510 Probabilistic Symmetries and Invariance Principlesψ, ψt, 416, 454Qk, Qs, Qprimes, 7, 71, 76, 112, 115, 459fQ+, Q[0,1], QI, 6, 36, 177tildewideQ,tildewideQd, 305, 309R, Ra, Rτ, 14, 35, 258, 460RJ, Symbol IndexAJ, A?J, 362Aτh, aτh, 297A, A1, 25, 422α, 43?, 50, 90, 137, 143, 272αh, α?π, 139, 142f, 353B, 43, 90B(·), 25β, βk, 49f, 90, 143, 272βJk, βJ, BJ, 229?β1, 133C, Cprime, 459CA, Ca,b, 3, 10, 3508 Probabilistic Symmetries and Invariance Principlesconvergent, 90, 110, 155distribution, 5, 15integrable, 83, 202randomization, 49, 205–6sampling, 279topology, 155truncation, 283–5uniqueness, 28, 8Subject Index 507coupling, 164shell σ-field, 308shift, 6, 69, 75, 162–4, 358optional, 6, 70, 75–6, 255-invariant σ-field, 25, 29, 128simplepoint process, 21, 46, 123predictable set, 177–9Skorohod topo506 Probabilistic Symmetries and Invariance Principlessymmetry, 406–7progressively measurable, 255, 261projection, 60, 362pseudo-isotropic, 466p-stable,5,60purely discontinuous, 83, 105–6, 175quadratiSubject Index 505integral, 353, 453L′evy, 4, 44, 158–9Markov, 290, 472Poisson, 52, 55, 112urn sequence, 5, 30mixing random measure, 166momentcomparison, 32, 149, 455estimate, 26existence, 363identity,504 Probabilistic Symmetries and Invariance Principlesintegration by parts, 214–5, 245interval sampling, 41invariantfunction, 312Palm measures, 112, 116set, 338, 440σ-field, 25, 29, 128, 272, 406, 443Subject Index 503distribution, 15Dol′eans, 200moments, 250scaling, 282truncation, 283–5extendedarray, 306, 319, 372, 385invariance, 42measure, 403process, 150, 282f, 387–96representation, 340extreme/a
M.Lang,E.D.Sontag/Automaticatheearlytheoryofhybrid(discrete/continuous)control,andworkedonlearningtheoryappliedtoneuralprocessingsystemsaswellasinfoundationsofanalogcomputing.Startingaround1999,hisworoforderN 1.IfsuchaservomechanismwouldadditionallybeNthorder0-invariant,itstransienterrordynamicswouldbecomeindependentofthecurrentpositionforN 1,ofthecurrentvelocity(N 2),andofthecurrentacceleration(M.Lang,E.D.Sontag/AutomaticaFor systems realizing second-order differential operators,(proper)characteristicinputs,(global)convergenceanddegener-acycanbedefinedsimilarlyasinDefinition11.Lemma 22 (SecoBy Theorem 7, the system is P Ts0U-equivariant with statetransformationsgivenbyR1Ts0U D f 1p V Z ! Zgp2R.Wesetzb.t/D 1 .T0w/.t/.za/andz .t/D 1 .Tpw/.t/.za/.Differentiatingbytime,theODEsforzbandz correM.Lang,E.D.Sontag/AutomaticaConsideraPT0U-equivariantsystem(3)excitedbytheinputuk1；k0.t/ D k1tCk0.Nu0/,withk0；k1 2 R andNu0 2 U.Sinceuk1；k0correspondstotheoutputoftheinputmodule(7)initializedatNp D k50 M.Lang,E.D.Sontag/AutomaticaFig. 2. Relationshipbetweenthecharacteristicmodel(6)ofaPTs0U-equivariantsystem,theinputmodule(7),andthesystemitself(3).Solidarrowsdepictsignaltransduction,anddashedarrow3.2.Example2 multipleinvariancesConsiderthesystem(compareShovaletal.,2011,Figure1c)ddtx1.t/Da.y.t/ y0/ bx1.t/ddtx2.t/Dcx2.t/.x1.t/Cy.t/ y0/ddty.t/Ddu.t/x2.t/ ey.t/；withparametersa；c；d；e 2 R>0 andb 2 R48 M.Lang,E.D.Sontag/AutomaticarespecttoP(inshort,isPTs0U-invariant),ifforallp2RandNz2ZthereexistsaNz02Zsuchthath. .t；Nz；u//Dh. .t；Nz0；t7! pexp.s0t/.u.t////； (4)forallu2Uandt 0.Remark 2. ForlD1,(2)isa
