Form Factors for Exclusive Semileptonic $B$--Decays

合集下载

The Subleading Isgur-Wise Form Factor $chi_3(vcdot v')$ to Order $alpha_s$ in QCD Sum Rules

a rXiv:h ep-ph/9212266v116Dec1992SLAC–PUB–6017WIS–92/99/Dec–PH December 1992T/E The Subleading Isgur-Wise Form Factor χ3(v ·v ′)to Order αs in QCD Sum Rules Matthias Neubert Stanford Linear Accelerator Center Stanford University,Stanford,California 94309Zoltan Ligeti and Yosef Nir Weizmann Institute of Science Physics Department,Rehovot 76100,Israel We calculate the contributions arising at order αs in the QCD sum rule for the spin-symmetry violating universal function χ3(v ·v ′),which appears at order 1/m Q in the heavy quark expansion of meson form factors.In particular,we derive the two-loop perturbative contribution to the sum rule.Over the kinematic range accessible in B →D (∗)ℓνdecays,we ﬁnd that χ3(v ·v ′)does not exceed the level of ∼1%,indicating that power corrections induced by the chromo-magnetic operator in the heavy quark expansion are small.(submitted to Physical Review D)I.INTRODUCTIONIn the heavy quark eﬀective theory(HQET),the hadronic matrix elements describing the semileptonic decays M(v)→M′(v′)ℓν,where M and M′are pseudoscalar or vector mesons containing a heavy quark,can be systematically expanded in inverse powers of the heavy quark masses[1–5].The coeﬃcients in this expansion are m Q-independent,universal functions of the kinematic variable y=v·v′.These so-called Isgur-Wise form factors characterize the properties of the cloud of light quarks and gluons surrounding the heavy quarks,which act as static color sources.At leading order,a single functionξ(y)suﬃces to parameterize all matrix elements[6].This is expressed in the compact trace formula[5,7] M′(v′)|J(0)|M(v) =−ξ(y)tr{(2)m M P+ −γ5;pseudoscalar meson/ǫ;vector mesonis a spin wave function that describes correctly the transformation properties(under boosts and heavy quark spin rotations)of the meson states in the eﬀective theory.P+=1g s2m Q O mag,O mag=M′(v′)ΓP+iσαβM(v) .(4)The mass parameter¯Λsets the canonical scale for power corrections in HQET.In the m Q→∞limit,it measures theﬁnite mass diﬀerence between a heavy meson and the heavy quark that it contains[11].By factoring out this parameter,χαβ(v,v′)becomes dimensionless.The most general decomposition of this form factor involves two real,scalar functionsχ2(y)andχ3(y)deﬁned by[10]χαβ(v,v′)=(v′αγβ−v′βγα)χ2(y)−2iσαβχ3(y).(5)Irrespective of the structure of the current J ,the form factor χ3(y )appears always in the following combination with ξ(y ):ξ(y )+2Z ¯Λ d M m Q ′ χ3(y ),(6)where d P =3for a pseudoscalar and d V =−1for a vector meson.It thus eﬀectively renormalizes the leading Isgur-Wise function,preserving its normalization at y =1since χ3(1)=0according to Luke’s theorem [10].Eq.(6)shows that knowledge of χ3(y )is needed if one wants to relate processes which are connected by the spin symmetry,such as B →D ℓνand B →D ∗ℓν.Being hadronic form factors,the universal functions in HQET can only be investigated using nonperturbative methods.QCD sum rules have become very popular for this purpose.They have been reformulated in the context of the eﬀective theory and have been applied to the study of meson decay constants and the Isgur-Wise functions both in leading and next-to-leading order in the 1/m Q expansion [12–21].In particular,it has been shown that very simple predictions for the spin-symmetry violating form factors are obtained when terms of order αs are neglected,namely [17]χ2(y )=0,χ3(y )∝ ¯q g s σαβG αβq [1−ξ(y )].(7)In this approach χ3(y )is proportional to the mixed quark-gluon condensate,and it was estimated that χ3(y )∼1%for large recoil (y ∼1.5).In a recent work we have reﬁned the prediction for χ2(y )by including contributions of order αs in the sum rule analysis [20].We found that these are as important as the contribution of the mixed condensate in (7).It is,therefore,worthwhile to include such eﬀects also in the analysis of χ3(y ).This is the purpose of this article.II.DERIV ATION OF THE SUM RULEThe QCD sum rule analysis of the functions χ2(y )and χ3(y )is very similar.We shall,therefore,only brieﬂy sketch the general procedure and refer for details to Refs.[17,20].Our starting point is the correlatord x d x ′d ze i (k ′·x ′−k ·x ) 0|T[¯q ΓM ′P ′+ΓP +iσαβP +ΓM+Ξ3(ω,ω′,y )tr 2σαβ2(1+/v ′),and we omit the velocity labels in h and h ′for simplicity.The heavy-light currents interpolate pseudoscalar or vector mesons,depending on the choice ΓM =−γ5or ΓM =γµ−v µ,respectively.The external momenta k and k ′in (8)are the “residual”oﬀ-shell momenta of the heavy quarks.Due to the phase redeﬁnition of the eﬀective heavy quark ﬁelds in HQET,they are related to the total momenta P and P ′by k =P −m Q v and k ′=P ′−m Q ′v ′[3].The coeﬃcient functions Ξi are analytic in ω=2v ·k and ω′=2v ′·k ′,with discontinuities for positive values of these variables.They can be saturated by intermediate states which couple to the heavy-light currents.In particular,there is a double-pole contribution from the ground-state mesons M and M ′.To leading order in the 1/m Q expansion the pole position is at ω=ω′=2¯Λ.In the case of Ξ2,the residue of the pole is proportional to the universal function χ2(y ).For Ξ3the situation is more complicated,however,since insertions of the chromo-magnetic operator not only renormalize the leading Isgur-Wise function,but also the coupling of the heavy mesons to the interpolating heavy-light currents (i.e.,the meson decay constants)and the physical meson masses,which deﬁne the position of the pole.1The correct expression for the pole contribution to Ξ3is [17]Ξpole 3(ω,ω′,y )=F 2(ω−2¯Λ+iǫ) .(9)Here F is the analog of the meson decay constant in the eﬀective theory (F ∼f M√m QδΛ2+... , 0|j (0)|M (v ) =iF2G 2tr 2σαβΓP +σαβM (v ) ,where the ellipses represent spin-symmetry conserving or higher order power corrections,and j =¯q Γh (v ).In terms of the vector–pseudoscalar mass splitting,the parameter δΛ2isgiven by m 2V −m 2P =−8¯ΛδΛ2.For not too small,negative values of ωand ω′,the coeﬃcient function Ξ3can be approx-imated as a perturbative series in αs ,supplemented by the leading power corrections in 1/ωand 1/ω′,which are proportional to vacuum expectation values of local quark-gluon opera-tors,the so-called condensates [22].This is how nonperturbative corrections are incorporated in this approach.The idea of QCD sum rules is to match this theoretical representation of Ξ3to the phenomenological pole contribution given in (9).To this end,one ﬁrst writes the theoretical expression in terms of a double dispersion integral,Ξth 3(ω,ω′,y )= d νd ν′ρth 3(ν,ν′,y )1Thereare no such additional terms for Ξ2because of the peculiar trace structure associated with this coeﬃcient function.possible subtraction terms.Because of theﬂavor symmetry it is natural to set the Borel parameters associated withωandω′equal:τ=τ′=2T.One then introduces new variables ω±=12T ξ(y) F2e−2¯Λ/T=ω0dω+e−ω+/T ρth3(ω+,y)≡K(T,ω0,y).(12)The eﬀective spectral density ρth3arises after integration of the double spectral density over ω−.Note that for each contribution to it the dependence onω+is known on dimensionalgrounds.It thus suﬃces to calculate directly the Borel transform of the individual con-tributions toΞth3,corresponding to the limitω0→∞in(12).Theω0-dependence can be recovered at the end of the calculation.When terms of orderαs are neglected,contributions to the sum rule forΞ3can only be proportional to condensates involving the gluonﬁeld,since there is no way to contract the gluon contained in O mag.The leading power correction of this type is represented by the diagram shown in Fig.1(d).It is proportional to the mixed quark-gluon condensate and,as shown in Ref.[17],leads to(7).Here we are interested in the additional contributions arising at orderαs.They are shown in Fig.1(a)-(c).Besides a two-loop perturbative contribution, one encounters further nonperturbative corrections proportional to the quark and the gluon condensate.Let usﬁrst present the result for the nonperturbative power corrections.WeﬁndK cond(T,ω0,y)=αs ¯q q TT + αs GG y+1− ¯q g sσαβGαβq√y2−1),δn(x)=1(4π)D×1dλλ1−D∞λd u1∞1/λd u2(u1u2−1)D/2−2where C F=(N2c−1)/2N c,and D is the dimension of space-time.For D=4,the integrand diverges asλ→0.To regulate the integral,we assume D<2and use a triple integration by parts inλto obtain an expression which can be analytically continued to the vicinity of D=4.Next we set D=4+2ǫ,expand inǫ,write the result as an integral overω+,and introduce back the continuum threshold.This givesK pert(T,ω0,y)=−αsy+1 2ω0dω+ω3+e−ω+/T(16)× 12−23∂µ+3αs9π¯Λ,(17)which shows that divergences arise at orderαs.At this order,the renormalization of the sum rule is thus accomplished by a renormalization of the“bare”parameter G2in(12).In the9π¯Λ 1µ2 +O(g3s).(18)Hence a counterterm proportional to¯Λξ(y)has to be added to the bracket on the left-hand side of the sum rule(12).To evaluate its eﬀect on the right-hand side,we note that in D dimensions[17]¯Λξ(y)F2e−2¯Λ/T=3y+1 2ω0dω+ω3+e−ω+/T(19)× 1+ǫ γE−ln4π+2lnω+−ln y+12T ξ(y) F2e−2¯Λ/T=αsy+1 2ω0dω+ω3+e−ω+/T 2lnµ6+ y r(y)−1+ln y+1According to Luke’stheorem,theuniversalfunction χ3(y )vanishes at zero recoil [10].Evaluating (20)for y =1,we thus obtain a sum rule for G 2(µ)and δΛ2.It reads G 2(µ)−¯ΛδΛ224π3ω00d ω+ω3+e −ω+/T ln µ12 +K cond (T,ω0,1),(21)where we have used that r (1)=1.Precisely this sum rule has been derived previously,starting from a two-current correlator,in Ref.[16].This provides a nontrivial check of our ing the fact that ξ(y )=[2/(y +1)]2+O (g s )according to (19),we ﬁnd that the µ-dependent terms cancel out when we eliminate G 2(µ)and δΛ2from the sum rule for χ3(y ).Before we present our ﬁnal result,there is one more eﬀect which has to be taken into account,namely a spin-symmetry violating correction to the continuum threshold ω0.Since the chromo-magnetic interaction changes the masses of the ground-state mesons [cf.(10)],it also changes the masses of higher resonance states.Expanding the physical threshold asωphys =ω0 1+d M8π3 22 δ3 ω032π2ω30e −ω0/T 26π2−r (y )−ξ(y ) δ0 ω096π 248T 1−ξ(y ).It explicitly exhibits the fact that χ3(1)=0.III.NUMERICAL ANALYSISLet us now turn to the evaluation of the sum rule (23).For the QCD parameters we take the standard values¯q q =−(0.23GeV)3,αs GG =0.04GeV4,¯q g sσαβGαβq =m20 ¯q q ,m20=0.8GeV2.(24) Furthermore,we useδω2=−0.1GeV from above,andαs/π=0.1corresponding to the scale µ=2¯Λ≃1GeV,which is appropriate for evaluating radiative corrections in the eﬀective theory[15].The sensitivity of our results to changes in these parameters will be discussed below.The dependence of the left-hand side of(23)on¯Λand F can be eliminated by using a QCD sum rule for these parameters,too.It reads[16]¯ΛF2e−2¯Λ/T=9T4T − ¯q g sσαβGαβq4π2 2T − ¯q q +(2y+1)4T2.(26) Combining(23),(25)and(26),we obtainχ3(y)as a function ofω0and T.These parameters can be determined from the analysis of a QCD sum rule for the correlator of two heavy-light currents in the eﬀective theory[16,18].Oneﬁnds good stability forω0=2.0±0.3GeV,and the consistency of the theoretical calculation requires that the Borel parameter be in the range0.6<T<1.0GeV.It supports the self-consistency of the approach that,as shown in Fig.2,weﬁnd stability of the sum rule(23)in the same region of parameter space.Note that it is in fact theδω2-term that stabilizes the sum rule.Without it there were no plateau.Over the kinematic range accessible in semileptonic B→D(∗)ℓνdecays,we show in Fig.3(a)the range of predictions forχ3(y)obtained for1.7<ω0<2.3GeV and0.7<T< 1.2GeV.From this we estimate a relative uncertainty of∼±25%,which is mainly due to the uncertainty in the continuum threshold.It is apparent that the form factor is small,not exceeding the level of1%.2Finally,we show in Fig.3(b)the contributions of the individual terms in the sum rule (23).Due to the large negative contribution proportional to the quark condensate,the terms of orderαs,which we have calculated in this paper,cancel each other to a large extent.As a consequence,ourﬁnal result forχ3(y)is not very diﬀerent from that obtained neglecting these terms[17].This is,however,an accident.For instance,the order-αs corrections would enhance the sum rule prediction by a factor of two if the ¯q q -term had the opposite sign. From thisﬁgure one can also deduce how changes in the values of the vacuum condensates would aﬀect the numerical results.As long as one stays within the standard limits,the sensitivity to such changes is in fact rather small.For instance,working with the larger value ¯q q =−(0.26GeV)3,or varying m20between0.6and1.0GeV2,changesχ3(y)by no more than±0.15%.In conclusion,we have presented the complete order-αs QCD sum rule analysis of the subleading Isgur-Wise functionχ3(y),including in particular the two-loop perturbative con-tribution.Weﬁnd that over the kinematic region accessible in semileptonic B decays this form factor is small,typically of the order of1%.When combined with our previous analysis [20],which predicted similarly small values for the universal functionχ2(y),these results strongly indicate that power corrections in the heavy quark expansion which are induced by the chromo-magnetic interaction between the gluonﬁeld and the heavy quark spin are small.ACKNOWLEDGMENTSIt is a pleasure to thank Michael Peskin for helpful discussions.M.N.gratefully acknowl-edgesﬁnancial support from the BASF Aktiengesellschaft and from the German National Scholarship Foundation.Y.N.is an incumbent of the Ruth E.Recu Career Development chair,and is supported in part by the Israel Commission for Basic Research and by the Minerva Foundation.This work was also supported by the Department of Energy,contract DE-AC03-76SF00515.REFERENCES[1]E.Eichten and B.Hill,Phys.Lett.B234,511(1990);243,427(1990).[2]B.Grinstein,Nucl.Phys.B339,253(1990).[3]H.Georgi,Phys.Lett.B240,447(1990).[4]T.Mannel,W.Roberts and Z.Ryzak,Nucl.Phys.B368,204(1992).[5]A.F.Falk,H.Georgi,B.Grinstein,and M.B.Wise,Nucl.Phys.B343,1(1990).[6]N.Isgur and M.B.Wise,Phys.Lett.B232,113(1989);237,527(1990).[7]J.D.Bjorken,Proceedings of the18th SLAC Summer Institute on Particle Physics,pp.167,Stanford,California,July1990,edited by J.F.Hawthorne(SLAC,Stanford,1991).[8]M.B.Voloshin and M.A.Shifman,Yad.Fiz.45,463(1987)[Sov.J.Nucl.Phys.45,292(1987)];47,801(1988)[47,511(1988)].[9]A.F.Falk,B.Grinstein,and M.E.Luke,Nucl.Phys.B357,185(1991).[10]M.E.Luke,Phys.Lett.B252,447(1990).[11]A.F.Falk,M.Neubert,and M.E.Luke,SLAC preprint SLAC–PUB–5771(1992),toappear in Nucl.Phys.B.[12]M.Neubert,V.Rieckert,B.Stech,and Q.P.Xu,in Heavy Flavours,edited by A.J.Buras and M.Lindner,Advanced Series on Directions in High Energy Physics(World Scientiﬁc,Singapore,1992).[13]A.V.Radyushkin,Phys.Lett.B271,218(1991).[14]D.J.Broadhurst and A.G.Grozin,Phys.Lett.B274,421(1992).[15]M.Neubert,Phys.Rev.D45,2451(1992).[16]M.Neubert,Phys.Rev.D46,1076(1992).[17]M.Neubert,Phys.Rev.D46,3914(1992).[18]E.Bagan,P.Ball,V.M.Braun,and H.G.Dosch,Phys.Lett.B278,457(1992);E.Bagan,P.Ball,and P.Gosdzinsky,Heidelberg preprint HD–THEP–92–40(1992).[19]B.Blok and M.Shifman,Santa Barbara preprint NSF–ITP–92–100(1992).[20]M.Neubert,Z.Ligeti,and Y.Nir,SLAC preprint SLAC–PUB–5915(1992).[21]M.Neubert,SLAC preprint SLAC–PUB–5992(1992).[22]M.A.Shifman,A.I.Vainshtein,and V.I.Zakharov,Nucl.Phys.B147,385(1979);B147,448(1979).FIGURESFIG.1.Diagrams contributing to the sum rule for the universal form factorχ3(v·v′):two-loop perturbative contribution(a),and nonperturbative contributions proportional to the quark con-densate(b),the gluon condensate(c),and the mixed condensate(d).Heavy quark propagators are drawn as double lines.The square represents the chromo-magnetic operator.FIG.2.Analysis of the stability region for the sum rule(23):The form factorχ3(y)is shown for y=1.5as a function of the Borel parameter.From top to bottom,the solid curves refer toω0=1.7,2.0,and2.3GeV.The dashes lines are obtained by neglecting the contribution proportional toδω2.FIG.3.(a)Prediction for the form factorχ3(v·v′)in the stability region1.7<ω0<2.3 GeV and0.7<T<1.2GeV.(b)Individual contributions toχ3(v·v′)for T=0.8GeV and ω0=2.0GeV:total(solid),mixed condensate(dashed-dotted),gluon condensate(wide dots), quark condensate(dashes).The perturbative contribution and theδω2-term are indistinguishable in thisﬁgure and are both represented by the narrow dots.11。

