Zero Mode and Symmetry Breaking on the Light Front

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Ecole Normale Sup'erieure de Lyon

Ecole Normale Sup'erieure de Lyon
M. Baillargeon1, G. Belanger2 and F. Boudjema2
1. Grupo Teorico de Altas Energias, Instituto Superior Tecnico Edif cio Ci^ncia (F sica) P-1096 Lisboa Codex, Portugal e 2. Laboratoire de Physique Theorique ENSLAPP Chemin de Bellevue, B.P. 110, F-74941 Annecy-le-Vieux, Cedex, France.
We calculate the process ! e? e ud with the full set of Feynman diagrams. We consider di erent implementations of the W width and di erent combinations of the photon polarisations. The results of an improved narrow width approximation based on ! W +W ? with full spincorrelations that takes into account di erent angular cuts as well as cuts on the invariant masses are compared to those of the complete calculation.
Groupe d'Annecy Laboratoire d'Annecy-le-Vieux de PhysiquGroupe de Lyon

倒易空间的г和m点与晶胞的价带和导带

倒易空间的г和m点与晶胞的价带和导带

倒易空间的г和m点与晶胞的价带和导带The question you are asking is about the г and m points of the reciprocal lattice in relation to the valence and conduction bands of a crystal.In crystallography, the reciprocal lattice is used to describe the periodicity of a crystal. It is essentially a mathematical construct that represents the Fourier transform of the direct lattice, with points corresponding to diffraction vectors. The reciprocal lattice's Brillouin zone, which is bounded by its corresponding direct lattice planes, plays a crucial role in understanding electronic properties of crystals.The г point (Gamma point) and m point are special high-symmetry points within the first Brillouin zone. The гpoint corresponds to zero wavevector (k=0) and lies at the center of the Brillouin zone. It represents the point where all plane waves interfere constructively and also serves as a reference for energy calculations. On an energy-momentum diagram, it usually marks the minimum of the valence bandor maximum of the conduction band.The m point, on the other hand, lies at one edge (boundary) between different planes in reciprocal space. It corresponds to finite wavevector (k≠0) and reflects translational symmetry breaking along that direction. In terms of electronic band structure, it often shows important features such as band crossings or strong dispersion - indicating transitions between different bands or regions with high density-of-states.In terms of valence and conduction bands, theirrelationship with these high-symmetry points can provide valuable insights into a material's electronic behavior. By mapping out how these bands evolve around г and m points, we can determine key information such as band gaps, band topology, and even predict collective phenomena like superconductivity or topological insulator properties.To summarize, studying the г and m points within acrystal's reciprocal lattice allows for an in-depth analysis of its electronic properties. These special high-symmetry points provide valuable information regarding the valence and conduction bands, allowing for a better understanding of the material's behavior and potential applications.我问题的关键点是:倒易空间的г和m点与晶胞的价带和导带之间的关系。

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

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.。

非厄米算符对易关系

非厄米算符对易关系

非厄米算符对易关系Quantum mechanics plays a fundamental role in understanding the behavior of particles at the tiny scales of atoms and subatomic particles. One of the key concepts in quantum mechanics is the notion of operators, which are mathematical objects that represent physical observables in the theory. In particular, the notion of a Hermitian operator, which corresponds to observable quantities in the physical world, is crucial to the theory. However, not all operators are Hermitian, and non-Hermitian operators can also play a role in quantum mechanics.量子力学在理解原子和亚原子粒子微观尺度行为方面起着基础性作用。

量子力学的一个关键概念是算符的概念,算符是代表理论中物理可观测量的数学对象。

特别是,埃尔米特算符的概念,它对应着物理世界中的可观测量,对于理论起着至关重要的作用。

然而,并非所有算符都是埃尔米特的,非埃尔米特算符也可以在量子力学中发挥作用。

In quantum mechanics, the observables are represented by Hermitian operators, which are essential for the statistical interpretation of the theory. Hermitian operators have realeigenvalues, which correspond to the possible measurement outcomes of physical quantities. However, non-Hermitian operators can also be used to describe certain physical systems, especially those involving open quantum systems or interactions with the environment. Although non-Hermitian operators do not correspond to observables in the same way as Hermitian operators, they can still provide important insights into the behavior of quantum systems.在量子力学中,可观测量由埃尔米特算符表示,这对于理论的统计解释至关重要。

执理性之刃破笼茧之门作文

执理性之刃破笼茧之门作文

执理性之刃破笼茧之门作文英文回答:The Blade of Resolve Breaking Through the Cocoon Gate.In the journey of life, we often find ourselves trapped in a cocoon of fear, doubt, and uncertainty. This cocoon can prevent us from reaching our full potential and experiencing the freedom and joy that comes with breaking through barriers. The Blade of Resolve is a powerful metaphor for the inner strength and determination we need to break free from the constraints of the cocoon.The cocoon gate represents the barriers and limitations that we face in our lives. It could be a fear of failure, a lack of self-confidence, or a feeling of being stuck in a rut. Whatever form it takes, the cocoon gate holds us back from achieving our goals and living our best lives.The Blade of Resolve is the tool we need to breakthrough the cocoon gate. It represents our inner strength, determination, and courage to face our fears and overcome obstacles. With the Blade of Resolve in hand, we can cut through the doubts and uncertainties that hold us back, and emerge on the other side as a stronger, more resilient version of ourselves.It takes courage to wield the Blade of Resolve and break through the cocoon gate. It requires us to confront our fears, push past our comfort zones, and embrace the unknown. But the rewards of doing so are immeasurable. By breaking through the cocoon gate, we can discover new opportunities, unleash our full potential, and live a life of purpose and fulfillment.The journey of breaking through the cocoon gate with the Blade of Resolve is not easy, but it is worth it. It requires patience, perseverance, and a willingness to face our inner demons. But with each swing of the blade, we can chip away at the barriers that hold us back and move closer to the freedom and joy that await us on the other side.中文回答:执理性之刃破笼茧之门。

initialize_parameters

initialize_parameters

initialize_parameters IntroductionIn machine learning, initializing parameters is a crucial step in building a model. Parameters are the values that are updated during the training process to minimize the loss function. The initialization of parameters is the process of setting their initial values before training the model. In this article, we will discuss the importance of initializing parameters, different methods used for initializing parameters, and their impact on the model's performance.Why is initializing parameters important?Initializing parameters is important because it helps the model converge faster during the training process. It is essential to start with parameter values that are close to the optimal values. If the initial values are too far from the optimal values, the model will take a longer time to converge. Additionally, initializing parameters prevents the model from getting stuck in local minima. Local minima is a situation where the model is not ableto find the global minimum and gets stuck in alocal minimum. Initializing the parameters helps the model escape the local minima and reach the global minima.Different methods for initializing parameters1. Zero InitializationIn this method, all the parameters are initialized to zero. This method is not recommended as it leads to symmetry in the network. Symmetry in the network means that all the neurons in a layer will have the same weight values. There will be no change in the weights during the training process, leading to no learning.2. Random InitializationIn this method, the parameters are randomly initialized. The random values are generated from a uniform distribution, typically between -1 and 1. This method is better than the zero initialization as it leads to different values for every parameter, which helps in breaking the symmetry in the network. However, random initialization can have some negative impact on the model'sperformance. If the values are too large or too small, it can result in exploding or vanishing gradients. Exploding gradients refer to the situation where the gradients get larger and larger during the backpropagation process, making the model unstable. Vanishing gradients refer to the opposite situation, where the gradients become smaller and smaller, making the model convergence slower.3. Xavier InitializationXavier initialization is a method that initializes the weights of the network using a Gaussian distribution with zero mean and standard deviation (std) of sqrt(2 / n_in + n_out). Here,n_in refers to the number of input neurons in the layer, and n_out refers to the number of output neurons in the layer. This method specifically addresses the issue of exploding and vanishing gradients. The random values are scaled, so they always have an appropriate range of values considering the input and output layers' size.4. He InitializationHe initialization is similar to Xavier Initialization, except the std is now sqrt(2 /n_in). This method is better suited for activation functions like ReLU, which can lead to dead neurons with Xavier initialization. Dead neurons are the neurons in the network that are not able to learn during training because they get stuck on the negative side of the ReLU activation function.ConclusionIn conclusion, initializing parameters is a crucial step in building a machine learning model. Random initialization, Xavier initialization, and He initialization are some of the methods used for initializing parameters. It is essential to choose the right method of initializing parameters to ensure fast convergence, break the symmetry in the network, and avoid issues such as exploding and vanishing gradients.。

The mysteries of the universe Dark matter

The mysteries of the universe Dark matter

The mysteries of the universe DarkmatterDark matter is one of the most enigmatic and perplexing concepts in the field of astrophysics and cosmology. It is a substance that makes up a significantportion of the universe, yet it remains largely elusive and mysterious to scientists. The existence of dark matter was first proposed in the 1930s by Swiss astronomer Fritz Zwicky, who observed that the visible matter in the Coma galaxy cluster could not account for the gravitational forces that were holding thecluster together. This led him to hypothesize the presence of unseen "dark" matter that was responsible for the gravitational effects. Since then, numerous observations and experiments have provided compelling evidence for the existenceof dark matter, but its true nature continues to elude researchers. One of the most compelling lines of evidence for dark matter comes from the study of the rotation curves of galaxies. When astronomers measure the velocities of stars and gas in a galaxy as a function of their distance from the galactic center, theyfind that the velocities do not decrease as expected with increasing distance. Instead, the velocities remain constant or even increase, indicating the presence of additional unseen mass that is providing the gravitational pull to keep thestars and gas in their orbits. This discrepancy between the observed motion of galactic objects and the visible matter in galaxies has led scientists to conclude that there must be a significant amount of dark matter present in galaxies, outweighing the visible matter by a factor of about six to one. Another piece of evidence for dark matter comes from the study of the large-scale structure of the universe. Observations of the cosmic microwave background radiation, the afterglow of the Big Bang, have revealed subtle patterns in the distribution of matter onthe largest scales. These patterns can be explained by the presence of dark matter, which exerts gravitational forces to shape the distribution of galaxies and galaxy clusters in the universe. Additionally, the gravitational lensing of distant galaxies by intervening mass concentrations, such as galaxy clusters, provides further evidence for the presence of dark matter. The bending of light from these distant galaxies can only be explained by the gravitational influence of unseenmass, which is consistent with the properties of dark matter. Despite the overwhelming evidence for the existence of dark matter, its true nature remains a profound mystery. Dark matter does not emit, absorb, or reflect light, making it invisible to telescopes and other instruments that rely on electromagnetic radiation for detection. This has made it incredibly challenging for scientists to directly observe and study dark matter, leading to a wide range of theoretical and experimental efforts to uncover its properties. One of the leading candidates for the identity of dark matter is a type of particle that interacts only weakly with ordinary matter and electromagnetic forces, known as a weakly interacting massive particle (WIMP). WIMPs are a theoretical class of particles that arise in extensions of the standard model of particle physics, and they are thought to have been produced in the early universe in sufficient quantities to account for the observed abundance of dark matter today. Numerous experiments around the world are dedicated to detecting WIMPs through their rare interactions with ordinary matter, such as through the recoil of atomic nuclei in underground detectors or the production of secondary particles in particle accelerators. Another potential explanation for dark matter is the existence of primordial black holes, which are hypothesized to have formed in the early universe from the gravitational collapse of overdense regions. These black holes would not emit significant amounts oflight or other radiation, making them difficult to detect directly. However, their gravitational influence on surrounding matter could betray their presence, and ongoing observational campaigns are searching for the signatures of primordial black holes in the universe. In addition to these particle-based and astrophysical explanations, some scientists have proposed modifications to the laws of gravity as an alternative to dark matter. These modified gravity theories seek to explain the observed gravitational effects in galaxies and galaxy clusters without invoking the presence of additional unseen mass. While these theories have had some success in reproducing certain observational data, they have yet to provide a comprehensive and consistent explanation for the full range of evidence for dark matter. The search for dark matter continues to be a vibrant and active area of research in astrophysics and particle physics. New generations of experiments are pushing the boundaries of sensitivity and precision in the huntfor dark matter particles, while astronomers are mapping the distribution ofmatter in the universe with ever-increasing detail. The discovery of dark matter would represent a profound breakthrough in our understanding of the fundamental constituents of the universe and the forces that govern its evolution. It would also have far-reaching implications for our understanding of the cosmos, from the formation of galaxies and galaxy clusters to the ultimate fate of the universe itself. The quest to unravel the mysteries of dark matter is not just ascientific endeavor, but also a deeply human one. It speaks to our innatecuriosity about the nature of the universe and our place within it. Therealization that the majority of the matter in the universe is invisible and fundamentally different from the matter we interact with on a daily basis is both humbling and awe-inspiring. It challenges our preconceived notions of the cosmos and forces us to confront the limits of our current understanding. The search for dark matter is a testament to the human spirit of exploration and discovery, as we strive to push the boundaries of knowledge and unlock the secrets of the universe. In conclusion, dark matter remains one of the most captivating and tantalizing mysteries in modern science. Its existence is supported by a wealth of observational evidence, yet its true nature continues to elude us. Whether it is composed of exotic particles, primordial black holes, or a modification of thelaws of gravity, the discovery of dark matter would revolutionize our understanding of the cosmos and our place within it. The ongoing quest to uncover the secrets of dark matter is a testament to the enduring human spirit ofcuriosity and exploration, as we continue to push the boundaries of knowledge and strive to unlock the mysteries of the universe.。

