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H[1].264_MPEG-4 Part 10 White Paper

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H.264 / MPEG-4 Part 10 : Overview
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An input frame Fn is presented for encoding. The frame is processed in units of a macroblock (corresponding to 16x16 pixels in the original image). Each macroblock is encoded in intra or inter mode. In either case, a prediction macroblock P is formed based on a reconstructed frame. In Intra mode, P is formed from samples in the current frame n that have previously encoded, decoded and reconstructed (uF’n in the Figures; note that the unfiltered samples are used to form P). In Inter mode, P is formed by motion-compensated prediction from one or more reference frame(s). In the Figures, the reference frame is shown as the previous encoded frame F’n-1 ; however, the predicton for each macroblock may be formed from one or two past or future frames (in time order) that have already been encoded and reconstructed. The prediction P is subtracted from the current macroblock to produce a residual or difference macroblock Dn. This is transformed (using a block transform) and quantized to give X, a set of quantized transform coefficients. These coefficients are re-ordered and entropy encoded. The entropyencoded coefficients, together with side information required to decode the macroblock (such as the macroblock prediction mode, quantizer step size, motion vector information describing how the macroblock was motion-compensated, etc) form the compressed bitstream. This is passed to a Network Abstraction Layer (NAL) for transmission or storage. 2.2 Encoder (reconstruction path)

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Singularity of the density of states in the two-dimensional Hubbard model from finite size

a r X i v :c o n d -m a t /9503139v 1 27 M a r 1995Singularity of the density of states in the two-dimensional Hubbard model from ﬁnitesize scaling of Yang-Lee zerosE.Abraham 1,I.M.Barbour 2,P.H.Cullen 1,E.G.Klepﬁsh 3,E.R.Pike 3and Sarben Sarkar 31Department of Physics,Heriot-Watt University,Edinburgh EH144AS,UK 2Department of Physics,University of Glasgow,Glasgow G128QQ,UK 3Department of Physics,King’s College London,London WC2R 2LS,UK(February 6,2008)A ﬁnite size scaling is applied to the Yang-Lee zeros of the grand canonical partition function for the 2-D Hubbard model in the complex chemical potential plane.The logarithmic scaling of the imaginary part of the zeros with the system size indicates a singular dependence of the carrier density on the chemical potential.Our analysis points to a second-order phase transition with critical exponent 12±1transition controlled by the chemical potential.As in order-disorder transitions,one would expect a symmetry breaking signalled by an order parameter.In this model,the particle-hole symmetry is broken by introducing an “external ﬁeld”which causes the particle density to be-come non-zero.Furthermore,the possibility of the free energy having a singularity at some ﬁnite value of the chemical potential is not excluded:in fact it can be a transition indicated by a divergence of the correlation length.A singularity of the free energy at ﬁnite “exter-nal ﬁeld”was found in ﬁnite-temperature lattice QCD by using theYang-Leeanalysisforthechiral phase tran-sition [14].A possible scenario for such a transition at ﬁnite chemical potential,is one in which the particle den-sity consists of two components derived from the regular and singular parts of the free energy.Since we are dealing with a grand canonical ensemble,the particle number can be calculated for a given chem-ical potential as opposed to constraining the chemical potential by a ﬁxed particle number.Hence the chem-ical potential can be thought of as an external ﬁeld for exploring the behaviour of the free energy.From the mi-croscopic point of view,the critical values of the chemical potential are associated with singularities of the density of states.Transitions related to the singularity of the density of states are known as Lifshitz transitions [15].In metals these transitions only take place at zero tem-perature,while at ﬁnite temperatures the singularities are rounded.However,for a small ratio of temperature to the deviation from the critical values of the chemical potential,the singularity can be traced even at ﬁnite tem-perature.Lifshitz transitions may result from topological changes of the Fermi surface,and may occur inside the Brillouin zone as well as on its boundaries [16].In the case of strongly correlated electron systems the shape of the Fermi surface is indeed aﬀected,which in turn may lead to an extension of the Lifshitz-type singularities into the ﬁnite-temperature regime.In relating the macroscopic quantity of the carrier den-sity to the density of quasiparticle states,we assumed the validity of a single particle excitation picture.Whether strong correlations completely distort this description is beyond the scope of the current study.However,the iden-tiﬁcation of the criticality using the Yang-Lee analysis,remains valid even if collective excitations prevail.The paper is organised as follows.In Section 2we out-line the essentials of the computational technique used to simulate the grand canonical partition function and present its expansion as a polynomial in the fugacity vari-able.In Section 3we present the Yang-Lee zeros of the partition function calculated on 62–102lattices and high-light their qualitative diﬀerences from the 42lattice.In Section 4we analyse the ﬁnite size scaling of the Yang-Lee zeros and compare it to the real-space renormaliza-tion group prediction for a second-order phase transition.Finally,in Section 5we present a summary of our resultsand an outlook for future work.II.SIMULATION ALGORITHM AND FUGACITY EXPANSION OF THE GRAND CANONICALPARTITION FUNCTIONThe model we are studying in this work is a two-dimensional single-band Hubbard HamiltonianˆH=−t <i,j>,σc †i,σc j,σ+U i n i +−12 −µi(n i ++n i −)(1)where the i,j denote the nearest neighbour spatial lat-tice sites,σis the spin degree of freedom and n iσis theelectron number operator c †iσc iσ.The constants t and U correspond to the hopping parameter and the on-site Coulomb repulsion respectively.The chemical potential µis introduced such that µ=0corresponds to half-ﬁlling,i.e.the actual chemical potential is shifted from µto µ−U412.(5)This transformation enables one to integrate out the fermionic degrees of freedom and the resulting partition function is written as an ensemble average of a product of two determinantsZ ={s i,l =±1}˜z = {s i,l =±1}det(M +)det(M −)(6)such thatM ±=I +P ± =I +n τ l =1B ±l(7)where the matrices B ±l are deﬁned asB ±l =e −(±dtV )e −dtK e dtµ(8)with V ij =δij s i,l and K ij =1if i,j are nearestneigh-boursand Kij=0otherwise.The matrices in (7)and (8)are of size (n x n y )×(n x n y ),corresponding to the spatial size of the lattice.The expectation value of a physical observable at chemical potential µ,<O >µ,is given by<O >µ=O ˜z (µ){s i,l =±1}˜z (µ,{s i,l })(9)where the sum over the conﬁgurations of Ising ﬁelds isdenoted by an integral.Since ˜z (µ)is not positive deﬁnite for Re(µ)=0we weight the ensemble of conﬁgurations by the absolute value of ˜z (µ)at some µ=µ0.Thus<O >µ= O ˜z (µ)˜z (µ)|˜z (µ0)|µ0|˜z (µ0)|µ0(10)The partition function Z (µ)is given byZ (µ)∝˜z (µ)N c˜z (µ0)|˜z (µ0)|×e µβ+e −µβ−e µ0β−e −µ0βn (16)When the average sign is near unity,it is safe to as-sume that the lattice conﬁgurations reﬂect accurately thequantum degrees of freedom.Following Blankenbecler et al.[1]the diagonal matrix elements of the equal-time Green’s operator G ±=(I +P ±)−1accurately describe the fermion density on a given conﬁguration.In this regime the adiabatic approximation,which is the basis of the ﬁnite-temperature algorithm,is valid.The situa-tion diﬀers strongly when the average sign becomes small.We are in this case sampling positive and negative ˜z (µ0)conﬁgurations with almost equal probability since the ac-ceptance criterion depends only on the absolute value of ˜z (µ0).In the simulations of the HSﬁelds the situation is dif-ferent from the case of fermions interacting with dynam-ical bosonﬁelds presented in Ref.[1].The auxilary HS ﬁelds do not have a kinetic energy term in the bosonic action which would suppress their rapidﬂuctuations and hence recover the adiabaticity.From the previous sim-ulations on a42lattice[3]we know that avoiding the sign problem,by updating at half-ﬁlling,results in high uncontrolledﬂuctuations of the expansion coeﬃcients for the statistical weight,thus severely limiting the range of validity of the expansion.It is therefore important to obtain the partition function for the widest range ofµ0 and observe the persistence of the hierarchy of the ex-pansion coeﬃcients of Z.An error analysis is required to establish the Gaussian distribution of the simulated observables.We present in the following section results of the bootstrap analysis[17]performed on our data for several values ofµ0.III.TEMPERATURE AND LATTICE-SIZEDEPENDENCE OF THE YANG-LEE ZEROS The simulations were performed in the intermediate on-site repulsion regime U=4t forβ=5,6,7.5on lat-tices42,62,82and forβ=5,6on a102lattice.The ex-pansion coeﬃcients given by eqn.(14)are obtained with relatively small errors and exhibit clear Gaussian distri-bution over the ensemble.This behaviour was recorded for a wide range ofµ0which makes our simulations reli-able in spite of the sign problem.In Fig.1(a-c)we present typical distributions of theﬁrst coeﬃcients correspond-ing to n=1−7in eqn.(14)(normalized with respect to the zeroth power coeﬃcient)forβ=5−7.5for diﬀer-entµ0.The coeﬃcients are obtained using the bootstrap method on over10000conﬁgurations forβ=5increasing to over30000forβ=7.5.In spite of diﬀerent values of the average sign in these simulations,the coeﬃcients of the expansion(16)indicate good correspondence between coeﬃcients obtained with diﬀerent values of the update chemical potentialµ0:the normalized coeﬃcients taken from diﬀerentµ0values and equal power of the expansion variable correspond within the statistical error estimated using the bootstrap analysis.(To compare these coeﬃ-cients we had to shift the expansion by2coshµ0β.)We also performed a bootstrap analysis of the zeros in theµplane which shows clear Gaussian distribution of their real and imaginary parts(see Fig.2).In addition, we observe overlapping results(i.e.same zeros)obtained with diﬀerent values ofµ0.The distribution of Yang-Lee zeros in the complexµ-plane is presented in Fig.3(a-c)for the zeros nearest to the real axis.We observe a gradual decrease of the imaginary part as the lattice size increases.The quantitative analysis of this behaviour is discussed in the next section.The critical domain can be identiﬁed by the behaviour of the density of Yang-Lee zeros’in the positive half-plane of the fugacity.We expect toﬁnd that this density is tem-perature and volume dependent as the system approaches the phase transition.If the temperature is much higher than the critical temperature,the zeros stay far from the positive real axis as it happens in the high-temperature limit of the one-dimensional Ising model(T c=0)in which,forβ=0,the points of singularity of the free energy lie at fugacity value−1.As the temperature de-creases we expect the zeros to migrate to the positive half-plane with their density,in this region,increasing with the system’s volume.Figures4(a-c)show the number N(θ)of zeros in the sector(0,θ)as a function of the angleθ.The zeros shown in theseﬁgures are those presented in Fig.3(a-c)in the chemical potential plane with other zeros lying further from the positive real half-axis added in.We included only the zeros having absolute value less than one which we are able to do because if y i is a zero in the fugacity plane,so is1/y i.The errors are shown where they were estimated using the bootstrap analysis(see Fig.2).Forβ=5,even for the largest simulated lattice102, all the zeros are in the negative half-plane.We notice a gradual movement of the pattern of the zeros towards the smallerθvalues with an increasing density of the zeros nearθ=πIV.FINITE SIZE SCALING AND THESINGULARITY OF THE DENSITY OF STATESAs a starting point for theﬁnite size analysis of theYang-Lee singularities we recall the scaling hypothesis forthe partition function singularities in the critical domain[11].Following this hypothesis,for a change of scale ofthe linear dimension LLL→−1),˜µ=(1−µT cδ(23)Following the real-space renormalization group treatmentof Ref.[11]and assuming that the change of scaleλisa continuous parameter,the exponentαθis related tothe critical exponentνof the correlation length asαθ=1ξ(θλ)=ξ(θ)αθwe obtain ξ∼|θ|−1|θ|ναµ)(26)where θλhas been scaled to ±1and ˜µλexpressed in terms of ˜µand θ.Diﬀerentiating this equation with respect to ˜µyields:<n >sing =(−θ)ν(d −αµ)∂F sing (X,Y )ν(d −αµ)singinto the ar-gument Y =˜µαµ(28)which deﬁnes the critical exponent 1αµin terms of the scaling exponent αµof the Yang-Lee zeros.Fig.5presents the scaling of the imaginary part of the µzeros for diﬀerent values of the temperature.The linear regression slope of the logarithm of the imaginary part of the zeros plotted against the logarithm of the inverse lin-ear dimension of the simulation volume,increases when the temperature decreases from β=5to β=6.The re-sults of β=7.5correspond to αµ=1.3within the errors of the zeros as the simulation volume increases from 62to 82.As it is seen from Fig.3,we can trace zeros with similar real part (Re (µ1)≈0.7which is also consistentwith the critical value of the chemical potential given in Ref.[22])as the lattice size increases,which allows us to examine only the scaling of the imaginary part.Table 1presents the values of αµand 1αµδ0.5±0.0560.5±0.21.3±0.3∂µ,as a function ofthe chemical potential on an 82lattice.The location of the peaks of the susceptibility,rounded by the ﬁnite size eﬀects,is in good agreement with the distribution of the real part of the Yang-Lee zeros in the complex µ-plane (see Fig.3)which is particularly evident in the β=7.5simulations (Fig.4(c)).The contribution of each zero to the susceptibility can be singled out by expressing the free energy as:F =2n x n yi =1(y −y i )(29)where y is the fugacity variable and y i is the correspond-ing zero of the partition function.The dotted lines on these plots correspond to the contribution of the nearby zeros while the full polynomial contribution is given by the solid lines.We see that the developing singularities are indeed governed by the zeros closest to the real axis.The sharpening of the singularity as the temperature de-creases is also in accordance with the dependence of the distribution of the zeros on the temperature.The singularities of the free energy and its derivative with respect to the chemical potential,can be related to the quasiparticle density of states.To do this we assume that single particle excitations accurately represent the spectrum of the system.The relationship between the average particle density and the density of states ρ(ω)is given by<n >=∞dω1dµ=ρsing (µ)∝1δ−1(32)and hence the rate of divergence of the density of states.As in the case of Lifshitz transitions the singularity of the particle number is rounded at ﬁnite temperature.However,for suﬃciently low temperatures,the singular-ity of the density of states remains manifest in the free energy,the average particle density,and particle suscep-tibility [15].The regular part of the density of states does not contribute to the criticality,so we can concentrate on the singular part only.Consider a behaviour of the typedensity of states diverging as the−1ρsing(ω)∝(ω−µc)1δ.(33)with the valueδfor the particle number governed by thedivergence of the density of states(at low temperatures)in spite of theﬁnite-temperature rounding of the singu-larity itself.This rounding of the singularity is indeedreﬂected in the diﬀerence between the values ofαµatβ=5andβ=6.V.DISCUSSION AND OUTLOOKWe note that in ourﬁnite size scaling analysis we donot include logarithmic corrections.In particular,thesecorrections may prove signiﬁcant when taking into ac-count the fact that we are dealing with a two-dimensionalsystem in which the pattern of the phase transition islikely to be of Kosterlitz-Thouless type[23].The loga-rithmic corrections to the scaling laws have been provenessential in a recent work of Kenna and Irving[24].In-clusion of these corrections would allow us to obtain thecritical exponents with higher accuracy.However,suchanalysis would require simulations on even larger lattices.The linearﬁts for the logarithmic scaling and the criti-cal exponents obtained,are to be viewed as approximatevalues reﬂecting the general behaviour of the Yang-Leezeros as the temperature and lattice size are varied.Al-though the bootstrap analysis provided us with accurateestimates of the statistical error on the values of the ex-pansion coeﬃcients and the Yang-Lee zeros,the smallnumber of zeros obtained with suﬃcient accuracy doesnot allow us to claim higher precision for the critical ex-ponents on the basis of more elaborateﬁttings of the scal-ing behaviour.Theﬁnite-size eﬀects may still be signiﬁ-cant,especially as the simulation temperature decreases,thus aﬀecting the scaling of the Yang-Lee zeros with thesystem rger lattice simulations will therefore berequired for an accurate evaluation of the critical expo-nent for the particle density and the density of states.Nevertheless,the onset of a singularity atﬁnite temper-ature,and its persistence as the lattice size increases,areevident.The estimate of the critical exponent for the diver-gence rate of the density of states of the quasiparticleexcitation spectrum is particularly relevant to the highT c superconductivity scenario based on the van Hove sin-gularities[25],[26],[27].It is emphasized in Ref.[25]thatthe logarithmic singularity of a two-dimensional electrongas can,due to electronic correlations,turn into a power-law divergence resulting in an extended saddle point atthe lattice momenta(π,0)and(0,π).In the case of the14.I.M.Barbour,A.J.Bell and E.G.Klepﬁsh,Nucl.Phys.B389,285(1993).15.I.M.Lifshitz,JETP38,1569(1960).16.A.A.Abrikosov,Fundamentals of the Theory ofMetals North-Holland(1988).17.P.Hall,The Bootstrap and Edgeworth expansion,Springer(1992).18.S.R.White et al.,Phys.Rev.B40,506(1989).19.J.E.Hirsch,Phys.Rev.B28,4059(1983).20.M.Suzuki,Prog.Theor.Phys.56,1454(1976).21.A.Moreo, D.Scalapino and E.Dagotto,Phys.Rev.B43,11442(1991).22.N.Furukawa and M.Imada,J.Phys.Soc.Japan61,3331(1992).23.J.Kosterlitz and D.Thouless,J.Phys.C6,1181(1973);J.Kosterlitz,J.Phys.C7,1046(1974).24.R.Kenna and A.C.Irving,unpublished.25.K.Gofron et al.,Phys.Rev.Lett.73,3302(1994).26.D.M.Newns,P.C.Pattnaik and C.C.Tsuei,Phys.Rev.B43,3075(1991);D.M.Newns et al.,Phys.Rev.Lett.24,1264(1992);D.M.Newns et al.,Phys.Rev.Lett.73,1264(1994).27.E.Dagotto,A.Nazarenko and A.Moreo,Phys.Rev.Lett.74,310(1995).28.A.A.Abrikosov,J.C.Campuzano and K.Gofron,Physica(Amsterdam)214C,73(1993).29.D.S.Dessau et al.,Phys.Rev.Lett.71,2781(1993);D.M.King et al.,Phys.Rev.Lett.73,3298(1994);P.Aebi et al.,Phys.Rev.Lett.72,2757(1994).30.E.Dagotto, A.Nazarenko and M.Boninsegni,Phys.Rev.Lett.73,728(1994).31.N.Bulut,D.J.Scalapino and S.R.White,Phys.Rev.Lett.73,748(1994).32.S.R.White,Phys.Rev.B44,4670(1991);M.Veki´c and S.R.White,Phys.Rev.B47,1160 (1993).33.C.E.Creﬃeld,E.G.Klepﬁsh,E.R.Pike and SarbenSarkar,unpublished.Figure CaptionsFigure1Bootstrap distribution of normalized coeﬃcients for ex-pansion(14)at diﬀerent update chemical potentialµ0for an82lattice.The corresponding power of expansion is indicated in the topﬁgure.(a)β=5,(b)β=6,(c)β=7.5.Figure2Bootstrap distributions for the Yang-Lee zeros in the complexµplane closest to the real axis.(a)102lat-tice atβ=5,(b)102lattice atβ=6,(c)82lattice at β=7.5.Figure3Yang-Lee zeros in the complexµplane closest to the real axis.(a)β=5,(b)β=6,(c)β=7.5.The correspond-ing lattice size is shown in the top right-hand corner. Figure4Angular distribution of the Yang-Lee zeros in the com-plex fugacity plane Error bars are drawn where esti-mated.(a)β=5,(b)β=6,(c)β=7.5.Figure5Scaling of the imaginary part ofµ1(Re(µ1)≈=0.7)as a function of lattice size.αm u indicates the theﬁt of the logarithmic scaling.Figure6Electronic susceptibility as a function of chemical poten-tial for an82lattice.The solid line represents the con-tribution of all the2n x n y zeros and the dotted line the contribution of the six zeros nearest to the real-µaxis.(a)β=5,(b)β=6,(c)β=7.5.。