ISFform_1

Attention:______________________________ Fax:_______________________________ From:_________________________________ Date:______________________________Re: Importer Security Filing request ISF 10+21.Seller. Name and address of the last known entity by whom the goods are sold or agreed to be sold. If the goods are to be imported otherwise than in pursuance of a purchase, the name and address of the owner of the goods must be provided. A widely recognized commercially accepted identification number (such as the Dun and Bradstreet Data Universal Numbering System (DUNS number) may be provided in lieu of the name and address.________________________________________________________________________________________________________________________________________________________2. Buyer. Name and address of the last known entity to whom the goods are sold or agreed to be sold. If the goods are to be imported otherwise than in pursuance of a purchase, the name and address of the owner of the goods must be provided. A widely recognized commercially accepted identification number (such as the DUNS number) for this party may be provided in lieu of the name and address.________________________________________________________________________________________________________________________________________________________3. Importer of record number/FTZ applicant identification number. Internal Revenue Service (IRS) number. Employer Identification Number (EIN), Social Security Number (SSN), or CBP assigned number of the entity liable for payment of al duties and responsible for meeting all statutory and regulatory requirements incurred as a result of importation. For goods intended to be delivered to a foreign trade zone, the IRS number, EIN, SSN, or CBP assigned number of the party filing the FTZ documentation with CBP must be provided._______________________________________4. Consignee number(s). IRS number, EIN, or CBP assigned number of the individual(s) or firm(s) in the U.S. On whose account the merchandise is shipped._______________________________________5. Manufacturer (or supplier). Name and address of the entity that last manufactures, assembles, produces, or grows the commodity or name and address of the party supplying the finished goods in the country from which the goods are leaving. In the alternative the name and address of the manufacturer (or supplier) that is currently required by the import laws, rules and regulations of the U.S. (i.e., entry procedures) may be provided (this is the information that is used to create the existing manufacturer identification (MID)number for entry purposes). A widely recognized commercially accepted identification number (such as the DUNS number) for this party may be provided in lieu of the name and address. CBP states that it will not accept the current MID as an alternative to the complete name and address of the manufacturer.________________________________________________________________________________________________________________________________________________________6. Ship to party. Name and address of the first deliver-to party scheduled to physically receive the goods after the goods have been released from customs custody. A widely recognized commercially accepted identification number for this party (such as the DUNS number or Facilities Information and Resources Management System (FIRMS) code, when applicable) may be provided in lieu of the name and address.________________________________________________________________________________________________________________________________________________________7. Country of origin. Country of manufacture, production, or growth of the article, based upon the import laws, rules and regulations of the U.S. ____________________________________8. Commodity HTS number. Duty/statistical reporting number under which the article is classified in the HTS. The HTS number must be provided to the six-digit level, but may be provided up to the 10-digit level. This data element can only be used for entry purposes if it is provided at the 10-digit level or greater by the importer of record or its licensed customs broker.__________________________________________________________________________________________________________________9. Container stuffing location. Name and address(es) where the goods were stuffed into the container. For break bulk shipments, as defined in 19 CFR 149.1, the name and address(es) of the physical location(s) where the goods were m ade “ship ready” must be provided. A widely recognized commercially accepted identification number (such as the DUNS number) for this element may be provided in lieu of the name and address.________________________________________________________________________________________________________________________________________________________10. Consolidator (stuffer).Name and address of the party who stuffed the container or arranged for the stuffing of the container. For break bulk shipments, as defined in 19CFR 149.1, the name and address of the party who made the goods “ship ready” or the party who arranged for the goods to bemade “ship ready” must be provided. A widely recognized commercially accepted identification number (such as the DUNS number) for this party may be provided in lieu of the name and address. ________________________________________________________________________________________________________________________________________________________Items 1 through 4 are required to be transmitted no later than 24 hours prior to vessel departure. Items 5 through 8 are required to be transmitted no later than 24 hours prior to vessel departure, but can be modified up to 24 hours prior to vessel arrival at a U.S. Port.Items 9 and 10 are required to be transmitted no later than 24 hours prior to vessel arrival at a U.S. Port.。

