Generalized Pinch Technique and the Background Field Method in General Gauges
A Fast and Practical Algorithm for Generalized Penetration Depth Computation
A Fast and Practical Algorithm for Generalized Penetration Depth Computation Liangjun Zhang1Young J.Kim2Dinesh Manocha1 1Dept.of Computer Science,University of North Carolina at Chapel Hill,USA,{zlj,dm}@2Dept.of Computer Science and Engineering,Ewha Womans University,Korea,kimy@ewha.ac.kr/PDGAbstract—We present an efficient algorithm to compute the generalized penetration depth(PD g)between rigid models.Given two overlapping objects,our algorithm attempts to compute the minimal translational and rotational motion that separates the two objects.We formulate the PD g computation based on model-dependent distance metrics using displacement vectors.As a result,our formulation is independent of the choice of inertial and body-fixed reference frames,as well as specific representation of the configuration space.Furthermore,we show that the optimum answer lies on the boundary of the contact space and pose the computation as a constrained optimization problem.We use global approaches tofind an initial guess and present efficient techniques to compute a local approximation of the contact space for iterative refinement.We highlight the performance of our algorithm on many complex models.I.I NTRODUCTIONPenetration depth(PD)is a distance measure that quantifies the amount of interpenetration between two overlapping objects. Along with collision detection and separation distance,PD is one of the proximity queries that is useful for many applications including dynamics simulation,haptics,motion planning,and CAD/CAM.Specifically,PD is important for computing collision response[1],estimating the time of con-tact in dynamics simulation[2],sampling for narrow passages in retraction-based motion planing[3],[4],and C-obstacle query in motion planning[5].There has been considerable work on PD computation,and good algorithms are known for convex polytopes.As for non-convex models,prior approaches on PD computation can be classified into local or global algorithms.The local algorithms only take into account the translational motion,i.e.transla-tional PD(PD t),and the results may be overly conservative. In many applications,including torque computation for6-DOF haptic rendering or motion planning for articulated models,it is important to compute a penetration measure that also takes into account the rotational motion,i.e.generalized penetra-tion depth(PD g).However,the computational complexity of global PD between non-convex models is high.For PD t,it can be computed using Minkowski sum formulation with the combinatorial complexity O(n6),where n is the number of features in the models[6].For PD g,it can be formulated by computing the arrangement of contact surfaces,and the combinatorial complexity of the arrangement is O(n12)[7]. As a result,prior algorithms for global PD only compute an approximate solution[5],[8].Moreover,these algorithms perform convex decomposition on non-convex models and can be rather slow for interactive applications.Overall,there are no good and practical solutions to compute the PD between non-convex models,thereby limiting their applications[4],[9],[10].A key issue in PD g computation is the choice of an appropriate distance metric.It is non-trivial to define a distance metric that can naturally combine the translational and rotational motion for an undergoing model,such that the resulting distance metric is bi-invariant with the choice of inertial and body-fixed reference frames,as well as of specific representations of the configuration space[11].Specifically,it is well-known that for the spatial rigid body motion group SE(3),it is impossible to define a bi-invariant distance metric unless the shape of the model is known a priori[12],[13].Finally,the distance metric should be easy to evaluate in order to devise an efficient PD g computation algorithm.A.Main ResultsWe present an efficient algorithm for computing PD g for rigid,non-convex models.We formulate PD g computation as a constrained optimization problem that minimizes an objective function defined by any proper distance metric that combines both translational and rotation motions,such as DISP[14]and object norm[15].We use global approaches, based on motion coherence and random sampling,to compute an initial guess and incrementally walk on the contact space along the maximally-decreasing direction of the objective function to refine the solution.The algorithm computes a local approximation of the contact space,and we present culling techniques to accelerate the computation.As compared to the prior approaches,our algorithm offers the following benefits:•Generality:Our approach is general and applicable to both convex and non-convex rigid models.The algorithm can be also extended to articulated or deformable models.•Practicality:Unlike the prior approaches,our algorithm is relatively simple to implement and useful for many applications requiring both translational and rotation mea-sures for inter-penetration.•Efficiency:We use a local optimization algorithm and reduce the problem of PD g computation to multiple collision detection and contact queries.As a result,our algorithm is efficient and can be used for interactive applications with high motion coherence.We have implemented our PD g algorithm and applied it to many non-convex polyhedra.In practice,our algorithm takes about a few hundred milli-seconds on models composed of a few thousand triangles.anizationThe rest of our paper is organized as follows.We providea brief survey of related work on PD g computations in Sec.2.In Sec.3,we present a formulation of PD g and give an overview of distance metrics.In Sec.4,we provide our optimization-based algorithm to compute PD g.We present its implementation and highlight its performance in Sec5.II.P REVIOUS W ORKThere has been considerable research work done on proximity queries including collision detection,separation distance,and PD computation[16],[17].In this section,we briefly discuss prior approaches to PD computation and distance metrics. A.PD ComputationMost of the work in PD computation has been restricted to PD t,and these algorithms are based on Minkowski sums[6],[18].A few good algorithms are known for convex polytopes[19],[20]and general polygonal models[8].Due to the difficulty of computing a global PD t between non-convex models,some local PD t algorithms have been proposed[9],[10],[21].A few authors have addressed the problem of PD g compu-tation.Ong’s work[22],[23]can be considered as one of the earliest attempts.The optimization-based method using a quadratic objective function can be regarded as implicitly computing PD g[24].Ortega et al.[25]presented a method to locally minimize the kinetic distance between the config-urations of a haptic probe and its proxy using constraint-based dynamics and continuous collision detection.Zhang et al.[5]proposed thefirst rigorous formulation of computing PD g.They presented an efficient algorithm to compute PD g for convex polytopes,and provide bounds on PD g of non-convex polyhedra.The problem of PD g computation is closely related to the containment problem[26].The notion of growth distance has been introduced to unify separation and penetra-tion distances[22].Recently,Nawratil et al.[27]have also described a constrained optimization based algorithm for PD g computation.B.Distance Metrics in Configuration SpaceThe distance metric in configuration space is used to measure the distance between two configurations in the space.It is well-known that model-independent metrics are not bi-invariant, and thus most approaches use model-dependent metrics for proximity computations[11],[14],[28].1)Distance Metrics in SE(3):The spatial rigid body displace-ments form a group of rigid body motion,SE(3).Throughout the rest of the paper,we will refer to a model-independent distance metric in SE(3)as a distance metric in SE(3).In theory,there is no natural choice for distance metrics in SE(3) [12],[13].Loncaric[29]showed that there is no bi-invariant Riemannian metric in SE(3).2)Model-dependent Distance Metrics:Using the notion ofa displacement vector for each point in the model,the DISP distance metric is defined as the maximum length over all the displacement vectors[14],[28],[30].The object norm, proposed by[15],is defined as an average squared length of all displacement vectors.Hofer and Pottmann[31]proposed a similar metric,but consider only a set of feature points in the model.All of these displacement vector-based metrics can be efficiently evaluated.The length of a trajectory travelled by a point on a moving model can be also used to define model-dependent metrics[5],[32].However,it is difficult to compute the exact value of these metrics.III.G ENERALIZED P ENETRATION D EPTH AND D ISTANCEM ETRICSIn this section,we introduce our notation and highlight issues in choosing an appropriate distance metric for defining PD g for polyhedral models.We then show that our metrics can naturally combine translational and rotational motions,have invariance properties,and can be rapidly calculated.We also show that the optimal solution for PD g computation with respect to each metric exists on the contact space.A.Notation and DefinitionsWefirst introduce some terms and notation used throughout the rest of the paper.We define the contact space,C contact, as a subset of the configuration space,C,that consists of the configurations at which a robot A only touches one or more obstacles without any penetration.The union of free space F and contact space constitutes the valid space,C valid,of the robot,and any configuration in C valid is a valid configuration. The complement of F in C is the C-obstacle space or O. PD g is a measure to quantify the amount of interpenetration between two overlapping models.Given a distance metricδin configuration space,PD g between two polyhedral models A and B can be defined as:PD gδ(A,B)={min{δ(q o,q)} interior(A(q))∩B=/0,q∈C},(1) where q o is the initial configuration of A,and q is any configuration in C.PD g can be formulated as an optimization problem under non-penetration constraints(Fig.1(a)),where the optimization objective is described by some distance metric to measure the extent of a model transformed from one configuration to another.Therefore,the computation of PD g is directly governed by the underlying distance metric.Fig. 1.PD g Definition and Contact Space Realization:(a)PD g is defined as the minimal distance between the initial collision configuration q o and any free or contact configuration,with respect to some distance metric.(b)The optimal configuration q b,which realizes PD g DISP or PD gσ, must be on the contact space C contact;otherwise,one can compute another contact configuration q b′,which further reduces the objective function.q b′is computed by applying the bisection method on the screw motion that interpolates q o and q b.B.Distance MetricWe address the issue of choosing an appropriate distance metric to define PD g.In principle,any distance metric in C-space can be used to define PD g.We mainly use two distance metrics for rigid models,displacement distance metric DISP [28],[30]and object norm[15].1)Displacement distance metric:Given a model A at two different configurations q a and q b,the displacement distance metric is defined as the longest length of the displacement vectors of all the points on A[28],[30]:DISP A(q a,q b)=maxx∈A||x(q b)−x(q a)||2.(2)2)Object norm:Also based on displacement vectors,Kazer-ounian and Rastegar[15]make use of an integral operator to define the object norm:σA(q a,q b)=1VAρ(x)||x(q b)−x(q a)||2dV,(3)where V andρ(x)are the volume and mass distribution of A, respectively.3)Properties of DISP andσ:Both metrics can combine the translational and rotational components of SE(3)without relying on the choice of any weighting factor to define PD g. Since both metrics are defined by using displacement vectors, they have some invariance properties;they are independent of the choice of inertial reference frame and body-fixed reference frame[11],and also independent of the representation of C. Moreover,DISP andσmetrics can be computed efficiently. In[14],we show that for a rigid model,the DISP distance is realized by a vertex on its convex hull.This leads to an efficient algorithm,C-DIST,to compute DISP.Forσ,by using a quaternion representation,we can further simplify the formula originally derived by Kazerounian and Rastegar[15] into:σA(q a,q b)=4V(I xx q21+I yy q22+I zz q23)+q24+q25+q26,(4)where diag(I xx,I yy,I zz)forms a diagonal matrix computed bydiagonalizing the inertia matrix I of A.(q0,q1,q2,q3)is thequaternion for the relative orientation of A between q a andq b,and(q4,q5,q6)is the relative translation.C.Properties of PD g DISP and PD gσGeometrically speaking,the generalized penetration depthunder DISP,PD g DISP,can be interpreted as the minimum ofthe maximum lengths of the displacement vectors for all thepoints on A,when A is placed at any collision-free or contactconfiguration.Also,the generalized penetration depth underσ,PD gσ,can be interpreted as the minimum cost to separateA from B,where the cost is related to the kinetic energy of A.Due to the underlying distance metric,both PD g DISP andPD gσare independent of the choice of inertial and body-fixed reference frames.In practice,these invariance propertiesare quite useful since one can choose any arbitrary referenceframe and representation of the configuration space to computePD g DISP and PD gσ.D.Contact Space RealizationFor rigid models,PD g DISP(or PD gσ)has a contact spacerealization property.This property implies that any validconfiguration q b that minimizes the objective DISP(orσ)forPD g must lie on the contact space of A and B,or equivalently,at this configuration q b,A and B just touch with each other.Theorem1(Contact Space Realization)For a rigid modelA placed at q o,and a rigid model B,if q b∈C valid andDISP A(q o,q b)=PD g DISP(A,B),then q b∈C contact.A similarproperty holds for PD gσ.Proof:We prove it by contradiction.Suppose the config-uration q b realizing PD g DISP does not lie on the contact spaceC contact.Then,q b must lie in the free space F((Fig.1(b)).We use Chasles’theorem in Screw theory[33],which statesthat a rigid body transformation between any two configura-tions can be realized by rotation about an axis followed bytranslation parallel to that axis,where the amount of rotationis within[0,π].The screw motion is a curve in C-space,andwe denote that curve between q o to q b as s(t),where s(0)=q oand s(1)=q b.Since q o is in O,and q b is in F,there is atleast one intersection between the curve{s(t)|t∈[0,1]}andthe contact space(Fig.1).We denote the intersection point asq b′.Based on Chasles theorem,we can compute the length of thedisplacement vector for any point x on A between q o and anyconfiguration on the screw motion s(t).Furthermore,we canshow that this length strictly increases with the parameter t.Therefore,for each point on A,the length of the displacementvector between q o and q b is less than the one between q o andq b′.Since DISP metric uses the maximum operator for thelength of the displacement vector over all points on A,we canFig.2.Optimization-based PD g Algorithm:ourcontact space C contact,i.e.from q a to q b,tofind a localany distance metric.infer that DISP A(q o,q b′)<DISP A(q o,q b).our assumption that q b is the realization forSimilarly,we can inferσA(q o,q b′)<σA(q o,prove the property for PD gσ.According to Thm.1,in order to compute PD g,to search only the contact space C contact,whichsion lower than that of C.Ourfor PD g uses this property.IV.PD g C OMPUTATION A LGORITHMIn this section,we present our PD g computation algorithm. Our algorithm can optimize any distance metric(or objective) presented in Sec.3by performing incremental refinement on the contact space.As Fig.2illustrates,our iterative optimiza-tion algorithm consists of three major steps:1)Given an initial contact configuration q a,the algorithmfirst computes a local approximation L qa of the contactspace around q a.2)The algorithm searches over the local approximation tofind a new configuration q b that minimizes the objective function.3)The algorithm assigns q b as a starting point for the nextiteration(i.e.walk from q a to q b)if q b is on the contact space with smaller value of the objective function as compared to q a’s.Otherwise,we compute a new contact configuration q b′based on q b.These steps are iterated until a local minimum configuration q m is found or the maximum number of iterations is reached. Next,we discuss each of these steps in more detail.Finally, we address the issue of computing an initial guess.A.Local Contact Space ApproximationSince it is computationally prohibitive to compute a global representation of the contact space C contact,our algorithm computes a local approximation.Given a configuration q a, where A is in contact with B,we enumerate all contact constraints according to the pairs of contact features[28],[34]. We further decompose each contact constraint into primitive contact constraints,i.e.vertex/face(v−f),face/vertex(f−v) or edge/edge(e−e).Conceptually,each primitive contact constraint represents a halfspace,and the set of all primitive constraints are used to characterize the local non-penetrationq a contactq a after concatenating all these primitive constraints{C i} using proper intersection or union operators{◦i}:L qa={C1◦1C2···◦n−1C n}.