EM Duality and Quasinormal Modes from Higher Derivatives ...downloads.hindawi.com/journals/ahep/2019/5472310.pdf · ReseaArticle EM Duality and Quasinormal Modes from Higher Derivatives

Research ArticleEM Duality and Quasinormal Modes from Higher Derivativeswith Homogeneous Disorder

Guoyang Fu1 and Jian-PinWu 12

1Center for Gravitation and Cosmology College of Physical Science and Technology Yangzhou University Yangzhou 225009 China2School of Aeronautics and Astronautics Shanghai Jiao Tong University Shanghai 200240 China

Correspondence should be addressed to Jian-Pin Wu jianpinwumailbnueducn

Received 14 April 2019 Revised 24 June 2019 Accepted 26 June 2019 Published 21 August 2019

Academic Editor Sally Seidel

Copyright copy 2019 Guoyang Fu and Jian-PinWuThis is an open access article distributed under the Creative Commons AttributionLicensewhichpermits unrestricteduse distribution and reproduction in anymedium provided the original work is properly citedThe publication of this article was funded by SCOAP3

We study the electromagnetic (EM) duality from 6 derivative theory with homogeneous disorder We find that with the change ofthe sign of the coupling parameter 1205741 of the 6 derivative theory the particle-vortex duality with homogeneous disorder holds betterthan that without homogeneous disorder The properties of quasinormal modes (QNMs) of this system are also explored Whenthe homogeneous disorder is introduced some modes emerge at the imaginary frequency axis for negative 1205741 but not for positive1205741 In particular with an increase in the magnitude of new branch cuts emerge for positive 1205741 These emerging modes violate theduality related to the change of the sign of 1205741 With the increase of this duality is getting violated more

1 Introduction

The electrical conductivity is one of the important featuresof real materials For the weakly coupled systems the opticalconductivity exhibits Drude-like peak at low frequencywhich can be described by the quantum Boltzmann theory[1] For the strongly coupled systems due to the absence of thepicture of the quasiparticle the Boltzmann theory is usuallyinvalid The quantum critical (QC) dynamics described byCFT [2] is just such strongly coupling systems A novelmechanism to deal with this kind of problem is the anti-deSitterconformal field theory (AdSCFT) correspondence [3ndash6] In holographic framework remarkable progress has beenmade [7ndash18]

Due to the electromagnetic (EM) self-duality the opticalconductivity in the neutral plasma dual to the Schwarzschild-AdS (SS-AdS) black brane is trivial [19] To break the EM self-duality an extra four-derivative interaction the Weyl tensor119862120583]120588120590 coupled to Maxwell field over the Schwarzschild-AdS (SS-AdS) black brane is introduced Since the EM self-duality is broke now the electrical conductivity is frequencydependent [7] The low frequency conductivity exhibits aDrude-like peak for the coupling parameter 120574 gt 0 which issimilar to the particle excitation [7] While for 120574 lt 0 a dip in

low frequency conductivity is observed which resembles thevortex case [7] It is a possible way to address the CFT of thesuperfluid-insulator quantum critical point (QCP) thoughthere is still a degree of freedom of the sign of 120574 In additionthe particle-vortex duality related by the change of the sign of120574 in the dual field theory is also found

We can also introduce higher derivative (HD) termsWhen higher derivative (HD) terms are introduced an arbi-trarily sharp DS peak can be observed at low frequency in theoptical conductivity depending on the coupling parametersand the bound of conductivity in 4 derivative theory isviolated such that a zero DC conductivity can be obtainedat specific parameter [14] In particular its behavior is quiteclosely analogous to that of the 119874(119873) superfluid-insulatormodel in the limit of large-119873 [1] Another progress is theconstruction of neutral scalar hair black brane by couplingWeyl tensor with neutral scalar field which provides aframework to describeQCdynamics and one away fromQCP[17 18 20 21]

Also we can introduce a mechanism of microscopictranslational symmetry broken for example [22ndash32] whichleads to the momentum dissipation of the dual field theoryand is also referred as homogeneous disorder into theholographic QC system [7ndash18] such that we can study the

HindawiAdvances in High Energy PhysicsVolume 2019 Article ID 5472310 19 pageshttpsdoiorg10115520195472310

2 Advances in High Energy Physics

effects from homogeneous disorder on these systems [33ndash36] We find that for 4 derivative theory studied in [33] thehomogeneous disorder drives the incoherent metallic statewithDS peak into the onewith a dip for 120574 gt 0 and an oppositescenario is found for 120574 lt 0 But for the 6 derivative theoryexplored in [34] the homogeneous disorder cannot make thepeak (gap) transform into its contrary Another interestingphenomena is that for 4 derivative theory there is a specificvalue of = 2radic3 for which the particle-vortex dualitycorresponding to EM duality in bulk almost exactly holdswith the change of the sign of 120574 [33]

On the other hand to study the holographic QC dynam-ics a good method we can use is the quasinormal modes(QNMs) of a gravitational theory on the bulk AdS spacetimeRecently the structure of the QNMs of 46 derivative theoryover SS-AdS has been explored [11 14] In particular theparticle-vortex duality has also been discussed Furtherwhen the homogeneous disorder in 4 derivative theory isintroduced more rich pole structures are exhibited [36] Inthis paper we shall study the EM duality and the QNMs of6 derivative theory with homogeneous disorder Our paperis organized as follows In Section 2 we present a briefintroduction on the 6 derivative theory with homogeneousdisorder EM duality is discussed in Section 3 In Section 4we study the properties of the QNMs A brief discussion ispresented in Section 5 In Appendix A we also give a briefdiscussion on the instabilities of gauge mode by the QNMsAppendixes B and C present the constraint on another 6derivative term and its properties of conductivity and QNMsrespectively

2 Holographic Setup

We consider the following neutral Einstein-axions (EA) the-ory [32]

1198780 = int1198894119909radicminus119892(119877 + 61198712 minus 12 sum119868=119909119910

(120597120601119868)2) (1)

In the above action we have introduced a pair of spatial lineardependent axionic fields 120601119868 = 120572119909119868 with 119868 = 119909 119910 and 120572being a constant which are responsible for dissipating themomentum of the dual boundary field 119871 is the radius of theAdS spacetimes

From the EA action (1) we have a neutral black branesolution [32]

1198891199042 = 11987121199062 (minus119891 (119906) 1198891199052 + 1119891 (119906)1198891199062 + 1198891199092 + 1198891199102) (2)

where

119891 (119906) = (1 minus 119906) 119901 (119906) 119901 (119906) = radic1 + 62 minus 22 minus 12 1199062 + 119906 + 1 (3)

TheAdSboundary is at119906 = 0 and the horizon locates at 119906 = 1 equiv 1205724120587119879 with the Hawking temperature 119879 = 119901(1)4120587

Since the translational symmetry breaks the momentumdissipates But the geometry is homogeneous and so we referto this mechanism as homogeneous disorder and denotesthe strength of disorderTherefore the background describesa specific thermal excited state with homogeneous disorder

Over this background we study the following HD actionfor gauge field coupling with Weyl tensor

119878119860 = int1198894119909radicminus119892(minus 181198922119865119865120583]119883120583]120588120590119865120588120590) (4)

where the tensor 119883 is an infinite family of HD terms [14]

119883120583]120588120590 = 119868120583]120588120590 minus 8120574111198712119862120583]120588120590 minus 41198714120574211198622119868120583]120588120590minus 8119871412057422119862120583]120572120573119862120572120573120588120590 minus 41198716120574311198623119868120583]120588120590minus 81198716120574321198622119862120583]120588120590minus 8119871612057433119862120583]12057211205731119862120572112057311205722120573211986212057221205732120588120590 +

(5)

Note that 119868120583]120588120590 = 120575120583120588120575]120590 minus 120575120583120590120575]120588 is an identity matrix and119862119899 = 119862120583]120572112057311198621205721120573112057221205732 119862120572119899minus1120573119899minus1120583] In the above equationsthe factor of 119871 is introduced such that the coupling parame-ters 119892119865 and 120574119894119895 are dimensionless But for later conveniencewe work in units where 119871 = 1 in what follows and set 119892119865 = 1in the numerical calculation When119883120583]120588120590 = 119868120583]120588120590 the theoryreduces to the standard Maxwell theory For convenience wedenote 12057411 = 120574 and 1205742119894 = 120574119894(119894 = 1 2) In the main body ofthis paper ourmain focus is the 1205741 derivative terms (themaineffect of both 1205741 and 1205742 terms is similar But there is still somedifference between them We shall present a brief discussionon the properties of the conductivity and the QNMs from1205742 higher derivative term in Appendixes B and C) Note thatwhen other parameters are turned off 1205741 is confined in theregion 1205741 le 148 over SS-AdS black brane background(In [34] they also study the instabilities of the gauge modeand the causality in CFT in addition to the constraint fromRe120590(120596) ge 0 they confirm that the constraint 1205741 le 148 alsoholds over EA-AdS geometry) [14] And then from the action(4) we can write down the equation of motion (EOM) as

nabla] (119883120583]120588120590119865120588120590) = 0 (6)

As [7] the dual EM theory of (4) can be constructedwhich is


119866120583] equiv 120597120583119861] minus 120597]119861120583 is the dual field strength and 119865 is thecoupling constant which relates 119892119865 as 2119865 equiv 11198922119865 is thedual tensor which satisfies

120583]120588120590 = minus14120576120583]120572120573(119883minus1)12057212057312057412058212057612057412058212058812059012(119883minus1)120583]120588120590119883120588120590120572120573 equiv 119868120583]120572120573(8)

Advances in High Energy Physics 3

And then the EOM of the dual theory (7) can be wrote downas

nabla] (120583]120588120590119866120588120590) = 0 (9)

For the four dimensional standard Maxwell theory120583]120588120590 = 119868120583]120588120590 At this moment both theories (4) and (7)are identical and so the standard Maxwell theory is self-dual Once the HD terms are introduced such self-duality isbroken However if 1205741 is small we have

(119883minus1)120583]

120588120590 = 119868120583]120588120590 + 412057411198622119868120583]120588120590 + O (12057421) (10)

120583]120588120590 = (119883minus1)120583]120588120590 + O (12057421) (11)

It indicates that for small 1205741 there is an approximate dualitybetween both theories (4) and (7) with the change of the signof 12057413 Conductivity and EM Duality

Turning on the perturbation 119860119910(119905 119906) sim 119890minus119894120596119905119860119910(119906) we canwrite down the EOM of gauge field [7 33 34]

11986010158401015840119910 + (1198911015840119891 + 119883101584061198836)1198601015840119910 +p221198912 11988321198836119860119910 = 0 (12)

where 119883119894 119894 = 1 6 are the components of 119883119860119861 definedas 119883119860119861 = 1198831(119906)1198832(119906)1198833(119906)1198834(119906)1198835(119906)1198836(119906) with119860 119861 isin 119905119909 119905119910 119905119906 119909119910 119909119906 119910119906 Due to the isotropy of back-ground 1198831(119906) = 1198832(119906) and 1198835(119906) = 1198836(119906) In additionwe have defined the dimensionless frequency in the aboveequation equiv 1205964120587119879 = 120596p with p equiv 119901(1) = 4120587119879 Letting119883119894 997888rarr 119894 = 1119883119894 we can obtain the dual EOM Solving theEOM (12) or the dual EOMwith ingoing boundary conditionat the horizon we can read off the conductivity in terms of

120590 () = 120597119906119860119910 (119906 )119894p119860119910 (119906 ) (13)

It has been illustrated in the last section that for verysmall 1205741 there is an approximate duality between both theoriginal EM theory and its dual theory with the change ofthe sign of 1205741 as that for 4 derivative theory [7 33] At thesame time there is an inverse relation for the conductivity ofthe original theory and its dual theory (this relation has beenproved in [7 11] Also it has been derived for a specific classof CFTs in [19]) [7 11]

120590lowast ( 1205741) = 1120590 ( 1205741) (14)

where 120590lowast is the conductivity of the dual theory Therefore forsmall 1205741 we conclude that [33]

120590lowast ( 1205741) asymp 120590 ( minus 1205741) 100381610038161003816100381612057411003816100381610038161003816 ≪ 1 (15)

It indicates that the dual optical conductivity is approximatelyequal to its original one with the change of the sign of 1205741 For

4 derivative theory it has been studied and found that theconductivity of the dual EM theory is not precisely equal tothat of its original theory with the change of the sign of 120574 [7]But when the homogeneous disorder is introduced we findthat this relation holds better for a specific value of = 2radic3than other values of Here we shall study the EM dualityfrom 6 derivative theory

Figure 1 exhibits the optical conductivity as the functionof for = 0 and various values of 1205741 We find that forsmall |1205741| (|1205741| = 0001) with the change of the sign of 1205741the relation (15) holds very well With the increase of |1205741|the relation (15) violates In particular for 1205741 = 002 thereal part of the conductivity at low frequency is a pseudogap-like behavior Correspondingly its dual conductivity Re120590lowast atlow frequency becomes sharper which is Drude-like But for1205741 = minus002 the real parts of the conductivity of the originaltheory and its dual one at low frequency are only peak anddip respectively At this moment the relation (15) is stronglyviolated

Next we explore the EM duality for 6 derivative theorywhen the homogeneous disorder is introduced At the firstsight we find that for the specific value of = 2radic3the optical conductivity of the original theory is almost(approximate but not exact) the same as that of the dual onewhen the sign of 1205741 changes ie the relation (15) holds verywell It is similar to that from 4 derivative theory [33] Forother values of the relation (15) also approximately holds Itseems that the homogeneous disorder makes the pseudogap-like behavior in the low frequency conductivity of the originaltheory become small dip while suppressing the sharp peak ofthe EM dual theory

It is hard to obtain an analytical understanding on thisphenomenon But we can follow the method of [33] andattempt to taste kind of analytical similarity between theEOM (B8) and its dual one which can be explicitly wrotedown

11986010158401015840119910 + (1198911015840119891 minus 119883101584061198836)1198601015840119910 +p21198912 11988361198832 2119860119910 = 0 (16)

Since11988321198836 = 11988361198832 for any the difference between (B8)and (16) is in the coefficients of 1198601015840119910 which are

X1 (1205741) equiv 119883101584061198836= minus (163) 1205741119906311989110158401015840 (119906)2 minus (83) 12057411199064119891(3) (119906) 11989110158401015840 (119906)1 minus (43) 1205741119906411989110158401015840 (119906)2

(17)

X2 (1205741) equiv minus119883101584061198836= minusminus (163) 1205741119906311989110158401015840 (119906)2 minus (83) 12057411199064119891(3) (119906) 11989110158401015840 (119906)1 minus (43) 1205741119906411989110158401015840 (119906)2

(18)

And then we explicitly plot the ration (X1X2)|1205741=002 asthe function of 119906 with different in Figure 3 For = 0the value of X1X2 is maximum departure from 1 nearthe horizon which indicates a most deviation between (B8)


00 02 04 06 08 10090

095

100

105

110

00 02 04 06 08 10 12 1400

05

10

15

20

00 02 04 06 08 10 12 1401234567

00 02 04 06 08 10minus010

minus005

000

005

010

00 02 04 06 08 10 12 14minus10

minus05

00

05

10

00 02 04 06 08 10 12 14minus4minus3minus2minus10123

=0 =0 =0

=0 =0 =0

1=00011=minus0001

1=0011=minus001

1=0021=minus002

1=00011=minus0001

1=0011=minus001

1=0021=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

Figure 1 The real part (the plot above) and the imaginary part (the plot below) of the optical conductivity as the function of for = 0 andvarious values of 1205741 The solid curves are the conductivity 120590 of the original EM theory (4) and the dashed curves display the conductivity 120590lowastof the EM dual theory (7)

and (16) Once the homogeneous disorder is introducedthis deviation becomes small Specially for = 2radic3 thisvalue approaches to 1 near the horizon which implies atmostsimilarity between (B8) and (16) On the other hand it is wellknown that the low frequency behavior of the conductivity ismainly controlled by the near horizon geometryTherefore tosome extent the above comparison in the original EOM andits dual one helps us understand the phenomenon shown inFigure 2

4 The Properties of QNMs

By definition QNMs are the poles of the Greenrsquos functionand are directly related to the conductivity Summing anincreasing number of QNMs and the corresponding residuesone should obtain a better and better approximation ofthe conductivity itself In this section we shall study theproperties of QNMs of the transverse gauge mode along 119910direction ie119860119910 from 6 derivative term with homogeneousdisorder In particular we shall also study the particle-vortexduality in the complex frequency panel in the holographicCFT

It is highly efficient to use the pseudospectral methodsoutlined in [37] to solve the QNM equations and obtainthe QNMs To this end we shall work in the advancedEddington-Finkelstein coordinate which is

1198891199042 = 11199062 (minus119891 (119906) 1198891199052 minus 2119889119905119889119906 + 1198891199092 + 1198891199102) (19)

Correspondingly the EOM (B8) can be changed as

1198601015840119910 (1198911015840119891 + 2119894p119891 + 119883101584061198836) + 119894p119860119910119883101584021198911198836 + 11986010158401015840119910 = 0 (20)

For the dual theory the corresponding EOM can be obtainedby letting 119860120583 997888rarr 119861120583 and 119883119894 997888rarr 119894 = 1119883119894 Imposing theingoing boundary at the horizon we numerically solve (20)to obtain QNMs

We first study the case without homogeneous disorderFigures 4 and 5 show QNMs of the gauge mode with = 0for |1205741| = 0001 and 002 respectively The left panels are theQNMs of the gauge mode and the right ones are that of thedual gauge mode We find that for small |1205741| for example|1205741| = 0001 in Figure 4 the qualitative correspondencebetween the poles of Re120590( 1205741) and the ones of Re120590lowast( minus1205741)holds well at low frequency But for the large frequencyregion there are some modes emerging at the imaginaryfrequency axis for negative 1205741 which results in the violationof the particle-vortex duality with the change of the sign of1205741 But with the increase of 1205741 this correspondence begins toviolate in complex frequency panel even in the low frequencyregion (see Figure 5 for |1205741| = 002) as that in real frequencyaxis We also plot the QNMs for 1205741 = minus1 in Figure 6 We seethat another branch cuts which are closer to the imaginaryfrequency axis emerge in the complex frequency panel

Now we turn to the study of the effects of homogeneousdisorder Figures 7 8 and 9 show the QNMs with repre-sentative for 120574 = 002 minus002 and minus1 (from left to right)in the complex frequency plane The panels above are the


00 05 10 15 200708091011121314

0 1 2 3 4 5 6

0994099609981000100210041006

0 1 2 3 4 5 60510152025

3530

00 05 10 15 20

minus02minus0100010203

0 1 2 4 5 6

minus0006minus0004minus00020000000200040006

0 1 2 33 4 5 6minus05

00

05

10

=23=12 =10

=23=12 =10

1=0021=minus002

1=0021=minus002

1=0021=minus002

1=0021=minus002

1=0021=minus002

1=0021=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

Figure 2The real part (the plot above) and the imaginary part (the plot below) of the optical conductivity as the function of for |1205741| = 002and various values of The solid curves are the conductivity 120590 of the original EM theory (4) and the dashed curves display the conductivity120590lowast of the EM dual theory (7)