Political Uncertainty and Corporate Investment Cycles

Political Uncertainty and Corporate InvestmentCyclesBRANDON JULIO and YOUNGSUK YOOK∗November5,2010ABSTRACTWe document cycles in corporate investment corresponding with the timing of na-tional elections around the world.During election years,ﬁrms reduce investment ex-penditures by an average of4.8%relative to non-election years,controlling for growthopportunities and economic conditions.The magnitude of the investment cycles varieswith different country and election characteristics.We investigate several potential expla-nations andﬁnd evidence supporting the hypothesis that political uncertainty leadsﬁrmsto reduce investment expenditures until the electoral uncertainty is resolved.Theseﬁnd-ings suggest that political uncertainty is an important channel through which the politicalprocess affects real economic outcomes.∗Julio is at London Business School and Yook is at the Graduate School of Business at Sungkyunkwan Uni-versity.Patrick Bolton,Murillo Campello,Ethan Cohen-Cole,Alex Edmans,Zsuzsanna Fluck,Paolo Fulghieri, Dirk Hackbarth,Cam Harvey(Editor),Li Jin,Tae-Young Kim,Stewart Myers,Bang Nguyen-Dang,Meijun Qian,Philipp Schnabl,Vikrant Vig,Michael Weisbach,Toni Whited,two anonymous referees,and an asso-ciate editor provided useful comments as did seminar participants at the China Europe International Business School,Georgetown University,Hong Kong Baptist University,Hong Kong Polytechnic University,Korea Uni-versity,London School of Economics,Nanyang Technological University,Norwegian School of Economics and Business Administration,Seoul National University,Sungkyunkwan University,University of North Carolina at Chapel Hill,the2008Chinese International Conference in Finance,the2008AsianFA-NFA International Con-ference,the2009Paris Spring Corporate Finance Conference,the2009Singapore International Conference on Finance,the2009European Finance Association Conference,the2010American Finance Association Confer-ence,and the2010Finance Down Under Conference.“...how unrealistic any theory of investment opportunity is which leaves the polit-ical factor out of account”.Joseph A.Schumpeter(1939)The relationship between politics and economic outcomes has a long history in research and public debate.One important way in which politics is hypothesized to inﬂuence real decisions is through the channel of uncertainty and instability.In particular,the incentives and uncer-tainties associated with possible changes in government policy or national leadership have implications for the behavior of both politicians andﬁrms.The effects of policy uncertainty are especially relevant in light of the recentﬁnancial crisis and recession.There is a great deal of uncertainty as to how governments will shape policy to stimulate investment in the short run and formulate regulatory and economic policy in the long run.It has been argued that this uncertainty itself may be hindering a recovery by inducingﬁrms to delay investment until the uncertainty related to futureﬁnancial regulation and macroeconomic policy is resolved.1In this paper,we examine the effects of political uncertainty on the investment behavior of ﬁrms in the context of national elections.Elections in which the national leader is determined provide an interesting setting to study the effects of political uncertainty on investment for two important reasons.First,while standard models of policy typically assume a single welfare maximizing planner that makes policy choices over the entire life of the economy,the real world is characterized by leaders who face limited terms and may be replaced by other lead-ers with different policy preferences.Election outcomes are relevant to corporate decisions as they have implications for industry regulation,monetary and trade policy,taxation,and,in more extreme cases,the possible expropriation or nationalization of privateﬁrms.Second, investigating the impact of political uncertainty on investment is a challenging task due to the potential endogeneity between uncertainty and economic growth as the economic downturn itself has arguably generated a great deal of political uncertainty.Elections around the world provide a natural experimental framework for studying political inﬂuences in corporate in-vestment,allowing us to disentangle some of the endogeneity between economic growth and political uncertainty.If political uncertainty is higher when changes in national leadership are more probable,elections provide a recurring event that helps isolate the impact of policy uncertainty on investment from other confounding factors.The timing of elections is out of the control of any individualﬁrm and evenﬁxed in time by constitutional rules for a largeproportion of observations in our sample.In addition,elections around the world take place in different years over time,allowing us to net out any global trends in corporate investment. Using national elections in48countries between1980and2005,we examine changes in cor-porate investment as political uncertaintyﬂuctuates by comparing corporate behavior in the year leading up to the national election outcomes with that in non-election years.The intuition underlying the relationship between electoral uncertainty and investment is simple:if an election can potentially result in a bad outcome from aﬁrm’s perspective,the option value of waiting to invest increases and theﬁrm may rationally delay investment until some or all of the policy uncertainty is resolved.The relationship between uncertainty and real investment has been modeled by Bernanke(1983)and Bloom,Bond and Van Reenen(2007), among others.In these models,ﬁrms become cautious and hold back on investment in the face of uncertainty.Others have modeled the effects of political uncertainty in a macroeco-nomic context.Rodrik(1991)and Pindyck and Solimano(1993)are prominent examples of this literature in which the uncertainty brought about by political factors leadsﬁrms to choose lower levels of investment expenditures.Chen and Funke(2003)model the private invest-ment decision in emerging markets in the face of policy uncertainty.More recently,Bloom, Flotoetto and Jaimovich(2009)model business cycles as a function of variation in levels of macroeconomic uncertainty.The idea that political instability can deter investment on the aggregate level is supported by empirical evidence.Barro(1991)and Alesina and Perotti(1996)ﬁnd that measures of political instability and violence are correlated with cross-country differences in investment rates.Pindyck and Solimano(1993)and Mauro(1995)ﬁnd evidence that political uncertainty and an index measuring bribery and corruption are negatively related to investment spending at the aggregate level.However,some difﬁculties arise in interpreting the aggregate evidence. First,it is not clear whether the various measures of political instability are exogenous to the economic conditions and the aggregate investment.Second,as discussed in Pindyck and Solimano(1993),the models of investment under uncertainty are less clear about how un-certainty affects long-run equilibrium investment rates,deﬁned as the ratio of investment to capital stock,as uncertainty affects both the optimal capital stock and investment in the long run.The predictions of the models are less ambiguous when there are temporary shocks to the level of uncertainty as the uncertainty mainly works through investment rather than capitalstock in the short run.Indeed,Bernanke(1983)shows that events whose long-run implica-tions are uncertain can generate investment cycles by increasing the returns to waiting for new information,particularly when the source of uncertainty periodically renews itself over time.A temporary increase in uncertainty surrounding national elections creates incentives that may induce immediate declines in investment expenditures.Our empirical investigation provides results consistent with the political uncertainty hy-pothesis.We document novel and robust evidence that political uncertainty around national elections induces cycles in corporate investment.In the period leading up to the election,in-vestment expenditures decline by an average of4.8%,controlling for growth opportunities, cashﬂows,and economic conditions.To address the concern that the results may be driven by elections that are notﬁxed in time by constitution,we repeat the analysis by estimating our investment regressions only for countries withﬁxed election timing andﬁnd similar re-sults.Additionally,we examine the determinants of early elections andﬁnd a strong positive correlation between economic growth and the probability of holding an early election.To the extent that aﬁrm’s investment expenditures are positively correlated with economic growth, this suggests that the inclusion of endogenously timed elections in the regressions has the net effect of reducing the dampening effect of electoral uncertainty on investment as the elections are generally called during periods of relatively high economic performance.Across countries,weﬁnd that the temporary decline in investment expenditures is larger in countries with civil law origins,fewer checks and balances,less stable governments,and in countries with a higher ratio of central government spending to GDP.Within countries,the cycles are more pronounced forﬁrms in industries considered to be more sensitive to political outcomes.Elections in which the outcome is“close”as measured by voting results lead to deeper investment cycles than elections in which the victor wins by a large margin.We also ﬁnd that investment rates drop more in election years in which the incumbent national leader is classiﬁed as“market-friendly”by the World Bank.We also show that the election-year drop in investment is followed by a small,temporary increase in investment in the year imme-diately following the election as the uncertainty over election outcomes subsides.However, the overall magnitude of the post-election increase in investment is smaller than that of the earlier decline.We also measure changes in cash holdings,ﬁnding temporary increases in cash balances in the year prior to the election in the amount of4.3%of the average cash toassets ratio,controlling forﬁrm and economic conditions.The increase in cash holdings is similar in magnitude to the election-year decline in investment,suggesting that the funds that would have been used as investment are temporarily held as cash until the election uncertainty is resolved.Political uncertainty is not the only mechanism whereby real outcomes can be affected around the timing of elections.There are two plausible alternative explanations in the lit-erature suggesting election-induced cycles in investment.Theﬁrst is the political business cycles hypothesis.Starting with Nordhaus’s(1975)model of political business cycles,there has been much debate over whether incumbents manipulateﬁscal and monetary policy instru-ments to inﬂuence the level of economic activity prior to an election in order to maximize the probability of reelection.Thus,one alternative explanation for our results is that corporate investment is reacting to changing macroeconomic fundamentals.While the political business cycle hypothesis predicts that average economic activity should be higher just before the elec-tion,the actions used to stimulate the economy could have a crowding-out effect on private investment.The second alternative explanation is related to the value of political connections. Someﬁrms may have incentives to change their investment behavior to help ensure that their political connections remain in ofﬁce through the election cycle.Bertrand,Kramarz,Schoar and Thesmar(2006)investigate the behavior of politically connectedﬁrms around municipal elections in France,andﬁnd that theﬁrms managed by connected CEOs boost their invest-ment during election years,particularly in politically contested areas,likely in an attempt to help their connection get re-elected.We conduct formal tests of these alternative hypotheses andﬁnd no evidence that they are operating in our sample ofﬁrms.We therefore view the political uncertainty hypothesis to be the explanation among existing theories that bestﬁts the patterns in the data.Theseﬁndings have two important contributions.First,we document a new stylized fact regarding corporate investment around the world.That is,there is a tendency forﬁrms to reduce investment in election years.These results demonstrate an important link between the political process and real outcomes.Second,the results suggest that political uncertainty mat-ters for aﬁrm’s real investment and savings decisions.This provides an interesting illustration of the impact of uncertainty in general as an important determinant of investment dynamics.As far as we know,we are theﬁrst to examine the effects of national elections onﬁrm-level investment behavior around the world.The remainder of the paper proceeds in the following manner.Section II develops the empirical predictions and discusses the identiﬁcation strategy.Section III summarizes the ﬁrm characteristics and the election data.Section IV presents our main empirical results related to corporate investment cycles around elections,including various subsample analyses, multiple robustness checks,and an examination of changes in corporate cash holdings around the election period.Section V concludes.I.Hypothesis Development and Empirical StrategyWhen a particular investment project is characterized by some degree of irreversibility and uncertainty over future cashﬂows or discount rates,the value of the investment project will be affected by the same factors that inﬂuence the pricing ofﬁnancial options,in particular,the volatility or uncertainty of the value of the underlying asset.The application of option pricing to capital budgeting has generated many empirical predictions and insights on how investment dynamics change in the face of uncertainty.Some classic examples include McDonald and Seigel(1986),who examine the valuation of operating options and the value of waiting to invest.They demonstrate that even moderate amounts of uncertainty can more than double the required rate of return for investment projects.Ingersoll and Ross(1992)model the timing decision in the face of interest rate uncertainty.They argue that,under the assumptions of irreversibility and uncertainty,the simple net present value(NPV)rule is not optimal from a value-maximizing perspective.Uncertainty increases the value of waiting to invest through what Bernanke(1983)termed the“bad news”principle.That is,an increase in uncertainty causes reductions in current investment only if there is some probability of a bad outcome.In the context of national elec-tions,this suggests thatﬁrms will delay investment in anticipation of possible negative changes in the country’s macroeconomic policy,taxation,monetary policy,or the general regulatory environment.However,in some cases,the outcome of an election could be construed as good news,regardless of who wins in the end.For example,if the current government is corruptor incompetent,ﬁrms could view a likely change in power as good news and hence may not reduce investment prior to the realization of the election outcome since any different outcome may be better than the current state of affairs.The bad news principle is more subtle in this case.For example,suppose aﬁrm is choosing among several mutually exclusive investment projects,each with a positive expected return.Also suppose that the outcome of an upcoming election will increase the expected return of each of the investment projects,regardless of the outcome.Theﬁrm still has an incentive to delay investment if the outcome would reorder the rankings of the individual projects in terms of expected returns.Thus,the bad news principle does not require the possibility of extreme policies such as nationalization of private assets to induce changes in investment.Even positive changes in policy may induce an incentive forﬁrms to wait to invest as the outcome will still have implications for howﬁrms allocate investment spending across various investment opportunities.If political uncertainty matters forﬁrms,then the recurring nature of the political uncer-tainty around elections can generate cycles in investment spending.This is an application of Bernanke’s bad news principal that the possibility of a bad election outcome induces aﬁrm to hold off on its investment projects.This leads to our primary hypothesis that investment expenditures are expected to decline in the year leading up to the election.That is,we expect the average effect of electoral uncertainty to be a temporary decline in the conditional mean investment rate for allﬁrms in the sample.The bad news principle also suggests that the value of waiting to invest will vary fromﬁrm toﬁrm and across countries.Within countries,the magnitude of the investment cycle may vary across elections,depending on the the degree of uncertainty regarding election outcomes.The spread between potential outcomes as well as the likelihood of each outcome will generate heterogeneity in the size of observed investment cycles.Across countries,we hypothesize that investment cycles will be more pronounced in countries with a higher probability of policy changes or a larger variation in possible policy outcomes after the election.Since we are investigating national elections,we expect the effect of elections to be larger for countries with more centralized governments.Political institutions may matter as well.Countries in which political decisions are more constrained by various checks and balances are less likely to experience large policy swings following a change of power.For example,presidential systems are typically considered to have greater checks and balances but lessﬂexibility inpolicy making relative to parliamentary systems,suggesting that perhaps large policy swings are more common in parliamentary systems.2We also expect that countries with less stable governments in general will experience larger changes in investment around elections.Within countries,we hypothesize that the drop in investment expenditures will be larger when the election outcome is more uncertain.In particular,we expect that cycles will be more pronounced for elections with close outcomes relative to those with large margins of victory. The amount of uncertainty regarding the impending election outcome is unobservable,but we do observe the election results and vote counts for each ing the size of the margin of victory as a proxy for the degree of outcome uncertainty in any given election, we examine whether investment cycles vary with the degree of uncertainty across elections within countries.We also investigate the political platform of the incumbent leader during the election year.The political platform of an incumbent with respect to economic policy may have asymmetric implications on investment cycles.Firms are likely to view a possible shift in leadership from a market-friendly leader to a socialist leader as worse news than a possible shift in the other direction.Our empirical strategy employs the timing of national elections around the world to test the political uncertainty hypothesis.It is important to note that the timing of elections is not a direct measure of political uncertainty.Hence,an important identiﬁcation assumption is that political uncertainty is indeed higher on average in the period leading up to an election compared to other time periods.There is evidence fromﬁnancial markets that the uncertainty related to elections and political changes are reﬂected in asset prices.Bialkowski,Gottschalk and Wisniewski(2008)and Boutchkova,Doshi,Durnev and Molchanov(2010)examine the stock market volatility around national elections andﬁnd that volatility is signiﬁcantly higher than normal during the election period.Boutchkova et al.(2010)ﬁnd that the return volatility is higher around elections forﬁrms operating in politically sensitive industries,suggesting that the increased volatility reﬂects a higher political risk.Bernhard and Leblang(2006)document changes in bond yields,exchange rates,and equity volatility around elections and other po-litical changes and show that these changes are larger during elections with less predictable outcomes.This evidence provides support for our identiﬁcation assumption that political un-certainty is higher than normal during elections.Our empirical analysis produces two broad sets of results.In ourﬁrst set of results,we employ the variation across elections,countries,andﬁrms to help us identify the uncertainty channel to explain the reduction in corporate investment.Our basic approach in thisﬁrst step is to examine variation in corporate investment around the timing of national elections and to demonstrate that these changes are larger for events in which the uncertainty related to election outcomes is higher.We recognize that other mechanisms may be at play during election periods that can lead to changes in investment behavior.Therefore,our second set of results attempts to distinguish the political uncertainty channel from other existing hypotheses, namely the political business cycle hypothesis and the possible effects of political connections.II.Data DescriptionA.Election DataThis study considers248national elections in48countries held between1980and2005in which the outcome determined the national leader directly or indirectly.The detailed election information is obtained from a variety of sources.The primary source for election and regime change data is the Polity IV database maintained by the Center for International Develop-ment and Conﬂict Management at the University of Maryland.This database contains annual information on the regime and authority characteristics of all independent states with total populations greater than500,000.The second major source of information is the World Bank Database of Political Institutions.This source provides information about electoral rules and the classiﬁcation of political platforms for the elected leaders and candidates.We supplement the election data with various internet sources3for cases in which the election information is missing from the Polity IV database or the Database of Political Institutions.Theﬁrst task for the election data collection is to identity the chief executive of each coun-try and the national elections associated with the selection of the chief executive.In a country with a presidential system,the supreme executive power is normally vested in the ofﬁce of the president.Thus,presidential elections are naturally considered in our analysis for coun-tries with presidential systems.In a parliamentary system,the executive power is normallyvested in a cabinet responsible to parliament.In such a country,the prime minister or pre-mier,being the head of the cabinet and leader of the parliament,functions as the actual chief executive of the nation.Thus,legislative elections are utilized for countries with parliamen-tary systems as the outcome of such legislative election has the foremost inﬂuence over the appointment of prime minister.4Some countries use a hybrid system combining elements of both parliamentary and presidential democracy;a president and a prime minister coexist with both presidential and legislative elections held nationally.In such cases,the constitutional framework and practice is examined in greater detail to understand how executive power is divided between the two leaders,and the election associated with the leader who exerts more power over executive decisions is selected for the study5.As a robustness check,we repeat our analysis excluding the four countries for which the classiﬁcation requires some discretion (Finland,France,Pakistan and Poland)andﬁnd that the results are unchanged.The resulting data set comprises31countries with legislative elections,16countries with presidential elections,and one country(Israel)with prime ministerial elections.6Table I presents the classiﬁcation of political systems and the number of elections utilized for each of the48countries in our sample.The table also shows the origin of each country’s legal system,as reported by La Porta,Lopez-de-Silanes,Shleifer,and Vishny(1998).[TABLE I HERE]Another important characteristic of national elections is whether the timing of the elec-tions is exogenously speciﬁed by electoral ernments under some electoral systems can be dissolved before the expiry of its full term for various reasons,and an election is then normally called to form a new government.This complicates the interpretation of our empirical results as the timing of elections may be endogenously connected to the country’s economic performance over time.Ito(1990),for example,documents that Japanese general elections have coincided with the periods of economic expansion,suggesting that the govern-ment opportunistically selected the timing of elections.To deal with the possible endogeneity of election timing,we classify countries as having either exogenous timing or endogenous timing.All countries with a record of early elections are classiﬁed as having endogenous tim-ing.All presidential elections,with the exception of Sri Lanka,are held on a regular basis and are classiﬁed as having exogenous timing.This leaves unclassiﬁed seven countries withparliamentary systems and one country with hybrid system.In order to classify those remain-ing countries,we refer to electoral laws and practices as well as the classiﬁcation provided by Alesina,Cohen and Roubini(1992)7.Accordingly,three of the remaining countries,Czech Republic,Finland,and New Zealand,are classiﬁed as having endogenous timing8and the rest are classiﬁed as having exogenous timing.Table I reports the election timing classiﬁcation for every country in our sample.Panel A of Table II summarizes the election data.Elections are held every3.8years on average and the average nominal term of a chief executive is4.4years.The next row reports the political platform of incumbent governments in the election years.The classiﬁcation is based on the World Bank Database of Political Institutions,which refers to various sources including Political Handbook yearbooks in order to identify party orientation with respect to economic policy.9The World Bank classiﬁes a government as being right-leaning if the political party is deﬁned as conservative,Christian democratic,or right-wing by these sources. Left-leaning parties are those that are deﬁned as communist,socialist,social democratic,or left-wing.Centrist parties are those that advocate strengthening private enterprise in a social-liberal context.We deﬁne the the incumbent political party as being”market-friendly”if the incumbent government in the election year is classiﬁed as right-leaning or centrist by the World Bank.Accordingly,63.3%of the incumbent administrations in the year leading up to an election are classiﬁed as market-friendly,and the remaining36.7%are classiﬁed as left-leaning.We also summarize the distribution of historical vote counts to give a sense for the degree of uncertainty surrounding a given election.On average,the winner of an election obtains41.9%of the total vote,followed by the runner-up at28.7%,and the third-place candidate receives12.2%of the total.The table also shows that45.6%of the elections are classiﬁed as having exogenous timing.Table II also shows that54%of the elections lead to the replacement of the national leader and43%of the elections result in change in the ruling party.[TABLE II HERE]B.Country-Level DataWe obtain institutional and macroeconomic data from various sources.The World Devel-opment Indicators from the World Bank is our primary source for the macroeconomic vari-ables including real GDP,central government spending,inﬂation,and real interest rate.We obtain data on the money supply(M1)from Political Risk Service’s International Country Risk Guide(ICRG).ICRG also reports the government stability ratings on a monthly basis for the countries in our sample.The government stability index assigns numbers between0and 12,where higher values indicate more stable governments.This time-varying index assesses the government’s ability to carry out its declared programs,and its ability to stay in ofﬁce.The Database of Political Institutions provides a measure of the effectiveness of checks and balances in each political system on an annual basis10.The basic idea is to capture the number of decision makers whose agreement is necessary for the approval of policy changes. The measure is a count of the number of veto players in the political system at a given point in time based on the prevailing electoral rules and laws.It also takes into account whether the executive and legislative branches of government are controlled by the same party,which effectively reduces the checks and balances relative to having different parties controlling different branches of government.In presidential systems,the count is increased by one for the president and increased by one for each additional legislative body.For parliamentary systems,the count is increased by one for the prime minister and increased by the number of parties included in the governing coalition.The number is reduced if the party of the executive is the same as the largest party in any particular chamber of government.Table II shows that the average of checks in the sample is3.95with the standard deviation of1.95.The index of central bank independence(CBI)measures the extent to which the central bank is independent from the political power.This annual,time-varying index is taken from Cukierman,Webb,and Neypati(1992)for the period between1980and1989,and from Polillo and Guillen(2005)for the period between1990and2000.Initially,Cukierman,Webb,and Neypati(1992)constructed the index for72industrial and developing countries for the period between1950and1989.11Later,Polillo and Guillen extended the index to the period between 1990and2000according to the deﬁnition of Cukierman,Webb,and Neypati(1992).The index is a continuous score ranging between zero and one,where one indicates maximum。