CP Asymmetries in Higgs decays to ZZ at the LHC

CP Asymmetries in Higgs decays to ZZ at the LHC

a r X i v :0708.3612v 1 [h e p -p h ] 27 A u g 2007CP Asymmetries in Higgs decays to ZZ at the LHCRohini M.Godbole 1,David ler 2,M.Margarete M¨u hlleitner 3,41Centre for High Energy Physics,Indian Institute of Science,Bangalore,560012,India.2Dept.of Physics and Astronomy,University of Glasgow,Glasgow G128QQ,U.K.3Theory Division,Physics Department,CERN,CH-1211Geneva 23,Switzerland.4Laboratoire d’Annecy-Le-Vieux de Physique Th´e orique,LAPTH,France.Abstract.We examine the effect of a general HZZ coupling through a study of the Higgs decay to leptons via Z bosons at the LHC.We discuss various methods for placing limits on additional couplings,including measurement of the partial width,threshold scans,and asymmetries constructed from angular observables.We find that only the asymmetries provide a definitive test of additional couplings.We further estimate the significances they provide.1.IntroductionThe verification of the Higgs mechanism as the cause of electroweak symmetry breaking and the discovery of the Higgs boson is the next big goal of particle physics.However,it is not enough to simply find a new resonance in the Higgs search channels at the next generation of colliders.One must ensure that this resonance is indeed the Higgs boson by measuring its properties:its CP and spin,to demonstrate its predicted scalar nature;its couplings to known particles,to verify that these couplings are proportional to the particle’s mass;and the Higgs self couplings,in order to reconstruct the Higgs potential itself.This will be a challenging programme and will not be fully realised at the Large Hadron Collider (LHC)(e.g.the quartic Higgs self coupling will be out of reach).However,such an analysis will be crucial in our investigation of electroweak symmetry breaking in scenarios where the suspected Higgs boson is all we find at the LHC,as well as scenarios where new physics is discovered.In the former case,testing for deviations from the Standard Model (SM)may provide clues to resolving some of the SM’s long standing problems;in the latter case,the Higgs boson properties will provide essential information on the nature of the new physics.It is interesting to note that the Higgs boson’s CP (and spin)is intimately related to its couplings to other SM particles,since its scalar or pseudoscalar nature allow or forbid certain tensor structures in the Higgs boson couplings.In this report,we investigate the tensor structure of the HZZ vertex in order to shed some light on the Higgs boson’s CP.We write down the most general tensor vertex for this coupling and investigate how the additional terms influence the decay H →ZZ (∗)→4leptons at the LHC.For a more detailed description of this analysis,see Ref.[1].The most general vertex for a spinless particle coupling to a pair of Z bosons,with four-momenta q 1and q 2,is given by,V µνHZZ=igm Zm 2Z+c ǫµναβp αk βwhere p=q1+q2and k=q1−q2,θW denotes the weak-mixing angle andǫµναβis the totally antisymmetric tensor withǫ0123=1.The CP conserving tree-level Standard Model coupling is recovered for a=1and b=c=0.Terms containing a and b are associated with the coupling of a CP-even Higgs,while that containing c is associated with that of a CP-odd Higgs boson.The simulanteous appearance of a non-zero a(and/or b)together with a non-zero c would lead to CP violation.In general these parameters can be momentum-dependent form factors that may be generated from loops containing new heavy particles or equivalently from the integration over heavy degrees of freedom giving rise to higher dimensional operators.The form factors b and c may,in general,be complex, but since an overall phase will not affect the observables studied here,we are free to adopt the convention that a is real.2.The total widthOne method of investigating the tensor structure of the HZZ coupling is to examine the threshold behaviour of the decay H→ZZ∗[2].Notice that the additional terms in the vertexwidth on the virtuality of the most virtual Z boson.width from the SM prediction.Alternatively,one could examine the magnitude of the total decay width for H→ZZ∗→4leptons to see if it differs from the SM.For the vertex of Equation1,the dependence on the coefficients a,b and c is given by,∂2ΓHm4H +|b|2m4H4+|c|28q21q22m2Zβ2 m4H ,(2)whereβis the usual Lorentz boost factor for the Z-bosons.(Notice that the only term with a linearβdependence(from the phase space)is proportional to a2,illustrating the principle described above for the threshold scan.)If additional terms are present one expects them toincrease or decrease the width according to this equation.We used the ATLAS study of Ref.[3,4] (including cuts and efficiencies)to estimate the number of signal and background events for the SM and CP-violating scenarios(scaling the signal according to Equation2).In Figure2we plot the number of standard deviations from the SM that the CP-violating scenario would imply,for a150GeV Higgs boson and an integrated luminosity of300fb−1(we set b=0for simplicity). The white area represents scenarios where the significance of the deviation is less than3σ,the light blue/grey region represents a3−5σdeviation,while the dark blue/grey region represents a greater than5σdeviation.This measurement would allow one to rule out much of the a−|c| parameter space,but does not allow one to definitively rule out(or place significant limits on) the CP-odd coupling|c|.A SM-like rate is perfectly consistent with a large value of|c|and a small value of a.3.Asymmetries as a probe of CP violationTo definitively ascertain whether or not extra tensor structures are present in the HZZ vertex one is better served by measuring asymmetries which vanish when such terms are absent.Such an asymmetry can be constructed from an observable,O,based on the angles of thefinal state leptons,Γ(O>0)−Γ(O<0)A=。

凝聚态物理英语

凝聚态物理英语

凝聚态物理英语全文共3篇示例,供读者参考篇1Condensed matter physics, also known as solid-state physics, is a branch of physics that deals with the physical properties of solids and liquids, such as conductors, semiconductors, and insulators. It is a highly multidisciplinary field that encompasses elements of quantum mechanics, statistical mechanics, and electromagnetism.One of the key concepts in condensed matter physics is the idea of collective behavior. In a solid or liquid, the individual atoms or molecules interact with each other in complex ways, leading to emergent phenomena that cannot be understood by studying the individual particles alone. For example, the behavior of electrons in a crystal lattice is governed by the collective interactions between them, leading to phenomena such as superconductivity and magnetism.Another important concept in condensed matter physics is symmetry breaking. In a system with particular symmetry, certain properties of the system are preserved under transformationssuch as rotations or translations. However, in a phase transition, the symmetry of the system is broken, leading to the emergence of new properties. For example, when a material undergoes a phase transition from a normal state to a superconducting state, the symmetry of the system is broken, leading to the emergence of perfect conductivity.Condensed matter physics is a highly active area of research, with ongoing investigation into phenomena such as topological insulators, quantum spin liquids, and high-temperature superconductors. These materials exhibit novel physical properties that could have profound implications for technology, such as faster computers and more efficient energy storage devices.In conclusion, condensed matter physics is a fascinating field that studies the physical properties of solids and liquids at the atomic and subatomic level. By understanding the complex interactions between particles in condensed matter systems, physicists can develop new materials with unique properties and advance our understanding of the fundamental laws of nature.篇2Condensed matter physics is a branch of physics that deals with the study of the physical properties of matter in its condensed phases, which include solids and liquids. This field of physics plays a crucial role in understanding the behavior of materials and their interactions at the atomic and subatomic level.One of the key concepts in condensed matter physics is the idea of symmetry breaking. Symmetry breaking refers to the process by which a system transitions from a symmetric state to a less symmetric state, leading to the emergence of new properties and phenomena. This phenomenon is responsible for the formation of structures such as crystals and magnets, which exhibit specific patterns and behaviors due to broken symmetries.Another important aspect of condensed matter physics is the study of phase transitions. Phase transitions occur when a material undergoes a change in its physical state, such as melting, freezing, or vaporization. Understanding the mechanisms and properties of phase transitions is crucial for developing new materials with unique properties and applications.One of the most exciting areas of research in condensed matter physics is the study of quantum materials. Quantummaterials display unique quantum mechanical properties, such as superconductivity and topological insulators, which have the potential to revolutionize technologies in fields such as electronics and energy storage.Overall, condensed matter physics is a diverse and dynamic field that continues to push the boundaries of our understanding of matter and its properties. By studying the behavior of materials at the atomic and subatomic level, physicists are able to develop new technologies and materials that have the potential to impact a wide range of industries. So yeah, Condensed matter physics is truly fascinating and holds immense potential for future discoveries.篇3Condensed matter physics is a branch of physics that deals with the physical properties of condensed phases of matter, such as solids and liquids. It is a highly interdisciplinary field that combines elements of physics, chemistry, materials science, and engineering to study the behavior of matter at the macroscopic level.One of the key goals of condensed matter physics is to understand the underlying principles that govern the behavior ofmaterials and to develop new materials with specific properties. This is important for a wide range of applications, from developing new electronic devices and sensors to designing new materials for energy storage and conversion.One of the key concepts in condensed matter physics is symmetry breaking. Symmetry breaking is a phenomenon in which the symmetry of a system is spontaneously broken, leading to the emergence of new properties and behaviors. For example, in a crystalline solid, the atoms are arranged in a regular, repeating pattern that exhibits translational symmetry. However, at low temperatures, the crystal may undergo a phase transition in which the symmetry is broken, leading to the emergence of new properties such as magnetic ordering or superconductivity.Another important concept in condensed matter physics is collective behavior. Condensed matter systems often exhibit complex collective behavior that emerges from the interactions between individual particles. For example, in a superconductor, the electrons pair up to form Cooper pairs, which then move through the material without resistance. This collective behavior is responsible for the unique properties of superconductors, such as zero electrical resistance and the expulsion of magnetic fields.Condensed matter physics also plays a critical role in advancing our understanding of quantum mechanics. Quantum mechanics is the branch of physics that describes the behavior of particles at the smallest scales, such as atoms and subatomic particles. In condensed matter systems, the behavior of electrons, atoms, and other particles is governed by the principles of quantum mechanics. Understanding how these principles manifest at the macroscopic level in condensed matter systems is essential for developing new materials with novel properties and applications.One of the key challenges in condensed matter physics is the complexity of the systems being studied. Condensed matter systems contain a large number of interacting particles, which can exhibit a wide range of behaviors and properties. This complexity makes it difficult to predict the behavior of condensed matter systems from first principles alone, and often requires the use of advanced theoretical and computational methods to model and simulate these systems.Despite these challenges, condensed matter physics has made significant contributions to our understanding of the physical world and has paved the way for numerous technological advances. From the discovery of new materialswith exotic properties to the development of new electronic devices and sensors, condensed matter physics continues to play a crucial role in shaping the future of science and technology.In conclusion, condensed matter physics is a vibrant and interdisciplinary field that studies the physical properties of condensed phases of matter. By exploring the principles of symmetry breaking, collective behavior, and quantum mechanics in condensed matter systems, researchers are able to develop new materials with novel properties and applications. As we continue to unlock the mysteries of condensed matter physics, we open up new possibilities for advancing science and technology in the 21st century.。