00 02 04 06 08 10

10

12

14

16

18

20

u

1(

1)2(minus

1)

=23

=12=0 =2

=10

Figure 3X1(1205741)X2(minus1205741) as the function of119906with different Herewe set 1205741 = 002

QNMs of the gauge mode while the ones below are that ofthe dual gauge mode Rich information and insights on thepole structures are exhibited At the first sight we find that allthe poles are in the lower-half plane (LHP) It indicates thatthe gaugemode is stable Further analysis on stability in termsof QNMs is presented in Appendix A

Next we shall study the main properties of the QNMswhen the homogeneous disorder is introduced in particularthe duality between the poles of Re120590( 1205741) and the ones ofRe120590lowast( minus1205741) They are briefly summarized as what followsfor |1205741| = 002

(i) For = 12 some modes emerge at the imaginaryfrequency axis for negative 1205741 as that without homo-geneous disorder but not for positive 1205741

(ii) For = 2radic3 there are still the modes locating in theimaginary frequency axis for negative 1205741 but not forpositive 1205741 In particular new branch cuts emerge forpositive 1205741

(iii) For = 2 the pole structures for = 2 are similarwith those for = 2radic3

(iv) These emerging modes violate the duality betweenthe poles of Re120590( 1205741) and the ones of Re120590lowast( minus1205741)With an increase in the magnitude of this duality isgetting violated more

We also exhibit the QNMs for 1205741 = minus1 with homogeneousdisorder (the third columns in Figures 7 8 and 9) Whenthe homogeneous disorder is introduced the newbranch cutsattach to the imaginary frequency axis Such pole structuresare interesting and are worthy of further study such that wecan understand the physics of these pole structures

The dominant poles reveal the late time dynamics ofthe system So we shall study the dominant pole behaviorssuch that we have a well understanding on the effect of thehomogeneous disorder

In our previous work [34] we have studied the conduc-tivity at the low frequency in real axis for 1205741 = minus1with homo-geneous disorder For small it exhibits a sharp DS peakThough it is not the Drude peak we can phenomenologicallydescribe this peak by the Drude formula (see Figure 5 in [34])

120590 () = 119870Γ minus 119894 (21)

where Γ is the relaxation rate which relates the relaxationtime 120591 as Γ = 1120591 This peak in the real axis corresponds to apurely imaginary mode in complex frequency panel (see the


minus6 minus4 minus2 minus6 minus4 minus2

minus6 minus4 minus2minus6 minus4 minus2

0 2 4 6

minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

0

0 2 4 6

0

0 2 4 6

0

0 2 4 6

0

Im

Im

Im

Im

Re Re

ReRe

Figure 4 QNMs (blue spots) of the gauge mode for |1205741| = 0001 (the panels above are for 1205741 = minus0001 and the ones below for 1205741 = 0001)and = 0 The left panels are the QNMs of the gauge mode and the right ones are that of the dual gauge mode

0 1 2

00

0 1 2

0 1 2

00

0 1 2

Im

Im

ReRe

ReRe

minus2 minus1minus14minus12minus10minus08minus06minus04minus02

minus2 minus1


00

00

Im

Im

minus14minus12minus10minus08minus06minus04minus02


minus2 minus1

Figure 5 QNMs (blue spots) of the gauge mode for |1205741| = 002 (the panels above are for 1205741 = minus002 and the ones below for 1205741 = 002) and = 0 The left panels are the QNMs of the gauge mode and the right ones are that of the dual gauge mode

0 5 10 15

0

0 5 10

0

Im

Re

Im

Reminus15 minus5minus10

minus8

minus6

minus4

minus2

minus10 minus5

minus6

minus4

minus2

Figure 6 QNMs (blue spots) of the gauge mode for 1205741 = minus1 and = 0 The left panels are the QNMs of the gauge mode and the right onesare that of the dual gauge mode


0 2 4 6

0

0 2 4 6 0 5 10

0

0 2 4 6

0

0 2 4 6 0 5 10

0

Im

Im

Im

Im

Re Re Re

Re Re Reminus6 minus2minus4

minus6 minus2minus4

minus6 minus2minus4

minus6 minus2minus4

minus10

minus8

minus6

minus4

minus2

0

Im

minus10

minus8

minus6

minus4

minus2

minus10 minus5

minus8

minus6

minus4

minus2

minus10

minus8

minus6

minus4

minus2

0

Im

minus10

minus8

minus6

minus4

minus2

minus10 minus5

minus10minus8minus6minus4minus2

Figure 7QNMs (blue spots) of the gaugemode for = 12The first second and third columns are for 1205741 = 002minus002 andminus1 respectivelyThe panels above are for the gauge mode while the ones below are for the dual gauge mode

0 2 4

0

0 2 4 0 5 10 15

0

0 2 4

0

0 2 4

0

0 5 10 15

0

Im

Im

Im

Im

Im

Re Re Re

Re Re Reminus4 minus2


0

Im


minus4 minus2 minus15 minus10 minus5minus20

minus15

minus10

minus5



minus15 minus10 minus5minus20

minus15

minus10

minus5

Figure 8 QNMs (blue spots) of the gauge mode for = 2radic3 The first second and third columns are for 1205741 = 002 minus002 and minus1respectivelyThe panels above are for the gauge mode while the ones below are for the dual gauge mode

left column in Figure 10) We also locate the position of thepurely imaginary mode which is listed in Table 1 We findthat the relaxation rates fitted by the Drude formula (21) arein agreement with that by locating the position of the purelyimaginary mode

More detailed evolution of the dominant QNMs with for 1205741 = minus1 is presented in Figure 10 Most of the dominantQNMs for the original theory except for that approximatelyin the region of isin (1 2radic3) are the purely imaginarymodes Correspondingly the dominant QNMs for the dualtheory except for that approximately in the region of isin(1 2radic3) are off axis

Further we study the evolution of the dominant QNMswith for |1205741| = 002 which is shown in Figure 11 Wedescribe the properties as what follows

(i) For 1205741 = minus002 all the dominant QNMs for theoriginal theory are purely imaginary modesWith theincrease of these modes firstly migrate downwardsand then migrate upwards finally approaching to acertain value The evolution of the dominant QNMsfor the dual theory with 1205741 = 002 is similar withthat for the original theory with 1205741 = minus002 Suchevolution is also similar with that for 4 derivativetheory studied in [36] For small the poles for thedual theory are closer to the real frequency axis thanthat for the original theory

(ii) For 1205741 = 002 the dominant poles for the originaltheory are off axis for small With the increase of the poles migrate downwards and are closer tothe real axis When reaches the value of ≃ 08


Table 1 The relaxation rate Γ119894 of the holographic system with 1205741 = minus1 for different Γ1 is fitted by the Drude formula (21) and Γ2 is obtainedby locating the position of the purely imaginary mode in complex frequency panel

0 001 01 05Γ1 00114 00115 00120 00284Γ2 00114 00115 00122 00370

0 2 4

0

0 2 4 minus30 minus20 minus10

minus30 minus20 minus10

0 10 20 30

0

0 2 4

0

0 2 4 0 10 20 30

0

Im

Im

Im

Im

Re Re Re

Re Re Reminus4 minus2 minus4 minus2

minus4 minus2


0

Im



0

Im


minus20

minus15

minus10

minus5

minus15

minus10

minus5

Figure 9 QNMs (blue spots) of the gauge mode for = 2 The first second and third columns are for 1205741 = 002 minus002 and minus1 respectivelyThe panels above are for the gauge mode while the ones below are for the dual gauge mode

00

01

02

03

04

05

Re

1 2 3 4 5 60

00051015202530

Re

1 2 3 4 5 60

minus06minus05minus04minus03minus02minus0100

Im

1 2 3 4 5 60

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im

Figure 10 Evolution of the dominant QNMs with for 1205741 = minus1 The left column is the QNMs for the original theory and the right one isthat for its dual theory

the poles merge into one purely imaginary pole Andthen when is beyond ≃ 14 the purely imaginarypole splits into two off-axis modes While for 1205741 =minus002 the evolution of the dominant poles for thedual theory is similar with that for the original theorywith 1205741 = 002 But when gt 35 the pole for the dualtheory with 1205741 = minus002 becomes a purely imaginaryone

(iii) In summary the correspondence between the evolu-tion of the poles as the function of for the originaltheory for 1205741 = minus002 and that for its dual theory for1205741 = 002 is violated

5 Conclusion and Discussion

In this paper we extend our previous work [34] whichstudied the optical conductivity from 6 derivative theory on


minus10

minus05

00

05

10

Re

1 2 3 4 5 60

00

01

02

03

04

Re

1 2 3 4 5 60

minus08

minus06

minus04

minus02

00

Im

1 2 3 4 5 60

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im

00

01

02

03

04

Re

1 2 3 4 5 60

minus10

minus05

00

05

10

Re

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im

1 2 3 4 5 60

1 2 3 4 5 60

minus08

minus06

minus04

minus02

00

Im

Figure 11 Evolution of the dominant QNMs with for |1205741| = 002 (the first two rows are for 1205741 = minus002 and the last two rows for 1205741 = 002)The left columns are the QNMs for original theory and the right ones are that for its dual theory

top of EA-AdS geometry to the holographic response of itsEM dual theory In particular we explore thoroughly theEM duality Also we study the QNMs and the EM duality incomplex frequency panel

In absence of the homogeneous disorder with the changeof the sign of 1205741 the particle-vortex duality only holds forsmall |1205741| With the increase of |1205741| this duality is violatedWhen the homogeneous disorder is introduced as foundin 4 derivative theory we find that for the specific value of = 2radic3 the optical conductivity of the original theoryis almost the same as that of the dual one when the signof 1205741 changes For other values of the particle-vortexalso approximately holds with the change of the sign of 1205741Therefore we can conclude that the homogeneous disordermakes the pseudogap-like of the low frequency conductivityof the original theory become small dip while suppressingthe sharp peak of the EM dual theory such that we have an

approximate particle-vortex duality with the change of thesign of 1205741

The properties of the QNMs are also analyzed In absenceof the homogeneous disorder the qualitative correspondencebetween the poles of Re120590( 1205741) and the ones of Re120590lowast( minus1205741)holds well at low frequency only for small 1205741 When 1205741becomes large this correspondence is also violated even atthe low frequency region For 1205741 = minus1 new branch cuts ofQNMs are observed

When the homogeneous disorder is introduced somemodes emerge at the imaginary frequency axis for negative1205741 but not for positive 1205741 In particular with an increase inthe magnitude of new branch cuts emerge for positive1205741 These emerging modes violate the duality between thepoles of Re120590( 1205741) and the ones of Re120590lowast( minus1205741) Withthe increase of this duality is getting violated more Inaddition for 1205741 = minus1 the homogeneous disorder drives the


Table 2 The coefficients of the power-law fit of the form 119886|1205741|119887 for the dominant poles for different 12 2radic3 2 10119886 0040 0957 0076 0072119887 0761 0816 0632 0662off-axis new branch cuts for = 0 to the purely imaginarymodes

The evolution of the dominant QNMs with is alsoexplored We find that all the dominant QNMs for theoriginal theory with 1205741 = minus002 and that for the dual theorywith 1205741 = 002 are purely imaginary modes Their evolutionsare also similar However for the evolution of the dominantQNMs for the original theory with 1205741 = 002 and that for thedual theory with 1205741 = minus002 the case is somewhat differentThemain difference is that the modes are not again the purelyimaginary ones except for some specific In summary thecorrespondence between the evolution of the poles as thefunction of for the original theory and that for its dualtheory is violated

There are lots of open questions worthy of further study

(i) We would like to carry out an analytical study onthe complex frequency conductivity by matchingmethod developed in [38] Also we can analyticallywork out the QNMs by WKB method as [11 14]These analytical analyses can surely provide morephysical insight and understanding into our presentobservation

(ii) At present the perturbative black brane solutionto the first order of the coupling parameter from4 derivative theory has been worked out in [3539ndash45] as well as the related exploration includingholographic metal-insulator transition holographicentanglement and holographic thermalization atfinite charge density on top of the perturbative back-ground with HD corrections In future it is alsointeresting to further study the QNMs includingscalar vector and tensor modes from HD theory atfinite charge density (the QNMs of massless scalarfield over the perturbative black hole with sphericalsymmetry horizon have been studied in [41])

(iii) The dispersing QNMswith finite momentum deservefurther studying It surely reveals richer physics of thesystem

(iv) The superconducting phase fromHD theory has beenwidely explored in [46ndash55] and references thereinIt is also interesting to study the QNMs in thesuperconducting phase of these models such that wecan get richer insight and understanding on the HDtheory

(v) It would be interesting to simultaneously investigatethe effects of both 4 and 6 derivative terms on theconductivity and the QNMs In this way the maineffect of higher derivative terms on EM duality couldbe revealed We shall continue to explore this projectin future

Appendix

A Brief Analysis on the Stability from 1205741Derivative Term

In [34] by analyzing the instabilities of gauge mode and thecausality in CFT in addition to the constraint from Re120590(120596) ge0 we confirm that the constraint 1205741 le 148 which hasbeen obtained over SS-AdS geometry in [14] also holds overEA-AdS geometry In this section we further examine thisconstraint by studying the QNMs Indeed we find that when1205741 le 148 all the QNMs are in the LHP

First it has been shown in Figures 10 and 11 that for1205741 = minus1 |1205741| = 002 and isin [0 6] all the poles and zeroswhich are the poles of the dual theory are in the LHP Inaddition we also see that when 997888rarr infin the imaginarymodes approach to a constant or even continue to migratedownwards Therefore we can conclude that for the selected1205741 the modes are stable for all

And then for representing we show the poles andzeros for larger region of 1205741 le 148 As that on top ofSS-AdS geometry in [14] the dominant poles from EA-AdSgeometry are purely imaginary modes and also asymptoti-cally approach the real -axis as 1205741 997888rarr minusinfin In additionthe data can be well fitted by a power-law formula as 119886|1205741|119887The coefficients 119886 and 119887 are listed in Table 2 We can seethat at least over the 3 decades we have studied the fit isvery well as shown in Figure 12 We would like to point outthat this result is consistent with the peak of the conductivityat low frequency becoming sharper with the decrease of 1205741and finally approaching a delta function as 1205741 997888rarr minusinfinFor the zeros we also see that they all locate at the LHPat least over the 3 decades we have studied here (see rightplots in Figure 12) Therefore by the analysis of QNMswe again confirm that the gauge mode is stable over EA-AdS geometry when the coupling parameter is confined to1205741 le 148B Constraint on the Coupling Parameter 1205742In this section we derive the constraint on the couplingparameter 1205742 over the EA-AdS background (1) by examiningthe bound from the positive definiteness of the DC conduc-tivity the instabilities and the causality of gauge mode andalso the QNMs

B1 Bound from DC Conductivity The DC conductivity forour present model can be expressed as

1205900 = radicminus119892119892119909119909radicminus119892119905119905119892119906119906119883111988351003816100381610038161003816100381610038161003816119906=1 (B1)


1=minus1

1=minus1

1=minus1

1=minus1

1=minus5

1=minus5

1=minus5

1=minus5

1=minus25

1=minus25

1=minus25

1=minus25

1=minus150

1=minus150

1=minus150

1=minus150

1=minus103

1=minus103

1=minus103

1=minus103

0001

0002

0005

00050010

00500100

Γ

0001

0002

0005

0010

0020

Γ

0001

0002

0005

0010

Γ

=12 =12

=2 =2

=23 =23

Γ

101 102 103

5 times 10minus4

2 times 10minus4minus03

minus02

minus01

00

Im

15 20 25 30 35 4010Reminus1

101 102 103

minus1

=10 =10

101 102 103

minus1

101 102 103

minus1


Im

02 03 04 0501Re


Im

45 50 55 60 65 7040Re


Im

07 08 09 10 1106Re

Figure 12 Left plot location of the dominate QNM for different Red dots are the numerical data and the blue line is the power-law fit ofthe form 119886|1205741|119887 Right plot location of the next QNM closet to the real axis a zero for different

It can be explicitly worked out as up to 6 derivatives1205900 = 1 minus 2312057411989110158401015840 (1) minus (431205741 + 191205742)11989110158401015840 (1)2 (B2)

with

11989110158401015840 (1) = minus2 minus 4 (minus1 minus 22 + radic1 + 62)2 (B3)

Here we separately consider the 1205742 term So we set 120574 = 0 and1205741 = 0 In the limit of 997888rarr 0 the DC conductivity reducesto

1205900 = 1 minus 41205742 (B4)

Immediately from the positive definiteness of the DC con-ductivity we obtain the bound of 1205742 as 1205742 le 14 in the SS-AdSbackground

Further we examine if this bound of 1205742 isin (minusinfin 14] isviolated when the background is EA-AdS To this end weplot the DC conductivity 1205900 as the function of 1205742 for somefixed (left plot) and as the function of for some fixed1205742 (right plot) From Figure 13 we clearly see that for thepositive 1205742 when le 2radic3 DC conductivity increases withthe increase of Once is beyond the value of 2radic3 DCconductivity decreases with the increase of but it tends tocertain positive constant value when 997888rarr +infin When 1205742 isnegative DC conductivity is larger than or equal to unity for


minus03 minus02 minus01 01 02 03

05

10

15

20

0 1 2 3 4 5 600051015202530

=0=10=05=23

0

2

0

=23

2=minus132=minus1482=0

2=1482=1242=14

Figure 13 Left plot the DC conductivity 1205900 versus 1205742 for some fixed Right plot the DC conductivity 1205900 versus for some fixed 1205742 Herewe have set other coupling parameters vanishing

all In summary the constraint of 1205742 isin (minusinfin 14] also holdsin the EA-AdS background

B2 Bound from Anomalies of Causality In this subsectionwe examine the bound from the instabilities and the causalityof the perturbative gauge modes To this end we turn onthe perturbations of the gauge field as 119860120583(119905 119909 119910 119906) sim119890119894qsdotx119860120583(119906 q) where q sdot x = minus120596119905 + 119902119909119909 + 119902119910119910 Without lossof generality we choose q120583 = (120596 119902 0) and fix the gauge as119860119906(119906q) = 0 And then we have a group of perturbativeequations as [33]

1198601015840119905 + 120596 119883511988331198601015840119909 = 0 (B5)

11986010158401015840119905 + 1198831015840311988331198601015840119905 minusp2119891 11988311198833 (119860 119905 + 119860119909) = 0 (B6)

11986010158401015840119909 + (1198911015840119891 + 119883101584051198835)1198601015840119909 +p21198912 11988311198835 (119860 119905 + 119860119909) = 0 (B7)

11986010158401015840119910 + (1198911015840119891 + 119883101584061198836)1198601015840119910 +p21198912 (211988321198836 minus 2119891

