Coupling in the singular limit of thin quantum waveguides

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coupling

couplingCoupling: An Essential Concept in Systems Design and AnalysisIntroductionIn the field of systems design and analysis, the concept of coupling plays a critical role in understanding and evaluating the complexity and efficiency of a software system. Coupling refers to the degree of interdependence between different modules or components within a system. It measures how closely one module is connected with another, and how much one module relies on another to perform its functionality. This document aims to provide a comprehensive overview of coupling, its types, impacts, and strategies to reduce coupling in software systems.Types of CouplingThere are several types of coupling that can exist in a software system. Each type represents a different level of interdependence between modules. Understanding thesetypes is crucial for analyzing software systems and identifying potential areas for improvement.1. Content Coupling: Content coupling refers to the highest level of interdependence between modules. In this type of coupling, one module directly accesses the internal data or methods of another module. This close dependency can make the system fragile and less maintainable, as any changes in one module may require modifications in other closely coupled modules.2. Common Coupling: Common coupling occurs when multiple modules share a global data item. Any changes in the shared data can impact the behavior of multiple modules. This type of coupling can make it difficult to maintain and reason about the system, as modifications in one module can have unintended consequences on others.3. Control Coupling: Control coupling happens when one module controls the flow of execution of another module by passing parameters, flags, or function calls. This type of coupling can lead to code complexity and reduce modularity, as changes in one module can have ripple effects on other modules that rely on the control flow.4. Stamp Coupling: Stamp coupling occurs when modules share a structure with multiple related elements, but only a subset of the elements is actually used by each module. This can lead to inefficiency and code bloat, as modules may require unnecessary data or functionality.5. Data Coupling: Data coupling represents a lower level of interdependence, where modules share data through parameters or return values. However, they do not rely on each other's internal implementation details. This is considered a desirable type of coupling, as it promotes modular and independent development of components.Impacts of CouplingHigh coupling in a software system can have various negative impacts on its maintainability, extensibility, and overall performance. Some of the key impacts of coupling include:1. Difficult Maintenance: When modules are tightly coupled, any changes or bug fixes in one module can have unintended consequences on other modules. This makes it more difficult to identify and fix issues, increasing the risk of introducing new bugs and errors.2. Limited Extensibility: In a system with high coupling, making changes or adding new features becomes a challenging task. The interdependence between modules can restrict the flexibility and modularity needed for easy extensibility. This can limit the ability to adapt to evolving requirements or integrate with new technologies.3. Testing Complexity: Closely coupled modules often require more extensive testing efforts. The high interdependence between modules makes it harder to isolate and test individual components. Any changes to one module may require retesting of multiple other modules, increasing the time and effort required for testing.Strategies to Reduce CouplingReducing coupling is crucial for developing and maintaining robust, flexible, and efficient software systems. Here are some strategies to minimize coupling and improve the overall quality of the system:1. Encapsulate Functionality: Encapsulating the functionality of modules and exposing only necessary interfaces canreduce the level of interdependence. This promotes modularity and allows changes to be localized within individual modules, minimizing the impact on other parts of the system.2. Use Abstraction and Interfaces: Instead of tightly coupling modules with concrete implementations, use abstractions and interfaces to define contracts between components. This allows for loose coupling, as modules can interact through well-defined interfaces without relying on the concrete implementation details.3. Apply Design Patterns: Design patterns like the Observer pattern, Dependency Injection, or Strategy pattern can help reduce coupling by decoupling modules and promoting more flexible interactions. These patterns provide standardized approaches for achieving loose coupling and modular designs.4. Follow Separation of Concerns: Ensure that each module or component has a clear and single responsibility. This helps to minimize the dependencies between modules and allows for better separation of concerns. By separating the functionality, changes or updates in one aspect of the system will have minimal impact on other components.5. Continuous Refactoring: Regularly review and refactor the codebase to identify and reduce coupling. Refactoring involves restructuring the code without changing its external functionality. By continuously refactoring, developers can identify opportunities to decouple modules and improve the overall code quality.ConclusionCoupling is a fundamental concept in systems design and analysis, influencing the complexity and maintainability of software systems. Understanding the different types of coupling and their impacts is essential for evaluating and improving the quality of a system. By adopting strategies like encapsulation, abstraction, design patterns, separation of concerns, and continuous refactoring, developers can minimize coupling and build more flexible, modular, and maintainable software systems.。

Compositeness, Triviality and Bounds on Critical Exponents for Fermions and Magnets

a rXiv:h ep-th/93127v11D ec1993ILL-(TH)-93-21COMPOSITENESS,TRIVIALITY AND BOUNDS ON CRITICAL EXPONENTS FOR FERMIONS AND MAGNETS Aleksandar KOCI ´C and John KOGUT Loomis Laboratory of Physics,University of Illinois,Urbana,Il 61801Abstract We argue that theories with fundamental fermions which undergo chiral symmetry breaking have several universal features which are qualitatively diﬀerent than those of theories with fundamental scalars.Several bounds on the critical indices δand ηfollow.We observe that in four dimensions the logarithmic scaling violations enter into the Equation of State of scalar theories,such as λφ4,and fermionic models,such as Nambu-Jona-Lasinio,in qualitatively diﬀerent ways.These observations lead to useful approaches for analyzing lattice simulations of a wide class of model ﬁeld theories.Our results imply that λφ4cannot be a good guide to understanding the possible triviality of spinor QED .1.IntroductionThere are two classes of theories in the literature that are used to model the Higgs sector of the Standard Model[1,2].One is based on the self-interactingφ4scalar theory in which the Higgs is elementary.The other is based on strongly interacting constituent fermiﬁelds in which the Higg’s particle is a fermion-antifermion bound state.A recent proposal for the second realization uses Nambu-Jona-Lasinio(NJL)models in which composite scalars emerge as a consequence of spontaneous chiral symmetry breaking[2].In four dimensions both types of models have a trivial continuum limit and are meaningful only as eﬀective theories with aﬁnite cutoﬀ.This restriction places constraints on the low-energy parameters e.g.bounds on the masses.We wish to point out in this letter that triviality in the two models is realized in diﬀerent ways.The diﬀerences between theories of composite and elementary mesons can be expressed in terms of the critical indicesδandη,and several inequalities and bounds on these indices will follow.These results should prove useful in theoretical,phenomenological and lattice simulation studies of a wide class of modelﬁeld theories.We begin with a few comments about the physics in each model.In a NJL model[3],as a result of spontaneous chiral symmetry breaking,the pion-fermion coupling is given by the Goldberger-Treiman relation gπN=M N/fπ,where M N is the fermion mass and fπis the pion decay constant.Being the wavefunction of the pion,fπdetermines its radius as well:rπ∼1/fπ.The Goldberger-Treiman can then be written in the suggestive form gπN∼M N rπ.Thus,the coupling between pions and fermions vanishes as the size of the pion shrinks to zero.The origin of triviality of the Nambu-Jona-Lasinio model is precisely the loss of compositeness of the