英文word版Robust one

英文word版Robust one

Robust one-way modes in gyromagnetic photonic crystal waveguideswith different interfacesJin-Xin Fu, Rong-Juan Liu, and Zhi-Yuan Li a_Laboratory of Optical Physics, National Laboratory for Condensed Matter Physics, Institute of Physics,Chinese Academy o f Science, P.O. Box 603, Beijing 100190, People’s Republic of China_Received 30 March 2010; accepted 5 July 2010; published online 27 July 2010_We experimentally realize a one-way waveguide made from a gyromagnetic photonic crystal _GPC_and a normal dielectric photonic crystal in their common photonic band gap. Due to the timereversal symmetry breaking of the GPC under a dc magnetic field, the edge modes exhibit aone-way property. We investigate the unidirectional propagation character originating from thechiral edge states at different widths of the waveguide. Both the forward modes, which are robustto various obstacles, and the backward modes show an obviousfrequency shift for differentwaveguide widths. Numerical simulations by means of the transfer-matrix method and multiplescattering methods agree well with the experimental data. © 2010 American Institute of Physics._doi:10.1063/1.3470873_Photonic crystal _PC_ has provided a fertile ground of research since 1980s due to its marvelous and attractive underlyingapplications.1–3 People dream of manipulating photonsin the same manner as electrons and have made goodefforts to realize various kinds of integrated optical devices, such as waveguides, cavities, switches, and so on. Very recently a unusual kind of optical waveguide made from gyromagnetic PC _GPC_ was proposed and realized, which exhibitsan interesting one-way propagation property.4–6 This phenomenon is analogous to the quantum Hall effect for the semiconductor in a strong magnetic field,7 which breaksdown the time reversal symmetry of the system. The chiral edge modes formed at the interface of the two-dimensional_2D_ GPC and air possess group velocities pointing in only one direction under an external dc magnetic field. Followingthe unique idea, several groups studied the peculiar characters in the GPC theoretically.8–13 In contrast, experimentalreports are relatively rare.The chiral edge states _CES_ that only exist at the boundaryof the GPC will easily radiate away without confinement.In Refs. 5 and 6 Wang et al.6 proposed two configurations: metal cladding and PC wall _normal PC without magnetic response_ confinement, to restrict the edge modes and transfer them into guided modes. The former configuration hasbeen demonstrated experimentally. In this paper, we experimentallyrealize and study the latter configuration, as depictedin Fig. 1. It is known that for a conventional waveguide formed by reciprocal PC, the guided modes often varywith the width of the waveguide.14–17 Something specialmight happen for the nonreciprocal system when the waveguide width changes.In our experiment, we fabricated a static magneticfield device using NdFeB magnets, which have both high coercivity and saturation magnetization. Two arrays ofstrip-shaped magnets that are compounded with each othervia epoxy resin were structured as the N-pole and S-pole,respectively. The GPC is a square-lattice array of microwave ferrites yttrium-iron-garnet _YIG_ rods embedded in foam with a refractive index close to 1. The lattice constant isa=1.33 cm, and the radius of the YIG rods is r1=0.16 cm.The upper PC consisting of Al2O3 rods is also a square lattice, but with a 45° rotation to the YIG lattice, i.e., the _-M direction of the PC along the waveguide. The Al2O3 lattice constant is a2=a _2, and the rod radius is r2=0.155 cm.All the rods are 1 cm long. The whole sample was then sandwiched between two parallel steel plates_in x-y plane_, which formed a chamber only with its left and right side open for the connection of input/output ports. This configuration insures a transverse magnetic _TM_ plane wave incidence and an equivalent two dimensional PC system.18 The dc magnetic field with an intensity of 2020 G was exerted along the +z direction, as depicted in Fig. 1. The microwave source and detector was supplied via AgilentN5230A network analyzer. Changing the width of the waveguidew _i.e., the center-center distance between the nearestAl2O3 rod and YIG rod comprising the waveguide_, we observe the transmission spectrum variance of the forward andbackward modes simultaneously.The permittivity of our YIG and Al2O3 rods are _1=15_0 and _2=9_0, respectively. The permeability of theAl2O3 _2 is always equals to _0 since it is nonmagnetic,a_Electronic mail: lizy@.Al2O3 PCS21Port1 S12 Port2z(H0)xywYIG PCFIG. 1. _Color online_ Schematic of the edge state waveguide formed by the interface between a GPC _bottom half_ and a normal Al2O3 PC _upper half_, which are surrounded by some microwave absorbing materials _the outerregions_. w is the width of the waveguide, and the applied dc static magneticfield H0 is along the +z direction. The forward and backward transmittancesare denoted by S21 and S12, respectively.APPLIED PHYSICS LETTERS 97, 041112 _2010_0003-6951/2010/97_4_/041112/3/$30.00 97, 041112-1 © 2010 American Institute of PhysicsDownloaded 28 Jul 2010 to 159.226.35.196. Redistribution subject to AIP license or copyright; see /apl/copyright.jsp however the permeability of YIG becomes a second-ranktensor19 under an applied dc field H0:_ = __r i_i 0−i_i _r 00 0 _0 _, where _r=1+__0_m/_02 −_2_, _i =−___m/_02 −_2_, withparameters as _0=2__H0, _m=2___4_m s_, _ =2.8_106_rad s−1 G−1_, and 4_m s=1780 G.We characterized the band gap of the Al2O3 PC and theYIG GPC separately by measuring the transmission spectrum of the perfect PCs in the _- and _-M direction, asshown in Figs. 2_a_ and 2_c_. A theoretical band structure calculation was performed with the plane-wave transfermatrix method20,21 in Figs. 2_b_ and 2_d_, and the resultsagree well with experimental data. Since the first band gap of the GPC is not influenced by the external magnetic field and the transmittance of the guided mode within the gap exhibits no asymmetry in the two directions, we only focus on the second band gap where our one-way waveguide works.5 Inzero magnetic field, the second and third photonic bands ofthe GPC are degenerate at the M-symmetry point at a frequency around 15 GHz, but the density of states is quitesmall. When the dc magnetic field is exerted, YIG becomes gyromagnetic due to the action of the off-diagonal element_iand the second band gap of the GPC shows up.From the photonic band structure diagram in Fig. 2_c_,we can see that the complete band gap of the Al2O3 PC ranges from 12 to 15.2 GHz, which covers the second gap of the YIG GPC that is located between 14.2 and 15.2 GHz under an external magnetic field of 2020 G. Therefore theAl2O3 PC can provide a good confinement for the generated CES which only occurs in the second gap. We continue totest the transmission property of our one-way waveguide using the apparatus as illustrated in Fig. 1. The microwavesignal coming from a wide spread horn is introduced intoport 1 and then coupled into the waveguide. After passing through the channel into port 2, it is finally collected by ahorn identical to the import one. Both horns are connected to our network analyzer with co-axial cables. This is the forward mode spectra S21 measurement process. The backwardmode spectra S12 measurement process is similar. Numerical simulation using 2D multiple scattering methods22 was performedto calculate the values of S21 and S12. The resultsagree very well, except for the absolute transmissive intensity difference between the calculation and experiments,which is caused by our calibration process during the measurement.It is generally assumed that a narrower waveguide supports fewer modes, so we start with w=0.3a, which is nearlythe closest distance controllable in our experiment. The calculatedand measured transmission spectra are shown inFigs. 3_a_ and 3_b_. We can observe a big contrast between the forward modes S21 _red lines_ and the backward modesS12 _blue lines_ in the region from 14.3 to 14.9 GHz. However, in the upper band gap between 14.9 and 15.2 GHz, both forward and backward waves show a relatively high transmittanceand the waveguide turns to be two-way conductive.One of the intrinsic properties of the CES wave is its immunity to various kinds of disorder and robustness against thescattering from strong obstacles, and this property can be used to distinguish our complex situation. For a better understandingof the difference between the two frequency regions,we inserted a metal slice to “cut off” the waveguideand measured S21 and S12 again. The results in Figs. 3_c_ and 3_d_ showed that the former transmitted backward wave in the second region was deeply suppressed by the metal intrusion, while the forward wave only slightly decreased by about 10 dB. This implies that the PC-GPC waveguide can support other propagating waves although it is fairly narrow. But these additional propagating waves, unlike the CES, cannot navigate round the external perturbation. With the intrusionof the metal, the PC-GPC waveguide now exhibits aone-way transmission in the entire band gap from 14.2 to15.2GHz.We continue to explore the influence of the waveguidewidth upon the one-way transmission. The measured spectra6 8 10 12 14 16 18 20-80-60-40-208 10 12 14 16-80-60-40-206 8 10 12 14 16 18 20 8 10 12 14 16_ (d)MX___________ ____________ _____XAl __M 2O3 PC YIG PC__X__M__________ ___(c)(b)_XM(a)FIG. 2. _Color online_ Measured transmission spectrum and calculated band structure for the two kinds of PC in our experiment. _a_ and _b_ are for the Al2O3 PC, where the dashed lines denote the complete gap. _c_ and _d_ describe the YIG GPC under a 2020 G magnetic field, and its complete gap locates between the two dashed lines.__ __ __ __ ____________ __ __ __ ____________ __ __ __ ______________ __ __ __ _____________ ____________ __ __________________ _______________________ !"#_ ________ !"#_FIG. 3. _Color online_ Calculated __a_ and _c__ and measured __b_ and _d__ spectra for the Al2O3 PC-YIG GPC waveguide with a width of 0.3a. _a_ and_b_ correspond to the pure waveguide; _c_ and _d_ are the case of the metal intrusion as shown in the inset. The metal plate is inserted 0.5a deep into theAl2O3 PC and 2a into the YIG GPC, which can efficiently block the direct propagating wave. The shadow region in _a_ and _c_ corresponds to thesecond band gap.041112-2 Fu, Liu, and Li Appl. Phys. Lett. 97, 041112 _2010_ Downloaded 28 Jul 2010 to 159.226.35.196. Redistribution subject to AIP license or copyright; see /apl/copyright.jsp for four different waveguide widths as 0.7a, 1a, 1.45a, and1.6a are displayed in Fig. 4. Compared with the 0.3a case above, the asymmetric transmission band in the 0.7a case shifts to a lower frequency region from 13.95 to 14.25 GHz, almost out of the second gap. The bandwidth decreases to 0.3GHz at the same time. Further enlarging w to 1a, we find that the waveguide becomes cut-off for a wide region, although the S21 shows a bit higher transmittance than S12 within the gap. For the 1.4a case, a narrow asymmetric band appears again just below the second gap. But inside the gap, the waveguide is almost a two-way waveguide, except a shadow basin around 15 GHz for the backward wave. Going on to the 1.6a situation, we can see the departure between S21 and S12 emerging again inside the gap, and the basin has transformedto a deep and wide valley. The bidirectional modesstill exist and the lower band edge of the forward mode shifts to 14.4 GHz.From the above discussions, we can see that the one-way waveguide composed of Al2O3 PC and YIG GPC exhibits a similar band-shifting effect as the conventional waveguide, especially for the backward modes. Our further experimental data _not shown here_ indicate that for different waveguide widths the CES plays a dominant role for the forward waves so that they are robust against intrusion of the metal plate. In contrast, the backward modes are strongly influenced and easily blocked by the intruded metal plate in all situations.We further notice that our waveguide works far awayfrom the ferromagnetic resonance frequency of YIG, so the strength of the gyromagnetic effect is rather smaller than the previous design.6 However, we also achieve a maximum relative bandwidth of 6.6% for the one-way propagation, which implies that a large Voigt number V= _i _r may notbe so important for the CES. This is really good news for the extension of the one-way waveguide to higher frequency becauseof a much weaker dc magnetic field requested and amuch lower cost for the magnetic devices.In summary, we have experimentally investigated microwave one-way waveguides formed between a YIG GPC anda dielectric Al2O3 PC based on the nonreciprocal property of the GPC. When the waveguide width changes, the forwardpropagatingwaves display a relative small change because ofthe immunity of the edge modes to the external influence. However, the backward-propagating waves are very sensitive to the width of the waveguide. Furthermore, the forward modes are very robust against intrusion of a metal platewhile the backward modes are strongly influenced and easily blocked by the intruded metal plate.This work was supported by the National Natural Science Foundation of China under Grant No. 10525419 and the National Basic Research Foundation of China under Grant No. 2006CB921702.1S. John, Phys. Rev. Lett. 58, 2486 _1987_.2E. Yablonovitch, Phys. Rev. Lett. 58, 2059 _1987_.3J. D. Joannopoulos, Photonic Crystals: Molding the Flow of Light, 2nd ed._Princeton University Press, Princeton, 2008_.4F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 _2008_.5Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, Phys. Rev.Lett. 100, 013905 _2008_.6Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, Nature _London _ 461, 772 _2009_.7S. Raghu and F. D. M. Haldane, Phys. Rev. A 78, 033834 _2008_.8X. Ao, Z. Lin, and C. T. Chan, Phys. Rev. B 80, 033105 _2009_.9S. Y. Liu, W. K. Chen, J. J. Du, Z. F. Lin, S. T. Chui, and C. T. Chan, Phys. Rev. Lett. 101, 157407 _2008_.10C. Huang and C. Jiang, J. Opt. Soc. Am. B 26, 1954 _2009_.11H. B. Zhu and C. Jiang, Opt. Express 18, 6914 _2010_.12C. He, X. L. Chen, M. H. Lu, X. F. Li, W. W. Wan, X. S. Qian, R. C. Yin, and Y. F. Chen, Appl. Phys. Lett. 96, 111111 _2010_.13W. Śmigaj, J. Romero-Vivas, B. Gralak, L. Magdenko, B. Dagens, and M. Vanwolleghem, Opt. Lett. 35, 568 _2010_.14Y. Z. Liu, R. J. Liu, C. Z. Zhou, D. Z. Zhang, and Z. Y. Li, Opt. Express16, 21483 _2008_.15M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, Electron. Lett. 37, 293 _2001_.16M. Notomi, A. Shinya, K. Yamada, J. Takahashi, C. Takahashi, and I. Yokohama, IEEE J. Quantum Electron. 38, 736 _2002_.17M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, Phys. Rev. Lett. 87, 253902 _2001_.18Z. Feng, X. Zhang, K. Ren, S. Feng, Z.-Y. Li, B. Cheng, and D. Zhang, Phys. Rev. B 73, 075118 _2006_.19D. M. Pozar, Microwave Engineering, 2nd ed. _Wiley, New York, 1997_.20Z. Y. Li and L. L. Lin, Phys. Rev. E 67, 046607 _2003_.21J. X. Fu, R. J. Liu, and Z. Y. Li, EPL 89, 64003 _2010_.22L. M. Li and Z. Q. Zhang, Phys. Rev. B 58, 9587 _1998_.__ __ __ __ ______________ __ __ __ ______________ __ __ __ ______________ __ __ __ _________________________ _____________________ ___ ___________ _!"#_______________ _!"#_______FIG. 4. _Color online_ Measured spectra for four different waveguide widthof _a_ 0.7a, _b_ 1a, _c_ 1.45a, and _d_ 1.6a, respectively.041112-3 Fu, Liu, and Li Appl. Phys. Lett. 97, 041112 _2010_Downloaded 28 Jul 2010 to 159.226.35.196. Redistribution subject to AIP license or copyright; see /apl/copyright.jsp。