11988341198836)119860119910= 0

(B8)

where equiv 1199024120587119879 = 119902p Equations (B5) and (B6) result in adecoupled equation for 119860 119905(119906 q) as

11986010158401015840119905 + (1198911015840119891 minus 119883101584011198831 + 2119883101584031198833)1198601015840119905 + (minus

p2211988311198911198833+ p22119883111989121198835 + 1198911015840119883101584031198911198833 minus

119883101584011198831015840311988311198833 +1198831015840101584031198833 )119860 119905 = 0

(B9)

Letting 119860120583 997888rarr 119861120583 and 119883119894 997888rarr 119894 in the above equations wecan obtain the equations for the dual EM theory

Further we recast (B9) and (B8) into the followingSchrodinger form

minus1205972119911120595119894 (119911) + 119881119894 (119906) 120595119894 (119911) = 2120595119894 (119911) (B10)

The potential 119881119894(119906) is119881119894 (119906) = 21198810119894 (119906) + 1198811119894 (119906) (B11)where

1198810119905 = 11989111988311198833 1198810119910 = 11989111988331198831

(B12)

1198811119905 = 1198914p211988321 [3119891 (11988310158401)2 minus 21198831 (11989111988310158401)1015840] (B13)

1198811119910 = 1198914p211988321 [minus119891 (11988310158401)2 + 21198831 (11989111988310158401)1015840] (B14)

Note that we have made the change of variables 119889119911119889119906 = p119891and wrote 119860 119894(119906) = 119866119894(119906)120595119894(119906) with 119894 = 119905 119910

When the potentials satisfy0 le 119881119894 (119906) le 1 (B15)

the modes are stable and also satisfy the causality of the dualboundary theory [56ndash58] If 119881119894(119906) violates the lower boundthe stability of the modes is determined by the zero energybound state of the potential

In the limit of the large momentum ( 997888rarr infin) 1198810119894 isthe dominant terms Figures 14 and 15 give 1198810119905(119906) and 1198810119910(119906)as the function of 119906 for sample 1205742 and respectively Bycareful analysis we give the allowed range of parameter as1205742 isin [minus13 124]

In the limit of the small momentum ( 997888rarr 0) 1198811119894becomes important We examine the potentials 1198811119905(119906) and1198811119910(119906) in the region of 1205742 isin [minus13 124] In Figure 16 weexhibit 1198811119894(119906) for sample 1205742 and We find that the upperbound of1198811119894(119906) (Eq (B15)) is satisfied But we note that thereis a negative minimum in 1198811119894 Therefore we need to furtherexamine the zero energy bound state of the potentials whichis [58]

1119905 = 119868120587 + 12 119868 equiv (119899 minus 12)120587 = int

1199061

1199060

p119891 (119906)radicminus1198811119905 (119906)119889119906(B16)


02 04 06 08 10u

minus05

05101520

02 04 06 08 10u

02

04

06

08

10

02 04 06 08 10u

0204060810

02 04 06 08 10u

02

04

06

08

10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

2=minus12=minus13

2=112 2=124V0 t(u)

V0 t(u)

V0 t(u)

V0 t(u)

Figure 14 The potentials1198810119905(119906) for sample 1205742 and which constrain 1205742 in the region of 1205742 isin [minusinfin 124]

02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

2=minus13 2=minus122=124V0y(u) V0y(u) V0y(u)

Figure 15 The potentials 1198810119910(119906) for sample 1205742 and which constrain 1205742 in the region of 1205742 isin [minus13 124]

where 119899 is a positive integer and the interval [1199060 1199061] definesthe negative well Both 1119905 and 1119910 as the function forsample 1205742 are shown in Figure 17 The detailed analysisindicates that when 1205742 isin [minus13 124] 1119894 are smaller thanunit and no unstable modes are developed Therefore theconstraint on 1205742 is

minus13 le 1205742 le 124 (B17)

B3 Examining QNMs In this subsection we examine theconstraint (B17) by studying theQNMs Figure 18 exhibits thedominate QNMs of the gauge modes and the dual ones Wefind that all the poles are in the LHP In particular when 997888rarrinfin the imaginary modes continue to migrate downwardsTherefore themodes are stable for all in the region of (B17)In Appendix Cwe shall study the conductivity QNMs andthe EM duality in the region of (B17)

C The Properties of Conductivity and QNMsfrom 1205742 Higher Derivative Term

In this section we shall present a brief exploration on theproperties of the conductivity and QNMs from the 1205742 higherderivative term

C1 e Properties of Conductivity Figure 19 exhibits theoptical conductivity as the function of for = 0 andvarious values of 1205742 When 1205742 is negative the low frequencyconductivity of the original theory is Drude-like While forpositive 1205742 it is a dip For small |1205742| with the change of the signof 1205742 the relation (15) holds very well (see the first column inFigure 19)With the increase of |1205742| the relation (15) graduallyviolates (see the second and third columns in Figure 20)Thisobservation is similar with that from 1205741 term Further we plotthe optical conductivity for the case of the lower bound ie1205742 = minus13 ( = 0) which is shown in Figure 20 We can see


02 04 06 08 10u

minus001

001002003004

02 04 06 08 10u

minus03

minus02

minus01

01

02 04 06 08 10u

minus004minus003minus002minus001

001

02 04 06 08 10u

01

02

03

=0=05=15

=0=05=15

=0=05=15

=0=05=15

2=124

2=124

2=minus13

2=minus13

V1 t(u)V1 t(u)

V1y(u)V1y(u)

Figure 16 The potentials 1198811119905(119906) (plots above) and 1198811119910(119906) (plots below) for sample 1205742 and

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

2=124

2=124

2=minus13

2=minus13n1y n1y

n1 t n1 t

Figure 17 Both 1119905 and 1119910 as the function for sample 1205742

that the Drude-like peak at low frequency becomes sharperBut in the allowed region of 1205742 we cannot see a pseudogap-like behavior at low frequency

When the homogenous disorder is introduced the resultsare similar with that from 1205741 higher derivative term that is

(i) for = 2radic3 the optical conductivity of the originaltheory is almost the same as that of the dual one when

the sign of 1205742 changes (see the second columns inFigures 21 and 22)

(ii) for other values of the relation (15) also approx-imately holds (the first and the third columns inFigures 21 and 22)

C2 e Properties of QNMs First of all we work out theQNMs from 1205742 higher derivative term without homogeneous


=5

=5=1

=1 =2=2

=10

=10

=23 =23

0 2 4 6 8 10 12minus30minus25minus20minus15minus10minus5


0

0 2 4 6 8 10 12minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

0

0 2 4 6 8 10 12

0

0 2 4 6 8 10 12

0

Im

Im

Im

Im

2=124 2=minus13

2=124 2=minus13

=5=1 =2

=10

=23

=5

=1

=2

=10

=23

Figure 18 The dominate QNMs of the gauge modes (plots above) and the dual modes (plots below) for sample 1205742 and

00 02 04 06 08 100990

0995

1000

1005

1010

00 02 04 06 08 10090

095

100

105

110

00 05 10 15 20 25 30080085090095100105110115120

00 02 04 06 08 10minus0010

minus0005

0000

0005

0010

00 02 04 06 08 10minus010

minus005

000

005

010

00 05 10 15 20 25 30minus02

minus01

00

01

02

ReRe

lowast

ReRe

lowast

ReRe

lowast

ImIm

lowast

ImIm

lowast

ImIm

lowast

=0 =0 =0

=0 =0 =0

2=00012=minus0001

2=00012=minus0001

2=0022=minus002

2=0022=minus002

2=1242=minus124

2=1242=minus124

Figure 19 The real part (the plot above) and the imaginary part (the plot below) of the optical conductivity as the function of for = 0and various values of 1205742 The solid curves are the conductivity 120590 of the original EM theory (4) and the dashed curves display the conductivity120590lowast of the EM dual theory (7)

disorder (Figure 23) In the region of 1205742 isin [minus124 124] theresult is closely similar with that from 1205741 term (see Figure 4)that is

(i) the qualitative correspondence between the poles ofRe120590( 1205742) and the ones of Re120590lowast( minus1205742) holds well atlow frequency

(ii) in the region of large frequency there are somemodesemerging at the imaginary frequency axis for negative

1205742 which violates the particle-vortex duality with thechange of the sign of 1205742

When the homogeneous disorder is introduced the mainresults are still similar with that from 1205741 termWe summarizethem as follows

(i) For small ( = 12 see Figure 24) the polestructure is qualitatively similar with that withouthomogeneous disorder (see Figure 23)


ReIm

ReIm

00 05 10 15 20 25 30

minus05

00

05

10

15

=02=minus13

ReRe

lowast

Figure 20The real part and the imaginary part of the optical conductivity as the function of for = 0 and 1205742 = minus13 The solid curves arethe conductivity 120590 of the original EM theory (4) and the dashed curves display the conductivity 120590lowast of the EM dual theory (7)

00 05 10 15 20096098100102104

0 1 2 3 4 5 609990

09995

10000

10005

10010

0 1 2 3 4 5 6090

095

100

105

110

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus00010

minus00005

00000

00005

00010

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast


00 05 10 15 20094096098100102104106

0 1 2 3 4 5 60998

0999

1000

1001

1002

0 2 4 6 8 10085090095100105110115

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus0002

minus0001

0000

0001

0002

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast



Im

Re

Im

Re

Im

Re

Im

Reminus4 minus2 0 2 4

minus4

minus3

minus2

minus1

0

minus6 minus4 minus2 0 minus6 minus4 minus2 02 4 6minus7minus6minus5minus4minus3minus2minus10

2 4 6minus7minus6minus5minus4minus3minus2minus10

minus4 minus2 0 2 4minus4

minus3

minus2

minus1

0

Figure 23 QNMs (blue spots) of the gauge mode for |1205742| = 124 (the first two panels are for 1205742 = minus124 and the second two panels for1205742 = 124) and = 0 The first and third panels are the QNMs of the gauge mode and the second and forth panels are that of the dual gaugemode

Im

Re

Im

Re

Im

Re

Im



minus4 minus2 0 2 4minus7minus6minus5minus4minus3minus2minus10



Figure 24 QNMs (blue spots) of the gauge mode for |1205742| = 124 (the first two panels are for 1205742 = minus124 and the second two panels for1205742 = 124) and = 12The first and third panels are the QNMs of the gauge mode and the second and forth panels are that of the dual gaugemode

Im

Re

Im

Re

Im

Re

Im



minus10 minus05 00 05 10minus12minus10minus8minus6minus4minus20



Figure 25 QNMs (blue spots) of the gauge mode for |1205742| = 124 (the first two panels are for 1205742 = minus124 and the second two panels for1205742 = 124) and = 2radic3 The first and third panels are the QNMs of the gauge mode and the second and forth panels are that of the dualgauge mode

(ii) With the increase of the correspondence betweenthe poles of Re120590( 1205742) and the ones of Re120590lowast( minus1205742)is seriously violated (see Figure 25)

Data Availability

This manuscript has no associated data

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the Natural Science Foundation ofChina under grant nos 11775036 and 11847313

References

[1] K Damle and S Sachdev ldquoNonzero-temperature transport nearquantum critical pointsrdquo Physical Review B Condensed Matterand Materials Physics vol 56 no 14 pp 8714ndash8733 1997

[2] S Sachdev Quantum Phase Transitions Cambridge UniversityPress England UK 2nd edition 2011

[3] J M Maldacena ldquoThe large N limit of superconformal fieldtheories and supergravityrdquo Advances in Theoretical and Math-ematical Physics vol 2 no 2 pp 231-252 1998

[4] S S Gubser I R Klebanov and A M Polyakov ldquoGauge theorycorrelators from non-critical string theoryrdquo Physics Letters Bvol 428 no 1-2 pp 105ndash114 1998

[5] E Witten ldquoAnti de Sitter space and holographyrdquo Advances ineoretical and Mathematical Physics vol 2 no 2 pp 253ndash2911998

[6] O Aharony S S Gubser J Maldacena H Ooguri and YOz ldquoLarge N field theories string theory and gravityrdquo PhysicsReports vol 323 no 3-4 pp 183ndash386 2000

[7] R C Myers S Sachdev and A Singh ldquoHolographic quantumcritical transport without self-dualityrdquo Physical Review D vol83 Article ID 066017 2011

[8] S Sachdev ldquoWhat can gauge-gravity duality teachus about con-densed matter physicsrdquo Annual Review of Condensed MatterPhysics vol 3 no 1 pp 9ndash33 2012

[9] S A Hartnoll A Lucas and S Sachdev Holographic quantummatter MIT Press Cambridge Mass USA 2018


[10] A Ritz and J Ward ldquoWeyl corrections to holographic conduc-tivityrdquo Physical Review D vol 79 Article ID 066003 2009

[11] W Witczak-Krempa and S Sachdev ldquoThe quasi-normal modesof quantum criticalityrdquo Physical Review B vol 86 Article ID235115 2012

[12] W Witczak-Krempa and S Sachdev ldquoDispersing quasinormalmodes in (2+1)-dimensional conformal field theoriesrdquo PhysicalReview B Condensed Matter and Materials Physics vol 87 no15 Article ID 155149 2013

[13] W Witczak-Krempa E S Soslashrensen and S Sachdev ldquoThedynamics of quantum criticality revealed by quantum MonteCarlo and holographyrdquo Nature Physics vol 10 no 5 pp 361ndash366 2014

[14] W Witczak-Krempa ldquoQuantum critical charge response fromhigher derivatives in holographyrdquo Physical Review B CondensedMatter and Materials Physics vol 89 no 16 Article ID 1611142014

[15] E Katz S Sachdev E S Soslashrensen and W Witczak-KrempaldquoConformal field theories at nonzero temperature Operatorproduct expansions Monte Carlo and holographyrdquo PhysicalReview B Condensed Matter and Materials Physics vol 90 no24 Article ID 245109 2014

[16] S Bai and D Pang ldquoHolographic charge transport in 2+1dimensions at finite Nrdquo International Journal of Modern PhysicsA vol 29 no 11n12 p 1450061 2014

[17] R C Myers T Sierens and W Witczak-Krempa ldquoA holo-graphic model for quantum critical responsesrdquo Journal of HighEnergy Physics vol 2016 no 05 article 073 2016

[18] A Lucas T Sierens and W Witczak-Krempa ldquoQuantumcritical response from conformal perturbation theory to holog-raphyrdquo Journal of High Energy Physics vol 2017 no 07 article149 2017

[19] C P Herzog P Kovtun S Sachdev and D T Son ldquoQuantumcritical transport duality and M-theoryrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 75 no 8Article ID 085020 2007

[20] J P Wu ldquoHolographic quantum critical conductivity fromhigher derivative electrodynamicsrdquo Physics Letters B vol 785p 296 2018

[21] J Wu ldquoHolographic quantum critical response from 6 deriva-tive theoryrdquo Physics Letters B vol 793 pp 348ndash353 2019

[22] A Donos and S A Hartnoll ldquoInteraction-driven localization inholographyrdquo Nature Physics vol 9 no 10 pp 649ndash655 2013

[23] A Donos and J P Gauntlett ldquoHolographic Q-latticesrdquo Journalof High Energy Physics vol 2014 no 40 2014

[24] A Donos and J P Gauntlett ldquoNovel metals and insulators fromholographyrdquo Journal of High Energy Physics vol 2014 no 6article 007 2014

[25] D Vegh ldquoHolography without translational symmetryrdquo httpsarxivorgabs13010537

[26] M Blake and D Tong ldquoUniversal resistivity from holographicmassive gravityrdquoPhysical Review D Particles Fields Gravitationand Cosmology vol 88 no 10 Article ID 106004 2013

[27] S Grozdanov A Lucas S Sachdev and K Schalm ldquoAbsenceof disorder-driven metal-insulator transitions in simple holo-graphic modelsrdquo Physical Review Letters vol 115 no 22 ArticleID 221601 2015

[28] Y Ling P Liu C Niu and J P Wu ldquoBuilding a doped Mottsystem by holographyrdquo Physical Review D vol 92 no 8 ArticleID 086003 2015

[29] Y Ling P Liu and J Wu ldquoA novel insulator by holographic Q-latticesrdquo Journal ofHigh Energy Physics vol 2016 Article ID 0752016

[30] XMKuang and J PWu ldquoThermal transport and quasi-normalmodes in GaussCBonnet-axions theoryrdquo Physics Letters B vol770 pp 117ndash123 2017

[31] X M Kuang J P Wu and Z Zhou ldquoHolographic transportsfrom Born-Infeld electrodynamics with momentum dissipa-tionrdquo httpsarxivorgabs180507904v2

[32] T Andrade and B Withers ldquoA simple holographic model ofmomentum relaxationrdquo Journal of High Energy Physics vol2014 no 5 article 101 2014

[33] J P Wu X M Kuang and G Fu ldquoMomentum dissipation andholographic transport without self-dualityrdquo European PhysicalJournal C vol 78 no 8 p 616 2018

[34] G Fu J P Wu B Xu and J Liu ldquoHolographic response fromhigher derivatives with homogeneous disorderrdquo Physics LettersB vol 769 p 569 2017

[35] J P Wu ldquoTransport phenomena and Weyl correction in effec-tive holographic theory of momentum dissipationrdquo EuropeanPhysical Journal C vol 78 no 4 p 292 2018

[36] J Wu and P Liu ldquoQuasi-normal modes of holographic systemwith Weyl correction and momentum dissipationrdquo PhysicsLetters B vol 780 pp 616ndash621 2018

[37] A Jansen ldquoOverdamped modes in Schwarzschild-de Sitterand a Mathematica package for the numerical computation ofquasinormal modesrdquo e European Physical Journal Plus vol132 no 12 p 546 2017

[38] T Faulkner H Liu J McGreevy and D Vegh ldquoEmergentquantum criticality Fermi surfaces and AdS2rdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 83 no 12Article ID 125002 2011

[39] Y Ling P Liu J P Wu and Z Zhou ldquoHolographic metal-insulator transition in higher derivative gravityrdquo Physics LettersB vol 766 p 41 2017

[40] W Li P Liu and JWu ldquoWeyl corrections to diffusion and chaosin holographyrdquo Journal of High Energy Physics vol 2018 ArticleID 115 2018

[41] S Mahapatra ldquoThermodynamics phase transition and quasi-normal modes with Weyl correctionsrdquo Journal of High EnergyPhysics vol 2016 no 4 Article ID 142 pp 1ndash33 2016

[42] A Dey S Mahapatra and T Sarkar ldquoThermodynamics andentanglement entropy with Weyl correctionsrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 94 no 2Article ID 026006 2016

[43] A Dey S Mahapatra and T Sarkar ldquoHolographic thermaliza-tion with Weyl correctionsrdquo Journal of High Energy Physics vol2016 no 1 Article ID 088 2016

[44] AMokhtari S AHMansoori andK B Fadafan ldquoDiffusivitiesbounds in the presence of Weyl correctionsrdquo Physics Letters Bvol 785 pp 591ndash604 2018

[45] J P Wu B Xu and G Fu ldquoHolographic fermionic spectrumwithWeyl correctionrdquoModern Physics Letters A vol 34 no 06Article ID 1950045 2019