mesons[3].The force between the fermions is so strong that the constituents collapse onto one another producing pointlike mesons and a noninteracting continuum theory.In a self-interacting scalar theory,likeφ4,the mesons are elementary and the reason for triviality is diﬀerent[4,5].At short distances the interaction is repulsive,so there is no collapse.The structure of the scalars,needed for the interaction to survive the continuum limit,should be built by weakly interacting bosons.In four dimensions, the short rangedλφ4interaction fails to provide a physical size for the mesons.It cannot be felt by the particles because of the short-distance repulsion–they cannot meet where collective behavior can set in and produce macroscopicﬂuctuations.In this way the cutoﬀremains the only scale and the continuum limit is trivial.Consider two simple,soluble examples:the large-N limits of the O(N)σ-and the four-fermi model [6,7].They exhibit a phase transition atﬁnite coupling for2<d<4.Their critical exponents are given in Table1.As is apparent from the Table,the two sets of critical indices evolve diﬀerently when d is reduced below4.Atﬁrst glance this might be surprising since both models break the same symmetry spontaneously and one expects that they describe the same low energy physics.The purpose of this paper is to show that this diﬀerence between the critical exponents is generally valid,irrespective of the approximations employed.As a consequence of this it will be possible to establish a bound on the exponentδwhich for scalar theories isδ≥3,and for fermionic theories isδ≤3.Although in four dimensions the two sets of exponents coincide,they are accompanied by logarithmic corrections due to scaling violations.Itconsequently the scaling violations have opposite signs in the two classes of theories.These bounds are a consequence of diﬀerent realizations of symmetry breaking,the essential diﬀerence being the fact that for scalar theories mesons are elementary,while in the case of the chiral transition in fermionic theories,they are composite.The bounds onδare just another way of expressing this diﬀerence in terms of universal quantities. Finally,we will discuss the implications of these results on triviality in both models in four dimensions.Table1Leading order critical exponents for the spherical and four-fermi modelexponentσ-model four-fermiβ1d−2ν1d−2δd+21d−2η04−d2.Mass ratios and bounds onδTo approach the problem,it is convenient to adopt a particular view of the phase transition[8].Instead of the order parameter we will use mass ratios to distinguish the two phases.While the order parameter is a useful quantity to parametrize the phase diagram,the spectrum carries direct information about the response of the system in it’s diﬀerent phases and its form does not change in the presence of an external symmetry-breakingﬁeld.In what follows we will switch from magnetic to chiral notation without notice.The correspondence is:magneticﬁeld(h)⇔bare mass(m);magnetization(M)⇔chiral condensate(<¯ψψ>); longitudinal and transverse modes⇔(σ,π);h→0⇔chiral limit.Theories that treat scalars as elementary will be referred to as’magnets’and those that give rise to composite mesons as a consequence of spontaneous chiral symmetry breaking will be refered to as’fermions’.Consider the eﬀect of spontaneous symmetry breaking on the spectrum from a physical point of view. In the symmetric phase,there is no preferred direction and symmetry requires the degeneracy between longitudinal and transverse modes(chiral partners).Therefore,in the zero-ﬁeld(chiral)limit the ratio R=M2T/M2L=M2π/M2σ→1.As the magneticﬁeld(bare mass)increases,the ratio decreases(because of level ordering,σis always heavier thenπ).In the broken phase,however,the ratio vanishes in the chiral limit because the pion is a Goldstone boson.This time,the ratio clearly increases away from the chiral limit.dynamics.The value of the ratio at large h is less sensitive to variations in the coupling.The qualitative behavior of the mass ratio is sketched in Fig.1.The important property of the mass ratio,in this context, is that its properties follow completely from the properties of the order parameter[8].This,after all,comes as no surprise since both quantities,M and R,contain the same physics and merely reﬂect two aspects of one phenomenon.The essential ingredients are the Equation of State(EOS)and the Ward identity which follows from it.h a=M a Mδ−1f t/M1/β ,χ−1T=h/M,χ−1L=∂hR =χ−1LOne way to determine the sign of the scaling violations in four dimensions is to proceed in the spirit of the ǫ-expansion i.e.to approach four dimensions from below [9].The transcription to the language of scaling violations is established by the replacement ǫ→1/log Λin the limit ǫ→0.Thus,the extension of the arguments made before for d <4can be made by simply taking the limit d →4.In this way we anticipate that the two inequalities prevail and suggest that the scaling violations have diﬀerent signs in the two theories.The diﬀerence in the sign of the scaling violations in fermions and magnets has a simple explanation and lies at the root of the diﬀerence between the patterns of symmetry breaking in the two systems.Imagine that we ﬁx the temperature to its critical value T =T c and approach the critical point (T =T c ,h =0)in the (T,h )plane from the large-h region.The possible similarity of the two models is related to their symmetry.This is apparent in the chiral (zero ﬁeld)limit where this symmetry is manifest.By going away from this limit chiral symmetry is violated and the two models diﬀer.Consider the behavior of the mass ratio for magnets in a strong magnetic ﬁeld,away from the scaling region.In this regime,the temperature factor can be neglected and the hamiltonian describes free spins in an external ﬁeld (H →h i S i ).The energy of longitudinal excitations is proportional to the ﬁeld-squared,χ−1L ∼h 2,while the transverse mass remains ﬁxed by the symmetry,namely χ−1T =h/M for any value of h .The eﬀect of the external ﬁeld is to introduce a preferred direction and its increase results in ampliﬁcation of the diﬀerence between the longitudinal and transverse dirrections.For large h the ratio scales as R ∼1/h .Therefore,an increase in magnetic ﬁeld reduces the ratio towards zero.The critical isotherm in this case bends down (Fig.3).For fermions,the mesons are fermion-antifermion composites.Close to the chiral limit,they are collec-tive.However,as the constituent mass increases,they turn into atomic states and the main contribution to the meson mass comes from the rest energy of its constituents.In the limit of inﬁnite bare mass,interactions are negligible and M →2m regardless of the channel.Thus,outside of the scaling region,an increase in m drives the ratio to 1(Fig.3).Thus,the scaling violations for magnetis and fermionis have opposite signs .They contain knowledge of the physics away from the chiral limit where the two models are quite diﬀerent and these diﬀerences remain as small corrections close to the chiral limit.To establish the connection between scaling violations and triviality,we introduce the renormalized cou-pling.It is a dimensionless low-energy quantity that contains information about the non-gaussian character of the theory.It is conventionally deﬁned as [10]g R =−χ(nl )∂h 3= 123<φ(0)φ(1)φ(2)φ(3)>c (4)The normalization fators,χ= x <φ(0)φ(x )>c and ξd ,in eq.(3)take care of the four ﬁelds and the three integrations.In a gaussian theory all higher-point functions factorize,so g R ing the hyperscalingg R∼ξ(2∆−γ−dν)/ν(5)where∆=β+γis the gap exponent.Being dimensionless,g R should be independent ofξifξis the only scale.Thus,the validity of hyperscaling requires that the exponent must vanish.It implies the relation, 2∆−γ−dν=0,between the critical indices.In general,it is known that the following inequality[11]holds2∆≤γ+dν(6)The exponent in the expression for g R is always non-positive,so that violations of hyperscaling imply that the resulting theory is non-interacting.Above four dimensions,the exponents are gaussian(γ=1,∆=3/2,ν=1/2).In this case,it is easy to verify the above inequality:3≤1+d/2,which amounts to d≥4.In four dimensions mostﬁeld theoretical models have mean-ﬁeld critical exponents,but with logarithmic corrections that drive g R to zero.Scaling violations in any thermodynamic quantity propagate into the renormalized coupling and,according to eq.