莫布勒复活英语作文

莫布勒复活英语作文

莫布勒复活英语作文Mobbler's ResurrectionThe world of technology has always been a rapidly evolving landscape, with new innovations and breakthroughs constantly reshaping the way we interact with our digital environments. One such pioneering figure in this ever-changing realm is Mobbler, a visionary whose contributions have left an indeluring mark on the industry. In the face of adversity and the constant challenges that come with pushing the boundaries of what's possible, Mobbler has managed to rise from the ashes, emerging stronger and more determined than ever before.Mobbler's journey began with a passion for pushing the limits of what was deemed achievable. As a young engineer, he was captivated by the potential of mobile technology, recognizing its ability to revolutionize the way we communicate, access information, and navigate our daily lives. With a keen eye for detail and an unwavering commitment to innovation, Mobbler set out to create products that would not only meet the needs of consumers but alsoredefine their expectations.One of Mobbler's most significant contributions was the development of a groundbreaking mobile operating system that challenged the status quo. At a time when the market was dominated by a handful of established players, Mobbler's vision for a more open, flexible, and user-centric platform quickly gained traction. His team of talented developers worked tirelessly to refine the system, adding features and capabilities that made it a compelling alternative to the existing options.However, Mobbler's path to success was not without its obstacles. The industry was fiercely competitive, and established players were not eager to cede their market share to an upstart challenger. Mobbler and his team faced intense scrutiny, skepticism, and even outright hostility from those who saw his vision as a threat to their dominance.Undeterred by the challenges, Mobbler doubled down on his commitment to his vision. He rallied his team, instilling in them a sense of purpose and a relentless drive to succeed. Together, they weathered the storms of criticism, working tirelessly to refine their product and demonstrate its superiority to the competition.As Mobbler's mobile operating system gained traction, the industrybegan to take notice. Consumers were drawn to the platform's intuitive design, robust functionality, and commitment to user privacy and security. Developers, too, were captivated by the opportunities the platform presented, flocking to create innovative applications that pushed the boundaries of what was possible on mobile devices.However, just as Mobbler's star was rising, a series of unexpected setbacks threatened to derail his progress. Technical issues, legal battles, and a shifting market landscape all conspired to undermine the momentum he had built. For a time, it seemed as though Mobbler's dream might be shattered, with the industry once again dominated by the established players.But Mobbler was not one to give up easily. Recognizing the gravity of the situation, he made the difficult decision to step away from the day-to-day operations of his company, taking time to regroup and reassess his strategy. It was a bold move, one that required immense courage and self-reflection, but Mobbler knew that it was necessary if he was to regain the momentum that had once propelled his vision forward.During this period of introspection, Mobbler immersed himself in the latest advancements in technology, studying the trends and patterns that were shaping the industry. He reconnected with his passion forinnovation, drawing inspiration from the pioneers who had come before him and the visionaries who were pushing the boundaries of what was possible.Gradually, Mobbler began to formulate a new plan, one that would leverage the lessons he had learned and the insights he had gained during his time away. He assembled a team of trusted advisors, tapping into their expertise and diverse perspectives to refine his strategy and identify new opportunities for growth.As Mobbler's vision began to take shape, he knew that he would need to confront the challenges that had nearly derailed his progress. With renewed determination and a clear sense of purpose, he set out to reclaim his place in the industry, using his platform to champion the values of innovation, user-centricity, and ethical technology.Through a series of strategic partnerships, bold product launches, and relentless advocacy, Mobbler slowly but surely regained the trust and admiration of the industry. His mobile operating system underwent a renaissance, with new features and capabilities that cemented its status as a viable alternative to the established players.But Mobbler's true triumph was not just in the technical achievements of his platform. It was in the way he had managed to inspire and empower a new generation of innovators, who saw inhim a model of what was possible when one refused to be defined by the limitations of the present.Today, Mobbler's name is synonymous with the spirit of innovation and the relentless pursuit of progress. His story has become a testament to the power of resilience, the importance of staying true to one's values, and the transformative potential of technology in the hands of those who dare to dream.As the industry continues to evolve and new challenges emerge, Mobbler's legacy serves as a guiding light for those who seek to create a more equitable, inclusive, and technologically advanced future. His journey is a reminder that even in the face of adversity, the power of human ingenuity and the unwavering commitment to a better tomorrow can overcome any obstacle.。

Symmetry Breaking

Symmetry Breaking

Symmetry BreakingToby WalshNational ICT Australia and School of CSE,University of New South Wales,Sydney,Australia,tw@.auSymmetry occurs in many problems in aritifical intelligence.For example,in the n-queens problem,the chessboard can be rotated90◦.As a second example, several machines in a factory might have the same capacity.In a production schedule,we might therefore be able to swap the jobs on machines with the same capacity.As a third example,two people within a company might have the same skills.Given a staffroster,we might therefore be able to interchange these two people.And as a fourth example,when configuring a computer,two memory boards might be identical.We might therefore be able to swap them around without changing the performance of the computer.Symmetries come in many different forms.For instance,there are rotation symmetries(e.g.the90◦rotations of the chessboard in the n-queens problem), reflection symmetries(e.g.the reflection of the chessboard along one of its di-agonals),and permutation symmetries(e.g.the interchange of equally skilled personnel in a staffroster).Problems may have many symmetries at once(e.g. the n-queens problem simultaneously has several rotation and reflection symme-tries).We can describe such symmetries using group theory.The symmetries of a problem form a group.Their actions are to map solutions(a n-queens solution, a production schedule,a staffroster,a configuration,etc.)onto solutions.We must deal with such symmetry or we will waste much time visiting symmetric solutions.In addition,we must deal with symmetry during our search for solu-tions otherwise we will visit many search states that are symmetric to those that we have already visited.One well known mechanism to deal with symmetry is to add constraints which eliminate symmetric solutions[1].I will describe a general method for breaking any type of symmetry[2,3].The basic idea is very simple.We pick an ordering on the variables,and then post symmetry breaking constraints to ensure that thefinal solution is lexicographically less than any symmetric re-ordering of the variables.That is,we select the“lex leader”assignment.In theory,this solves the problem of symmetries,eliminating all symmetric solutions and pruning many symmetric states.Unfortunately,the set of symmetries might be exponentially large(for example,there are n!symmetries if we have n identical machines in a factory).By exploting properties of the set of symmetries we can sometimes overcome this problem.I will describe two special cases where we can deal with an expo-nential number of symmetries in polynomial time.These two cases frequently occur in practice.In thefirst case,we have a matrix(or array)of decisions vari-ables in which the rows and/or columns are symmetric and can be permuted. In this case,we can lexicographical order the rows and/or columns[4,5].In the second case,we have values for our decision variables which are symmetric and can be interchanged.I show how so called“value precedence”can be used to break all such symmetry[6,7].Finally,I will end with a discussion of open research questions in this area. Are they efficient ways to combine together these symmetry breaking constraints for particular types of symmetries?Are there useful subsets of these symmetry breaking constraints when there are too many to post individually?Are there problems where the symmetry breaking constraints can be simplified?Can these symmetry breaking methods be used outside of constraint programming? References1.Puget,J.F.:On the satisfiability of symmetrical constrained satisfaction problems.In Komorowski,J.,Ras,Z.,eds.:Proceedings of ISMIS’93.LNAI689,Springer-Verlag(1993)350–3612.Crawford,J.,Luks,G.,Ginsberg,M.,Roy,A.:Symmetry breaking predicates forsearch problems.In:Proceedings of the5th International Conference on Knowledge Representation and Reasoning,(KR’96).(1996)148–1593.Walsh,T.:General symmetry breaking constraints.In:12th International Confer-ence on Principles and Practices of Constraint Programming(CP-2006),Springer-Verlag(2006)4.Shlyakhter,I.:Generating effective symmetry-breaking predicates for search prob-lems.In:Proceedings of LICS workshop on Theory and Applications of Satisfiability Testing(SAT2001).(2001)5.Flener,P.,Frisch,A.,Hnich,B.,Kiziltan,Z.,Miguel,I.,Pearson,J.,Walsh,T.:Breaking row and column symmetry in matrix models.In:8th International Con-ference on Principles and Practices of Constraint Programming(CP-2002),Springer (2002)w,Y.,Lee,J.:Global constraints for integer and set value precedence.In:Pro-ceedings of10th International Conference on Principles and Practice of Constraint Programming(CP2004),Springer(2004)362–3767.Walsh,T.:Symmetry breaking using value precedence.In:Proceedings of the17thECAI,European Conference on Artificial Intelligence,IOS Press(2006)。