[46] J PWu Y Cao XM Kuang andW J Li ldquoThe 3+1 holographicsuperconductor with Weyl correctionsrdquo Physics Letters B vol697 no 2 pp 153ndash158 2011

[47] D Z Ma Y Cao and J P Wu ldquoThe Stuckelberg holographicsuperconductors with Weyl correctionsrdquo Physics Letters B vol704 no 5 pp 604ndash611 2011


[48] J PWu and P Liu ldquoHolographic superconductivity fromhigherderivative theoryrdquoPhysics Letters BPhysics Letters B vol 774 pp527ndash532 2017

[49] Y Ling and X Zheng ldquoHolographic superconductor withmomentum relaxation andWeyl correctionrdquoNuclear Physics Bvol 917 pp 1ndash18 2017

[50] D Momeni andM R Setare ldquoA note on holographic supercon-ductors with Weyl correctionsrdquo Modern Physics Letters A vol26 no 38 pp 2889ndash2898 2011

[51] D Momeni N Majd and R Myrzakulov ldquop-Wave holographicsuperconductors with Weyl correctionsrdquo EPL (Europhysics Let-ters) vol 97 no 6 Article ID 61001 2012

[52] Z Zhao Q Pan and J Jing ldquoHolographic insulatorsupercon-ductor phase transition with Weyl correctionsrdquo Physics LettersB vol 719 no 4-5 pp 440ndash447 2013

[53] D Momeni R Myrzakulov and M Raza ldquoHolographic super-conductors with Weyl Corrections via gaugegravity dualityrdquoInternational Journal of Modern Physics A vol 28 no 19 ArticleID 1350096 2013

[54] D Momeni M Raza and R Myrzakulov ldquoHolographic super-conductors with Weyl correctionsrdquo International Journal ofGeometric Methods in Modern Physics vol 13 no 01 p 15501312016

[55] L Zhang Q Pan and J Jing ldquoHolographic p-wave supercon-ductormodelswithWeyl correctionsrdquoPhysics Letters B vol 743pp 104ndash111 2015

[56] A Buchel and R C Myers ldquoCausality of holographic hydrody-namicsrdquo Journal of High Energy Physics vol 2009 no 8 article16 2009

[57] M Brigante H Liu R C Myers S Shenker and S YaidaldquoViscosity bound and causality violationrdquo Physical ReviewLetters vol 100 Article ID 191601 2008

[58] R C Myers A O Starinets and R MThomson ldquoHolographicspectral functions and diffusion constants for fundamentalmatterrdquo Journal of High Energy Physics A SISSA Journal vol2007 no 11 article 091 2007

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effects from homogeneous disorder on these systems [33ndash36] We find that for 4 derivative theory studied in [33] thehomogeneous disorder drives the incoherent metallic statewithDS peak into the onewith a dip for 120574 gt 0 and an oppositescenario is found for 120574 lt 0 But for the 6 derivative theoryexplored in [34] the homogeneous disorder cannot make thepeak (gap) transform into its contrary Another interestingphenomena is that for 4 derivative theory there is a specificvalue of = 2radic3 for which the particle-vortex dualitycorresponding to EM duality in bulk almost exactly holdswith the change of the sign of 120574 [33]

On the other hand to study the holographic QC dynam-ics a good method we can use is the quasinormal modes(QNMs) of a gravitational theory on the bulk AdS spacetimeRecently the structure of the QNMs of 46 derivative theoryover SS-AdS has been explored [11 14] In particular theparticle-vortex duality has also been discussed Furtherwhen the homogeneous disorder in 4 derivative theory isintroduced more rich pole structures are exhibited [36] Inthis paper we shall study the EM duality and the QNMs of6 derivative theory with homogeneous disorder Our paperis organized as follows In Section 2 we present a briefintroduction on the 6 derivative theory with homogeneousdisorder EM duality is discussed in Section 3 In Section 4we study the properties of the QNMs A brief discussion ispresented in Section 5 In Appendix A we also give a briefdiscussion on the instabilities of gauge mode by the QNMsAppendixes B and C present the constraint on another 6derivative term and its properties of conductivity and QNMsrespectively

2 Holographic Setup

We consider the following neutral Einstein-axions (EA) the-ory [32]

1198780 = int1198894119909radicminus119892(119877 + 61198712 minus 12 sum119868=119909119910

(120597120601119868)2) (1)

In the above action we have introduced a pair of spatial lineardependent axionic fields 120601119868 = 120572119909119868 with 119868 = 119909 119910 and 120572being a constant which are responsible for dissipating themomentum of the dual boundary field 119871 is the radius of theAdS spacetimes

From the EA action (1) we have a neutral black branesolution [32]

1198891199042 = 11987121199062 (minus119891 (119906) 1198891199052 + 1119891 (119906)1198891199062 + 1198891199092 + 1198891199102) (2)

where

119891 (119906) = (1 minus 119906) 119901 (119906) 119901 (119906) = radic1 + 62 minus 22 minus 12 1199062 + 119906 + 1 (3)

TheAdSboundary is at119906 = 0 and the horizon locates at 119906 = 1 equiv 1205724120587119879 with the Hawking temperature 119879 = 119901(1)4120587

Since the translational symmetry breaks the momentumdissipates But the geometry is homogeneous and so we referto this mechanism as homogeneous disorder and denotesthe strength of disorderTherefore the background describesa specific thermal excited state with homogeneous disorder

Over this background we study the following HD actionfor gauge field coupling with Weyl tensor


where the tensor 119883 is an infinite family of HD terms [14]

119883120583]120588120590 = 119868120583]120588120590 minus 8120574111198712119862120583]120588120590 minus 41198714120574211198622119868120583]120588120590minus 8119871412057422119862120583]120572120573119862120572120573120588120590 minus 41198716120574311198623119868120583]120588120590minus 81198716120574321198622119862120583]120588120590minus 8119871612057433119862120583]12057211205731119862120572112057311205722120573211986212057221205732120588120590 +

(5)

Note that 119868120583]120588120590 = 120575120583120588120575]120590 minus 120575120583120590120575]120588 is an identity matrix and119862119899 = 119862120583]120572112057311198621205721120573112057221205732 119862120572119899minus1120573119899minus1120583] In the above equationsthe factor of 119871 is introduced such that the coupling parame-ters 119892119865 and 120574119894119895 are dimensionless But for later conveniencewe work in units where 119871 = 1 in what follows and set 119892119865 = 1in the numerical calculation When119883120583]120588120590 = 119868120583]120588120590 the theoryreduces to the standard Maxwell theory For convenience wedenote 12057411 = 120574 and 1205742119894 = 120574119894(119894 = 1 2) In the main body ofthis paper ourmain focus is the 1205741 derivative terms (themaineffect of both 1205741 and 1205742 terms is similar But there is still somedifference between them We shall present a brief discussionon the properties of the conductivity and the QNMs from1205742 higher derivative term in Appendixes B and C) Note thatwhen other parameters are turned off 1205741 is confined in theregion 1205741 le 148 over SS-AdS black brane background(In [34] they also study the instabilities of the gauge modeand the causality in CFT in addition to the constraint fromRe120590(120596) ge 0 they confirm that the constraint 1205741 le 148 alsoholds over EA-AdS geometry) [14] And then from the action(4) we can write down the equation of motion (EOM) as

nabla] (119883120583]120588120590119865120588120590) = 0 (6)

As [7] the dual EM theory of (4) can be constructedwhich is


119866120583] equiv 120597120583119861] minus 120597]119861120583 is the dual field strength and 119865 is thecoupling constant which relates 119892119865 as 2119865 equiv 11198922119865 is thedual tensor which satisfies

120583]120588120590 = minus14120576120583]120572120573(119883minus1)12057212057312057412058212057612057412058212058812059012(119883minus1)120583]120588120590119883120588120590120572120573 equiv 119868120583]120572120573(8)



nabla] (120583]120588120590119866120588120590) = 0 (9)


(119883minus1)120583]

120588120590 = 119868120583]120588120590 + 412057411198622119868120583]120588120590 + O (12057421) (10)

120583]120588120590 = (119883minus1)120583]120588120590 + O (12057421) (11)



11986010158401015840119910 + (1198911015840119891 + 119883101584061198836)1198601015840119910 +p221198912 11988321198836119860119910 = 0 (12)


120590 () = 120597119906119860119910 (119906 )119894p119860119910 (119906 ) (13)


120590lowast ( 1205741) = 1120590 ( 1205741) (14)








11986010158401015840119910 + (1198911015840119891 minus 119883101584061198836)1198601015840119910 +p21198912 11988361198832 2119860119910 = 0 (16)



(17)


(18)



00 02 04 06 08 10090

095

100

105

110

00 02 04 06 08 10 12 1400

05

10

15

20

00 02 04 06 08 10 12 1401234567

00 02 04 06 08 10minus010

minus005

000

005

010

00 02 04 06 08 10 12 14minus10

minus05

00

05

10


=0 =0 =0

=0 =0 =0

1=00011=minus0001

1=0011=minus001

1=0021=minus002

1=00011=minus0001

1=0011=minus001

1=0021=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast






1198891199042 = 11199062 (minus119891 (119906) 1198891199052 minus 2119889119905119889119906 + 1198891199092 + 1198891199102) (19)


1198601015840119910 (1198911015840119891 + 2119894p119891 + 119883101584061198836) + 119894p119860119910119883101584021198911198836 + 11986010158401015840119910 = 0 (20)





00 05 10 15 200708091011121314

0 1 2 3 4 5 6

0994099609981000100210041006

0 1 2 3 4 5 60510152025

3530

00 05 10 15 20


0 1 2 4 5 6


0 1 2 33 4 5 6minus05

00

05

10

=23=12 =10

=23=12 =10

1=0021=minus002

1=0021=minus002

1=0021=minus002

1=0021=minus002

1=0021=minus002

1=0021=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast


00 02 04 06 08 10

10

12

14

16

18

20

u

1(

1)2(minus

1)

=23

=12=0 =2

=10











120590 () = 119870Γ minus 119894 (21)





0 2 4 6

minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

0

0 2 4 6

0

0 2 4 6

0

0 2 4 6

0

Im

Im

Im

Im

Re Re

ReRe


0 1 2

00

0 1 2

0 1 2

00

0 1 2

Im

Im

ReRe

ReRe


minus2 minus1


00

00

Im

Im



minus2 minus1


0 5 10 15

0

0 5 10

0

Im

Re

Im


minus8

minus6

minus4

minus2

minus10 minus5

minus6

minus4

minus2



0 2 4 6

0

0 2 4 6 0 5 10

0

0 2 4 6

0

0 2 4 6 0 5 10

0

Im

Im

Im

Im

Re Re Re


minus6 minus2minus4

minus6 minus2minus4

minus6 minus2minus4

minus10

minus8

minus6

minus4

minus2

0

Im

minus10

minus8

minus6

minus4

minus2

minus10 minus5

minus8

minus6

minus4

minus2

minus10

minus8

minus6

minus4

minus2

0

Im

minus10

minus8

minus6

minus4

minus2

minus10 minus5



0 2 4

0

0 2 4 0 5 10 15

0

0 2 4

0

0 2 4

0

0 5 10 15

0

Im

Im

Im

Im

Im

Re Re Re



0

Im



minus15

minus10

minus5




minus15

minus10

minus5









0 001 01 05Γ1 00114 00115 00120 00284Γ2 00114 00115 00122 00370

0 2 4

0



0 10 20 30

0

0 2 4

0

0 2 4 0 10 20 30

0

Im

Im

Im

Im

Re Re Re


minus4 minus2


0

Im



0

Im


minus20

minus15

minus10

minus5

minus15

minus10

minus5


00

01

02

03

04

05

Re

1 2 3 4 5 60

00051015202530

Re

1 2 3 4 5 60


Im

1 2 3 4 5 60

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im







minus10

minus05

00

05

10

Re

1 2 3 4 5 60

00

01

02

03

04

Re

1 2 3 4 5 60

minus08

minus06

minus04

minus02

00

Im

1 2 3 4 5 60

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im

00

01

02

03

04

Re

1 2 3 4 5 60

minus10

minus05

00

05

10

Re

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im

1 2 3 4 5 60

1 2 3 4 5 60

minus08

minus06

minus04

minus02

00

Im
















Appendix








1=minus1

1=minus1

1=minus1

1=minus1

1=minus5

1=minus5

1=minus5

1=minus5

1=minus25

1=minus25

1=minus25

1=minus25

1=minus150

1=minus150

1=minus150

1=minus150

1=minus103

1=minus103

1=minus103

1=minus103

0001

0002

0005

00050010

00500100

Γ

0001

0002

0005

0010

0020

Γ

0001

0002

0005

0010

Γ

=12 =12

=2 =2

=23 =23

Γ

101 102 103

5 times 10minus4


minus02

minus01

00

Im

15 20 25 30 35 4010Reminus1

101 102 103

minus1

=10 =10

101 102 103

minus1

101 102 103

minus1


Im

02 03 04 0501Re


Im

45 50 55 60 65 7040Re


Im

07 08 09 10 1106Re



with



1205900 = 1 minus 41205742 (B4)





05

10

15

20

0 1 2 3 4 5 600051015202530

=0=10=05=23

0

2

0

=23


2=1482=1242=14




1198601015840119905 + 120596 119883511988331198601015840119909 = 0 (B5)

11986010158401015840119905 + 1198831015840311988331198601015840119905 minusp2119891 11988311198833 (119860 119905 + 119860119909) = 0 (B6)

11986010158401015840119909 + (1198911015840119891 + 119883101584051198835)1198601015840119909 +p21198912 11988311198835 (119860 119905 + 119860119909) = 0 (B7)

11986010158401015840119910 + (1198911015840119891 + 119883101584061198836)1198601015840119910 +p21198912 (211988321198836 minus 2119891

11988341198836)119860119910= 0

(B8)


11986010158401015840119905 + (1198911015840119891 minus 119883101584011198831 + 2119883101584031198833)1198601015840119905 + (minus

p2211988311198911198833+ p22119883111989121198835 + 1198911015840119883101584031198911198833 minus

119883101584011198831015840311988311198833 +1198831015840101584031198833 )119860 119905 = 0

(B9)



minus1205972119911120595119894 (119911) + 119881119894 (119906) 120595119894 (119911) = 2120595119894 (119911) (B10)


1198810119905 = 11989111988311198833 1198810119910 = 11989111988331198831

(B12)

1198811119905 = 1198914p211988321 [3119891 (11988310158401)2 minus 21198831 (11989111988310158401)1015840] (B13)

1198811119910 = 1198914p211988321 [minus119891 (11988310158401)2 + 21198831 (11989111988310158401)1015840] (B14)






1119905 = 119868120587 + 12 119868 equiv (119899 minus 12)120587 = int

1199061

1199060

p119891 (119906)radicminus1198811119905 (119906)119889119906(B16)


02 04 06 08 10u

minus05

05101520

02 04 06 08 10u

02

04

06

08

10

02 04 06 08 10u

0204060810

02 04 06 08 10u

02

04

06

08

10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

2=minus12=minus13

2=112 2=124V0 t(u)

V0 t(u)

V0 t(u)

V0 t(u)


02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10




minus13 le 1205742 le 124 (B17)






02 04 06 08 10u

minus001

001002003004

02 04 06 08 10u

minus03

minus02

minus01

01

02 04 06 08 10u


001

02 04 06 08 10u

01

02

03

=0=05=15

=0=05=15

=0=05=15

=0=05=15

2=124

2=124

2=minus13

2=minus13

V1 t(u)V1 t(u)

V1y(u)V1y(u)


0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

2=124

2=124

2=minus13

2=minus13n1y n1y

n1 t n1 t









=5

=5=1

=1 =2=2

=10

=10

=23 =23



0

0 2 4 6 8 10 12minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

0

0 2 4 6 8 10 12

0

0 2 4 6 8 10 12

0

Im

Im

Im

Im

2=124 2=minus13

2=124 2=minus13

=5=1 =2

=10

=23

=5

=1

=2

=10

=23


00 02 04 06 08 100990

0995

1000

1005

1010

00 02 04 06 08 10090

095

100

105

110

00 05 10 15 20 25 30080085090095100105110115120

00 02 04 06 08 10minus0010

minus0005

0000

0005

0010

00 02 04 06 08 10minus010

minus005

000

005

010

00 05 10 15 20 25 30minus02

minus01

00

01

02

ReRe

lowast

ReRe

lowast

ReRe

lowast

ImIm

lowast

ImIm

lowast

ImIm

lowast

=0 =0 =0

=0 =0 =0

2=00012=minus0001

2=00012=minus0001

2=0022=minus002

2=0022=minus002

2=1242=minus124

2=1242=minus124









ReIm

ReIm

00 05 10 15 20 25 30

minus05

00

05

10

15

=02=minus13

ReRe

lowast


00 05 10 15 20096098100102104

0 1 2 3 4 5 609990

09995

10000

10005

10010

0 1 2 3 4 5 6090

095

100

105

110

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus00010

minus00005

00000

00005

00010

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast


00 05 10 15 20094096098100102104106

0 1 2 3 4 5 60998

0999

1000

1001

1002

0 2 4 6 8 10085090095100105110115

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus0002

minus0001

0000

0001

0002

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast



Im

Re

Im

Re

Im

Re

Im


minus4

minus3

minus2

minus1

0




minus3

minus2

minus1

0


Im

Re

Im

Re

Im

Re

Im







Im

Re

Im

Re

Im

Re

Im








Data Availability




Acknowledgments


References
































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery







nabla] (120583]120588120590119866120588120590) = 0 (9)


(119883minus1)120583]

120588120590 = 119868120583]120588120590 + 412057411198622119868120583]120588120590 + O (12057421) (10)

120583]120588120590 = (119883minus1)120583]120588120590 + O (12057421) (11)



11986010158401015840119910 + (1198911015840119891 + 119883101584061198836)1198601015840119910 +p221198912 11988321198836119860119910 = 0 (12)


120590 () = 120597119906119860119910 (119906 )119894p119860119910 (119906 ) (13)


120590lowast ( 1205741) = 1120590 ( 1205741) (14)








11986010158401015840119910 + (1198911015840119891 minus 119883101584061198836)1198601015840119910 +p21198912 11988361198832 2119860119910 = 0 (16)



(17)


(18)



00 02 04 06 08 10090

095

100

105

110

00 02 04 06 08 10 12 1400

05

10

15

20

00 02 04 06 08 10 12 1401234567

00 02 04 06 08 10minus010

minus005

000

005

010

00 02 04 06 08 10 12 14minus10

minus05

00

05

10


=0 =0 =0

=0 =0 =0

1=00011=minus0001

1=0011=minus001

1=0021=minus002

1=00011=minus0001

1=0011=minus001

1=0021=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast






1198891199042 = 11199062 (minus119891 (119906) 1198891199052 minus 2119889119905119889119906 + 1198891199092 + 1198891199102) (19)