(6), these violations lead to triviality.Instead of using the ideas of theǫ-expansion where scaling is always respected and where equalities between exponents hold,we willﬁx d=4and compute the logarithmic corrections to the critical exponents. In order to focus on the problem in question,we analyze two simple models:φ4and(¯ψψ)2theories both in the large-N limit.The results that will be discussed are completely general and the two models are chosen just to make the argument simple.The eﬀective actions for the two models are[12,13]V(M)=−14M42t<¯ψψ>2+<¯ψψ>4log(1/<¯ψψ>)(7b) This is the leading log contribution only.In theﬁrst example,it is clear how log-corrections lead to triviality. The logarithm can simply be thought of as coming from the running coupling–quantum corrections lead to the replacementλ→λR.The vanishing of the renormalized coupling is then manifest from eq.(7a).In the case of fermions,eq.(7b),the details are completely diﬀerent–the analogous reasoning would lead to an erroneous conclusion that the renormalized coupling increases in the infrared.In eq.(7b)the explicit coupling is absent from theﬂuctuating term–it is already absorbed in the curvature.Once the curvature isﬁxed,the eﬀective coupling is independent of the bare one.The vanishing of the renormalized coupling here follows from the wave funciton renormalization constant Z∼1/ln(1/<¯ψψ>)[13].In both cases the renormalized coupling is obtained through the nonlinear susceptibility.For simplicity, we work in the symmetric phase where the odd-point functions vanish.The correlation length is related to the susceptibility byξ2=χ/Z.For magnets the folowing relations holdχ(nl)∼χ4λln(1/M)∼1For fermions,on the other hand,we have1χ(nl)∼χ4ln(1/<¯ψψ>),Z∼(9b)ln(1/<¯ψψ>)In this context the following point should be made.The nonlinear susceptibility is a connected four-point function for the composite¯ψψﬁeld.The free fermionic theory is not gaussian in¯ψψ,so even in freeﬁeld theory g R does not vanish.The fact that g R→0near the critical point indicates that the resulting theory is indeed gaussian in the compositeﬁeld which results in a free bosonic theory in the continuum.The Equation of State(EOS)is obtained from the eﬀective potential by simple diﬀerentiation.To make the connection withδ,we take t=0.The critical EOS for the magnets is[12]M3h∼(11)log(1/M)Thus,on the ratio plot,Fig.2a,the critical isotherm is no longerﬂat,but goes down,as the h-ﬁeld(order parameter)increases.The result of eq.(11)is well known in the literature and has been obtained in the past using theǫ-expansion:δ=3+ǫ[12],where the correspondence with eq.(10)is made after the replacement 1/ǫ→log.Such corrections toδ,as eq.(11),can never occur in the case of the chiral transition.For the four-fermi model the critical EOS[14]reads,m∼<¯ψψ>3log(1/<¯ψψ>)(12) Unlike scalar theories,the log’s appear in the numerator–the right hand side in eq.(12)vanishes slower then the pure power and the”eﬀective”δis smaller then the(pure)mean-ﬁeld value.˜δ=3−1We have seen that the two diﬀerent realizations of spontaneous symmetry breaking can be expressed simply in terms of universal quantities.In this way,many apparently complicated dynamical questions become transparent.In addition,some of our observations lead to practical applications–they can be used for extracting the properties of the continuum limit of theories with newﬁxed points,especially when clear theoretical ideas about the low-energy physics of the theory are missing.Of special importance is the knowledge of the position of the logarithms when triviality is studied on the lattice.It is extremely diﬃcult to establish the presence of the logarithms for aﬁnite system and to disentangle them fromﬁnite size eﬀects. The bounds obtained in this paper establish some criteria in this direction as far as chiral transitions are concerned.Recently,they were proven to be decisive in studies of the chiral transition of QED[15]and in establishing triviality of the NJL model in four dimensions by computer simulations[16].The literature on fermionic QED abounds with loose statements such as”QED is ultimately trivial and reduces toλφ4”.If QED suﬀers from complete screening(the Moscow zero),then we expect the NJL model to describe its triviality.One result of this paper is thatλφ4cannot be a good guide to fermionic QED under any circumstances!There are several physical implications that the two bounds onδimply and we discuss some of them brieﬂy below.The wavefunction renormalization constant respects the Lehmann bound:0≤Z≤1.Roughly speaking, Z is the probability that the scalarﬁeld creates a single particle from the vacuum.The limit Z=1 corresponds to a noninteracting theory whereas the compositeness condition,Z=0,sets an upper bound on the eﬀective coupling[17].The anomalous dimensionηdetermines the scaling of Z in the critical region Z∼ξ−η.It describes the scaling of the correlation function in the massless limit:D(x)∼1/|x|d−2+η.Small ηis associated with weak coupling andη=O(1)with the strong coupling limit of the theory.It is related to the exponentδthrough the hyperscaling relationd+2−ηδ=especially visible from the behavior of the correlation functions in the two theories.In the scalar theory, where anomalous dimensions are small,the scaling of the correlation functions is weakly aﬀected at short distances.They behave almost like free particle propagators,D(x)∼1/|x|d−2.For fermions,however, there are nontrivial changes in scaling due to strongly interacting dynamics at short distances.In four-fermi theory,for example,at large-Nη=4−d and D(x)∼1/|x|2irrespective of d.Another consequence of the diﬀerence in the bounds onδconcerns the physics in the broken phase. As an eﬀect of spontaneous symmetry breaking a trilinear coupling is generated and the decayσ→ππis a dominant decay mode in the Goldstone phase.The pole in theσpropagator is buried in the continuum states and the pion state saturates all the correlation functions.This is especially visible in lower dimensions and persists even atﬁnite h.For fermions,however,this is not necessarily the case for the simple reason that theπ−σmass-squared ratio in the broken phase is bounded by1/δ>1/3,away from the chiral limit and for theσ→ππdecay to occur the masses must satisfy M2π/M2σ≤1/4.Thus,for appropriately chosen couplings and bare fermion masses,the decay becomes kinematically forbidden even in the broken phase.Regarding the usage of perturbation theory,the rules are diﬀerent for magnets and fermions.The applicability of perturbation theory to magnets was noted long ago[18].Its success near two and four dimensions is not surprising.Below four dimensions,φ4posesses an infraredﬁxed point at coupling g c∼O(ǫ). This coupling is an upper bound on the renormalized coupling i.e.g R∼ǫ.So,theǫ-expansion is in eﬀect renormalized perturbation theory.The critical exponents receive corrections of the typeδ=3+O(g R),β= 1/2+O(g R),etc..Thus,inφ4the success of perturbation theory is a consequence of the fact that the infraredﬁxed point moves to the origin as d→4.In the non-linearσ-model the critical coupling is an ultravioletﬁxed point that moves to the origin as d→2.The weak coupling phase is at low-temperatures. Due to the presence of Goldstone bosons,all the correlation functions are saturated with massless states and the entire low-temperature phase is massless.Every point is a critical point in the limit of vanishing magneticﬁeld.Thus,the low-temperature expansion is an expansion in powers of T.Terms of the form exp(−M/T)are absent and there is no danger that they will be omitted by using perturbation theory.In this way,in principle,the critical region can be accessed through perturbation theory[18].Clearly,such reasoning can not be applied to fermions simply because the weak coupling phase is symmetric.Thus,no matter how small the coupling is,perturbation theory omits the Goldstone physics as a matter of principle. It can not produce bound states that accompany the chiral transition and its applicability is questionable in general.Especially,it is diﬃcult to imagine how perturbation theory could give a mass ratio that is constant, independent of the bare parameters,once the bare coupling is tuned to the critical value.Even if this were possible,the renormalized coupling would be sensitive to the variation of the bare mass leading to conﬂicting renormalization group trajectories as a consequence.Finally,we comment on one possible use of theδ<3bound for controllingﬁnite size eﬀects in lattice studies of chiral transitions.As was argued in[8],it is convenient to introduce a plotχ−1πversus<¯ψψ>2. The usefulness of this plot becomes clear if we write the critical EOS.From m∼<¯ψψ>δandχ−1π=m/<¯ψψ>,it follows thatχ−1π∼(<¯ψψ>2)(δ−1)/2(15)concave downwards.On a small lattice the order parameter is smaller and pion mass is bigger then in the thermodynamic limit.Thus,small volume distortions are always in the direction of opposite concavity of the plot and the wrong concavity of this plot is a clear sign of the presence ofﬁnite size eﬀects[8].Acknowledgement We wish to acknowledge the discussions with E.Fradkin,A.Patrascioiou and E.Seiler. This work is supported by NSF-PHY92-00148.References[1]See for example The Standard Model Higgs Boson,Edited by M.Einhorn,(North-Holland,Amsterdam, 1991).