KeyTek EMCPro PLUS 高级EMC测试系统用户指南说明书

Analyze •Detect •Measure •Control TMKeyTek EMCPro ®PLUSAdvanced EMC test system forcompliance testing to 6 IEC/EN standardsThe newly configured KeyTek EMCPro ®PLUS test system features resident capabilities for EMC CE Mark compliance testing to 6 IEC/EN standards,and fully addresses new requirements for a 100 kHz burst rate per IEC 61000-4-4, Edition 2 (EFT) and 80%dip per IEC 61000-4-11, Edition 2 (PQF ™).Portable and low cost, the KeyTek EMCPro PLUS is the answer to manufacturers’ demand for a mid-range, multi-capability EMC immunity tester. It’s ideal for companies who require flexibility, versatility, and the highest test level-to-cost ratio instrument on the market.Portable, mid-range EMC test systemResident capabilities for compliance testing to 6 IEC/EN standards Addresses ANSI/IEEE, ITU, ETSI & UL standardsSurge testing to 6.6kV with the combination, telecom, & ring wavesMonitors surge voltage & current at the output terminals Monitors output of the coupling unit & automatically switches connections according to coupling modeHighest test levels, widest selection of tests & lowest in-use costsUpgradable as standards changeTechnical Specifications Model PRO-BASEEMCPro PLUS Base UnitSystem Voltage90-240VAC, 50/60HzINTEGRATED EUT MAINS COUPLER/DECOUPLER AC Voltage 1 phase, 50 - 250VAC. 50/60Hz AC Current 16A max.**DC Voltage 100VDC max.DC Current:10A max.Frequency 50/60Hz EUT Connectors Nema, British, SchukoCONTROL INTERFACE Interface RS232 Fiber-optic SAFETY FEATURESExternal Interlock for usersInterlock for CCL connectorExternal stop input ENVIRONMENTAL OPERATING CONDITIONS Temperature 15°- 40°C Humidity 10-75%, non-condensing Altitude 8000 ft. max.PHYSICAL Height 22.9cm (8.7 in)Width 43.4cm (17.1 in)Depth 64.8cm (25.5 in)Weight 39kg (85 lbs.)CE MARKINGSafety and EMC Directives 1981Model PRO-ESDESD per IEC 61000-4-2 and EN 61000-4-2Trigger Modes One shot manual, multi-shot tripodRepetition Rate Single shot, 1pps or 20ppsAir Discharge Voltage500V - 8.8kV ±10%Contact Discharge Voltage500V - 4.4kV ±10%Discharge Capacitor150pF ±10%Discharge Resistance330Ω±10%Charging Resistance50MΩ- 100MΩPolarity Front panel or software controlledShot Counter 1 - 999 dischargesEnergy Storage***********Model PRO-EFTEFT per IEC 61000-4-4 Edition 2, EN 61000-4-4 and ANSI C62.41Voltage Waveform5/50ns ±30%Peak Voltage250V - 4.4kV ±5%Burst Period300ms ±10%Burst Duration15ms ±20%, for pulse frequencies uo to 5kHz, 0.75msabove 5kHzFrequency1-100kHz, in 0.5kHz steps, ±10%DC Blocking Capacitor10nF (internal)Options Model CM-3CD-16/32:16 or 32 Amp, 3 phase EFT &surge coupler/decouplerModel CM-CCL: Capacitive coupling clampModel CM-CCLC: Coupling clamp coverModel EFT-ATTN:EFT attenuator for oscilloscopemonitoringModel PRO-SURGESurge for compliant testing per IEC 61000-4-5, EN 61000-4-5, ANSI C62.41 Category B and UL 1449Voltage Waveform 1.2/50µsPeak Voltage250V - 6.6kV ±5%, 12Ωmode250V - 6.0kV ±5%, 2ΩmodePeak Current125A - 3.3kA ±10%Additional 10ΩResistor Software selectableRepetition Rate Up to 4 per minuteOpen-circuit Voltage Front time: 1.2µs ±30%Duration: 50 µs ±20%1Undershoot: ≤30%Short-circuit Current Front time: 8.0µs ±20%Duration*50µs ±20%Undershoot≤30%Line sync accuracy±15%, 50 - 277VACOptions ModelCM-3CD-16/32:16 or 32 Amp, 3 phase EFT& surge coupler/decouplerModel CM-I/OCD: External 8 line coupler/decouplerfor I/O signal linesModel CM-I/OCD-HS:High speed I/OCD option fortesting data rates to >100kHz*Durations are reduced in 12Ωmode and when coupling multiple lines to PE Model PRO-RING**Ring Wave Surge per ANSI C62.41 Cat. A, B, and UL 864Voltage Waveform100kHz damped cosinePeak Voltage250 - 6.6kV ±5%Repetition Rate<4/minute at 6kV, faster at lower voltagesOpen-circuit Voltage Rise Time: 0.5µs ±30%Short-circuit Current Vp/Ip: 12Ω±3Ωor 30Ω±8Ωsoftware selectable Options ModelCM-3CD-16/32:16 or 32 Amp, 3 phase EFT& surge coupler/decouplerModel PRO-TELECOM**Surge Telecom compliant testing per IEC 61000-4-5, EN 61000-4-5, FCC Part 68, ITU K.17, K.20, K.21 and ETSIVoltage Waveform10/700µs (9/720µs FCC Part 68)Peak Voltage250V - 6.6kV ±5%Peak Current 6.25 - 165A +10/-0%, 40ΩmodeRepetition Rate Up to 4 per minuteOpen-circuit Voltage Front time: 7.0µs to 11.7µsDuration: 576µs to 840µsShort-circuit Current Front time: 3.5µs to 6.5µsDuration: 256µs to 384µsOptions Model CM-TELCD: External coupler for telecom linesSurge Waveform MonitoringLines Monitored Monitors are automatically switched to matchgenerator coupling modeOpen-circuit Voltage1000:1 ±10%Short-circuit 200:1 ±7%Current AttenuationModel PRO-HPOWERPower Frequency Magnetic Field for compliant testing per IEC 61000-4-8 and EN 61000-4-8 Field Frequency50Hz/60HzField Amplitude0.5 - 4A/m, in 0.25A steps, ±10% (with CM-HCOIL)up to 100A/m with optional external HPOWER-EXT AC Source InternalResolution0.25A minimumCoil Factor0.65 to 1.00Coil Resistance0.05ΩmaximumOptions ModelCM-HMON:Measurement probe for powerfrequency magnetic fieldsModel CM-HCOIL: 1m x1m magnetic field coilModel HPOWER - EXT: External generator for powerfrequency magnetic field to 30A/m**PRO-TELECOM and PRO-RING can not be installed in same unit.Model PRO-HPULSEPulse Magnetic Field for compliant testing per IEC 61000-4-9 and EN 61000-4-9Field Pulse8/20µsField Amplitude100A/m - 1000A/m, ±10%Resolution5A/mCoil Factor0.65 to 1.00Options ModelCM-HMON:Measurement probe for powerfrequency magnetic fieldsModel CM-HCOIL: 1m x 1m magnetic field coilModel PRO-PQFDips and Interrupts for compliant testing IEC 61000-4-11 Edition 2, and EN 61000-4-11 Dips40%, 70%, 80%Interrupts0% (short and open)Transition Time1µs - 5µsInrush Minimum 250Amps @ 100 - 120V,Minimum 500Amps @ 220 - 240VAC Voltage50 - 250VAC, 50/60HzAC Current16A max.**PQF Sync Output5V signal occurs at each dip or interrupt transition Options Model PQF-QUAL:Circuit per IEC 61000-4-11 fortesting PQF generator inrush capabilityPQF Waveform MonitoringVoltage Input Connection Fixed, L1 to L2Voltage Attenuation100:1 ±5%Current Input Connection Fixed, L1Peak Current Minimum 500A inrush into 1700µFCurrent Attenuation200:1 ±5%Model CM-3CD-16 & CM-3CD-32*Semi-automatic, stand alone, three-phase AC/DC mains coupler/decouplers for EFT & Surge per IEC 61000-4-4, Edition 2 and IEC 61000-4-5ELECTRICALWaveforms EFT: 5/50ns, per IEC 61000-4-4Surge: Combination wave: 1.2/50µs open-circuitvoltage, 8/20µs short-circuit current, per IEC 61000-4-5 Maximum Surge 6.6kV, 3.3kAVoltage & CurrentMaximum EFT Voltage 4.4kVCoupling Modes EFT: L1, L2, L3, N or PESurge Hi: L1, L2, L3 or NSurge Lo: L1, L2, L3, N or PE* Not available for delivery until October 2004COUPLER/DECOUPLERSAC Voltage50 to 250V, 50/60Hz line to ground, 50 to 433Vline to lineAC Current CM-3CD-16:16A/phase continuousCM-3CD-32:32A/phase continuousDC Current CM-3CD-16:16A up to 48V8A up to 110V1.2A up to 220V0.3A up to 440VCM-3CD-32:25A up to 48V8A up to 220V1.2A up to 220V0.3A up to 440VEUT Mains Safety SocketsOutput ConnectorsPOWER REQUIREMENTSInput Voltage90-250VAC, 50/60HzInput Current1A at 120VAC; 0.5A at 240VACModel CM-I/OCDI/O coupler/decoupler - provides the ability to couple surges from EMCPro PLUS or any surge simulator, to I/O or data lines per IEC 61000-4-5ELECTRICALWaveforms Designed to couple combination waves of 1.2/50µsopen-circuit voltage, 8/20µs short-circuit currentsupplied by option PRO-SURGE with the KeyTekEMCPro PLUSRepetition Rate Up to 5 per minute at 4.4kVData Line Frequency To greater than 100kHz without significant degradationwhen CM-I/OCD-HS is installed. Option CM-I/OCD-HSis recommended for data line frequencies greater than1kHzNumber of Lines Eight lines - any line can be surged to any other line orgroundMaximum Surge Voltage 4.4kVMaximum Signal Line Voltage200VMaximum Signal Line Current1A AC or DCClamping Selectable built-in clamps of 20V and 220V; externalbias input for other clamp levelsAvailable Options CM-I/OCD-HS:Internally-installed option providesselectable parallel resistors (400s, 200s, 100s) - highlyrecommended for data line frequencies greater than1kHz.Model CM-TELCDTelecom line coupler/decoupler - provides the ability to couple both the telecom wave and combination wave per IEC 61000-4-5ELECTRICALWaveforms Designed to couple 1.2/50µs combination or 10/700µstelecom wavesTelecom Line Frequency To 100kHz without significant degradationNumber of Lines Up to four lines - one or two pairs of balanced TelecomlinesMaximum Surge Voltage 4.4kVMaximum Signal Line Voltage200VMaximum Signal Line Current1A AC or DCClamping Selectable built-in clamps of 20V and 225V: externalbias input for other clamp levelsExperience the many benefits of working with recognized experts in the field of EMC (Electromagnetic Compatibility)testing. Our commitment to the discipline is wide ranging; we actively participate on global standards committees, andhave helped define test methodologies to achieve regulatory standards such as CE Mark requirements, as well ascompany and market-driven product quality objectives,.Our goal is to support you with lifelong service — from applications support, calibration services and preventativemaintenance scheduling to full tactical field support.Thermo can help you reach the next level of success.Please see the KeyTek EMC Test System Options & Accessories data sheet for additional KeyTek EMCPro PLUS testsystem options and accessories.Specialists who understand the challenges you face. Innovative ideas. Leading technologies. Breadth of EMC test equipment.Thermo–your EMC test solutions partner. Contact us today for details.This sheet is for informational purpose only and is subject to change without notice.© 2004 Thermo Electron Corporation. All rights reserved. Thermo Electron Corporation, Question everything, and Analyze. Detect. Measure. Control.are trademarks of Thermo Electron Corporation.Control Technologies Division One Lowell Research Center(978) /esdLowell, MA, 01852(978) 275-0850 faxFormerly doing business as Thermo KeyTek。