(5)It should be noted that we do not explicitly compute a geometric representation of L qa.Instead,it is algebraically represented,and each primitive constraint is simply recorded as a pair of IDs,identifying the contact features from A and B,respectively.When decomposing each constraint into primitive constraints, we need to choose proper Boolean operators to concatenate the resulting primitive constraints.This issue has been addressed in the area of dynamics simulation[35]and we address it in a similar manner for PD g computation.Fig.3shows a2D ex-ample with a triangle-shaped robot A touching a notch-shaped obstacle B.When decomposing a v−v contact constraint into two v−e constraints C1and C2,if both of the contact vertices of A and B are convex(Fig.3(a)),we use a union operator, because if either constraint C1or C2is enforced,there is no local penetration.Otherwise,if one contact vertex is non-convex(Fig.3(b)),the intersection operation is used.For3D models,a similar analysis is performed by identifying the convexity of edges based on their dihedral angles.In case of multiple contacts,one canfirst use intersection operations to concatenate all the constraints.Each individual constraint is then further decomposed into primitive constraints.B.Searching over Local Contact SpaceGiven a local contact space approximation L of the contact configuration q a,we search over L tofind q b that minimizes the objective function.Since the contact space is a non-linear subspace of C,we use two different search methods:random sampling in L and optimization over afirst-order approxima-tion of L.Each of them can be performed independently.C1C2L12{}aqc cFFig.4.Sampling in Local Contact Space:L qa is a local approximationof contact space around q a,represented by the intersection of its contact constraints C1and C2.Our algorithm randomly generates samples on C1and C2.Many potentially infeasible samples,such as q l,can be discarded since they are lying outside the halfspace of L qa.1)Sampling in Local Contact Space:Our algorithm randomlygenerates samples on the local contact approximation L qa around q a(Fig.4),by placing samples on each primitivecontact constraint C i as well as on their intersections[36].We discard any generated sample q if it lies outside of thehalfspace formulated by L qa by simply checking the signof L qa (q).Since L qais a local contact space approximationbuilt from all contact constraints,this checking of L allows us to cull potentially many infeasible colliding configurations. For the rest of the configuration samples,we evaluate their distancesδto the initial configuration q o,and compute the minimum.These samples are efficiently generated for each non-linear contact constraint C i.First,we generate random values for the rotation parameters.By plugging these values into a non-linear contact constraint,we formulate a linear constraint for the additional translation parameters.Under the formulated linear constraint,random values are generated for these translation parameters.In practice,an optimal solution for PD g may correspond to multiple contacts,suggesting that one needs to generate more samples on the boundary formed by multiple contact constraints.As a result,we set up a system of non-linear equations for each combination of these constraints,generate random values for the rotation parameters in the system (thereby making the system linear),and sample the resulting linear system for the translation parameters.2)Linearizing the Local Contact Space:We search for a configuration with smaller distance to the contact space by lin-early approximating the contact space.For each basic contact constraint C i,we compute its Jacobian,which is the normal of the corresponding parameterized configuration ing this normal,we obtain a half-plane,which is a linearization of the contact surface[21],[37].By concatenating the half-planes using Boolean operators◦i,we generate a non-convex polyhedral cone,which serves as a local linear approximation of C contact.3)Local Search:The sampling-based method is general for any distance metric.Moreover,we can generate samples on each non-linear contact constraint efficiently.Finally,using the local contact space approximation,our method can cull many potentially infeasible samples.On the other hand,the method of linearizing the contact space Algorithm1Optimization-based Local PD g Algorithm Input:two intersecting polyhedra:A-movable,B-static. q o:=the initial collision configuration of A,q o∈O.q a:=a seed contact configuration of A,q a∈C contact. Output:PD g(A,B)1:repeat2:i++;3:L q a:=Local contact space approximation at q a;4:q b:=argmin{δ(q o,q),q∈L q a};5:ifδ(q o,q b)==δ(q o,q a)then6:returnδ(q o,q a);7:else if q b∈C contact then8:q a:=q b;9:else if q b∈F then10:q a:=CCD Bisection(q o,q b);11:else12:q b′:=CCD(q a,q b);13:L q a:=L q a L′q b;14:goto3;15:end if16:until i<MAX IT ERAT IONis suitable for optimizing PD g,if the underlying objective has a closed form.For example,for the object norm,we transform the coordinate in the quadratic function in Eq.(4),from an elliptic form to a circular one.Now,the problem of searching over L reduces tofinding the closest point in the Euclidean space from q a to the non-convex polyhedral cone,formulated using the linearization of L.Since the polyhedral cone is formulated as a local approximation of C contact,it typically has a small size.Therefore,the closest point query can be performed by explicitly computing the non-convex polyhedral cone.C.RefinementAlthough searching over the local contact space L around q a can yield a new configuration q b that improves the opti-mization objective of q a,we still need to check whether q b is a valid contact configuration before advancing to it because q b is computed based upon a local approximation of contact space and q b may not be on the contact space.For instance,the new configuration q b may be a collision-free configuration due to thefirst-order approximation.To handle this case,we project q b back to C contact by computing the intersection q b′between the contact space and a curve interpolating from q o to q b using screw motion(Fig.1).Since q o is in O and q b is free,the intersection q b′can be efficiently computed by bisection(CCD Bisection in Alg.1).Also, according to the contact space realization theorem in Sec. III.D,δ(q o,q b′)<δ(q o,q b).Therefore,we are guaranteed to obtain a new configuration q b′,which is closer to q o,and thus it can be used for successive iterations.It is also possible the new configuration q b may be a colliding configuration.As Fig.5on the left shows,when moving fromqFig.5.Refinement.Left:using the local contact space representation ofq a,which includes only one constraint C1,we obtain new configuration q b.Though q b is still on C1,it may not be on the contact space any more,sinceit will violate other constraint,such as C2here.The rightfigure shows a dualexample happening in the workspace.When A slides on B,i.e.from q a to q b,a collision can be created by other portions of the models.Our algorithmuses CCD to compute a correct,new contact configuration q b′.q a to q b,the contact constraint C1is maintained.However,q b is a colliding configuration as it does not satisfy the newconstraint C2.Thefigure on the right highlights this scenarioin the workspace.When A moves from q a to q b,the contact isstill maintained.In order to handle this case,we use continuouscollision detection(CCD)to detect the time offirst collisionwhen an object continuously moves from one configurationto another using a linearly interpolating motion in C[38].Inour case,when A moves from q a to q b,we ignore the slidingcontact of q a,and use CCD to report thefirst contact q b′before the collision[39].The new configuration q b′can beused to update the local approximation of q a.This yields amore accurate contact space approximation and consequentlyimproves the local search,e.g.culling away additional invalidsamples.D.Initial GuessThe performance of the PD g algorithm depends on a goodinitial guess.For many applications,including dynamic sim-ulation and haptic rendering,the motion coherence can beused to compute a good initial guess.Since no such motioncoherence could be exploited in some other applications(e.g.sample-based motion planning),we propose a heuristic.Ourmethod generates a set of samples on the contact space asa preprocess.At runtime,given a query configuration q o,our algorithm searches for the K nearest neighbors from theset of precomputed samples,and imposes the inter-distancebetween any pair of these K samples should be greater thansome threshold.The distance metric used for nearest neighborsearch is the same as the one to define PD g.The resulting Ksamples serve as initial guesses for our PD g algorithms.Togenerate samples on the contact space,we randomly samplethe configuration space and enumerate all pairs of free andcollision samples.For each pair,a contact configuration canbe computed by a bisection method(Fig.1(b)).V.I MPLEMENTATION AND P ERFORMANCEWe have implemented our PD g algorithm using local contactspace sampling for general non-convex polyhedra.In thissection,we discuss some important implementation issues andhighlight the performance of our algorithm on a set of complexpolyhedral models.All the timeings reported here were takenon a Windows PC,with2.8GHZ of CPU and2GB of memory.A.ImplementationSince our PD g formulation is independent of the representationof the configuration space,we use a quaternion to representthe rotation because of its simplicity and efficiency.In ourPD g algorithm,any proximity query package supporting col-lision detection or contact determination can be employed.Inour current implementation,we use the SWIFT++collisiondetection library,because of its efficiency and it providesboth these proximity queries[40].Based on SWIFT++,ouralgorithm computes all the contacts between A at a contactconfiguration q a with B.We sample the contact space locallyaround q a.For each primitive contact constraint C i,we deriveits implicit equation with respect to the parameters of a rotationcomponent(a quaternion)and a translation component(a3-vector).In order to sample on a constraint C i,wefirst slightlyperturb its rotational component by multiplying a randomquaternion with a small rotational angle.The resulting rota-tional component is plugged back into the constraint C i.Thisyields a linear constraint with only translational components,and therefore can be used to generate additional samples.To linearize C i,we compute the Jacobian of its implicitequation for C i.For other types of contacts,we decomposethem into primitive contact constraints.Proper operators toconcatenate them are identified by computing the dihedralangle of contacting edges,thereby determining whether thecontact features are convex or not.In the refinement step of the algorithm,we perform collisiondetection using SWIFT++to check whether q b from the localsearch step still lies on the contact space.When q b is oncontact space,our algorithm proceeds to the next iteration.Otherwise,when q b is free,a new contact configuration q b′iscomputed for the next iteration by performing recursive bisec-tions(Fig.1(b))on the screw motion interpolating between q oand q b.Finally,when q b is in C-obstacle space,we computea new contact configuration q b′by using CCD.In our currentimplementation,we check for collision detection on a set ofdiscrete samples on a linear motion between q a and q b.Inorder to ignore the old contact during CCD query,the idea ofsecurity distance is used[39].After computing a new contactconfiguration q b′from the CCD query,our algorithm updatesthe local approximation around q a and resumes a local searchagain.B.PerformanceWe use different benchmarks to test the performance of ouralgorithm.Fig.6(a)shows a typical setup of our experimentincluding two overlapping models,where A(‘Pawn’)is mov-able and B(‘CAD Part’)is stationary.In(b),our algorithmcomputes PD g DISP or PD gσto separate the model A,initiallyplaced at A0,from the model B.The three images on theright highlight the intermediate configurations of A1and A2and a PD g DISP solution A3with yellow color.The sequence ofimages(b,c,d,e)illustrates that our algorithm successfullyfinds。
rhetorical techniques
修辞技巧（Rhetorical Techniques）词义修辞格(Lexical Stylistic Devices)明喻(Simile)He jumped back as if he had been stung, and the blood rushed into his wrinkled face.(他往后一跳，好像被什么东西叮了一下似的，他那张布满皱纹的脸顿时涨得通红。
)在《品尝家》一文中老人对“我”的慷慨施舍的反应如同被蜜蜂叮过一样，生动地刻画出一个处境凄凉内心却极度敏感的可怜老人的形象。
The cheque fluttered to the floor like a bird with a broken wing. (支票跌落到地上，像一只断了翅膀的小鸟。
)《礼物》一文中，老太太喜迎八十大寿，大女儿不来庆祝，只寄来一张支票。
作者把这张支票比作断了翅膀的小鸟，形象地表达出此刻老太太希望破灭，极度伤心的心情。
暗喻(Metaphor)What will parents do without the electronic baby-sitter? (如果没有这位电子保姆，父母该怎么办呢？)形象地说明了电视机的保姆功用。
... while most of us are only too ready to apply to others the cold wind of criticism, we are somehow reluctant to give our fellows the warm sunshine of praise.(……但是我们中的很多人太容易给别人批评的冷风，而不愿意给自己的同伴赞扬的阳光。
)作者把批评比作冷风，把赞扬比作温暖的阳光，生动形象，寓意隽永。
转喻(Metonymy) 即借代, 是通过相近的联想，借喻体代替本体。
My 15 students read Emerson, Thoreau, and Huxley.(我的十五位学生读了爱默生、梭罗和赫胥黎的作品。
On Generalized Uncertainty Principle
a rXiv:081.342v1[he p-th]22Ja n28On Generalized Uncertainty Principle Bhupendra Nath Tiwari ∗Department of Physics,Indian Institute of Technology,Kanpur-208016,India.Abstract We study generalized uncertainty principle through the basic concepts of limit and Fourier transformation to analyze the quantum theory of grav-ity or string theory from the perspective of a complex function.Motivated from the noncommutative nature of string theory,we have proposed a UV/IR mixing dependent function ˜δ(∆x,∆k,ǫ).We arrived at the string uncertainty principle from the analyticity condition of a newly introduced complex function which depends upon the UV cut-off.This non trivially modifies the quantum measurements,black hole physics and short distance geometries.Present analysis is based on the postulate that the Planck scale is the minimal length scale in nature and is in good agreement with the existance of maximum length scale in the nature.Both of these rely only on the analysis of the complex function and do not directly make use of any theory or the specific structure of the Hamiltonian.The Regge behaviour of the string spectrum with the quantization of area is also a natural conse-quence of our new complex function which may contain all the corrections operating in nature and reveal important clues to find the origins of the M-theory.Keywords :generalized uncertainity priciple;string theory;quantum gravityPACS:04.60.-m Quantum gravity;04.60.Nc Lattice and discrete methods;02.30.-f Function theory,analysis.1IntroductionThe analysis of the uncertainty principle in string theory have led to many new insights into the classical and quantum aspects of the general relativity,black hole physics and string theory.The question of the ultimate foundations and the ultimate reality of physics remain open.It is not known in what direction there will be it’sfinal solution or even whether afinal objective answer can at all be expected.We know that the short distance physics is not well understood.So in order to describe the small scale structure of spacetime adequately,we need to modify the usual classical continuum geometry for example,the Connes NCG[1].In other words,an extension of quantum mechanics might be required in order to accomodate the gravity.Moreover,the existence of duality symmetries in non-perturbative string theory indicates that strings do not distinguish small spacetime scales from the large ones which requires a modification of the usual Heisenbergs uncertainty principle where beyond Planck scale energies the size of the string grows with momenta instead of the falling off.An introduction on this T-dual description of spacetime as a result of string theory is given by Witten [2],[3]where below the Planck length the very concept of spacetime changes it’s meaning and the Heisenberg uncertainty principle needs to be modified.The thrust of the present work is to explain the completely generalized uncertainty relations from the viewpoint of a complex function whosefirst order faithful interpolation in terms of the fundamental string length l s is the following formula:∆xµ≥¯h¯h∆pµ,where henceforth the speed of light is taken to be c=1.On the same lines,Carlos Castro has conjectured[4]that the special theory of scale relativity recently proposed by Nottale[5]must play a fundamental role in string theory,specially it demonstrates that there is a universal,absolute and impassable scale in nature,which is invariant under dilatations and it’s lower limit is the Planck scale.The fundamental scales of nature are determined by constraints which are set at both small and large scales in perfect agreement with the string duality principles.Applying the scale relativity principle to the universe,one arrives at the proposition that there must exist an absolute,im-passable,upper scale in nature which is invariant under dilatations(particularly the expansion of the universe)that holds all the properties of the infinity.This upper scale L defines the radius of the universe and when it is seen at its own resolution,it becomes invariant under dilations.In recent years,the measurements in the quantum gravity are goverened by generalized uncertainty principle.Evidences from string theory,quantum geome-try and black hole physics suggest that the usual Heisenburg uncertainty principle needs certain modification(s).These evidences have an origin on the quantum fluctuations of the background metric.The generalized uncertainty principle pro-vides the existance of a minimal length scale to the nature which is of the order of the Planck length.Adler et.al.