[2]Y.Nambu,in New Trends in Physics,proceedings of the XI International Symposium on Elementary Particle Physics,Kazimierz,Poland,1988,edited by Z.Ajduk S.Pokorski and A.Trautman(World Scientiﬁc, Singapore,1989);V.Miransky,M.Tanabashi and K.Yamawaki,Mod.Phys.Lett.A4,1043(1989);W. Bardeen,C.Hill and M.Lindner,Phys.Rev.D41,1647(1990).[3]Y.Nambu and G.Jona-Lasinio,Phys.Rev.122,345(1961).[4]M.Aizenman,Comm.Math.Phys.86,1(1982);C.Arag˜a o de Carvalho,S.Caracciolo and J.Fr¨o hlich, Nuc.Phys.B215[FS7],209(1983).[5]M.L¨u scher and P.Weisz,Nuc.Phys.B290[FS20],25(1987)[6]S.K.Ma,in Phase Transitions and Critical Phenomena Vol.6,eds.C.Domb and M.Green(Academic Press,London,1976).[7]S.Hands,A.Koci´c and J.B.Kogut,Phys.Lett.B273(1991)111.[8]A.Koci´c,J.B.Kogut and M.-P.Lombardo,Nuc.Phys.B398,376(1993).[9]K.Wilson and J.Kogut,Phys.Rep.12C,75(1974).[10]See,for example,C.Itzykson and J.-M.Drouﬀe,Statistical Field Theory(Cambridge University Press, 1989);V.Privman,P.C.Hohenberg and A.Aharony,in Phase Transitions and Critical Phenomena Vol.14, eds.C.Domb and J.L.Lebowitz(Academic Press,London,1991).[11]B.Freedman and G.A.Baker Jr,J.Phys.A15(1982)L715;R.Schrader,Phys.Rev.B14(1976)172;B.D.Josephson,Proc.Phys.Soc.92(1967)269,276.[12]E.Brezin,J.-C.Le Guillou and J.Zinn-Justin,in Phase Transitions and Critical Phenomena Vol.6,eds.C.Domb and M.Green(Academic Press,London,1976).[13]T.Eguchi,Phys.Rev.D17,611(1978).[14]S.Hands,A.Koci´c and J.B.Kogut,Ann.Phys.224,29(1993).[15]A.Koci´c,J.B.Kogut and K.C.Wang,Nucl.Phys.B398(1993)405.[16]S.Kim,A.Koci´c and J.Kogut(unpublished)[17]S.Weinberg,Phys.Rev.130,776(1963).[18]E.Brezin and J.Zinn-Justin,Phys.Rev.B14,3110(1976).Figure captions1.Susceptibility ratio as a function of magneticﬁeld(bare mass)forﬁxed values of the temperature (coupling).2.The behavior of the critical mass ratio,R(t=0,h)=1/δ,for diﬀerent values of d,in a)magnets and b) in the case of chiral transition.3.Critical mass ratio,R(t=0,h),for fermions and magnets in four dimensions over extended range of magneticﬁeld(mass).This figure "fig1-1.png" is available in "png" format from: /ps/hep-th/9312007v1This figure "fig1-2.png" is available in "png" format from: /ps/hep-th/9312007v1This figure "fig1-3.png" is available in "png" format from: /ps/hep-th/9312007v1。

半导体器件双语词汇

Doping profile（掺杂分布）Step junction（突变结）One-side Step junction（单边突变结)Diffussion(扩散)Graded junction (缓变结）Gradient（梯度）Net charge（净电荷）Depletion（耗尽层）Space charge region（空间电荷区）Potential barrier region（势垒区）Electric field(电场）Built-in potential（内建电场）Space charge region width（空间电荷区宽度）Quantative calculation（定量的)Qualitative(定性的)Substrate (衬底的)Forward bias(正偏)Reverse bias(反偏)Non-uniform doping（非均匀掺杂）Linearly graded junction（线性缓变结）Ideal-diode equation （理想二极管方程）Ideal pn junction model（理想pn结模型）Using boltgmann approximation（波尔兹曼近似）No generation and recombination inside the deletion layer（耗尽区内没有产生与复合）Low injection（小注入）Step junction with abrupt depletion layer approximation(突变结耗尽层近似)Mathmatical model(数学模型)Reverse saturation current(反向饱和电流)High junction(大注入)Small-signal model of pn junction(小信号)pn（结模型）Diffusion capacitance（扩散电容)Depletion layer capacitance（势垒电容）Junction capacitance（结电容）Breakdown voltage of pn junction pn(结击穿电压) Avalanche Breakdown (雪崩击穿)Tunnel Breakdown(隧道击穿)Transient of pn junction pn(结瞬态特性)Model and model parameters of pn junction diode (二极管模型和模型参数)Base width modulation and early voltage(基区宽变效应和厄利电压）Cutoff frequency（截止频率）JFET （junction field effect transistor）MESFET（metal semiconductor)Enhancement(增强型)Depletion (耗尽型)Flat band voltage(平带电压)11111。

GE Inspection Technologies CL5 超音波精确厚度测量仪说明书

GEInspection TechnologiesCL5gUltrasonic Precision Thickness GaugeMicrometer Precision in a Rugged PackageThe CL5 is an easy-to-use precision thickness measuring solution for components used in the automotive and aerospace industries, such as:•Cast and stamped metal components made of aluminum, steel, copper, bronze • Machined workpieces • Chemically milled components • Metal strips, metal plates • Plastics and composites • GlassThe instrument can be held in one hand or placed on flat workpieces, making the CL5 a compact way to test your material for the required thickness or checking for sheet corrosion.Compact Solution With a Full Range of Functionality The CL5 precision thickness gauge offers a full range offunctionality in an easy to use, compact and rugged package.Three soft keys directly under the display activate the functions shown on the display menus. Four directional keys help make menu changes and navigation of the text entry screen simple and efficient.The graphical display presents the user with seven different operation modes. The user can select Normal, Minimum Scan,Maximum Scan, Differential/Rate of Reduction, Thk+A-Scan (option), Velocity (option) or Quality View. The CL5 uses aprogrammable data recorder for easy set up of data files from the PC. The SD Card memory system places all the datarecording and set-up information on a removable SD memory card. The files are formatted allowing drag and drop files when plugged directly into the PC. Other data such as digitalphotographs can also be stored on the same SD card. The CL5allows direct connection to the PC, using a serial or USB port (with optional cable).Simple OperationThe CL5 is a very straightforward instrument to operate. The MODE key progresses the user through a series of selection and set-up menus and back to the measurement mode. One press of the MODE key displays a table of standard probes and up to five special set-ups. Another press of the MODE key displays a set-up menu where the user can easily scroll through the menu, see the current settings and make fast changes to any of the displayed settings.A supervisor lock-out function enables a knowledgeable user to set up all the specific measuring functions and settings of the CL5 and lock the settings so critical settings cannot be changed by a subordinate user.Additional advantages offered by this compact, multifunctional instrument include:•Enhanced measurement performance produces stable and repeatable thickness values•Seven measurement and display modes: Normal, Minimum Capture, Maximum Capture, Differential and Rate of Reduction,Velocity (with CL5 VL option), Thickness+A-Scan (with Live A-Scan Option) and Quality View Mode (with Data Recorder option).•Snapshot A-Scan on all models•Hollow/Fill thickness digits showing coupling or non-coupling status•Visual LED alarm to alert user when measurements are exceeding the user selectable limit values•Customer parameter set-ups for special configurations and quick instrument set-up•Flexible power system via standard AA batteries or rechargeable battery pack system (standard)•Multi-language user interface•Automatic ultrasonic performance (gain and gate controls)•Wide variety of standard probes (sold separately)Filled digits indicate successful couplingLive A-Scan for more precise evaluationsRate of reductionData recorderSnapshot A-ScanCL5––Simply reliable, reliably simpleThe Velocity Option: Performance and FlexibilityThe CL5 Velocity option gives the user an added measurement mode used for determining the velocity of a known thickness of material. Material thickness can be entered manually via the CL5keyboard or a digital caliper can be connected, allowing the thickness value to be sent electronically from the caliper to the CL5. The user simply places the probe on the part, and the CL5displays the material velocity of the test object. Both the thickness and the velocity value can be stored in the Data Recorder and downloaded to the PC.The Live A-Scan OptionThe optional Live A-Scan feature gives the user a real time view of the echoes being digitally measured by the CL5.Viewing the Live A-Scan can aid users when attempting toproperly align the probe and the test object to achieve the best measurement values. Viewing the Live A-Scan enables the user to ensure the proper echoes are being measured and the digital value is correct.The Data Recorder OptionThe Data Recorder option permits the quick and easy storage of thickness values in file form. Fully user-programmable, it stores up to 10,000 measured values or as many as 500 values with attached A-Scan.The programmable data recorder allows creation of data recorder files directly from the CL5 keypad, or from the PC using the flexible UltraMATE ®or UltraMATE ®Lite software program. The Data Recorder supports the use of alphanumeric file names,standard linear and grid files and custom linear files.Extended file types store the thickness values, velocity settings and other critical data for each measurement point, making the CL5 and UltraMATE ®ideal for test data management.Achieve More Precision With Quality ViewQuality View Mode permits Data Recorder-driven control and capture of thickness measurements. It is