Fractional order nonlinear systems with delay in iterativelearning control

Fractional order nonlinear systems with delay in iterative learning controlqLi Yan,Jiang Wei ⇑School of Mathematical Science,AnHui University,Hefei,Anhui 230601,PR Chinaa r t i c l e i n f o Keywords:Fractional order with delayGeneralized Gronwall inequality Iterative learning controla b s t r a c tIn this paper,we discuss a P-type iterative learning control (ILC)scheme for a class of frac-tional-order nonlinear systems with delay.By introducing the k -norm and using Gronwall inequality,the sufﬁcient condition for the robust convergence of the tracking errors is obtained.Based on this convergence,the P-type ILC updating laws can be determined.Ó2015Elsevier Inc.All rights reserved.1.IntroductionIterative learning control is suitable for repetitive movements of the controlled system.Its goal is to achieve full range of tracking tasks on ﬁnite interval.Iterative learning control (ILC)is one of the most active ﬁelds in control theories.ILC belongs to the intelligent control methodology,is an approach for improving the transient performance of systems that operate repetitively over a ﬁxed time interval.The combination of ILC and fractional calculus was ﬁrst propose in 2001.In the following ten years,many fractional-order ILC problems were presented aiming at enhancing the performance of ILC scheme for linear or nonlinear systems [2–6].In recent years,the application of ILC to the fractional-order system has become a new topic.The objective of ILC is to determine a control input iteratively,resulting in plant’s ability to track the given reference signal or the output trajectory over a ﬁxed time interval.Owing to its simplicity and effectiveness,ILC has been found to be a good alternative in many areas and applications,see recent surveys for detailed results [7–13].This paper is on the basis of [14].We discuss the tracking problem through the open-loop P-type iterative learning control and the closed-loop P-type iterative learning control.The sufﬁcient condition errors with respect to initial positioning error under P-type ILC is obtained by introducing the k -norm and using Gronwall inequality.The delay part of this paper,we reference [15–20].In Section 2some basic deﬁnitions of fractional calculus used in this paper are mentioned.Section 3the main results are shown.Finally,some conclusions are drawn in Section 4.Throughout this paper,the 2-norm for the n -dimensional vector w ¼ðw 1;w 2;...;w n Þis deﬁned as k w k ¼ðP n r ¼0w 2i Þ12,while the k -norm for a function is deﬁned as k Ák k ¼sup t 2½0;T f e Àk t j Ájg ,where k >0./10.1016/j.amc.2015.01.0140096-3003/Ó2015Elsevier Inc.All rights reserved.qThis research had been supported by National Natural Science Foundation of China (nos.11371027and 11471015);Doctoral Fund of Ministry of Education of China (no.20093401110001);Major Program of Educational Commission of Anhui Province of China (no.KJ2010ZD02);National Natural Science Tianyuan Specialty Foundation of China (11326115),Research Foundation for the Doctoral Program of Higher Education of China (20133401120013),Natural Science Program of Higher Colleges of Anhui Province (KJ2013A032).⇑Corresponding author.E-mail addresses:liyanahu@ (L.Yan),jiangwei@ (J.Wei).2.PreliminariesIn this section,some basic deﬁnitions and lemmas are introduced,which will be used in the following discussions.Deﬁnition 2.1.Riemann–Liouville’s fractional integral of order a >0for a function f :R +!R is deﬁned ast 0D Àat f ðt Þ¼1a Ztt 0ðt Às Þa À1f ðs Þds :ð2:1ÞDeﬁnition 2.2.The Caputo derivatives is deﬁned asC t 0D a t f ðt Þ¼t 0D a Àm tD mf ðt Þ;a 2½m À1;m Þ;ð2:2Þwhere m 2Z þ;D m is the classical m -order integral derivative.Deﬁnition 2.3.The deﬁnition of the two-parameter function of the Mittag–Lefﬂer type is described byE a ;b ðz Þ¼X1k ¼0z kða k þb Þ;a >0;b >0;z 2C :ð2:3ÞFor b ¼1we obtain the Mittag–Lefﬂer function of one-parameter:E a ðz Þ¼X1k ¼0z kC a ;a >0;z 2C :ð2:4ÞLemma 2.4.The fractional-order differentiation of the Mittag–Lefﬂer function ist 0D c t ½tb À1E a ;b ðk t a Þ ¼t b Àc À1E a ;b Àc ðk t a Þ;c <b :ð2:5ÞLemma 2.5.If the function f ðt ;x Þis continuous,then the initial value problemC t 0D at x ðt Þ¼f ðt ;x t Þ;0<a <1;x ðt 0Þ¼u :(ð2:6Þis equivalent to the following nonlinear Volterra integral equationx ðt Þ¼x ðt 0Þþ1C ða ÞZtt 0ðt Às Þa À1f ðs ;x s Þds :ð2:7Þand its solutions are continuous [12].Lemma 2.6.([14]Generalized Gronwall Inequality).Let u ðt Þbe a continuous function on t 2½0;T and let v ðt Às Þbe continuous and nonnegative on the triangle 06s 6T.Moreover,let w ðt Þbe a positive continuous and non-decreasing function on t 2½0;T .Ifu ðt Þ6w ðt ÞþZtv ðt Às Þu ðs Þds ;t 2½0;T ;ð2:8Þthenu ðt Þ6w ðt ÞeR tv ðt Às Þds;t 2½0;T :ð2:9Þ3.P-type iterative learning controlConsider the following SISO fractional-order nonlinear system with delayD a t x k ðt Þ¼f ðx k t ;u k ;t Þ;y k ðt Þ¼g ðx k t ;u k ;t Þ;(ð3:1Þwhere k is the number of iterations,k 2f 0;1;2;...g ;t 2½0;T ;a 2ð0;1Þ,L.Yan,J.Wei /Applied Mathematics and Computation 257(2015)546–5525470<d 16j g u j ¼@g ðx k t ;u k ;t Þ@u k6d 2;0<d 36g x t ¼@g ðx k t ;u k ;t Þ@x t6d 4;k f ðx k t ;u k ;t ÞÀf ð x t k ; u k ;t Þk 6f 0j u k À u k j þf 0k x k t À x t k k ;and d 1;d 2;d 3;d 4and f 0are positive constants.x k t 2R nis the state variables in a time interval of length t,that isx k t ðh Þ¼x kðt þh Þ;ð3:2Þh 2½Às ;0 ;s >0;u k ðt Þ2R and y k ðt Þ2R are the control input and output,respectively.D a t denotes the Caputo derivative oforder a .The purpose of this paper is to ﬁnd an iterative control law to generate the control input u k ðt Þsuch that the system output y k ðt Þtracks the desired output trajectory y d ðt Þas accurately as possible when k goes to inﬁnity for all t 2½0;T .We discuss the tracking problem through the open-loop P-type iterative learning control and the closed-loop P-type iterative learning control,respectively.3.1.Open-loop P-type iterative learning controlFor the fractional-order nonlinear system with (3.1),we ﬁrst consider the following open-loop ILC updating law with initial state learning:x k þ1t 0¼x k t 0þLe k ðt 0Þ;u k þ1ðt Þ¼u k ðt Þþg 1e k ðt Þ;(ð3:3Þwhere e k ðt Þ¼y d ðt ÞÀy k ðt Þdenotes the tracking error.L and g 1are unknown parameters to be determined.In order to obtain our main result,the lemma is ﬁrst proved.Lemma 3.1.For fractional-order nonlinear system with delay (3.1)and a given reference y d ðt Þ,denote thatD u k ðt Þ¼u k þ1ðt ÞÀu k ðt Þ;D x k t ¼x k þ1t Àx k t ,ifmax 1Àg 1d 1ÀL d 3j j ;1Àg 1d 1ÀL d 4j j ;1Àg 1d 2ÀL d 3j j ;1Àg 1d 2ÀL d 4j j f g 6q 0<1;ð3:4Þwhere q 0is constant,then for all t 2½0;T ,and arbitrary input u 0ðt Þ,the open-loop-type ILC updating low (3.3)guarantees thatlim k !1j e k ðt 0Þj k ¼0:Proof.It follows from e k ðt Þ¼y d ðt ÞÀy k ðt Þand the mean value theorem thate k þ1ðt 0Þ¼e k ðt 0Þþy k ðt 0ÞÀy k þ1ðt 0Þ¼e k ðt 0Þþg ðx k t 0;u k ðt 0Þ;t 0ÞÀg ðx k þ1t 0;u k þ1ðt 0Þ;t 0Þ¼e k ðt 0ÞÀg x t ðn ÞD x k t 0Àg u ðn ÞD u kðt 0Þ;where n 2½x k t 0þr D x k t 0;u k ðt 0Þþr D u kðt 0Þ;t ,r 2ð0;1Þ.Then we havee k þ1ðt 0Þ¼e k ðt 0ÞÀLg x t ðn Þe k ðt 0ÞÀg 1g u ðn Þe k ðt 0Þ:ð3:5ÞTaking the k -norm,we obtainj e k þ1ðt 0Þj k 6j 1ÀLg x t ðn ÞÀg 1g u ðn Þjj e k ðt 0Þj k 6q 0j e k ðt 0Þj k :ð3:6ÞIf the condition (3.4)is satisﬁed,then q 0<1,we can obtainlim k !1j e k ðt 0Þj k ¼0:ð3:7ÞThe proof is complete.Then we give the main conclusion of this article.Theorem 3.2.For fractional-order nonlinear system (3.1)and a given reference y d ðt Þ,ifmax fj 1Àg 1d 1j ;j 1Àg 1d 2jg 6q 1<1;ð3:8Þwhere q 1is constant,then for all t 2½0;T ,and arbitrary input u 0ðt Þ,the open-loop-type ILC updating low (3.3)guarantees thatlim k !1y k ðt Þ¼y d ðt Þ:ð3:9Þ548L.Yan,J.Wei /Applied Mathematics and Computation 257(2015)546–552Proof.It can be proved thate k þ1ðt Þ¼e k ðt Þþy k ðt ÞÀy k þ1ðt Þ¼e k ðt Þþg ðx k t ;u k ;t ÞÀg ðx k þ1t ;u k þ1;t Þ¼e k ðt ÞÀg x t ðn ÞD x k t Àg u ðn ÞD u kðt Þ;ð3:10Þwhere n ðt Þ2½x k t þr D x k t ;u k þr D u k;t ;r 2ð0;1Þ.Applying the 2-norm to Eq.(3.10)and following from (3.3),we havej e k þ1ðt Þj 6j 1Àg 1g u ðn Þjj e k ðt Þj þj g x t ðn Þjk D x k t k 6j 1Àg 1g u ðn Þjj e k ðt Þj þd 4k D x kt k :ð3:11ÞIn the next,we give an estimation for the upper bound of k D x k t k .It follows from Lemma 2.5and in accordance with the prop-erty of the fractional-order 0<a <1,we can prove that for any t 2½0;T .xk þ1ðt Þ¼xk þ1ðt 0Þþ1C ða ÞZtt 0ðt Às Þa À1f ðs ;x k þ1s;u k þ1ðs ÞÞds ;thusx kðt Þ¼x kðt 0Þþ1C ða ÞZtt 0ðt Às Þa À1f ðs ;x k s ;u kðs ÞÞds :k D x kðt Þk 6k D x kðt 0Þk þ1ða ÞZ t t 0ðt Às Þa À1½f ðs ;x k þ1s ;u k þ1ðs ÞÞÀf ðs ;x k s ;u kðs ÞÞ ds6k D x kðt 0Þk þfC a Z t t 0ðt Às Þa À1k D x k s k dsþfC a Z tt 0ðt Às Þa À1j D u k ðs Þj ds 6k D x k ðt 0Þk þf 0C ða ÞZt t 0ðt Às Þa À1k D x k s k ds þf 0C ða ÞZtt 0ðt Às Þa À1e k s ds j D u k j k :We deﬁnek D x k ðt þh Þk ¼sup h 2½Àr ;0k D x k ðt þh Þk :ð3:12Þk D x kðt þh Þk 6k D x kðt 0Þk þfC a Zt þht 0ðt þh Às Þa À1k D x k s k dsþfC a Zt þht 0ðt þh Às Þa À1e k s ds j D u k j k :It follows from [1],t 0D Àat f ðt Þ¼lim h !0nh ¼t h aX n r ¼0a r f ðt Àrh Þ¼1C ða ÞZ t t 0ðt Às Þa À1f ðs ;x s Þds ;if f ðt Þ>0,lim h !0nh ¼thaX n r ¼0a rf ðt Àrh Þ;increases,when t increases.Then1C ða ÞZtt 0ðt Às Þa À1f ðs ;x s Þds ;is an increasing function.k D x k t j6k D x kðt 0Þk þfC a Ztt 0ðt Às Þa À1k D x k s k dsþfC a Ztt 0ðt Às Þa À1e k s ds j D u k j k :On the other hand,it follows from Lemma 2.4and the deﬁnition of the Mittag–Lefﬂer function that for k >0,we havedt aE 1;1þa ðk t Þdt¼t a À1E 1;a ðk t Þ>0:Therefore,f 0C a Ztðt Às Þa À1e k s ds ¼f 0t a E 1;1þa ðk t Þ;is an increasing function.Settingv ðt Às Þ¼f 0C a ðt Às Þa À1;L.Yan,J.Wei /Applied Mathematics and Computation 257(2015)546–552549w ðt Þ¼k D x kðt 0Þk þfa Ztt 0ðt Às Þa À1e k s ds j D u k j k ;and using Lemma 2.6,we havek D x k t k6ef 0T aC ða þ1Þk D x k ðt 0Þk þf 0e f 0T aC ða þ1ÞC ða ÞZtt 0ðt Às Þa À1e k s ds j D u k j k :Then fromZtt 0ðt Às Þa À1e k s ds <e k tk aC ða Þ;it yieldsk D x k t k 6c k D x kðt 0Þk þe k tf 0c ka j D u kj k ;ð3:13Þwhere c ¼e f 0T aC ða þ1Þis a constant.Now,substituting (3.13)into (3.11).we obtainj ek þ1ðt Þj 6j 1Àg 1g u ðn Þjj e kðt Þj þd 4c k D x kðt 0Þk þe k t f 0c d 4k aj D u k j k 6j 1Àg 1g u ðn Þjj e k ðt Þj þd 4c k D x k t 0Þk þe k tf 0c d 4k aj D u k j k :ð3:14ÞUsing (3.3)and multiplying both sides of the above inequality (3.14)by e Àk t and taking the k -norm,we havej ek þ1j k 6j 1Àg 1g u ðn Þjj e kj k þd 4cL j e kðt 0Þj k þf 0c g 1d 4k a j e k j k 6j 1Àg 1g u ðn Þj þf 0c g 1d 4kaj e k j k þd 4cL j e k ðt 0Þj k :By the assumption that max fj 1Àg 1d 1j ;j 1Àg 1d 2jg 6q 1<1,then there exists a sufﬁciently large k such thatj 1Àg 1g u ðn Þj þf 0cg 1d 4a<1:Therefore,from the inequality upon and Lemma 3.1,we have lim k !1j e k j k ¼0.It follows from that the equivalence of norms,we get that lim k !1j e k ðt Þj ¼0.which completes the proof.Next let us discuss the closed-loop P-type iterative control.3.2.Closed-loop P-type iterative controlThe above techniques that we used can be extended to the following closed-loop P-type ILC updating law:x k þ1t 0¼x k t 0þLe k ðt 0Þ;u k þ1ðt Þ¼u k ðt Þþg 2e k þ1ðt Þ;(ð3:15Þwhere L and g 2are unknown parameters to be determined.Lemma 3.3.For fractional-order nonlinear system with delay (3.1)and a given reference y d ðt Þ,denote thatD u k ðt Þ¼u k þ1ðt ÞÀu k ðt Þ;D x k t ¼x k þ1t Àx k t ,ifmax 1ÀL d 3g 21;1ÀL d 3g 22 ;1ÀL d 4g 21 ;1ÀL d 4g 226q 2<1;ð3:16Þwhere q 2is constant,then for all t 2½0;T ,and arbitrary input u 0ðt Þ,the closed-loop-type ILC updating low (3.15)guaranteesthatlim k !1j e k ðt 0Þj k ¼0:Proof.Similar to the proof of Lemma 3.1,we havee k þ1ðt 0Þ¼e k ðt 0ÞÀLg x ðf Þe k ðt 0ÞÀg 2g u ðf Þe k þ1ðt 0Þ:Thus,from the above equation,we gete k þ1ðt 0Þ¼1ÀLg x ðf Þ1þg 2g u ðf Þe kðt 0Þ:550L.Yan,J.Wei /Applied Mathematics and Computation 257(2015)546–552Taking the k -norm,we can obtaine k þ1ðt 0Þ k¼1ÀLg x ðf Þ1þg 2g uðf Þj e k ðt 0Þj k 6q 2j e k ðt 0Þj k :ð3:17ÞBecause q 2<1,which yields lim k !1j e k ðt 0Þj k ¼0.The proof is complete.Theorem 3.4.For fractional-order nonlinear system with delay (3.1)and a given reference y d ðt Þ,if the assumptions in Lemma 3.5are met,and ifmax 11þg 2d 1;11þg 2d 2 6q 3<1;where q 3is constant,then for all t 2½0;T ,and arbitrary input u 0ðt Þ,the closed-loop P-type ILC updating low (3.15)guaran-tees thatlim k !1y k ðt Þ¼y d ðt Þ:Proof.With the same argument as Theorem 3.2and Lemma 3.3,it can be prove easily thatj ek þ1ðt Þj 61g 2u j e k ðt Þj þd 4g 2u k D x k t k :ð3:18ÞSubstituting (3.13)into (3.18),we obtainj ek þ1ðt Þj 611þg 2g u ðf Þ j e k ðt Þj þd 4c j 1þg 2g u ðf Þj k D x k ðt 0Þk þd 4f 0ce k t j 1þg 2g u ðf Þj k a j D u k j k 61g 2uj e k ðt Þj þd 4c g 2u k D x k t 0k þd 4f 0ce k t j 1þg 2g u ðf Þj k a j D u k j k ;where c ¼e f 0T aC a is a constant.j e k þ1j k 61j 1þg 2g u ðf Þjj e k j k þd 4cL j 1þg 2g u ðf Þj je k ðt 0Þj k þd 4f 0c g 2j 1þg 2g u ðf Þj k aj e k þ1j k :From the above equation,we getj e k þ1j k 61j 1þg 2g u ðf Þj Àd 4f 0c g 2k a j e k j k þd 4cL j 1þg 2g u ðf Þj Àd 4f 0c g 2kaj e k ðt 0Þj k :Similar to the Theorem 3.2,thus we omitted it.4.Numerical exampleIn this section,We have presented a numerical example to demonstrate our main conclusion.Consider the fractional-order nonlinear systemD 12t x kðt Þ¼3½x k t 2þ1u k ðt Þ;y k ðt Þ¼x k t þ910u k ðt Þ;(ð4:1ÞThe iterative learning control law is chosenx k þ1t 0¼x k t 0þ12e k ðt 0Þ;u k þ1ðt Þ¼u k ðt Þþe k ðt Þ;(ð4:2ÞIn this case,let the initial condition is x k ðt 0Þ¼À3,the initial control is u 0ðt Þ¼0and the reference is y d ðt Þ¼12t 2ð1Àt Þ.It can be easily proved that the condition of the Theorem 3.2is satisﬁed.Next,let the iterative learning control law be chosenx k þ1t 0¼x k t 0þ1e k ðt 0Þ;u k þ1ðt Þ¼u k ðt Þþe k þ1ðt Þ;(ð4:3ÞLet the initial control and the reference be the same.From numerical example,we can prove our main conclusion is satisﬁed.L.Yan,J.Wei /Applied Mathematics and Computation 257(2015)546–552551552L.Yan,J.Wei/Applied Mathematics and Computation257(2015)546–5525.ConclusionThis paper based on the property of fractional order and the generalized Gronwall inequality.This paper mainly solves iterative learning control problems for a class of fractional with delay.Based on the robust convergence of the tracking errors, the P-type ILC updating laws can be determined.This is our main conclusion.Our future work will discuss the output delay. References[1]I.Podlubny,Fractional Differential Equations,Academic Press,San Diego,1999.[2]Yong Zhou,Basic Theory of Fractional Differential Equations,World Scientiﬁc,Singapore,2014.[3]J.X.Xu,Y.Tan,Linear and Nonlinear Iterative Learning Control,Spring-Verlag,Berlin,2003.[4]zarevic´,PD a-type iterative learning control for fractional LTI system,in:Proceedings of the16th International Congress of Chemical andProceedings Engineering,Praha,Czech Republic,2004,pp.22–26.[5]Y.Li,Y.Q.Chen,H.-S.Ahh,Fractional order iterative learning control,in:Int.Joint Conference,2009,pp.3106–3110.[6]Li Jia,Tian Yang,Minsen Chiu,An integrated iterative learning control strategy with model identiﬁcation and dynamic R-parameter for batch processes,J.Process Control vol.23(no.9)(2013)1332–1341.[7]Y.Li,Y.Q.Chen,H.-S.Ahh,On the PD a-type iterative learning control for the fractional-order nonlinear systems,in:American Control Conference,2011,pp.4320–4325.[8]Yong-Hong Lan,Yong Zhou,D-type iterative learning control for fractional-order linear time-delay systems,Asian J.Control15(3)(2013)669–677.[9]Y.Li,Y.Q.Chen,H.-S.Ahn,Fractional-order iterative learning control for fractional-order linear systems,Asian J.Control13(1)(2011)1–10.[10]Fali Ma,Chuandong Li,Tingwen Huang,Iterative learning control design of nonlinear multiple time-delay systems,put.vol.218(2011)4333–4340.[11]Ming-Tzong Lin,Chung-Liang Yen,Meng-Shiun Tsai,Hong-Tzong Yau,Application of robust iterative learning algorithm in motion control system,Mechatronics vol.23(no.5)(2013)530–540.[12]Y.Ye,A.Tayebi,X.Liu,All-passﬁltering in iterative learning control,Automatica45(1)(2009)257–264.[13]Yong Zhou,Feng Jiao,J.Pecaric,On the Cauchy problem for fractional functional differential equations in banach spaces,Topol.Methods NonlinearAnal.42(2013)119–136.[14]n,Iterative learning control with initial state learning for fractional order nonlinear systems,Comput.Math.Appl.64(2012)3210–3216.[15]Yong Zhou,Xiao Hui,Lu Zhang,Cauchy problem for fractional evolution equations with Caputo derivative,Eur.Phys.J.Special Topics222(2013)1747–1764.[16]Yong Zhou,Lu Zhang,Xiao Hui,Existence of mild solutions for fractional evolution equations,J.Int.Equ.Appl.25(2013)557–585.[17]Jiang Wei,The controllability of fractional control systems with control delay,Comput.Math.Appl.64(10)(2012)3153–3159.[18]Jiang Wei,The constant variation formulae for singular fractional differential systems with delay,Comput.Math.Appl.59(3)(2010)1184–1190.[19]Zhang Zhixin,Jiang Wei,Some results of the degenerate fractional differential system with delay,Comput.Math.Appl.62(3)(2011)1264–1291.[20]S.J.Sadati,D.Baleanu,A.Ranjbar,R.Ghaderi,T.Abdeljawad(Maraaba),Mittag–Lefﬂer stability theorem for fractional nonlinear system with delay,Abstr.Appl.Anal.vol.2010(2010)7.。