自主创新和依靠外力的英语作文素材

自主创新和依靠外力的英语作文素材

全文分为作者个人简介和正文两个部分:作者个人简介:Hello everyone, I am an author dedicated to creating and sharing high-quality document templates. In this era of information overload, accurate and efficient communication has become especially important. I firmly believe that good communication can build bridges between people, playing an indispensable role in academia, career, and daily life. Therefore, I decided to invest my knowledge and skills into creating valuable documents to help people find inspiration and direction when needed.正文:自主创新和依靠外力的英语作文素材全文共3篇示例,供读者参考篇1Self-Innovation vs. Reliance on External ForcesAs a student, I often find myself caught between the desire to innovate and create something entirely my own, and the temptation to rely on external forces and pre-existing resources.It's a constant tug-of-war, and finding the right balance can be challenging.On one hand, self-innovation is an incredibly rewarding and empowering experience. When you conceive an idea, nurture it, and bring it to fruition through your own efforts, there's an unparalleled sense of accomplishment and pride. It's a testament to your creativity, problem-solving skills, and perseverance. Moreover, self-innovation often leads to truly unique and groundbreaking solutions, as you're not constrained by the limitations or biases of external sources.Take, for instance, the case of Mark Zuckerberg and the inception of Facebook. What started as a college project born out of Zuckerberg's own ingenuity and determination eventually transformed into a global phenomenon that revolutionized the way we communicate and share information. Had he simply relied on existing social media platforms or followed conventional wisdom, we might have never witnessed such a game-changing innovation.However, self-innovation is not without its challenges. It requires an immense amount of time, effort, and resilience, as you're essentially starting from scratch and navigating uncharted territory. There's a high risk of failure, as you may encounterunforeseen obstacles or realize that your idea is not as viable as initially thought. Additionally, self-innovation often demands a diverse set of skills and knowledge, which can be difficult to acquire and master independently.This is where the allure of relying on external forces comes into play. By leveraging pre-existing resources, tools, and knowledge, we can potentially save time, minimize risks, and gain access to expertise and support that would otherwise be out of reach. For instance, as a student pursuing a degree in computer science, I could rely on established programming languages, frameworks, and libraries to build my projects, rather than attempting to create everything from the ground up.Moreover, external forces can provide valuable guidance and mentorship, helping us refine our ideas, identify potential pitfalls, and learn from the experiences of others who have walked a similar path. Collaborating with peers, seeking advice from professors, or participating in internships or research projects can expose us to new perspectives and insights that we might have otherwise overlooked.However, an overreliance on external forces can also be detrimental. When we become too dependent on pre-existing solutions or blindly follow the advice of others, we risk stiflingour own creativity and critical thinking skills. We may find ourselves trapped in a cycle of imitation, unable to break free from established norms and conventions.Furthermore, external forces can sometimes be biased, outdated, or simply ill-suited for our specific needs or goals. By relying too heavily on them, we may inadvertently perpetuate flawed approaches or miss out on opportunities for genuine innovation.So, what's the solution? In my opinion, the key lies in striking a balance between self-innovation and judicious reliance on external forces.As students, we should embrace self-innovation as a means of cultivating our creativity, problem-solving abilities, and intellectual independence. We should actively seek out opportunities to conceive and develop our own ideas, whether it's through personal projects, research initiatives, or entrepreneurial endeavors. By doing so, we not only contribute to the advancement of knowledge and progress but also develop invaluable skills that will serve us well in our future careers and endeavors.At the same time, we must recognize the immense value that external forces can provide. We should leverage existingresources, knowledge, and mentorship judiciously, using them as a foundation upon which to build our own innovations. By standing on the shoulders of giants, we can accelerate our learning curve, avoid reinventing the wheel, and benefit from the collective wisdom of those who have come before us.Crucially, however, we must maintain a critical mindset and avoid blindly accepting external forces as gospel. We should approach them with a healthy dose of skepticism, questioning their underlying assumptions, evaluating their relevance and applicability to our specific contexts, and identifying potential biases or limitations.Ultimately, the true path to innovation lies in the synthesis of self-innovation and external forces. By combining our own creativity and determination with the guidance and resources provided by external sources, we can create something truly remarkable – something that pushes the boundaries of what is possible while still remaining grounded in established principles and best practices.As I navigate my academic journey and beyond, I strive to embody this balance. I aim to nurture my own innovative spirit, constantly challenging myself to think outside the box and develop novel solutions to complex problems. At the same time,I remain open to learning from others, seeking out mentors and collaborators who can provide valuable insights and support.It's a delicate dance, one that requires constant recalibration and self-awareness. But by embracing both self-innovation and the judicious use of external forces, I believe we can unlock our full potential as students, scholars, and future leaders, paving the way for truly transformative and impactful innovations.篇2Self-Innovation vs. Reliance on External ForcesThroughout human history, there has been an ongoing debate around whether true progress and achievement stem from self-motivated innovation or from relying on external forces and influences. As a student, I have grappled with this question in various aspects of my academic journey. Do I succeed through my own hard work and creativity, or do I depend on the guidance and resources provided by teachers, institutions, and society at large?On one hand, the power of self-innovation cannot be overstated. Some of humanity's greatest accomplishments have come from individuals who dared to think outside the box and challenge conventional wisdom. Galileo Galilei's pioneering workin astronomy, for instance, faced immense opposition from the Catholic Church and the academic establishment of his time. Yet, his unwavering curiosity and willingness to question authority ultimately revolutionized our understanding of the universe.Similarly, in the realm of technology, many groundbreaking inventions have emerged from independent tinkerers and entrepreneurs working in garages or home labs. The personal computer revolution, spearheaded by visionaries like Steve Jobs and Steve Wozniak, is a prime example. By eschewing traditional corporate structures and embracing a DIY ethos, these innovators reshaped the entire computing landscape.From a student's perspective, self-innovation can manifest in various ways. It might involve exploring unconventional study methods, developing unique problem-solving strategies, or pursuing independent research projects outside of the classroom. By thinking critically and challenging established norms, we can push the boundaries of our understanding and potentially make novel contributions to our respective fields.However, it would be naive to discount the importance of external forces and support systems in driving progress and achievement. Even the most brilliant minds and pioneering thinkers have relied, to some extent, on the knowledge,resources, and infrastructure provided by larger institutions and societal frameworks.Universities, for instance, serve as crucibles of innovation by fostering collaborative environments, providing access to cutting-edge research facilities, and facilitating the exchange of ideas across disciplines. Many groundbreaking discoveries and innovations have emerged from the collective efforts of teams of researchers working within these academic institutions.Moreover, government funding and policies can play a pivotal role in shaping the direction and pace of innovation. The space race between the United States and the Soviet Union during the Cold War era, for example, catalyzed tremendous advancements in aerospace engineering and related fields. Without the backing of national resources and political will, such endeavors would have been nearly impossible to undertake.From a student's perspective, relying on external forces can manifest in various forms. It might involve seeking mentorship from experienced professors, utilizing campus resources and facilities, or participating in collaborative research projects. By leveraging these external support systems, we can gain invaluable knowledge, guidance, and opportunities that can propel our academic and professional growth.Ultimately, the most effective approach may lie in striking a balance between self-innovation and strategic reliance on external forces. While independent thinking and creativity are essential for pushing boundaries and challenging the status quo, it is often through the synergy of individual efforts and institutional support that true transformative change can occur.As students, we should strive to cultivate our innate curiosity, critical thinking skills, and problem-solving abilities. At the same time, we must recognize the value of seeking guidance, collaborating with peers and mentors, and leveraging the resources and infrastructure provided by our educational institutions. By combining our individual drive for self-innovation with the power of external support systems, we can maximize our potential for personal growth, academic achievement, and, ultimately, contributing to the advancement of knowledge and society.In conclusion, the interplay between self-innovation and reliance on external forces is a complex and nuanced dynamic. While individual ingenuity and independent thinking are indispensable drivers of progress, true transformative change often requires the convergence of personal efforts with institutional resources and societal support systems. As students,our task is to navigate this balance judiciously, nurturing our intrinsic thirst for knowledge and creativity while strategically leveraging the guidance and opportunities afforded by the world around us.篇3Independent Innovation vs. Reliance on External ForcesEver since I was a kid, I've been really into science and technology. I remember spending hours tinkering with old electronics, trying to figure out how they worked and if I could modify or improve them in any way. My parents didn't always understand my geeky hobbies, but they encouraged my curiosity and let me set up a little workshop in the garage.As I got older and started learning about the history of innovation, I became fascinated by the stories of independent inventors and tinkerers whose creative sparks led to groundbreaking inventions and advancements. People like Thomas Edison, Nikola Tesla, the Wright Brothers, and Steve Wozniak. They didn't have big research labs or corporate backing - they followed their passions, thought outside the box, and persevered through failure after failure until they achieved something revolutionary.At the same time, I recognized the incredible resources, infrastructure, and brain power that large organizations and institutions can bring to research and development efforts. Juggernauts like NASA, Silicon Valley tech giants, international research consortiums, and major universities have capabilities that no lone individual could ever match. Their work has undeniably propelled humanity forward in countless fields.So which approach is better: independent, free-wheeling innovation driven by maverick thinkers? Or intensive R&D backed by the vast resources of corporations, governments, and academia? In my opinion, both models are valuable and the greatest advances often come from a synthesis of the two.The archetypal garage tinkerer has the freedom to explore weird ideas without being constrained by bureaucracy or the pressure to show constant, measurable progress. Their offbeat perspectives can spark wholly original lines of thinking that transcend conventional wisdom. When Thomas Edison was developing the light bulb, the scientific consensus was that his approach would never work. If he had given up or followed the crowd, we might still be sitting in the dark.Steve Jobs and Steve Wozniak started Apple Computer in a suburban garage, fueled by wild ideas about making computersaccessible to ordinary people. Their scrappy startup completely revolutionized the tech industry. Imagine if they had just taken regular jobs at established companies instead of taking that entrepreneurial leap.These lone innovators have the liberty to indulge their curiosities and pursue passion projects that big organizations would never greenlight because they seem too eccentric or impractical. But that openness to unconventional thinking is precisely what allows them to achieve such unconventional results. As the saying goes, "no guts, no glory."On the flip side, few individual amateurs could accomplish what large, well-funded teams can with their specialized knowledge, cutting-edge tools, and pooled brainpower. Good luck building a particle accelerator or sending a rover to Mars out of your garden shed! The immense scale and complexity of modern scientific and technological development often requires institutional resources, collaboration among experts across disciplines, and expensive capital.For example, CERN's Large Hadron Collider was an audaciously ambitious project that cost over 9 billion and involved over 10,000 researchers from 100+ countries. Coordinating such a massive, multinational effort would beutterly impossible for a lone eccentric. An individual might spark the initial idea, but executing on something of that magnitude necessitates government funding, political cooperation, and bringing together diverse technical teams with specialized facility.We see a similar dynamic in the world of tech startups. While a visionary like Steve Jobs started in his garage, Apple couldn't have grown into the juggernaut it is today without eventually achieving corporate scale. Entrepreneurial zeal and creative thinking are vital sparks, but to truly change the world, you ultimately need institutional backing.Large organizations also benefit from economy of scale, extensive infrastructure, and accumulated expertise that loners lack access to. This allows them to iterate on existing innovations at a rapid pace without having to start from square one each time. Corporations can take a cool concept developed in a university lab and use their vast R&D resources to streamline manufacturing, work through practical issues, and optimize the product for the mass market.So in an ideal world, the two approaches feed into each other. Quirky outsiders come up with the moonshot ideas that mainstream science or industry scoffs at...until those blue-skyvisions are eventually validated and absorbed by universities, national labs, or corporate R&D departments who have the means to thoroughly explore and harness those novel concepts at a larger scale.The alternating current system that powers our cities? Pioneered by the maverick Nikola Tesla, but its development and widespread adoption was driven by industrial titans like George Westinghouse. The Apollo space program that landed men on the moon? Sparked by backyard amateur rocket clubs, then amplified by NASA's eye-popping resources. Moderne-commerce? An eccentric concept - shopping from home via computer networks - until infrastructure from internet and logistics giants finally made it mainstream.So in retrospect, it seems that the greatest advancements arise from a productive cycle of individual ingenuity catalyzing institutional efforts that further develop, scale, and proliferate those original insights in ways a lone tinkerer could never accomplish alone. Independent innovation disrupts; institutional resources transform those disruptions into concrete progress. This symbiotic relationship has hopefully become clear.So then where do I hope to fit into this dynamic landscape of innovation? Well, in the short term, I thoroughly plan to embracemy identity as an impassioned amateur daydreamer. I intend to indulge every weird notion that pops into my head, unencumbered by practical considerations or judgments about what's "realistic."My rule is: if a batty idea captivates my imagination, I'll follow it down the rabbit hole of research and tinkering to see what unexpected insights or prototypes I can produce. Who knows, maybe I'll stumble into something significant. Even if not, the process of unfettered exploration itself yields personal growth, creative confidence, and honed critical thinking skills.Or perhaps some of my weird concepts will be radical enough to disrupt conventional thinking in a certain field, planting the seeds for the next big institutional research effort to run with and build upon. I may not cure cancer or Land on Mars myself, but I could instigate fresh thinking that sets the stage for teams of experts to accomplish such feats down the line.After all, we'd never have the wonders of semiconductors, cellular networks, nuclear energy, airplanes, or the internet today without the weird ideas and tireless obsessions of quirky individuals who dared to reimagine what's possible. Perhaps I can play a small part in provoking the next paradigm shift that improves the human condition.Simultaneously though, I aim to prepare myself for more institutional environments like university research labs or forward-thinking companies where I could one day integrate into collaborative teams working on complex, capital-intensive R&D initiatives. By immersing myself in the realms of both independent innovation and institutional resources during different phases of my journey, I believe I can best position myself to make a tangible impact and advance crucial fields in sustainable ways.So in summary, I don't think we have to decisively choose between free-spirited dreamers and deep-pocketed R&D programs when it comes to driving progress. We need both iconoclastic innovators AND institutional resources working in reciprocal harmony for society to keep evolving in positive directions. While my own path may oscillate between the two poles over time, I believe integrating the spirit of both independent and institutional efforts is key to making a lasting mark.Perhaps the most vital skill going forward will be the ability to translate between those two worlds - to gather unconventional ideas from the fringes and find ways to germinate, refine, and scale them up through largercollaborative systems and infrastructures. I hope to play a bridging role in facilitating that fertile exchange of creativity and resources. With open-minded humility, passion, critical thinking, and dedication, I believe our civilization's greatest ambitions can keep getting realized through the dynamism of this balanced approach.。