[6],[7]considers the issue of black hole re-manants in the framework of generalized uncertainity priciple and shows that the generalized uncertainty principle may prevent small black hole’s total evap-oration whereas in the Bekenstein Hawking approach,the total evaporation of a micro black hole is possible.The generalized uncertainty principle indicates the quantum gravitational corrections to the black hole thermodynamics.The coordinates and the corresponding momenta cannot be simultaneously specified after the quantization due to the uncertainty relation.Moreover,phase space as the total physical states space must be modified for nonzero Planck constant.It is the Hilbert space of wave functions in quantum mechanics that is the space of states characterized by the principle of superposition of quantum mechanical states.Similarly,in view of the validity of the spacetime uncertainty relation,we may expect certain modifications of the notion of the spacetime. Thus,string theory may be taken as some sort of‘quantum geometry’along this line of thought but at present it is difficult to formulate this kind of ideas concretely.As before string theory,there have been several attempts to gener-alize localfield theories based on similar ideas.For example,let the spacetime coordinates be operators instead of ordinary numbers,then the coordinates and momenta can be treated as operators in the quantum mechanics acting on certain Hilbert space.This idea in a certain limit has been seen recently in the context of string theory with some assumptions on certain backgroundfields as a‘non-commutativefield theory’[8],[9],[10].However,these limits neglect the crucial extendedness of the strings along the longitudinal directions so we do not have todate notable new insight on the spacetime uncertainties characterized by the string length parameter l s.As a matter of fact,due to the noncommutative nature of spacetime at Plank scale,the usual Heisenberg uncertainty principle should be reformulated.So as a consequence,there exists a minimal observable distance of the order of the Plank length where all the measurements in the limit of extreme quantum gravity are governed.In the context of string theory,this observable minimal distance is known as generalized uncertainty principle:∆x≥¯h+α′l2p∆pgives a simple qualitative characterization of nonlocal and/or noncommutative nature of short distance spacetime structure in string theory.For example,Tami-aki Yoneya[23]considers spacetime uncertainty and approaches to D-branefield theory where the recent approaches towardfield theories for D-branes are briefly outlined and some key ideas lying in the background are putted on the emphasis [24]and references therein.Further motivation comes from the quantum propaga-tor of a bosonic p-brane obtained in the quenched minisuperspace approximation which suggests a possibilty of novel and unified description of p-branes with dif-ferent dimensionality[25].In this case,the background metric has emerged as a quadratic form on a Clifford manifold where the substitution of Lorenzian metric with the Clifford line element changes the very structure of the spacetime fabric as the new metric is built out of a minimum length below which it is impossible to resolve the distance between two points.Furthermore the introduction of the Clifford line element extends the usual relativity of the motion to the case of relative dimensionalities of all p-branes which makes up the spacetime manifold near the Plank scale.The stringy corrections to the original Heisenberg’s uncertainty principle also follow directly from the quantum mechanical wave equations on noncommutative Clifford manifolds where all dimensions and signatures of spacetime are on the same footing[26].Castro has considered the new relativity principle tofind a fully covariant formulation of the p-brane quantum mechanical loop wave equa-tions where the string uncertainty relations arrises naturally.Infact,there is one to one correspondence between the nested hierarchy of p-loop histories encoded in terms of hypermatrices and wave equations written in terms of Clifford alge-bra valued multivector quantities which allows to write the quantum mechanical wave equations associated with the hierarchy of nested p-loop histories being em-bedded in a D dimensional target spacetime with a single quantum mechanical functional wave equation whose lines live in a noncommutative Clifford manifold of2D dimensions having p=0,1,2,3...D−1where D−1is the maximum value of p that saturates the dimension of embedding spacetime[27].In this C-spaces the x,p must not be interpreted as ordinary vectors of spacetime but as one of the many components of the Clifford algebra valued multivectors that “coordinatize”the noncommutative Clifford manifold.The noncommutativity is encoded in the effective Plancks constant¯h eff which modifies the Heisenberg Weyl x, p commutation algebra and consequently on keeping thefirst two terms in the expansion of¯h eff(k)generates the ordinary string uncertainty relation[27]:∆x≥¯h4¯h (∆p).This is an inherent noncommutative nature of the Cliffordmanifold which reshuffles a loop history into a membrane history,a membrane history into a p-brane history or more generally it can transform a p-brane history into suitable combinations of the other p-brane histories as building blocks.This bootstrap idea is taken from the point particle case to the p-branes case with each brane made out of all the other p-branes where the Lorentz transformationsin C-spaces involve hypermatrix changes of coordinates in the p-brane quantum mechanics[26].The present paper is an effort to bridge up both string theory and quantum machanics within the framework of Heisenbergs uncertainty principle and give a more general expression from the complex analysis than the one presented in past years[9],[11],[12],[13],[14],[16],[17],[19],[20],[23],[24],[25],[27],[28],[29],[30],[31], [32],[33].We shall notice,why the upper scale of nature must appear in the fundamental equation?The importance of noncontinuous maps in string theory has been discussed by Borde and Lizzi[34].The space of string configurations in string theory required both continuous and noncontinuous square integrable maps in order to reproduce the results from the dual models.The size and shape of strings in their ground state in the lightcone gauge has been investigated long time ago by Susskind et.al.[35]where it is found that in two dimensions the extrinsic curvature is divergent.A regularization scheme is needed where the string is kept continuous.As the dimensionality of spacetime increases the string become smoother and have divergent average size.This is unphysical since their size cannot exceed the size of the Universe.It is because of this reason that the upper scale of nature must also appear in the fumndamental equation.The four dimensional average curvature diverges due to kinks and cusps on a string and so it is important to further study and analyze the properties of correcting functions in the uncertainity principle.Having presented the reasons why uncer-tainity principle is an important relevant issue,we shall explain the importace of the stringy uncertainity principle and generalize it from the perspective of the complex function theory.In the present article,we study the effects of higher derivative terms on the uncertainty principle from the analyticity condition of a complex function and give a simple explanation of the string uncertainty principle from the analysis of a holomorphic or anti-holomorphic function.Present work has been organized in several sections.Thefirst section introduces the problem and it’s motivation. In section2,we have reviewed usual uncertainty principle infinite dimensional quantum mechanics.On taking the account of shape and size we have illustrated the well known Heisenburg uncertainty principle for any arbitrary L2function. In section3,motivated from the string theory with the emphasis on the con-cept of limit and Fourier transform of any complex function,we have proposed a function˜δ(∆x,∆k,ǫ)and given a resolution resolution criteria for the UV/IR mixings.Furthermore,we have perturbatively proved our proposition and out-line a generalization on an arbitrary manifold.In section4,we have explained that our completely generalized uncertainty principle renders at the string un-certainty principle with all order perturbative corrections,a resolution criteria for the UV/IR mixings,physics of quantum gravity,black hole physics,existance of minimal and maximal length scales in nature,short dictance geometry versus string theory,Fourier transformation versus distribution and discretization of the spacetime.Moreover,our completely generalized uncertainty principle reveles allthese known physical and mathematical concepts nicely from a newly proposed single function˜δ(∆x,∆k,ǫ)that may reveal tofind the geometric origin of the fundamental M-theory.Finally,the section5contains some concluding issues and remarks for the future.2Measurements and Quantum Mechanics.We begin by considering some needful basic features of quantum mechanics and measuremants of certain physical observables[36]needed for further develop-ments in the later section.On the basis of dual nature of matter,it is well known that macroscopically it is possible to measure exactly the position of a moving particle at any instant and momentum of the particle at that position,but micro-scopically it is not possible to measure exactly(or with certainity)the position of a particle and it’s momentum simultaneously.According to the usual quantum mechanics,the behaviour of a moving particle can be defined by a wave packet moving with a group velocity v g=dωλ,whereλis wave length.Let the equations of the waves beϕ1=Asin(ωt−kx),ϕ2= Asin((ω+∆ω)t−(k+∆k)x).Then the wave packet obtained by their super-position is:ϕ=ϕ1+ϕ2=2Asin((ω+∆ω2)x)·cos((∆ω2)x).Orϕ=2Acos((∆ω2)x)·sin(ωt−kx).So the amplitude of the resultantwave is A res=2Acos((∆ω2)x)and the spread of each wave packet is equalto half of the wave lengthλm of the resultant wave.In other words,uncertaintyin the position of particle is∆x=λm2and so with k m=2πλandλ=hhor∆k=2π∆p2.In particular we consider the Heisenburg’s uncertainty principle in the regoroussetting for a particle moving in R1.The usual uncertainty principle for the one dimensional quantum mechanicalparticle can be seen by the calculation of the uncertainties in the position andin the momentum from the standard statistical deviation method as follows:Let ψ(x)be a normalized wave function of a particle in R1then probability offindingthe particle in between the position x and x+dx is defined byψ⋆(x)ψ(x)dx.The expectation value of x for normalizedψ(x)is defined by<x>:= ψ⋆(x)xψ(x)dx. Then uncertainty in position of a particle in the x direction is given by∆x:=[<{x−<x>}2>]1/2.Similarly the uncertainty in momentum is given by ∆p:=[<{p−<p>}2>]1/2where<p>:= ψ⋆(x)(−i¯h∂dx (xψ(x)).Integrating by parts from−∞to+∞with boundry conditionψ⋆ψ|x=±∞=0, we have,i¯h dψ⋆dx(xψ(x))dx=−i¯h[ ψ⋆x dψdx.xψdx+i¯h ψ⋆dψdx xψdx)=−i¯h.Now taking modulus of both side and squaringyields,4|Im( i¯h dψ⋆z|≥|y|,∀complex number z=x+iy. So| i¯h dψ⋆dx xψdx)|.In otherwords,4| i¯h dψ⋆dx xψdx|2≤( i¯h dψ⋆dx)dx)( xψxψ⋆dx).Hence above equation reads,i¯h dψ⋆dx)dx ψ⋆x2ψdx≥¯h2dx|2dx.One then obtains,(∆x)2(∆p)2≥¯h22(∆p) which is usual one dimensional Heisenburg’s uncertainty principle in ordinaryquantum mechanics.On other hands,in short distance regime,the notion of continuum spacetimeis dratistically modified by the string theory which mayfinally unify the general relativity with quantum theory.In this article,we discuss the possible signifi-cance of short distance aspect of string theory from a perspective focusing on the uncertainty principles and complex analysis.One of the distinguishing feature ofany quantum theory compared to classical physics is that there exists non zero quantumfluctuations.In classical physics,in principle a physical state can be exactly determined with sufficient knowledge of the state at a given time and one can exactly predict the precise values of various physical quantities at any other time just by solving the equations of motion,whereas in quantum theory,one can predict only the probabilities of possible values of physical quantities,eventhough one know the state at a given time as precisely as possible in the theory,which is well known Heisenberg uncertainty principle.Precisely,we can never make both uncertainties either∆x and∆p or∆t and∆E small beyond the following restriction for these errors:∆x∆p≥h or∆t∆E≥h which simply follow from the Heisenberg Weyl algebra:∆x∆p≥|<[ x, p]>|with[ x, p]:=i¯h.On other hand,in the general case of any L2function,we can arrive at the Heisenburg’s uncertainty principle of quantum mechanics by using the standared deviation of |f|2as a measure∆(f)of the spread of f and the same measure∆( f)for f defined by f,g =12π Rf(x) 2then the Heisenburg’s uncertaintyprinciple is just∆(f)∆( f)≥hmass-shell scattering amplitudes,one need to develop a theory whose ingredients can be deduced from certain quantumfield theory.For example unitarity and maximal analyticity of the S matrix which basically encode the requirements of causality and non-negative probabilities.Actually,there are two possible ques-tions:(i)Is it only the failure of the renormalization methods based on perturba-tion theory and not the failure of the general relativity theory itself?