Gradient-based learning applied to document recognition

Gradient-Based Learning Appliedto Document RecognitionYANN LECUN,MEMBER,IEEE,L´EON BOTTOU,YOSHUA BENGIO,AND PATRICK HAFFNER Invited PaperMultilayer neural networks trained with the back-propagation algorithm constitute the best example of a successful gradient-based learning technique.Given an appropriate network architecture,gradient-based learning algorithms can be used to synthesize a complex decision surface that can classify high-dimensional patterns,such as handwritten characters,with minimal preprocessing.This paper reviews various methods applied to handwritten character recognition and compares them on a standard handwritten digit recognition task.Convolutional neural networks,which are speciﬁcally designed to deal with the variability of two dimensional(2-D)shapes,are shown to outperform all other techniques.Real-life document recognition systems are composed of multiple modules includingﬁeld extraction,segmentation,recognition, and language modeling.A new learning paradigm,called graph transformer networks(GTN’s),allows such multimodule systems to be trained globally using gradient-based methods so as to minimize an overall performance measure.Two systems for online handwriting recognition are described. Experiments demonstrate the advantage of global training,and theﬂexibility of graph transformer networks.A graph transformer network for reading a bank check is also described.It uses convolutional neural network character recognizers combined with global training techniques to provide record accuracy on business and personal checks.It is deployed commercially and reads several million checks per day. Keywords—Convolutional neural networks,document recog-nition,ﬁnite state transducers,gradient-based learning,graphtransformer networks,machine learning,neural networks,optical character recognition(OCR).N OMENCLATUREGT Graph transformer.GTN Graph transformer network.HMM Hidden Markov model.HOS Heuristic oversegmentation.K-NN K-nearest neighbor.Manuscript received November1,1997;revised April17,1998.Y.LeCun,L.Bottou,and P.Haffner are with the Speech and Image Processing Services Research Laboratory,AT&T Labs-Research,Red Bank,NJ07701USA.Y.Bengio is with the D´e partement d’Informatique et de Recherche Op´e rationelle,Universit´e de Montr´e al,Montr´e al,Qu´e bec H3C3J7Canada. Publisher Item Identiﬁer S0018-9219(98)07863-3.NN Neural network.OCR Optical character recognition.PCA Principal component analysis.RBF Radial basis function.RS-SVM Reduced-set support vector method. SDNN Space displacement neural network.SVM Support vector method.TDNN Time delay neural network.V-SVM Virtual support vector method.I.I NTRODUCTIONOver the last several years,machine learning techniques, particularly when applied to NN’s,have played an increas-ingly important role in the design of pattern recognition systems.In fact,it could be argued that the availability of learning techniques has been a crucial factor in the recent success of pattern recognition applications such as continuous speech recognition and handwriting recognition. The main message of this paper is that better pattern recognition systems can be built by relying more on auto-matic learning and less on hand-designed heuristics.This is made possible by recent progress in machine learning and computer ing character recognition as a case study,we show that hand-crafted feature extraction can be advantageously replaced by carefully designed learning machines that operate directly on pixel ing document understanding as a case study,we show that the traditional way of building recognition systems by manually integrating individually designed modules can be replaced by a uniﬁed and well-principled design paradigm,called GTN’s,which allows training all the modules to optimize a global performance criterion.Since the early days of pattern recognition it has been known that the variability and richness of natural data, be it speech,glyphs,or other types of patterns,make it almost impossible to build an accurate recognition system entirely by hand.Consequently,most pattern recognition systems are built using a combination of automatic learning techniques and hand-crafted algorithms.The usual method0018–9219/98$10.00©1998IEEE2278PROCEEDINGS OF THE IEEE,VOL.86,NO.11,NOVEMBER1998Fig.1.Traditional pattern recognition is performed with two modules:aﬁxed feature extractor and a trainable classiﬁer.of recognizing individual patterns consists in dividing the system into two main modules shown in Fig.1.Theﬁrst module,called the feature extractor,transforms the input patterns so that they can be represented by low-dimensional vectors or short strings of symbols that:1)can be easily matched or compared and2)are relatively invariant with respect to transformations and distortions of the input pat-terns that do not change their nature.The feature extractor contains most of the prior knowledge and is rather speciﬁc to the task.It is also the focus of most of the design effort, because it is often entirely hand crafted.The classiﬁer, on the other hand,is often general purpose and trainable. One of the main problems with this approach is that the recognition accuracy is largely determined by the ability of the designer to come up with an appropriate set of features. This turns out to be a daunting task which,unfortunately, must be redone for each new problem.A large amount of the pattern recognition literature is devoted to describing and comparing the relative merits of different feature sets for particular tasks.Historically,the need for appropriate feature extractors was due to the fact that the learning techniques used by the classiﬁers were limited to low-dimensional spaces with easily separable classes[1].A combination of three factors has changed this vision over the last decade.First, the availability of low-cost machines with fast arithmetic units allows for reliance on more brute-force“numerical”methods than on algorithmic reﬁnements.Second,the avail-ability of large databases for problems with a large market and wide interest,such as handwriting recognition,has enabled designers to rely more on real data and less on hand-crafted feature extraction to build recognition systems. The third and very important factor is the availability of powerful machine learning techniques that can handle high-dimensional inputs and can generate intricate decision functions when fed with these large data sets.It can be argued that the recent progress in the accuracy of speech and handwriting recognition systems can be attributed in large part to an increased reliance on learning techniques and large training data sets.As evidence of this fact,a large proportion of modern commercial OCR systems use some form of multilayer NN trained with back propagation.In this study,we consider the tasks of handwritten character recognition(Sections I and II)and compare the performance of several learning techniques on a benchmark data set for handwritten digit recognition(Section III). While more automatic learning is beneﬁcial,no learning technique can succeed without a minimal amount of prior knowledge about the task.In the case of multilayer NN’s, a good way to incorporate knowledge is to tailor its archi-tecture to the task.Convolutional NN’s[2],introduced in Section II,are an example of specialized NN architectures which incorporate knowledge about the invariances of two-dimensional(2-D)shapes by using local connection patterns and by imposing constraints on the weights.A comparison of several methods for isolated handwritten digit recogni-tion is presented in Section III.To go from the recognition of individual characters to the recognition of words and sentences in documents,the idea of combining multiple modules trained to reduce the overall error is introduced in Section IV.Recognizing variable-length objects such as handwritten words using multimodule systems is best done if the modules manipulate directed graphs.This leads to the concept of trainable GTN,also introduced in Section IV. Section V describes the now classical method of HOS for recognizing words or other character strings.Discriminative and nondiscriminative gradient-based techniques for train-ing a recognizer at the word level without requiring manual segmentation and labeling are presented in Section VI. Section VII presents the promising space-displacement NN approach that eliminates the need for segmentation heuris-tics by scanning a recognizer at all possible locations on the input.In Section VIII,it is shown that trainable GTN’s can be formulated as multiple generalized transductions based on a general graph composition algorithm.The connections between GTN’s and HMM’s,commonly used in speech recognition,is also treated.Section IX describes a globally trained GTN system for recognizing handwriting entered in a pen computer.This problem is known as “online”handwriting recognition since the machine must produce immediate feedback as the user writes.The core of the system is a convolutional NN.The results clearly demonstrate the advantages of training a recognizer at the word level,rather than training it on presegmented, hand-labeled,isolated characters.Section X describes a complete GTN-based system for reading handwritten and machine-printed bank checks.The core of the system is the convolutional NN called LeNet-5,which is described in Section II.This system is in commercial use in the NCR Corporation line of check recognition systems for the banking industry.It is reading millions of checks per month in several banks across the United States.A.Learning from DataThere are several approaches to automatic machine learn-ing,but one of the most successful approaches,popularized in recent years by the NN community,can be called“nu-merical”or gradient-based learning.The learning machine computes afunction th input pattern,andtheoutputthatminimizesand the error rate on the trainingset decreases with the number of training samplesapproximatelyasis the number of trainingsamples,is a number between0.5and1.0,andincreases,decreases.Therefore,when increasing thecapacitythat achieves the lowest generalizationerror Mostlearning algorithms attempt tominimize as well assome estimate of the gap.A formal version of this is calledstructural risk minimization[6],[7],and it is based on deﬁn-ing a sequence of learning machines of increasing capacity,corresponding to a sequence of subsets of the parameterspace such that each subset is a superset of the previoussubset.In practical terms,structural risk minimization isimplemented byminimizingisaconstant.that belong to high-capacity subsets ofthe parameter space.Minimizingis a real-valuedvector,with respect towhichis iteratively adjusted asfollows:is updated on the basis of a singlesampleof several layers of processing,i.e.,the back-propagation algorithm.The third event was the demonstration that the back-propagation procedure applied to multilayer NN’s with sigmoidal units can solve complicated learning tasks. The basic idea of back propagation is that gradients can be computed efﬁciently by propagation from the output to the input.This idea was described in the control theory literature of the early1960’s[16],but its application to ma-chine learning was not generally realized then.Interestingly, the early derivations of back propagation in the context of NN learning did not use gradients but“virtual targets”for units in intermediate layers[17],[18],or minimal disturbance arguments[19].The Lagrange formalism used in the control theory literature provides perhaps the best rigorous method for deriving back propagation[20]and for deriving generalizations of back propagation to recurrent networks[21]and networks of heterogeneous modules[22].A simple derivation for generic multilayer systems is given in Section I-E.The fact that local minima do not seem to be a problem for multilayer NN’s is somewhat of a theoretical mystery. It is conjectured that if the network is oversized for the task(as is usually the case in practice),the presence of “extra dimensions”in parameter space reduces the risk of unattainable regions.Back propagation is by far the most widely used neural-network learning algorithm,and probably the most widely used learning algorithm of any form.D.Learning in Real Handwriting Recognition Systems Isolated handwritten character recognition has been ex-tensively studied in the literature(see[23]and[24]for reviews),and it was one of the early successful applications of NN’s[25].Comparative experiments on recognition of individual handwritten digits are reported in Section III. They show that NN’s trained with gradient-based learning perform better than all other methods tested here on the same data.The best NN’s,called convolutional networks, are designed to learn to extract relevant features directly from pixel images(see Section II).One of the most difﬁcult problems in handwriting recog-nition,however,is not only to recognize individual charac-ters,but also to separate out characters from their neighbors within the word or sentence,a process known as seg-mentation.The technique for doing this that has become the“standard”is called HOS.It consists of generating a large number of potential cuts between characters using heuristic image processing techniques,and subsequently selecting the best combination of cuts based on scores given for each candidate character by the recognizer.In such a model,the accuracy of the system depends upon the quality of the cuts generated by the heuristics,and on the ability of the recognizer to distinguish correctly segmented characters from pieces of characters,multiple characters, or otherwise incorrectly segmented characters.Training a recognizer to perform this task poses a major challenge because of the difﬁculty in creating a labeled database of incorrectly segmented characters.The simplest solution consists of running the images of character strings through the segmenter and then manually labeling all the character hypotheses.Unfortunately,not only is this an extremely tedious and costly task,it is also difﬁcult to do the labeling consistently.For example,should the right half of a cut-up four be labeled as a one or as a noncharacter?Should the right half of a cut-up eight be labeled as a three?Theﬁrst solution,described in Section V,consists of training the system at the level of whole strings of char-acters rather than at the character level.The notion of gradient-based learning can be used for this purpose.The system is trained to minimize an overall loss function which measures the probability of an erroneous answer.Section V explores various ways to ensure that the loss function is differentiable and therefore lends itself to the use of gradient-based learning methods.Section V introduces the use of directed acyclic graphs whose arcs carry numerical information as a way to represent the alternative hypotheses and introduces the idea of GTN.The second solution,described in Section VII,is to eliminate segmentation altogether.The idea is to sweep the recognizer over every possible location on the input image,and to rely on the“character spotting”property of the recognizer,i.e.,its ability to correctly recognize a well-centered character in its inputﬁeld,even in the presence of other characters besides it,while rejecting images containing no centered characters[26],[27].The sequence of recognizer outputs obtained by sweeping the recognizer over the input is then fed to a GTN that takes linguistic constraints into account andﬁnally extracts the most likely interpretation.This GTN is somewhat similar to HMM’s,which makes the approach reminiscent of the classical speech recognition[28],[29].While this technique would be quite expensive in the general case,the use of convolutional NN’s makes it particularly attractive because it allows signiﬁcant savings in computational cost.E.Globally Trainable SystemsAs stated earlier,most practical pattern recognition sys-tems are composed of multiple modules.For example,a document recognition system is composed of aﬁeld loca-tor(which extracts regions of interest),aﬁeld segmenter (which cuts the input image into images of candidate characters),a recognizer(which classiﬁes and scores each candidate character),and a contextual postprocessor,gen-erally based on a stochastic grammar(which selects the best grammatically correct answer from the hypotheses generated by the recognizer).In most cases,the information carried from module to module is best represented as graphs with numerical information attached to the arcs. For example,the output of the recognizer module can be represented as an acyclic graph where each arc contains the label and the score of a candidate character,and where each path represents an alternative interpretation of the input string.Typically,each module is manually optimized,or sometimes trained,outside of its context.For example,the character recognizer would be trained on labeled images of presegmented characters.Then the complete system isLECUN et al.:GRADIENT-BASED LEARNING APPLIED TO DOCUMENT RECOGNITION2281assembled,and a subset of the parameters of the modules is manually adjusted to maximize the overall performance. This last step is extremely tedious,time consuming,and almost certainly suboptimal.A better alternative would be to somehow train the entire system so as to minimize a global error measure such as the probability of character misclassiﬁcations at the document level.Ideally,we would want toﬁnd a good minimum of this global loss function with respect to all theparameters in the system.If the loss functionusing gradient-based learning.However,at ﬁrst glance,it appears that the sheer size and complexity of the system would make this intractable.To ensure that the global loss functionwithrespect towith respect toFig.2.Architecture of LeNet-5,a convolutional NN,here used for digits recognition.Each plane is a feature map,i.e.,a set of units whose weights are constrained to be identical.or other2-D or one-dimensional(1-D)signals,must be approximately size normalized and centered in the input ﬁeld.Unfortunately,no such preprocessing can be perfect: handwriting is often normalized at the word level,which can cause size,slant,and position variations for individual characters.This,combined with variability in writing style, will cause variations in the position of distinctive features in input objects.In principle,a fully connected network of sufﬁcient size could learn to produce outputs that are invari-ant with respect to such variations.However,learning such a task would probably result in multiple units with similar weight patterns positioned at various locations in the input so as to detect distinctive features wherever they appear on the input.Learning these weight conﬁgurations requires a very large number of training instances to cover the space of possible variations.In convolutional networks,as described below,shift invariance is automatically obtained by forcing the replication of weight conﬁgurations across space. Secondly,a deﬁciency of fully connected architectures is that the topology of the input is entirely ignored.The input variables can be presented in any(ﬁxed)order without af-fecting the outcome of the training.On the contrary,images (or time-frequency representations of speech)have a strong 2-D local structure:variables(or pixels)that are spatially or temporally nearby are highly correlated.Local correlations are the reasons for the well-known advantages of extracting and combining local features before recognizing spatial or temporal objects,because conﬁgurations of neighboring variables can be classiﬁed into a small number of categories (e.g.,edges,corners,etc.).Convolutional networks force the extraction of local features by restricting the receptive ﬁelds of hidden units to be local.A.Convolutional NetworksConvolutional networks combine three architectural ideas to ensure some degree of shift,scale,and distortion in-variance:1)local receptiveﬁelds;2)shared weights(or weight replication);and3)spatial or temporal subsampling.A typical convolutional network for recognizing characters, dubbed LeNet-5,is shown in Fig.2.The input plane receives images of characters that are approximately size normalized and centered.Each unit in a layer receives inputs from a set of units located in a small neighborhood in the previous layer.The idea of connecting units to local receptiveﬁelds on the input goes back to the perceptron in the early1960’s,and it was almost simultaneous with Hubel and Wiesel’s discovery of locally sensitive,orientation-selective neurons in the cat’s visual system[30].Local connections have been used many times in neural models of visual learning[2],[18],[31]–[34].With local receptive ﬁelds neurons can extract elementary visual features such as oriented edges,endpoints,corners(or similar features in other signals such as speech spectrograms).These features are then combined by the subsequent layers in order to detect higher order features.As stated earlier,distortions or shifts of the input can cause the position of salient features to vary.In addition,elementary feature detectors that are useful on one part of the image are likely to be useful across the entire image.This knowledge can be applied by forcing a set of units,whose receptiveﬁelds are located at different places on the image,to have identical weight vectors[15], [32],[34].Units in a layer are organized in planes within which all the units share the same set of weights.The set of outputs of the units in such a plane is called a feature map. Units in a feature map are all constrained to perform the same operation on different parts of the image.A complete convolutional layer is composed of several feature maps (with different weight vectors),so that multiple features can be extracted at each location.A concrete example of this is theﬁrst layer of LeNet-5shown in Fig.2.Units in theﬁrst hidden layer of LeNet-5are organized in six planes,each of which is a feature map.A unit in a feature map has25inputs connected to a5case of LeNet-5,at each input location six different types of features are extracted by six units in identical locations in the six feature maps.A sequential implementation of a feature map would scan the input image with a single unit that has a local receptive ﬁeld and store the states of this unit at corresponding locations in the feature map.This operation is equivalent to a convolution,followed by an additive bias and squashing function,hence the name convolutional network.The kernel of the convolution is theOnce a feature has been detected,its exact location becomes less important.Only its approximate position relative to other features is relevant.For example,once we know that the input image contains the endpoint of a roughly horizontal segment in the upper left area,a corner in the upper right area,and the endpoint of a roughly vertical segment in the lower portion of the image,we can tell the input image is a seven.Not only is the precise position of each of those features irrelevant for identifying the pattern,it is potentially harmful because the positions are likely to vary for different instances of the character.A simple way to reduce the precision with which the position of distinctive features are encoded in a feature map is to reduce the spatial resolution of the feature map.This can be achieved with a so-called subsampling layer,which performs a local averaging and a subsampling,thereby reducing the resolution of the feature map and reducing the sensitivity of the output to shifts and distortions.The second hidden layer of LeNet-5is a subsampling layer.This layer comprises six feature maps,one for each feature map in the previous layer.The receptive ﬁeld of each unit is a 232p i x e l i m a g e .T h i s i s s i g n i ﬁc a n tt h e l a r g e s t c h a r a c t e r i n t h e d a t a b a s e (a t28ﬁe l d ).T h e r e a s o n i s t h a t i t it h a t p o t e n t i a l d i s t i n c t i v e f e a t u r e s s u c h o r c o r n e r c a n a p p e a r i n t h e c e n t e r o f t h o f t h e h i g h e s t l e v e l f e a t u r e d e t e c t o r s .o f c e n t e r s o f t h e r e c e p t i v e ﬁe l d s o f t h e l a y e r (C 3,s e e b e l o w )f o r m a 2032i n p u t .T h e v a l u e s o f t h e i n p u t p i x e l s o t h a t t h e b a c k g r o u n d l e v e l (w h i t e )c o ro fa n d t h e f o r e g r o u n d (b l ac k )c o r r e s p T h i s m a k e s t h e m e a n i n p u t r o u g h l y z e r o r o u g h l y o n e ,w h i c h a c c e l e r a t e s l e a r n i n g I n t h e f o l l o w i n g ,c o n v o l u t i o n a l l a y e r s u b s a m p l i n g l a y e r s a r e l a b e l ed S x ,a n d l a ye r s a r e l a b e l e d F x ,w h e r e x i s t h e l a y L a y e r C 1i s a c o n v o l u t i o n a l l a y e r w i t h E a c h u n i t i n e a c hf e a t u r e m a p i s c o n n e c t28w h i c h p r e v e n t s c o n n e c t i o n f r o m t h e i n p t h e b o u n d a r y .C 1c o n t a i n s 156t r a i n a b l 122304c o n n e c t i o n s .L a y e r S 2i s a s u b s a m p l i n g l a y e r w i t h s i s i z e 142n e i g h b o r h o o d i n t h e c o r r e s p o n d i n g f T h e f o u r i n p u t s t o a u n i t i n S 2a r e a d d e d ,2284P R O C E E D I N G S O F T H E I E E E ,V O L .86,N O .11,N O VTable 1Each Column Indicates Which Feature Map in S2Are Combined by the Units in a Particular Feature Map ofC3a trainable coefﬁcient,and then added to a trainable bias.The result is passed through a sigmoidal function.The25neighborhoods at identical locations in a subset of S2’s feature maps.Table 1shows the set of S2feature maps combined by each C3feature map.Why not connect every S2feature map to every C3feature map?The reason is twofold.First,a noncomplete connection scheme keeps the number of connections within reasonable bounds.More importantly,it forces a break of symmetry in the network.Different feature maps are forced to extract dif-ferent (hopefully complementary)features because they get different sets of inputs.The rationale behind the connection scheme in Table 1is the following.The ﬁrst six C3feature maps take inputs from every contiguous subsets of three feature maps in S2.The next six take input from every contiguous subset of four.The next three take input from some discontinuous subsets of four.Finally,the last one takes input from all S2feature yer C3has 1516trainable parameters and 156000connections.Layer S4is a subsampling layer with 16feature maps of size52neighborhood in the corresponding feature map in C3,in a similar way as C1and yer S4has 32trainable parameters and 2000connections.Layer C5is a convolutional layer with 120feature maps.Each unit is connected to a55,the size of C5’s feature maps is11.This process of dynamically increasing thesize of a convolutional network is described in Section yer C5has 48120trainable connections.Layer F6contains 84units (the reason for this number comes from the design of the output layer,explained below)and is fully connected to C5.It has 10164trainable parameters.As in classical NN’s,units in layers up to F6compute a dot product between their input vector and their weight vector,to which a bias is added.This weighted sum,denotedforunit (6)wheredeterminesits slope at the origin.Thefunctionis chosen to be1.7159.The rationale for this choice of a squashing function is given in Appendix A.Finally,the output layer is composed of Euclidean RBF units,one for each class,with 84inputs each.The outputs of each RBFunit(7)In other words,each output RBF unit computes the Eu-clidean distance between its input vector and its parameter vector.The further away the input is from the parameter vector,the larger the RBF output.The output of a particular RBF can be interpreted as a penalty term measuring the ﬁt between the input pattern and a model of the class associated with the RBF.In probabilistic terms,the RBF output can be interpreted as the unnormalized negative log-likelihood of a Gaussian distribution in the space of conﬁgurations of layer F6.Given an input pattern,the loss function should be designed so as to get the conﬁguration of F6as close as possible to the parameter vector of the RBF that corresponds to the pattern’s desired class.The parameter vectors of these units were chosen by hand and kept ﬁxed (at least initially).The components of thoseparameters vectors were set to1.While they could have been chosen at random with equal probabilities for1,or even chosen to form an error correctingcode as suggested by [47],they were instead designed to represent a stylized image of the corresponding character class drawn on a7。