1198601015840119910 (1198911015840119891 + 2119894p119891 + 119883101584061198836) + 119894p119860119910119883101584021198911198836 + 11986010158401015840119910 = 0 (20)





00 05 10 15 200708091011121314

0 1 2 3 4 5 6

0994099609981000100210041006

0 1 2 3 4 5 60510152025

3530

00 05 10 15 20


0 1 2 4 5 6


0 1 2 33 4 5 6minus05

00

05

10

=23=12 =10

=23=12 =10

1=0021=minus002

1=0021=minus002

1=0021=minus002

1=0021=minus002

1=0021=minus002

1=0021=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast


00 02 04 06 08 10

10

12

14

16

18

20

u

1(

1)2(minus

1)

=23

=12=0 =2

=10











120590 () = 119870Γ minus 119894 (21)





0 2 4 6

minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

0

0 2 4 6

0

0 2 4 6

0

0 2 4 6

0

Im

Im

Im

Im

Re Re

ReRe


0 1 2

00

0 1 2

0 1 2

00

0 1 2

Im

Im

ReRe

ReRe


minus2 minus1


00

00

Im

Im



minus2 minus1


0 5 10 15

0

0 5 10

0

Im

Re

Im


minus8

minus6

minus4

minus2

minus10 minus5

minus6

minus4

minus2



0 2 4 6

0

0 2 4 6 0 5 10

0

0 2 4 6

0

0 2 4 6 0 5 10

0

Im

Im

Im

Im

Re Re Re


minus6 minus2minus4

minus6 minus2minus4

minus6 minus2minus4

minus10

minus8

minus6

minus4

minus2

0

Im

minus10

minus8

minus6

minus4

minus2

minus10 minus5

minus8

minus6

minus4

minus2

minus10

minus8

minus6

minus4

minus2

0

Im

minus10

minus8

minus6

minus4

minus2

minus10 minus5



0 2 4

0

0 2 4 0 5 10 15

0

0 2 4

0

0 2 4

0

0 5 10 15

0

Im

Im

Im

Im

Im

Re Re Re



0

Im



minus15

minus10

minus5




minus15

minus10

minus5









0 001 01 05Γ1 00114 00115 00120 00284Γ2 00114 00115 00122 00370

0 2 4

0



0 10 20 30

0

0 2 4

0

0 2 4 0 10 20 30

0

Im

Im

Im

Im

Re Re Re


minus4 minus2


0

Im



0

Im


minus20

minus15

minus10

minus5

minus15

minus10

minus5


00

01

02

03

04

05

Re

1 2 3 4 5 60

00051015202530

Re

1 2 3 4 5 60


Im

1 2 3 4 5 60

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im







minus10

minus05

00

05

10

Re

1 2 3 4 5 60

00

01

02

03

04

Re

1 2 3 4 5 60

minus08

minus06

minus04

minus02

00

Im

1 2 3 4 5 60

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im

00

01

02

03

04

Re

1 2 3 4 5 60

minus10

minus05

00

05

10

Re

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im

1 2 3 4 5 60

1 2 3 4 5 60

minus08

minus06

minus04

minus02

00

Im
















Appendix








1=minus1

1=minus1

1=minus1

1=minus1

1=minus5

1=minus5

1=minus5

1=minus5

1=minus25

1=minus25

1=minus25

1=minus25

1=minus150

1=minus150

1=minus150

1=minus150

1=minus103

1=minus103

1=minus103

1=minus103

0001

0002

0005

00050010

00500100

Γ

0001

0002

0005

0010

0020

Γ

0001

0002

0005

0010

Γ

=12 =12

=2 =2

=23 =23

Γ

101 102 103

5 times 10minus4


minus02

minus01

00

Im

15 20 25 30 35 4010Reminus1

101 102 103

minus1

=10 =10

101 102 103

minus1

101 102 103

minus1


Im

02 03 04 0501Re


Im

45 50 55 60 65 7040Re


Im

07 08 09 10 1106Re



with



1205900 = 1 minus 41205742 (B4)





05

10

15

20

0 1 2 3 4 5 600051015202530

=0=10=05=23

0

2

0

=23


2=1482=1242=14




1198601015840119905 + 120596 119883511988331198601015840119909 = 0 (B5)

11986010158401015840119905 + 1198831015840311988331198601015840119905 minusp2119891 11988311198833 (119860 119905 + 119860119909) = 0 (B6)

11986010158401015840119909 + (1198911015840119891 + 119883101584051198835)1198601015840119909 +p21198912 11988311198835 (119860 119905 + 119860119909) = 0 (B7)

11986010158401015840119910 + (1198911015840119891 + 119883101584061198836)1198601015840119910 +p21198912 (211988321198836 minus 2119891

11988341198836)119860119910= 0

(B8)


11986010158401015840119905 + (1198911015840119891 minus 119883101584011198831 + 2119883101584031198833)1198601015840119905 + (minus

p2211988311198911198833+ p22119883111989121198835 + 1198911015840119883101584031198911198833 minus

119883101584011198831015840311988311198833 +1198831015840101584031198833 )119860 119905 = 0

(B9)



minus1205972119911120595119894 (119911) + 119881119894 (119906) 120595119894 (119911) = 2120595119894 (119911) (B10)


1198810119905 = 11989111988311198833 1198810119910 = 11989111988331198831

(B12)

1198811119905 = 1198914p211988321 [3119891 (11988310158401)2 minus 21198831 (11989111988310158401)1015840] (B13)

1198811119910 = 1198914p211988321 [minus119891 (11988310158401)2 + 21198831 (11989111988310158401)1015840] (B14)






1119905 = 119868120587 + 12 119868 equiv (119899 minus 12)120587 = int

1199061

1199060

p119891 (119906)radicminus1198811119905 (119906)119889119906(B16)


02 04 06 08 10u

minus05

05101520

02 04 06 08 10u

02

04

06

08

10

02 04 06 08 10u

0204060810

02 04 06 08 10u

02

04

06

08

10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

2=minus12=minus13

2=112 2=124V0 t(u)

V0 t(u)

V0 t(u)

V0 t(u)


02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10




minus13 le 1205742 le 124 (B17)






02 04 06 08 10u

minus001

001002003004

02 04 06 08 10u

minus03

minus02

minus01

01

02 04 06 08 10u


001

02 04 06 08 10u

01

02

03

=0=05=15

=0=05=15

=0=05=15

=0=05=15

2=124

2=124

2=minus13

2=minus13

V1 t(u)V1 t(u)

V1y(u)V1y(u)


0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

2=124

2=124

2=minus13

2=minus13n1y n1y

n1 t n1 t









=5

=5=1

=1 =2=2

=10

=10

=23 =23



0

0 2 4 6 8 10 12minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

0

0 2 4 6 8 10 12

0

0 2 4 6 8 10 12

0

Im

Im

Im

Im

2=124 2=minus13

2=124 2=minus13

=5=1 =2

=10

=23

=5

=1

=2

=10

=23


00 02 04 06 08 100990

0995

1000

1005

1010

00 02 04 06 08 10090

095

100

105

110

00 05 10 15 20 25 30080085090095100105110115120

00 02 04 06 08 10minus0010

minus0005

0000

0005

0010

00 02 04 06 08 10minus010

minus005

000

005

010

00 05 10 15 20 25 30minus02

minus01

00

01

02

ReRe

lowast

ReRe

lowast

ReRe

lowast

ImIm

lowast

ImIm

lowast

ImIm

lowast

=0 =0 =0

=0 =0 =0

2=00012=minus0001

2=00012=minus0001

2=0022=minus002

2=0022=minus002

2=1242=minus124

2=1242=minus124









ReIm

ReIm

00 05 10 15 20 25 30

minus05

00

05

10

15

=02=minus13

ReRe

lowast


00 05 10 15 20096098100102104

0 1 2 3 4 5 609990

09995

10000

10005

10010

0 1 2 3 4 5 6090

095

100

105

110

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus00010

minus00005

00000

00005

00010

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast


00 05 10 15 20094096098100102104106

0 1 2 3 4 5 60998

0999

1000

1001

1002

0 2 4 6 8 10085090095100105110115

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus0002

minus0001

0000

0001

0002

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast



Im

Re

Im

Re

Im

Re

Im


minus4

minus3

minus2

minus1

0




minus3

minus2

minus1

0


Im

Re

Im

Re

Im

Re

Im







Im

Re

Im

Re

Im

Re

Im








Data Availability




Acknowledgments


References
































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






00 02 04 06 08 10090

095

100

105

110

00 02 04 06 08 10 12 1400

05

10

15

20

00 02 04 06 08 10 12 1401234567

00 02 04 06 08 10minus010

minus005

000

005

010

00 02 04 06 08 10 12 14minus10

minus05

00

05

10


=0 =0 =0

=0 =0 =0

1=00011=minus0001

1=0011=minus001

1=0021=minus002

1=00011=minus0001

1=0011=minus001

1=0021=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast






1198891199042 = 11199062 (minus119891 (119906) 1198891199052 minus 2119889119905119889119906 + 1198891199092 + 1198891199102) (19)


1198601015840119910 (1198911015840119891 + 2119894p119891 + 119883101584061198836) + 119894p119860119910119883101584021198911198836 + 11986010158401015840119910 = 0 (20)





00 05 10 15 200708091011121314

0 1 2 3 4 5 6

0994099609981000100210041006

0 1 2 3 4 5 60510152025

3530

00 05 10 15 20


0 1 2 4 5 6


0 1 2 33 4 5 6minus05

00

05

10

=23=12 =10

=23=12 =10

1=0021=minus002

1=0021=minus002

1=0021=minus002

1=0021=minus002

1=0021=minus002

1=0021=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast


00 02 04 06 08 10

10

12

14

16

18

20

u

1(

1)2(minus

1)

=23

=12=0 =2

=10











120590 () = 119870Γ minus 119894 (21)





0 2 4 6

minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

0

0 2 4 6

0

0 2 4 6

0

0 2 4 6

0

Im

Im

Im

Im

Re Re

ReRe


0 1 2

00

0 1 2

0 1 2

00

0 1 2

Im

Im

ReRe

ReRe


minus2 minus1


00

00

Im

Im



minus2 minus1


0 5 10 15

0

0 5 10

0

Im

Re

Im


minus8

minus6

minus4

minus2

minus10 minus5

minus6

minus4

minus2



0 2 4 6

0

0 2 4 6 0 5 10

0

0 2 4 6

0

0 2 4 6 0 5 10

0

Im

Im

Im

Im

Re Re Re


minus6 minus2minus4

minus6 minus2minus4

minus6 minus2minus4

minus10

minus8

minus6

minus4

minus2

0

Im

minus10

minus8

minus6

minus4

minus2

minus10 minus5

minus8

minus6

minus4

minus2

minus10

minus8

minus6

minus4

minus2

0

Im

minus10

minus8

minus6

minus4

minus2

minus10 minus5



0 2 4

0

0 2 4 0 5 10 15

0

0 2 4

0

0 2 4

0

0 5 10 15

0

Im

Im

Im

Im

Im

Re Re Re



0

Im



minus15

minus10

minus5




minus15

minus10

minus5









0 001 01 05Γ1 00114 00115 00120 00284Γ2 00114 00115 00122 00370

0 2 4

0



0 10 20 30

0

0 2 4

0

0 2 4 0 10 20 30

0

Im

Im

Im

Im

Re Re Re


minus4 minus2


0

Im



0

Im


minus20

minus15

minus10

minus5

minus15

minus10

minus5


00

01

02

03

04

05

Re

1 2 3 4 5 60

00051015202530

Re

1 2 3 4 5 60


Im

1 2 3 4 5 60

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im







minus10

minus05

00

05

10

Re

1 2 3 4 5 60

00

01

02

03

04

Re

1 2 3 4 5 60

minus08

minus06

minus04

minus02

00

Im

1 2 3 4 5 60

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im

00

01

02

03

04

Re

1 2 3 4 5 60

minus10

minus05

00

05

10

Re

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im

1 2 3 4 5 60

1 2 3 4 5 60

minus08

minus06

minus04

minus02

00

Im
















Appendix








1=minus1

1=minus1

1=minus1

1=minus1

1=minus5

1=minus5

1=minus5

1=minus5

1=minus25

1=minus25

1=minus25

1=minus25

1=minus150

1=minus150

1=minus150

1=minus150

1=minus103

1=minus103

1=minus103

1=minus103

0001

0002

0005

00050010

00500100

Γ

0001

0002

0005

0010

0020

Γ

0001

0002

0005

0010

Γ

=12 =12

=2 =2

=23 =23

Γ

101 102 103

5 times 10minus4


minus02

minus01

00

Im

15 20 25 30 35 4010Reminus1

101 102 103

minus1

=10 =10

101 102 103

minus1

101 102 103

minus1


Im

02 03 04 0501Re


Im

45 50 55 60 65 7040Re


Im

07 08 09 10 1106Re



with



1205900 = 1 minus 41205742 (B4)





05

10

15

20

0 1 2 3 4 5 600051015202530

=0=10=05=23

0

2

0

=23


2=1482=1242=14




1198601015840119905 + 120596 119883511988331198601015840119909 = 0 (B5)

11986010158401015840119905 + 1198831015840311988331198601015840119905 minusp2119891 11988311198833 (119860 119905 + 119860119909) = 0 (B6)

11986010158401015840119909 + (1198911015840119891 + 119883101584051198835)1198601015840119909 +p21198912 11988311198835 (119860 119905 + 119860119909) = 0 (B7)

11986010158401015840119910 + (1198911015840119891 + 119883101584061198836)1198601015840119910 +p21198912 (211988321198836 minus 2119891

11988341198836)119860119910= 0

(B8)


11986010158401015840119905 + (1198911015840119891 minus 119883101584011198831 + 2119883101584031198833)1198601015840119905 + (minus

p2211988311198911198833+ p22119883111989121198835 + 1198911015840119883101584031198911198833 minus

119883101584011198831015840311988311198833 +1198831015840101584031198833 )119860 119905 = 0

(B9)



minus1205972119911120595119894 (119911) + 119881119894 (119906) 120595119894 (119911) = 2120595119894 (119911) (B10)


1198810119905 = 11989111988311198833 1198810119910 = 11989111988331198831

(B12)

1198811119905 = 1198914p211988321 [3119891 (11988310158401)2 minus 21198831 (11989111988310158401)1015840] (B13)

1198811119910 = 1198914p211988321 [minus119891 (11988310158401)2 + 21198831 (11989111988310158401)1015840] (B14)






1119905 = 119868120587 + 12 119868 equiv (119899 minus 12)120587 = int

1199061

1199060

p119891 (119906)radicminus1198811119905 (119906)119889119906(B16)


02 04 06 08 10u

minus05

05101520

02 04 06 08 10u

02

04

06

08

10

02 04 06 08 10u

0204060810

02 04 06 08 10u

02

04

06

08

10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

2=minus12=minus13

2=112 2=124V0 t(u)

V0 t(u)

V0 t(u)

V0 t(u)


02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10




minus13 le 1205742 le 124 (B17)






02 04 06 08 10u

minus001

001002003004

02 04 06 08 10u

minus03

minus02

minus01

01

02 04 06 08 10u


001

02 04 06 08 10u

01

02

03

=0=05=15

=0=05=15

=0=05=15

=0=05=15

2=124

2=124

2=minus13

2=minus13

V1 t(u)V1 t(u)

V1y(u)V1y(u)


0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

2=124

2=124

2=minus13

2=minus13n1y n1y

n1 t n1 t









=5

=5=1

=1 =2=2

=10

=10

=23 =23



0

0 2 4 6 8 10 12minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

0

0 2 4 6 8 10 12

0

0 2 4 6 8 10 12

0

Im

Im

Im

Im

2=124 2=minus13

2=124 2=minus13

=5=1 =2

=10

=23

=5

=1

=2

=10

=23


00 02 04 06 08 100990

0995

1000

1005

1010

00 02 04 06 08 10090

095

100

105

110

00 05 10 15 20 25 30080085090095100105110115120

00 02 04 06 08 10minus0010

minus0005

0000

0005

0010

00 02 04 06 08 10minus010

minus005

000

005

010

00 05 10 15 20 25 30minus02

minus01

00

01

02

ReRe

lowast

ReRe

lowast

ReRe

lowast

ImIm

lowast

ImIm

lowast

ImIm

lowast

=0 =0 =0

=0 =0 =0

2=00012=minus0001

2=00012=minus0001

2=0022=minus002

2=0022=minus002

2=1242=minus124

2=1242=minus124









ReIm

ReIm

00 05 10 15 20 25 30

minus05

00

05

10

15

=02=minus13

ReRe

lowast


00 05 10 15 20096098100102104

0 1 2 3 4 5 609990

09995

10000

10005

10010

0 1 2 3 4 5 6090

095

100

105

110

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus00010

minus00005

00000

00005

00010

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast


00 05 10 15 20094096098100102104106

0 1 2 3 4 5 60998

0999

1000

1001

1002

0 2 4 6 8 10085090095100105110115

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus0002

minus0001

0000

0001

0002

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast



Im

Re

Im

Re

Im

Re

Im


minus4

minus3

minus2

minus1

0




minus3

minus2

minus1

0


Im

Re

Im

Re

Im

Re

Im







Im

Re

Im

Re

Im

Re

Im








Data Availability




Acknowledgments


References
































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






00 05 10 15 200708091011121314

0 1 2 3 4 5 6

0994099609981000100210041006

0 1 2 3 4 5 60510152025

3530

00 05 10 15 20


0 1 2 4 5 6


0 1 2 33 4 5 6minus05

00

05

10

=23=12 =10

=23=12 =10

1=0021=minus002

1=0021=minus002

1=0021=minus002

1=0021=minus002

1=0021=minus002

1=0021=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast


00 02 04 06 08 10

10

12

14

16

18

20

u

1(

1)2(minus

1)

=23

=12=0 =2

=10











120590 () = 119870Γ minus 119894 (21)