DETERMINATION OF jV ub j

CITATION: K. Hagiwara
et al.
(Particle Data Group), Phys. Rev. D 66, 010001 (2002) (URL: ) June 19, 2002 st be provided. The measurements to date have made use of detector hermeticity and the well-determined beam parameters to de ne a missing momentum that is de ned as the neutrino momentum. Signal-to-background ratios (S/B) of order one have been obtained in these channels. To extract jVub j from an exclusive channel, the form factors for that channel must be known. The form factor normalization dominates the uncertainty on jVubj. The q2 dependence of the form factors, which is needed to determine the experimental e ciency, also contributes to the uncertainty, but at a much reduced level. For example, the requirement of a sti lepton for background reduction in these analyses introduces a q2 dependence to the e ciency. In the limit of a massless charged lepton (a reasonable limit for the electron and muon decay channels), the B ! ` decay depends on one form factor f1(q2 ): 2 d?(B0 ! ?`+ ) = jV j2 G2 p3 MB sin2 jf (q2)j2; (1) F ` 1 ub dy d cos ` 32 3 2 where y = q2=MB , and ` is the angle between the charged lepton direction in the virtual W (` + ) rest frame and the direction of the virtual W . For the vector meson nal states and !, three form factors, A1 , A2 and V , are necessary (see e.g., Ref. 6). Calculation of these form factors constitutes a considerable theoretical industry, with a variety of techniques now being employed. Form factors based on lattice calculations 11{23], and on light cone sum rules 24{32], currently have uncertainties in the 15% to 20% range. A variety of quark model calculations exist 33{47]. Finally, a number of other approaches 48{53], such as dispersive bounds and experimentally-constrained models based on Heavy Quark Symmetry, seek to improve the q2 range over which the form factors can be estimated without introduction of a signi cant model dependence. Unfortunately, all these calculations currently have contributions to the uncertainty that remain uncontrolled. The light cone sum rules calculations assume quark-hadron duality, o ering a \canonical" contribution to the uncertainty of 10%, but with no known means of rigorously limiting that uncertainty. The lattice calculations to date remain in the \quenched" approximation (no

The Light-Cone Fock Expansion in Quantum Chromodynamics

Abstract
A fundamental question in QCD is the non-perturbative structure of hadrons at the amplitude level—not just the single-particle ﬂavor, momentum, and helicity distributions of the quark constituents, but also the multi-quark, gluonic, and hiddencolor correlations intrinsic to hadronic and nuclear wavefunctions. The light-cone Fock-state representation of QCD encodes the properties of a hadrons in terms of frame-independent wavefunctions. A number of applications are discussed, including semileptonic B decays, deeply virtual Compton scattering, and dynamical higher twist eﬀects in inclusive reactions. A new type of jet production reaction, “selfresolving diﬀractive interactions” can provide direct information on the light-cone wavefunctions of hadrons in terms of their quark and gluon degrees of freedom as well as the composition of nuclei in terms of their nucleon and mesonic degrees of freedom. The relation of the intrinsic sea to the light-cone wavefunctions is discussed. The physics of light-cone wavefunctions is illustrated for the quantum ﬂuctuations of an electron.

Micro Focus Unified Functional Testing Pro (LeanFT

OverviewOne of the world’s leading financial market institutions relies on speed, reliability and stateof-the-art technology to help its cli-ents build wealth. With Micro Focus Unified Functional T esting Pro (LeanFT), the company has established an efficient application test -ing platform capable of integrating testing from application development through production to speed quality and time-to-market.ChallengeOne of the world’s leading financial market institutions offers a full suite of capital market services. With a highly-skilled team of people, the company works with a broad range of B2B and B2C customers. Operating in a highlyregu-lated environment, the company helps clientsbuild wealth and manage investment and op-erating risk. The company’s network and data center are connected to leading financial hubs. Speed, reliability and state-of-the-art technol-ogy are fundamental to its success.Running a combination of 200+ in-house, ac-quired and modified applications, the com-pany is focused on delivering quality software to meet the needs of its users. A few years ago, it began to embrace agile methodolo-gies to improve quality and speed-to-mar-ket. Having outgrown its use of Micro Focus Unified Functional T esting (UFT), the company decided to look for a testing solution that pro-vided automation across Application Program Interface (API) and Graphical User Interface (GUI) components.“We wanted to align test automation to de-velopment practices to feed into continuous integration,” says a spokesperson for the com-pany. “But, we didn’t want to add complexity to our testing ecosystem by having another tool-set—we wanted a single application that would satisfy our requirements.SolutionAfter investigating potential testing applica-tions, the company selected UFT Pro (LeanFT), a powerful functional testing solution builtFinancial Market ExchangeLeading financial market institution maximizes developmentspeed with Micro Focus UFT Pro.At a Glance■Industry Financial Services ■Location Sydney, Australia ■ChallengeAutomate testing, provide cross-platform and cross-browser support for API and GUI components, and integrate with standard IDEs. ■Products and Services UFT Pro (LeanFT) ■Success Highlights+Enables rapid software development and deployment to enhance agility.+Reduces risks through efficient testing practices and scripts implementations.+Standardizes functional testing on one platform. +Facilitates greater collaboration between testing and development teamsCase StudyApplication Delivery Management“We chose UFT Pro (LeanFT) due to its functionality, particularly with respect to cross-platform, cross-browser and desktop applications support, as well as its integration capabilities with API and GUI components.”SPOKESPERSONLeading Financial Market Institution AustraliaCase StudyFinancial Market Exchangespecifically for continuous testing and continu-ous integration.The spokesperson says, “We chose UFT Pro (LeanFT) due to its functionality, particularly with respect to cross-platform, cross-browser and desktop applications support, as well as its integration capabilities with API and GUI com-ponents. Its support for the most common Applications Under T est (AUT) technologies was important to us as we needed a solution that could handle both desktop and web ap-plications. Having a solution that works inside standard Integrated Development Environment (IDE) using modern scripting languages was also a deciding factor. The solution’s Object Identification Center (OIC) meant that we could model AUT s and objects with ease as we cre-ate robust scripts.”T o test the suitability of UFT Pro (LeanFT), the company ran a pilot on a live project.“We took the opportunity to investigate the functionality of UFT Pro (LeanFT) by testing one of our web browser based applications,” explains the spokesperson.“After a few months of using the solution, we realized it was the right tool for us. We gained a good understanding of how to use applica-tion models, share code resources, integrate with Continuous Integration (CI) / Continuous Development (CD) tools, and support com-mon AUT technologies including Java, .NET and AngularJS.“Using UFT Pro (LeanFT) on this project gave us a roadmap for how we could leverage the solution’s functionality more broadly. It showed us what it would take to create a company-wide foundation for integration from develop-ment through to production with end-to-end traceability.”Following the success of this pilot, the com-pany rolled out UFT Pro (LeanFT) across itstesting environment. It is now being used totest seven core applications.ResultsHaving the ability in UFT Pro (LeanFT) to cre-ate Application Models that serve as a sharedobjects repository, and to easily identify ob-jects via the OIC feature, so that the objectswill not break from one build to the other, aretwo key benefits the company is realizing fromthe solution.The spokesperson explains, “We can nowcreate abstractions of our AUTs, and in turnprovide our tests with an interface to the ap-plications and their objects. This allows us tomaintain our test objects in a single location foruse across our testing suite, which in turn helpsour developers to write code more quickly,without the need to write manual programmaticdescriptions for each object.“In addition, we now have access to multiplevisual relational identifiers to discover fields inour AUT s based on neighboring objects. Thisfeature alone has helped us many times whenobjects were changing depending on optionsset in the application.”The UFT Pro (LeanFT) integration capabilitiesfacilitate greater collaboration between test-ing and development teams, the spokesper-son continues, “This allows for robust softwaretesting that easily accommodates changesto applications.“Our investment in UFT Pro (LeanFT) is for thelong-term. We want to reduce our automationtoolkit to a common AUT like Java and handle amyriad of third-party plug-ins. In due time, we’llbe able to hook into all the tools our develop-ment teams use, and write test scripts in thesame language. This will significantly reducethe time between development and testing,and enhance the quality of our applications,not to mention speed-to-market. “We’re cur-rently leveraging the interoperability of UFT Pro(LeanFT) to create closer alignment with the CI/CD process.”While the company is still in the infancy stagesof using UFT Pro (LeanFT), it has recognizedthe potential for improving the efficiency ofits testing.The spokesperson says, “UFT Pro (LeanFT) is apowerful solution that provides openness andallows us to use our object-oriented program-ming knowledge and advanced coding tech-niques, such as polymorphism and inheritance.This has opened up so many options for us tomake our test scripts more efficient.“Once we mature our testing processes and getmore familiar with UFT Pro (LeanFT), we will be-come more productive. Certainly, as we migratemore and more of our 700+ UFT test scripts—automate them, store them in a central reposi-tory, and re-use them—we will see dramaticimprovements in speed and efficiency.“In the long-run, having a single tool for testautomation will make it easier for us to moveour resources to different projects as required.”Looking ahead, the company is focused ongetting up to speed with UFT Pro (LeanFT)and evolving its approach to functional test-ing as part of a broader commitment to qualityassurance.“To maximize the return on our investment,we’re working towards getting the most outof the functionality of UFT Pro (LeanFT),” saysthe spokesperson. “Our progress is a little slowbecause this is a brand new way of handlingtesting for us, particularly as we have amassed more than 10 years of experience with UFT, but we are becoming quicker and more confident every day.”The company has a lot of systems, includ-ing legacy applications, that use different technologies.The priority is to automate test cases for both API and GUI across multiple programming lan-guages, and leverage modern industry stan-dards. Centralization and standardization of test scripts will also help drive further efficiency and productivity gains.“We want to get the basics right first in terms of test creation and automation,” concludes the spokesperson.“Once we’ve achieved that, we’ll look at lever-aging other functionality such as analytics.“We’re playing the long game with UFT Pro (LeanFT) and have chosen to implement the solution as we recognize the potential to achieve closer alignment between testing and development. We know this will ultimately help us release applications rapidly and confidently.”“The UFT Pro (LeanFT) integration capabilities facilitate greater collaboration between our testing and development teams. It allows for robust software testing that easily accommodates changes to the application.”SPOKESPERSONLeading Financial Market InstitutionAustralia。

form generator的表格

form generator的表格Form Generator: Simplifying Data Collection and AnalysisIn today's fast-paced world, data collection and analysis have become essential for businesses, organizations, and individuals alike. Whether it is for market research, customer feedback, or employee surveys, gathering accurate and reliable data is crucial for making informed decisions. However, the process of creating and distributing forms can be time-consuming and tedious. This is where a form generator comes into play.A form generator is a powerful tool thatsimplifies the process of creating online forms. It allows users to design custom forms with ease, without the need for any coding or technical skills. With just a few clicks, users can select from awide range of pre-designed templates or createtheir own unique form layout.One of the key advantages of using a form generator is its user-friendly interface. The drag-and-drop functionality makes it incredibly easy to add various form elements such as text fields, checkboxes, radio buttons, dropdown menus, and more. Users can also customize the appearance of their forms by choosing from different color schemes and fonts.Another significant benefit of using a form generator is its ability to collect dataefficiently. Once the form design is complete, users can easily share it with their target audience through various channels such as email, social media platforms, or embedding it on their website. The forms are mobile-responsive, ensuring that respondents can access them from any device.Furthermore, a form generator offers advanced features that enhance data collection and analysis. Users can set up conditional logic to display specific questions based on previous responses. This ensures that respondents only see relevant questions tailored to their needs. Additionally, users can enable validation rules to ensure that respondents provide accurate information by setting required fields or specific data formats.The collected data is automatically stored in a secure database provided by the form generator platform. Users have access to real-time analytics and reports that provide valuable insights into the collected data. They can easily visualize trends and patterns, export data for further analysis, and generate comprehensive reports.The benefits of using a form generator extend beyond data collection and analysis. It also saves time and resources. Instead of manually creating and distributing paper forms or designing complex online forms from scratch, users can rely on the form generator to streamline the entire process. This allows businesses to focus on their coreactivities while ensuring efficient data collection. In conclusion, a form generator is a valuabletool that simplifies the process of creating, distributing, and analyzing online forms. Its user-friendly interface, advanced features, andefficient data collection capabilities make it an indispensable asset for businesses, organizations, and individuals. By utilizing a form generator, users can save time, resources, and make informed decisions based on accurate data.。

fractional factorial conjoint

$fractional factorial conjoint$

fractional factorial conjoint
我们要了解fractional factorial conjoint的含义。

首先，我们需要了解什么是fractional factorial design和conjoint analysis。

Fractional factorial design是一种实验设计方法，用于在有限的资源下进行多因素实验。

Conjoint analysis是一种统计方法，用于研究产品或服务的不同属性如何影响消费者对产品的整体评价。

当我们结合这两种方法时，我们得到fractional factorial conjoint。

这种方法允许我们同时考虑多个因素及其交互作用，并确定哪些因素对产品的整体评价有显著影响。

为了更好地理解fractional factorial conjoint，我们可以考虑一个简单的例子：
假设我们正在测试两种不同品牌的手机，每种手机都有三个属性：价格、屏幕大小和电池寿命。

使用fractional factorial conjoint，我们可以设计实验来同时测试所有可能的属性组合，并确定哪些属性组合对消费者的购买决策有最大影响。

总结：Fractional factorial conjoint是一种结合了fractional factorial design和conjoint analysis的方法，用于在有限的资源下研究多个因素及其交互作用对产品整体评价的影响。