(ii)Should general relativity be modified in the short distance regime irrespective of the va-lidity of perturbation theory such that the quantumfluctuations of energy and momentum tensor becomes large?The significance of the above conflict between general relativity and quantum theory is so profound that we can not prevent ourselves from these difficulties for concentrating towards its resolution.The res-olution as by now understood is the string theory which can be regarded as a sort of thefinal outcome of many essential ideas springing from various attempts towards the fundamental theory of all the interactions.Although thefinal answer to the above fundamental question has not yet been obtained,it is at least true by exploring string theory that we are uncovering a multitude of facets of the theory which is useful for strengthing out our understanding of gauge theory and general relativity in quite unexpected way apart from the general understanding of string theories themselves.The basic reason to make it possible is that the both of gaugefield theory and general relativity are inextricably intertwined in the same framework of string theory.3Stringy Uncertainty Principle.What follows in this sections,we consider the stringyα′corrections from the perspective of complex analysis and show that there exists corrections from the holomorphic and anti-holomorphic sectors.It is well known fact that all string theories automatically contain gravity which has already provided a remarkable arena where various physical ideas and mathematical structures that were re-garded as being entirely unrelated,are unified.So it seems that there is a great possibility to achieve the ultimate unifiedfinal theory of the nature.The string theory is astonishingly rich and has many features which are desirable in an ul-timate unified theory.This is because after the establishment of the existence of gravity in any string theory,Scherk and Schwarz suggested[37]that string theory should be regarded as a fundamental theory.The idea of string theory as the fundamental theory was taken more seriously after the failure of various attempts towards building a consistent theory of quantum gravity within the framework of the ordinary localfield theories.Further the extreme self consistency of the string theory is not a defect but it is interpreted as the most important signature for the ultimate unification.As string theory[38],[39]includes gravity,gauge like forces etc which in the low energy limit,when the length of the string can be ignored,are approximatelydescribed by appropriate gauge theories of ordinary type like Maxwells electro-magnetism.The gravitational interaction contained in string theory is described in the low energy limit by the usual supergravities which are actually constructed in the attempts towards a generalization of the general relativity by extending symmetries.The mathematical structure of the theory shows that all the pa-rameters of the theory,apart from the fundamental string length,including the space time geometry itself can in principle be determined by the dynamics of the theory itself.The appearance of the critical spacetime dimensions can be regarded as a special case of this general feature of the theory.Unfortunately, one doesn’t actually think that the meaning and the content of string theory are fully grasped at the present stage of the developments.The string theory resolved the problem of the divergences associated with the earlier attempts at quantizing gravity but it is needed to understand non perturbatively along with the limita-tions of perturbative quantum gravity.This shows how deep string theory could be in general and how difficult it is tofind the really appropriate mathematical language to formulate the principles behind the string theory.In my opinion,we need probably a new mathematical framework in order to satisfactorily express the whole content of the string theory and the principles behind it without using perturbation theory.It is known,the fundamental difficulty of the divergences of quantum gravity are related to the quantum uncertainties which are resolved in string theory.In order to see the basic nature of the string dynamics,we need to understand a string which is simply a one dimensional extended object where energy density along a string in the fundamental string theory is assumed to be a universal constant given as1the ordinary Feynmann diagrams of the gauge theories[38],[39].These string interactions are splitting or joining at the end points of open strings and the rejoining of two closed strings at arbitrary points along with both open and closed strings which amounts to the statement that sufficiently small portion of each worldsheet at an arbitrary point on it is always dynamically equivalent to the segment of a one sheeted plane.The uniformity of the worldsheet in this sense is mathematically formulated by a characteristic conformal invariance which is intimately connected to the universal nature of the energy density of the string.One of the important proposal due to Chew and Frautschi is maximal analyt-icity in angular momentum[40],[41].According to this proposal one can uniquely extend the partial wave amplitudes a l(s)to an analytic function a(l,s)of l with isolated poles,the so called Regge poles.The Mandelstam invariant s is the square of the invariant energy of the scattering reaction.The position of a Regge pole is given by a Regge trajectory l=α(s).The physical hadron states are determined by the values of s for which l takes a physical values.The necessity of branch points in the l plane associated with Regge cuts has been established by Mandelstam[42].Phenomenologically there are many new hadrons discovered in experiments for which mass squared versus angular momentum plot withfixed values of other quantum numbers shows the Regge trajectories that are approxi-mately linear with a common slopeα(s)=α(0)+α′s,α∼1.0(GeV)2.One argue on the basis of crossing symmetry properties of analytically continued scatter-ing amplitudes that the exchange of Regge poles in the t-channel,controlled at high-energy withfixed momentum transfer is given by the following asymptotic behavior of physical amplitudes:A(s,t)∼β(t)(sΓ(−α(s)−α(t))[43].In string theory,there are severalremarkable discoveries of an N-particle generalization of the Veneziano formulaor Virasoro formula having a consistent factorization on a spectrum of single particle states which is described by an infinite number of harmonic oscillators{aµm},µ:=1,2,...,d−1;m=1,2,...with one set of such oscillators in the Veneziano case and two sets in the Virasoro case[44],[45],[46],[47].These resultsmay be interpreted as describing the scattering modes of a relativistic string:open strings in thefirst case and closed strings in the second case.Furthermore, the branch points become poles forα(0)=1and d=26[48].These poles are interpreted as closed-string modes in a one-loop open-string amplitude which is referred to as open string-closed string duality.On other hands,the theory of renormalization group is based on the factthat it is possible to organize physical phenomena according to the energy(or distance)scale,i.e.the short distance physics is not directly affected by thequalitative features of the long distance physics and vice versa.This sort of separation of ultraviolet versus infrared physics holds good in usual quantum field theories.But there exists interrelations between UV and IR physics for the generalizations such as noncommutative field theory and quantum gravity particularly the string theory where one can explicitly demonstrate the UV/IR mixings[10],[49],[50],[51],[52].From the viewpoint of probing the short distance spacetime structure,the most decisive directions of distances or momentum in the string dynamics are along the strings themselves where the physical pictures are united in the properties of the string worldsheet which can be analyzed just by using complex analysis.Since the simplest model does not contain gravity explicitly,so the generalized uncertainty principle arises as a consequence of the discretization of space which may or may not be a property of the full quantum gravity.But such explanation of the generalized uncertainty principle in a simple models may be useful in understanding how the generalized uncertainty principle arises in more realistic physical situations.In particular,it is the complex anal-ysis from which we have shown that our theorem is a resolution criteria of the UV/IR mixing problem with the existance of certain functions in our following proposition which contains all the effects of the quantum gravity at the all scales of nature.Let us now turn to the analysis of the generalized uncertainty relations as-sociated with any quantum mechanical physical system.Recall the concept of ǫ,δlimit and usual Fourier transformation for any complex function f (x )[53].For given any ǫ>0,∃δ>0such that |f (x )−f (x 0)|<ǫwhenever |x −x 0|<δ.Consider a fourier pair (f, f)for a fourier conjugates (x,k )with f (x )=1∆k where a is some constant depending on the size and shape of the wave packet.Let us now consider various step sizes to be {εi }N i =1for some given εas a certain sequence.Consider real lattice with variable step sizes {εi −εj }.Let ǫ=max i,j ∈Λ{|εi −εj |}be the maximum step size and the Λ={1,2,...,N }be some index set.That is the equation ∆x =a(Existance):There exist a function ˜δ(∆x,∆k,ǫ)such that ˜δ(∆x,∆k,ǫ)→0whenever ǫ→0,then following equation holds:∆x =a:For any function f (x,y )∀x,y ∈R ∃a complex valued function。
英语做饭的过程作文
英语做饭的过程作文Title: The Art of Cooking: A Journey in English。
Cooking is not merely a task; it's an art that engages all senses and creativity. From selecting the ingredients to presenting the final dish, each step in the process contributes to the culinary masterpiece. Let's embark on a journey through the process of cooking in English.1. Preparing the Ingredients:The first step in cooking any dish is gathering the ingredients. Whether it's vegetables, meats, or spices, each component plays a crucial role in defining the flavors and textures of the final dish. For instance, if we're preparing a classic spaghetti Bolognese, we'll need ripe tomatoes, minced garlic, onions, ground beef, herbs like basil and oregano, and of course, pasta.2. Chopping and Slicing:Once the ingredients are assembled, it's time to chop and slice them with precision. This step requires careful attention to detail to ensure uniformity in size, which promotes even cooking and enhances the dish's aesthetics. The sound of the knife rhythmically slicing through vegetables is like music to a cook's ears, signaling progress toward a delicious outcome.3. Heating the Cookware:Before we start cooking, it's essential to heat the cookware properly. Whether it's a skillet, saucepan, or pot, preheating ensures that the ingredients cook evenly and develop rich flavors. The sizzle of oil hitting the hot surface signifies the beginning of the culinary journey, awakening the senses with the promise of delectable aromas.4. Sautéing and Stirring:With the cookware he ated, it's time to sauté the aromatics and other ingredients. The aroma of garlic andonions sizzling in olive oil fills the kitchen, creating an appetizing ambiance. As we stir the ingredients, we observe them gradually transforming, releasing their flavors and infusing the dish with depth and complexity.5. Adding Seasonings and Spices:Seasonings and spices are the soul of any dish, elevating its taste profile and adding character. Whether it's a pinch of salt, a dash of pepper, or a sprinkle of paprika, each ingredient contributes to the symphony of flavors. The act of seasoning requires intuition and experimentation, as we adjust the amounts to achieve the perfect balance.6. Simmering and Reducing:After adding the liquid components, such as broth or sauce, it's time to let the dish simmer and reduce. This step allows the flavors to meld together while intensifying the dish's richness. As we watch the liquid gradually evaporate, we anticipate the moment when the sauce reachesthe desired consistency, signaling that it's ready to be served.7. Cooking the Main Component:While the sauce simmers, we focus on cooking the main component of the dish, whether it's pasta, meat, or vegetables. Each ingredient requires different cooking techniques and times, demanding careful monitoring to prevent overcooking or undercooking. As we tend to the main component, we visualize how it will complement the flavors of the sauce and complete the dish.8. Plating and Presentation:The final step in the cooking process is plating and presentation. With attention to detail and creativity, we arrange the components on the plate, creating a visually appealing masterpiece. The garnishes add color and texture, enhancing the dish's aesthetic appeal and inviting the diner to indulge not only in taste but also in visual delight.In conclusion, cooking is a multifaceted journey that engages all senses and requires both skill and creativity. From selecting the freshest ingredients to presenting the final dish with flair, each step in the process contributes to the culinary masterpiece. Through the language of English, we can articulate this journey and share the joy of cooking with others.。
阿姆河盆地油气评估
World Petroleum Resources Project
Assessment of Undiscovered Oil and Gas Resources of the Amu Darya Basin and Afghan–Tajik Basin Provinces, Afghanistan, Iran, Tajikistan, Turkmenistan, and Uzbekistan, 2011
GENERALIZED ANTI-HEBBIAN LEARNING FOR SOURCE SEPARATION
L
å x(t) = H(k)s(t – k)
(1)
k=1
where H(k) parameterizes a series of mixing matrices composed by the transforms of the echo and interference paths, and t denotes time. The problem is to reconstruct s(t) in the form of
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ïï ýý
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(4)
ïï
ïï
î îi = 1
þþ
where f(.) denotes the probability density function.
4. GENERALIZED GAUSSIAN DISTRIBUTION
In Equation (4), it is obvious that we need the a priori knowledge of the probability density function, i.e., the statistical model of the estimator. The generalized Gaussian distribution model [3, 4] can be a good candidate. That specific family of symmetric distribution can be characterized by a two parameters set (α, β) as
However, in the more realistic case of convolutive mixtures, the characterization of the joint probability density function is not available. The simple relationship between the output and input through a Jacobian determinant does not exist any more [2]. In addition, the nonlinearity restricts the separation performance due to the mismatch problem [2]. Hence, in this paper, we utilize the minimum entropy principle applied to the marginal probability den-
注射用美曲普汀(metreleptin for injection)说明书翻译对照
FULL PRESCRIBING INFORMATION1.1 Patients with Generalized Lipodystrophy 全身性脂肪代谢障碍患者MYALEPT (metreleptin for injection) is indicated as an adjunct to diet as replacement therapy to treat the complications of leptin deficiency in patients with congenital or acquired generalized lipodystrophy.本药用于辅助饮食，用作补充疗法治疗先天性或获得性全身性脂肪代谢障碍患者瘦素缺乏的并发症。
Limitations of Use 使用限制The safety and effectiveness of MYALEPT for the treatment of complications of partial lipodystrophy have not been established.本药用于治疗局部脂肪代谢障碍的安全性及有效性尚未建立。
The safety and effectiveness of MYALEPT for the treatment of liver disease, including nonalcoholic steatohepatitis (NASH), have not been established.本药用于治疗肝病，包括非酒精性脂肪肝炎（NASH）的安全性及有效性尚未建立。
MYALEPT is not indicated for use in patients with HIV-related lipodystrophy.本药不用于HIV感染相关的脂肪代谢障碍。
MYALEPT is not indicated for use in patients with metabolic disease, including diabetes mellitus and hypertriglyceridemia, without concurrent evidence of congenital or acquired generalized lipodystrophy.在没有并发先天性或获得性全身脂肪代谢障碍证据时，本药不用于代谢疾病，包括糖尿病和高三酰甘油血症。
规律和技艺感悟英语作文800字
规律和技艺感悟英语作文800字The Profound Insight of Discipline and Mastery.Throughout the annals of human history, the pursuit of discipline and mastery has served as a cornerstone for extraordinary achievements and profound personal growth. From the intricate brushstrokes of master painters to the groundbreaking discoveries of scientific pioneers, the unwavering commitment to rigorous practice and relentless exploration has been the common thread weaving together the tapestry of human progress.To unravel the essence of discipline, one must delve into its intrinsic properties. It is the unwavering adherence to a set of principles, a voluntary surrender to the constraints that ultimately liberate the spirit. Discipline fosters a heightened sense of organization, clarity, and efficiency. It instills the habit of punctuality, orderliness, and meticulous attention to detail. By consistently applying oneself to structuredroutines and schedules, one develops the mental fortitudeto overcome procrastination, distraction, and the allure of instant gratification.Mastery, on the other hand, transcends mere proficiency. It represents the pinnacle of skill and expertise, a stateof profound comprehension and unwavering execution. It isthe culmination of countless hours of deliberate practice, intense focus, and an unyielding determination to refineone's craft. Mastery entails not only command of the technical aspects of a discipline but also the ability to think critically, adapt creatively, and innovateeffectively. It elevates the individual from being apassive recipient of knowledge to an active generator of ideas and solutions.The pursuit of discipline and mastery is an arduous endeavor, fraught with challenges and setbacks. There will be moments of self-doubt, frustration, and the temptationto abandon the path. However, it is precisely in these moments that the true value of these virtues is revealed.By persevering through adversity, one cultivates resilience,perseverance, and an indomitable spirit.To fully appreciate the transformative power of discipline and mastery, consider the following examples:A concert pianist who spends countless hourspracticing scales and études, relentlessly honing their technique and musicality, ultimately delivering breathtaking performances that stir the emotions.A renowned surgeon who has dedicated years to mastering their craft, performing intricate procedures with unparalleled precision and dexterity, saving countless lives.A brilliant scientist who has devoted their existence to unraveling the mysteries of the universe, conducting groundbreaking experiments and formulating innovative theories that expand the boundaries of human knowledge.These individuals embody the transformative power of discipline and mastery. They have not merely achievedtechnical proficiency but have ascended to the pinnacle of their respective fields, becoming beacons of inspirationand excellence for others.The benefits of discipline and mastery extend far beyond personal accomplishment. A society that values and cultivates these virtues is one that thrives in all aspects. Discipline fosters order, accountability, and productivity, while mastery drives innovation, creativity, and progress. By encouraging our citizens to embrace discipline and mastery, we invest in the future, laying the foundation for a more prosperous, equitable, and fulfilling world.In conclusion, the pursuit of discipline and mastery is an essential ingredient for personal growth, societal progress, and enduring legacy. It is a path fraught with challenges but ultimately paved with immense rewards. By embracing these virtues, we unlock our true potential, inspire others, and contribute to a harmonious and flourishing world.。
