discrete like above.

• Could you post your approach for non-discrete sets? – G. Bach Feb 2 '13 at 12:53
• Just pick the point closest to the median (your problem is the "geometric distance" problem which reduces to median in 1-d case). – Vor Feb 2 '13 at 13:02
• There is a trivial $\Theta(n^2)$-time and -space algorithm. What are your non-functional restrictions? – Raphael Feb 2 '13 at 21:35
• @vor The median is in the set. – Yuval Filmus Feb 3 '13 at 3:07

For a point $x$, let $d(x)$ be the sum of distances between $x$ and points in $A$. For $x \notin A$, the derivative $d'(x)$ has the nice formula $$d'(x) = |\{y \in A : y < x\}| - |\{y \in A : y > x\}|.$$ This shows why the median is the best answer when you don't have to select a point from $A$. For a point $x \in A$, $d(x)$ is the same as your objective function, hence the solution is to choose the median. You can find the median in linear time, as described in Wikipedia and various other resources.

• If A is a contiguous set, I know why the point is the median. But here A is a discrete set, I don't quite understand your proof. – avocado Feb 3 '13 at 6:28
• In short, if $x$ is any point to the left of the median, then $d'(x) < 0$, and so it is always better to slightly increase $x$. If $x$ is any point to the right of the median, then $d'(x) > 0$, and so it is always better to slightly decrease $x$. So the median is the unique optimum. – Yuval Filmus Feb 3 '13 at 6:45
• For a uniformly spread set, I immediately agree. But when there are clusters of close points, the argument becomes less clear (to me), that is why $d'$ has this form. Also, the set $\{-2,-1,1,2\}$ has no unique median; any value from $[-1,1]$ suffices, with $0$ the canonical choice. Furthermore, the median is not unique when chosen from the set (in the example, both $-1$ and $1$ work). For the question, we can just choose either one of them (if "the" median is indeed the correct solution). – Raphael Feb 3 '13 at 9:48
• @Raphael, if A is a uniformly spread set, I also agree that the point should be the median. What I am not sure about is just that when A is not a uniformly spread discrete set. – avocado Feb 3 '13 at 11:57
• You're both right that the median is defined only for sets of odd size, while for sets of even size it's really an interval. – Yuval Filmus Feb 3 '13 at 17:01


text:
# Linear matrix inequalities in system and control theory

Updated:

@book{boyd1994,
title={Linear matrix inequalities in system and control theory},
author={Boyd, S and El Ghaoui, L and Feron, E and Balakrishnan, V},
volume={15},
year={1994},
publisher={SIAM}
}