ANSYS错误提示及其含义

ANSYS错误提示及其含义1 在Ansys中出现“Shape testing revealed that 450 of the 1500 new or modified elements violate shape warning limits.”，是什么原因造成的呢？单元网格质量不够好，尽量用规则化网格，或者再较为细密一点。

2 在Ansys中，用Area Fillet对两空间曲面进行倒角时出现以下错误：Area 6 offset could not fully converge to offset distance 10. Maximum error between the two surfaces is 1% of offset distance.请问这是什么错误？怎么解决？其中一个是圆柱接管表面，一个是碟形封头表面。

ansys的布尔操作能力比较弱。

如果一定要在ansys里面做的话，那么你试试看先对线进行倒角，然后由倒角后的线形成倒角的面。

建议最好用UG、PRO/E这类软件生成实体模型然后导入到ansys。

3 在Ansys中，出现错误“There are 21 small equation solver pivot terms。

”，是否是在建立接触contact时出现的错误？不是建立接触对的错误，一般是单元形状质量太差（例如有接近零度的锐角或者接近180度的钝角）造成small equation solver pivot terms4 在Ansys中，出现警告“SOLID45 wedges are recommended only in regions of relatively low stress gradients.”，是什么意思？"这只是一个警告，它告诉你：推荐SOLID45单元只用在应力梯度较低的区域。

它只是告诉你注意这个问题，如果应力梯度较高，则可能计算结果不可信。

"5 ansys向adams导的过程中，出现如下问题“There is not enough memory for the Sparse Matrix Solver to proceed.Please shut down other applications that may be running or increase the virtual memory on your system and return ANSYS.Memory currently allocated for the Sparse Matrix Solver=50MB.Memory currently required for the Sparse Matrix Solver to continue=25MB”，是什么原因造成的？不清楚你ansys导入adams过程中怎么还需要使用Sparse Matrix Solver（稀疏矩阵求解器）。

Lab 2

Lab 2VelocityOverviewIn this lab, we will explore the relationship between the position function and the velocity function of a moving object. In the lectures, we have seen that the velocity function v(t) is the derivative of the position function s(t) (and therefore s(t) is an antiderivative ofv(t)). We will be examining these statements in this lab by using a motion detector to measure the velocity and position of a person walking.Vocabulary used in this lab•Distance: This is the distance an object is from the motion detector. The words distance and position will be used interchangeably in this lab.•Displacement: This is “change of position,” that is, the difference between the initial and terminal coordinates the object. Displacement can be either positive or negative. (Think about what the difference between positive and negativedisplacement is.)•Velocity and speed: The important thing to note here is that velocity can be either positive or negative, depending on whether motion is in the positive ornegative direction. But speed = |v(t)| is never negative: it is the absolute value of the velocity. For example, a car’s speedometer is measuring speed and alwaysreads positive, regardless of the direction you’re traveling.•Total distance traveled: Perhaps this is best illustrated by an example. If you walk 2 meters forward and then 2 meters backward then your displacement willzero (the initial and terminal positions are the same so therefore the difference of coordinates is zero). But, you actually traveled 4 meters. In this case, the totaldistance traveled would be 4 meters. The “total distance traveled” is nevernegative.Mathematics for this labThe mathematics that we will need is the relationship between displacement and velocity: •Displacement is equal to the definite integral of velocity v(t) over a time interval.•Total distance traveled is equal to the definite integral of speed |v(t)|over a time interval.These ideas are discussed in the textbook on pages 374-375.Materials•Computer with Vernier computer interface.•Vernier motion detector. This will plug into the LoggerPro computer interface.•Separate “Lab Report 2” sheet. This is what you will turn in. It is due one week after your scheduled lab day, by 4:30 pm, in the Mathematics Office (Cupples I,room 100).Comments on the Motion DetectorThe motion detector works by emitting short bursts of ultrasonic sound waves. You will hear a clicking sound when the detector is operating. The detector “listens” for the ech o of these ultrasonic waves returning to it. The motion detector measures the time it takes for the sound waves to make the trip from the detector to the object and back to the detector. Knowing the speed of sound, the detector is then able to calculate the distance to the object.The specifications on the detector state that it has a minimum range of 0.4 m. and a maximum range of 6.0 m. The detector seems to work best if there is a smooth flat surface for the sound waves to bounce off of.Practically speaking, this means that as you collect data•Your motion needs to be parallel to the sensor and not perpendicular to it (i.e., you will need to walk either away from or towards the sensor, or both).•You will probably want to be holding a book or piece of cardboard in the sensor’s “line of sight” while you walk.•If neighboring lab teams are moving while you are collecting data, your motion detector might pick up their motion as well. You will have to cooperate with your neighboring teams so that your motion detector is only picking up data from your walking.•It is very important that the motion detector does not move. (What would the motion detector measure if it was moving?)ProcedureThe first thing to do is get the computer and motion detector ready.Connect the motion detector to the Dig/Sonic 2 port of the Lab Pro. Make sure the Lab Pro is connected to the USB port of the computer. (It is likely that these have already been done for you.)•Set the detector upright on the edge of the desk, facing out in a direction in which you have a clear path at least 2 m. long.•Open the LoggerPro program. You will want to select “Lab Pro USB” for the port in the Setup Interface. It is possible (hopefully not likely) that LoggerProwill not recognize the hardware. If this happens, unplugging the Lab Pro andplugging it back in should take care of the problem.•Open the experiment file for this lab. (Click on the “Open” button and open the file labeled “lab2_velocity.” (It is possible that LoggerPro will complain aboutthis file – just ignore any complaint and click “OK.”)•At this point, you should be able to collect distance and velocity data. Try playing around with the motion detector by pressing the “collect button” on thetoolbar. (For example: turn the sensor to face the nearest wall and collect data—what happens? Turn the sensor around and collect data by moving your handback-and-forth in front of it.) When you start collecting data, notice that velocity and distance data is graphed and the actual data appears in a table to the right ofthe graphs. Notice also that each time you start collecting new data, the data from the previous run is cleared. (Another way to clear recently collected data is toselect Data, Delete Run from the menu.)You are now ready to start collecting data that you will analyze.•Decide who will monitor the computer (and hit the collect button) and who will be the “mover”—the mover will create distance and velocity data bywalking back and forth in front of the motion detector.•Your goal is to generate a data set that exhibits both positive and negative velocity during the five seconds of movement.•There is a limited amount of free space in the lab for straight-line movement, so you will have to coordinate your data collection with your neighbors.Make sure you have a clear path before you start collecting your data.•The mover should take an initial position standing in front of the motion detector, holding a book or a piece of cardboard in the detector’s “line ofsight.” This will help to make a smoother velocity graph as you collect thedata.Before beginning, note the mover’s initial position. When the datacollection stops, the mover should remain in position while his/her partnernotes the terminal position. (An acceptable method to note these positionsis to put a marker, like a pen, on the floor where the mover’s feet start.)Measure and write down these positions (number of meters in front of thesensor); you’ll need this information later. (There will be a tape measurein the lab for you to measure the distances. You’re not looking for greataccuracy—just something in the right ballpark. Remember that 1 meter isapproximately 3.28 feet.).•Click on the “Collect” button and gather the data. When the mover hears the motion detector making the clicking sound, s/he should start moving along astraight line in front of the motion detector. Remember your goal to createdata with positive and negative velocity.After collecting your distance and velocity data, you should make sure your data looks “reasonable”: for example,•Is the graph of the velocity reasonably smooth?•Does the distance data in the table look like what you were expecting? Do the distances more-or-less match up with what you noted for the initial and finalpositions?•Is the velocity positive and negative when you were expecting it to be positive and negative?•If your data looks questionable, collect another set of data. (The good news is that it doesn’t take much time to collect another data set if necessary.) •If your data looks good, save it so that you can finish your report after the lab period if necessary. (If you didn’t bring a diskette, you could e-mail the datato yourself.)At this point, you are ready to analyze your data.•You should have two graphs –position (labeled “Distance vs. Time”) and velocity (labeled “Velocity vs. Time). You should print both of these graphs and also the table: activate each graph or table window before you print (so that youget a nice printout of each one separately).•Click on the “Examine” button (Analyze – Examine in the menu). This should allow you to more easily read both the graph and the data (watch what happenswhen you move the mouse to the graphs and the data table). Using this, answer#1-3 on Lab Report 2.DisplacementWe want to calculate the displacement of the mover after 5 seconds. We can do this in three different ways:a) by having LoggerPro estimate the integral of the velocityfunction v(t) for 0 <= t <= 5b) by using our data and the Fundamental Theorem of Calculus (Part II) to evaluate the same integralc) by using the measurements we made “by hand” during the data collection process.In detail:a) Using LoggerPro to estimate the integral of the velocity:•Click on the left-hand side of the velocity graph (Time=0) and drag the black line across to the right-hand side of the graph. While you are doing this, the values in the table window should be highlighted. Make sure you actually click on thegraph on not outside the graph (otherwise you will see the pop-up box asking you about the y-axis and y-scale).•Click the Integral button or choose Analyze – Integrate from the menu. A floating box with LoggerPro’s estimate of the value of the integral appears in the Veloci ty graph window. Note the units in this box: m/s * s, which is just meters (theseconds cancel out); don’t misread the units as “m/sec2” !! Print a copy of thegraph with the box. Record this value in #4a on the Lab Report.b) According the Fundamental Theorem of Calculus (Part II), we can compute the integral of v(t) over the time interval [0,5] using an antiderivative. In this situation, the numeric result can be worked out using subtracting two numbers taken from your data tables. In the Lab Report #4b state what numbers you are using and give the resulting value for the integral.c) Finally, calculate the displacement using the distances from the sensor that you measured during the data collection process. Record the result of the calculation in Lab Report #4c.In Lab Report #4d, you are asked to offer some reasons why the results in a), b) and c) are (probably) not equal.Total Distance TraveledTo find the total distance traveled, we need to integrate the speed |v(t)| during our 5 second time interval. To have LoggerPro do this,we need the speed data—so we create it in a new column:•Activate the velocity graph window by clicking on the velocity graph. Select View – Graph Options from the menu.•Modify the Graph Title by typing “Speed/” before Velocity. It should now read Speed/Velocity vs. Time. Click Apply, then click OK.•Select Data - New Column, Formula from the Data menu. The New Column window will appear.•Under the Options tab, in the Labels section, type “Speed” for Long Name, “sp”for Short Name and “m/s” for Units.•Click the Definition tab. In the Equation box, enter the formula(abs(“Velocity”)). You can (and should) do this by selecting “abs()” from the Functions list followed by selecting “Velocity” from the Variables list.•Click Try New Column. If it new data looks as it should (how should it look?), then click “OK”.•Notice that the new graph (Speed) coincides with the original velocity graph when the velocity is positive. How do these graphs compare when velocity is negative? •Click on the left-hand side of the speed/velocity graph (Time = 0) and drag the black line across to the right-hand side of the graph. All of the values in the table window should be highlighted.•Click the Integrate button. An Integral Selection window appears. Remove the check next to Velocity by clicking in its box. Make sure that Speed is checked.Click OK.• A floating box with LoggerPro’s estimate of the value of the integral appears in the Speed/Velocity graph window. Print a copy of the graph with the box. Report the total distance traveled on Lab Report, #5.Answer #6 and #7 on the Lab Report sheet.。

自考英语二(00015)Unit6 TextA 练习

原则上,理论上
Bridging the Gap(P218)
basic grow invest
ideally called
now that wasteful accumulated where direct
将单词分类：动词：grow invest called accumulated direct 形容词：basic wasteful direct 副词：ideally 连词：now that where
Section A (P219)
1. To obtain objective findings, scientists ___d_r_e_w_t_h_e_____ _c_o_n_c_lu__si_o_n_s_o_n__t_h_e_b_a_s_is__o_f_e_x_p_e_r_im__e_n_t_s_ (根据实验得出结论).(on the basis of) 2. She is such a self-disciplined person that _sh_e__h_a_s_n_e_v_e_r _b_e_e_n_i_n_d_u_l_g_e_d_i_n__a_lc_o_h_o_l_(她从不沾酒).(indulge in) 3. When he loses his temper, he _is__n_o_t_r_e_sp_o_n__si_b_l_e_f_o_r_h_i_s_ _b_e_h_a_v_i_o_r_s_(对自己的行为不负责任).(be responsible for) 4. The result of being employed proves that m__y__te_a_c_h_i_n_g _e_x_p_e_r_ie_n_c_e__st_a_n_d_s__m_e__in__g_o_o_d_s_t_e_a_d_(我的从教经验对我利). (stand...in good stead) 5. Figures show that fifty percent of road accidents _h_a_v_e__re_s_u_l_te_d__in__h_e_a_d__in_j_u_r_i_es__(导致头部受伤).(result in)

Robust Principal Component Analysis

Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Matrices via Convex Optimization
∗ John Wright† , Arvind Ganesh† , Shankar Rao† , and Yi Ma†
Department of Electrical Engineering University of Illinois at Urbana-Champaign Visual Computing Group Microsoft Research Asia
Abstract. Principal component analysis is a fundamental operation in computational data analysis, with myriad applications ranging from web search to bioinformatics to computer vision and image analysis. However, its performance and applicability in real scenarios are limited by a lack of robustness to outlying or corrupted observations. This paper considers the idealized “robust principal component analysis” problem of recovering a low rank matrix A from corrupted observations D = A + E . Here, the error entries E can be arbitrarily large (modeling grossly corrupted observations common in visual and bioinformatic data), but are assumed to be sparse. We prove that most matrices A can be eﬃciently and exactly recovered from most error sign-and-support patterns, by solving a simple convex program. Our result holds even when the rank of A grows nearly proportionally (up to a logarithmic factor) to the dimensionality of the observation space and the number of errors E grows in proportion to the total number of entries in the matrix. A by-product of our analysis is the ﬁrst proportional growth results for the related but somewhat easier problem of completing a low-rank matrix from a small fraction of its entries. We propose an algorithm based on iterative thresholding that, for large matrices, is signiﬁcantly faster and more scalable than general-purpose solvers. We give simulations and real-data examples corroborating the theoretical results.