xyFig. 6. Field of baby-Skyrmions with no relative iso-rotation.25(a)xy(b)xyFig. 5. Representation of the baby-Skyrmion field using Manton’s notation of dipole regions.The regions where the field φ1 dominates are drawn in white, while those where φ2 dominates aredran = 7n = 6n = 5n = 4n = 3n = 2n = 0,1 n = 0,1n = 2n = 3n = 4n = 5n = 6n = 7Rigid Rotor Deformed RotorEnergyFig. 4. Comparison of the quantum rotational energy spectra for a baby-Skyrmion with N = 1and1 2 3-0.20.20.40.60.811.2rEnergy(r) /piFig. 3. Plot of the radial energy distribution of a baby-Skyrmion spinning at angular velocities0 (long dashed line), 0.25 (dashed line) and ωmax (solid line), f0.1 0.2 0.31.51.71.81.922.12.2Angular velocitypiEnergy /ωFig. 2. Plots for N = 1 and μ2 = 0.1 of the energies of a baby-Skyrmion spinning at angularvelocity ω ranging from 0.001 to ωmax, in the rigid FIGURES0 0.5 1 1.5 2 2.5 3 3.5r00.511.521 - Cos f(r)Fig. 1. Profile of the function f(r) of a baby-Skyrmion spinning at the angular velocity 0 (longdashed line), 0.25 (dashed line) and ωmax (solid linReferences1. B.M.A.G. Piette, B.J. Schroers, W.J. Zakrzewski, Nucl. Phys. B439, 205 (1995)2. B.M.A.G. Piette, H.J.W. Mueller-Kirsten, D.H. Tchrakian, W.J. Zakrzewski, Phys.Lett. B320, 294 (1994)3. B.ME = 2πN2R2 + 4πμ2R2 (49)which is always higher in energy than the baby-Skyrmion. We see that there is no phasetransition between the two states as R changes values.We thank R. Mackenzie and W.J. ZakrzThis solution has again compact support and finite size and can only exist for RradicalBigμ/|N|≥ 1in order for θ0 to be real. Rradical
Box, G. E. P. (1980). Sampling and Bayes inference in scientic modeling androbustness (with discussion). J. R. Statist. Soc. A 143, 383-430.Box,G.E.P. and Cox, D. R. (1964). An analysis of transformaDerivation of S2Again, we assume that , 0and (and hence ) are known. Then, the log-likelihood function L(；；2；；) of (1.1)-(1.2) with the AR(1) error structure isL(；；2u；；)=const: ；12log j Using equations (9) and (10) of Henderson and Searle (1981), we haveI11；1=(T；V0F；1V)；1；(T；V0F；1V)；1V0F；1；F；1V(T；V0F；1V)；1(F；VT；1V0)；1#:Further algebraic simplications yieldT；V0F；1V =(X)0(X)=^2；；aAppendixDerivation of S1For the purpose of simplicity,weassumethat, 0and (and hence ) are known.As for the tedious computations of the score test under the full parameters , 0,, , 2and Suppose that V1and I11；I12I；122I21in (3.1) computed from the model with mea-surement errors are the same as constants m and m2multiplied respectively by V1and I11； I12I；122I21computed from the corresp3.3 Approximate Score Test for the Box-Cox transformationFinally,we consider the Box-Cox transformation model (BoxandCox, 1964), and itsassociated score statistic. Letyi()=0+ 0xi+ i(i =1；；；；It has the same form as that for testing heteroscedasticity in the classical weightedregression model, but the computations are quite dierent (see the derivation of S1in the appendix and the result iUnder the null hypothesis, the score test statistic asymptotically has a chi-squareddistribution with p1degrees of freedom, where p1is the dimension of 1. The advan-tage of the score test is that the