我长大了想当厨师的英语作文
我长大了想当厨师的英语作文As a child, my days were filled with the aroma ofsizzling pans and the clatter of dishes in the kitchen. My mother, a remarkable cook herself, would often let me stand on a stool beside her, watching with wide eyes as she transformed simple ingredients into culinary masterpieces. It was during these formative years that the dream of becoming a chef was first ignited within me.The kitchen was my playground, and the pots and pans, my toys. I was fascinated by the alchemy of cooking, how a dash of this and a pinch of that could create a symphony offlavors on the palate. As I grew older, my interest evolved from mere observation to active participation. I began experimenting with recipes, learning the art of balancing flavors, and discovering the joy of creating dishes that brought smiles to people's faces.My aspiration to become a chef is not just about the love for food; it's about the passion for storytelling through cuisine. Every culture has a story to tell, and food is the universal language that brings those stories to life. I dream of traveling the world, learning about different culinary traditions, and incorporating those learnings into my own unique style of cooking.The journey to becoming a chef is a challenging one,filled with long hours, hot kitchens, and the pressure ofperfection. But for me, the reward is worth the effort. The ability to create, to innovate, and to share my creations with others is a privilege that I cherish deeply. I aspire to not only satisfy the hunger of my guests but also to touch their hearts with the stories and flavors of my dishes.In the future, I envision myself leading a team of talented chefs, each bringing their own passion andcreativity to the table. Together, we will create an environment where food is not just served but experienced. I will continue to learn, to grow, and to push the boundaries of what is possible in the culinary world.Becoming a chef is more than just a career choice for me; it's a lifelong commitment to excellence, to learning, and to sharing the joy of food with others. It's a journey I eagerly anticipate, one that I will embark on with an open heart and an even more open mind.。
过年我学会了包饺子的英语作文六年级
过年我学会了包饺子的英语作文六年级Learning to Make Dumplings for Chinese New YearWinter break was finally here! I was so excited for all the fun activities and traditions that come with Chinese New Year. My favorite part is always the delicious food. Dumplings are an absolute must for ringing in the new year. The cute little bundles filled with savory goodness just make me so happy.This year, I really wanted to learn how to make dumplings myself instead of just eating them. I've watched my mom and grandma folding the dumplings so many times. It seems simple enough - just some dough wrapped around a filling. But when I've tried before, they always ended up looking lopsided and leaking filling everywhere. Definitely not as elegant as the plump little purses my family creates.I decided to ask my mom to teach me her technique during our New Year preparations. "Mom, can you please show me how to wrap dumplings properly this year? I really want to learn," I requested. She smiled and said she would be delighted to pass down the skill.The first step was making the dough. My mom showed me how to mix together the flour, water, and a bit of salt until itformed a shaggy dough. Then we needed to knead it for quite a while until it became smooth, elastic, and a little bit springy when poked. Kneading is hard work! My arms started to feel sore after just a few minutes of pushing and folding the dough over and over. But mom said this step was crucial for getting the right texture.Once the dough was ready, we let it rest for about 30 minutes wrapped in plastic. Then it was time to roll out the dough into thin circular wrappers. Mom dusted the counter with flour and used a small rolling pin to roll the dough into a long cylinder. She cut off a piece and rolled it into a perfect flat circle, pivoting the dough a quarter turn after each roll. Her movements were so practiced and efficient. I tried to mimic her actions, but my circles ended up lumpy and oval-shaped. "Don't worry, it takes practice," she encouraged with a warm smile.The filling was next. This year, we made a traditional pork and cabbage filling. We chopped ingredients like ginger, garlic, green onions, and cabbage into a mix with ground pork, soy sauce, sesame oil, and other seasonings. The vibrant flavors and fragrant aromas reminded me of past New Year's celebrations.Now came the fun part - wrapping the dumplings! Mom pinched off a small piece of dough and rolled it into a sphere,then used a rolling pin to flatten it into a round wrapper. She placed a small spoonful of filling into the middle. Using her thumb and index finger, she made a series of folds to stretch the wrapper over the filling and pinched it at the top, creating a little purse shape.I tried my first one and...it was a mess. The filling immediately burst out the side as I tried to fold and pinch the dough. Mom laughed gently, "Don't worry, the first ones are always ugly." She showed me again how to seal the dumpling by pinching the edges together firmly all the way around. She did it slowly so I could see every motion.My second and third attempts were better, but still looked a bit sloppy compared to mom's perfect specimens. The key things I learned were: 1) Use just a small amount of filling so the dumpling doesn't get overstuffed. 2) Stretch and turn the wrapper as I folded it over to prevent rips. And 3) Pinch firmly to seal really well while making sure not to twist the dough.As I made more dumplings, I started to get the hang of it. My finger motions became more sure and smooth. It was sort of like tying shoes - at first you really have to think through each step, but eventually it becomes muscle memory. Mom sat with mepatiently as I wrapped dozens of dumplings, offering little tips like "That's it! Nice pinch." or "Don't overstuff this one."By the end of our session, I had a small mountain ofnot-too-shabby looking dumplings that I had wrapped all by myself. They were definitely rougher and plumper than mom's pristine ones. But I felt so proud of my creations. I couldn't wait to eat the dumplings I had worked so hard on.That night, mom cooked my dumplings along with her batches in boiling water. She also pan-fried some for that crispy bottom. When they were ready, the whole family sat around the table to enjoy the feast. I took my first big bite of a dumpling I had made from scratch. The chewy wrapper gave way to that explosion of savory pork and vegetable flavor. It was absolute perfection."Ces mom, your dumplings are amazing as always," my dad complimented. "But I have to say, I think Mei's are just as delicious this year!" He bit into one of my lumpy creations with a smile. I beamed from ear to ear, feeling immensely proud.My first foray into the centuries-old art of dumpling making was a success! Even though it took some patience and practice, I could now cross off this crucial life skill. In that moment, I felt soconnected to my cultural heritage that has been lovingly passed down through generations.I looked around our warm family room, the air filled with笑声和好吃的香气。
奶奶在厨房做饭写一篇英语作文3种句型
奶奶在厨房做饭写一篇英语作文3种句型Here is an English essay with more than 1000 words, without a title, and using three different sentence structures. The topic is "Grandma Cooking in the Kitchen".Grandma stood at the kitchen counter chopping vegetables with her worn but nimble hands. The rhythmic sound of the knife against the wooden cutting board filled the air as she deftly sliced the carrots, onions, and celery into even pieces. Her brow furrowed in concentration as she worked, her focus solely on the task at hand.As Grandma cooked, the aromas of simmering broth, sautéing aromatics, and herbs wafted through the kitchen, tantalizing the senses. She moved with a practiced ease, her movements fluid and efficient from years of experience. Reaching for a jar of dried thyme, she sprinkled it into the pot, her hand guided by muscle memory.The kitchen was Grandma's domain, a place where she reigned supreme. She took pride in nourishing her family with wholesome, homemade meals, just as her mother and grandmother had done before her. There was an undeniable sense of comfort and security that came from Grandma's cooking, a feeling of being cared for andloved.Grandma had learned to cook from a young age, watching her own mother prepare meals for their large family. As a child, she would stand on a stool, peering over the counter, eager to assist and learn. The smell of freshly baked bread or simmering stew would fill the air, and Grandma would inhale deeply, committing each aroma to memory.Now, as Grandma stirred the bubbling pot on the stove, she couldn't help but reflect on the countless meals she had prepared over the years. She remembered the times when her own children were young, coming home from school to the welcoming scent of a home-cooked meal. The kitchen had always been a gathering place, where the family would congregate to share stories, laughter, and nourishment.As Grandma added a pinch of salt to the simmering broth, she smiled to herself, thinking of the joy she found in feeding her loved ones. There was something inherently satisfying about providing sustenance, about nurturing the bodies and souls of those she cared for. It was a legacy she was proud to carry on, a tradition that connected her to her ancestors and her family.The sound of the kitchen timer suddenly broke the tranquil rhythmof Grandma's cooking. Glancing at the clock, she realized the meal was nearly ready. Quickly, she checked the seasoning, tasting the broth to ensure it was perfectly balanced. Satisfied with the result, she began to set the table, arranging the plates and utensils with the same care and precision she had applied to her culinary creations.As the family gathered around the table, Grandma watched with a warm heart, taking in the sight of her loved ones enjoying the fruits of her labor. She listened intently as they shared their days, their voices mingling with the clinking of silverware and the occasional laughter. In these moments, Grandma felt a profound sense of purpose and contentment, knowing that her efforts had brought them together in this shared experience of nourishment and connection.For Grandma, the kitchen was more than just a place to prepare food; it was a sanctuary, a space where she could express her love and nurture her family. The act of cooking was a form of care-giving, a way to nourish not only the body but the soul. Each dish she created was imbued with her love, her memories, and her desire to provide for those she held dear.As the meal drew to a close, Grandma began to clear the table, her movements unhurried and deliberate. She took pleasure in this ritual, knowing that the dishes she had prepared would sustain her familyuntil the next time she cooked. With a contented sigh, she made her way back to the kitchen, ready to begin the process anew, for there was always more love to be shared through the art of cooking.。
制作花卷步骤的英语作文
制作花卷步骤的英语作文Flower rolls are a delightful and visually appealing type of Chinese steamed bun that are often enjoyed as a snack or part of a meal. The process of making them can be a bit intricate but the end result is well worth the effort. In this essay, I will walk you through the step-by-step process of creating these beautiful flower-shaped buns.The first step in making flower rolls is to prepare the dough. In a large mixing bowl, combine flour, sugar, salt, and yeast. Slowly add warm water and knead the mixture until it forms a smooth and elastic dough. Cover the bowl with a damp cloth or plastic wrap and let the dough rest for about an hour to allow the yeast to activate and the gluten to develop.Next, it's time to make the filling. A common filling for flower rolls is a sweet red bean paste, though you can get creative and use other fillings as well such as lotus seed paste, black sesame, or even savory options like pork or vegetable. Regardless of the filling, the key is to make sure it is thick and not too wet, as this will help the flower shape hold its form. Once your filling is ready, set it aside.After the dough has rested, it's time to start shaping the flower rolls. Divide the dough into equal-sized pieces, about the size of a ping pong ball. Using your hands, gently roll each piece into a smooth ball. Then, use a rolling pin to roll each ball into a flat circular wrapper, about 4 inches in diameter. Be careful not to roll them too thin or they may tear.Now comes the fun part - creating the flower shape. Place a tablespoon or so of the filling into the center of the wrapper. Wet the edges of the wrapper with a bit of water, then gather the edges up and pinch them together, creating pleats as you go. Continue pleating all the way around until you have a flower-like shape with 8-10 petals. Gently squeeze the bottom to seal in the filling.Once all the flower rolls are shaped, it's time to steam them. Line a steamer basket with parchment paper or cabbage leaves to prevent sticking. Carefully place the flower rolls in the basket, making sure they are not touching each other. Cover and steam for 12-15 minutes until the dough is soft and cooked through.After steaming, the flower rolls are ready to be enjoyed. They can be served warm or at room temperature. The delicate dough should have a soft and pillowy texture, while the filling provides a sweet and satisfying contrast. Flower rolls are often garnished with sesameseeds, chopped scallions, or a drizzle of honey for added flavor and visual appeal.One of the great things about flower rolls is their versatility. While the traditional red bean paste filling is delicious, you can get creative and experiment with different fillings to suit your tastes. Some popular variations include pork and vegetable, chicken and mushroom, or even sweet fillings like custard or chocolate. You can also play around with the size and shape, making them bite-sized for a snack or larger for a more substantial dish.Another appealing aspect of flower rolls is their beautiful presentation. The pleated flower shape not only looks stunning but also adds a touch of elegance to any meal or gathering. They make a wonderful addition to a dim sum spread, a festive holiday table, or even a casual afternoon tea. The intricate folds and vibrant colors of the flower rolls are sure to impress your guests and delight their taste buds.In conclusion, the process of making flower rolls may seem a bit daunting at first, but with a little practice, it becomes quite straightforward. The key is to take your time, work gently with the dough, and have fun with the creative shaping process. Whether you're an experienced baker or a novice in the kitchen, mastering the art of flower rolls is a rewarding and satisfying experience. So gatheryour ingredients, put on your apron, and get ready to create these beautiful and delicious steamed buns that will bring a touch of elegance to any occasion.。
Chinese+Painting+in+High+School+English+Basic+Writ
CHAPTER
The Composition and Color of Chinese Painting
Chinese painting often uses balance and symmetry in composition to create a sense of harmony and stability This can be achieved by arranging objects or figures in a balanced way, or using symmetric patterns
Although traditional Chinese painting does not rely on health on perspective like Western painting, it still uses techniques such as overlaying and size variation to create a sense of depth and distance
Paper
High quality paper is essential for Chinese painting, as it can absorb the ink without cracking or writing Different types of paper are used for different techniques and effects
Structure
A well structured essay or piece of writing is easy to read and understand Students should learn how to organize their ideas into a logical flow that is easy for the reader to follow
重庆小面的做法的英语作文