ideal for singular parts or structures with numerous measurement points that havedifferent target thicknesses and/or varying upper and lower limits or tolerances.Uses of Quality View Mode include:1.Fast collection of thickness measurement data for statistical analysis during variation control and quality assurance.2.Digitally capturing thickness measurement data for quality records and traceability.3.Variation control of work in progress on the manufacturing or workshop floor.Quality View Mode displays the current measurement location name, a bar graphic of the thickness measurement that shows the lower specified limit value, the nominal/target value, the upper specified limit and a numerical readout of the measurement.Selection of Quality View Mode displaysTo work in Quality View Mode, custom four-point linear files are created in either Microsoft®Excel or UltraMate ®software applications on a PC and downloaded to the CL5 using the optional serial or USB cable. Measurements can also be uploaded into a PC for processing and analysis using Microsoft ®Excel, UltraMate ®or a third party statistics and/or quality software application.Quality View Mode Out of tolerance dialogue Measurement Review ModeNumerical value of thickness is filled whenprobe is coupled to the location of measurementAlpha 2 DFR/CLF4Standard Delay Line 15 MHz 0.30 in (7.6 mm)0.007 to 1.0 in (0.18 to 25.4 mm)Alpha 2 F/CLF5Fingertip Contact 10 MHz 0.38 in (9.5 mm)0.060 to 10.0 in (1.52 to 254 mm)Mini DFR Thin Range Delay Line20 MHz0.19 in (4.8 mm)0.006 to 0.2 in (0.16 to 5.1 mm)Alpha DFR-P Delay Line for Plastic Materials 22 MHz 0.30 in (7.6 mm)0.005 to 0.15 in (0.13 to 3.8 mm) in plastic materials K-Pen Delay Line Pencil Probe 20 MHz 0.065 or 0.090 in (1.7 or 2.3 mm)0.008 to 0.175 in (0.20 to 4.4 mm)CA211AStandard Contact5 MHz0.75 in (19.1 mm)0.060 to 20.0 in (1.52 to 508 mm)gGE Inspection Technologies: productivity through inspection solutionsGE Inspection Technologies provides technology-driven inspection solutions that deliver productivity, quality and safety. We design,manufacture and service ultrasonic, remote visual, radiographic and eddy current equipment and systems. We offerspecialized solutions that will help you improve productivity in your applications in the aerospace, power generation, oil & gas,automotive or metals Industries./inspectiontechnologies© 2007 General Electric Company. All Rights Reserved. We reserve the right to technical modifications without prior notice. GE ®is a registered trademark of General Electric Co. Other company or product names mentioned in this document may be trademarks or registered trademarks of their respective companies, which are not affiliated with GE.GEIT-20206EN (09/07)CL5 Compatible Transducer SpecificationsEnvironmental Sealing Impact resistant, dust and splash proof, gasket-sealed, case-tested to IP54Weight 0.92 lb (420 g) with batteries Size7.1 in H × 3.7 in W × 1.8 in D (180 mm × 94 mm × 46 mm)Temperature Range Operating: –10 ºC to +60 ºC Storage: –20 ºC to +70 ºCOperating Languages English, German, French, Spanish, Italian, Russian, Japanese, ChineseApplication SoftwareUltraMATE ®Lite and UltraMATE ®Base Instrument PackageCL5 precision thickness gauge Lithium poly battery pack AC power supply Plastic carry case Wire standXL couplant sample, 4 oz Firmware upgrade CD-ROM Operating manualOperating instruction card Certificate of ConformityOptionsCL5 AS OPT – Live A-Scan option CL5 DR OPT – Data Recorder option CL5 VL – Velocity option AccessoriesPCCBL-690 USB PC cable PCCBL-419 serial PC cableLi-135 lithium poly battery pack AC-296 AC power supplyUltraMATE ®Lite or UltraMATE ®Data Management softwareMeasuring Range.005 in to 20.00 in (0.13 mm to 500 mm): depends on material, probe, surface condition and temperatureUnits and Measuring Resolution Inch – 0.0001, 0.001, 0.01 Millimeter – 0.001, 0.01, 0.1Material Velocity Range 0.03937 to 0.78736 in/μs 1000 to 19999 m/s Receiver Bandwidth of 1.0 to 16 MHz at –6 dB Update Rate User selectable 4 or 8 Hz, up to 32 Hz in Min Cap orMax Cap modeDisplay Type Graphical LCD 64 × 128 pixels2.25 in × 2.56 in (40 mm × 57 mm) with backlight and adjustable contrastThickness Display Five-digit display with 0.75 in (19.5 mm) height digitsin standard mode and 0.25 in (6.35 mm) height digits in Thickness + A-Scan mode, solid or hollow digits coupling indicator, A-Scan view – R.F. mode onlyDisplay Modes Thickness (includes Snapshot A-Scan), Thickness +Live A-Scan (optional), Minimum Capture, Maximum Capture, Differential and Rate of Reduction, Velocity Mode (optional), Quality View Mode (optional)Supervisor Lockout Alphanumeric password lockout for calibrations,set-up and Data RecorderI/O Port Bi-directional serial RS-232: baud rate 1200, 9600,57600 and 115200Data Recorder Programmable Data Recorder, 120 files max. oneach 64 MB SD cardFile Formats Grid created from instrument keypad. Grid andCustom Linear files accepted from UltraMATE ®software.Power Supply Three AA batteries (Alkaline, NiMH or NiCad) orcustom rechargeable battery packTechnical Data。

漫谈微分几何、多复变函数与代数几何(Differential geometry, functions

漫谈微分几何、多复变函数与代数几何（Differential geometry, functions of complex variable and algebraic geometry）Differential geometry and tensor analysis, developed with the development of differential geometry, are the basic tools for mastering general relativity. Because general relativity's success, to always obscure differential geometry has become one of the central discipline of mathematics.Since the invention of differential calculus, the birth of differential geometry was born. But the work of Euler, Clairaut and Monge really made differential geometry an independent discipline. In the work of geodesy, Euler has gradually obtained important research, and obtained the famous Euler formula for the calculation of normal curvature. The Clairaut curve of the curvature and torsion, Monge published "analysis is applied to the geometry of the loose leaf paper", the important properties of curves and surfaces are represented by differential equations, which makes the development of classical differential geometry to reach a peak. Gauss in the study of geodesic, through complicated calculation, in 1827 found two main curvature surfaces and its product in the periphery of the Euclidean shape of the space not only depends on its first fundamental form, the result is Gauss proudly called the wonderful theorem, created from the intrinsic geometry. The free surface of space from the periphery, the surface itself as a space to study. In 1854, Riemann made the hypothesis about geometric foundation, and extended the intrinsic geometry of Gauss in 2 dimensional curved surface, thus developing n-dimensional Riemann geometry, with the development of complex functions. A group of excellentmathematicians extended the research objects of differential geometry to complex manifolds and extended them to the complex analytic space theory including singularities. Each step of differential geometry faces not only the deepening of knowledge, but also the continuous expansion of the field of knowledge. Here, differential geometry and complex functions, Lie group theory, algebraic geometry, and PDE all interact profoundly with one another. Mathematics is constantly dividing and blending with each other.By shining the charming glory and the differential geometric function theory of several complex variables, unit circle and the upper half plane (the two conformal mapping establishment) defined on Poincare metric, complex function theory and the differential geometric relationships can be seen distinctly. Poincare metric is conformal invariant. The famous Schwarz theorem can be explained as follows: the Poincare metric on the unit circle does not increase under analytic mapping; if and only if the mapping is a fractional linear transformation, the Poincare metric does not change Poincare. Applying the hyperbolic geometry of Poincare metric, we can easily prove the famous Picard theorem. The proof of Picard theorem to modular function theory is hard to use, if using the differential geometric point of view, can also be in a very simple way to prove. Differential geometry permeates deep into the theory of complex functions. In the theory of multiple complex functions, the curvature of the real differential geometry and other series of calculations are followed by the analysis of the region definition metric of the complex affine space. In complex situations, all of the singular discrete distribution, and in more complex situations, because of the famous Hartogsdevelopment phenomenon, all isolated singularities are engulfed by a continuous region even in singularity formation is often destroyed, only the formation of real codimension 1 manifold can avoid the bad luck. But even this situation requires other