0 2 4 6

minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

0

0 2 4 6

0

0 2 4 6

0

0 2 4 6

0

Im

Im

Im

Im

Re Re

ReRe


0 1 2

00

0 1 2

0 1 2

00

0 1 2

Im

Im

ReRe

ReRe


minus2 minus1


00

00

Im

Im



minus2 minus1


0 5 10 15

0

0 5 10

0

Im

Re

Im


minus8

minus6

minus4

minus2

minus10 minus5

minus6

minus4

minus2



0 2 4 6

0

0 2 4 6 0 5 10

0

0 2 4 6

0

0 2 4 6 0 5 10

0

Im

Im

Im

Im

Re Re Re


minus6 minus2minus4

minus6 minus2minus4

minus6 minus2minus4

minus10

minus8

minus6

minus4

minus2

0

Im

minus10

minus8

minus6

minus4

minus2

minus10 minus5

minus8

minus6

minus4

minus2

minus10

minus8

minus6

minus4

minus2

0

Im

minus10

minus8

minus6

minus4

minus2

minus10 minus5



0 2 4

0

0 2 4 0 5 10 15

0

0 2 4

0

0 2 4

0

0 5 10 15

0

Im

Im

Im

Im

Im

Re Re Re



0

Im



minus15

minus10

minus5




minus15

minus10

minus5









0 001 01 05Γ1 00114 00115 00120 00284Γ2 00114 00115 00122 00370

0 2 4

0



0 10 20 30

0

0 2 4

0

0 2 4 0 10 20 30

0

Im

Im

Im

Im

Re Re Re


minus4 minus2


0

Im



0

Im


minus20

minus15

minus10

minus5

minus15

minus10

minus5


00

01

02

03

04

05

Re

1 2 3 4 5 60

00051015202530

Re

1 2 3 4 5 60


Im

1 2 3 4 5 60

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im







minus10

minus05

00

05

10

Re

1 2 3 4 5 60

00

01

02

03

04

Re

1 2 3 4 5 60

minus08

minus06

minus04

minus02

00

Im

1 2 3 4 5 60

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im

00

01

02

03

04

Re

1 2 3 4 5 60

minus10

minus05

00

05

10

Re

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im

1 2 3 4 5 60

1 2 3 4 5 60

minus08

minus06

minus04

minus02

00

Im
















Appendix








1=minus1

1=minus1

1=minus1

1=minus1

1=minus5

1=minus5

1=minus5

1=minus5

1=minus25

1=minus25

1=minus25

1=minus25

1=minus150

1=minus150

1=minus150

1=minus150

1=minus103

1=minus103

1=minus103

1=minus103

0001

0002

0005

00050010

00500100

Γ

0001

0002

0005

0010

0020

Γ

0001

0002

0005

0010

Γ

=12 =12

=2 =2

=23 =23

Γ

101 102 103

5 times 10minus4


minus02

minus01

00

Im

15 20 25 30 35 4010Reminus1

101 102 103

minus1

=10 =10

101 102 103

minus1

101 102 103

minus1


Im

02 03 04 0501Re


Im

45 50 55 60 65 7040Re


Im

07 08 09 10 1106Re



with



1205900 = 1 minus 41205742 (B4)





05

10

15

20

0 1 2 3 4 5 600051015202530

=0=10=05=23

0

2

0

=23


2=1482=1242=14




1198601015840119905 + 120596 119883511988331198601015840119909 = 0 (B5)

11986010158401015840119905 + 1198831015840311988331198601015840119905 minusp2119891 11988311198833 (119860 119905 + 119860119909) = 0 (B6)

11986010158401015840119909 + (1198911015840119891 + 119883101584051198835)1198601015840119909 +p21198912 11988311198835 (119860 119905 + 119860119909) = 0 (B7)

11986010158401015840119910 + (1198911015840119891 + 119883101584061198836)1198601015840119910 +p21198912 (211988321198836 minus 2119891

11988341198836)119860119910= 0

(B8)


11986010158401015840119905 + (1198911015840119891 minus 119883101584011198831 + 2119883101584031198833)1198601015840119905 + (minus

p2211988311198911198833+ p22119883111989121198835 + 1198911015840119883101584031198911198833 minus

119883101584011198831015840311988311198833 +1198831015840101584031198833 )119860 119905 = 0

(B9)



minus1205972119911120595119894 (119911) + 119881119894 (119906) 120595119894 (119911) = 2120595119894 (119911) (B10)


1198810119905 = 11989111988311198833 1198810119910 = 11989111988331198831

(B12)

1198811119905 = 1198914p211988321 [3119891 (11988310158401)2 minus 21198831 (11989111988310158401)1015840] (B13)

1198811119910 = 1198914p211988321 [minus119891 (11988310158401)2 + 21198831 (11989111988310158401)1015840] (B14)






1119905 = 119868120587 + 12 119868 equiv (119899 minus 12)120587 = int

1199061

1199060

p119891 (119906)radicminus1198811119905 (119906)119889119906(B16)


02 04 06 08 10u

minus05

05101520

02 04 06 08 10u

02

04

06

08

10

02 04 06 08 10u

0204060810

02 04 06 08 10u

02

04

06

08

10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

2=minus12=minus13

2=112 2=124V0 t(u)

V0 t(u)

V0 t(u)

V0 t(u)


02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10




minus13 le 1205742 le 124 (B17)






02 04 06 08 10u

minus001

001002003004

02 04 06 08 10u

minus03

minus02

minus01

01

02 04 06 08 10u


001

02 04 06 08 10u

01

02

03

=0=05=15

=0=05=15

=0=05=15

=0=05=15

2=124

2=124

2=minus13

2=minus13

V1 t(u)V1 t(u)

V1y(u)V1y(u)


0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

2=124

2=124

2=minus13

2=minus13n1y n1y

n1 t n1 t









=5

=5=1

=1 =2=2

=10

=10

=23 =23



0

0 2 4 6 8 10 12minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

0

0 2 4 6 8 10 12

0

0 2 4 6 8 10 12

0

Im

Im

Im

Im

2=124 2=minus13

2=124 2=minus13

=5=1 =2

=10

=23

=5

=1

=2

=10

=23


00 02 04 06 08 100990

0995

1000

1005

1010

00 02 04 06 08 10090

095

100

105

110

00 05 10 15 20 25 30080085090095100105110115120

00 02 04 06 08 10minus0010

minus0005

0000

0005

0010

00 02 04 06 08 10minus010

minus005

000

005

010

00 05 10 15 20 25 30minus02

minus01

00

01

02

ReRe

lowast

ReRe

lowast

ReRe

lowast

ImIm

lowast

ImIm

lowast

ImIm

lowast

=0 =0 =0

=0 =0 =0

2=00012=minus0001

2=00012=minus0001

2=0022=minus002

2=0022=minus002

2=1242=minus124

2=1242=minus124









ReIm

ReIm

00 05 10 15 20 25 30

minus05

00

05

10

15

=02=minus13

ReRe

lowast


00 05 10 15 20096098100102104

0 1 2 3 4 5 609990

09995

10000

10005

10010

0 1 2 3 4 5 6090

095

100

105

110

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus00010

minus00005

00000

00005

00010

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast


00 05 10 15 20094096098100102104106

0 1 2 3 4 5 60998

0999

1000

1001

1002

0 2 4 6 8 10085090095100105110115

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus0002

minus0001

0000

0001

0002

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast



Im

Re

Im

Re

Im

Re

Im


minus4

minus3

minus2

minus1

0




minus3

minus2

minus1

0


Im

Re

Im

Re

Im

Re

Im







Im

Re

Im

Re

Im

Re

Im








Data Availability




Acknowledgments


References
































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery








0 2 4 6

minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

0

0 2 4 6

0

0 2 4 6

0

0 2 4 6

0

Im

Im

Im

Im

Re Re

ReRe


0 1 2

00

0 1 2

0 1 2

00

0 1 2

Im

Im

ReRe

ReRe


minus2 minus1


00

00

Im

Im



minus2 minus1


0 5 10 15

0

0 5 10

0

Im

Re

Im


minus8

minus6

minus4

minus2

minus10 minus5

minus6

minus4

minus2



0 2 4 6

0

0 2 4 6 0 5 10

0

0 2 4 6

0

0 2 4 6 0 5 10

0

Im

Im

Im

Im

Re Re Re


minus6 minus2minus4

minus6 minus2minus4

minus6 minus2minus4

minus10

minus8

minus6

minus4

minus2

0

Im

minus10

minus8

minus6

minus4

minus2

minus10 minus5

minus8

minus6

minus4

minus2

minus10

minus8

minus6

minus4

minus2

0

Im

minus10

minus8

minus6

minus4

minus2

minus10 minus5



0 2 4

0

0 2 4 0 5 10 15

0

0 2 4

0

0 2 4

0

0 5 10 15

0

Im

Im

Im

Im

Im

Re Re Re



0

Im



minus15

minus10

minus5




minus15

minus10

minus5









0 001 01 05Γ1 00114 00115 00120 00284Γ2 00114 00115 00122 00370

0 2 4

0



0 10 20 30

0

0 2 4

0

0 2 4 0 10 20 30

0

Im

Im

Im

Im

Re Re Re


minus4 minus2


0

Im



0

Im


minus20

minus15

minus10

minus5

minus15

minus10

minus5


00

01

02

03

04

05

Re

1 2 3 4 5 60

00051015202530

Re

1 2 3 4 5 60


Im

1 2 3 4 5 60

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im







minus10

minus05

00

05

10

Re

1 2 3 4 5 60

00

01

02

03

04

Re

1 2 3 4 5 60

minus08

minus06

minus04

minus02

00

Im

1 2 3 4 5 60

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im

00

01

02

03

04

Re

1 2 3 4 5 60

minus10

minus05

00

05

10

Re

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im

1 2 3 4 5 60

1 2 3 4 5 60

minus08

minus06

minus04

minus02

00

Im
















Appendix








1=minus1

1=minus1

1=minus1

1=minus1

1=minus5

1=minus5

1=minus5

1=minus5

1=minus25

1=minus25

1=minus25

1=minus25

1=minus150

1=minus150

1=minus150

1=minus150

1=minus103

1=minus103

1=minus103

1=minus103

0001

0002

0005

00050010

00500100

Γ

0001

0002

0005

0010

0020

Γ

0001

0002

0005

0010

Γ

=12 =12

=2 =2

=23 =23

Γ

101 102 103

5 times 10minus4


minus02

minus01

00

Im

15 20 25 30 35 4010Reminus1

101 102 103

minus1

=10 =10

101 102 103

minus1

101 102 103

minus1


Im

02 03 04 0501Re


Im

45 50 55 60 65 7040Re


Im

07 08 09 10 1106Re



with



1205900 = 1 minus 41205742 (B4)





05

10

15

20

0 1 2 3 4 5 600051015202530

=0=10=05=23

0

2

0

=23


2=1482=1242=14




1198601015840119905 + 120596 119883511988331198601015840119909 = 0 (B5)

11986010158401015840119905 + 1198831015840311988331198601015840119905 minusp2119891 11988311198833 (119860 119905 + 119860119909) = 0 (B6)

11986010158401015840119909 + (1198911015840119891 + 119883101584051198835)1198601015840119909 +p21198912 11988311198835 (119860 119905 + 119860119909) = 0 (B7)

11986010158401015840119910 + (1198911015840119891 + 119883101584061198836)1198601015840119910 +p21198912 (211988321198836 minus 2119891

11988341198836)119860119910= 0

(B8)


11986010158401015840119905 + (1198911015840119891 minus 119883101584011198831 + 2119883101584031198833)1198601015840119905 + (minus

p2211988311198911198833+ p22119883111989121198835 + 1198911015840119883101584031198911198833 minus

119883101584011198831015840311988311198833 +1198831015840101584031198833 )119860 119905 = 0

(B9)



minus1205972119911120595119894 (119911) + 119881119894 (119906) 120595119894 (119911) = 2120595119894 (119911) (B10)


1198810119905 = 11989111988311198833 1198810119910 = 11989111988331198831

(B12)

1198811119905 = 1198914p211988321 [3119891 (11988310158401)2 minus 21198831 (11989111988310158401)1015840] (B13)

1198811119910 = 1198914p211988321 [minus119891 (11988310158401)2 + 21198831 (11989111988310158401)1015840] (B14)






1119905 = 119868120587 + 12 119868 equiv (119899 minus 12)120587 = int

1199061

1199060

p119891 (119906)radicminus1198811119905 (119906)119889119906(B16)


02 04 06 08 10u

minus05

05101520

02 04 06 08 10u

02

04

06

08

10

02 04 06 08 10u

0204060810

02 04 06 08 10u

02

04

06

08

10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

2=minus12=minus13

2=112 2=124V0 t(u)

V0 t(u)

V0 t(u)

V0 t(u)


02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10




minus13 le 1205742 le 124 (B17)






02 04 06 08 10u

minus001

001002003004

02 04 06 08 10u

minus03

minus02

minus01

01

02 04 06 08 10u


001

02 04 06 08 10u

01

02

03

=0=05=15

=0=05=15

=0=05=15

=0=05=15

2=124

2=124

2=minus13

2=minus13

V1 t(u)V1 t(u)

V1y(u)V1y(u)


0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

2=124

2=124

2=minus13

2=minus13n1y n1y

n1 t n1 t









=5

=5=1

=1 =2=2

=10

=10

=23 =23



0

0 2 4 6 8 10 12minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

0

0 2 4 6 8 10 12

0

0 2 4 6 8 10 12

0

Im

Im

Im

Im

2=124 2=minus13

2=124 2=minus13

=5=1 =2

=10

=23

=5

=1

=2

=10

=23


00 02 04 06 08 100990

0995

1000

1005

1010

00 02 04 06 08 10090

095

100

105

110

00 05 10 15 20 25 30080085090095100105110115120

00 02 04 06 08 10minus0010

minus0005

0000

0005

0010

00 02 04 06 08 10minus010

minus005

000

005

010

00 05 10 15 20 25 30minus02

minus01

00

01

02

ReRe

lowast

ReRe

lowast

ReRe

lowast

ImIm

lowast

ImIm

lowast

ImIm

lowast

=0 =0 =0

=0 =0 =0

2=00012=minus0001

2=00012=minus0001

2=0022=minus002

2=0022=minus002

2=1242=minus124

2=1242=minus124









ReIm

ReIm

00 05 10 15 20 25 30

minus05

00

05

10

15

=02=minus13

ReRe

lowast


00 05 10 15 20096098100102104

0 1 2 3 4 5 609990

09995

10000

10005

10010

0 1 2 3 4 5 6090

095

100

105

110

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus00010

minus00005

00000

00005

00010

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast


00 05 10 15 20094096098100102104106

0 1 2 3 4 5 60998

0999

1000

1001

1002

0 2 4 6 8 10085090095100105110115

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus0002

minus0001

0000

0001

0002

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast



Im

Re

Im

Re

Im

Re

Im


minus4

minus3

minus2

minus1

0




minus3

minus2

minus1

0


Im

Re

Im

Re

Im

Re

Im







Im

Re

Im

Re

Im

Re

Im








Data Availability




Acknowledgments


References
































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






0 2 4 6

0

0 2 4 6 0 5 10

0

0 2 4 6

0

0 2 4 6 0 5 10

0

Im

Im

Im

Im

Re Re Re


minus6 minus2minus4

minus6 minus2minus4

minus6 minus2minus4

minus10

minus8

minus6

minus4

minus2

0

Im

minus10

minus8

minus6

minus4

minus2

minus10 minus5

minus8

minus6

minus4

minus2

minus10

minus8

minus6

minus4

minus2

0

Im

minus10

minus8

minus6

minus4

minus2

minus10 minus5



0 2 4

0

0 2 4 0 5 10 15

0

0 2 4

0

0 2 4

0

0 5 10 15

0

Im

Im

Im

Im

Im

Re Re Re



0

Im



minus15

minus10

minus5




minus15

minus10

minus5









0 001 01 05Γ1 00114 00115 00120 00284Γ2 00114 00115 00122 00370

0 2 4

0



0 10 20 30

0

0 2 4

0

0 2 4 0 10 20 30

0

Im

Im

Im

Im

Re Re Re


minus4 minus2


0

Im



0

Im


minus20

minus15

minus10

minus5

minus15

minus10

minus5


00

01

02

03

04

05

Re

1 2 3 4 5 60

00051015202530

Re

1 2 3 4 5 60


Im

1 2 3 4 5 60

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im







minus10

minus05

00

05

10

Re

1 2 3 4 5 60

00

01

02

03

04

Re

1 2 3 4 5 60

minus08

minus06

minus04

minus02

00

Im

1 2 3 4 5 60

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im

00

01

02

03

04

Re

1 2 3 4 5 60

minus10

minus05

00

05

10

Re

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im

1 2 3 4 5 60

1 2 3 4 5 60

minus08

minus06

minus04

minus02

00

Im
















Appendix








1=minus1

1=minus1

1=minus1

1=minus1

1=minus5

1=minus5

1=minus5

1=minus5

1=minus25

1=minus25

1=minus25

1=minus25

1=minus150

1=minus150

1=minus150

1=minus150

1=minus103

1=minus103

1=minus103

1=minus103

0001

0002

0005

00050010

00500100

Γ

0001

0002

0005

0010

0020

Γ

0001

0002

0005

0010

Γ

=12 =12

=2 =2

=23 =23

Γ

101 102 103

5 times 10minus4


minus02

minus01

00

Im

15 20 25 30 35 4010Reminus1

101 102 103

minus1

=10 =10

101 102 103

minus1

101 102 103

minus1


Im

02 03 04 0501Re


Im

45 50 55 60 65 7040Re


Im

07 08 09 10 1106Re



with



1205900 = 1 minus 41205742 (B4)





05

10

15

20

0 1 2 3 4 5 600051015202530

=0=10=05=23

0

2

0

=23


2=1482=1242=14




1198601015840119905 + 120596 119883511988331198601015840119909 = 0 (B5)

11986010158401015840119905 + 1198831015840311988331198601015840119905 minusp2119891 11988311198833 (119860 119905 + 119860119909) = 0 (B6)

11986010158401015840119909 + (1198911015840119891 + 119883101584051198835)1198601015840119909 +p21198912 11988311198835 (119860 119905 + 119860119909) = 0 (B7)

11986010158401015840119910 + (1198911015840119891 + 119883101584061198836)1198601015840119910 +p21198912 (211988321198836 minus 2119891

11988341198836)119860119910= 0

(B8)


11986010158401015840119905 + (1198911015840119891 minus 119883101584011198831 + 2119883101584031198833)1198601015840119905 + (minus

p2211988311198911198833+ p22119883111989121198835 + 1198911015840119883101584031198911198833 minus

119883101584011198831015840311988311198833 +1198831015840101584031198833 )119860 119905 = 0

(B9)



minus1205972119911120595119894 (119911) + 119881119894 (119906) 120595119894 (119911) = 2120595119894 (119911) (B10)


1198810119905 = 11989111988311198833 1198810119910 = 11989111988331198831

(B12)

1198811119905 = 1198914p211988321 [3119891 (11988310158401)2 minus 21198831 (11989111988310158401)1015840] (B13)

1198811119910 = 1198914p211988321 [minus119891 (11988310158401)2 + 21198831 (11989111988310158401)1015840] (B14)






1119905 = 119868120587 + 12 119868 equiv (119899 minus 12)120587 = int

1199061

1199060

p119891 (119906)radicminus1198811119905 (119906)119889119906(B16)


02 04 06 08 10u

minus05

05101520

02 04 06 08 10u

02

04

06

08

10

02 04 06 08 10u

0204060810

02 04 06 08 10u

02

04

06

08

10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

2=minus12=minus13

2=112 2=124V0 t(u)

V0 t(u)

V0 t(u)

V0 t(u)


02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10




minus13 le 1205742 le 124 (B17)






02 04 06 08 10u

minus001

001002003004

02 04 06 08 10u

minus03

minus02

minus01

01

02 04 06 08 10u


001

02 04 06 08 10u

01

02

03

=0=05=15

=0=05=15

=0=05=15

=0=05=15

2=124

2=124

2=minus13

2=minus13

V1 t(u)V1 t(u)