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Multiobjective Optimization Problems With Complicated Pareto Sets, MOEA_D and NSGA-II

H. Li was with the Department of Computing and Electronic Systems, University of Essex, Colchester CO4 3SQ, U.K. He is now with the School of Computer Science, University of Nottingham, Nottingham NG8 1BB, U.K. (e-mail: hzl@).
I. INTRODUCTION
A multiobjective optimization problem (MOP) can be stated as follows:
minimize
subject to
(1)
where is the decision (variable) space, is the objective
MOP.
Let
,
be two
vectors, is said to dominate if
for all
,
and
.1 A point
is called (globally) Pareto optimal
if there is no
such that
Hale Waihona Puke dominates. The set
of all the Pareto optimal points, denoted by , is called the
Pareto set. The set of all the Pareto objective vectors,
, is called the Pareto front [1].

Transition from Mott insulator to superconductor in GaNb$_{4}$Se$_{8}$ and GaTa$_{4}$Se$_{8

$Transition from Mott insulator to superconductor in GaNb$_{4}$Se$_{8}$ and GaTa$_{4}$Se$_{8$

Transition from Mott insulator to superconductor in GaNb4Se8and GaTa4Se8under high pressureM. M. Abd-Elmeguid1, B. Ni1, D. I. Khomskii1,a, R. Pocha2, D. Johrendt2, X. Wang3, and K. Syassen31II. Physikalisches Institut, Universität zu Köln, Zülpicher Str. 77, 50937 Köln, Germany2Department Chemie, Ludwig-Maximilians-Universität München, Butenandtstr. 5-13 (Haus D), 81377 München, Germany3Max-Planck-Institut für Festkörperforschung, Heisenbergstr. 1, 70569 Stuttgart, GermanyElectronic conduction in Ga M4Se8(M= Nb, Ta) compounds with the fcc GaMo4S8-type structure originates from hopping of localized unpaired electrons (S= ½) among widely separated tetrahedral M4 metal clusters. We show that under pressure these systems transformT= 2.9 K and 5.8 K at 13 from Mott insulators to a metallic and superconducting state withCGPa and 11.5 GPa for GaNb4Se8and GaTa4Se8, respectively. The occurrence of superconductivity is shown to be connected with a pressure-induced decrease of the M Se6 octahedral distortion and simultaneous softening of the phonon associated with M−Se bonds.PACS number(s): 71.30.+h, 74.10.+v, 74.25.Kc, 74.62.FjSuperconductivity in the presence of strong electron correlations has attracted considerable attention, especially after the discovery of high-C T superconductors. Usually superconductivity is obtained in such systems by doping Mott insulators, like in cuprates [1] or in Na x CoO 2 . yH 2O [2]. Another option is to study the occurrence of superconductivity under high pressure in stoichiometric systems in the proximity to a Mott transition. The advantage in this case is the absence of disorder. Unfortunately, there are very few such systems known (e.g. β-Na 0.33V 2O 5 [3] and recent theoretical discussion [4]).In this work we show that cluster compounds Ga M 4Se 8 (M = Nb, Ta), which are nonmagnetic Mott insulators at ambient pressure, transform to a metallic and superconducting state at pressures of 13 GPa and 11.5 GPa with critical temperatures C T = 2.9 K and 5.8 K, respectively. We show that the Mott transition itself is apparently connected with internal distortions of the clusters rather than a change of the lattice symmetry. We also observed a rather strong softening of one of the phonon modes, which correlates with the appearance of superconductivity.Ternary chalcogenides AM 4X 8 (A = Ga, Ge; M = V, Mo, Nb, Ta; X = S, Se) belong to an interesting class of transition metal systems which exhibit strong electronic correlation effects. The origin of the electronic correlation in these systems is a consequence of their peculiar crystal structure, shown in Fig. 1(a). This fcc structure (GaMo 4S 8-type) can be described as a deficient spinel A 0.5M 2X 4 [5,6], in which the ordering of the tetrahedral A -ions reduces the symmetry from m Fd 3 to m F 34. As a result, the M (transition metal)-atoms are shifted off the centers of the S / Se octahedral, see Fig. 1(b), forming tetrahedral 4M -clusters with typical intracluster M M − distances ()M d of ≤ 3 Å. At the same time the M M − distances ()C d between the 4M -clusters become large (> 4 Å), which results in a formation of localized electronic states in the clusters. This leads to unusual transport and magnetic properties. None of these compounds show metallic conductivity; instead the electronic conduction takes place by hopping of carriers between the clusters [7-10]. Simultaneously, magnetic susceptibility is typical for localized spins. Thus, this class of systems can be considered as Mott-insulators.The ground state properties of these compounds strongly depend on the local electronic structure of the 4M -cluster (actually 44X M -clusters) which is mainly determined by the number of valence electrons per cluster [9-11]. According to molecular orbital (MO) calculations, the d -orbitals within the 4M -clusters can be described by MO’s which consist of three energetically different bonding states (for cubic T d symmetry) [12]: a nondegeneratelevel (1a ), followed by twofold (e ) and threefold (2t ) degenerated levels (see Fig. 1(c)). For cluster compounds of the type Ga 3+(M 3.25+)4(S 2−,Se 2−)8, we have 7 valence electrons per cluster with M = V, Nb, Ta and 11 electrons with M = Mo. In both cases, the occupation of the cluster orbitals leads to one unpaired electron (i.e. S = ½) per cluster. This is in agreement with the values of the magnetic moments obtained from magnetic susceptibility measurements and is also consistent with spin polarized band structure calculations [10,11]. In one respect, however, these systems are different from the conventional Mott insulators such as transition metal oxides: in contrast to the latter, the correlated units are 4M clusters which may have extra internal degrees of freedom. As we will show below, this leads to a high sensitivity of these systems to external pressure.Single phase polycrystalline samples of GaTa 4Se 8 and GaNb 4Se 8 were prepared as described in Ref. [9]. X-ray powder patterns were completely indexed using the structural data obtained from single crystal experiments [13]. The pressure dependence of the lattice constants at 300 K up to about 26 GPa was measured on powdered samples by energy dispersive x-ray diffraction (EDX) at HASYLAB using the diamond anvil cell (DAC) technique. The same type of DAC has been used for conventional four-terminal electrical resistance measurements up to about 29 GPa between 1.6 K and 300 K. Single crystal x-ray diffraction measurements (MoK α1) of GaTa 4Se 8 were performed at 300 K up to p = 15 GPa using a special DAC. Raman spectra were recorded in back-scattering geometry using a micro-spectrometer.Before discussing the high pressure results we briefly mention some experimental data at ambient pressure. The values of the lattice parameter a as determined from x-ray diffraction measurements at 300 K are found to be 10.440(1) Å and 10.358(1) Å for GaNb 4Se 8 and GaTa 4Se 8, respectively, in agreement with previous results [6]. From single crystal x-ray data we obtained the values of the characteristic intra- and intercluster distances: M d = 3.051(3) Å, 3.015(2) Å and C d = 4.332(3) Å, 4.338(2) Å, for GaNb 4Se 8 and GaTa 4Se 8, respectively. Measurements of the temperature dependence of electrical resistivity (1.6 K ≤ T ≤ 300 K) show for both samples a semiconductor-like behavior with activation energies of 0.14 eV (GaNb 4Se 8) and 0.1 eV (GaTa 4Se 8). Actually, the activation energy decreases with decreasing temperature, in agreement with that reported for GaMo 4S 8 and GaV 4S 8 [10]. The magnetic susceptibility of the two samples shows Curie-Weiss behavior (100 K ≤ T ≤ 300 K), indicating the existence of magnetic correlations, but no magnetic ordering is found down to1.6 K in agreement with Ref. [6]. The estimated values of the effective magnetic moments are1.6B µ per Nb 4-cluster (close to theoretical value 1.73 B µ for S = ½) and 0.7 B µ per Ta 4-cluster. Detailed analysis of the results at ambient pressure are presented elsewhere [13,14], in the present paper we focus on high pressure results.Fig. 2(a, b) displays the temperature dependence of the normalized electrical resistance )297(/)(K R T R R n = in the temperature range 1.6 K ≤ T ≤ 300 K as a function of pressure for GaNb 4Se 8 and GaTa 4Se 8, respectively. Considering first the overall behavior of ()p T R n , in both samples, one finds with increasing pressure a gradual change from the semiconducting to a metallic-like behavior and a sudden drop of n R at low temperatures above a critical pressure (c p ), indicative of a superconducting transition. While the metallic behavior 0/>dT dR is observed at rather high pressures (p ≥ 19 GPa (GaNb 4Se 8) and p ≥ 15 GPa (GaTa 4Se 8)), superconductivity already sets in at lower pressures where the temperature dependence of n R is still semiconducting-like; C T = 2.9 K at 13 GPa for GaNb 4Se 8 and 5.8 K at 11.5 GPa for GaTa 4Se 8. This type of behavior is usually observed in the superconducting state of polycrystalline sintered samples, e.g. at ambient pressure in La 1-x Sr x CuO 4 [15] and under high pressure in the Chevrel phase compound Eu 1.2Mo 6(S,Se)8 [16, 17], and is known to be due to a coexistence of superconducting and semiconducting phases (granular superconductivity), in the bulk and surface of the grains of such samples, respectively. This explains the finite value of the resistivity observed in the superconducting state of our samples (cm Ω≈−4010ρat T = 1.6 K and p ≈ 20 GPa) despite their single phase purity as well as the increase of the drop of )(T R with increasing pressure (see Fig. 2). We note, however, that the drop of )(T R is substantial (~ 70 %) at p ≈ 20 GPa and 1.6 K and is expected to further increase at lower temperatures resulting in a lower value of the resistivity. This indicates an increase of the fraction of superconductivity in the samples with increasing pressure. Fig. 3(c) shows the pressure dependence of C T for both samples. The value of C T increases remarkably with increasing pressure up to p ≈ 22 GPa, p T C ∂∂≈ 0.4 KGPa -1 and ≈ 0.2 KGPa -1, for GaNb 4Se 8 and GaTa 4Se 8, respectively. For GaNb 4Se 8 we observe a decrease of C T with increasing pressure at p > 22 GPa. As we show below this decrease is probably connected with a pressure-induced structural distortion in GaNb 4Se 8.To prove that the observed behavior is indeed a superconducting transition, we investigated the effect of external magnetic fields ()ex B on the temperature dependence of the electrical resistance at pressures of p = 20 GPa (GaNb 4Se 8) and p = 22 GPa (GaTa 4Se 8). Fig. 3(a, b) shows the temperature dependence of the electrical resistance as a function of ex B . Asexpected, C T is clearly shifted to lower temperatures with increasing ex B ; the overall behavior of ()ex B T R , is indeed typical for a bulk superconducting transition: we find a linear decrease of C T with ex B that can be described by the well known WHH theory for “dirty superconductors” [18]. This excludes the existence of weak links and/or a minority phase superconductivity. We obtain for GaNb 4Se 8 and GaTa 4Se 8 values of the upper critical field 2C B (0→T ) of 1.7 T at 20 GPa and 8 T at 17 GPa and a corresponding coherence length ζ of 130 Å and 61 Å, respectively. Thus, despite the fact that without measurements of the Meissner effect under high pressure, we cannot give a value for the superconducting fraction in our samples, the above mentioned experimental results clearly verify a pressure-induced transition from a Mott-insulating state to a metallic and superconducting state.Next, we investigate whether the observed pressure-induced superconductivity is connected with a structural instability and/or a change of the lattice dynamics under high pressure. The pressure dependence of the lattice parameter a of the two samples is displayed in Fig. 4(a). As evident from the figure, there is no indication of a structural phase transition up to ~ 20 GPa within the resolution of the EDX measurements. Thus, the onset of superconductivity in both systems is not connected with a structural transition [19]. From the analysis of the obtained data (Fig. 4(a)), we find that the intercluster distance C d decreases by only about 0.20 Å at p ≈ 20 GPa (s. Fig. 4(b)), resulting in values of C d ≈ 4.10 Å for both compounds. This is still much larger than the Nb-Nb or Ta-Ta intracluster distances (≈ 3 Å) and most likely too long for any noticeable overlap of d-orbitals of metal atoms of neighbouring cluster units. Thus, the change of direct metal-metal intercluster hopping with increasing pressure cannot account for the observed metallic and superconducting state.To obtain specific information about a possible change of the local structure with increasing pressure, we have performed single crystal x-ray structure determinations of GaTa 4Se 8 up to p = 15 GPa. Fig. 4(c) shows the pressure-induced change of the two characteristic Ta-Se bond lengths: Ta −Se1 within the Ta 4Se 4 clusters and the bridging Ta −Se2 bonds between the clusters (see Fig. 1(b)). As clearly shown in Fig. 4(c), the intercluster bond length Ta −Se2 decreases with increasing pressure more than 3 times stronger than that of the intracluster Ta −Se1. Consequently, the Ta atoms move towards the centre of the TaSe 6 octahedra with increasing pressure, which results in a corresponding decrease of the distortion of these octahedral ((see Fig. 1(b)). Apparently, this leads to a strong increase of the hybridization of the 5d -states of Ta with the p -states of the bridging Se2-ions, and to a consequent increase ofthe effective intercluster hopping. We believe that such a change leads to the observed pressure-induced metallic and superconducting state in GaNb4Se8 and GaTa4Se8.The pronounced changes of the local structure under pressure are expected to affect the vibrational properties, with a possible relationship to the occurrence of superconductivity. Indeed, we find highly unusual changes in phonon frequencies. Selected Raman spectra of GaTa4Se8 measured up to 15 GPa at 300 K are shown in Fig. 5(a). According to group theory, 12 zone-center Raman-active modes (3A1+ 3E + 6T2) are expected, but fewer modes are observed. The frequencies of the two well-resolved features seen at ambient pressure are 236 cm-1and 272 cm-1. Additional Raman lines become clearly observable with increasing pressure. The most notable effect in the present context is the pressure-driven softening of the strongest Raman band (Fig. 5(b)). The initial shift of -7.3 cm-1 saturates near 15 GPa, where the mode frequency has dropped by 20 %. GaNb4Se8 exhibits qualitatively similar pressure effects; the frequency of the dominant mode decreases in a nonlinear fashion from 234 cm-1 at ambient pressure to 180 cm-1 at 20 GPa. The soft mode is attributed to vibrations involving the stretching of Ta(Nb)-Se bonds. Polarized single-crystal Raman spectra are needed for a detailed mode assignment.The pressure-driven mode softening indicates the emergence of a strongly anharmonic potential energy for atomic displacements. The anharmonicity is induced here by the continuous suppression of the octahedral distortion under pressure. Similar, but less pronounced pressure effects on phonon modes have been reported for ionic compounds with distorted octahedral coordination (see e.g. [20]). In view of the changes in chemical bonding discussed above, one may speculate that this soft mode enhances the electron-phonon coupling and may contribute to Cooper pairing.In conclusion, we observed in cluster compounds GaNb4Se8and GaTa4Se8a rather rare phenomenon - pressure-induced transition from Mott insulator to a superconductor. We have shown that this transition is not connected with a structural phase transition, but is accompanied, and may be is driven by the internal structural modifications - a reduction of the distortion of M Se6 octahedra. The transition to a metallic and superconducting state under pressure is accompanied by strong softening of one of the phonon modes. However, the existing data are not yet sufficient to determine the exact type and mechanism of superconductivity; this requires further study. But the results already obtained make this system a very interesting object for the experimental investigation of the interplay between strong electron correlations and superconductivity in stoichiometric materials without disorder as well as for testing related theoretical models.M. M. A. would like to thank L. H. Tjeng and K. Westerholt for fruitful discussions. This work was supported by the Deutsche Forschungsgemeinschaft through SFB 608 and JO257/2.a Also at Groningen University, Nijenborgh 4, 9722 AG Groningen, The NetherlandsREFERENCES:[1]J. G. Bednorz and K. A. Müller, Z. Phys. B 64, 189 (1986). [2]K. Takada et al., Nature 422, 53 (2003). [3]T. Yamauchi et al., Phys. Rev. Lett. 89, 57002 (2002). [4]G. Baskaran, Phys. Rev. Lett. 90, 197007 (2003). [5]J. M. Vandenberg and D. Brasen, J. Solid State Chem. 14, 203 (1975). [6]H. B. Yaich et al., J. Less-Common Met. 102, 9 (1984). [7]Y. Sahoo, and A. K. Rastogi, J. Phys.: Condens. Matter 5, 5953 (1993). [8]A. K. Rastogi, and A. Niazi, Physica B 223 & 224, 588 (1996). [9]D. Johrendt, Z. Anorg. Allg. Chem. 624, 952 (1998). [10]R. Pocha et al., Chem. Mater. 12, 2882 (2000). [11]N. Shanthi and D. D. Sarma, J. Solid State Chem. 148, 143 (1999). [12]S. Harris, Polyhedron, 8, 2843 (1989). [13]R. Pocha and D. Johrendt, unpublished results. [14]B. Ni, Ph.D thesis, University of Cologne (Shaker Verlag Aachen) (2001). [15]H. Takagi et al., Phys. Rev. B 40, 2254 (1989). [16]C. W. Chu et al., Phys. Rev. Lett. 46, 276 (1981). [17]Y. S. Yao et al., Phys. Rev. B 37, 5032 (1988). [18]N. R. Werthamer et al., Phys. Rev. 147, 295 (1966). [19] We find at higher pressures, p ≈ 22 GPa (GaNb 4Se 8) and p ≈ 23 GPa (GaTa 4Se 8), asmall but sharp decrease of a (≤∆a a 1.8 %) without any change of the lattice symmetry. The kink in )(p T C at p ≈ 22 GPa in GaNb 4Se 8 seems to correlate with this structural transition.[20]U. Schwarz et al., J. Solid State Chem. 118, 20 (1995).Fig. 1: (a) Linkage of the Ta 4Se 4 cluster units via bridging Se2 atoms and their connection with the GaSe 4 tetrahedra in the fcc GaMo 4S 8 structure. (b) (Ta,Nb) atoms shifted off the centres of distorted edge-sharing Se 6 octahedra (d Ta-Se1 = 2.508 Å; d Ta −Se2 = 2.643 Å). (c) MO-scheme for the M −M bonding orbitals of a M 4 cluster with ideal T d symmetry for 7 electrons per cluster.Fig. 2: Temperature dependence of the normalized electrical resistance ()()[]K R T R R n 297= of GaNb 4Se 8 (a) and GaTa 4Se 8 (b) at different pressures up to 28.5 GPa. Insets show the drop of n R at high pressures and low temperatures.Fig. 3: Temperature dependence of the electrical resistance as a function of magnetic field at pressures of 22 GPa and 20 GPa for GaTa4Se8(a) and GaNb4Se8(b), respectively. (c)T in GaTa4Se8 and GaNb4Se8.Pressure dependence ofCFig. 4: (a) Pressure dependence of the lattice parameter a of the unit cell of Ga(Nb,Ta)4Se8 at 300 K. (b) Pressure variation of the intercluster distance (d) of Ga(Nb,Ta)4Se8 as deducedCfrom the experimental data in (a). (c) Pressure-induced change of Ta-Se1 and Ta-Se2 bondlengths in GaTa4Se8 relative to their values at ambient pressure ()0d.Fig. 5: Raman scattering results for GaTa4Se8: (a) Selected Raman spectra and (b) frequencies of Raman features as a function of pressure. Asterisks in (a) indicate laser plasma lines. Linesin (b) are guides to the eye.。