As a high school student with a keen interest in culinary arts, Ive always been fascinated by the diversity of flavors that different cuisines offer. One of the most memorable dishes Ive had the pleasure of learning to make is the famous Chongqing Xiao Mian, a simple yet incredibly flavorful noodle dish from the southwestern Chinese city of Chongqing.The first time I encountered Chongqing Xiao Mian was during a family trip to Chongqing. The aroma wafting from the street vendors was irresistible, and the taste was a revelation. It was a harmonious blend of spicy, sour, and savory flavors that left a lasting impression. Determined to recreate this dish at home, I embarked on a culinary adventure that taught me not only how to make the dish but also about the importance of authenticity in cooking.The process of making Chongqing Xiao Mian is a dance of flavors and textures. It starts with the preparation of the noodles, which are typically made from a simple dough of wheat flour, water, and a pinch of salt. The dough is kneaded until its smooth and elastic, then rolled out into thin sheets and cut into ribbonlike strips. The noodles are boiled until theyre al dente, a process that requires a keen eye and a sense of timing to achieve the perfect texture.While the noodles are cooking, the real magic happens in the preparation of the sauce. The sauce is the soul of Chongqing Xiao Mian, and its a complex mixture of ingredients that includes chili oil, Sichuan peppercorns, garlic, ginger, soy sauce, and a splash of vinegar. Each ingredient plays a crucial role in creating the dishs signature flavor profile. The chili oil adds afiery kick, the Sichuan peppercorns provide a numbing sensation, and the garlic and ginger offer a pungent aroma thats both comforting and invigorating.Once the sauce is prepared, its time to assemble the dish. The cooked noodles are drained and placed in a bowl, then topped with a generous ladle of the sauce. The dish is finished with a sprinkle of chopped green onions and a drizzle of sesame oil, which adds a nutty richness to the overall flavor.The first time I made Chongqing Xiao Mian at home, I was nervous about getting the flavors right. But as I followed the recipe and tasted each component, I could feel the dish coming together. The moment I took the first bite, I was transported back to the bustling streets of Chongqing, with the flavors evoking memories of my trip.Learning to make Chongqing Xiao Mian has been a rewarding experience. Its not just about mastering a recipe its about understanding the cultural significance of food and the stories it tells. Each bowl of noodles is a testament to the ingenuity and creativity of Chongqings culinary tradition.In a world where fast food and instant meals are increasingly popular, taking the time to learn and appreciate traditional dishes like Chongqing Xiao Mian is a refreshing reminder of the joy and satisfaction that come from cooking with love and care. Its a reminder that food is more than just sustenance its a connection to our past, a celebration of our present, and a bridge to our future.As I continue to explore the world of cooking, Im excited to learn more about the diverse flavors and techniques that different cuisines offer. But for now, Im content with the knowledge that I can recreate a piece of Chongqings culinary heritage in my own kitchen. And every time I make Chongqing Xiao Mian, Im not just cooking a dish Im carrying on a tradition and honoring the art of cooking.。
如何有好的书法英语作文
Writing a good calligraphy in English involves a combination of artistic skill and technical precision. Here are some steps to help you achieve a beautiful and legible script:1. Choose the Right Tools: Select a pen that suits your writing style. Fountain pens and calligraphy pens are popular choices for their ability to create varying line widths.2. Understand the Basics: Learn about the different styles of English calligraphy, such as Copperplate, Spencerian, or Modern Calligraphy. Each has its own unique characteristics and techniques.3. Practice Basic Strokes: Start by practicing basic strokes like the downstroke, upstroke, and the oval. These are the building blocks of more complex letters.4. Learn Letter Formation: Study the formation of each letter in the style you have chosen. Pay attention to the angles, curves, and the way the letters connect to each other.5. Consistency is Key: Ensure that your letters are consistent in size, slant, and spacing. This will make your writing look more professional and aesthetically pleasing.6. Guidelines and Grids: Use guidelines or a grid to help maintain consistent letter height and spacing. This is especially important for beginners.7. Hold the Pen Correctly: The way you hold your pen can greatly affect your writing. Hold it at a slight angle and use a relaxed grip to allow for fluid movement.8. Posture Matters: Sit up straight and position your paper at a comfortable angle. This will help you maintain control and reduce strain on your hand and wrist.9. Practice Regularly: Like any skill, calligraphy requires practice. Dedicate time each day to writing and refining your technique.10. Experiment with Ink: Different inks can affect the look of your calligraphy. Experiment with various types to see which one you prefer.11. Study Examples: Look at examples of good calligraphy to understand the flow and rhythm of the script. This can inspire your own writing.12. Personalize Your Style: Once you have mastered the basics, feel free to add your own personal touches to your calligraphy. This could be in the form of unique letter connections or flourishes.13. Patience: Developing good calligraphy takes time and patience. Dont be discouraged if your writing doesnt look perfect right away.14. Seek Feedback: Share your work with others and ask for constructive criticism. This can help you identify areas for improvement.15. Enjoy the Process: Calligraphy is an art form. Enjoy the process of creating something beautiful with your own hands.By following these steps and dedicating time to practice, you can develop your skills and create impressive calligraphy in English. Remember, the key to good calligraphy is patience, practice, and a keen eye for detail.。
折叠的英语作文
折叠的英语作文Origami, the ancient Japanese art of paper folding, is a captivating craft that has been cherished for centuries. It is an art form that requires patience, precision, and creativity, allowing individuals to transform a simple sheet of paper into intricate and beautiful shapes.The History of OrigamiOrigami's roots can be traced back to the 6th century in Japan, where it was initially used for religious purposes. Over time, it evolved into a popular pastime and a means of artistic expression. The word "origami" itself comes from "oru" meaning "to fold" and "kami" meaning "paper" in Japanese.Techniques and StylesThere are various techniques used in origami, ranging from simple folds to complex patterns that require a high level of skill. Some of the basic techniques include valley folds, mountain folds, and petal folds. The most iconic origami model is arguably the crane, which is often associated with good fortune and longevity.Cultural SignificanceOrigami has a deep cultural significance in Japan, where itis often used in traditional ceremonies and celebrations. For example, during the Boys' Festival in May, families fly koinobori (carp-shaped streamers) which are a symbol of strength and courage. Similarly, origami cranes are often given as gifts for their symbolic meaning.Educational ValueOrigami is not just an art form; it also has educational benefits. It helps to develop fine motor skills, hand-eye coordination, and spatial awareness. Moreover, it encourages problem-solving and fosters a sense of accomplishment when a complex model is successfully completed.Modern OrigamiIn the modern era, origami has expanded beyond itstraditional boundaries. Artists around the world are pushing the boundaries of what can be achieved with paper, creating lifelike animals, geometric designs, and even sculptures. Origami is also being used in fields such as architecture, fashion, and even space technology, where it has been employed to design solar panels for satellites.ConclusionThe art of origami is a testament to the beauty andversatility of paper. It is a timeless craft that continues to inspire and challenge people of all ages. Whether you are a beginner looking to create a simple crane or an experiencedartist exploring complex designs, origami offers a world of possibilities for creativity and expression.。
煮面英语作文
煮面英语作文Boiling Noodles: An English CompositionIn the bustling world of culinary arts, there exists a simple yet universally cherished dish: noodles. The process of boiling noodles is not only a fundamental cooking skill but also a metaphor for life's journey, where each ingredient and step adds flavor and texture to the final dish.Introduction:The art of cooking noodles begins with selecting the right type of noodles. There are countless varieties, from the thin and delicate rice noodles to the hearty and robust wheat-based pasta. Each type has its unique characteristics and culinary applications.Preparation:Before boiling, noodles must be prepared. This involves checking for any broken pieces and ensuring they are free from any foreign matter. The noodles should be rinsed under cold water to remove any dust or residue.Boiling Process:The next step is to bring a large pot of water to a rolling boil. It is crucial to use plenty of water to prevent the noodles from sticking together. A pinch of salt can be added to the water to enhance the flavor of the noodles.Cooking:Once the water is boiling, the noodles are gently loweredinto the pot. Stirring them occasionally ensures they don't stick together. The cooking time varies depending on the type of noodles and the desired texture. Al dente, which means 'to the tooth' in Italian, is a popular choice, indicating that the noodles should be cooked until they are tender yet still firm when bitten.Drain and Rinse:After the noodles have reached the desired consistency, they are drained using a colander. Rinsing them under cold water stops the cooking process and prevents them from becoming too soft.Serving:The final step is to serve the noodles either hot or cold, depending on the recipe. They can be dressed with a variety of sauces, from a simple garlic and oil mixture to a rich and creamy Alfredo sauce. Toppings such as vegetables, meats, or seafood can be added to create a complete meal.Conclusion:Boiling noodles is more than just a cooking technique; it is an opportunity to practice patience and precision. Each step, from preparation to serving, requires attention to detail and a respect for the ingredients. The result is a dish that is both satisfying and nourishing, a testament to the simple pleasures of a well-cooked meal.In conclusion, the process of boiling noodles is a culinarydance that, when executed with care, can yield a dish that is both comforting and delicious. It is a skill that every aspiring chef should master, as it forms the foundation for many more complex dishes.。
剪纸艺术文化英语作文初一
Paper cutting is a traditional Chinese art form that has been cherished for centuries. It involves cutting intricate designs and patterns into paper using scissors or a knife.This art form is not only a way to express creativity but also a means to preserve and pass on cultural heritage.The history of paper cutting dates back to the6th century during the Han Dynasty. However,it gained popularity during the Tang and Song Dynasties.Traditionally,paper cutting was used for various occasions such as festivals,weddings,and funerals.Red paper was the most commonly used color,symbolizing good fortune and happiness.There are several techniques used in paper cutting,including folding,cutting,and layering.The artist first sketches the design on the paper and then carefully cuts along the lines.The most common method is to fold the paper in half or multiple times,creating a symmetrical pattern.The paper is then cut using scissors,with the artists skill and precision determining the intricacy of the design.Paper cutting often features various motifs and symbols that hold cultural significance. For example,dragons and phoenixes represent power and nobility,while fish symbolize abundance and prosperity.Flowers,birds,and animals are also common themes, reflecting the beauty and harmony of nature.In modern times,paper cutting has evolved to include contemporary themes and styles. Artists have experimented with different materials,such as tissue paper and cardboard, and have incorporated digital technology to create innovative designs.Despite these changes,the essence of paper cutting as a cultural art form remains.To promote and preserve this art form,many schools and cultural centers offer paper cutting workshops and classes.These programs not only teach the technical skills but also instill an appreciation for the history and cultural significance of paper cutting.In conclusion,paper cutting is a unique and beautiful art form that reflects the rich cultural heritage of China.It is a testament to the creativity,skill,and craftsmanship of its artists.By learning and practicing this art,we can help ensure that the tradition of paper cutting continues to thrive for generations to come.。
水墨画英语作文
Ink wash painting,also known as literati painting,is a traditional Chinese art form that has been cherished for centuries.It is characterized by its use of black ink and water to create a variety of shades and textures on rice paper or silk.Here are some key aspects to consider when writing an essay about ink wash painting:1.Historical Background:Begin by discussing the origins of ink wash painting,which dates back to the Tang Dynasty618907AD.Mention how it evolved over time, becoming a significant part of Chinese culture and a means of expression for scholars and literati.2.Artistic Techniques:Describe the basic techniques used in ink wash painting,such as the control of ink and water to create different shades,the use of a brush for both broad strokes and fine details,and the importance of the bone method bone structure of the painting and sinew method the softness and flexibility.3.Philosophical Significance:Ink wash painting is not just about visual aesthetics it is deeply rooted in Chinese philosophy.Discuss how it reflects the principles of Daoism, emphasizing the harmony between humans and nature,and the concept of wu wei nonaction or effortless action.4.Cultural Symbolism:Explain the symbolic meanings behind common elements in ink wash paintings,such as mountains,water,bamboo,and birds,which often represent virtues like strength,purity,resilience,and freedom.5.Famous Artists and Works:Mention some of the renowned ink wash painters,such as Mi Fu,Huang Gongwang,and Bada Shanren,and describe some of their notable works. Discuss their unique styles and contributions to the art form.6.Contemporary Relevance:Address how ink wash painting has been adapted and reinterpreted in modern times.Discuss contemporary artists who have incorporated traditional techniques into their work while also introducing new themes and styles.7.Personal Reflection:Conclude your essay with a personal reflection on the impact of ink wash painting.You might consider how it has influenced your understanding of Chinese culture,art,and philosophy,or how it has inspired you in your own creative endeavors.Remember to use descriptive language to help readers visualize the delicate beauty of ink wash paintings and to convey the depth of their cultural and philosophical significance.。
拉康派英文著作一览表（二）