V1y(u)V1y(u)


0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

2=124

2=124

2=minus13

2=minus13n1y n1y

n1 t n1 t









=5

=5=1

=1 =2=2

=10

=10

=23 =23



0

0 2 4 6 8 10 12minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

0

0 2 4 6 8 10 12

0

0 2 4 6 8 10 12

0

Im

Im

Im

Im

2=124 2=minus13

2=124 2=minus13

=5=1 =2

=10

=23

=5

=1

=2

=10

=23


00 02 04 06 08 100990

0995

1000

1005

1010

00 02 04 06 08 10090

095

100

105

110

00 05 10 15 20 25 30080085090095100105110115120

00 02 04 06 08 10minus0010

minus0005

0000

0005

0010

00 02 04 06 08 10minus010

minus005

000

005

010

00 05 10 15 20 25 30minus02

minus01

00

01

02

ReRe

lowast

ReRe

lowast

ReRe

lowast

ImIm

lowast

ImIm

lowast

ImIm

lowast

=0 =0 =0

=0 =0 =0

2=00012=minus0001

2=00012=minus0001

2=0022=minus002

2=0022=minus002

2=1242=minus124

2=1242=minus124









ReIm

ReIm

00 05 10 15 20 25 30

minus05

00

05

10

15

=02=minus13

ReRe

lowast


00 05 10 15 20096098100102104

0 1 2 3 4 5 609990

09995

10000

10005

10010

0 1 2 3 4 5 6090

095

100

105

110

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus00010

minus00005

00000

00005

00010

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast


00 05 10 15 20094096098100102104106

0 1 2 3 4 5 60998

0999

1000

1001

1002

0 2 4 6 8 10085090095100105110115

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus0002

minus0001

0000

0001

0002

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast



Im

Re

Im

Re

Im

Re

Im


minus4

minus3

minus2

minus1

0




minus3

minus2

minus1

0


Im

Re

Im

Re

Im

Re

Im







Im

Re

Im

Re

Im

Re

Im








Data Availability




Acknowledgments


References
































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery







0 001 01 05Γ1 00114 00115 00120 00284Γ2 00114 00115 00122 00370

0 2 4

0



0 10 20 30

0

0 2 4

0

0 2 4 0 10 20 30

0

Im

Im

Im

Im

Re Re Re


minus4 minus2


0

Im



0

Im


minus20

minus15

minus10

minus5

minus15

minus10

minus5


00

01

02

03

04

05

Re

1 2 3 4 5 60

00051015202530

Re

1 2 3 4 5 60


Im

1 2 3 4 5 60

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im







minus10

minus05

00

05

10

Re

1 2 3 4 5 60

00

01

02

03

04

Re

1 2 3 4 5 60

minus08

minus06

minus04

minus02

00

Im

1 2 3 4 5 60

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im

00

01

02

03

04

Re

1 2 3 4 5 60

minus10

minus05

00

05

10

Re

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im

1 2 3 4 5 60

1 2 3 4 5 60

minus08

minus06

minus04

minus02

00

Im
















Appendix








1=minus1

1=minus1

1=minus1

1=minus1

1=minus5

1=minus5

1=minus5

1=minus5

1=minus25

1=minus25

1=minus25

1=minus25

1=minus150

1=minus150

1=minus150

1=minus150

1=minus103

1=minus103

1=minus103

1=minus103

0001

0002

0005

00050010

00500100

Γ

0001

0002

0005

0010

0020

Γ

0001

0002

0005

0010

Γ

=12 =12

=2 =2

=23 =23

Γ

101 102 103

5 times 10minus4


minus02

minus01

00

Im

15 20 25 30 35 4010Reminus1

101 102 103

minus1

=10 =10

101 102 103

minus1

101 102 103

minus1


Im

02 03 04 0501Re


Im

45 50 55 60 65 7040Re


Im

07 08 09 10 1106Re



with



1205900 = 1 minus 41205742 (B4)





05

10

15

20

0 1 2 3 4 5 600051015202530

=0=10=05=23

0

2

0

=23


2=1482=1242=14




1198601015840119905 + 120596 119883511988331198601015840119909 = 0 (B5)

11986010158401015840119905 + 1198831015840311988331198601015840119905 minusp2119891 11988311198833 (119860 119905 + 119860119909) = 0 (B6)

11986010158401015840119909 + (1198911015840119891 + 119883101584051198835)1198601015840119909 +p21198912 11988311198835 (119860 119905 + 119860119909) = 0 (B7)

11986010158401015840119910 + (1198911015840119891 + 119883101584061198836)1198601015840119910 +p21198912 (211988321198836 minus 2119891

11988341198836)119860119910= 0

(B8)


11986010158401015840119905 + (1198911015840119891 minus 119883101584011198831 + 2119883101584031198833)1198601015840119905 + (minus

p2211988311198911198833+ p22119883111989121198835 + 1198911015840119883101584031198911198833 minus

119883101584011198831015840311988311198833 +1198831015840101584031198833 )119860 119905 = 0

(B9)



minus1205972119911120595119894 (119911) + 119881119894 (119906) 120595119894 (119911) = 2120595119894 (119911) (B10)


1198810119905 = 11989111988311198833 1198810119910 = 11989111988331198831

(B12)

1198811119905 = 1198914p211988321 [3119891 (11988310158401)2 minus 21198831 (11989111988310158401)1015840] (B13)

1198811119910 = 1198914p211988321 [minus119891 (11988310158401)2 + 21198831 (11989111988310158401)1015840] (B14)






1119905 = 119868120587 + 12 119868 equiv (119899 minus 12)120587 = int

1199061

1199060

p119891 (119906)radicminus1198811119905 (119906)119889119906(B16)


02 04 06 08 10u

minus05

05101520

02 04 06 08 10u

02

04

06

08

10

02 04 06 08 10u

0204060810

02 04 06 08 10u

02

04

06

08

10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

2=minus12=minus13

2=112 2=124V0 t(u)

V0 t(u)

V0 t(u)

V0 t(u)


02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10




minus13 le 1205742 le 124 (B17)






02 04 06 08 10u

minus001

001002003004

02 04 06 08 10u

minus03

minus02

minus01

01

02 04 06 08 10u


001

02 04 06 08 10u

01

02

03

=0=05=15

=0=05=15

=0=05=15

=0=05=15

2=124

2=124

2=minus13

2=minus13

V1 t(u)V1 t(u)

V1y(u)V1y(u)


0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

2=124

2=124

2=minus13

2=minus13n1y n1y

n1 t n1 t









=5

=5=1

=1 =2=2

=10

=10

=23 =23



0

0 2 4 6 8 10 12minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

0

0 2 4 6 8 10 12

0

0 2 4 6 8 10 12

0

Im

Im

Im

Im

2=124 2=minus13

2=124 2=minus13

=5=1 =2

=10

=23

=5

=1

=2

=10

=23


00 02 04 06 08 100990

0995

1000

1005

1010

00 02 04 06 08 10090

095

100

105

110

00 05 10 15 20 25 30080085090095100105110115120

00 02 04 06 08 10minus0010

minus0005

0000

0005

0010

00 02 04 06 08 10minus010

minus005

000

005

010

00 05 10 15 20 25 30minus02

minus01

00

01

02

ReRe

lowast

ReRe

lowast

ReRe

lowast

ImIm

lowast

ImIm

lowast

ImIm

lowast

=0 =0 =0

=0 =0 =0

2=00012=minus0001

2=00012=minus0001

2=0022=minus002

2=0022=minus002

2=1242=minus124

2=1242=minus124









ReIm

ReIm

00 05 10 15 20 25 30

minus05

00

05

10

15

=02=minus13

ReRe

lowast


00 05 10 15 20096098100102104

0 1 2 3 4 5 609990

09995

10000

10005

10010

0 1 2 3 4 5 6090

095

100

105

110

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus00010

minus00005

00000

00005

00010

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast


00 05 10 15 20094096098100102104106

0 1 2 3 4 5 60998

0999

1000

1001

1002

0 2 4 6 8 10085090095100105110115

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus0002

minus0001

0000

0001

0002

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast



Im

Re

Im

Re

Im

Re

Im


minus4

minus3

minus2

minus1

0




minus3

minus2

minus1

0


Im

Re

Im

Re

Im

Re

Im







Im

Re

Im

Re

Im

Re

Im








Data Availability




Acknowledgments


References
































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






minus10

minus05

00

05

10

Re

1 2 3 4 5 60

00

01

02

03

04

Re

1 2 3 4 5 60

minus08

minus06

minus04

minus02

00

Im

1 2 3 4 5 60

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im

00

01

02

03

04

Re

1 2 3 4 5 60

minus10

minus05

00

05

10

Re

1 2 3 4 5 60

minus05

minus04

minus03

minus02

minus01

00

Im

1 2 3 4 5 60

1 2 3 4 5 60

minus08

minus06

minus04

minus02

00

Im
















Appendix








1=minus1

1=minus1

1=minus1

1=minus1

1=minus5

1=minus5

1=minus5

1=minus5

1=minus25

1=minus25

1=minus25

1=minus25

1=minus150

1=minus150

1=minus150

1=minus150

1=minus103

1=minus103

1=minus103

1=minus103

0001

0002

0005

00050010

00500100

Γ

0001

0002

0005

0010

0020

Γ

0001

0002

0005

0010

Γ

=12 =12

=2 =2

=23 =23

Γ

101 102 103

5 times 10minus4


minus02

minus01

00

Im

15 20 25 30 35 4010Reminus1

101 102 103

minus1

=10 =10

101 102 103

minus1

101 102 103

minus1


Im

02 03 04 0501Re


Im

45 50 55 60 65 7040Re


Im

07 08 09 10 1106Re



with



1205900 = 1 minus 41205742 (B4)





05

10

15

20

0 1 2 3 4 5 600051015202530

=0=10=05=23

0

2

0

=23


2=1482=1242=14




1198601015840119905 + 120596 119883511988331198601015840119909 = 0 (B5)

11986010158401015840119905 + 1198831015840311988331198601015840119905 minusp2119891 11988311198833 (119860 119905 + 119860119909) = 0 (B6)

11986010158401015840119909 + (1198911015840119891 + 119883101584051198835)1198601015840119909 +p21198912 11988311198835 (119860 119905 + 119860119909) = 0 (B7)

11986010158401015840119910 + (1198911015840119891 + 119883101584061198836)1198601015840119910 +p21198912 (211988321198836 minus 2119891

11988341198836)119860119910= 0

(B8)


11986010158401015840119905 + (1198911015840119891 minus 119883101584011198831 + 2119883101584031198833)1198601015840119905 + (minus

p2211988311198911198833+ p22119883111989121198835 + 1198911015840119883101584031198911198833 minus

119883101584011198831015840311988311198833 +1198831015840101584031198833 )119860 119905 = 0

(B9)



minus1205972119911120595119894 (119911) + 119881119894 (119906) 120595119894 (119911) = 2120595119894 (119911) (B10)


1198810119905 = 11989111988311198833 1198810119910 = 11989111988331198831

(B12)

1198811119905 = 1198914p211988321 [3119891 (11988310158401)2 minus 21198831 (11989111988310158401)1015840] (B13)

1198811119910 = 1198914p211988321 [minus119891 (11988310158401)2 + 21198831 (11989111988310158401)1015840] (B14)






1119905 = 119868120587 + 12 119868 equiv (119899 minus 12)120587 = int

1199061

1199060

p119891 (119906)radicminus1198811119905 (119906)119889119906(B16)


02 04 06 08 10u

minus05

05101520

02 04 06 08 10u

02

04

06

08

10

02 04 06 08 10u

0204060810

02 04 06 08 10u

02

04

06

08

10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

2=minus12=minus13

2=112 2=124V0 t(u)

V0 t(u)

V0 t(u)

V0 t(u)


02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10




minus13 le 1205742 le 124 (B17)






02 04 06 08 10u

minus001

001002003004

02 04 06 08 10u

minus03

minus02

minus01

01

02 04 06 08 10u


001

02 04 06 08 10u

01

02

03

=0=05=15

=0=05=15

=0=05=15

=0=05=15

2=124

2=124

2=minus13

2=minus13

V1 t(u)V1 t(u)

V1y(u)V1y(u)


0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

2=124

2=124

2=minus13

2=minus13n1y n1y

n1 t n1 t









=5

=5=1

=1 =2=2

=10

=10

=23 =23



0

0 2 4 6 8 10 12minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

0

0 2 4 6 8 10 12

0

0 2 4 6 8 10 12

0

Im

Im

Im

Im

2=124 2=minus13

2=124 2=minus13

=5=1 =2

=10

=23

=5

=1

=2

=10

=23


00 02 04 06 08 100990

0995

1000

1005

1010

00 02 04 06 08 10090

095

100

105

110

00 05 10 15 20 25 30080085090095100105110115120

00 02 04 06 08 10minus0010

minus0005

0000

0005

0010

00 02 04 06 08 10minus010

minus005

000

005

010

00 05 10 15 20 25 30minus02

minus01

00

01

02

ReRe

lowast

ReRe

lowast

ReRe

lowast

ImIm

lowast

ImIm

lowast

ImIm

lowast

=0 =0 =0

=0 =0 =0

2=00012=minus0001

2=00012=minus0001

2=0022=minus002

2=0022=minus002

2=1242=minus124

2=1242=minus124









ReIm

ReIm

00 05 10 15 20 25 30

minus05

00

05

10

15

=02=minus13

ReRe

lowast


00 05 10 15 20096098100102104

0 1 2 3 4 5 609990

09995

10000

10005

10010

0 1 2 3 4 5 6090

095

100

105

110

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus00010

minus00005

00000

00005

00010

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast


00 05 10 15 20094096098100102104106

0 1 2 3 4 5 60998

0999

1000

1001

1002

0 2 4 6 8 10085090095100105110115

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus0002

minus0001

0000

0001

0002

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast



Im

Re

Im

Re

Im

Re

Im


minus4

minus3

minus2

minus1

0




minus3

minus2

minus1

0


Im

Re

Im

Re

Im

Re

Im







Im

Re

Im

Re

Im

Re

Im








Data Availability




Acknowledgments


References
































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery














Appendix








1=minus1

1=minus1

1=minus1

1=minus1

1=minus5

1=minus5

1=minus5

1=minus5

1=minus25

1=minus25

1=minus25

1=minus25

1=minus150

1=minus150

1=minus150

1=minus150

1=minus103

1=minus103

1=minus103

1=minus103

0001

0002

0005

00050010

00500100

Γ

0001

0002

0005

0010

0020

Γ

0001

0002

0005

0010

Γ

=12 =12

=2 =2

=23 =23

Γ

101 102 103

5 times 10minus4


minus02

minus01

00

Im

15 20 25 30 35 4010Reminus1

101 102 103

minus1

=10 =10

101 102 103

minus1

101 102 103

minus1


Im

02 03 04 0501Re


Im

45 50 55 60 65 7040Re


Im

07 08 09 10 1106Re



with



1205900 = 1 minus 41205742 (B4)





05

10

15

20

0 1 2 3 4 5 600051015202530

=0=10=05=23

0

2

0

=23


2=1482=1242=14




1198601015840119905 + 120596 119883511988331198601015840119909 = 0 (B5)

11986010158401015840119905 + 1198831015840311988331198601015840119905 minusp2119891 11988311198833 (119860 119905 + 119860119909) = 0 (B6)

11986010158401015840119909 + (1198911015840119891 + 119883101584051198835)1198601015840119909 +p21198912 11988311198835 (119860 119905 + 119860119909) = 0 (B7)

11986010158401015840119910 + (1198911015840119891 + 119883101584061198836)1198601015840119910 +p21198912 (211988321198836 minus 2119891

11988341198836)119860119910= 0

(B8)


11986010158401015840119905 + (1198911015840119891 minus 119883101584011198831 + 2119883101584031198833)1198601015840119905 + (minus

p2211988311198911198833+ p22119883111989121198835 + 1198911015840119883101584031198911198833 minus

119883101584011198831015840311988311198833 +1198831015840101584031198833 )119860 119905 = 0

(B9)



minus1205972119911120595119894 (119911) + 119881119894 (119906) 120595119894 (119911) = 2120595119894 (119911) (B10)


1198810119905 = 11989111988311198833 1198810119910 = 11989111988331198831

(B12)

1198811119905 = 1198914p211988321 [3119891 (11988310158401)2 minus 21198831 (11989111988310158401)1015840] (B13)

1198811119910 = 1198914p211988321 [minus119891 (11988310158401)2 + 21198831 (11989111988310158401)1015840] (B14)






1119905 = 119868120587 + 12 119868 equiv (119899 minus 12)120587 = int

1199061

1199060

p119891 (119906)radicminus1198811119905 (119906)119889119906(B16)


02 04 06 08 10u

minus05

05101520

02 04 06 08 10u

02

04

06

08

10

02 04 06 08 10u

0204060810

02 04 06 08 10u

02

04

06

08

10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

2=minus12=minus13

2=112 2=124V0 t(u)

V0 t(u)

V0 t(u)

V0 t(u)


02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10




minus13 le 1205742 le 124 (B17)






02 04 06 08 10u

minus001

001002003004

02 04 06 08 10u

minus03

minus02

minus01

01

02 04 06 08 10u


001

02 04 06 08 10u

01

02

03

=0=05=15

=0=05=15

=0=05=15

=0=05=15

2=124

2=124

2=minus13

2=minus13

V1 t(u)V1 t(u)

V1y(u)V1y(u)


0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

2=124

2=124

2=minus13

2=minus13n1y n1y

n1 t n1 t









=5

=5=1

=1 =2=2

=10

=10

=23 =23



0

0 2 4 6 8 10 12minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

0

0 2 4 6 8 10 12

0

0 2 4 6 8 10 12

0

Im

Im

Im

Im

2=124 2=minus13

2=124 2=minus13

=5=1 =2

=10

=23

=5

=1

=2

=10

=23


00 02 04 06 08 100990

0995

1000

1005

1010

00 02 04 06 08 10090

095

100

105

110

00 05 10 15 20 25 30080085090095100105110115120

00 02 04 06 08 10minus0010

minus0005

0000

0005

0010

00 02 04 06 08 10minus010

minus005

000

005

010

00 05 10 15 20 25 30minus02

minus01

00

01

02

ReRe

lowast

ReRe

lowast

ReRe

lowast

ImIm

lowast

ImIm

lowast

ImIm

lowast

=0 =0 =0

=0 =0 =0

2=00012=minus0001

2=00012=minus0001

2=0022=minus002

2=0022=minus002

2=1242=minus124

2=1242=minus124









ReIm

ReIm

00 05 10 15 20 25 30

minus05

00

05

10

15

=02=minus13

ReRe

lowast


00 05 10 15 20096098100102104

0 1 2 3 4 5 609990

09995

10000

10005

10010

0 1 2 3 4 5 6090

095

100

105

110

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus00010

minus00005

00000

00005

00010

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast


00 05 10 15 20094096098100102104106

0 1 2 3 4 5 60998

0999

1000

1001

1002

0 2 4 6 8 10085090095100105110115

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus0002

minus0001

0000

0001

0002

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast



Im

Re

Im

Re

Im

Re

Im


minus4

minus3

minus2

minus1

0




minus3

minus2

minus1

0


Im

Re

Im

Re

Im

Re

Im







Im

Re

Im

Re

Im

Re

Im








Data Availability




Acknowledgments


References
































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






1=minus1

1=minus1

1=minus1

1=minus1

1=minus5

1=minus5

1=minus5

1=minus5

1=minus25

1=minus25

1=minus25

1=minus25

1=minus150

1=minus150

1=minus150

1=minus150

1=minus103

1=minus103

1=minus103

1=minus103

0001

0002

0005

00050010

00500100

Γ

0001

0002

0005

0010

0020

Γ

0001

0002

0005

0010

Γ

=12 =12

=2 =2

=23 =23

Γ

101 102 103

5 times 10minus4


minus02

minus01

00

Im

15 20 25 30 35 4010Reminus1

101 102 103

minus1

=10 =10

101 102 103

minus1

101 102 103

minus1


Im

02 03 04 0501Re


Im

45 50 55 60 65 7040Re


Im

07 08 09 10 1106Re



with



1205900 = 1 minus 41205742 (B4)