2nd IEEE International Conference on Cloud Computing Technology and Science

2nd IEEE International Conference on Cloud Computing Technology and ScienceSelf-Organizing Agents for Service Composition in Cloud Computing (59)J. Octavio Gutierrez-Garcia and Kwang-Mong SimelasticLM: A Novel Approach for Software Licensing in DistributedComputing Infrastructures (67)Claudio Cacciari, Francesco D’Andria, Miriam Gozalo, Björn Hagemeier,Daniel Mallmann, Josep Martrat, David García Peréz, Angela Rumpl,Wolfgang Ziegler, and Csilla ZsigriA Mechanism of Flexible Memory Exchange in Cloud Computing Environments (75)Takeshi Okuda, Eiji Kawai, and Suguru YamaguchiA Novel Approach for Cooperative Overlay-Maintenance in Multi-overlay Environments (81)Chin-Jung Hsu, Wu-Chun Chung, Kuan-Chou Lai, Kuan-Ching Li, and Yeh-Ching ChungFine-Grained Data Access Control Systems with User Accountability in Cloud Computing (89)Jin Li, Gansen Zhao, Xiaofeng Chen, Dongqing Xie, Chunming Rong, Wenjun Li,Lianzhang Tang, and Yong TangTrusted Data Sharing over Untrusted Cloud Storage Providers (97)Gansen Zhao, Chunming Rong, Jin Li, Feng Zhang, and Yong TangA Token-Based Access Control System for RDF Data in the Clouds (104)Arindam Khaled, Mohammad Farhan Husain, Latifur Khan, Kevin W. 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Chebyshev method

Time-domain steady-state simulation of frequency-dependent components using multi-intervalChebyshev methodBaolin Y ang Cadence Design Systems byang@Joel Phillips Cadence Berkeley Laboratories jrp@ABSTRACTSimulation of RF circuits often demands analysis of distributed component models that are described via frequency-dependent multi-port Y,Z,or S parameters.Frequency-domain methods such as harmonic balance are able to handle these components without dif-ﬁculty,while they are more difﬁcult for time-domain simulation methods to treat.In this paper,we propose a hybrid frequency-time approach to treat these components in steady-state time-domain simulations.Efﬁciency is achieved through the use of the multi-interval Chebyshev(MIC)simulation method and a low-order rational-ﬁtting model for preconditioning matrix-implicit Krylov-subspace solvers.Categories and Subject DescriptorsI.6[Computing Methodologies]:Simulation and ModelingGeneral TermsAlgorithmsKeywordsRF circuit simulation,frequency dependent,S parameter1.INTRODUCTIONSpecial-purpose RF circuit simulators exploit the sparsity of the spectrum in order to make the computations tractable[1].RF com-munication circuits often contain components(particularly passive components)such as transmission lines,integrated inductors,and SAWﬁlters,where distributed effects are important.Distributed components are those that are not conveniently described by aﬁnite-dimensional(i.e.lumped)state-space model,for example because they have an inﬁnite-dimensional state space.We will use the term to mean any component more conveniently described by a frequency-domain or convolutional representation.The author is now with Celestry Design Technologies,Inc. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for proﬁt or commercial advantage and that copies bear this notice and the full citation on theﬁrst page.To copy otherwise,to republish,to post on servers or to redistribute to lists,requires prior speciﬁc permission and/or a fee.DAC2002,June10-14,2002,New Orleans,Louisiana,USA.Copyright2002ACM1-58113-461-4/02/0006...$5.00.Circuit simulation methods solve circuit equations in either the time domain or frequency domain.Time-domain simulation meth-ods discretize circuit equations usingﬁnite difference methods such as the second-order Gear method.The advantage of the time-domain methods is that they can select time-points based on localized er-ror estimates and as a result can easily handle strongly nonlinear phenomena and sharp transitions in circuit waveforms.Frequency-domain simulation methods,such as the harmonic balance method, are popular for RF simulation and have the advantage of attaining spectral accuracy for smooth waveforms.Development of matrix-free Krylov-subspace algorithms[2,3,4]has made dedicated RF simulation tools even more popular as they can now be used to an-alyze circuits with thousands of devices.In contrast to frequency-domain techniques,which have no more difﬁculty simulating distributed devices than lumped devices,a draw-back of time-domain methods is that the simulation of distributed or passive components is more difﬁcult.There are generally two ap-proaches.Theﬁrst approach is to apply a model generation tool[5, 6]to generateﬁnite-dimensional state-space models for those com-ponents.These state-space models are easily simulated in the time domain.The second approach is to compute the impulse response of the distributed element,and apply a convolution approach at each timestep to obtain the time-domain response of the element. In transient analysis,one may use direct convolution approaches to simulate these components[7]without signiﬁcant difﬁculty.In RF simulation,however,such as periodic steady-state analy-sis via shooting-Newton methods[3],problems arise due to the fact that distributed components have inﬁnite-dimensional state.First, it is not sufﬁcient to calculate the state of the circuit at a single point of the periodic time interval in order to describe the peri-odic steady-state,the distributed state of the devices must be com-puted[8].This in turn implies that the sensitivity calculation in shooting methods involves more than the two ending points of the periodic interval.Second,the distributed components destroy the block-banded structure of the Jacobian that is exploited in precon-ditioner computation[3].In this paper,we propose a novel hybrid frequency-time method to include these distributed passive components in an RF simulator. Because we will take a matrix-implicit approach to equation solu-tion,the distributed devices can be treated as black-boxes with a port-to-port operator given most naturally by frequency-dependent S/Y/Z-parameters.Since it is only necessary to evaluate this opera-tor,not to write an explicit matrix,we propose an approach that is a dual to traditional harmonic balance.We formulate the equations in the time-domain,then use Fourier analysis to switch between can-didate time-domain waveform solutions and the frequency-domain spectrum where we evaluate the frequency-domain devices.Withthis combined frequency/time-domain approach,we can inherit the advantages of both time-domain methods in describing hard non-linearities,and harmonic balance methods in conveniently treating frequency-domain devices.In principle,any time domain method can be used,but for tra-ditional low-order polynomial methods such as second-order Gear, the Fourier transformation step is expensive.Therefore we advo-cate the multi-interval Chebyshev(MIC)method[9]as our time-domain simulation method,as we can preserve the accuracy of the frequency-domain model description,but with less computational time.Within the multi-interval Chebyshev context,the remain-der of the paper gives details of the formulation,describes optimal partitioning of distributed device evaluation between time and fre-quency domains,and considers practical aspects such as effective preconditioner construction.2.CIRCUIT EQUATIONS2.1Formulating Distributed Component Equa-tionsLumped circuit behavior is usually described by a set of N non-linear differential algebraic equations(DAEs)that can be written, without loss of generality,asdQ v tdtI v t u t0(1)where Q v t n represents contribution of the dynamic ele-ments such as inductors and capacitors,I v t n is the vector representing static elements such as resistors,u t n is the vec-tor of inputs,and the vector v t n contains state variables such as node voltages.Distributed components must be described us-ing a more general form.Suppose that there are K linear,time-invariant,but distributed devices,each of which possess a convolu-tional(impulse response)representation.Then we may formulate the time-domain circuit equations asdQ v tdt I v tK∑k1P kt∞H k tτP T k vτdτu t0(2)Here H:∞t p k p k is the p k p k matrix of impulse re-sponses for the k th distributed component.It is related to the restof the circuit equations via a matrix P k n p.P T k is an opera-tor that extracts the relevant portion of the state vector.That is,P kis one in the i j entry if the j th connection to the k th device is through the state variable with index i.In RF simulation,we are usually interested in solving steady-state problems.The periodic steady-state problem is the simplest and serves as a model for all the others.It is also important to study because it forms the basis of more advanced analyses,such as cy-clostationary noise analysis,multi-frequency distortion analysis,or envelope simulation.The periodic steady-state solution is the solu-tion of Eq.(2)that also satisﬁes the two-point boundary conditionv t T v t0for some period T and all t.It is therefore suf-ﬁcient to calculate the periodic steady-state solution in the interval0T,i.e.we seek v0T n.This is useful because we may utilize the(discrete)Fourier transform F v:,and its inverseF1=F T,in representing and manipulating signals.F maps v on0T to its frequency-domain representation,the Fourier series coefﬁcients.Without loss of generality,we can describe any physically-realizable distributed linear element as an N-port component described by anN N frequency-dependent scattering(S)parameter matrix.As the convolution operation is diagonalized by the Fourier transform,we may write the circuit equations in terms of the frequency-dependent matrix HωasdQ v tdtI v tN∑k1P k F T H kωF P T k v u t0(3)Here H kωis the(continuous)Fourier transform of the k th impulse response matrix.The relation between the scattering parameter ma-trix Sωof the underlying element and the matrix Hωdepends on the way the circuit equations are stamped.For simplicity of pre-sentation,we consider a2-port device with S-parameter description to be stamped in the MNA formulation.We denote the potential at theﬁrst port as e1and e1and the second port as e2and e2.The cur-rents at the two ports are denoted as I1and I2.The port resistance is R1and R2.If we ignore contributions from other circuit elements (*),formally the circuit equation that describes the behavior of the 2-port and deﬁnes Hωis:0000100000100000010000011S11S111S12R1R2S12R1R2S111R1S12R1R2S21R2R1S21R2R11S22S221S21R1R2S221R2e1e1e2e2i1i200(4) 2.2DiscretizationIn frequency-domain simulation methods one solves for the Fourier spectrum of the waveform instead of function values at a set of time points.It is straightforward to include distributed or passive components that may be described using frequency-dependent tab-ulated S-parameter.With the S-parameter information,one knows how to load the Jacobian of the circuit equation in frequency do-main and set up the linear system.A linear solver,such as a Krylov-subspace iterative solver,can be used to solve the linear system.In time-domain methods,on the other hand we solve for the waveform values,u t j,at a set of time steps:t j j01M.To perform the discretization at t t k,we seek an interpolating poly-nomial of degree p based on p1time points.The time derivative is approximated by evaluating the derivative of the interpolating polynomial at t t k.Thus the time-derivative of the approximate solution is required to match the differential equation at the point t k.The time steps are not uniform and their variation can be big.It is difﬁcult to simulate these components on this set of time points without loss of efﬁciency.We propose an approach that makes this happen.In our novel approach,advanced Krylov-subspace iterative solver and multi-domain Chebyshev method are used to achieve this purpose.Here we give the basics of MIC method.Given the values,u t j, at the Chebyshev Gauss-Lobatto collocation points,t j cosπj M j0M in one subinterval,we seek an interpolating polynomial of degree M:I M u tM∑k0˜u k T k t(5)and the derivative of the polynomial can be computed by using backward recurrence relation for Chebyshev polynomials T k t.We can also obtain a differentiation matrix D by noting that the interpo-lating polynomial can be expressed in terms of a Lagrange interpo-lation polynomial g j t.The entries of the matrix D are computed by taking the analytical derivative of g j t,D k j g j t k.Refer to[9]for the exact formulation of the matrix.As in harmonic balance, high accuracy is achievable because in each approximation interval all the desirable approximation properties of Chebyshev polynomi-als can be exploited.3.HYBRID FREQUENCY-TIME DOMAINSIMULATION3.1Matrix-Implicit Krylov Subspace SolutionTechniqueThe major computational task for large-scale RF simulation is to solve the large linear systems generated from the circuit Jacobian matrix.In the nonlinear steady-state computation,a system must be solved at each step of Newton’s method.The Jacobian J is com-posed of three pieces,the dynamic piece J C,the static piece J G,and the distributed piece J D.With the Chebyshev discretization andG t k ∂I∂v v tkC t k∂Q∂v v tk(6)we haveJ C D1I1D10I1...D20I2D2I2...C t1C t2C t3...(7)where D k is part of the Chebyshev differentiation matrix D in the k th interval(D i j i j1M),D k0is D i j i1M;j0,and I k the identity matrix of dimension equal to the number of timepoints in the k th interval,J GG t1G t2G t3...and J DN∑k1P k F T H kωF P T k(8) In small-signal and noise analysis,a related system must be solvedat each frequency point of analysis.One notes that because of thefrequency-dependent property of the distributed components,thetransformation F is dependent on all time points in the simulation time interval.From Equation3,it is clear that every p-port dis-tributed element introduces p rows and columns into the Jacobianthat are full.This can be seen by noting that the matrices P F T and F P T are dense.It is not possible to form and solve large systems of such a form efﬁciently using direct linear algebra.In our ap-proach,as is now standard,a Krylov-subspace iterative solver such as GMRES is used.The use of these methods in circuit simulation are treated in detail elsewhere[2,3,4,10].We focus on the matrix-implicit treatment of the distributed com-ponents.To evaluate the term J D,we need a procedure that takestime-domain waveforms as input,applies the Fourier transforma-tion to transform them into the frequency domain,applies the trans-fer function of the distributed components,in terms of frequency-dependent S-parameters,to the frequency-domain result,and inverse-transforms to the time-domain.Such a procedure has as muchﬂex-ibility in dealing with distributed-components as harmonic balance methods.In fact,there is moreﬂexibility.We can extract any part of the transfer function that is best suited to evaluation in the time-domain and perform that part of the port-to-port transformation in time domain.For example,we can perform the delay transforma-tion of transmission lines in the time domain and obtain much more accurate answers than possible by representing the delay using a sum of Fourier harmonics.All these operationsﬁt in the matrix-vector product calculation of iterative solvers.3.2Time-Frequency DecompositionTaking a two-port system as an example,we now explain the partitioning of the distributed component properties between the time-and frequency-domain representations.For each port in the distributed component,there is a port equation in the circuit equa-tions.For a two-port system,we have1S11e1S111e1S12R1R2e2S12R1R2e2S111R1i1S12R1R2i20(9) for one port and a similar equation for the other port.Note that the S-parameter in this formula is frequency dependent.Hence the corresponding equations in time domain are more complicated and can be obtained using frequency-time-domain transformations.For periodic steady state analysis,the spectrum of solution,which we show how to compute in the next section,is at integer multiples of the fundamental frequency.We need the S-parameters at those frequencies and then we can obtain the equations in time domain. Now we discuss decomposing the port equations into a frequency-independent part and a frequency-dependent S-parameter part.We decompose theﬁrst port equation intoA11¯S11e11¯S11e1¯S12R1R2e2¯S12R1R2e21¯S11R1i1¯S12R1R2i2(10) andB1¯S11S11e1S11¯S11e1S12¯S12R1R2e2S12¯S12R1R2e2S11¯S11R1i1S12¯S12R1R2i2(11)and enforce A1B10.Likewise we can decompose the second port equation.In general,s-parameters with bar’s in the split equations are the S-parameters of some time-domain effect,such as delay,DC offset, or linear scale.These time-domain effect can be calculated in time domain separately.One simple example is to choose constants for the approximation when S-parameters at high frequencies approach a constant.Thus,the frequency-dependent parts,B1and B2,do not have any high-frequency effects,which can be calculated in frequency domain through Fourier transforms without exhibiting oscillations in sharp-transition regions of time-domain waveform, known as Gibbs phenomena.3.3Chebyshev Fourier QuadraturesIt should be clear that,while the above procedure can be per-formed with time-domain periodic-steady-state solution method en-force periodicity explicitly,one key to efﬁcient implementation is efﬁcient evaluation of the Fourier transforms.The Chebyshev dis-cretization achieves high accuracy with relatively few timepoints. With this property,one can afford to apply Fourier transformation and inverse Fourier transformation to transform the waveforms be-tween frequency domain and time domain.The computational cost is O M2,where M is the number of time steps.Because of the non-uniform timepoint distribution,we cannot base an implemen-tation on the standard FFT,though we could use more sophisticated O M log M non-uniform Fourier analysis techniques[11].In order to use the frequency-dependent S-parameters directly, the time-domain waveform is transformed into frequency domain. We use Chebyshev quadrature formulas to obtain the Fourier spec-trum with high-order accuracy.The order of accuracy is basicallyequal to the number of Chebyshev collocation points,which in turn given by the number time steps in one Chebyshev time interval. Unlike low-order polynomial approximations such as Gear meth-ods,the Chebyshev quadratures maintain spectral accuracy in this process.Assume the time steps we have are t i i0i M. The integration formula ist M t0v t dtM∑i0w i v t i(12)wherew i1M21i02M11i1M21∑M21k1214k2cos2kiπMi M2w M i M2i M(13)Note that we are considering a periodic function.The Fourier spec-trum can be calculated using the following formula:t M t0v t e jk2πT t dtM∑i0w i v t i e jk2πT t i(14)With the Fourier spectrum known,it is possible to perform thenecessary part of the Jacobian and equation evaluation in the fre-quency domain,just as in harmonic balance.After we obtain theresult of the operation in frequency domain,we then transform itback to the time domain using inverse Fourier transformation.Ourtime-domain simulation needs to have the result at M time pointst i i0M.For all these transformations,the computational complexity of the multi-interval Chebyshev method and traditional low-order methods are the same,generally proportional to M orM2.Because M for the Chebyshev methods is much less than thatof traditional time-domain methods,the Chebyshev methods havea large advantage over traditional low-order methods.This Fourier weight calculation method is subject to aliasing er-ror.When a large k is required,the integration formula(14)may not have enough resolution,because M needs to be at least2k for the formula to have any accuracy.Spectral accuracy is achievedwhen Mk is large.However,one can interpolate the solution att i i0M to aﬁner Chebyshev grid.This interpolation keeps the high-order accuracy of the time-domain solution.Then one can perform integration of(14)on theﬁne grid to guarantee the accuracy.In practice,it is not necessary to really interpolate the solution.One only needs to pre-correct the integration weights,w i i0M,to realize this reﬁnement in integration.This sit-uation can occur when the time-domain waveform in an interval is much smoother than the Fourier harmonic to be evaluated.4.PRECONDITIONING4.1Block-Lower Triangular Preconditioners Rapid convergence of Krylov-subspace iterative methods requires construction of effective preconditioners.Shooting-based time-domain RF simulation methods[2,3,10]typically construct precondition-ers by observing that that block-lower-triangular portion of the Ja-cobian is easily inverted,given that the diagonal blocks can be ef-ﬁciently factored via sparse direct techniques.That is,to solve the preconditioned system P1Jx P1b,the Jacobian isﬁrst decom-posed into upper-and lower-block triangular pieces,J L U,withU0D10I1...C t1C t2C t3...(15) L˜J C J G,where˜JCD1I1...D20I2D2I2...C t1C t2C t3...(16)The preconditioner is then P˜J C J G L.As P is block-lower-triangular,inversion is easy as long as the diagonal blocks can be factored or inverted efﬁciently,which is the case here.The problem with applying a similar technique,when distributed elements are present,is the term J D.In the matrix-implicit context,it is not even clear how to construct the lower-diagonal piece,nor how to invert it,much less that this would be an effective strategy.In general,solutions at nodes of the distributed components are coupled in time because of the Fourier transformation done in the periodic time interval.Existing preconditioning techniques for pe-riodic steady-state simulation can not solve this difﬁculty.In this paper,we propose to use low-order lumped models to construct the distributed part of the preconditioner.4.2Rational FittingTheﬁrst idea in our preconditioning strategy is toﬁrst construct an approximate˜Hωthat is close to Hω.One possibility to con-struct an approximate˜Hωis to use rational approximation[5]of the underlying device parameters Sω.We propose constructing a rational approximation˜Sωof Sω.Since,because the ratio-nal approximation will be used for a preconditioner,it does not have to be extremely accurate,we may approximate Sωwith a low-order model,which is cheap to construct.This approach has the advantage that we may use the block-lower-triangular precon-ditioners discussed in the previous section.Constructing a rational function model is equivalent toﬁnding a state-space model of the formdxdtAx Bu t y Cx Du(17)such that the transfer function˜SωD C iωI A1B approx-imates Sω.We do this by requiring that the mean-square errorE∑kWωk2˜Sωk Sωk2(18)be minimized[5].In the above,Wωk represent the weighting,at frequency point k,for the least-squares minimization.4.3Lumped Model PreconditionerIdeally,the preconditioner would then be P˜J C J G˜J D,where˜JDN∑k1P k F T˜H kωF P T k v(19)Unfortunately,even with the rational model technique,we cannot invert matrices of this form efﬁciently.However,as state-space models haveﬁnite-dimensional state space,the rational function models may be included in an extended version of the circuit equa-tions.The extended circuit equations will include the states of the rationalﬁtting model,and so the Jacobian of the extended system will be the Jacobian of the original system treating the distributedcomponent as the rationalﬁtting model.Adopting the J L U nota-tion,letJ R L R U R(20) be the Jacobian for the circuit equations including the rationalﬁt-ting model,where R denotes rationalﬁtting,and L R and U R de-note the upper-and lower-,respectively,block-triangular pieces as deﬁned above.If there are M timesteps in the Chebyshev dis-cretization of the periodic interval,and n circuit equations,we have J Mn Mn but J R M n q M n q where q is the total number of additional state space variables introduced by the rational ap-proximation of the frequency response of all the distributed com-ponents.Thus we cannot simply take P L R,as L R and J have different dimensions.Deﬁne an operator Q:Mn M n q that embeds the dimen-sion M n q space into the Mn one–it is a section of the identity matrix.Q T likewise extracts the original state space from the ex-tended one.The relation that the Jacobian of the extended system will be the Jacobian of the original system treating the distributed component as the rationalﬁtting model can be expressed asJ Q T J R Q(21) To use L R as the preconditioner for Jacobian J,we may use Q T to extract the relevant components,so thatP1Q T L1R Q(22) Note that the inverse of L R can be easily computed because it is a block lower-triangular system.Note that the preconditioner will also include an approximation to˜H.If the rationalﬁtting model is a fairly good approximation of the original system,the GM-RES solver should be efﬁcient whenever the block-lower-triangular strategy is also effective.Note that,because we use the rational model as a preconditioner, its accuracy does not affect the accuracy of theﬁnal solution.If we were to use rational approximation to construct an equivalent lumped model,instead of evaluating the distributed components, the rational model would need to very closely track the frequency response,meaning it would often be of very high order.We would also have to introduce physical constraints on the rational model–such as passivity–that are computationally expensive to enforce for high-order multi-port models.Both the model generation pro-cess,and the later time-domain simulation,would be more compu-tationally expensive than the strategy proposed here,where rational approximations of orders two or three are often enough to give ef-fective preconditioners.5.NUMERICAL RESULTSIn this section we show the effectiveness of the hybrid frequency-time multi-interval Chebyshev(MIC)method in periodic steady state simulation of distributed elements including a spiral induc-tor,a transmission line,and a SAWﬁlter.We compare the new approach with the pure rational-ﬁtting model generation approach. Efﬁciency of the low-order rationalﬁtting model in building a pre-conditioner for the hybrid frequency-time approach is shown through comparing the convergence history of GMRES solvers with and without the preconditioner.In theﬁrst example,we simulate the periodic steady state of a spiral inductor driven by a pulse source.The spiral inductor is de-scribed with an S-parameter table.The frequency range of the S-parameters is from50MHz to10GHz.The fundamental frequency of the periodic steady state is1GHz.We show the numerical re-sult,waveform at the output,of three simulations.First,we use Figure1:Comparison of low-order rationalﬁtting,high-order rationalﬁtting,and hybrid frequency-time MIC with low-orderFigure2:Comparison of GMRES convergence with and with-a third-order rationalﬁtting model and shooting-Newton method to obtain the periodic steady state.The solution is very inaccu-rate compared to other results.Clearly,the third-order rationalﬁt-ting model is inaccurate by itself.Second,we use a20th order rationalﬁtting model instead.The result is much more accurate than that of theﬁrst st,we use the hybrid frequency-time MIC method with the third-order rationalﬁtting model as the preconditioner.With no regard to the inaccuracy of the3rd-order model,the result of the new method is satisfactory.There is only a small discrepancy between the second and third simulation result, which may be due to the fact that the treatment of the S-parameters is done in fundamentally different ways and there is no guarantee that the two methods will agree in the frequency range outside of 50MHz10GHz.Although the accuracy of rationalﬁtting model is satisfactory in this experiment,it is inefﬁcient because of the high order used in doing rationalﬁtting.The additional state-space equations added when the high-order rationalﬁt model is used means that the cir-cuit equations we need to solve are of much larger size.Indeed, the overall simulation time of the hybrid frequency-time domain method is5seconds while that of the20th-order rationalﬁtting method is30seconds.In Fig.2we plot the convergence history of the GMRES solver.We compare GMRES convergence with the 3rd-orderﬁtting model preconditioner and without the precondi-tioner.One can clearly see that GMRES convergence without the preconditioner is very slow while GMRES converges in a few itera-。