[16] G. Amelino-Camilia et al., Nature 393 (1998) 763.[17] D.A. Kirzhnits and V.A. Chechin, Yad. Fiz. 15 (1972) 1051.[18] S. Coleman and S.L. Glashow, Phys. Rev. D59, (1999) 116008.[19] F.W. Stecker ais likely to be closely tied to the star formation rate for various reasons.If the GZK effect is confirmed, eq. (1) places a very strong constraint on theLIV parameter δppi. If the pair-production “bishow the HiRes [8], Fly’s Eye [7] and AGASA [6] data.4 These spectra arenormalized at an energy of 3 EeV above which energy the extragalactic cosmicray component is assumed to be dominant and below wh 101001000100001000001e+061e+071e+081e+17 1e+18 1e+19 1e+20 1e+21J(E) x E2 (m-2 sr-1 s-1 eV)Energy (eV)Fig. 6. Predicted spectra for an E?2.6 source spectrum with no evolution (see text)shown 101001000100001000001e+061e+071e+081e+17 1e+18 1e+19 1e+20 1e+21J(E) x E2 (m-2 sr-1 s-1 eV)Energy (eV)Fig. 5. Predicted spectra for an E?2.6 source spectrum with no evolution (see text)shown 101001000100001000001e+061e+071e+081e+17 1e+18 1e+19 1e+20 1e+21J(E) x E2 (m-2 sr-1 s-1 eV)Energy (eV)Fig. 4. Predicted spectra for an E?2.6 source spectrum with source evolution (seetext) sh 101001000100001000001e+061e+071e+081e+17 1e+18 1e+19 1e+20 1e+21J(E) x E2 (m-2 sr-1 s-1 eV)Energy (eV)Fig. 3. Predicted spectra for an E?2.6 source spectrum with source evolution (seetext) shWe first calculate the initial energy Ei(z) at which a proton is created at aredshift z whose observed energy today is E following the methods detailedin Refs. [29] and [30]. We neglect the effect of
[MS2] G. Moore and N. Seiberg, Polynomial equations for rational con-formal field theories, Phys. Lett. B212 (1988), 451–460.[S1] G. Segal, The definition of conformal field theory, in: Differentialge[H11] Y.-Z. Huang, Rigidity and modularity of vertex tensor categories,in preparation.[HL1] Y.-Z. Huang and J. Lepowsky, A theory of tensor products formodule categories for a vertex operator algebra,[FLM] I. B. Frenkel, J. Lepowsky, and A. Meurman, Vertex operator alge-bras and the Monster, Pure and Appl. Math., Vol. 134, AcademicPress, New York, 1988.[H1] Y.-Z. Huang, A theory of tensor productsRemark 7.5 If we replace Conditions 1 and 3 by the conditions that forn < 0, V(n) = 0 and every V-module is C2-cofinite, then the conclusions ofTheorems 7.2 and 7.3 are also true.Remark 7.6 Geometricais inFw1,...,wn?1,Pr( ?Yn(wn,zn?zn+1)wn+1).Thus by Theorem 4.3 again,summationdisplayr∈RFY1,...,Yn?1,?Yn+1((γτ +δ)?L(0)w1,...,(γτ +δ)?L(0)wn?1,(γτ +δ)?L(0)Pr(?Y(wn,zn ?zn+1)wn+1)；z′1,...,z′n?1,z′n+1；τFrom (7.16), (7.17) and (7.20), we obtainCK?1,l,0(u?w1) = CK,l,0(w1 ?u),CK?1,l,0( ?O(W1)) = 0,CK?1,l,0parenleftbiggparenleftBigω? c24 ?rlparenrightBig2?w1parenrightbigg= 0.The same argument as we haveis of the formN(s)summationdisplay1=1summationdisplaym∈NC(s)K,l,m(w1)qr(s)l +mτ ,where for l = 1,...,N(s), r(s)l are larger than the lowest weights of all ir-reducible V-modules. But this happens onlywhere rl for l = 1,...,N are real numbers such that rl1 ? rl2 negationslash∈ Z whenl1 negationslash= l2. From (7.6)–(7.12), we obtainCK,l,0(u?w1) = CK,l,0(w1 ?u), (7.13)CK,l,0( ?O(W1)) = 0, (7.14)CK,l
the time interval between the arrows in the years 1994 and 1996 in Fig. 1.The middle period in Fig. 2 corresponds to the time interval between the1996 arrow and the early 1997 arrow in Fig. 1. The leftmost portion of Fig.2 corresponds to part of the much narrower interval between the two 1997arrows in Fig. 1, This portion is close to the October event and, as in pastfits [1, 2, 3], the statistical fluctuations are larger in this region.Monitoring log-periodic oscillations in the index over the three year periodpreceding the recent 1997 correction clearly indicated the impending ruptureevent. Whether and how these oscillations are related to the time-evolutionof events – e.g. the Asian currency devaluations – currently believed to havecaused this correction is an interesting open problem. Such events may haveprovided the small perturbation which unleashed the rupture signaled by thelog-periodic oscillation pattern.We wish to thank Mircea Pigli for his interest in this work. This workwas supported in part by NSF Grant No. PHY-9123780-A3.References[1] J.A. Feigenbaum and P.G.O. Freund, Int. J. Mod. Phys. B 10, 3737(1996).