莫兰蒂作文

莫兰蒂作文

莫兰蒂作文英文回答:Morandi is a captivating and influential aesthetic movement that has left an enduring legacy in the art world. It is characterized by its muted, earthy tones, simple forms, and nuanced compositions, creating a sense of tranquility and timeless elegance.The movement is named after Italian painter Giorgio Morandi, who pioneered the use of these muted tones and simplified forms in the early 20th century. Morandi's work often depicted ordinary objects such as bottles, vases, and bowls, arranging them in geometric and harmonious compositions.One of the key characteristics of Morandi's work is the use of a limited palette, typically consisting of shades of gray, beige, brown, and blue. These colors create a sense of unity and coherence, allowing the subtle variations intone and texture to take center stage.In Morandi's still lifes, objects are carefully arranged to create a sense of balance and harmony. He often used simple, geometric forms, such as circles, squares, and rectangles, which contribute to the overall sense of order and tranquility.Morandi's work has inspired countless artists around the world, and the Morandi aesthetic has found expression in various mediums, including painting, photography, interior design, and fashion. The movement's focus on simplicity, muted tones, and harmonious compositions has made it a timeless and enduring inspiration.中文回答:莫兰蒂是一种迷人而有影响力的美学运动,在艺术界留下了持久的遗产。

丑陋的食物作文英文

丑陋的食物作文英文

丑陋的食物作文英文The Aesthetics of Ugliness in Food.In a world where beauty is often equated withperfection and symmetry, ugliness often finds itself on the fringes, overlooked and underappreciated. This isespecially true when it comes to food, where appearance often plays a crucial role in determining our perception of taste and quality. However, ugly food, despite its unattractive exterior, can sometimes hold surprisingflavors and values that go beyond mere aesthetics.Ugly food is not just about appearance; it's about breaking the mold and challenging our preconceived notions of what makes a dish appealing. It's the squash that's been neglected in the garden, the fruits that have been misshapen by the elements, or the vegetables that have been bruised in the harvesting process. These are the foods that don't quite fit the standard of beauty but are nonetheless rich in nutrition and flavor.One of the most fascinating aspects of ugly food is its ability to surprise us. Just because a dish may not lookits best doesn't mean it lacks in taste or texture. In fact, sometimes, it's the ugliest foods that pack the most punch. Think of the deeply wrinkled and misshapen tomatoes thatare often the most flavorful, or the oddly shaped potatoes that can make the most delicious mashed potatoes. It's a testament to the fact that beauty is not always skin-deep.Ugly food also carries a social message. In a world where appearance is often prized above all else, ugly food reminds us that beauty is not the sole determinant of value. It challenges us to look beyond the superficial and appreciate the intrinsic qualities of a thing. This is especially relevant in today's society, where social media and advertising often promote a narrow definition of beauty that excludes many. By embracing ugly food, we can learn to appreciate diversity and inclusivity, values that arecrucial in building a more equitable and compassionate world.Moreover, ugly food has a significant role to play in sustainable food systems. In an era where food waste is becoming an increasingly pressing issue, ugly food presents an opportunity to reduce waste and make the most of our resources. By embracing foods that may not be visually appealing, we can help divert them from the waste stream and instead turn them into delicious and nutritious meals. This not only reduces waste but also helps to preserve our precious natural resources, making for a more sustainable and environmentally friendly food system.In conclusion, ugly food is not just about appearance; it's about embracing diversity, challenging preconceptions, and valuing what's truly important – flavor, nutrition, and sustainability. It's a reminder that beauty is not always skin-deep and that true value lies within. By opening our minds and hearts to ugly food, we can learn to appreciate its unique beauty and contribute to a more inclusive and sustainable world.。

ansys惯性释放方法

ansys惯性释放方法

IRLFIRLF, KEYSpecifies that inertia relief calculations are to be performed. SOLUTION: InertiaKEYCalculation key:0 — No inertia relief calculations.1 — Counterbalance loads with inertia relief forces.-1 —Command DefaultNo inertia relief calculations.NotesThe IRLF command specifies that the program is to counterbalance the applied loads (inertia relief).for a 2-D structure and 6 for a 3-D structureconstraint points will be zero. Accelerations are calculated This option applies only to the static (ANTYPE,STATIC)plane strain elements are not allowed.analysismay be input as usual. Displacements and stresses are IRLIST to print inertia relief calculation results. The mass summary printed before the solution is accurateprocedural details.must specify EMATWRITE,YES in the initial load step for needed to perform the calculations to be available.5)In the finite element implementation, the position vector {r} and the moment in theapplied load vector are taken with respect to the origin. Considering further specialization for finite elements, Equation 15–5 is rewritten in equivalent form as:6)beasROTATIONAL7)8)all 批注 [T3]: 位置矢量{r}和力矩{Fr a}是基于原点计算的。

发育生物学体轴与胚层

发育生物学体轴与胚层
Compaction at the late 8-cell stage, With apical-basal polarization of cells
哺乳动物卵裂球的compaction: 发生第三次卵裂后不久
紧密化Compaction: 8细胞分裂后不久,卵裂球突然挤在一起, 卵裂球之间的接触面增大,形成一个紧密的细胞球体.
人类的原肠作用
上胚层和下胚层的形成:在原肠作用开始时,内细胞团分裂为两层.与囊胚 腔接触的一层为下胚层,将用于形成yolk sac;另一层为上胚层.上胚层细胞 间的缝隙将合并、扩大成为羊膜腔,腔中的液体可防止胚胎脱水和保护胚胎受 振荡.上胚层将发育为胚胎的本体.
原肠作用前的胚泡,示内细胞团和滋养层细胞.
注射E-Cadherin抗体的胚胎不能够致密化. 在4细胞期激活 蛋白激酶C导致compaction发生.因此,Compaction可能始于蛋 白激酶C的活化,它引起细胞骨架的重排,在膜上均匀分布的ECadherin重新定位在胞间相交处.
哺乳动物囊胚细胞命运的早期分化-位置决定论
16细胞期桑椹胚morula:位于内部的少数细胞产生的子细胞将组 成内细胞团inner cell mass,而位于外部的细胞产生的子细胞大多 构成滋养外胚层trophoblast.
哺乳动物胚轴形成与胚层分化
Vertebrate embryos show differences in form before gastrulation
After gastrulation, rather similar at
Phylotypic
stage 种系型 期: head structure, neural tube, notochord, somites

《星际迷航斯波克》经典台词

《星际迷航斯波克》经典台词

《星际迷航斯波克》经典台词On the environment: Judging by the pollution content of the atmosphere, I believe we have arrived at the late twentieth century.关于环境:根据大气污染物来判断,我认为我们来到了20世纪末。

On violence: Where there is no emotion, there is no motive for violence.关于暴力:没有情绪,也就不会有使用暴力的念头。

On technology: Computers make excellent and efficient servants, but I have no wish to serve under them.关于科技:计算机是优秀高效的仆人,但我不希望听他们指挥。