拉康派英文著作一览表（二）拉康派英文著作一览表（二）一、理论引介类：Lacan and the Subject of Language (RLE: Lacan)Edited by Ellie Ragland-Sullivan, Mark Bracher【难度：可，概念有简化，有些混乱，仍推荐】Jacques Lacan and the Logic of Structure: Lacanian Structures and Language in Psychoanalysisby Ellie Ragland【难度：可，概念有简化，有些混乱，仍推荐】••Figuring Lacan (RLE: Lacan): Criticism and the Unconscious•by Juliet Flower MacCannell••Death and Desire (RLE: Lacan): Psychoanalytic Theory in Lacan's Return to Freud•by Richard Boothby•【经典-难度较难，强烈推荐】Four Lessons of Psychoanalysisby Moustafa Safouan拉康文本阐释导读类：The Title of the Letter作者: Jean-Luc Nancy / Philippe Lacoue-Labarthe 出版社: State University of New York Press【难度：较难有中文译本，拉康自己曾在S20中推荐，很不错的书，关于能指、字母，男女性，压抑的建构】Lacan to the Letter: Reading Ecrits CloselyAuthor(s) : Bruce Fink【难度较难，推荐】Antigone, in Her Unbearable Splendor: New Essays on Jacques Lacan's The Ethics of Psychoanalysisby Charles FreelandReading Seminar XX: Lacan's Major Work onLove,Knowledge, and FemininEditor : Suzanne Barnard, Editor : Bruce Fink【难度较难】The Law of Desire: On Lacan's 'Kant with Sade' Author(s) : Dany Nobus【难度尚可】Jacques Lacan: A Critical IntroductionAuthor(s) : Martin MurrayThe Works of Jacques Lacan: An Introductionby Bice Benvenuto, Richard Kennedy【难度可推荐】Interpreting Lacanby Joseph H. Smith, William J. KerriganBeyond the psychoanalytic dyad: Developmental semiotics in Freud, Peirce and Lacanby John Muller【难度可推荐精神病和自闭症与指号学的拉康】Having a Life: Self Pathology After Lacanby Lewis A. Kirshner【难度可】Eros and Ethics: Reading Jacques Lacan's Seminar VIIby Marc De Kesel【难度可】Lacan's Four Fundamental Concepts of Psychoanalysis: An Introductionby Roberto Harari【难度可推荐】How James Joyce Made His Name: A Reading of the Final Lacanby Roberto Harari【难度极难】Irrepressible Truth: On Lacan's 'the Freudian Thing'by Adrian JohnstonSix Moments in Lacan: Communication and identification in psychology and psychoanalysisby Derek HookOutside the Dream (RLE: Lacan): Lacan and French Styles of Psychoanalysisby Martin Stanton跨学派对话：Working with Trauma: Lessons from Bion and Lacan Author(s) : Marilyn CharlesBetween Winnicott and Lacan: A Clinical EngagementEdited by Lewis A. KirshnerFrom Klein to Kristeva: Psychoanalytic Feminism and the Search for the 'Good Enough Mother'by Janice Doane二、实践Fundamentals of Psychoanalytic Technique: A Lacanian Approach for Practitionersby Bruce Fink【难度可推荐】Jacques Lacan and the Freudian Practice of Psychoanalysisby Dany Nobus【难度可强烈推荐】More Lacanian Coordinates: On Love, Psychoanalytic Clinic, and the Ends of Analysisby Bogdan WolfConfessions from the Couch: Psychoanalytical Notions Illustrated with Extracts from SessionsAuthor(s) : Valerie Blanco, Translator : Jane Hodgson-McCrohan精神病：After Lacan: Clinical Practice and the Subject of the UnconsciousEdited by Willy Apollon, Lucie Cantin【难度可】Lacan on Madness: Madness, Yes You Can'tEdited by Patricia Gherovici, Manya Steinkoler【难度可】神经症：A Non-Oedipal Psychoanalysis? A Clinical Anthropology of Hysteria in the Work of Freud and LacanAuthor(s) : Philippe Van Haute, Author(s) : Tomas Geyskens Psychoanalyzing: On the Order of the Unconscious and the Practice of the Letter (Hardback)by Serge Leclaire【难度可推荐】The Clinical Lacanby Joel Dor【难度可推荐】儿童分析：What is a Child?: Childhood, Psychoanalysis, and DiscourseAuthor(s) : Michael Gerard Plastow特殊临床：Perversion and the Social Relationby Mikel A. Rothenberg, Dennis A. FosterPeversion Now!Edited by Diana Caine, Colin Wright实践类：The Constitution of the Psychoanalytic Clinic: A History of its Structure and Powerby Christian DunkerEvolving Lacanian Perspectives for Clinical Psychoanalysis: On Narcissism, Sexuation, and the Phases of Analysis in Contemporary Cultureby Raul Moncayo【难度可推荐有拉康派的抑郁症阐释】A Psychoanalyst on the Couchby Juan-David Nasio【有中文译本】Psychology After the Unconscious: From Freud to Lacan by Ian ParkerPsychology After Lacan: Connecting the Clinic and Researchby Ian Parker三、分析的政策The Trainings of the Psychoanalystby Annie Tardits四、爱情、性别身份、科学哲学、社会文化、哲学对话等：The Structures of Love: Art and Politics beyond the Transferenceby James PenneyLacan, Discourse, Event: New Psychoanalytic Approaches to Textual IndeterminacyEdited by Ian Parker, David Pavon CuellarAn Introduction to Electronic Art Through the Teaching of Jacques Lacan: Strangest Thingby David SchwarzPsychoanalysis is An Antiphilosophyby Justin ClemensThe Singularity of Being: Lacan and the Immortal Within by Mari RutiPsychoanalytic Reflections on Politics: Fatherlands in Mothers' Handsby Eszter SalgoThe Weary Sons of Freudby Catherine ClementJacques Lacan: Between Psychoanalysis and PoliticsEdited by Samo T omsic, Andreja ZevnikJoyce and Lacan: Reading, Writing and Psychoanalysisby Daniel BristowDeath and Mastery: Psychoanalytic Drive Theory and the Subject of Late Capitalismby Benjamin Y. FongLacan's Return to Antiquity: Between Nature and the Gods by Oliver HarrisThe Subject of Liberation: Zizek, Politics, Psychoanalysis by Charles WellsApropos of Nothing: Deconstruction, Psychoanalysis, and the Coen Brothersby Clark BucknerBetween Levinas and Lacan: Self, Other, Ethicsby Mari RutiThe Trouble with Pleasure: Deleuze and Psychoanalysisby Aaron SchusterThe Not-Two: Logic and God in Lacanby Lorenzo ChiesaLacan, Psychoanalysis and ComedyEdited by Patricia Gherovici, Manya SteinkolerReading Alice Munro with Jacques Lacanby Jennifer MurrayFeminine SexualityEdited by Juliet Mitchell, Jacqueline Rose【难度难】Knowing Nothing, Staying Stupid: Elements for a Psychoanalytic Epistemologyby Dany Nobus, Malcolm Quinn【难度可】The Book of Love and Pain: Thinking at the Limit with Freud and Lacanby Juan-David NasioThe Law of Kinship: Anthropology, Psychoanalysis, andthe Family in Franceby Camille RobcisArt, Death and Lacanian Psychoanalysisby Shirley Sharon-ZisserLuce Irigaray: Key Writingsby Luce IrigarayThe Logic of Sexuation: From Aristotle to Lacanby Ellie Ragland【难度可】Jacques Lacan and Feminist Epistemologyby Kirsten CampbellLacan in the German-Speaking WorldEdited by Elizabeth Stewart, Richard FeldsteinElemental Passionsby Luce IrigarayWhat Lacan Said About Women: A Psychoanalytic Study by Colette Soler【难度可推荐】What Is Sex?by Alenka ZupancicThe Way of Loveby Luce IrigarayLacan Today: Psychoanalysis, Science, Religionby Alexandre LeupinPsychoanalytic Film Theory and the Rules of the Game by Todd McGowanLacan's Ethics and Nietzsche's Critique of PlatonismRead My Desire: Lacan Against the Historicistsby Joan Copjec【难度难不推荐】The Tenderness of Terrorists: And Other Letters written by Jacques-Alain Miller to an Enlightened Publicby Jacques-Alain Miller, Josefina AyerzaTrauma and the Ontology of the Modern Subject: Historical Studies in Philosophy, Psychology, and PsychoanalysisAuthor(s) : John L. RobertsThe End of Dissatisfaction? Jacques Lacan and the Emerging Society of EnjoymentAuthor(s) : Todd McGowanStop Making Sense: Music from the Perspective of the RealAuthor(s) : Scott WilsonPink Herrings: Fantasy, Object Choice, and Sexuation Author(s) : Damien W. RiggsLost in Cognition: Psychoanalysis and the Cognitive SciencesAuthor(s) : Eric Laurent, Translator : Adrian Price【难度可】The Pathology of Democracy: A Letter to Bernard Accoyer and to Enlightened Opinion - JLS Supplement (Ex-tensions) Lacan and Contemporary Filmby Todd McGowanWithout Ground: Lacanian Ethics and the Assumption of SubjectivityLacan and Religionby Aron DunlapJacques Lacan and Cinema: Imaginary, Gaze, Formalisationby Pietro BianchiLacan: In Spite of Everythingby Elisabeth RoudinescoThe Failed Assassination of Psychoanalysis: The Rise and Fall of Cognitivismby Agnes Aflalo传记：Jacques Lacan: Outline of a Life, History of a System of Thoughtby Elisabeth RoudinescoJacques Lacan (Volume II) (RLE: Lacan): An Annotated Bibliography: Volume Iby Michael P. ClarkJacques Lacan (Volume I) (RLE: Lacan): An Annotated Bibliography: Volume Iby Michael P. ClarkJacques Lacan, Past and Present: A Dialogueby Alain Badiou, Elisabeth Roudinesco。
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a rXiv:h ep-ph/967451v217Dec1996RAL-TR/96–055August 1996Generalized Pinch Technique and the Background Field Method in General Gauges Apostolos Pilaftsis ∗Rutherford Appleton Laboratory,Chilton,Didcot,Oxon,OX110QX,UK ABSTRACT It is shown that Cornwall’s pinch technique can be extended in a consistent diagrammatic way,so as to describe general background field gauges in Yang-Mills theories.The resulting one-loop Green’s functions are found to obey Ward identities identical to those derived from the classical action at the tree level.This generalization of the pinch technique may hence be related to the background field method implemented with novel gauge-fixing conditions invariant under background field gauge transformations.To one loop,the connection between the generalized pinch technique and the background field method in covariant and in non-covariant gauges is explicitly demonstrated.1IntroductionThe pinch technique(PT),as has originally been introduced by Cornwall[1],is a power-fulfield-theoretical algorithm,which re-arranges the S-matrix elements of gauge theories, such that the resulting proper two-point,three-point,...,n-point correlation functions satisfy Ward identities(WIs)identical to those derived from the classical Lagrangian at the tree level[1,2,3].Within this framework,the one-loop effective PT Green’s functions can further be shown to be independent[1,2,3,4,5]of the gauge-fixing conditions imposed in a rigorous way[6].In Ref.[4],an approach has been suggested for the construction of high-order self-energies,which are gauge independent within the PT.The authors of Ref.[7]have independently tested the gauge invariance and consistency of this approach in a two-loop example.Apart from gauge independence,most importantly,basicfield theo-retical requirements based on unitarity,analyticity and renormalizability are satisfied for the off-shell PT correlation functions[6].These conditions are deduced from resummation considerations[4,6],which naturally emanate from describing the underlying dynamics of unstable particles in spontaneous symmetry breaking(SSB)theories,such as the Standard Model(SM)and/or its renormalizable extensions.From that point of view,we think that a physical meaning may be assigned to an off-shell PT Green’s function[6].One may now raise the question whether the diagrammatic approach of the PT can be formulated on the basis of the path integral,through which the quantized Lagrangian can give rise to Green’s functions exhibiting the very same properties mentioned above.If one relaxes the requirements of unitarity and gauge independence,such a quantized action, for which the derived Green’s functions obey tree-level WIs,can be found with the help of the backgroundfield method(BFM)[8].The focal idea of the BFM may be explained as follows.First,one decomposes linearly the gaugefield appearing in the classical action in terms of a backgroundfield,ˆAµ,and the quantumfield,Aµ,which is a variable of integra-tion in the path integral.In the Fadeev-Popov quantization method[9],it is then necessary to eliminate the unphysical degrees of the gaugefield by breaking the gauge invariance of the classical Lagrangian through a gauge-fixing condition,which is usually taken to be of covariant form,even though such a choice of gaugefixing may not be unique.Most impor-tantly,the gauge-fixing condition is chosen to be invariant under gauge transformations of the backgroundfieldˆAµ.Thus,the whole Lagrangian possesses a background-field gauge invariance with respect to thefieldˆAµ,which only appears outside the loops.However,the gauge symmetry is explicitly broken by the quantumfield Aµ,which occurs in the loop only. In SSB theories,the latter leads toξQ-dependent unphysical thresholds in the resummed off-shell self-energies,thus spoiling the physical requirement of unitarity[4,6].Only for thespecific choice of the gauge-fixing parameterξQ=1,unitarity cuts of the one-loop Green’s functions are found to correspond to physical Landau singularities[6].As a consequence, the one-loop analytic results obtained by the PT coincide with those calculated in the BFM quantized in a covariant gauge withξQ=1[10].In this paper,we shall present a different point of view.Given the above connection between PT and BFM forξQ=1,one may now ask the question whether it is possible to generalize the algorithm of the PT so as to obtain an explicit relation between the new diagrammatic approach and the BFM for any value ofξQ.In Section2,we shall present an extended version of the PT,which is here called the generalized PT(GPT),and address the above question in the affirmative.It is also worth emphasizing that our generalization of the PT will be based on a gauge-dependent procedure and so will give rise to Green’s functions that will in turn depend explicitly on the gaugefixing of the procedure chosen. Obviously,this should be considered as a fundamental departure from the primary aim of Cornwall’s PT,which is to produce gauge-invariant Green’s functions.After gaining some insight of the GPT in the covariant gauges in Section2.1,we will extend our considerations into non-covariant gauges in Section2.2,such as axial [11,12,13,14,15]or Coulomb gauges.Again,the effective Green’s functions derived with the GPT will satisfy the usual PT or BFM WIs.In Section3,the Lagrangian is quan-tized via the BFM in covariant and non-covariant gauges.The one-loop analytic results obtained for the Green’s functions are shown to be identical to those found by the GPT in the corresponding gauge.The latter is demonstrated in the scattering q¯q→q′¯q′in Section 4.This establishes an explicit connection between the GPT and the BFM in a wide class of gauges.Section5contains our conclusions.2Generalized pinch techniqueWe shall briefly outline the main features of the PT and present the crucial modifications pertaining to the GPT in an arbitrary gauge.We restrict ourselves to pure Yang-Mills theories.For a comprehensive discussion on the PT,the reader is referred to[16].Consider the scattering q(p1)¯q(p2)→q′(k1)¯q′(k2)in the covariant Rξgauges.The one-loop transition amplitude can conveniently be written down asq′¯q′|T|q¯q =Γρ∆(ξ)ρµ(q)Π(ξ)µν(q)∆(ξ)νλ(q)Γ∗λ+Γ(ξ)1µ(q,p1,p2)∆(ξ)µλ(q)Γ∗λ(−q,k1,k2)+B(ξ)(p1,p2,−k1,−k2).(2.1)+Γµ∆(ξ)µλ(q)Γ(ξ)∗1λHere,p1+p2=−q=k1+k2andΓµ=(g/2)¯uγµλa v is the tree gluon-quark-quark vertex (Aq¯q),where g is the strong coupling constant andλa are the SU(N)generators in thefundamental representation.Furthermore,the gluon propagator in the covariant gauges is given by∆(ξ)µν(q)= tµν(q)−ξℓµν(q) 1q2,ℓµν(q)=qµqνThe PT accomplishes to arrive at Eq.(2.4)starting from Eq.(2.1).The observation to be made is that the residualξdependence ofΠ(ξ)µν(q),for example,is hidden in the vertices and boxes and should therefore be“pinched”out.These“pinching”(or PT)terms are kinematically indistinguishable with that ofΠ(ξ)µν(q)and should therefore be added to it, yielding the gauge-independent PT self-energy Πµν(q).Similarly,the PT effective vertex Γµ(q,p1,p2)receives PT contributions from the box graphs.In order to extract the PT terms,one has to employ some elementary“WIs”at the S-matrix level.Each time a loop momentum kµgets contracted with aγµof an internal quark line,it gives rise to identities of the kindk=(k1+k−m q)−(k1−m q).(2.7) In general,such pinching momenta can originate either from the kµ-dependent part of the gluon propagator in the loop or from the tri-gluon vertex depicted in Fig.1.This is diagrammatically shown in Fig.2.In a typical gauge-dependent vertex graph,thefirst term of the RHS of Eq.(2.7)cancels the virtual fermion propagator,whereas the second one vanishes for external on-shell quarks.This algorithm produces self-energy-type graphs, which we call self-energy PT parts,shown in Fig.2(b).These self-energy PT terms should be allotted to the proper two-point correlation function evaluated in the given gauge.In this context,we must remark that the gauge dependence in the QED-like part ofΓ(ξ)µ(q) vanishes identically.As can be seen from Fig.3in a diagrammatic manner,even for a propagator in the axial gauge[11]∆(η)µν(q)= −gµν+qµην+qνηµ(q·η)2 1++−+−Fig.3:Vanishing of the(G)PT terms in the axial gauge for the Abelian part ofΓ1µ(q,p1,p2).Apart from self-energy PT terms originating from vertex graphs,a box diagram can also contribute PT terms to Πµν(q)forξ=1,as has been displayed in Fig.4.Since the PT self-energy is independent of the gauge-fixing parameterξ[16],the Feynman-’t Hooft gauge provides a great computational simplification.In this gauge,the only pinching momenta are contained in the tri-gluon vertex,Γabcµνλ(q,p,k)=gf abc (p−k)µgνλ+(k−q)νgµλ+(q−p)λgµν ,(2.9) where q+p+k=0and all four-momenta are incoming,as shown in Fig.1.To make this explicit,wefirst decompose f abcΓµνλ(q,p,k)in the following way:Γµνλ(q,p,k)=ΓFµνλ(q,p,k)+ΓPµνλ(q,p,k),(2.10) withΓFµνλ(q,p,k)=g (p−k)µgνλ−2qνgµλ+2qλgµν ,(2.11)ΓPµνλ(q,p,k)=g(kλgµν−pνgµλ).(2.12) Notice that the splitting of the tri-gauge coupling in Eq.(2.10)makes reference to the external gluon,A aµsay.This explicitly breaks the cyclic symmetry ofΓµνλ(q,p,k),whichis not present neither in ΓF µνλ(q,p,k )nor in ΓP µνλ(q,p,k ).To underline this feature,weshall diagrammatically represent the gluons in the loop by wavy lines,e.g.,see Figs.2(a)and 2(b).It is now easy to recognize that ΓP µνλin Eq.(2.12)contains the pinching loop momenta k and p .In addition,we have1g q µΓF µνλ(q,p,k )=−g νλp 2+g νλk2=∆F −1νλ(p )−∆F −1νλ(k ),(2.14)where U −1µν(q )=t µν(q )q 2and ∆F −1µν(q )=−g µνq 2are the inverse gluon propagators in theunitary and the ξ=1gauge,respectively.Note that U −1νλ(q )does not have any inverse,unless a fictitious SSB mass is introduced for the massless gluon in order to cope with infra-red (IR)infinities.2′)2)=(a)+(b)Fig.4:(G)PT decomposition of the box graph.It is important to notice that the decomposition of Γµνλ(q,p,k )in Eq.(2.10)willimply the transversality of Πµν(q ).After extracting the self-energy PT terms induced by ΓP µνλ(q,p,k )and considering the ghost loop,one obtains the analytic expression [16]Πµν(q )=1i (2π)n 1on account of Eq.(2.14)and the fact that massless tadpoles vanish in DR.Furthermore, it has been noticed[10]that the result given in Eq.(2.15)is identical to that obtained by the BFM for the gauge-fixing parameter valueξQ=1in a covariant gauge condition [8].Note that this gauge-fixing condition(see Eq.