[2] D. Sornette, A. Johansen and J.-P. Bouchaud, J. Phys.I (France) 6,167(1996).[3] D. Sornette, A. Johansen, cond-mat/9704127.4-6-4-2024683.0 3.5 4.0 4.5 5.0 5.5sln xFigure2:Theperiodicfunctions(lnx),determinedasexplainedinthetext.3Calendar timeS&P500199819971996199519941000900800700600500400Figure 1: Plot of the S&P 500 index. The arrows indicate the troughs of theapparent log-periodic fluctuations.If the discrete-scaling hypothesis applies in this case, the
[17] S. Weinstein: Gravity and Gauge Theory in Philosophy of Science, Vol. 66, No. 3, Supplement. Pro-ceedings of the 1998 Biennial Meetings of the Philosophy of Science Association. Part I: Contributkernel of ?p if k(p) = ω, (3) a non-zero vector Zp and a (positive definite) scalar product h?p in thekernel of Zp in dual space TpM? if k(p) ∈ 0, and (4) an Euclidean scalar product if k(p) ∈ (0,∞).T? A bound for the number of possible parallel Universes minimally similar to ours (evenselected by the antropic principle).? Additionally, for adherents to the Mathematical Universe Hypothesis [14] (o? they are based on minimal symmetries from the experimental viewpoint, expressed in afundamental (reasonably baggage free) sense.2. Gauge invariance is not an a priori imposition for mathematical bea(ii)3 Compatibility of sections and actions: given two sections on σ1,σ2 : U ? M → P, definegU(p) ∈ Gp ? Gm by means of σ2(p) = σ1(p)gU(p), for all p ∈ U； then both, the mapgU : U →Gm andpi?1(U)(?P) →Example 5.1 Put M = (?∞,1),V = R2 and E = M ×V, and construct P as follows. For anyv ∈V!@#{0}, let Bv the (ordered) baseBv = (v,w) univocally determined by: v·w = 0,bardblv bardbl=bardblw bardbl,and Bv (PGI)? implies the existence of a connection in the fiber bundle GM(M,Oω(4,R)), or Galileanconnection. Such a connection ? parallelizes the Leibnizian structure (?? = 0,?h = 0). Galileanconnections aldifference between these two “baggage” interpretations might yield consequences for quantities asthe Lagrangianterm associated to the connection (footnote 9). At any case, Levi-Civita connec
6would signal a violation of Lorentz invariance and not ofCPT symmetry.Acknowledgment: This work was supported in partby US DOE Grant number DE-FG-02-91ER40685 andby FONDECYT-Chile grants 1050114, 10653. Bounds for α (Lo
References[1] J Aczel and J Dhombres. Functional equations in several variables,volume 31 of Encyclopaedia of Mathematics and its Applications. Cam-bridge University Press, Cambridge, 1989.[2] R R Baso by Corollary 46Qp() = (?1)pp+ 1p(?1)pp+ 1p?1P(0；1)p(2?1)= P(0；1)p(2?1)Corollary 48.Qp() =pXs=0(?1)s(2p+ 1?s)!s!(p?s)!