Microeconomic Probset2_solution

(Remark: that is, prove that if h(p, V (p, m)) minimizes the expenditure to reach the utility level u = V (p, m), then we must have x∗(p, m) = h(p, V (p, m)).) Solution: Suppose x∗ = x∗(p, m) maximizes the utility given (p, m), then we have V (p, m) = u(x∗) ≥ u(x) for all x ∈ X. If x∗ does not minimize the cost to reach V (p, m), then we must have another bundle x which solves the minimization problem, that is, we have px < px∗ and u(x ) ≥ u(x∗). However, by local non-satiation, there always exists a bundle x which is
x−1
2 3
−1
2
x23
=
−
2 9
x−1
5 3
2
x23
<
0
So the marginal utility M U1(x) is decreasing in x1.
Similarly
we
can
compute
M U2(x) =
2 3
1
x13
x−2
1 3
and
show
that
it

FORM-FLEXMETALDISCFLEXIBLECOUPLINGS-…

FORM-FLEX METAL DISC FLEXIBLE COUPLINGSForm-Flex couplings transmit torque while compensating for angular, parallel and axial misalignment between two connected shafts. Flexible disc couplings minimize the misalignment forces on the connected equipment.The Basic flex coupling consists of two hubs, a spacer and two flexible discs. The flex disc is an assembly of thin metal laminations. In figure shown below, flex disc holes A & C are bolted to the hub and holes B & D are bolted to the spacer. Torque is transmitted in direct tensions from A to B and from C to D through the flex disc.Misalignment is taken through bending in the link between the bolt holes.COOLING TOWER DRIVESForm-Flex metal disc couplings are widely used incooling fan drive applications. Form-Flex 4 bolt disccouplings offer more misalignment capacity than anycompeting metal disc design.For smaller towers up to about 100 inches DBSE, TB Wood ’s offers steel and composite spacer tubingoptions. TrueTube composite torque tubes are lighter than steel and eliminate thermal growth and vibration problems.Form-Flex composite floating shaft couplings are recommended as a replacement for oldermulti-section drivelines. Composite couplings can span up to 240 inches without high maintenance center support bearings.VERTICALPUMP DRIVESForm-Flex floating shaft couplings are a cost-effective, maintenance free alternative to cardanU-joints for vertical pump drivelines. Form-Flexcouplings are available with either steel or composite spacer tubing. Composite spacer tubing can reduce total cost by eliminating the need for bearings and support structures.FORM-FLEX METAL DISC FLEXIBLE COUPLINGSCOMPLETE PRODUCT OFFERING-TORQUE CAPACITY TO 3175 HP/100 RPM-CLOSE COUPLE, SPACER AND FLOATING SHAFT DESIGNSHIGH STRENGTH STEEL FASTENERS-NO MOVING PARTS -ZERO BACKLASHOVER 30 YEARS EXPERIENCE IN METAL DISC COUPLINGSAPPLICATIONS-PUMPS-FANS AND BLOWERS -COMPRESSORS -FOOD PROCESSING -PRINTING-MACHINE TOOLSALL METAL CONSTRUCTION-WIDE TEMPERATURE RANGE-AVAILABLE IN CARBON OR STAINLESS STEEL -COMPOSITE MATERIALS NOW AVAILABLEHIGH STRENGTH STAINLESS STEEL FLEX DISCS-HIGH TORSIONAL STIFFNESS -NO LUBRICATION REQUIREDTYPICAL APPLICATIONSForm-Flex Flex Disc DesignsMATERIAL CLASS BY COMPONENTCOUPLING HUB SPACER ASSY REPAIR KITA A A A Mild steel hubs and spacer, alloy steel hardware, 300 series SS flex discB B B A Zinc plated steel hubs and spacer, alloy steel hardware, 300 series SS flex discC B C E Zinc plated steel hubs and spacer, 300 series SS flex disc and hardware EEEEAll 300 series stainless steel constructionDISC STYLEDESIGN FEATURESWHERE USEDIdeal for general industrial applications with motor or turbine drivers and smooth to moderate load conditions.Low to moderate speed ranges.Serve or stepper driven positioning systems.Applications where misalignment may be a problem. 4 bold designs offer the highest misalignment capacity of any metal disc design.Not recommended for engine driven applications.Straight sided flex disc.1 degree angular misalignment.Torque range: 35 LB. IN. to 30,240 LB. IN.Zero backlash.All machined steel construction.Stainless steel flex discs.Steel or stainless steel materials.Minimum reaction forces.4 BOLT(A, M SERIES)6 BOLT (B SERIES)8 BOLT(D, F, H SERIES)Ideal for motor or turbine drivers with any load conditions.Use for reversing, reciprocating or other rough load conditions.May be used with industrial engines driving smooth loads.Moderate to high speed ranges and applications where dynamic balancing is required.Consider 6 bolt where 4 bolt size requiresincreasing coupling size to meet bore size requirements.Straight sided disc.0.7 degree angular misalignment.Torque range: 3050 LB. IN. to 233,000 LB. IN.Suitable for precision balancing.Zero backlash.All machined steel construction.Stainless steel flex discs.Steel or stainless steel materials.High torque-low speed applications.Industrial engines driving reciprocating equipment.Heavy-duty reversing applications.Custom designs for high torque applications.Round disc design.0.3 degree angular misalignment.Torque range: 9500 LB. IN. to 2,000,000 LB. IN.Zero backlash.Heavy duty cast construction.Alloy or stainless steel flex discs.Flywheel mount designs.MATERIAL CLASSESAPPLIES TO 4 AND 6 BOLT DESIGNSPRODUCT FEATURES AND OPTIONS*Alloy steel flex disc is standard for DA and DP series. Stainless steel is optional.DESCRIPTIONFEATURE AR, AK, APBH, BP, BYBF BA, DA*A5, A7B5HFTH HH, HSH, FSHAX, AY DP*STANDARD BORE FIT CLEARANCE INTERFERENCE CLEARANCE INTERFERENCE SET SCREWS STANDARD OPTIONAL STANDARD OPTIONALPULLER HOLES OPTIONALSTANDARDOPTIONALSTANDARD OPTIONAL STANDARD FLEX DISCS 300 SERIES STAINLESS STEEL*ALLOY STEELBALANCE CLASS AGMA 7AGMA 8AGMA 9AGMA 7N/AN/A DYNAMIC BALANCEOPTIONALPER TBW COMMERCIAL STANDARDN/A。

form factor标准

在物理学中，Form Factor是一个描述粒子间相互作用矩阵元中含有的洛伦兹标量函数因子的参数，它反映了相互作用振幅随动量转移的变化关系。

Form Factor可以分为形状因子和散射因子，分别对应于粒子的初始和最终状态。

在计算Form Factor时，通常采用不同的方法，例如利用质谱学方法进行测量或者利用量子力学方法进行计算。

对于不同的粒子，Form Factor的大小和形式可能会有所不同。

此外，Form Factor还被用于描述原子或分子在受到电场、磁场或热能等外部能量作用时，其形状和大小的变化情况。

对于不同的物质，Form Factor可能会有所不同，这也会影响物质在电学、光学、热学等方面的性质。

总之，Form Factor是物理学和化学领域中非常重要的一个参数，它可以帮助我们更好地理解粒子的性质以及它们之间的相互作用机制。

formelQ培训教材

发展历程
1. 第一版
2. 第二版
3. 经全面修订的第三版
4. 经全面修订的第四版
5. 经全面修订的第五版
6.
经全面修订的第六版
7.
经全面修订的第七版
1991 年 1994 年 1 月 1997 年 1 月 2000 年 4 月 2005 年 1 月
2009 年 8 月 2012 年 1 月
务5
15
Copyrights LISI AUTOMOTIVE – Confidential and proprietary information
16
Copyrights LISI AUTOMOTIVE – Confidential and proprietary information
四项基本出发点（没有单独提问，来
Copyrights LISI AUTOMOTIVE – Confidential and proprietary information
8
在产品生命周期中 Q 质量能力的应用范围
问题分析潜在供应商评价自我审核有存档责任的 / 技术准则文件供应商技术评审分供方管理过程审核（包括产品审核）
A部分
产品/ 过程开发
P2: 项目管理 7（新增） P3: 产品和过程开发的策划 5
P4: 产品和过程开发的实现 9
Copyrights LISI AUTOMOTIVE – Confidential and proprietary information
E5原材料和供方管理 7
P5: 供方管理 7
质量体系没有通过（ 16949或 6.1)认证产品审核发现B缺陷或一个系统性C缺陷供应链审核（)结果为“黄色”，将导致直接供应商的降级