restrictions to ensure safety". The singular properties of singularities in the theory of functions of complex functions make them destined to be manifolds. In 1922, Bergman introduced the famous Bergman kernel function, the more complex function or Weyl said its era, in addition to the famous Hartogs, Poincare, Levi of Cousin and several predecessors almost no substantive progress, injected a dynamic Bergman work will undoubtedly give this dead area. In many complex function domains in the Bergman metric metric in the one-dimensional case is the unit circle and Poincare on the upper half plane of the Poincare, which doomed the importance of the work of Bergman.The basic object of algebraic geometry is the properties of the common zeros (algebraic families) of any dimension, affine space, or algebraic equations of a projective space (defined equations),The definitions of algebraic clusters, the coefficients of equations, and the domains in which the points of an algebraic cluster are located are called base domains. An irreducible algebraic variety is a finite sub extension of its base domain. In our numerical domain, the linear space is the extension of the base field in the number field, and the dimension of the linear space is the number of the expansion. From this point of view, algebraic geometry can be viewed as a study of finite extension fields. The properties of algebraic clusters areclosely related to their base domains. The algebraic domain of complex affine space or complex projective space, the research process is not only a large number of concepts and differential geometry and complex function theory and applied to a large number of coincidence, the similar tools in the process of research. Every step of the complex manifold and the complex analytic space has the same influence on these subjects. Many masters in related fields, although they seem to study only one field, have consequences for other areas. For example: the Lerey study of algebraic topology that it has little effect on layer, in algebraic topology, but because of Serre, Weil and H? Cartan (E? Cartan, eldest son) introduction, has a profound impact on algebraic geometry and complex function theory. Chern studies the categories of Hermite spaces, but it also affects algebraic geometry, differential geometry and complex functions. Hironaka studies the singular point resolution in algebraic geometry, but the modification of complex manifold to complex analytic space and blow up affect the theory of complex analytic space. Yau proves that the Calabi conjecture not only affects algebraic geometry and differential geometry, but also affects classical general relativity. At the same time, we can see the important position of nonlinear ordinary differential equations and partial differential equations in differential geometry. Cartan study of symmetric Riemann space, the classification theorem is important, given 1, 2 and 3 dimensional space of a Homogeneous Bounded Domain complete classification, prove that they are all homogeneous symmetric domains at the same time, he guessed: This is also true in the n-dimensional equivalent relation. In 1959, Piatetski-Shapiro has two counterexample and find the domain theory of automorphic function study in symmetry, in the 4 and 5dimensional cases each find a homogeneous bounded domain, which is not a homogeneous symmetric domain, the domain he named Siegel domain, to commemorate the profound work on Siegel in 1943 of automorphic function. The results of Piatetski-Shapiro has profound impact on the theory of complex variable functions and automorphic function theory, and have a profound impact on the symmetry space theory and a series of topics. As we know, Cartan transforms the study of symmetric spaces into the study of Lie groups and Lie algebras, which is directly influenced by Klein and greatly develops the initial idea of Klein. Then it is Cartan developed the concept of Levi-Civita connection, the development of differential geometry in general contact theory, isomorphic mapping through tangent space at each point on the manifold, realize the dream of Klein and greatly promote the development of differential geometry. Cartan is the same, and concluded that the importance of the research in the holonomy manifold twists and turns, finally after his death in thirty years has proved to be correct. Here, we see the vast beauty of differential geometry.As we know, geodesic ties are associated with ODE (ordinary differential equations), minimal surfaces and high dimensional submanifolds are associated with PDE (partial differential equations). These equations are nonlinear equations, so they have high requirements for analysis. Complex PDE and complex analysis the relationship between Cauchy-Riemann equations coupling the famous function theory, in the complex case, the Cauchy- Riemann equations not only deepen the unprecedented contact and the qualitative super Cauchy-Riemann equations (the number of variables is greater than the number of equations) led to a strange phenomenon. This makes PDE and the theory ofmultiple complex functions closely integrated with differential geometry.Most of the scholars have been studying the differential geometry of the intrinsic geometry of the Gauss and Riemann extremely deep stun, by Cartan's method of moving frames is beautiful and concise dumping, by Chern's theory of characteristic classes of the broad and profound admiration, Yau deep exquisite geometric analysis skills to deter.When the young Chern faced the whole differentiation, he said he was like a mountain facing the shining golden light, but he couldn't reach the summit at one time. But then he was cast as a master in this field before Hopf and Weil.If the differential geometry Cartan development to gradually change the general relativistic geometric model, then the differential geometry of Chern et al not only affect the continuation of Cartan and to promote the development of fiber bundle in the form of gauge field theory. Differential geometry is still closely bound up with physics as in the age of Einstein and continues to acquire research topics from physicsWhy does the three-dimensional sphere not give flatness gauge, but can give conformal flatness gauge? Because 3D balls and other dimension as the ball to establish flat space isometric mapping, so it is impossible to establish a flatness gauge; and n-dimensional balls are usually single curvature space, thus can establish a conformal flat metric. In differential geometry, isometry means that the distance between the points on the manifold before and after the mapping remains the same. Whena manifold is equidistant from a flat space, the curvature of its Riemann cross section is always zero. Since the curvature of all spheres is positive constant, the n-dimensional sphere and other manifolds whose sectional curvature is nonzero can not be assigned to local flatness gauge.But there are locally conformally flat manifolds for this concept, two gauge G and G, if G=exp{is called G, P}? G between a and G transform is a conformal transformation. Weyl conformal curvature tensor remains unchanged under conformal transformation. It is a tensor field of (1,3) type on a manifold. When the Weyl conformal curvature tensor is zero, the curvature tensor of the manifold can be represented by the Ricci curvature tensor and the scalar curvature, so Penrose always emphasizes the curvature =Ricci+Weyl.The metric tensor g of an n-dimensional Riemann manifold is conformally equivalent to the flatness gauge locally, and is called conformally flat manifold. All Manifolds (constant curvature manifolds) whose curvature is constant are conformally flat, so they can be given conformal conformal metric. And all dimensions of the sphere (including thethree-dimensional sphere) are manifold of constant curvature, so they must be given conformal conformal metric. Conversely, conformally flat manifolds are not necessarily manifolds of constant curvature. But a wonderful result related to Einstein manifolds can make up for this regret: conformally conformally Einstein manifolds over 3 dimensions must be manifolds of constant curvature. That is to say, if we want conformally conformally flat manifolds to be manifolds of constant curvature, we must call Ric= lambda g, and this is thedefinition of Einstein manifolds. In the formula, Ric is the Ricci curvature tensor, G is the metric tensor, and lambda is constant. The scalar curvature S=m of Einstein manifolds is constant. Moreover, if S is nonzero, there is no nonzero parallel tangent vector field over it. Einstein introduction of the cosmological constant. So he missed the great achievements that the expansion of the universe, so Hubble is successful in the official career; but the vacuum gravitational field equation