(p+ 1?s)!p?sProof. By Lemma 13 and Lemma 14 we haveP(0；1)p(2?1)= (?Proof. We have the radial inner productZ10Lp()Lp0()d = cppp0We thus have polynomials of order p, orthogonal with respect to weightingfunction = 2?1(1?)2?2and hence want the Courant-Hilbert JacDenition 17. The Orthogonal Fourier-Mellon radial polynomial Qpis de-ned to be the polynomial of order p arising out from Gram-Schmidt orthog-onalization of the functions1；；2；:::For normalization, Proof. By Lemma 13 and Lemma 14 we haveP(0；2q+1)p?q(2?1)= (?1)p?qP(2q+1；0)p?q(1?2)= (?1)p?q1(p?q)!p?qX=0p?q(p?q + 2q + 1 + 1)(p?q + 2q + 1 +)(2q + 1 + + 1)(2q + 1 +p?q)1?2?12= (?1)Proof. Follows from Proposition 9 and Corollary 36.Corollary 38.Spq() = (?1)p?qp+q + 1p?qqGp?q(2q + 2；2q + 2；)Proof. With the constant bqpsolved, we have a direct equality.Corollary 39. Spq() = We may hence restrict our attention to q 0.The calculationof the explict form of the pseudo-Zernike radial polynomi-als may be found in Section 4 of [3]. Here we indicate the key ideas. Becauseof thProposition 30. The functions specied in Denition 15 satisfy the require-ments of Denition 12. Furthermore, the resulting set is unique.Proof. Theorem 2 of [3] is cited by [3] as being the method.C
Markram, H., Lu¨bke, J., Frotscher, M. & Sakmann, B. (1997) Regulation ofsynaptic efficacy by coincidence of postsynaptic APs and EPSPs. Science,275, 213–215.Mel, B.W. (1997) SEEMORE: combining color,could provide the foundation for the thresholded competitive learningused in our model.We use a ‘max norm’ to integrate the input from the middle layercells onto the top layer cells. Whereas some studby cells with simple cell-like bottom-up connections and recurrentconnections from other simple cells. However, their study does notaddress how the necessary recurrent connectivity could be acquired.Iespecially the middle layer cells favour a sparse code, although itremains to be shown, that their sparseness is indeed optimized by thegiven learning rule.Besides sparseness, temporal slowness (equattheir learning did not require temporal continuity, while top layercells no longer exhibited complex cell type properties. In the q–r-diagrams of the top layer cells after 450 simulated hours shown inconnections to a top cell all share the same preferred orientation butdiffer in position. This indicates that each top layer cell codes for onespecific orientation regardless of position and thus exhiIn order to control for the influence of network size and the choseninput, we simulated networks with 30 or 120 instead of 60 neurons inthe middle layer. Furthermore we trained the network with inputdapproaches 1.0 for perfectly orientated structures. We observed amean bar-ness of 0.16, thus only a small percentage of the naturallyobtained stimuli were dominated by orientated structures (
(M. Levy et. al. eds.) Plenum Press, New York 1980, p687； R.N. Mohapatra and G.Senjanovic. Phys. Rev. Lett. 44, 912 (1980).[15] A.Ilakovac and A.Pilaftisis, Nucl. Phys. B437 (1995) 491[16] Tai-Pei Che[2] T. Huber et al., Phys. Rev. D 41, 2709 (1990)； B. Matthias et al., Phys. Rev. Lett. 66,2716 (1991)； R. Abela et al., Phys. Rev. Lett. 77, 1950 (1996).[3] L. Willmann et al, Phys. Rev. Lett 82, 49 Appendix: Proof of Identity (22)Using the definition of the mixing matrix VaA = summationtext3c=1(A?1L )acUcA, one can write6summationdisplayA=1VaAVbAmνA =6summationdisplayA=1????????3summationdisplayCase2 : |ReG ¯MM| ～ G2F M4WM2R lnMRMW , A = 4, 5, 6, B = 4, 5, 6Case3 : |ReG ¯MM| ～ G2F M6WM4R lnMRMW , A = 1, 2, 3, B = 4, 5, 6 (50)Case 1 and case 2 give the same order MR dependence, while case 3 iMM ¯M = radicalbig, M ¯MM = radicalbig(44)Since the neutrino sector is expected to be CP violating, these will be independent, complexmatrix elements. If the neutrino sector conserves CP, with |M > anwithL(xA, xB) = 4 ? xAxB2(xA ? 1)+ xA(2xB ? xAxB ? 4)2(xA ? 1)2ln xA (38)The T-matrix element of graph (c) is thus secured asTc = G2F M2W8π2 [¯µ(3)γµ(1 ?γ5)e(2)][¯µ(4)γµ(1 ?γ5)e(1)]·bracketleftBig 6sua3Tc3 =6summationdisplayA=16summationdisplayB=1integraldisplay ∞0dtB(xA, xB,t)·parenleftBig?t· 1(t + ξ)2 + 1t + ξparenrightBiga4Tc4 =6summationdisplayA=16summationdisplayB=1integraldisplay ∞0dtB(xA, xa1νBW WνAeµeµa1νBW φνAeµeµ(c1) (c2)a1νBφ WνAeµeµa1νBφ φνAeµeµ(c3) (c4)Fig.3 Feynman graphs o