1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。

a rXiv:h ep-ph/956202v11J un1995YUMS 95–11KAIST-CHEP-95/04SNUTP 95–039Form Factors for Exclusive Semileptonic B –Decays C.S.Kim ∗Department of Physics,Yonsei University,Seoul 120–749,KOREA Jae Kwan Kim,Yeong Gyun Kim †and Kang Young Lee ‡Department of Physics,KAIST,Taejon 305–701,KOREA (February 1,2008)Abstract We developed the new parton model approach for exclusive semileptonic de-cays of B -meson to D,D ∗by extending the inclusive parton model,and by combining with the results of the HQET,motivated by Drell-Yan process.Without the nearest pole dominance ans¨a tze,we derived the dependences of hadronic form factors on q 2.We also calculated numerically the slope of theIsgur-Wise function,which is consistent with the experimental results.Typeset using REVT E XI.INTRODUCTIONIn exclusive weak decay processes of hadrons,the eﬀects of strong interaction are encoded in hadronic form factors.These decay form factors are Lorentz invariant functions which depend on the momentum transfer q2,and their behaviors with varying q2are dominated by non-perturbative eﬀects of QCD.B–meson decay processes have been studied in detail as providing many interesting infor-mations on the interplay of electroweak and strong interactions and as a source extracting the parameters of weak interactions,such as|V cb|and|V ub|.As more data will be accu-mulated from the asymmetric B–factories in near future,the theoretical and experimental studies on exclusive B–meson decays would also give better understandings on the Standard Model and its possible extensions.Over the past few years,a great progress has been achieved in our understanding of the exclusive semileptonic decays of heavyﬂavors to heavyﬂavors[1].In the limit where the mass of the heavy quark is taken to inﬁnity,its stong interactions become independent of its mass and spin,and depend only on its velocity.This provides a new SU(2N f)spin–ﬂavor symmetry,which is not manifest in the theory of QCD.However,this new symmetry has been made explicit in a framework of the heavy quark eﬀective theory(HQET)[2]. In practice,the HQET and this new symmetry relate all the hadronic matrix elements of B→D and B→D∗semileptonic decays,and all the form factors can be reduced to a single universal function,the so-called Isgur-Wise function[2,3],which represents the common non-perturbative dynamics of weak decays of heavy mesons.However,the HQET cannot predict the values of the Isgur-Wise function over the whole q2range,though the normalization of the Isgur-Wise function is precisely known in the zero recoil limit.Hence the extrapolation of q2dependences of the Isgur-Wise function and of all form factors is still model dependent and the source of uncertainties in any theoretical model.Therefore,it is strongly recommended to determine hadronic form factors of B–meson decay more reliably, when we think of their importance in theoretical and experimental analyses.2In this paper we developed the parton model approach for exclusive semileptonic B decays to D,D∗,and predicted the q2dependences of all form factors.Previously the parton model approach has been established to describe inclusive semileptonic B decays [4,5],and found to give excellent agreements with experiments for electron energy spectrum at all energies.While many attempts describing exclusive B decays often take the pole-dominance ans¨a tze as behaviors of form factors with varying q2[6,7],in our approach they are derived by the kinematical relations between initial b quark andﬁnal c quark.According to the Wirbel et.al.model[6],which is one of the most popular model to describe exclusive decays of B mesons,the hadronic form factors are related to the meson wavefunctions’overlap-integral in the inﬁnite momentum frame,but in our model they are determined by integral of the fragmentation functions,which are experimentally measuable.For completeness,here we brieﬂy review the parton model approach for inclusive semilep-tonic decays of B meson.The parton model approach pictures the mesonic decay as the decay of the partons in analogy to deep inelastic scattering process.The probability of ﬁnding a b-quark in a B meson carrying a fraction x of the meson momentum in the inﬁnite momentum frame is given by the distribution function f(x).Then we write the Lorentz invariant decay width as follows:E B dΓ(B→X q eν)= dx f(x)E b dΓ(b→qeν),(1) with the relation p b=xp ing standard deﬁnitions of the structure functions for the hadronic tensor WµνWµν=−gµνW1+pµB pνB W22m2B+qµqνW42m2B+i(pµB qν−pνB qµ)W6 2(f(x+)+f(x−)),W2m2B(x+−x−)(x+f(x+)−x−f(x−)),3W3m2B(x+−x−)(f(x+)−f(x−)),and W4=W5=W6=0,whereq0± m B,x±=with theﬁnal state quark mass m q.Hence the double diﬀerential decay rate is given bydΓy{x+f(x+)(x+−y−m2q/m2B/x+)−(x+↔x−)},(3)8π3where y=2Eν/m B.In the limit f(z)=δ(1−z),we reproduce the HQET leading term except that m b is replaced by m B.In Section II,we develop the parton model approach for exclusive semileptonic decays of B meson,and give all the theoretical details for B→Dlνand B→D∗lν.Section III contains discussions and conclusions of this paper.II.PARTON MODEL APPROACH FOR EXCLUSIVE DECAYS OF B MESON We now develop the parton model approach for exclusive semileptonic decays of B meson by extending the previously explained inclusive parton model,and by combining with the results of the HQET.Theoretical formulation of this approach is,in a sence,closely related to Drell-Yan process,while the parton model of inclusive B decays is motivated by deep inelastic scattering process.And the bound state eﬀects of exclusive B decays are encoded into the hadronic distribution functions of partons inside an initial B meson and of partons of aﬁnal state resonance hadron.Then,the Lorentz invariant hadronic decay width can be obtained using the structure functions,as in Eq.(1),E B·dΓ(B→D(D∗)eν)= dx dy f B(x)E b·dΓ(b→ceν)f D(y).(4) Theﬁrst integral represents the eﬀects of motion of b quark within B meson and the second integral those of c quark within D meson.The variables x and y are fractions of momenta of partons to momenta of mesons,4p b=xp B,p c=yp D,(5) in the inﬁnite momentum frame.The functions f B(x)and f D(y)are the distribution func-tion of b quark inside B meson,and the fragmentation function of c quark to D meson respectively.Since the momentum fractions and the distribution functions are all deﬁned in the inﬁnite momentum frame,we have to consider the Lorentz invariant quantity,E·dΓas deﬁned in Eq.(1)and(4),to use at any other frame.For a heavy quark(Q=t,b,c)the distribution and fragmentation functions in a heavy meson(Qq),which are closely related by a time reversal transformation,are of similar functional forms,and peak both at large value of x.Therefore,we follow the previous work of Paschos et.al.[4]to use the Peterson’s fragmentation function[9]for both distributions, f B(x)and f D(y).It has the functional form:f Q(z)=Nz−1 1−11−z −2,(6) where N is a normalization constant,and Q denotes b or c quark.In the Drell-Yan process,the rest degrees of freedom of initial nucleons which do not take part in the scattering make incoherentﬁnal states,see in Fig.1(a).In the exclusive semileptonic decay of a heavy meson into aﬁnal state heavy meson,however,two sets of left-over light-degrees of freedom are summed to have the connection,|<D(or D∗)|J|B>|2∼ spin,X1,X2|{<X2|<c|}J{|b>|X1>}|2.(7) For more explicit meaning of Eq.(7),see Fig.1(b)and Section II A and B.Two sets of states,|X1>,|X2>are not independent here,in fact.And to connect them we need a relation between x and y from the decay kinematics.The momentum transfer of the decay between mesons is deﬁned asq≡p B−p D.(8) On the other hand,the momentum transfer of the partonic subprocess is given byq(parton)=p b−p c=xp B−yp D.(9)5In fact,the heavy meson’s momentum would be p H=p Q+k+O(1/m Q),where H=B,D or D∗,and Q=b or c.And k denotes the momentum of the light-degrees of freedom,and is related to the eﬀective mass of a common light quark,¯Λ.Therefore we have q=q(parton) up to the common part of the1/m Q corrections.With these kinematic relations we derived the following relationy(x,q2)=1x(x−1)m2B+(1−x)q2+xm2D.(10)Substituting y of Eq.(4)for y(x)of Eq.(10),the double integral of Eq.(4)is reduced to the single integral over ing this relation,we can sum the intermediate states in Fig. 1(b),as in(7).In Fig.1,we show the schematic diagrams of Drell-Yan process and the related exlusive semileptonic decay of B meson.We note here that the connection(7)and the kinematic relation(10)are valid approxi-mations for the heavy-to-heavy resonance decays,with the common light-degrees of freedom of the size O(1/m Q).As explained before,in the limit where f Q(x)=δ(1−x)by increasing m Q to inﬁnity,we can reproduce the HQET leading term.By comparison,the inclusive parton model approach is more reliable for the heavy-to-light non-resonant decays toﬁnal states of many particles.A.B→DeνFrom Lorentz invariance we write the matrix element of the decay¯B→De¯νin the form <D|Jµ|B>=f+(q2)(p B+p D)µ+f−(q2)(p B−p D)µ,(11) and in terms of the HQET<D(v′)|Jµ|B(v)>=√Hµν=<D|Jµ|B><D|Jν|B>∗=2|f+(q2)|2(p Bµp Dν+p Bνp Dµ),(13) and can be expressed by the Isgur-Wise function,Hµν=R−1|ξ(v·v′)|2(p Bµp Dν+p Bνp Dµ) 1+O(1m B m D2(2π)4δ4(p b−p c−q)·2G F2|V cb|2H(parton)µνLµν×d3p c(2π)32Eed3pνspin-inert parts.Therefore,the parameter N is introduced to estimate the size of spin-inert part which contributes to B→D process.Generally we can write the hadronic tensor of any exclusive semileptonic decay modes for B mesons as=H1(q2)(p bµp cν+p bνp cµ)+H2(q2)gµνp b·p c+iH3(q2)ǫµναβp bαp cβ,(17) H(parton)µνwhich is expressed in the form motivated by that of inclusive B→X c eνdecays.With the expression of Eq.(17),weﬁndH1=N,H2=H3=0.in the case of B→Deνdecay.The constant N will be later determined by the zero recoil limit of the Isgur-Wise function.Using the relation(5),we can write the hadronic tensor in the parton level as followsH(parton)=N xy(p Bµp Dν+p Bνp Dµ).(18)µνThe momentum conservation of the partonic subprocess corresponds to the momentum conservation in the hadronic level in our model,as explained before.So we can substitute the Dirac delta functionδ4(p b−p c−q)forδ4(p B−p D−q)in Eq.(15)with no loss of generality.Therefore,we write the decay width of¯B→De¯ν,E B·dΓ(B→Deν)= dx f B(x)f D(y(x,q2))E b dΓ(b→ceν)= dx f B(x)f D(y(x,q2))(2π)4δ4(p B−p D−q)×G F2|V cb|2N xy(x,q2)(p Bµp Dν+p Bνp Dµ)Lµνd3pν×y2(x,q2)d3p D(2π)32Eewhere we deﬁned the function F(q2)asF(q2)≡ dx f B(x)f D(y(x,q2))xy3(x,q2).(21) For given q2in our parton picture,the function F(q2)measures the weighted transition amplitude,which is explicitly given by the overlap integral of distribution functions of initial andﬁnal state hadrons.Comparing(20)with the Eq.(14),the Isgur-Wise function is calculated within the parton model approach|ξ(v·v′)|2 1+O(1dq2=G F2|V cb|2<D∗|Vµ+Aµ|B>=2im B+m D∗(p B+p D∗)µA2(q2)−2m D∗ǫ∗·q q2qµA0(q2),(24) whereA3(q2)=m B+m D∗2m D∗A2(q2),and in terms of the HQET<D∗(v′)|Vµ+Aµ|B(v)>=√(m B+m D∗)2(p Bµp D∗ν+p Bνp D∗µ)(1+O(1(m B+m D∗)2gµνp B·p D∗(1+O(1M Q)) , where R∗=2√(m B+m D∗)2N1(p bµp cν+p bνp cµ)−2 1−q2The parameters N i’s give the relative size of form factors and overall normalization.In general they are not constants and have the q2dependences.In the heavy quark limit, N i’s become constants and the values are equal to that of the normalization constant N in B→Deνprocess.In order to investigate the procedure more conveniently,we deﬁne the ratios of form factors as follows:R1≡ 1−q2A1(q2),R2≡ 1−q2A1(q2),(28) where V(q2),A1(q2)and A2(q2)denote vector and axial vector form factors respectively. Then we can write the relations among form factors and the Isgur-Wise function asA1(q2)= 1−q2Hereafter we also take them to be constants for simplicity.In our model the parameters N 1,N 2and N 3are represented in terms of R 1and R 2as follows,N 1=N/2(m B +m D ∗)2·2R 21+ 1−q 22+m 2B +q 22m 2D ∗(1−R 22) ,N 2=Nm 2B +m 2D ∗−q 2(1−R 21) ,N 3=NR 1.(32)In the heavy quark limit,we know that R 1=R 2=ing the expression R i =1+O (1/m Q )we can separate the leading contributions and 1/m Q corrections in N i ’s:N 1=N2(1+R 21)+O (1−R 1),N 3=NR 1.(33)When R 1,R 2→1,we explicitly see that N 1=N 2=N 3→N .Now substituting ξ(v ·v ′)for F (q 2)with the Eq.(22),the hadronic tensor for the decay ¯B →D ∗e ¯νis given byH µν(B →D ∗)=F (q 2) (1−2q 2(m B +m D ∗)2)(N 2g µνp B ·p D ∗−iN 3ǫµναβp αB p βD ∗) .(34)The q 2dependences of form factors are mainly determined by the function F (q 2),instead of the commonly used pole-dominance ans¨a tze.When we calculate the hadronic tensor within the HQET framework,we have generally some parameters parametrizing non-perturbative eﬀects,which are obtained in model de-pendent ways.The slope parameter is such a characteristic parameter of the Isgur-Wise function,which represents the common behaviors of form factors.We calculated it,andﬁnd that the value of the slope parameter is related to the parameters ǫb and ǫc in ourapproach.The HQET also contains the parameter λa ∼− k 2Q which is related to the ki-netic energy of the heavy quark inside the heavy meson,and spin-symmetry breaking term λ2=1dq 2=G 2F |V cb |22m 2B W 2(q 2)(m 2B −m 2D ∗+q 2)+3m 2B W 3(q 2) ,(35)whereW 1(q 2)=−N 1 1−2q 2(m B +m D ∗)2−2N 3q 2 1−q 2(m B +m D ∗)2+N 3q 2(m 2B −m 2D ∗+q 2) 1−q 2(m B +m D ∗)2 ,(36)and F (q 2)is deﬁned in (21).The result is plotted in Fig.2(b),also compared with the CLEO data[10].The thick solid line is our model prediction with the parameters (ǫb =0.004,ǫc =0.04),the thin solid line the Wirbel et.al.model prediction [6],and the dotted line the K¨o rner et.al.model prediction [7].III.DISCUSSIONS AND CONCLUSIONSAll form factors show the same behavior for varying q2,which is described by the Isgur-Wise function of the HQET,which represents the common non-perturbative dynamics of weak decays of heavy mesons.Ever since Fakirov and Stech[12],the nearest pole dominance has been usually adopted as the dependence of common behaviors on q2.In our approach, their q2dependences are derived from the kinematic relations of b-and c-quark.When b-quark decays to c-quark,the momentum transfer to leptonic sector is equal to the diﬀerence between b-quark momentum and c-quark momentum in the parton picture.The b-and c-quark momenta within the B and D mesons have some speciﬁc distributions.For given momentum transfer q2,there exist possible conﬁgurations of b-and c-quark momentum pairs (p b,p c),and each pair is appropriately weighted with the momentum distributions of the quarks.Our F(q2)function in(21)measures the weighted transition amplitude for given q2 in the parton picture;it is explicitly given by the overlap integral of distribution functions of initial andﬁnal state hadrons.This is common to all form factors,as explained in Section II.As mentioned earlier,all non-perturbative strong interaction eﬀects are considered through the distribution functions in our model,soξ(v·v′)obtained from Eq.(22)corre-sponds to the hadronic form factorˆξ(v·v′)deﬁned in the Ref.[8],including1/m b corrections rather than the lowest order Isgur-Wise function.And the slope parameter of our results in Table1also corresponds toˆρ2related toˆξ(v·v′).We obtain the values of the slope parameterρwithin the parton model framework,as in Table1,ρ2=0.582−0.896,which are compatible with the Neubert’s prediction[8],ˆρ2≃ρ2±0.2=0.7±0.2.Our result is rather smaller than the predictions of other models,ρ2=1.29±0.28[13],ρ2=0.99±0.04[14],but it is consistent with the average value measured by experiments[15],ˆρ2=0.87±0.12.In calculating the numerical values,we still have two free parametersǫb andǫc of Eq.(6),i.e.of the heavy quark fragmentation functions.Their values can be determined independently from the various experimental and theoretical methods1.For the parton model to be consistent with the HQET,we require that with theﬁxed value of the parameter ǫQ,all the appropriate results of the parton model approach agree with those of the HQET.In other words,the value of parameterǫQ should be determined to give all the phenomenological results to coincide with those of the HQET.In this point of view,we have previously studied the parton model approach for inclusive semileptonic decays of B meson in the Ref.[16], and showed that the valueǫb≈0.004gives consistent results with those of the HQET. In this paper we use the value ofǫb as0.004or0.006.The latter value is given by the experiments for the determination of the Peterson fragmentation function[11].Weﬁnd that our prediction of the slope parameterρ2with the parameterǫb=0.004andǫc=0.04gives the best agreed value0.705with that of the HQET,ˆρ2=0.7±0.2.In this context,we conclude that our model with the parameterǫb=0.004gives consistent predictions with the HQET.Phenomenologically,our model prediction on q2spectrum in the B→D∗eνdecay shows a good agreement with the result of the CLEO[10],as shown in Fig.2(b).If we let f Q(z)=δ(1−z)and R1=R2=1in our model,we can reproduce the lowest order results of the HQET,and obtain the similar plot with those of other models in Fig.2(b).For the B→Deνdecay,our results agrees with those of other models,as in Fig.2(a).Finally we obtain the ratio of integrated total widthsΓ(B→D∗)/Γ(B→D)≈2.66, which agrees with the experimental results[17].It may be a phenomenological support of our model because this quantity is independent of the CKM elements|V cb|,which has uncer-tainties in determining its value yet.The perturbative QCD corrections can be factorized in the decay width calculation[8,18],which does not aﬀect the ratioΓ(B→D∗)/Γ(B→D).To summarize,we developed the new parton model approach for exclusive semileptonic decays of B-meson by extending the inclusive parton model,and by combining with the results of the HQET,motivated by Drell-Yan process.Without the nearest pole dominance ans¨a tze,we derived the dependences of hadronic form factors on q2.We also calculated numerically the slope of the Isgur-Wise function,which is consistent with the experimental results.ACKNOWLEDGMENTSWe thank Pyungwon Ko and E.Paschos for their careful reading of manuscript and their valuable comments.The work was supported in part by the Korean Science and Engineering Foundation,Project No.951-0207-008-2,in part by Non-Directed-Research-Fund,Korea Research Foundation1993,in part by the CTP,Seoul National University,in part by Yonsei University Faculty Research Grant1995,and in part by the Basic Science Research Institute Program,Ministry of Education,1994,Project No.BSRI-94-2425.REFERENCES[1]M.Neubert,SLAC preprint SLAC–PUB–6263(1993),to appear in Phys.Rep.andreferences therein.[2]M.A.Shifman and M.B.Voloshin,Yad.Fiz.41,187(1985);N.Isgur and M.B.Wise,Phys.Lett.B232,113(1989);B237,527(1990);H.Georgi,Phys.Lett.B240,447 (1990).[3]A.F.Falk,H.Georgi,B.Grinstein and M.B.Wise,Nucl.Phys.B343,1(1990).[4]A.Bareiss and E.A.Paschos,Nucl.Phys.B327,353(1989).[5]C.H.Jin,W.F.Palmer and E.A.Paschos,preprint DO–TH93/21,unpublished;C.H.Jin,W.F.Palmer and E.A.Paschos,Phys.Lett.B329,364(1994);C.H.Jin,W.F.Palmer and E.A.Paschos,preprint DO–TH94/12.[6]M.Wirbel,B.Stech and M.Bauer,Z.Phys.C29,637(1985).[7]J.G.K¨o rner and G.A.Schuler,Z.Phys.C46,93(1990).[8]M.Neubert,Phys.Lett.B338,84(1994).[9]C.Peterson,D.Schlatter,I.Schmitt and P.M.Zerwas,Phys.Rev.D27,1051(1983);J.Chrin,Z.Phys.C36,163(1987).[10]CLEO Collaboration:P.Avery el.al.,preprint CLEO–CONF–94–7.[11]Heavy Flavours in Z Physics at LEP1,ed.by G.Altarelli,R.Kleiss and C.Verzegnassi,CERN89–08vol.1.[12]D.Fakirov and B.Stech,Nucl.Phys.B133,315(1978).[13]J.L.Rosner,Phys.Rev.D42,3732(1990).[14]T.Mannel,W.Roberts and Z.Ryzak,Phys.Lett.B254,274(1991).[15]CLEO Collaboration:B.Barish et.al.,preprint CLNS94/1285(1994);ARGUS Col-laboration:H.Albrecht et.al.,Z.Phys.C57,533(1993).[16]K.Y.Lee,Y.G.Kim and J.K.Kim,KAIST preprint KAIST–CHEP94/07(1994),hep-ph9411404.[17]K.Hikasa et.al.,Particle Data Group,Phys.Rev.D45,S1(1992).[18]M.Neubert,Phys.Rev.D46,2212(1992).TABLE I.The normalization constant N and the slope parameterρare shown with several values of the parametersǫb,ǫc.ǫb=0.004ǫb=0.006ǫc=0.040.060.080.10.040.060.080.1FIG.1.(a)The diagram of Drell-Yan Process.(b)The schematic diagram of semileptonic exclusive decay of B-meson in the parton model.FIG.2.(a)q2spectrum in the B→Deνdecays.The solid line is our model prediction and the dotted line the Wirbel et.al.model prediction[6].We used the values of parameters, (ǫb=0.004,ǫc=0.04).(b)q2spectrum in the B→D∗eνdecays.The thick solid line is our model prediction and the thin solid line the Wirbel et.al.model prediction[6]and the dotted line the K¨o rner el.al.model prediction[7].We used the values of parameters,(ǫb=0.004,ǫc=0.04).The data are quoted from Ref.[10].-Fig. 1 (a)P Pγl l -+qq BFig. 1 (b)νlD c b W )y-y(x,q )2δ(,D*...。