of cosmological term with had a Einstein manifold, which provides a new stage for mathematicians wit.For the 3 dimensional connected Einstein manifold, even if does not require the conformal flat, it is also the automatic constant curvature manifolds, other dimensions do not set up this wonderful nature, I only know that this is the tensor analysis summer learning, the feeling is a kind of enjoyment. The sectional curvature in the real manifold is different from the curvature of the Holomorphic cross section in the Kahler manifold, and thus produces different results. If the curvature of holomorphic section is constant, the Ricci curvature of the manifold must be constant, so it must be Einstein manifold, called Kahler- Einstein manifold, Kahler. Kahler manifolds are Kahler- Einstein manifolds, if and only if they are Riemann manifolds, Einstein manifolds. N dimensional complex vector space, complex projective space, complex torus and complex hyperbolic space are Kahler- and Einstein manifolds. The study of Kahler-Einstein manifolds becomes the intellectual enjoyment of geometer.Let's go back to an important result of isometric mapping.In this paper, we consider the isometric mapping between M and N and the mapping of the cut space between the two Riemann manifolds, take P at any point on M, and select two non tangent tangent vectors in its tangent space, and obtain its sectional curvature. In the mapping, the two tangent vectors on the P point and its tangent space are transformed into two other tangent vectors under the mapping, and the sectional curvature of the vector is also obtained. If the mapping is isometric mapping, then the curvature of the two cross sections is equal. Or, to be vague, isometric mapping does not change the curvature of the section.Conversely, if the arbitrary points are set, the curvature of the section does not change in nature, then the mapping is not isometric mapping The answer was No. Even in thethree-dimensional Euclidean space on the surface can not set up this property. In some cases, the limit of the geodesic line must be added, and the properties of the Jacobi field can be used to do so. This is the famous Cartan isometry theorem. This theorem is a wonderful application of the Jacobi field. Its wide range of promotion is made by Ambrose and Hicks, known as the Cartan-Ambrose-Hicks theorem.Differential geometry is full of infinite charm. We classify pseudo-Riemannian spaces by using Weyl conformal curvature tensor, which can be classified by Ricci curvature tensor, or classified into 9 types by Bianchi. And these things are all can be attributed to the study of differential geometry, this distant view Riemann and slightly closer to the Klein point of the perfect combination, it can be seen that the great wisdom Cartan, here you can see the profound influence of Einstein.From the Hermite symmetry space to the Kahler-Hodge manifold, differential geometry is not only closely linked with the Lie group, but also connected with algebra, geometry and topologyThink of the great 1895 Poicare wrote the great "position analysis" was founded combination topology unabashedly said differential geometry in high dimensional space is of little importance to this subject, he said: "the home has beautiful scenery, where Xuyuan for." (Chern) topology is the beauty of the home. Why do you have to work hard to compute the curvature of surfaces or even manifolds of high dimensions? But this versatile mathematician is wrong, but we can not say that the mathematical genius no major contribution to differential geometry? Can not. Let's see today's close relation between differential geometry and topology, we'll see. When is a closed form the proper form? The inverse of the Poicare lemma in the region of the homotopy point (the single connected region) tells us that it is automatically established. In the non simply connected region is de famous Rham theorem tells us how to set up, that is the integral differential form in all closed on zero.Even in the field of differential geometry ignored by Poicare, he is still in a casual way deeply affected by the subject, or rather is affecting the whole mathematics.The nature of any discipline that seeks to be generalized after its creation, as is differential geometry. From the curvature, Euclidean curvature of space straight to zero, geometry extended to normal curvature number (narrow Riemann space) andnegative constant space (Lobachevskii space), we know that the greatness of non Euclidean geometry is that it not only independent of the fifth postulate and other alternative to the new geometry. It can be the founder of triangle analysis on it. But the famous mathematician Milnor said that before differential geometry went into non Euclidean geometry, non Euclidean geometry was only the torso with no hands and no feet. The non Euclidean geometry is born only when the curvature is computed uniformly after the metric is defined. In his speech in 1854, Riemann wrote only one formula: that is, this formula unifies the positive curvature, negative curvature and zero curvature geometry. Most people think that the formula for "Riemann" is based on intuition. In fact, later people found the draft paper that he used to calculate the formula. Only then did he realize that talent should be diligent. Riemann has explored the curvature of manifolds of arbitrary curvature of any dimension, but the quantitative calculations go beyond the mathematical tools of that time, and he can only write the unified formula for manifolds of constant curvature. But we know,Even today, this result is still important, differential geometry "comparison theorem" a multitude of names are in constant curvature manifolds for comparison model.When Riemann had considered two differential forms the root of two, this is what we are familiar with the Riemann metric Riemannnian, derived from geometry, he specifically mentioned another case, is the root of four four differential forms (equivalent to four yuan product and four times square). This is the contact and the difference between the two. But he saidthat for this situation and the previous case, the study does not require substantially different methods. It also says that such studies are time consuming and that new insights cannot be added to space, and the results of calculations lack geometric meaning. So Riemann studied only what is now called Riemann metric. Why are future generations of Finsler interested in promoting the Riemann's not wanting to study? It may be that mathematicians are so good that they become a hobby. Cartan in Finsler geometry made efforts, but the effect was little, Chern on the geometric really high hopes also developed some achievements. But I still and general view on the international consensus, that is the Finsler geometry bleak. This is also the essential reason of Finsler geometry has been unable to enter the mainstream of differential geometry, it no beautiful properties really worth geometers to struggle, also do not have what big application value. Later K- exhibition space, Cartan space will not become mainstream, although they are the extension of Riemannnian geometry, but did not get what the big development.In fact, sometimes the promotion of things to get new content is not much, differential geometry is the same, not the object of study, the more ordinary the better, but should be appropriate to the special good. For example, in Riemann manifold, homogeneous Riemann manifold is more special, beautiful nature, homogeneous Riemann manifolds, symmetric Riemann manifold is more special, so nature more beautiful. This is from the analysis of manifold Lie group action angle.From the point of view of metric, the complex structure is given on the even dimensional Riemann manifold, and the complexmanifold is very elegant. Near complex manifolds are complex manifolds only when the near complex structure is integrable. The complex manifold must be orientable, because it is easy to find that its Jacobian must be nonnegative, whereas the real manifold does not have this property in general. To narrow the scope of the Kahler manifold has more good properties, all complex Submanifolds of Kahler manifolds are Kahler manifolds, and minimal submanifolds (Wirtinger theorem), the beautiful results captured the hearts of many differential geometry and algebraic geometry, because other more general manifolds do not set up this beautiful results. If the first Chern number of a three-dimensional Kahler manifold is zero, the Calabi-Yau manifold can be obtained, which is a very interesting manifold for theoretical physicists. The manifold of mirrors of Calabi-Yau manifolds is also a common subject of differential geometry in algebraic geometry. The popular Hodge structure is a subject of endless appeal.Differential geometry, an endless topic. Just as algebraic geometry requires double - rational equivalence as a luxury, differential geometry requires isometric transformations to be difficult. Taxonomy is an eternal subject of mathematics. In group theory, there are single group classification, multi complex function theory, regional classification, algebraic geometry in the classification of algebraic clusters, differential geometry is also classified.The hard question has led to a dash of young geometry and old scholars, and the prospect of differential geometry is very bright.。