De nition 7.2: (KLL stability with one measure) Let ! :K ! R?0 be continuous. The set of hybrid trajectories Sis said to be KLL-stable with respect to ! if it is forwardcomplete on K and there exists for some set U ‰ O s
20 E. P. RYANREFERENCES[1] J. P. Aubin and A. Cellina, Dierential Inclusions, Springer-Verlag, New York, 1984.[2] C. I. Byrnes and C. F. Martin, An integral-invariance principle for nonlinear systemsAN INTEGRAL INVARIANCE PRINCIPLE AND ADAPTIVE CONTROL 19Proof. Let Fcand Vc, parameterized by c > 0, be dened as in the proof ofLemma 3.10. By an argument essentially the same as that adopted in the 18 E. P. RYANAssumption H. (i) g : R RN! RNis locally Lipschitz； (ii) system (18)is input-to-state stable (ISS)； (iii) h : R RN! R is continuous； (iv) there exista function 0: R+! R+and a scalar 1AN INTEGRAL INVARIANCE PRINCIPLE AND ADAPTIVE CONTROL 17that, for all (p；r；v；w) ~p2~K K [?R；R]3,j~f(~p；e)j+ kD1~f(~p；e)k jf(p；e+r)j+jvj+ jwj+ kDf(p；e+r)k++1K(je+rj)+(1+)(1+R)KR(jej)+(116 E. P. RYANTherefore, by Theorem 2.10, x() approaches the largest weakly-invariant (relativeto the autonomous dierential inclusion (39)) set W in f(y；z；)j y = 0g. Let w =(0；z；) 2W. By denitiAN INTEGRAL INVARIANCE PRINCIPLE AND ADAPTIVE CONTROL 15for almost all t2 [0；!), wherein we have used the fact that?(d(t)+)y(t)z(t) jy(t)z(t)j z2(t)+24y2(t) z2(t)+24(jy(t)j)jy(t)j:with 14 E. P. RYANX is upper semicontinuousonRR3and takes non-empty,convex and compact valuesin R3. Therefore, for each x02 R3, the initial-value problem (28) has a solution andevery solution can be extenAN INTEGRAL INVARIANCE PRINCIPLE AND ADAPTIVE CONTROL 13constant (but unknown) natural dampingd quantied by a known parameter in thesense that d 0 and let : R+! R+be a continuous,positive-den
thors mainly consider nonminimal coupling.13 The kinetic energy K must not be confused with the "timelike energy" p0which isnot conserved except in the trivial case of a static universe.In the particu5 N.D.Birrell and P.C.W. Davies, Quantum elds in curved space, Cambridge UniversityPress, (1982).6 In the physical literature, they are often called "Weinberg two-point functions" andintroduced as exSince @0ij= 0, we notice that@0(ij@iXj) = ij@0@iXjhence the contraction@0(ij@iXj)+ij@j@iX0=_SSij@jXi(A14)Insert (A13) into (A14)； we nally get(S?_S2S)X0+2qij@j@iX0= 0As the second term doesnot depenOn the other hand we haveriX0= @iX0??ij0XjAccording to (A5) we getriX0= @iX0?_S2SXi(A9)The sum of (A8)(A9) and condition that r(iX0)vanishes give@0Xi+@iX0=_SSXi(A10)Then, we must express that r(iXj)vaThe question is whether X0may be dierent from zero. Christoel symbols are as follows[18]?000= ?00i= ?0i0= 0?i0j=12@0gij(A3)?0ij=12gik@0gkj(A4)But gij= ?S?1ij, hencegik@0gkj=_SSijand nally?0ij=_S2Sand "out" asymptotic conditions.In contradistinction, there is no conceptual discrepancy between the interpretation pro-posed here and various eorts made in the past [5] and recently [8][17] in orderwhole theory of free elds； this point is important because isometric invariance plays incurved spacetime the same role as Poincare invariance in special Relativity.We have left aside the special castwo-dimensional, and corresponds to the possibility of performing an arbitrary Bogoliubovtransformation. Indeed the expansion (20) of D+in terms of the eigenfunctions of