ACM-GIS%202006-A%20Peer-to-Peer%20Spatial%20Cloaking%20Algorithm%20for%20Anonymous%20Location-based%

A Peer-to-Peer Spatial Cloaking Algorithm for AnonymousLocation-based Services∗Chi-Yin Chow Department of Computer Science and Engineering University of Minnesota Minneapolis,MN cchow@ Mohamed F.MokbelDepartment of ComputerScience and EngineeringUniversity of MinnesotaMinneapolis,MNmokbel@Xuan LiuIBM Thomas J.WatsonResearch CenterHawthorne,NYxuanliu@ABSTRACTThis paper tackles a major privacy threat in current location-based services where users have to report their ex-act locations to the database server in order to obtain their desired services.For example,a mobile user asking about her nearest restaurant has to report her exact location.With untrusted service providers,reporting private location in-formation may lead to several privacy threats.In this pa-per,we present a peer-to-peer(P2P)spatial cloaking algo-rithm in which mobile and stationary users can entertain location-based services without revealing their exact loca-tion information.The main idea is that before requesting any location-based service,the mobile user will form a group from her peers via single-hop communication and/or multi-hop routing.Then,the spatial cloaked area is computed as the region that covers the entire group of peers.Two modes of operations are supported within the proposed P2P spa-tial cloaking algorithm,namely,the on-demand mode and the proactive mode.Experimental results show that the P2P spatial cloaking algorithm operated in the on-demand mode has lower communication cost and better quality of services than the proactive mode,but the on-demand incurs longer response time.Categories and Subject Descriptors:H.2.8[Database Applications]:Spatial databases and GISGeneral Terms:Algorithms and Experimentation. Keywords:Mobile computing,location-based services,lo-cation privacy and spatial cloaking.1.INTRODUCTIONThe emergence of state-of-the-art location-detection de-vices,e.g.,cellular phones,global positioning system(GPS) devices,and radio-frequency identiﬁcation(RFID)chips re-sults in a location-dependent information access paradigm,∗This work is supported in part by the Grants-in-Aid of Re-search,Artistry,and Scholarship,University of Minnesota. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for proﬁt or commercial advantage and that copies bear this notice and the full citation on theﬁrst page.To copy otherwise,to republish,to post on servers or to redistribute to lists,requires prior speciﬁc permission and/or a fee.ACM-GIS’06,November10-11,2006,Arlington,Virginia,USA. Copyright2006ACM1-59593-529-0/06/0011...$5.00.known as location-based services(LBS)[30].In LBS,mobile users have the ability to issue location-based queries to the location-based database server.Examples of such queries include“where is my nearest gas station”,“what are the restaurants within one mile of my location”,and“what is the traﬃc condition within ten minutes of my route”.To get the precise answer of these queries,the user has to pro-vide her exact location information to the database server. With untrustworthy servers,adversaries may access sensi-tive information about speciﬁc individuals based on their location information and issued queries.For example,an adversary may check a user’s habit and interest by knowing the places she visits and the time of each visit,or someone can track the locations of his ex-friends.In fact,in many cases,GPS devices have been used in stalking personal lo-cations[12,39].To tackle this major privacy concern,three centralized privacy-preserving frameworks are proposed for LBS[13,14,31],in which a trusted third party is used as a middleware to blur user locations into spatial regions to achieve k-anonymity,i.e.,a user is indistinguishable among other k−1users.The centralized privacy-preserving frame-work possesses the following shortcomings:1)The central-ized trusted third party could be the system bottleneck or single point of failure.2)Since the centralized third party has the complete knowledge of the location information and queries of all users,it may pose a serious privacy threat when the third party is attacked by adversaries.In this paper,we propose a peer-to-peer(P2P)spatial cloaking algorithm.Mobile users adopting the P2P spatial cloaking algorithm can protect their privacy without seeking help from any centralized third party.Other than the short-comings of the centralized approach,our work is also moti-vated by the following facts:1)The computation power and storage capacity of most mobile devices have been improv-ing at a fast pace.2)P2P communication technologies,such as IEEE802.11and Bluetooth,have been widely deployed.3)Many new applications based on P2P information shar-ing have rapidly taken shape,e.g.,cooperative information access[9,32]and P2P spatio-temporal query processing[20, 24].Figure1gives an illustrative example of P2P spatial cloak-ing.The mobile user A wants toﬁnd her nearest gas station while beingﬁve anonymous,i.e.,the user is indistinguish-able amongﬁve users.Thus,the mobile user A has to look around andﬁnd other four peers to collaborate as a group. In this example,the four peers are B,C,D,and E.Then, the mobile user A cloaks her exact location into a spatialA B CDEBase Stationregion that covers the entire group of mobile users A ,B ,C ,D ,and E .The mobile user A randomly selects one of the mobile users within the group as an agent .In the ex-ample given in Figure 1,the mobile user D is selected as an agent.Then,the mobile user A sends her query (i.e.,what is the nearest gas station)along with her cloaked spa-tial region to the agent.The agent forwards the query to the location-based database server through a base station.Since the location-based database server processes the query based on the cloaked spatial region,it can only give a list of candidate answers that includes the actual answers and some false positives.After the agent receives the candidate answers,it forwards the candidate answers to the mobile user A .Finally,the mobile user A gets the actual answer by ﬁltering out all the false positives.The proposed P2P spatial cloaking algorithm can operate in two modes:on-demand and proactive .In the on-demand mode,mobile clients execute the cloaking algorithm when they need to access information from the location-based database server.On the other side,in the proactive mode,mobile clients periodically look around to ﬁnd the desired number of peers.Thus,they can cloak their exact locations into spatial regions whenever they want to retrieve informa-tion from the location-based database server.In general,the contributions of this paper can be summarized as follows:1.We introduce a distributed system architecture for pro-viding anonymous location-based services (LBS)for mobile users.2.We propose the ﬁrst P2P spatial cloaking algorithm for mobile users to entertain high quality location-based services without compromising their privacy.3.We provide experimental evidence that our proposed algorithm is eﬃcient in terms of the response time,is scalable to large numbers of mobile clients,and is eﬀective as it provides high-quality services for mobile clients without the need of exact location information.The rest of this paper is organized as follows.Section 2highlights the related work.The system model of the P2P spatial cloaking algorithm is presented in Section 3.The P2P spatial cloaking algorithm is described in Section 4.Section 5discusses the integration of the P2P spatial cloak-ing algorithm with privacy-aware location-based database servers.Section 6depicts the experimental evaluation of the P2P spatial cloaking algorithm.Finally,Section 7con-cludes this paper.2.RELATED WORKThe k -anonymity model [37,38]has been widely used in maintaining privacy in databases [5,26,27,28].The main idea is to have each tuple in the table as k -anonymous,i.e.,indistinguishable among other k −1tuples.Although we aim for the similar k -anonymity model for the P2P spatial cloaking algorithm,none of these techniques can be applied to protect user privacy for LBS,mainly for the following four reasons:1)These techniques preserve the privacy of the stored data.In our model,we aim not to store the data at all.Instead,we store perturbed versions of the data.Thus,data privacy is managed before storing the data.2)These approaches protect the data not the queries.In anonymous LBS,we aim to protect the user who issues the query to the location-based database server.For example,a mobile user who wants to ask about her nearest gas station needs to pro-tect her location while the location information of the gas station is not protected.3)These approaches guarantee the k -anonymity for a snapshot of the database.In LBS,the user location is continuously changing.Such dynamic be-havior calls for continuous maintenance of the k -anonymity model.(4)These approaches assume a uniﬁed k -anonymity requirement for all the stored records.In our P2P spatial cloaking algorithm,k -anonymity is a user-speciﬁed privacy requirement which may have a diﬀerent value for each user.Motivated by the privacy threats of location-detection de-vices [1,4,6,40],several research eﬀorts are dedicated to protect the locations of mobile users (e.g.,false dummies [23],landmark objects [18],and location perturbation [10,13,14]).The most closed approaches to ours are two centralized spatial cloaking algorithms,namely,the spatio-temporal cloaking [14]and the CliqueCloak algorithm [13],and one decentralized privacy-preserving algorithm [23].The spatio-temporal cloaking algorithm [14]assumes that all users have the same k -anonymity requirements.Furthermore,it lacks the scalability because it deals with each single request of each user individually.The CliqueCloak algorithm [13]as-sumes a diﬀerent k -anonymity requirement for each user.However,since it has large computation overhead,it is lim-ited to a small k -anonymity requirement,i.e.,k is from 5to 10.A decentralized privacy-preserving algorithm is proposed for LBS [23].The main idea is that the mobile client sends a set of false locations,called dummies ,along with its true location to the location-based database server.However,the disadvantages of using dummies are threefold.First,the user has to generate realistic dummies to pre-vent the adversary from guessing its true location.Second,the location-based database server wastes a lot of resources to process the dummies.Finally,the adversary may esti-mate the user location by using cellular positioning tech-niques [34],e.g.,the time-of-arrival (TOA),the time diﬀer-ence of arrival (TDOA)and the direction of arrival (DOA).Although several existing distributed group formation al-gorithms can be used to ﬁnd peers in a mobile environment,they are not designed for privacy preserving in LBS.Some algorithms are limited to only ﬁnding the neighboring peers,e.g.,lowest-ID [11],largest-connectivity (degree)[33]and mobility-based clustering algorithms [2,25].When a mo-bile user with a strict privacy requirement,i.e.,the value of k −1is larger than the number of neighboring peers,it has to enlist other peers for help via multi-hop routing.Other algorithms do not have this limitation,but they are designed for grouping stable mobile clients together to facil-Location-based Database ServerDatabase ServerDatabase ServerFigure 2:The system architectureitate eﬃcient data replica allocation,e.g.,dynamic connec-tivity based group algorithm [16]and mobility-based clus-tering algorithm,called DRAM [19].Our work is diﬀerent from these approaches in that we propose a P2P spatial cloaking algorithm that is dedicated for mobile users to dis-cover other k −1peers via single-hop communication and/or via multi-hop routing,in order to preserve user privacy in LBS.3.SYSTEM MODELFigure 2depicts the system architecture for the pro-posed P2P spatial cloaking algorithm which contains two main components:mobile clients and location-based data-base server .Each mobile client has its own privacy proﬁle that speciﬁes its desired level of privacy.A privacy proﬁle includes two parameters,k and A min ,k indicates that the user wants to be k -anonymous,i.e.,indistinguishable among k users,while A min speciﬁes the minimum resolution of the cloaked spatial region.The larger the value of k and A min ,the more strict privacy requirements a user needs.Mobile users have the ability to change their privacy proﬁle at any time.Our employed privacy proﬁle matches the privacy re-quirements of mobiles users as depicted by several social science studies (e.g.,see [4,15,17,22,29]).In this architecture,each mobile user is equipped with two wireless network interface cards;one of them is dedicated to communicate with the location-based database server through the base station,while the other one is devoted to the communication with other peers.A similar multi-interface technique has been used to implement IP multi-homing for stream control transmission protocol (SCTP),in which a machine is installed with multiple network in-terface cards,and each assigned a diﬀerent IP address [36].Similarly,in mobile P2P cooperation environment,mobile users have a network connection to access information from the server,e.g.,through a wireless modem or a base station,and the mobile users also have the ability to communicate with other peers via a wireless LAN,e.g.,IEEE 802.11or Bluetooth [9,24,32].Furthermore,each mobile client is equipped with a positioning device, e.g.,GPS or sensor-based local positioning systems,to determine its current lo-cation information.4.P2P SPATIAL CLOAKINGIn this section,we present the data structure and the P2P spatial cloaking algorithm.Then,we describe two operation modes of the algorithm:on-demand and proactive .4.1Data StructureThe entire system area is divided into grid.The mobile client communicates with each other to discover other k −1peers,in order to achieve the k -anonymity requirement.TheAlgorithm 1P2P Spatial Cloaking:Request Originator m 1:Function P2PCloaking-Originator (h ,k )2://Phase 1:Peer searching phase 3:The hop distance h is set to h4:The set of discovered peers T is set to {∅},and the number ofdiscovered peers k =|T |=05:while k <k −1do6:Broadcast a FORM GROUP request with the parameter h (Al-gorithm 2gives the response of each peer p that receives this request)7:T is the set of peers that respond back to m by executingAlgorithm 28:k =|T |;9:if k <k −1then 10:if T =T then 11:Suspend the request 12:end if 13:h ←h +1;14:T ←T ;15:end if 16:end while17://Phase 2:Location adjustment phase 18:for all T i ∈T do19:|mT i .p |←the greatest possible distance between m and T i .pby considering the timestamp of T i .p ’s reply and maximum speed20:end for21://Phase 3:Spatial cloaking phase22:Form a group with k −1peers having the smallest |mp |23:h ←the largest hop distance h p of the selected k −1peers 24:Determine a grid area A that covers the entire group 25:if A <A min then26:Extend the area of A till it covers A min 27:end if28:Randomly select a mobile client of the group as an agent 29:Forward the query and A to the agentmobile client can thus blur its exact location into a cloaked spatial region that is the minimum grid area covering the k −1peers and itself,and satisﬁes A min as well.The grid area is represented by the ID of the left-bottom and right-top cells,i.e.,(l,b )and (r,t ).In addition,each mobile client maintains a parameter h that is the required hop distance of the last peer searching.The initial value of h is equal to one.4.2AlgorithmFigure 3gives a running example for the P2P spatial cloaking algorithm.There are 15mobile clients,m 1to m 15,represented as solid circles.m 8is the request originator,other black circles represent the mobile clients received the request from m 8.The dotted circles represent the commu-nication range of the mobile client,and the arrow represents the movement direction.Algorithms 1and 2give the pseudo code for the request originator (denoted as m )and the re-quest receivers (denoted as p ),respectively.In general,the algorithm consists of the following three phases:Phase 1:Peer searching phase .The request origina-tor m wants to retrieve information from the location-based database server.m ﬁrst sets h to h ,a set of discovered peers T to {∅}and the number of discovered peers k to zero,i.e.,|T |.(Lines 3to 4in Algorithm 1).Then,m broadcasts a FORM GROUP request along with a message sequence ID and the hop distance h to its neighboring peers (Line 6in Algorithm 1).m listens to the network and waits for the reply from its neighboring peers.Algorithm 2describes how a peer p responds to the FORM GROUP request along with a hop distance h and aFigure3:P2P spatial cloaking algorithm.Algorithm2P2P Spatial Cloaking:Request Receiver p1:Function P2PCloaking-Receiver(h)2://Let r be the request forwarder3:if the request is duplicate then4:Reply r with an ACK message5:return;6:end if7:h p←1;8:if h=1then9:Send the tuple T=<p,(x p,y p),v maxp ,t p,h p>to r10:else11:h←h−1;12:Broadcast a FORM GROUP request with the parameter h 13:T p is the set of peers that respond back to p14:for all T i∈T p do15:T i.h p←T i.h p+1;16:end for17:T p←T p∪{<p,(x p,y p),v maxp ,t p,h p>};18:Send T p back to r19:end ifmessage sequence ID from another peer(denoted as r)that is either the request originator or the forwarder of the re-quest.First,p checks if it is a duplicate request based on the message sequence ID.If it is a duplicate request,it sim-ply replies r with an ACK message without processing the request.Otherwise,p processes the request based on the value of h:Case1:h= 1.p turns in a tuple that contains its ID,current location,maximum movement speed,a timestamp and a hop distance(it is set to one),i.e.,< p,(x p,y p),v max p,t p,h p>,to r(Line9in Algorithm2). Case2:h> 1.p decrements h and broadcasts the FORM GROUP request with the updated h and the origi-nal message sequence ID to its neighboring peers.p keeps listening to the network,until it collects the replies from all its neighboring peers.After that,p increments the h p of each collected tuple,and then it appends its own tuple to the collected tuples T p.Finally,it sends T p back to r (Lines11to18in Algorithm2).After m collects the tuples T from its neighboring peers, if m cannotﬁnd other k−1peers with a hop distance of h,it increments h and re-broadcasts the FORM GROUP request along with a new message sequence ID and h.m repeatedly increments h till itﬁnds other k−1peers(Lines6to14in Algorithm1).However,if mﬁnds the same set of peers in two consecutive broadcasts,i.e.,with hop distances h and h+1,there are not enough connected peers for m.Thus, m has to relax its privacy proﬁle,i.e.,use a smaller value of k,or to be suspended for a period of time(Line11in Algorithm1).Figures3(a)and3(b)depict single-hop and multi-hop peer searching in our running example,respectively.In Fig-ure3(a),the request originator,m8,(e.g.,k=5)canﬁnd k−1peers via single-hop communication,so m8sets h=1. Since h=1,its neighboring peers,m5,m6,m7,m9,m10, and m11,will not further broadcast the FORM GROUP re-quest.On the other hand,in Figure3(b),m8does not connect to k−1peers directly,so it has to set h>1.Thus, its neighboring peers,m7,m10,and m11,will broadcast the FORM GROUP request along with a decremented hop dis-tance,i.e.,h=h−1,and the original message sequence ID to their neighboring peers.Phase2:Location adjustment phase.Since the peer keeps moving,we have to capture the movement between the time when the peer sends its tuple and the current time. For each received tuple from a peer p,the request originator, m,determines the greatest possible distance between them by an equation,|mp |=|mp|+(t c−t p)×v max p,where |mp|is the Euclidean distance between m and p at time t p,i.e.,|mp|=(x m−x p)2+(y m−y p)2,t c is the currenttime,t p is the timestamp of the tuple and v maxpis the maximum speed of p(Lines18to20in Algorithm1).In this paper,a conservative approach is used to determine the distance,because we assume that the peer will move with the maximum speed in any direction.If p gives its movement direction,m has the ability to determine a more precise distance between them.Figure3(c)illustrates that,for each discovered peer,the circle represents the largest region where the peer can lo-cate at time t c.The greatest possible distance between the request originator m8and its discovered peer,m5,m6,m7, m9,m10,or m11is represented by a dotted line.For exam-ple,the distance of the line m8m 11is the greatest possible distance between m8and m11at time t c,i.e.,|m8m 11|. Phase3:Spatial cloaking phase.In this phase,the request originator,m,forms a virtual group with the k−1 nearest peers,based on the greatest possible distance be-tween them(Line22in Algorithm1).To adapt to the dynamic network topology and k-anonymity requirement, m sets h to the largest value of h p of the selected k−1 peers(Line15in Algorithm1).Then,m determines the minimum grid area A covering the entire group(Line24in Algorithm1).If the area of A is less than A min,m extends A,until it satisﬁes A min(Lines25to27in Algorithm1). Figure3(c)gives the k−1nearest peers,m6,m7,m10,and m11to the request originator,m8.For example,the privacy proﬁle of m8is(k=5,A min=20cells),and the required cloaked spatial region of m8is represented by a bold rectan-gle,as depicted in Figure3(d).To issue the query to the location-based database server anonymously,m randomly selects a mobile client in the group as an agent(Line28in Algorithm1).Then,m sendsthe query along with the cloaked spatial region,i.e.,A,to the agent(Line29in Algorithm1).The agent forwards thequery to the location-based database server.After the serverprocesses the query with respect to the cloaked spatial re-gion,it sends a list of candidate answers back to the agent.The agent forwards the candidate answer to m,and then mﬁlters out the false positives from the candidate answers. 4.3Modes of OperationsThe P2P spatial cloaking algorithm can operate in twomodes,on-demand and proactive.The on-demand mode:The mobile client only executesthe algorithm when it needs to retrieve information from the location-based database server.The algorithm operatedin the on-demand mode generally incurs less communica-tion overhead than the proactive mode,because the mobileclient only executes the algorithm when necessary.However,it suﬀers from a longer response time than the algorithm op-erated in the proactive mode.The proactive mode:The mobile client adopting theproactive mode periodically executes the algorithm in back-ground.The mobile client can cloak its location into a spa-tial region immediately,once it wants to communicate withthe location-based database server.The proactive mode pro-vides a better response time than the on-demand mode,but it generally incurs higher communication overhead and giveslower quality of service than the on-demand mode.5.ANONYMOUS LOCATION-BASEDSERVICESHaving the spatial cloaked region as an output form Algo-rithm1,the mobile user m sends her request to the location-based server through an agent p that is randomly selected.Existing location-based database servers can support onlyexact point locations rather than cloaked regions.In or-der to be able to work with a spatial region,location-basedservers need to be equipped with a privacy-aware queryprocessor(e.g.,see[29,31]).The main idea of the privacy-aware query processor is to return a list of candidate answerrather than the exact query answer.Then,the mobile user m willﬁlter the candidate list to eliminate its false positives andﬁnd its exact answer.The tighter the spatial cloaked re-gion,the lower is the size of the candidate answer,and hencethe better is the performance of the privacy-aware query processor.However,tight cloaked regions may represent re-laxed privacy constrained.Thus,a trade-oﬀbetween the user privacy and the quality of service can be achieved[31]. Figure4(a)depicts such scenario by showing the data stored at the server side.There are32target objects,i.e., gas stations,T1to T32represented as black circles,the shaded area represents the spatial cloaked area of the mo-bile client who issued the query.For clariﬁcation,the actual mobile client location is plotted in Figure4(a)as a black square inside the cloaked area.However,such information is neither stored at the server side nor revealed to the server. The privacy-aware query processor determines a range that includes all target objects that are possibly contributing to the answer given that the actual location of the mobile client could be anywhere within the shaded area.The range is rep-resented as a bold rectangle,as depicted in Figure4(b).The server sends a list of candidate answers,i.e.,T8,T12,T13, T16,T17,T21,and T22,back to the agent.The agent next for-(a)Server Side(b)Client SideFigure4:Anonymous location-based services wards the candidate answers to the requesting mobile client either through single-hop communication or through multi-hop routing.Finally,the mobile client can get the actualanswer,i.e.,T13,byﬁltering out the false positives from thecandidate answers.The algorithmic details of the privacy-aware query proces-sor is beyond the scope of this paper.Interested readers are referred to[31]for more details.6.EXPERIMENTAL RESULTSIn this section,we evaluate and compare the scalabilityand eﬃciency of the P2P spatial cloaking algorithm in boththe on-demand and proactive modes with respect to the av-erage response time per query,the average number of mes-sages per query,and the size of the returned candidate an-swers from the location-based database server.The queryresponse time in the on-demand mode is deﬁned as the timeelapsed between a mobile client starting to search k−1peersand receiving the candidate answers from the agent.On theother hand,the query response time in the proactive mode is deﬁned as the time elapsed between a mobile client startingto forward its query along with the cloaked spatial regionto the agent and receiving the candidate answers from theagent.The simulation model is implemented in C++usingCSIM[35].In all the experiments in this section,we consider an in-dividual random walk model that is based on“random way-point”model[7,8].At the beginning,the mobile clientsare randomly distributed in a spatial space of1,000×1,000square meters,in which a uniform grid structure of100×100cells is constructed.Each mobile client randomly chooses itsown destination in the space with a randomly determined speed s from a uniform distribution U(v min,v max).When the mobile client reaches the destination,it comes to a stand-still for one second to determine its next destination.Afterthat,the mobile client moves towards its new destinationwith another speed.All the mobile clients repeat this move-ment behavior during the simulation.The time interval be-tween two consecutive queries generated by a mobile client follows an exponential distribution with a mean of ten sec-onds.All the experiments consider one half-duplex wirelesschannel for a mobile client to communicate with its peers with a total bandwidth of2Mbps and a transmission range of250meters.When a mobile client wants to communicate with other peers or the location-based database server,it has to wait if the requested channel is busy.In the simulated mobile environment,there is a centralized location-based database server,and one wireless communication channel between the location-based database server and the mobile。