On vanity: Beauty is transitory.关于浮华:美是短暂的。

On impulse buying: It is curious how often you humans manage to obtain that which you do not want.关于冲动消费:真是奇怪,你们人类多长时间就要想法设法,得到自己不想要的东西。

On women: I have never understood the female capacity to avoid a direct answer to any question.关于女人:我永远也搞不明白女士们那种回避直接回答问题的能力。

On desire: After a time, you may find that having is not so pleasing a thing, after all, as wanting. It is not logical, but it is so often true.关于欲望:过些时候,你也许会发现拥有一件东西毕竟不像需要时那样令人开心。

青岛2024年11版小学5年级第三次英语第一单元期中试卷

青岛2024年11版小学5年级第三次英语第一单元期中试卷

青岛2024年11版小学5年级英语第一单元期中试卷考试时间:80分钟(总分:140)A卷考试人:_________题号一二三四五总分得分一、综合题(共计100题)1、听力题:The _____ (信封) is addressed.2、听力题:A chemical reaction that releases heat is called an ________ reaction.3、填空题:The ________ (生态恢复行动) is ongoing.4、听力题:The puppy is ___. (cute)5、填空题:Many plants need ______ (水) to survive.6、ts can grow in ______, while others need rich soil.(有些植物可以在贫瘠的土壤中生长,而其他植物需要肥沃的土壤。

) 填空题:Some pla7、填空题:I have a robot _______ that can dance and sing.8、听力题:In ancient times, people used the stars for ______.9、填空题:The first human to break the sound barrier was ______ (查尔斯·杨)。

10、听力题:An endothermic reaction requires _____ from its surroundings.The _____ (植物维护) requires knowledge and dedication.12、填空题:The ________ makes a nest in the trees.13、听力题:A solution that can no longer dissolve any more solute is called a _______ solution.14、填空题:The weather is _______ (非常宜人).15、听力题:The process of separating a solid from a liquid using a filter is called _______.16、What do we call the main road through a city?A. PathB. AlleyC. AvenueD. Street答案:C17、听力题:My friend is very ________.18、Which animal says "woof"?A. CatB. DogC. CowD. Sheep答案:B19、听力题:The __________ Ocean is located between Europe and Asia.20、填空题:My birthday is in ______ (十月).21、What is the main ingredient in bread?A. RiceB. FlourC. SugarD. Salt答案:B__________ are used in the food industry for flavor enhancement.23、听力题:The chemical formula for sodium sulfate is __________.24、听力题:She is _______ (sharing) her toys with others.25、听力题:The process of melting involves adding _____ to a solid.26、What do you call a person who cures animals?A. VeterinarianB. DoctorC. ScientistD. Farmer27、听力题:I have a _____ (ticket) for the show.28、听力题:Gravity is weaker on the moon than on _______.29、选择题:What do we call a person who studies plants?A. BotanistB. ZoologistC. GeologistD. Ecologist30、选择题:Which animal is known for its stripes?A. LionB. TigerC. BearD. Elephant31、听力题:My friend is ______ (tall) and funny.32、填空题:The __________ (古代王国) often waged wars for territory.33、填空题:The capital of Djibouti is ________ (吉布提).The chemical structure of a compound can be represented by a ______ formula.35、听力题:The age of rocks can be determined using ______ dating methods.36、填空题:The ________ (山) is very tall and covered with trees.37、填空题:My sister loves to _______ (动词) in the park. 我们常常一起 _______ (动词).38、听力题:The car is parked _____ (in front/behind) the house.39、What do you call a story that is written to teach a lesson?A. FableB. NovelC. BiographyD. Poem答案: A40、What is the name of the process by which plants lose water?A. TranspirationB. PhotosynthesisC. RespirationD. Germination答案:A41、填空题:The discovery of ________ has had wide-ranging impacts on industry.42、听力题:We play basketball in the ______. (gym)43、What do we call the place where we go to watch movies?A. TheaterB. CinemaC. AuditoriumD. Both A and B答案:D44、听力题:The chemical formula for acetic anhydride is ______.The chemical formula for sodium phosphate is _____.46、听力题:A __________ is a solid formed from a solution.47、听力题:A __________ is a young horse.48、听力题:The weather is _____ (nice/bad) today.49、填空题:When I go to the movies, I like to bring my favorite ________ (玩偶) along.50、听力题:A shooting star is actually a _______ that burns up in the atmosphere.51、听力题:I want to _____ (join/start) a club.52、填空题:Chemists use _______ to predict the outcome of chemical reactions. (模型)53、What is the main ingredient in guacamole?A. TomatoB. AvocadoC. BananaD. Pepper答案: B54、Road connected China with _______. (欧洲) 填空题:The Silk55、填空题:The tortoise is very _________ (耐心).56、听力题:A __________ is a large area of sand.57、填空题:The horse gallops across the _________. (草原)58、What do we wear on our feet?A. HatB. SocksC. GlovesD. Scarf答案:B59、听力题:The baby is ______ in her crib. (sleeping)60、填空题:I have a _______ (新朋友) in school.61、听力题:__________ are important for the production of synthetic materials.62、听力题:The chemical formula for potassium nitrate is __________.63、填空题:My sister enjoys __________ (参加) creative workshops.64、听力题:My cousin plays the ____ (violin) very well.65、听力题:A shark is a type of __________.66、听力题:The __________ is a famous historical building.67、What is the capital of Vietnam?A. Ho Chi Minh CityB. HanoiC. Da NangD. Nha Trang68、填空题:The ancient Greeks used ______ (戏剧) for entertainment.69、填空题:I can ______ (制定) plans for the future.70、填空题:We need to water the ________ (花园).71、填空题:My family has a ________ (宠物) cat.72、What is the name of the fairy tale character who lives in a tower?A. CinderellaB. RapunzelC. Snow WhiteD. Belle73、What color do you get when you mix red and white?A. PinkB. PurpleC. OrangeD. Brown74、填空题:My dog is very ____.75、填空题:在古代的________ (kingdoms) 中,王子通常接受良好的教育。