[39] E. J. Williams. Cauchy-distributed functions and a characterization of the Cauchydistribution. The Annals of Mathematical Statistics, 40:1083 { 1085, 1969.17[25] T. Nakata and M. Nakamura. On the Julia sets of rational functions of degree twowith two real parameters. Hiroshima Math. J., 26(2):253{275, 1996.[26] J. H. Neuwirth. Ergodicity of some mappings [13] M. E. H. Ismail and J. Pitman. Algebraic evaluations of some Euler integrals,duplication formulae for Appell''s hypergeometric function F1, and Brownian vari-ations. Technical Report 554, Dept. StReferences[1] J. Aaronson. Ergodic theory for inner functions of the upper half plane. Ann. Inst.Henri Poincare, 14:233{253, 1978.[2] J. Aaronson. An introduction to innite ergodic theory. American Formula (37) can be veried in another way as follows. By integration with respectto h(x)dx, formula (37) is equivalent to the following identity: for all non-negativemeasurable functions hs2E"ZX0dxxwhere X and Y are independent standard Gaussian. But this is the well known resultof Levy[20] that the distribution of 1=X2is stable with index12. The same argumentyields directly the following multiwhich is veried by observing that both sides vanish at = 1 and have the samederivative with respect to at each > 0. Alternatively, (26) can be checked as follows,using the Cauchy representation5 Some integral identitiesLet (Bt；t 0) denote a standard one-dimensional Brownian motion. Let''(z) :=1p2e?12z2；(x) :=Z1x''(z)dz = P(B1> x):According to formula (13) of [29], the following identity nd that f is continuous
?505?4?20240123yξzFigure 4: Three dimensional view of the trajectory inside the invariance region G. Thesimulation begins at and ends at 130 1 2 3 4 5 60123z0 1 2 3 4 5 6?1?0.50ξ0 1 2 3 4 5 6012ytFigure 2: Response under Invariance Control Showing States , y, and z.0 0.02 0.04 0.06 0.08?4?20x 10?3Φ0 0.02 0.04 0.06 0.08510α0 0.02 0.04 0.[10] R. Sepulchre, M. Jankovic, and P. Kokotovic, Constructive Nonlinear Control. London:Springer{Verlag, 1 ed., 1997.[11] Spong, M.W., and Praly, L., !@#Control of Underactuated Mechanical Systems Usof the gains. It appears, at least in this example, that the switching controller automaticallyadjusts the output gain to nd a nonpeaking controller.One of the additional advantages of Invariance CThe relative degree and minimum phase assumptions are satised by any positive gains k1,k2. We next dene W(z) =12z2and set = W(z) +122+12y2?12r2(46)to dene the candidate invariance region G = fzj Corollary 2 Suppose the system_z = f(z；0；0) (35)is GAS. Let W(z) be a Lyapunov function for (35) and dene 0according to (21). Suppose,for every C > 0 there exists k such that (A；b；kT) is minimum phapositively invariant for t > T, hence the trajectory will not intersect the boundary, @G, fort > T and no further switching will occur. Therefore, the sequence iis nite and the proofis complete.Coroi) (t) is a piecewise constant, monotonically increasing sequence.ii) (t) is eventually constant, i.e., the control switches only nitely many times.iii) the Invariance Region G is positively invariThe function (z；；y) is called an Invariance Fun