05

10

15

20

0 1 2 3 4 5 600051015202530

=0=10=05=23

0

2

0

=23


2=1482=1242=14




1198601015840119905 + 120596 119883511988331198601015840119909 = 0 (B5)

11986010158401015840119905 + 1198831015840311988331198601015840119905 minusp2119891 11988311198833 (119860 119905 + 119860119909) = 0 (B6)

11986010158401015840119909 + (1198911015840119891 + 119883101584051198835)1198601015840119909 +p21198912 11988311198835 (119860 119905 + 119860119909) = 0 (B7)

11986010158401015840119910 + (1198911015840119891 + 119883101584061198836)1198601015840119910 +p21198912 (211988321198836 minus 2119891

11988341198836)119860119910= 0

(B8)


11986010158401015840119905 + (1198911015840119891 minus 119883101584011198831 + 2119883101584031198833)1198601015840119905 + (minus

p2211988311198911198833+ p22119883111989121198835 + 1198911015840119883101584031198911198833 minus

119883101584011198831015840311988311198833 +1198831015840101584031198833 )119860 119905 = 0

(B9)



minus1205972119911120595119894 (119911) + 119881119894 (119906) 120595119894 (119911) = 2120595119894 (119911) (B10)


1198810119905 = 11989111988311198833 1198810119910 = 11989111988331198831

(B12)

1198811119905 = 1198914p211988321 [3119891 (11988310158401)2 minus 21198831 (11989111988310158401)1015840] (B13)

1198811119910 = 1198914p211988321 [minus119891 (11988310158401)2 + 21198831 (11989111988310158401)1015840] (B14)






1119905 = 119868120587 + 12 119868 equiv (119899 minus 12)120587 = int

1199061

1199060

p119891 (119906)radicminus1198811119905 (119906)119889119906(B16)


02 04 06 08 10u

minus05

05101520

02 04 06 08 10u

02

04

06

08

10

02 04 06 08 10u

0204060810

02 04 06 08 10u

02

04

06

08

10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

2=minus12=minus13

2=112 2=124V0 t(u)

V0 t(u)

V0 t(u)

V0 t(u)


02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10




minus13 le 1205742 le 124 (B17)






02 04 06 08 10u

minus001

001002003004

02 04 06 08 10u

minus03

minus02

minus01

01

02 04 06 08 10u


001

02 04 06 08 10u

01

02

03

=0=05=15

=0=05=15

=0=05=15

=0=05=15

2=124

2=124

2=minus13

2=minus13

V1 t(u)V1 t(u)

V1y(u)V1y(u)


0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

2=124

2=124

2=minus13

2=minus13n1y n1y

n1 t n1 t









=5

=5=1

=1 =2=2

=10

=10

=23 =23



0

0 2 4 6 8 10 12minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

0

0 2 4 6 8 10 12

0

0 2 4 6 8 10 12

0

Im

Im

Im

Im

2=124 2=minus13

2=124 2=minus13

=5=1 =2

=10

=23

=5

=1

=2

=10

=23


00 02 04 06 08 100990

0995

1000

1005

1010

00 02 04 06 08 10090

095

100

105

110

00 05 10 15 20 25 30080085090095100105110115120

00 02 04 06 08 10minus0010

minus0005

0000

0005

0010

00 02 04 06 08 10minus010

minus005

000

005

010

00 05 10 15 20 25 30minus02

minus01

00

01

02

ReRe

lowast

ReRe

lowast

ReRe

lowast

ImIm

lowast

ImIm

lowast

ImIm

lowast

=0 =0 =0

=0 =0 =0

2=00012=minus0001

2=00012=minus0001

2=0022=minus002

2=0022=minus002

2=1242=minus124

2=1242=minus124









ReIm

ReIm

00 05 10 15 20 25 30

minus05

00

05

10

15

=02=minus13

ReRe

lowast


00 05 10 15 20096098100102104

0 1 2 3 4 5 609990

09995

10000

10005

10010

0 1 2 3 4 5 6090

095

100

105

110

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus00010

minus00005

00000

00005

00010

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast


00 05 10 15 20094096098100102104106

0 1 2 3 4 5 60998

0999

1000

1001

1002

0 2 4 6 8 10085090095100105110115

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus0002

minus0001

0000

0001

0002

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast



Im

Re

Im

Re

Im

Re

Im


minus4

minus3

minus2

minus1

0




minus3

minus2

minus1

0


Im

Re

Im

Re

Im

Re

Im







Im

Re

Im

Re

Im

Re

Im








Data Availability




Acknowledgments


References
































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery







05

10

15

20

0 1 2 3 4 5 600051015202530

=0=10=05=23

0

2

0

=23


2=1482=1242=14




1198601015840119905 + 120596 119883511988331198601015840119909 = 0 (B5)

11986010158401015840119905 + 1198831015840311988331198601015840119905 minusp2119891 11988311198833 (119860 119905 + 119860119909) = 0 (B6)

11986010158401015840119909 + (1198911015840119891 + 119883101584051198835)1198601015840119909 +p21198912 11988311198835 (119860 119905 + 119860119909) = 0 (B7)

11986010158401015840119910 + (1198911015840119891 + 119883101584061198836)1198601015840119910 +p21198912 (211988321198836 minus 2119891

11988341198836)119860119910= 0

(B8)


11986010158401015840119905 + (1198911015840119891 minus 119883101584011198831 + 2119883101584031198833)1198601015840119905 + (minus

p2211988311198911198833+ p22119883111989121198835 + 1198911015840119883101584031198911198833 minus

119883101584011198831015840311988311198833 +1198831015840101584031198833 )119860 119905 = 0

(B9)



minus1205972119911120595119894 (119911) + 119881119894 (119906) 120595119894 (119911) = 2120595119894 (119911) (B10)


1198810119905 = 11989111988311198833 1198810119910 = 11989111988331198831

(B12)

1198811119905 = 1198914p211988321 [3119891 (11988310158401)2 minus 21198831 (11989111988310158401)1015840] (B13)

1198811119910 = 1198914p211988321 [minus119891 (11988310158401)2 + 21198831 (11989111988310158401)1015840] (B14)






1119905 = 119868120587 + 12 119868 equiv (119899 minus 12)120587 = int

1199061

1199060

p119891 (119906)radicminus1198811119905 (119906)119889119906(B16)


02 04 06 08 10u

minus05

05101520

02 04 06 08 10u

02

04

06

08

10

02 04 06 08 10u

0204060810

02 04 06 08 10u

02

04

06

08

10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

2=minus12=minus13

2=112 2=124V0 t(u)

V0 t(u)

V0 t(u)

V0 t(u)


02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10




minus13 le 1205742 le 124 (B17)






02 04 06 08 10u

minus001

001002003004

02 04 06 08 10u

minus03

minus02

minus01

01

02 04 06 08 10u


001

02 04 06 08 10u

01

02

03

=0=05=15

=0=05=15

=0=05=15

=0=05=15

2=124

2=124

2=minus13

2=minus13

V1 t(u)V1 t(u)

V1y(u)V1y(u)


0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

2=124

2=124

2=minus13

2=minus13n1y n1y

n1 t n1 t









=5

=5=1

=1 =2=2

=10

=10

=23 =23



0

0 2 4 6 8 10 12minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

0

0 2 4 6 8 10 12

0

0 2 4 6 8 10 12

0

Im

Im

Im

Im

2=124 2=minus13

2=124 2=minus13

=5=1 =2

=10

=23

=5

=1

=2

=10

=23


00 02 04 06 08 100990

0995

1000

1005

1010

00 02 04 06 08 10090

095

100

105

110

00 05 10 15 20 25 30080085090095100105110115120

00 02 04 06 08 10minus0010

minus0005

0000

0005

0010

00 02 04 06 08 10minus010

minus005

000

005

010

00 05 10 15 20 25 30minus02

minus01

00

01

02

ReRe

lowast

ReRe

lowast

ReRe

lowast

ImIm

lowast

ImIm

lowast

ImIm

lowast

=0 =0 =0

=0 =0 =0

2=00012=minus0001

2=00012=minus0001

2=0022=minus002

2=0022=minus002

2=1242=minus124

2=1242=minus124









ReIm

ReIm

00 05 10 15 20 25 30

minus05

00

05

10

15

=02=minus13

ReRe

lowast


00 05 10 15 20096098100102104

0 1 2 3 4 5 609990

09995

10000

10005

10010

0 1 2 3 4 5 6090

095

100

105

110

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus00010

minus00005

00000

00005

00010

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast


00 05 10 15 20094096098100102104106

0 1 2 3 4 5 60998

0999

1000

1001

1002

0 2 4 6 8 10085090095100105110115

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus0002

minus0001

0000

0001

0002

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast



Im

Re

Im

Re

Im

Re

Im


minus4

minus3

minus2

minus1

0




minus3

minus2

minus1

0


Im

Re

Im

Re

Im

Re

Im







Im

Re

Im

Re

Im

Re

Im








Data Availability




Acknowledgments


References
































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






02 04 06 08 10u

minus05

05101520

02 04 06 08 10u

02

04

06

08

10

02 04 06 08 10u

0204060810

02 04 06 08 10u

02

04

06

08

10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10

2=minus12=minus13

2=112 2=124V0 t(u)

V0 t(u)

V0 t(u)

V0 t(u)


02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

02 04 06 08 10u

0204060810

=0=510

=2=10

=0=510

=2=10

=0=510

=2=10




minus13 le 1205742 le 124 (B17)






02 04 06 08 10u

minus001

001002003004

02 04 06 08 10u

minus03

minus02

minus01

01

02 04 06 08 10u


001

02 04 06 08 10u

01

02

03

=0=05=15

=0=05=15

=0=05=15

=0=05=15

2=124

2=124

2=minus13

2=minus13

V1 t(u)V1 t(u)

V1y(u)V1y(u)


0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

2=124

2=124

2=minus13

2=minus13n1y n1y

n1 t n1 t









=5

=5=1

=1 =2=2

=10

=10

=23 =23



0

0 2 4 6 8 10 12minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

0

0 2 4 6 8 10 12

0

0 2 4 6 8 10 12

0

Im

Im

Im

Im

2=124 2=minus13

2=124 2=minus13

=5=1 =2

=10

=23

=5

=1

=2

=10

=23


00 02 04 06 08 100990

0995

1000

1005

1010

00 02 04 06 08 10090

095

100

105

110

00 05 10 15 20 25 30080085090095100105110115120

00 02 04 06 08 10minus0010

minus0005

0000

0005

0010

00 02 04 06 08 10minus010

minus005

000

005

010

00 05 10 15 20 25 30minus02

minus01

00

01

02

ReRe

lowast

ReRe

lowast

ReRe

lowast

ImIm

lowast

ImIm

lowast

ImIm

lowast

=0 =0 =0

=0 =0 =0

2=00012=minus0001

2=00012=minus0001

2=0022=minus002

2=0022=minus002

2=1242=minus124

2=1242=minus124









ReIm

ReIm

00 05 10 15 20 25 30

minus05

00

05

10

15

=02=minus13

ReRe

lowast


00 05 10 15 20096098100102104

0 1 2 3 4 5 609990

09995

10000

10005

10010

0 1 2 3 4 5 6090

095

100

105

110

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus00010

minus00005

00000

00005

00010

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast


00 05 10 15 20094096098100102104106

0 1 2 3 4 5 60998

0999

1000

1001

1002

0 2 4 6 8 10085090095100105110115

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus0002

minus0001

0000

0001

0002

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast



Im

Re

Im

Re

Im

Re

Im


minus4

minus3

minus2

minus1

0




minus3

minus2

minus1

0


Im

Re

Im

Re

Im

Re

Im







Im

Re

Im

Re

Im

Re

Im








Data Availability




Acknowledgments


References
































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






02 04 06 08 10u

minus001

001002003004

02 04 06 08 10u

minus03

minus02

minus01

01

02 04 06 08 10u


001

02 04 06 08 10u

01

02

03

=0=05=15

=0=05=15

=0=05=15

=0=05=15

2=124

2=124

2=minus13

2=minus13

V1 t(u)V1 t(u)

V1y(u)V1y(u)


0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

0 2 4 6 8 1000

02

04

06

08

10

2=124

2=124

2=minus13

2=minus13n1y n1y

n1 t n1 t









=5

=5=1

=1 =2=2

=10

=10

=23 =23



0

0 2 4 6 8 10 12minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

0

0 2 4 6 8 10 12

0

0 2 4 6 8 10 12

0

Im

Im

Im

Im

2=124 2=minus13

2=124 2=minus13

=5=1 =2

=10

=23

=5

=1

=2

=10

=23


00 02 04 06 08 100990

0995

1000

1005

1010

00 02 04 06 08 10090

095

100

105

110

00 05 10 15 20 25 30080085090095100105110115120

00 02 04 06 08 10minus0010

minus0005

0000

0005

0010

00 02 04 06 08 10minus010

minus005

000

005

010

00 05 10 15 20 25 30minus02

minus01

00

01

02

ReRe

lowast

ReRe

lowast

ReRe

lowast

ImIm

lowast

ImIm

lowast

ImIm

lowast

=0 =0 =0

=0 =0 =0

2=00012=minus0001

2=00012=minus0001

2=0022=minus002

2=0022=minus002

2=1242=minus124

2=1242=minus124









ReIm

ReIm

00 05 10 15 20 25 30

minus05

00

05

10

15

=02=minus13

ReRe

lowast


00 05 10 15 20096098100102104

0 1 2 3 4 5 609990

09995

10000

10005

10010

0 1 2 3 4 5 6090

095

100

105

110

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus00010

minus00005

00000

00005

00010

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast


00 05 10 15 20094096098100102104106

0 1 2 3 4 5 60998

0999

1000

1001

1002

0 2 4 6 8 10085090095100105110115

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus0002

minus0001

0000

0001

0002

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast



Im

Re

Im

Re

Im

Re

Im


minus4

minus3

minus2

minus1

0




minus3

minus2

minus1

0


Im

Re

Im

Re

Im

Re

Im







Im

Re

Im

Re

Im

Re

Im








Data Availability




Acknowledgments


References
































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






=5

=5=1

=1 =2=2

=10

=10

=23 =23



0

0 2 4 6 8 10 12minus8

minus6

minus4

minus2

minus8

minus6

minus4

minus2

0

0 2 4 6 8 10 12

0

0 2 4 6 8 10 12

0

Im

Im

Im

Im

2=124 2=minus13

2=124 2=minus13

=5=1 =2

=10

=23

=5

=1

=2

=10

=23


00 02 04 06 08 100990

0995

1000

1005

1010

00 02 04 06 08 10090

095

100

105

110

00 05 10 15 20 25 30080085090095100105110115120

00 02 04 06 08 10minus0010

minus0005

0000

0005

0010

00 02 04 06 08 10minus010

minus005

000

005

010

00 05 10 15 20 25 30minus02

minus01

00

01

02

ReRe

lowast

ReRe

lowast

ReRe

lowast

ImIm

lowast

ImIm

lowast

ImIm

lowast

=0 =0 =0

=0 =0 =0

2=00012=minus0001

2=00012=minus0001

2=0022=minus002

2=0022=minus002

2=1242=minus124

2=1242=minus124









ReIm

ReIm

00 05 10 15 20 25 30

minus05

00

05

10

15

=02=minus13

ReRe

lowast


00 05 10 15 20096098100102104

0 1 2 3 4 5 609990

09995

10000

10005

10010

0 1 2 3 4 5 6090

095

100

105

110

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus00010

minus00005

00000

00005

00010

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast


00 05 10 15 20094096098100102104106

0 1 2 3 4 5 60998

0999

1000

1001

1002

0 2 4 6 8 10085090095100105110115

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus0002

minus0001

0000

0001

0002

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast



Im

Re

Im

Re

Im

Re

Im


minus4

minus3

minus2

minus1

0




minus3

minus2

minus1

0


Im

Re

Im

Re

Im

Re

Im







Im

Re

Im

Re

Im

Re

Im








Data Availability




Acknowledgments


References
































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






ReIm

ReIm

00 05 10 15 20 25 30

minus05

00

05

10

15

=02=minus13

ReRe

lowast


00 05 10 15 20096098100102104

0 1 2 3 4 5 609990

09995

10000

10005

10010

0 1 2 3 4 5 6090

095

100

105

110

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus00010

minus00005

00000

00005

00010

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

2=0022=minus002

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast


00 05 10 15 20094096098100102104106

0 1 2 3 4 5 60998

0999

1000

1001

1002

0 2 4 6 8 10085090095100105110115

00 05 10 15 20minus004minus002000002004

0 1 2 3 4 5 6minus0002

minus0001

0000

0001

0002

0 2 4 6 8 10minus010

minus005

000

005

010

=12 =10=23

=12 =10=23

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

2=1242=minus124

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast

ReRe

lowastIm

Im

lowast



Im

Re

Im

Re

Im

Re

Im


minus4

minus3

minus2

minus1

0




minus3

minus2

minus1

0


Im

Re

Im

Re

Im

Re

Im







Im

Re

Im

Re

Im

Re

Im








Data Availability




Acknowledgments


References
































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






Im

Re

Im

Re

Im

Re

Im


minus4

minus3

minus2

minus1

0




minus3

minus2

minus1

0


Im

Re

Im

Re

Im

Re

Im







Im

Re

Im

Re

Im

Re

Im








Data Availability




Acknowledgments


References
































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery



























































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery




















Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery








Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery





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EM Duality and Quasinormal Modes from Higher Derivatives ...downloads.hindawi.com/journals/ahep/2019/5472310.pdf · ReseaArticle EM Duality and Quasinormal Modes from Higher Derivatives