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a rXiv:h ep-th/977141v221J ul1997DPNU-97-33hep-th/9707141Zero Mode and Symmetry Breaking on the Light Front 1Koichi Yamawaki 2Department of Physics,Nagoya University,Nagoya 464-01,Japan Abstract We discuss the spontaneous symmetry breaking (SSB)on the light front (LF)in view of the zero mode.We first demonstrate impossibility to remove the zero mode in the continuum LF theory by two examples:The Lorentz invariance forbids even a free theory on the LF and the trivial LF vacuum is lost in the SSB phase,both due to the zero mode as the accumulating point causing uncontrollable infrared sin-gularity.We then adopt the Discretized Light-Cone Quantization (DLCQ)which was first introduced by Maskawa and Yamawaki to establish the trivial LF vacuum and was re-discovered by Pauli and Brodsky in a different context.It is shown in DLCQ that the SSB phase can be realized on the trivial LF vacuum only when an explicit symmetry-breaking mass of the Nambu-Goldstone (NG)boson m πis introduced as an infrared regulator.The NG-boson zero mode integrated over the LF must exhibit singular behavior ∼1/m 2πin the symmetric limit m π→0in such a way that the LF charge is not conserved even in the symmetric limit;˙Q =0.There exists no NG theo-rem on the LF.Instead,this singular behavior establishes existence of the massless NG boson coupled to the current whose charge satisfies Q |0 =0and ˙Q =0,in much the same as the NG theorem in the equal-time quantization which ensures existence ofthemassless NG boson coupled to the current whose charge satisfies Q |0 =0and ˙Q =0.We demonstrate such a peculiarity in a concrete model,the linear sigma model,where the role of zero-mode constraint is clarified.1INTRODUCTIONMuch attention has recently been paid to the light-front(LF)quantization[1]as a promising approach to solve the nonperturbative dynamics.The most important aspect of the LF quantization is that the physical LF vacuum is simple,or even trivial[2].However,such a trivial vacuum,which is vital to the whole LF approach,can be realized only if we can remove the so-called zero mode out of the physical Fock space(“zero mode problem”[3]).Actually,the Discretized Light-Cone Quantization(DLCQ)wasfirst introduced by Maskawa and Yamawaki[3]in1976to resolve the zero mode problem and was re-discovered by Pauli and Brodsky[4]in1985in a different context.The zero mode in DLCQ is clearly isolated from other modes and hence can be treated in a well-defined manner without ambiguity, in sharp contrast to the continuum theory where the zero mode is the accumulating point and hard to be controlled in isolation[5].In DLCQ,Maskawa and Yamawaki[3]in fact discovered a constraint equation for the zero mode(“zero-mode constraint”)through which the zero mode becomes dependent on other modes and then they observed that the zero mode can be removed from the physical Fock space by solving the zero-mode constraint, thus establishing the trivial LF vacuum in DLCQ.Such a trivial vacuum,on the other hand,might confront the usual picture of complicated nonperturbative vacuum structure in the equal-time quantization corresponding to confine-ment,spontaneous symmetry breaking(SSB),etc..Since the vacuum is proved trivial in DLCQ[3],the only possibility to realize such phenomena would be through the compli-cated structure of the operator and only such an operator would be the zero mode.In fact the zero-mode constraint itself implies that the zero mode carries essential information on the complicated dynamics.One might thus expect that explicit solution of the zero-mode constraint in DLCQ would give rise to the SSB while preserving the trivial LF vacuum.Ac-tually,several authors have recently argued in(1+1)dimensional models that the solution to the zero-mode constraint might induce spontaneous breaking of the discrete symmetries[6]. However,the most outstanding feature of the SSB is the existence of the Nambu-Goldstone(NG)boson for the continuous symmetry breaking in(3+1)dimensions.In this talk I shall explain,within the canonical DLCQ[3],how the NG phenomenon is realized through the zero modes in(3+1)dimensions while keeping the vacuum trivial.The talk is based on recent works done in collaboration with Yoonbai Kim and Sho Tsujimaru [7,8].We encounter a striking feature of the zero mode of the NG boson:Naive use of the zero-mode constraints does not lead to the NG phase at all((false)“no-go theorem”)in contrast to the current expectation mentioned above.It is inevitable to introduce an infrared regularization by explicit-breaking mass of the NG boson mπ.The NG phase can only be realized via peculiar behavior of the zero mode of the NG-bosonfields:The NG-boson zero mode,when integrated over the LF,must have a singular behavior∼1/m2πin the symmetric limit m2π→0.This we demonstrate both in a general framework of the LSZ reductionformula and in a concretefield theoretical model,the linearσmodel.The NG phase is in fact realized in such a way that the vacuum is trivial while the LF charge is not conserved in the symmetric limit m2π→0.2ZERO MODE PROBLEM IN THE CONTINUUM THEORYBefore discussing the zero mode in DLCQ,we here comment that it is impossible to remove the zero mode in the continuum theory in a manner consistent with the trivial vacuum,as far as we use the canonical commutator3:[φ(x),φ(y)]x+=y+=−iπP +∞−∞dp+3We choose the LF“time”as x+=x−≡12(x0+x3)and the longitudinal and the transverse coordinates are denoted by x≡(x−,x⊥),with x−≡12(x0−x3)and x⊥≡(x1,x2),respectively.through the principal value prescription and hence has no p+≡0mode.The point is that the real problem with the zero mode in the continuum theory is not a single mode with p+≡0,which is just measure zero,but actually the accumulating point p+→0[5].To demonstrate this,let usfirst illustrate a no-go theorem found by Nakanishi and Yamawaki[5].The LF canonical commutator(1)gives explicit expression of Wightman two-point function on LF:0|φ(x)φ(0)|0 |x+=0=12p+e−ip+x−·δ(2)(x⊥),(3) which is logarithmically divergent at p+=0and local in x⊥and,more importantly,is independent of the interaction and the mass.It is easy to see that this is wrong,for example, in the free theory where the Lorentz-invariant two-point Wightman function is given at any point x by∆(+)(x)=12p+e−ip−x+−ip+x−+ip⊥x⊥=m−x2K1(m√4π2√x2⊥),which isfinite(positive definite),nonlocal in x⊥and dependent on mass,in obvious contra-diction to the above result(3).Actually,the Lorentz-invariant result(4)is a consequence of the mass-dependent regularization of1/p+singularity by the infinitely oscillating(mass-dependent)phase factor e−ip−x+=e−i(m2+p2⊥)/2p+·x+before taking the LF restriction x+=0. Namely,there exists no free theory on the LF!This difficulty also applies to the interacting theory satisfying the Wightman axioms[5].Thus the LF restriction from the beginning loses all the information of dynamics carried by the zero mode as the accumulating point.Next we show another difficulty of the continuum LF theory,that is,as far as the sign function is used,the LF charge does not annihilate the vacuum in the SSB phase and hence the trivial LF vacuum is never realized[8].Let me illustrate this in the O(2)σmodel where thefieldsσ,πobey the canonical commutator(1),which yields[Q,σ′(x)]=−iπ(x),[Q,π(x)]=iσ′(x)+iwhere the LF charge is defined as usual by Q= d3 x(π∂−σ′−σ′∂−π−v∂−π),and the anti-periodic boundary condition(B.C.)is imposed onπand the shiftedfieldσ′≡σ−v (v= σ )in order to eliminate the surface terms in(5),φ(x−=∞)+φ(x−=−∞)=0. The non-zero constant term on the R.H.S.of(5)has its origin in the commutation relation (1),which is consistent with the anti-periodic B.C.,π(x−=∞)=−π(x−=−∞)=04. Then we have0|[Q,π(x)]|0 =i 0|σ′(x)|0 +i2v=0.(6) Namely,the LF charge does not annihilate the vacuum,Q|0 =0,due to the zero mode as the accumulating point,even though we have“removed”exact zero mode p+≡0by shifting thefieldσtoσ′.This implies that there in fact exists a zero-mode state|α ≡Q|0 with zero eigenvalue of P+such that P+|α =[P+,Q]|0 =0(due to P+conservation).Our result disagrees with Wilson et al.[10]who claim to have eliminated the zero mode(to be compensated by“unusual counter terms”)in the continuum LF theory.3NAMBU-GOLDSTONE BOSON ON THE LIGHT FRONTIn contrast to the continuum LF theory mentioned above,we already mentioned in Intro-duction that the trivial vacuum is always realized in DLCQ[3].We now discuss how such a trivial vacuum can be reconciled with the SSB phenomena.Here we use DLCQ[3,4], x−∈[−L,L],with a periodic boundary condition in the x−direction,and then take the continuum limit L→∞in the end of whole calculation.We should mention here that the above no-go theorem[5]cannot be solved by simply taking the continuum limit of the DLCQ nor by any other existing method and would be a future problem to be solved in a more profound way.3.1“NO-GO THEOREM”(F ALSE)Let usfirst prove a“no-go theorem”(which will turn out to be false later)that the naive LF restriction of the NG-bosonfield leads to vanishing of both the NG-boson emission vertex and the corresponding current vertex;namely,the NG phase is not realized in the LF quantization[7].Based on the LSZ reduction formula,the NG-boson emission vertex A→B+πmay be written asd4xe iqx B|2π(x)|ABπ(q)|A =i=i(2π)4δ(p−A−p−B−q−)δ(3)( p A− p B− q) B|jπ(0)|A ,(7) whereπ(x)and jπ(x)=2π(x)=(2∂+∂−−∂2⊥)π(x)are the interpolatingfield and the source function of the NG boson,respectively,and qµ=pµA−pµB are the NG-boson four-momenta and q≡(q+,q⊥).It is customary[11]to take the collinear momentum frame, q=0and q−=0(not a soft momentum),for the emission vertex of the exactly massless NG boson with q2=0.Then the NG-boson emission vertex should vanish on the LF due to the periodic boundary condition:(2π)3δ(3)( p A− p B) B|jπ(0)|A= d2x⊥lim L→∞ B| L−L dx−2∂−∂+π |A =0.(8) Another symptom of this disease is the vanishing of the current vertex(analogue of g A in the nucleon matrix element).When the continuous symmetry is spontaneously broken, the NG theorem requires that the corresponding current Jµcontains an interpolatingfield of the NG bosonπ(x),that is,Jµ=−fπ∂µπ+ Jµ,where fπis the“decay constant”of the NG boson and Jµdenotes the non-pole term.Then the current conservation∂µJµ=0leads to0= B| d3 x∂µ Jµ(x)|A x+=0m2A−m2B =−i(2π)3δ(3)( q)theπzero mode(with P+=0)ωπ≡1m2π(m2π→0).(11) This situation may be clarified when the PCAC relation is written in the momentumspace:m2πfπjπ(q2)m2π−q2+∂µ Jµ(q).(12) What we have done when we reached the“no-go theorem”can be summarized as follows: Wefirst set L.H.S of(12)to zero(or equivalently,assumed implicitly the regular behavior of d3 xωπ(x))in the symmetric limit in accord with the current conservation∂µJµ=0.Then in the LF formalism with q=0(q2=0),thefirst term(NG-boson pole term)of R.H.S. was also zero due to the periodic boundary condition or the zero-mode constraint.Thus we arrived at∂µ Jµ(q)=0.However,this procedure is equivalent to playing a nonsense game:1=lim m2π,q2→0(m2π−q2i [Q,P−]=∂µJµ| q=0=fπlim m2π→0m2πd3 xωπ=0.(13)4THE SIGMA MODELLet us now demonstrate [7]that(11)and(13)indeedtake place as the solution of the constrained zero modes in the NG phase of the O (2)linear σmodel:L =12(∂µπ)2−14(σ2+π2)2+cσ,(14)where the last term is the explicit breaking which regularizes the NG-boson zero mode.In the DLCQ we can clearly separate the zero modes (with P +=0),π0≡14{ǫ(x −−y −)−x −−y −2LL −L dx − (µ2−∂2⊥)π+λπ(π2+σ2) =0,(16)and similarly,χσ≡12LL −L dx −2∂+∂−π=0,(similarly for σ)which we have used to prove the “no-go theorem”for the case of m 2π≡0.Thus the ‘no-go”theorem is a consequence of the zero-mode constraint itself in the case of m 2π≡ly,solving the zero-mode constraint does not give rise to SSB at all in the exact symmetric case m 2π≡0,in contradiction to the naive expectation [6].Actually,in the NG phase(µ2<0)the equation of motion ofπreads(2+m2π)π(x)=−λ(π3+πσ′2+2vπσ′)≡jπ(x),withσ′=σ−v and m2π=µ2+λv2=c/v,where v≡ σ isthe classical vacuum solution determined byµ2v+λv3=c.Integrating the above equation of motion over x,we haved3 x jπ(x)−m2π d3 xωπ(x)= d3 x2π(x)=− d3 xχπ=0,(17)where d3 xωπ(x)= d3 xπ(x).Were it not for the singular behavior(11)for theπzero modeωπ,we would have concluded(2π)3δ(3)( q) π|jπ(0)|σ =− π| d3 xχπ|σ =0in the symmetric limit m2π→ly,the NG-boson vertex at q2=0would have vanished, which is exactly what we called“no-go theorem”now related to the zero-mode constraint χπ.On the contrary,direct evaluation of the matrix element of jπ=−λ(π3+πσ′2+2vπσ′) in the lowest order perturbation yields non-zero result even in the symmetric limit m2π→0: π|jπ(0)|σ =−2λv π|ϕσϕπ|σ =−2λv=0( q=0),which is in agreement with the usual equal-time formulation.Thus we have seen that naive use of the zero-mode constraints by setting m2π≡0leads to the internal inconsistency in the NG phase.The“no-go theorem”is again false.The same conclusion can be obtained more systematically by solving the zero-mode constraints in the perturbation around the classical(tree level)solution to the zero-mode constraints which is nothing but the minimum of the classical potential:vπ=0and vσ≡v, where we have divided the zero modesπ0(orσ0)into classical constant piece vπ(or vσ)and operator partωπ(orωσ).The operator zero modes are solved perturbatively by substituting the expansionωi= k=1λkω(k)i intoχπ,χσunder the Weyl ordering.The lowest order solution of the zero-mode constraintsχπandχσforωπtakes the form:λ(−m2π+∂2⊥)ωπ=This actually ensures non-zeroσ→ππvertex through(17): π|jπ(0)|σ =−2λv,which agrees with the previous direct evaluation as it should.Let us next discuss the LF charge operator corresponding to the current Jµ=∂µσπ−∂µπσ.The LF charge Q= Q= d3 x(∂−ϕσϕπ−∂−ϕπϕσ)contains no zero modes and hencenoπ-pole term which was dropped by the integration due to the periodic boundary condition and the∂−,so that Q is well defined even in the NG phase and hence annihilates the vacuum simply by the P+conservation[3]:Q|0 =0.(20) This is also consistent with explicit computation of the commutators: [Q,ϕσ] =−i ϕπ =0 and [Q,ϕπ] =i ϕσ =05,which are contrasted to(6)in the continuum theory.They are also to be compared with those in the usual equal-time case where the SSB charge does not annihilate the vacuum Q et|0 =0: [Q et,σ] =−i π =0, [Q et,π] =i σ =0.Since the PCAC relation is now an operator relation for the canonicalfieldπ(x)with fπ=v in this model,(19)ensures[Q,P−]=0or a non-zero current vertex π| J+|σ =0 (q2=0)in the symmetric limit.Noting that Q= Q,we conclude that the regularized zero-mode constraints indeed lead to non-conservation of the LF charge in the symmetric limit m2π→0:˙Q=15By explicit calculation with a careful treatment of the zero-modes contribution we can also show that [Q,σ] = [Q,π] =0[8].behavior of the NG-boson zero mode(11)(or(19))may be understood as a remnant of the Lagrangian symmetry,an analogue of the NG theorem in the equal-time quantization. AcknowledgmentsI would like to thank Y.Kim and S.Tsujimaru for collaboration.This work was supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education,Science and Culture(No.08640365).References[1]P.A.M.Dirac,Rev.Mod.Phys.21(1949)392.[2]H.Leutwyler,J.R.Klauder,and L.Streit,Nuovo Cim.66A,536(1970).[3]T.Maskawa and K.Yamawaki,Prog.Theor.Phys.56,270(1976).[4]H.C.Pauli and S.J.Brodsky,Phys.Rev.D32,1993(1985);ibid2001(1985).[5]N.Nakanishi and K.Yamawaki,Nucl.Phys.B122,15(1977).[6]T.Heinzl,S.Krusche,S.Simb¨u rger and E.Werner,Z.Phys.C56,415(1992);D.G.Robertson,Phys.Rev.D47,2549(1993);C.M.Bender,S.Pinsky,and B.Van de Sande,Phys.Rev.D48,816(1993).[7]Y.Kim,S.Tsujimaru and K.Yamawaki,Phys.Rev.Lett.74(1995)4771.[8]S.Tsujimaru and K.Yamawaki,Heidelberg/Nagoya Preprint,hep-th/9704171.[9]P.Steinhardt,Ann.Phys.32,425(1980).[10]K.G.Wilson,T.S.Walhout,A.Harindranath,W.Zhang,R.J.Perry and S.D.Glazek,Phys.Rev.D49,6720(1994).[11]S.Weinberg,Phys.Rev.150,1313(1966).[12]G.McCartor and D.G.Robetson,Z.Phys.C53,679(1992).。

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