REGIME SWITCHING GARCH MODELS

CORE DISCUSSION PAPER 2006/11
REGIME SWITCHING GARCH MODELS
Luc Bauwens1 , Arie Preminger,2 and Jeroen V.K. Rombouts3 June 2005, corrected July 17, 2006
ห้องสมุดไป่ตู้
1
Introduction
Over the past two decades there has been a large amount of theoretical and empirical research on modelling volatility in ﬁnancial markets. Since volatility is commonly used as a measure of risk associated with ﬁnancial returns, it is important to portfolio managers, option traders and market makers among others. Further, portfolio optimization, derivative pricing and risk management, such as Value-at-Risk (VaR), use volatility estimates as inputs. So far in the literature, the most widespread approach to modeling volatility consists of the GARCH model of Bollerslev (1986) and its numerous extensions that can account for the volatility clustering and excess kurtosis found in the data (see e.g. Bollerslev and Wooldridge (1992) for an overview of the GARCH literature). The accumulated evidence from empirical research suggests that the volatility of ﬁnancial markets displays some type of persistence that cannot be appropriately captured by classical GARCH models. In particular, these models usually indicate high persistence in the conditional volatility. This persistence, as was noted by Hamilton and Susmel (1994), Gray (1996), and Klaassen (2002), is not compatible with the poor forecasting results of these models. Furthermore, Diebold (1986) and Lamoureux and Lastrapes (1990), among others, argue that the near integrated behavior of the conditional variance may originate from structural changes in the variance process, which are not accounted for by standard GARCH models. Mikosch and Starica (2004) show that estimating a GARCH(1,1) model on a sample displaying structural changes in the unconditional volatility does indeed create an integrated GARCH (IGARCH) eﬀect. These ﬁndings clearly indicate a potential source of misspeciﬁcation, to the extent that the structural form of the conditional mean and variance is relatively inﬂexible and held ﬁxed throughout the entire sample period. For example, the existence of shifts in the variance process over time can induce volatility persistence (see Wong and Li (2001) and Lanne and Saikkonen (2003)). Hence the estimates of a GARCH model suﬀer from a substantial upward bias in the persistence parameter. Therefore, models in which the parameters are allowed to change over time may be more appropriate for modelling volatility. In this perspective, several models that are based on a mixture of distributions have been proposed. Schwert (1989) considers a model in which returns may have either a high or a low variance, and switches between these states are determined by a two-state Markov process. Hamilton and Susmel (1994) and Cai (1994) introduce an ARCH model with regime-switching 1

The use of optical flow for the analysis of non-rigid motions

Introduction
The majority of available algorithms for the analysis and recovery of 3D motion of moving objects from image sequences, makes the assumption of opacity and rigidity (Fennema & Thompson, 1979; Hildreth, 1984; Longuet-Higgins, 1984; Nagel, 1983; Francois & Bouthemy, 1990). As these objects are seen by an imaging device, the 3D motion eld of moving objects in the scene is transformed into a 2D motion eld in the image plane (Gibson, 1950; Horn & Schunck, 1981). The assumption of opacity implies that at any location in the image plane, the 2D motion eld is single valued, that is the 2D motion eld is uniquely determined. In the case of transparent objects, the 2D motion eld is not single valued and two di erent velocities can be assigned to the same location in the image plane. In the case of opaque objects, the 2D motion eld is a planar vector eld, which

Simulating the growth of viruses

SIMULATING THE GROWTH OF VIRUSESLINGCHONG YOU AND JOHN YIN* Department of Chemical Engineering, University of Wisconsin-Madison 1415 Engineering Drive, Madison, WI 53706-1691 USAToFigure 2. Intracellular growth cycle of phage T7. The solid lines with half arrows indicate tr lation, the dashed lines represent reactions, and the solid lines with full arrows indicate the three classes of T7 DNA.(a) Infection initiation, class I gene expression.(b) Class II gene expression, phage DNA replication. (c) Class III gene expression,prsminutes post infectionT 7 p a r t i c l e s p e r b a c t e r i u mF O riod between infection initiation and the tim is the slope of the straight line starting from the end of the eclipse period and ending at the end of lysis, and the burs of phage progeny being produced from a single infection. (b) Comparison of the simulated (lines) and ex1.2 doublings per hour. Simulated results are shown for =1.2 and 1.5 doublings per hour. In the eminutes post infectionp h a g e p a r t i c l e s p e r c e l lTo study how host physiology affects ph0.51 1.52 2.5300.511.522.53host growth rate, doublings per hourn o r m a l i z e d v a r i a b l e seclipse timerise rateFigure 4. Effect of the host growth rate on T7 growth. The eclipse time and rise rate as thercurrent WFurther, the simulation predicts well the eclipse time for ecto12,gene 1 postion(kb)gene 1 postion(kb)Fe(minutes) for the eclipse time and 6.13 (particles/minute) for the rise rate. Experimental data are n the eclipse time and 6.35 (particles/minute) for the rise rate.wiToBy compFigure 6. A protein correlation matrix (PCM) for 21 essential T7 proteins. Each off-diagonal matrix element represents the correlation coefficient, propotional to the diameter of the filled cwfor viral-grown base for information on the specified virus. If it locates the information, it will continue to the next step; if not, it will prompt thebase or directly from the usto accept the output and updameE-virus is being developed as a generalization of T7v2.5, which itself has been desi11. Kutter, E., et a27. Buchholtz, F. and F.W. Schneider, Co。