(3.6)below)is different from the usual one imposed in Rξgauges.In particular,it is easy to recognize[17]that forξQ=1,the coupling of the gluon backgroundfield,ˆA,to quantum gluons,A,and ghosts,c g,are equal to f abcΓFµνλ(q,p,k)and f abc S Fµ(p,k),respectively.We will return to this point in Section3.In the derivation of Eq.(2.17),the WI in Eq.(2.14)has been crucial.One could therefore ask the question whether the PT can be generalized by modifying the conventional decomposition of Eq.(2.10),so that an elementary Abelian-type WI analogous to Eq.(2.14) is satisfied.Suppose we make the decomposition of the tri-gluon vertex,Γµνλ(q,p,k)=Γ(ξQ)µνλ(q,p,k)+ΓP(ξQ)µνλ(q,p,k),(2.18)for afixed givenξ,ξ=ξQ say,such that1ξQℓµν(q) .(2.20) Moreover,we require that Cornwall’s PT be recovered forξQ= 1.∗Then,pinchingmomenta will arise fromΓP(ξQ)µνλ(q,p,k)and from the difference of propagators∆(ξ)µν(q)−∆(ξQ)µν(q)=ℓµν(q)(ξQ−ξ)/q2.This generalized version of the PT,the GPT,will be for-mulated in the covariant Rξgauges in Section2.1and in non-covariant gauges in Section 2.2.2.1GPT in covariant gaugesIn this section,we shall formulate the GPT in the covariant Rξgauges and argue that this extended version of the PT has very similar features with Cornwall’s PT[1].We start again from the decomposition in Eq.(2.18),Γµνλ(q,p,k)=Γ(ξQ)µνλ(q,p,k)+ΓP(ξQ)µνλ(q,p,k).It is not difficult tofind that the termΓ(ξQ)µνλ(q,p,k)in compliance with the WI of Eq.(2.19) may be given byΓ(ξQ)µνλ(q,p,k)=g (p−k)µgνλ−2qνgµλ+2qλgµν+ 1−1ξQ pνgµλ ,(2.21) andΓP(ξQ)µνλ(q,p,k)=g1g ΓP(ξQ)µνλ(q,p,k)∆(ξQ)νρ(p)∆(ξQ)λσ(k)=−kσp2∆(ξQ)µσ(k).(2.23)Indeed,the RHS of Eq.(2.23)has the correct structure to provide the self-energy GPT terms for a givenξQ by means of Eq.(2.7).Again,settingξQ=1in Eq.(2.23),Cornwall’s pinching procedure is fully recovered.The remaining tri-gauge couplingΓ(ξQ)µνλ(q,p,k),to-gether with the QED-like graphs shown in Fig.3,give rise to the one-loop effective GPT Aq¯q-coupling,denoted by Γ(ξQ)µ(q,p1,p2).In addition,we haveqµ Γ(ξQ)µ(q,p1,p2)=g Σ(ξQ)(p1)− Σ(ξQ)(p2) .(2.24) which is exactly the WI of Eq.(2.6).Correspondingly, Σ(ξQ)(p)is the GPT quark self-energy,which coincides with the usual quark self-energy evaluated in the gaugeξ=ξQ. After all the self-energy GPT terms induced by the generalized pinching momenta in Eq.(2.23)have been identified and added toΠ(ξQ)µν(q),as shown in Fig.5,the effective GPT self-energy, Π(ξQ)µν(q),takes on the analytic formΠ(ξQ)µν(q)=c A i(2π)n ∆(ξQ)(p)∆(ξQ)(k)Γ(ξQ)µ(q,p,k)Γ(ξQ)ν(q,p,k)−2where all contracted Lorentz indices on the RHS of Eq.(2.25)are not explicitly displayed.At this stage,one can already see the connection between the GPT and the BFM in thecovariant gauges,when comparing Π(ξQ )µν(q )with the BFM gluon self-energy given in Ref.[10,17].This relation will be further elaborated in Section 4.++++=+Fig.5:Connection between GPT and BFM in a ξ=ξQ gauge for the self-energy Π(ξQ )µν.Even though we have worked in a gauge,in which the virtual gluon propagators have been gauge-fixed in ξ=ξQ ,one can,however,check that our results would have remained unaffected if we had chosen another gauge.The algorithm of the GPT in the ξQ gauge is completely specified,as long as the steps contained in Eqs.(2.18)–(2.22)are explicitly given.In fact,these would not change,even if the gluon propagators were taken in the axial gauge given in Eq.(2.8).In this case,it is important to identify what the generalized pinching momenta are.These GPT momenta originate from the gluon propagators difference∆(η)µν(q )−∆(ξQ )µν(q )= ξQ −η2q 2q 2+q µην+q νηµq 2(2.26)and the tri-gauge generalized pinching part of the vertex,ΓP (ξQ )µνλ.In particular,the process[18]and gauge independence [5]of the GPT may be shown rigorously by virtue of Becchi-Rouet-Stora identities[19],as was done in Ref.[6].We will not pursue this topic here any further.Instead,we shall apply the GPT in non-covariant gauges.2.2GPT in non-covariant gaugesFollowing the method developed in the previous section,we shall extend the GPT beyond the covariant gauges,i.e.,the non-covariant gauges[11,12,13].Let usfirst consider the general non-covariant gauge-fixing condition[12]G a[A]=ηµην2ξ(η2)2(ηµην∂µA aν)2,(2.28) whereηµis an arbitrary but constant four-vector.In general,we can classify the non-covariant gauges from the different values ofη2,i.e.,η2<0(axial gauge),η2=0(light-cone gauge),η2>0(Hamilton or time-like gauge).Forξ=0,one may use Lagrange multipliers L a and write LηasLη=−L a(ηµην∂µA aν),(2.29) where L a is an auxiliaryfield that mixes with the gluon A a.This leads to a proliferation of Feynman rules.Therefore,it may be more convenient to work withξ=0and then take the limitξ→0.In this limit,the gauge-fixing condition(2.27)leads to the gluon propagator in Eq.(2.8).To avoid excessive complication,we setξ=1in the following, unless it is explicitly stated otherwise.The latter,however,does not confine the generality of our formulation concerning the GPT in the non-covariant gauges.Considering the gauge-fixing term in Eq.(2.28),the inverse propagator is written down∆(η)−1µν(q)=q2 −gµν+qµqνη2 ,(2.30) withα=(q·η)2/(q2η2)andξ=1.This leads to the propagator∆(η)µν(q)=1q·η−βqµqνso that 1(η2)2[(k −p )·η] ,(2.34)ΓP (η)µνλ(q,p,k )=g ηµηνηλ(q ·η)3,(2.36)we can readily see that ΓP (η)µνλ(q,p,k )contains generalized pinching momenta,viz.1(p ·η)3(k ·η)3.(2.37)In these non-covariant gauges,it can be shown that ghosts decouple from S -matrix elements completely in the DR [12].To give an example,we consider the ghost contribution to the gluon self-energy.The interaction Lagrangian containing the ghosts,c a g ,may be derived from L ghost =c a †gδG a [A ]g∂µθa +f abc θb A c µ.(2.39)In this way,we find the ghost propagator,D ab (q )=δab η2η2ηµ,(2.41)with q +p +k =0.With the aid of the Feynman rules in Eqs.(2.40)and (2.41),it is straightforward to show the vanishing of the ghost loop in the gluon self-energy.Indeed,12one hasΠ(ghost )µν(q )∝ηµην d n k 1[(k +q )·η]2(k ·η)=ηµην d n k 1q ·η d n k 1(k +q )·η .(2.42)The last integral in Eq.(2.42)vanishes through a shift of the loop-momentum variable to the origin.By analogy,one can show that ghosts do not contribute to three-gluon,four-gluon,etc.,vertices at one loop.Taking the afore-mentioned decoupling property of the ghosts into account,we find that the effective GPT self-energy in non-covariant gauges is given byΠ(η)µν(q )=c A i (2π)n ∆(η)(p )∆(η)(k )Γ(η)µ(q,p,k )Γ(η)ν(q,p,k )+T (tad.)µν ,(2.43)where T (tad.)µνis a tadpole contribution having the formd n k T (tad.)µν∝ηµην d n k [(k−p )·η]2(k ·η)2+1Beyond one loop,these gauges may pose some computational difficulties[13],since the higher-order correlation functions will contain unphysical poles of the kind1/(k·η).Nev-ertheless,at one loop,Mandelstam–Leibbrandt prescriptions for regularization of these poles can lead to meaningful results[14].Another and,perhaps,more familiar form of the gauge-fixing condition in non-covariant gauges is[11,12]1G a[A]=,(2.50)ξη2which yields∆Aµν(q)= −gµν+qµην+qνηµ(q·η)2−ξη2q2qµqνq2.(2.51) Forξ=0,the propagator in this class of gauges shows a bad high-energy unitarity be-haviour coming from theξ-dependent term in Eq.(2.51),which will affect multiplicative renormalization.The characteristic feature of these gauges is the complete absence of a(q,p,k),within the framework of the GPT.It is easy to see pinching tri-gauge term,ΓP,Aµνλthat1∂µA aν,(2.53)η2withηµ=(1,0,0,0).Again,one can evaluate effective GPT two-point,three-point,etc., correlation functions,using the extended PT algorithm outlined above.In Section2.1,it has become apparent that it exists a connection between the GPT and the BFM in covariant gauges.One may therefore attempt to investigate if there is a14similar analogue for the GPT in non-covariant gauges.This will be our main concern in the next section.3Backgroundfield method in general gaugesFirst,we shall briefly review the main features of the BFM in pure Yang-Mills theories,such as quark-less QCD with N colours.For more details,the reader is referred to[17].Then, we shall consider the BFM in general non-covariant gauges and compare our results for the two-point Green’s functions with those obtained by the GPT in the respective gauge.The BFM relies on the linear expansion of the gaugefield,A aµ,about the background field,ˆA aµ,which amounts to replacingA aµ→ˆA aµ+A aµ,(3.1) in the classical Yang-Mills Lagrangian1L Y M[ˆA+A]=−exp i d4x L Y M−1δθb∂µθa+f abcθb(ˆA cµ+A cµ).(3.5)gOne of the main advantages of the BFM is that one can maintain gauge invariance in Z[J,ˆA]with respect to the backgroundfieldˆA aµ.In covariant gauges,one usually chooses the backgroundfield gauge-fixing conditionG a[ˆA,A]=∂µA a,µ+gf abcˆA bµA c,µ.(3.6)15With the gauge-fixing condition(3.6),one can show that Z[J,ˆA]is invariant under the infinitesimal transformations[17]:δˆA aµ=−1δθb G a[ˆA+δˆA,Aθ(ˆA+δˆA)]=O T,ac(ˆθ)δwith¯A a µ=δW[J,ˆA]η2D abµ[ˆA]A′bν,(3.20) 17leaves Z[J,ˆA]invariant under backgroundfield gauge transformations.In fact,in order to get from the conventional approach in covariant and non-covariant gauges to the corre-sponding BFM quantized action,it is sufficient to make the replacementδab∂µ→D abµ[ˆA], in the gauge-fixing term of the former.Thus,the BFM analogue of the general Coulomb gauge in Eq.(2.53)will be given byG a(Coul.)[ˆA]= −gµν+ηµην(η2)1/2ηµA aµ,which trivially satisfies Eqs.(3.11)and(3.12).Consequently,the effective action quantized via the gauge-fixing condition of Eq.(2.49)possesses a background-field-gauge invariance inherently[15].As has also been noticed in[15],the n-point BFM correlation functions (for n≥2)will be identical to the Green’s function’s derived from the usual approach in the same gauge.This fact is consistent with our earlier observation of the complete absence of generalized pinching parts in the tri-gauge couplingΓµνλ(q,p,k)through the WI of Eq.(2.52),which renders GPT trivial in these gauges.Clearly,linear combinations[15]of Eqs.(2.49),(3.6)and(3.20)will constitute acceptable forms of gauge-fixing conditions within the BFM applied to a renormalizable Yang-Mills theory.Instead of studying the most general gauge-fixing condition mentioned above,we can show that the results obtained in the BFM in the non-covariant gauge(3.20)are equal to those found with the GPT in the corresponding gauge(cf.Eq.(2.27)).Assumingξ=1for simplicity,the quantum gluon propagator,δab∆(η)µν(q),and the ghost propagator,D ab(q), do not differ from those calculated by the conventional method in Eqs.(2.31)and(2.40), respectively.The same holds true for the vertexˆA aµ(q)−A bν(p)−A cλ(k),which equals f abcΓ(η)µνλ(q,p,k)in Eq.(2.34).Therefore,it is obvious that the one-loopˆAq¯q-coupling will be the same in both approaches.To calculate the gluon self-energy in the BFM in non-covariant gauges,we need the coupling ofˆA to ghosts,i.e.,ˆA aµ(q)−c b g(p)−c c g(k).This is given byS(η)abc µ(p,k)=−gf abc(p−k)·ηFurthermore,we must remark that the equivalence established between GPT and BFM in axial gauges will persist,even if the gluon propagators are taken in some other gauge,e.g.,covariant Rξgauge.For the one-loop example of quark-quark scattering pre-sented in Section2,one thenfinds that box-graphs will produce self-energy-and vertex-like GPT terms induced from propagators differences of the form(2.26).These GPT terms will cancel against corresponding terms present in the one-loop coupling Aµq¯q and the vacuum polarization,leading to a unique result.The connection between GPT and BFM will be elaborated further on in Section4.Finally,it may be worth commenting on the fact[12]that the UV divergent part of Π(η)µν(q)evaluated in gauges(2.49)and(3.20)is in general proportional to b1=11c A/3only in the pure axial-gauge limitξ→0,although the WIs in Eqs.(2.45)and(2.46)are satisfied. The reason is that Π(η)µν(q)may contain UV divergences proportional to nµν(q,η).These UV infinities will eventually induce non-covariant counter-terms[12]of the gauge-invariant formηµηνF aµλF a,λ.ν4Connection between GPT and BFMIn this section,we shall demonstrate the relation between the GPT and the BFM in the non-covariant gauge(2.27)forξ=1,by analyzing a typical quark-quark scattering,e.g., q¯q→q′¯q′.Recently,analogous considerations based on BRS identities have been applied to show that basicfield-theoretical requirements necessary for a resummation formalism are satisfied by the PT[6].In this context,we wish to briefly address the issue of how unique is the tri-gauge decomposition in Eqs.(2.18)and(2.32)within the framework of the GPT in covariant and non-covariant gauges.Furthermore,we will present a way to deduce the effective tree-level four-gluon vertex present in a PT one-loop amplitude,in which two of the gluons are in the loop while the other two are external.The effective four-gluon coupling can be isolated from a proper one-loop four-point function by resorting to the intrinsic property[16]of the PT.The analytic result so-derived is found to be the same with the respective vertex,ˆAˆAAA,in the BFM forξQ=1.In the conventional PT,the four-gauge coupling does not contain any pinching momenta.However,one could define a ‘pinching’term by decomposing the tree-level four-gluon vertex into two parts,so that the ‘pinching’and non-‘pinching’part separately satisfy the same BRS identity.Let us consider the one-loop transition amplitude T(2)= q′¯q′|T(2)|q¯q ,where the superscript on T will denote the order of expansion in powers of g2.Following Ref.[6],we will calculate the absorptive part of T(2).Then,one may rely on the analyticity property19of the S matrix to obtain its dispersive or Hermitian part.After carrying out all possible unitarity cuts,one can see that the absorptive amplitude depends on two tree-level matrix elements,where thefirst involves quarks in the intermediate state,i.e., q′′¯q′′|T(1)|q¯q ,and the second gluons,e.g., gg|T(1)|q¯q .More explicitly,we have12 dX LIP Sq′′q′¯q′|T(1)|q′′¯q′′ q¯q|T(1)|q′′¯q′′ ∗+ 1q·η−η2qµqν2i q′¯q′|(T(2)−T(2)†)|q¯q bos.=1where M(η)is the absorptive amplitude one obtains within the GPT or BFM in the ghost-free gauge mentioned above,M(η)=14Pµρ(l1,η)Pνσ(l2,η) −T s P(η)µνT s P(η)∗ρσ+T s P(η)µνT∗ρσ+TµνT s P(η)∗ρσ .(4.7) On account of the WI in Eq.(4.2)and the fact thatηµPµν(l,η)=0,δM vanishes identically. This example explicitly demonstrates the connection between the GPT and the BFM in the non-covariant gauge(2.27).Following a line of similar arguments,one can reach the same conclusion for the GPT and the BFM in covariant or in more general gauges discussed in Section3.It is now interesting to analyze briefly to what extend the splitting of the three-gluon vertex,Γµνλ(q,p,k),given in Eq.(2.18),is uniquely determined,provided the non-pinching part ofΓµνλ(q,p,k)satisfies the WI in Eq.(2.19).For example,another possible expansion ofΓµνλ(q,p,k)in covariant gauges would be¯Γ(ξQ)µνλ(q,p,k)=g (p−k)µ gνλ+pνpλ−kνkλg qµ¯Γ(ξQ)µνλ(q,p,k)=∆(ξQ)−1νλ(p)−∆(ξQ)−1νλ(k).(4.9)The generalized pinching part ofΓµνλ(q,p,k)would then be defined as¯ΓP(ξQ)µνλ(q,p,k)=g(p−k)µpνpλ−kνkλ(2.32)appears not to admit a non-local solution beyond the local one found in Eqs.(2.34) and(2.35).So far,we have focused our attention on the tri-gauge coupling,Γµνλ(q,p,k).In the PT,the four-gluon coupling,A aµA bνA cνA dλ,given by=−ig2 f abx f xcd(gµλgνρ−gµρgνλ)+f dax f xbc(gµλgνρ−gµνgλρ)Γabcdµνλρ+f acx f xbd(gµνgλρ−gµρgνλ) ,(4.11) does not contain any pinching part,as opposed toΓµνλ(q,p,k).In the BFM in covari-appearing in the classical La-ant gauges,the couplingˆA aµˆA bνA cλA dρis different fromΓabcdµνλρgrangian.In a generalξQ gauge in the BFM,the four-gauge couplingˆA aµˆA bνA cλA dρmay be written as[17]=−ig2 f acx f xbd(gµνgλρ−gµρgνλ+1Γ(ξQ)abcdµνλρgµρgνλ)ξQ+f abx f xcd(gµνgλρ−gµρgνλ) ,(4.12) This might make one think that the connection established for the tri-gauge coupling may get spoiled for the four-gluon vertex,especially when one compares the proper one-loop PT four-point function q¯q A aµA bν,shown in Fig.6,with the corresponding one obtained by the BFM.To show that this relation still exists,we intend to isolate the effective four-,from the graph in Fig.6(d),after including the relevant pinchinggluon vertex,ΓF,abcdµνλρcontributions originating from the one-loop transition amplitude q¯q A aµA bν.Consequently, this analysis presented here will equally carry over to the GPT in general gauges.Let us consider the1PI four-point Green’s function,q(p1)¯q(p2)→A aµ(k1)A bν(k2)shown in Fig.6,where the external quarks and gluons are taken to be on-shell.According to the diagrammatic approach of the intrinsic PT[16],we will only keep PT terms that are akin to the graph in Fig.6(d).For simplicity,we will work in the Feynman-’t-Hooft gauge.To,may be deduced from Fig.6(b),by including the be specific,the effective vertex,ΓF,abcdµνλρpinching parts resulting from the two graphs in Fig.6(a).Up to overall factors,external quark spinors and gluon polarizations,we haveFig.6(c)∼ d n kΓc,λS(p1−k)Γd,ρ Γabcdµνλρ−ΓP,abcdµνλρ ,(4.14)k2p2。