ABAQUS单元类型

ABAQUS单元类型Advanced Finite Element Analysis–And ApplicationsDaming Zhang, Ph.D.Associate Professor of Transportation SystemsDepartment of Industrial TechnologyCollege of Agricultural Sciences and TechnologyCalifornia State University, FresnoMay 27, 2009Dr. Daming Zhang -Cal State Univ Fresno1Advanced Finite Element Analysis -And ApplicationsLecture 4:ABAQUS Element LibraryDr. Daming Zhang -Cal State Univ Fresno Advanced Finite Element Analysis -And Applications2ABAQUS Elements OverviewA wide range of elements available for solving different problemsFive characteristics of an element:–Family–Degree of freedom–Number of nodes–Formulation–IntegrationUnique name: T2D2, S4R, C3D8Iused on the *ELEMENT option, TYPE parameterDr. Daming Zhang -Cal State Univ Fresno Advanced Finite Element Analysis -And Applications3FamilyUsed to distinguish the geometryIndicated by first letter or letters: S4R, C3D8I, CINPE4 Dr. Daming Zhang -Cal State Univ Fresno Advanced Finite Element Analysis -And Applications4Degree of FreedomDegree of Freedom (DOF) are the fundamental variables 1Translation in direction 12Translation in direction 23Translation in direction 34Rotation about the 1-axis5Rotation about the 2-axis6Rotation about the 3-axis7Warping in open-section beam elements8Acoustic pressure or pore pressure9Electric potential11Temperature12+Temperature at other points through the thickness of beamsand shellsDr. Daming Zhang -Cal State Univ Fresno Advanced Finite Element Analysis -And Applications5Number of Nodes ?Determines the interpolation orderfirst order, second order, …Clearly identified in the name: C3D8, S8RBeam family indicating order of interpolation: B31, B32 Dr. Daming Zhang -Cal State Univ Fresno Advanced Finite Element Analysis -And Applications6FormulationRefers to the mathematical theoryLagrangian:material descriptionEulerian:Spatial descriptionShell family has 3 classes:General purposethin-onlythick-onlyAlternative formulations (end of element name)Hybrid formulation: C3D8H, B31HIncompatible formulation: C3D8IDr. Daming Zhang -Cal State Univ Fresno Advanced Finite Element Analysis -And Applications7IntegrationGaussian Quadrature to integrate quantities over the volume of each elementFull or Reduced integrationUse “R” at the end of element name to distinguish the reduced-integration elements: CAX4, CAX4RWill significantly affect the accuracyDr. Daming Zhang -Cal State Univ Fresno Advanced Finite Element Analysis -And Applications8Continuum Elements (1)Used to model the widest variety of components ?Element names begin with “C”Next two indicate the dimensionality: 3D, PE, PS, AX ?The last shows the degree of freedom3D continuum elements:hexa, penta, tetra2D continuum elements:plane strainplane stressaxisymmetricshape: quadrilateralor triangularDr. Daming Zhang -Cal State Univ Fresno Advanced Finite Element Analysis -And Applications 9Continuum Elements (2)2D continuum elements must be defined in 1-2 plane Node order should be counterclockwiseElement normals must all pointed at same direction ?Degree of freedom: translational DOFsElement properties: *SOLID SECTIONFormulation & Integration: Incompatible, Hybrid, Reduced ?Output variable: default in global coordinate system *ORIENTATION to define local coordinate systemDr. Daming Zhang -Cal State Univ Fresno Advanced Finite Element Analysis -And Applications10Shell Elements (1)Used to model structure with one dimension small ?Element name begins with: S, SAX, SAXAThe first number indicates the number of nodesIf the last character is “5”, the element doesn’t use the rotational DOF around normal of middle plane ?Quadrilateral or triangular; Linear or quadratic elements ?Three different formulationsDr. Daming Zhang -Cal State Univ Fresno Advanced Finite Element Analysis -And Applications 11Shell Elements (2)Degree of freedom: default 6 DOF, but also 5 DOF: S4R5 Axisymmetric shells have 3DOF:1Translation in the r-direction.2Translation in the z-direction.6Rotation in the r-z plane.*SHELL GENERAL SECTION: you define the properties*SHELL SECTION: ABAQUS calculates section properties ?Formulation & Integration: complicated, check before use ?Output variable: defined in the local material directions Lie on the surface of each shell elementAxes rotate with the element’s deformation in large-displacement simulationsDr. Daming Zhang -Cal State Univ Fresno Advanced Finite Element Analysis -And Applications12Beam ElementsUsed to model structure with one dimension is quite large ?Element name begins with: B (e.g. B31)The first number indicates the dimensionalityThe third character indicates the interpolation order ?Degree of freedom: default 6 DOF, 2D beams have 3 DOF Open-section beams (e.g. B31OS) have DOF 7 for warping ?*BEAM GENERAL SECTION: you define the properties *BEAM SECTION: ABAQUS calculates section properties ?Formulation & Integration: Hybrid for very slender beams;B21, B31, B22, B32: shear deformable, and finite axial strain; B23 and B33 are not; Open section: B31OS, B32OS ?Output variable: axial stress (s 11), shear stress (s 12)Dr. Daming Zhang -Cal State Univ FresnoAdvanced Finite Element Analysis -And Applications13Truss ElementsModel rods that can carry only tensile or compressive loads ?Element name begins with: T (e.g. T2D3, T3D2) The next two characters indicates the dimensionality ?Thefinal character indicates the number of nodes ?Degree of freedom: has only translational DOFs ?*SOLID SECTION: specify the material properties The cross-sectional area is given on the data line ?Formulation & Integration: Hybrid for very rigid links ?Output variable: Axial stress and strainDr. Daming Zhang -Cal State Univ FresnoAdvanced Finite Element Analysis -And Applications14Rigid ElementsElement name begins with: R (e.g. R3D4, R3D3)The next two characters indicates the dimensionality ?The final character indicates the number of nodes ?The nodes have no independent degrees of freedomThe nodes defining rigid elements can have loads applied to them or can be connected to other elements but they cannot have any boundary conditions ?*RIGID BODY defines the rigid body reference node ?Pay attention to the ‘sides’ of the rigid body elements ?Formulation & Integration: none ?Output: motion onlyDr. Daming Zhang -Cal State Univ FresnoAdvanced Finite Element Analysis -And Applications15Continuum Elements OverviewThe biggest family with over 20 just for 3D models ?3D: Hexa, Penta, Tetra; 2D: triangles and quadrilaterals ?Linear and quadratic versions for each of these shapes ?Full-and reduced-integration elements for hexa and quad ?Standard or hybrid element formulationFor linear hexa or quad: incompatible mode formulation ?For quadratic tria or tetra: "modified" formulationThe accuracy of your simulation will depend strongly on the type of element you use in your modelDr. Daming Zhang -Cal State Univ Fresno Advanced Finite Element Analysis -And Applications 16Full IntegrationThe accuracy of Gaussian Quadrature is (2n-1) for n=4?The Element Stiffness Matrix is calculated by:Fully integrated linear elements use two integration points in each directionFully integrated quadratic elements use three integration points in each directionDr. Daming Zhang -Cal State Univ Fresno Advanced Finite Element Analysis -And Applications 17k []m e=B []V ∫TD []B []dVFull Integration ExampleUse a cantilever beam to show the accuracy of analysisDr. Daming Zhang -Cal State Univ Fresno Advanced Finite Element Analysis -And Applications 18Shear LockingHappened on fully integrated, first-order, solid elements causes the elements to be too stiff in bending ?Deformation of material subjected to bending moment MDeformation of a fully integrated, linear element Dr. Daming Zhang -Cal State Univ Fresno Advanced Finite Element Analysis -And Applications 19Reduced IntegrationOnly quadrilateral and hexahedral elements can use a reduced-integration schemeuse one fewer integration point in each direction than the fully integrated elementsDr. Daming Zhang -Cal State Univ Fresno Advanced Finite Element Analysis -And Applications 20Reduced Integration ResultsLinear reduced-integration elements tend to be too flexible But fine mesh will produce acceptable results ?Deformation of a linear element with reduced integrationDr. Daming Zhang -Cal State Univ Fresno Advanced Finite Element Analysis -And Applications21Incompatible mode elementsAn attempt to overcome the problems of shear locking in fully integrated first-order elementsAdditional degrees of freedom enhance the element's deformation gradients as linear variationcan produce results in bending problems that are comparable to quadratic elements but at significantly lower computational costThe mesh distortion should be minimized as much as possible to improve the accuracy of the resultsDr. Daming Zhang -Cal State Univ Fresno Advanced Finite Element Analysis -And Applications22Hybrid ElementsHybrid elements have the letter "H" in their names ?Hybrid elements are used when the material behavior is incompressible (Poisson's ratio = 0.5)The volume cannot change if thematerial is incompressibleThe pressure stress cannot becomputed from the displacementsof the nodesHybrid elements include an additional degree of freedom that determines the pressure stress in the element directly Dr. Daming Zhang -Cal State Univ Fresno Advanced Finite Element Analysis -And Applications23Selecting Continuum ElementsUse quadratic, reduced-integration elements (CAX8R, CPE8R, CPS8R, C3D20R, etc.) for general analysis workUse quadratic, fully integrated elements (CAX8, CPE8, CPS8, C3D20, etc.) locally where stress concentrations may exist Use a fine mesh of linear, reduced-integration elements (CAX4R, CPE4R, CPS4R, C3D8R, etc.) for large-strain analysis For contact problems use a fine mesh of linear, reduced-integration elements or incompatible elements (CAX4I, CPE4I, CPS4I, C3D8I, etc.)?Minimize the mesh distortion as much as possibleIn three dimensions use hexahedral (brick-shaped) elements wherever possible; Use C3D6 and C3D4 only when necessary ?modified quadratic tetrahedral element (C3D10M) is robust for large-deformation and contact problems and exhibits minimal shear and volumetric lockingDr. Daming Zhang -Cal State Univ Fresno Advanced Finite Element Analysis -And Applications24Thank You Dr. Daming Zhang -Cal State Univ Fresno Advanced Finite Element Analysis -And Applications25。

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Simulation of Computer on the Relationship between the Characteristics and Link Equation of the Longitudinal Differential Protection of TransformerFang Ying1, Li Hua2, Yang Kaiyu1(1 Test Center for Examination Science of Jilin University, Changchun 130025, China;2 Transportation College of Jilin University, Changchun 130025, China)Abstract: Differentiation protection is the successful operation of the main protection of all the electrical equipments, it has the characteristics of high sensitivity, better selectivity etc. The paper is on the basis of the whole statement on the theory of longitudinal differential protection of transformer, when the transformer in which the current at both ends have phase-angle difference cuts short outside the protection area, the reason why the difference of transformer won’t move by wrong is ignored in practice. This paper discusses the concept of relative equation and discusses that the phases difference of different phase current will not affect the establishment of the equation when external short circuit happens. F rom the view that the power exchange point works relatively independently. At the same time, it points out the relationship between whether the protection moves or not and the relative equation when internal short circuit happens. This paper analyzes the amplitude of collection current, and put forward the resolutions of the imbalanced current in the longitudinal differential protection of transformer, it draws the conclusion that it won’t move.Key words: longitudinal differential protection of transformer; external short circuit; fixed connection; link equation; phase difference analysis; amplitude analysis; simulation experimentI. INTRODUCTIONThis paper discusses a unattracly electrical equipment problem, which is the main protection, and researched a kind of auxiliary analysis method for communication.Different phase current at both ends of the transformer longitudinal differential protection, in order to make sure not malfunction under normal conditions, using phase compensation, which can compensate based on symmetrical current difference 120 degrees in each phase. But sometimes there may be some problem, the current phase difference maybe 0 or 180 degree not 120 degree, when it is short circuit between phases in outside.A t the same time, different current amplitude, is it also can compensate and not malfunction?This is a basic problem needed to be answered. For this question, with inherent connection relation and short circuit between phases for external, it required as follows:1. A ngle changing between different phase current will not damage the protection of balanced.2. Amplitude value changing between different phase current will not damage the protection of balance.3. The former can be referred to as phase analysis; the later one can be referred to as amplitude analysis.Longitudinal differential protection has many features and difficulties which is the main protection for internal fault of transformer. Circuit breaker for longitudinal differential protection parameters are quite different as the transformer has two or more voltage steps, so it caused quite more unbalanced current than generator. This paper attempts to analysis the problem from the transformer and inherent link protection, which is installation and manufacturing, to simplify the problem and help to explain the reasons.II. THE LINK EQUATION OF DIFFERENTIALPROTECTION AND THE ANALYSIS OF PHASEDIFFERENCEIn order to provide convenience for the link equation of differential protection and introduce the link equation of differential protection, and show its significance by the way.A. Differential protection link equationDrawing 1 is wiring diagram, of which a typical Y/Ƹ-11 longitudinal differential protection of wiring transformer.The so-called Differential protection link equation (referred to as link equation) is defined as an equation based on transformer and protection inherent link, it express the secondary circuit of Y/ transformerƸdifferential protection and transformer also with itsFig.1 The Wiring Diagram of the Longitudinal Differential Protectionof Y/Ƹ-11 Transformer___________________________________ 978-1-4244-8160-6/11/$26.00 ©2011 IEEETake A phase differential as an example. Providing that the direction is from transformer side Y to side Ƹ. You can see from the drawing, in the protection of balance point, the balanced conditions are that the sizes of the collection current I aY,I aǻ are the same and the direction of them are contrary of the differential electrical relay KDa.No matter how to choose the change of CT and how to realize phase compensation, it both should realize this point.[1][2]Take phase A as example; establish link equation for the balanced protection.Take the transformer as the connection point; the point is the exchange point of primary side and secondary power. Ignore the field current, providing that U AY=UΊAƸ, U BY=UΊBƸ, and the connection current I AY=IΊAƸ, I BY=IΊBƸ. Providing all the standard transformation ratios of CT are 1. And it is:I aY=IΊaY-IΊbY=I AY-I BY =IΊAƸ-IΊBƸ=I AƸ=IΊaƸ=I(1) Or:I aƸ=IΊaƸ= I AƸ=IΊAƸ-IΊBƸ=I AY-I BY=IΊaY-IΊbY=I aY (2) That is, I aY=I aƸˈI KDa=I aY-I aƸ=0ˈthe protection doesn’t move.This is the link equation of the protection. If the transformer voltage ratio of the transformer is 1, it also can work out the link equation, the intrinsic quality doesn’t change. The practical transformer doesn’t meet U AY=UΊAƸand U BY=UΊBƸ, the transformation ratio on both sides of mutual inductor are different. But it can find balance with non-vectorial factor and other methods; they both are included in the transformation ratio of CT design and balanced measures.The most important is: the relationship of current exchange between exchange connections of every phase current is independent, and it has no relationship with the current separation angle between phase current.No matter how the phase separation between I AY and I BY, IΊAƸand IΊBƸchange, the relationships between original secondary current are tightly locked. It won’t affect the relationship of link equation because of the changes of phase angle.That is, IΊaY-IΊbY =I AY-I BY =IΊAƸ-IΊBƸ, they change synchronously, and has no relationship with phase separation.It expands the whole transformer and: if only there is no new connection point or broken wire occurring in the link involved in the link equation, in the extent of 360 degree, any changes of current included angle between threads and phases won’t make the link equation unequal.B. Significance of Link EquationA s for the transformer which is located at equinoctial point, are as follows:1. The link equations which have phase compensation are the description of electromagnetic relation which has intrinsic link, so long as the link equation is not broken, the protection won’t move forever.Link equation is not broken, which means that the power transfer relation of power exchange crunode isn’t broken, there is no new crunode in three phase link equation and scope, including that there is no crunode in the power transformation process, there is no broken wire in first and secondary circuit, there is no exceptional link or changes of transformation ratio in CT.2. All external interphase short circuits do not break the equal relation of link equation, so the protection won’t move. For example, the short circuit at the point of K3 in Chart 1.3. The side Y protection area of YN crunode transformer links with ground, although there is the zero-sequence current path linking with the ground, the Ƹ link and the secondary Ƹ link on Y side still don’t respond to the protection, spill current is 0-0=0. And so, the protection area on side Ƹ of transformer links with the ground, because there is no zero-sequence current path, the protection doesn’t respond. It both can be explained because the synthesis effect of three phase link equation isn’t broken and the protection doesn’t move.4. Open-phase operation and unsymmetrical operation, because the synthesis effect of three phase link equation isn’t broken, the protection doesn’t move.5. Internal interphase short circuit, the only condition for protection action is that there is new crunode or broken wire in the link involved in link equation. For example, the short circuit at the point of K1 and K2 in Chart 1.6. Throw out the problems of sensitivity, the internal grounding short circuit on Y N side, the conditions that the protection may move, it can be explained with that there is new crunode in the link involved in link equation.The above introduces the relation between link equation and protection action from different views, it can be more simple and direct. For the parts of new learners, it can provide assistant explanation.7. The link equation can conveniently calculate exceptional collection of CT. For example:Y side of CT reverses connection:I aY=-IΊaY-IΊbY=(-2IΊaY+ IΊaY)-IΊbY=-2IΊaY+(IΊaY-IΊbY)=-2IΊaY+ I AY-I BY=-2IΊaY+(IΊAƸ-IΊBƸ)=-2IΊaY+ I aƸ=-2IΊaY+ IΊaƸI aƸ=IΊaƸI KDa = I aY- I aƸ=-2IΊaYC. The Analyses about the Policy of Rear Difference Current Amplitude of External Interphase ShortCircuitFig 2 is the well-known current distribution about short-circuit of the rear two-phase of Y/Ƹ-11 connectiontransformer [1]. The condition is that the transformation ratio of transformer n conventionally is 1, and the[4].Fig.2 The Current Distribution of the Rear Two-Phase Short-Circuit ofY/Ƹ-11 Connection TransformerA lmost all teaching material talk about that when longitudinal differential protection of short circuit cut short outside, spill current is 0, so for the transformer which has turn angle, it should give explanation.Therefore, it should complementally analyze the amplitude.Take phase A as an example, phase separation of phase A and phase B current on side Y is 180 degree, the difference value of interphase current calculated from primary current is:I AY-I BY==I K(2)/3+2I K(2)/3=3I K(2) (3)Because of CT transformation ratio on Y side of Y/Ƹ-11 transformer differential protection is chosen to be enlarged by 3 timesǒ1Ǔ, so get the collection current through dividing 3by the result of equation(3):I aY= I K(2)ǄWhile the collection current on Ƹ side of transformer is I K(2).I KDa= I aY-I aƸ= I K(2) - I K(2)=0The protection doesn’t move. In a similar way, differential motion of B phase and C phase also can confirm that I KD is 0 with the same method.It also can calculate the two-phase short current outside the side Y protection area, forgive for not unfolding.Current setting principle components: reduce the current element value to meet the sensitivity requirements I DZ=1 076 AV oltage setting principle components: Umax=39.68V, can’t meet the requirements of order. It can be seen from the above calculation, not all double lines running parallel are suitable for installation phase transverse differential current protection and phase current direction of instantaneous trip delay protection. To systems large and small greatly different operating mode, heavy duty, long lines of double circuit line configurations such protection cannot be used. So, the protection won’t act by wrong because the different augmentations of current when external short circuit pass through.D. ConclusionsWhen construct longitudinal differential protection, if appropriate measures are not taken, imbalanced current which inpour differential relays will become very big, and its sensibility of protection will decrease to a degree, if taking the effects of Inrush exciting current of transformer into consideration, the protection will not work. Therefore, how to conquer imbalanced current, get rid of its effects on the protection and improve the sensitivity of the protection has become the core problem of longitudinal differential protection.In the analyses of performance characteristic of bus differential protection, in the terms of the traditional analysis method, unconventionality occurred in the scope and process for protection devices to collect primary current is collection unconventionality.[5] Because the quantitative analysis method for it isn’t found in the traditional analysis method, general differential method is put forward and demonstrated. This method can calculate simultaneously many unknown quantities a time, it also includes all information about first and second, it expands the traditional analysis method. A nd draw three conclusions that starting aside generatrix at the same time with non-preference, the protection refuses to move and back up protection skips a grade to trip operation, and the conclusion that it trips operation with selectivity, it also shows the computational process of this method. The reverse polarity of CT and the exceptional transformation ratio of current are analyzed on the basis of this method.When the imbalanced current is caused because of the external fault of transformer transient traversing short cut, in the transient process of the external fault of transformer, the main reasons for the imbalanced current caused by the longitudinal differential protection is that differential relay with different characteristics usually is adopted by the non- periodic-component of the short cut current in primary system to get rid of its effects on the longitudinal differential protection of transformer.The relationship between the link equation and the characteristics of protection moving is discussed, and answers the problem that the fault out of the protection area won’t make the protection move by wrong from the different views. A lthough the link equation isn’t discussed, it hopes that the external characteristics of the longitudinal differential protection are discussed and applied in practice, which is another opinion in the paper.REFERENCES[1] WA NG W J. A pplication of generator transformer relayprotection[M]. Beijing: China Electric Power Publishing House,2005: 58-103.[2] CHEN D SH. Principle and technique of computer relayprotection[M]. Beijing: Hydraulic and Electric Power Press,1992:42-89.[3] YA NG J W. The effects on the longitudinal differentialprotection of transient transmission characteristic of currenttransformer[J]. Shanxi Electric Power, 2008, 36(9): 46-49.[4] CAO T J. A new method of monitoring the circuit of longitudinaldifferential protection channel [J]. A utomation of Electric Systems, 2008, 32(21):65-68.[5] DA I J P. Recognition on several parameters of differentialprotection principle based ratio-restraint character of transformer[J]. Protection and Control of Electric Power System,2008, 36(21):73-76.[6] DA I J. False operation of transformer longitudinal differentialprotection caused by excitation surge current[J]. JiangsuElectrical Engineering, 2008, 27(5):32-34.AUTHOR BIOGRAPHYFang Ying was born in 1966, female, graduating from Harbin Institute of Technology and getting the Master's Degree, she is from Changchun city, Jilin Province, and is an engineer of Test Center for Examination Science of Jilin University, the research fields are relay protection, computer emulation.Li Hua was born in 1986, female, her research field is transportation computer emulation.Yang Kaiyu was born in 1971, male, is corresponding author from Changchun city, Jilin Province, and is an associate professor of Test Center for Examination Science of Jilin University.。