机械设计专业术语的英语翻译

如有你有帮助，请购买下载，谢谢！阿基米德蜗杆 Archimedes worm安全系数 safety factor; factor of safety安全载荷 safe load凹面、凹度 concavity扳手 wrench板簧 flat leaf spring半圆键 woodruff key变形 deformation摆杆 oscillating bar摆动从动件 oscillating follower摆动从动件凸轮机构 cam with oscillating follower 摆动导杆机构 oscillating guide-bar mechanism摆线齿轮 cycloidal gear摆线齿形 cycloidal tooth profile摆线运动规律 cycloidal motion摆线针轮 cycloidal-pin wheel包角 angle of contact保持架 cage背对背安装 back-to-back arrangement背锥 back cone ；normal cone背锥角 back angle背锥距 back cone distance比例尺 scale比热容 specific heat capacity闭式链 closed kinematic chain闭链机构 closed chain mechanism臂部 arm变频器 frequency converters变频调速 frequency control of motor speed变速 speed change变速齿轮 change gear ; change wheel变位齿轮 modified gear变位系数 modification coefficient标准齿轮 standard gear标准直齿轮 standard spur gear表面质量系数 superficial mass factor表面传热系数 surface coefficient of heat transfer 表面粗糙度 surface roughness并联式组合 combination in parallel并联机构 parallel mechanism并联组合机构 parallel combined mechanism并行工程 concurrent engineering并行设计 concurred design, CD不平衡相位 phase angle of unbalance不平衡 imbalance (or unbalance)不平衡量 amount of unbalance 不完全齿轮机构 intermittent gearing波发生器 wave generator波数 number of waves补偿 compensation参数化设计 parameterization design, PD残余应力 residual stress操纵及控制装置 operation control device槽轮 Geneva wheel槽轮机构 Geneva mechanism ；Maltese cross槽数 Geneva numerate槽凸轮 groove cam侧隙 backlash差动轮系 differential gear train差动螺旋机构 differential screw mechanism差速器 differential常用机构 conventional mechanism; mechanism in common use 车床 lathe承载量系数 bearing capacity factor承载能力 bearing capacity成对安装 paired mounting尺寸系列 dimension series齿槽 tooth space齿槽宽 spacewidth齿侧间隙 backlash齿顶高 addendum齿顶圆 addendum circle齿根高 dedendum齿根圆 dedendum circle齿厚 tooth thickness齿距 circular pitch齿宽 face width齿廓 tooth profile齿廓曲线 tooth curve齿轮 gear齿轮变速箱 speed-changing gear boxes齿轮齿条机构 pinion and rack齿轮插刀 pinion cutter; pinion-shaped shaper cutter齿轮滚刀 hob ,hobbing cutter齿轮机构 gear齿轮轮坯 blank齿轮传动系 pinion unit齿轮联轴器 gear coupling齿条传动 rack gear齿数 tooth number齿数比 gear ratio齿条 rack如有你有帮助，请购买下载，谢谢！齿条插刀 rack cutter; rack-shaped shaper cutter齿形链、无声链 silent chain齿形系数 form factor齿式棘轮机构 tooth ratchet mechanism插齿机 gear shaper重合点 coincident points重合度 contact ratio冲床 punch传动比 transmission ratio, speed ratio传动装置 gearing; transmission gear传动系统 driven system传动角 transmission angle传动轴 transmission shaft串联式组合 combination in series串联式组合机构 series combined mechanism串级调速 cascade speed control创新 innovation ; creation创新设计 creation design垂直载荷、法向载荷 normal load唇形橡胶密封 lip rubber seal磁流体轴承 magnetic fluid bearing从动带轮 driven pulley从动件 driven link, follower从动件平底宽度 width of flat-face从动件停歇 follower dwell从动件运动规律 follower motion从动轮 driven gear粗线 bold line粗牙螺纹 coarse thread大齿轮 gear wheel打包机 packer打滑 slipping带传动 belt driving带轮 belt pulley带式制动器 band brake单列轴承 single row bearing单向推力轴承 single-direction thrust bearing单万向联轴节 single universal joint单位矢量 unit vector当量齿轮 equivalent spur gear; virtual gear当量齿数 equivalent teeth number; virtual number of teeth 当量摩擦系数 equivalent coefficient of friction当量载荷 equivalent load刀具 cutter导数 derivative倒角 chamfer 导热性 conduction of heat导程 lead导程角 lead angle等加等减速运动规律 parabolic motion; constant acceleration and deceleration motion等速运动规律 uniform motion; constant velocity motion等径凸轮 conjugate yoke radial cam等宽凸轮 constant-breadth cam等效构件 equivalent link等效力 equivalent force等效力矩 equivalent moment of force等效量 equivalent等效质量 equivalent mass等效转动惯量 equivalent moment of inertia等效动力学模型 dynamically equivalent model底座 chassis低副 lower pair点划线 chain dotted line（疲劳）点蚀 pitting垫圈 gasket垫片密封 gasket seal碟形弹簧 belleville spring顶隙 bottom clearance定轴轮系 ordinary gear train; gear train with fixed axes动力学 dynamics动密封 kinematical seal动能 dynamic energy动力粘度 dynamic viscosity动力润滑 dynamic lubrication动平衡 dynamic balance动平衡机 dynamic balancing machine动态特性 dynamic characteristics动态分析设计 dynamic analysis design动压力 dynamic reaction动载荷 dynamic load端面 transverse plane端面参数 transverse parameters端面齿距 transverse circular pitch端面齿廓 transverse tooth profile端面重合度 transverse contact ratio端面模数 transverse module端面压力角 transverse pressure angle锻造 forge对称循环应力 symmetry circulating stress对心滚子从动件 radial (or in-line ) roller follower对心直动从动件 radial (or in-line ) translating follower如有你有帮助，请购买下载，谢谢！对心移动从动件 radial reciprocating follower对心曲柄滑块机构 in-line slider-crank (or crank-slider) mechanism 多列轴承 multi-row bearing多楔带 poly V-belt多项式运动规律 polynomial motion多质量转子 rotor with several masses惰轮 idle gear额定寿命 rating life额定载荷 load ratingII 级杆组 dyad发生线 generating line发生面 generating plane法面 normal plane法面参数 normal parameters法面齿距 normal circular pitch法面模数 normal module法面压力角 normal pressure angle法向齿距 normal pitch法向齿廓 normal tooth profile法向直廓蜗杆 straight sided normal worm法向力 normal force反馈式组合 feedback combining反向运动学 inverse ( or backward) kinematics反转法 kinematic inversion反正切 Arctan范成法 generating cutting仿形法 form cutting方案设计、概念设计 concept design, CD防振装置 shockproof device飞轮 flywheel飞轮矩 moment of flywheel非标准齿轮 nonstandard gear非接触式密封 non-contact seal非周期性速度波动 aperiodic speed fluctuation非圆齿轮 non-circular gear粉末合金 powder metallurgy分度线 reference line; standard pitch line分度圆 reference circle; standard (cutting) pitch circle分度圆柱导程角 lead angle at reference cylinder分度圆柱螺旋角 helix angle at reference cylinder分母 denominator分子 numerator分度圆锥 reference cone; standard pitch cone分析法 analytical method封闭差动轮系 planetary differential复合铰链 compound hinge 复合式组合 compound combining复合轮系 compound (or combined) gear train 复合平带 compound flat belt复合应力 combined stress复式螺旋机构 Compound screw mechanism 复杂机构 complex mechanism杆组 Assur group干涉 interference刚度系数 stiffness coefficient刚轮 rigid circular spline钢丝软轴 wire soft shaft刚体导引机构 body guidance mechanism刚性冲击 rigid impulse (shock)刚性转子 rigid rotor刚性轴承 rigid bearing刚性联轴器 rigid coupling高度系列 height series高速带 high speed belt高副 higher pair格拉晓夫定理 Grashoff`s law根切 undercutting公称直径 nominal diameter高度系列 height series功 work工况系数 application factor工艺设计 technological design工作循环图 working cycle diagram工作机构 operation mechanism工作载荷 external loads工作空间 working space工作应力 working stress工作阻力 effective resistance工作阻力矩 effective resistance moment公法线 common normal line公共约束 general constraint公制齿轮 metric gears功率 power功能分析设计 function analyses design共轭齿廓 conjugate profiles共轭凸轮 conjugate cam构件 link鼓风机 blower固定构件 fixed link; frame固体润滑剂 solid lubricant关节型操作器 jointed manipulator惯性力 inertia force如有你有帮助，请购买下载，谢谢！惯性力矩 moment of inertia ,shaking moment惯性力平衡 balance of shaking force惯性力完全平衡 full balance of shaking force惯性力部分平衡 partial balance of shaking force 惯性主矩 resultant moment of inertia惯性主失 resultant vector of inertia冠轮 crown gear广义机构 generation mechanism广义坐标 generalized coordinate轨迹生成 path generation轨迹发生器 path generator滚刀 hob滚道 raceway滚动体 rolling element滚动轴承 rolling bearing滚动轴承代号 rolling bearing identification code 滚针 needle roller滚针轴承 needle roller bearing滚子 roller滚子轴承 roller bearing滚子半径 radius of roller滚子从动件 roller follower滚子链 roller chain滚子链联轴器 double roller chain coupling滚珠丝杆 ball screw滚柱式单向超越离合器 roller clutch过度切割 undercutting函数发生器 function generator函数生成 function generation含油轴承 oil bearing耗油量 oil consumption耗油量系数 oil consumption factor赫兹公式 H. Hertz equation合成弯矩 resultant bending moment合力 resultant force合力矩 resultant moment of force黑箱 black box横坐标 abscissa互换性齿轮 interchangeable gears花键 spline滑键、导键 feather key滑动轴承 sliding bearing滑动率 sliding ratio滑块 slider环面蜗杆 toroid helicoids worm环形弹簧 annular spring 缓冲装置 shocks; shock-absorber灰铸铁 grey cast iron回程 return回转体平衡 balance of rotors混合轮系 compound gear train积分 integrate机电一体化系统设计 mechanical-electrical integration system design 机构 mechanism机构分析 analysis of mechanism机构平衡 balance of mechanism机构学 mechanism机构运动设计 kinematic design of mechanism机构运动简图 kinematic sketch of mechanism机构综合 synthesis of mechanism机构组成 constitution of mechanism机架 frame, fixed link机架变换 kinematic inversion机器 machine机器人 robot机器人操作器 manipulator机器人学 robotics技术过程 technique process技术经济评价 technical and economic evaluation技术系统 technique system机械 machinery机械创新设计 mechanical creation design, MCD机械系统设计 mechanical system design, MSD机械动力分析 dynamic analysis of machinery机械动力设计 dynamic design of machinery机械动力学 dynamics of machinery机械的现代设计 modern machine design机械系统 mechanical system机械利益 mechanical advantage机械平衡 balance of machinery机械手 manipulator机械设计 machine design; mechanical design机械特性 mechanical behavior机械调速 mechanical speed governors机械效率 mechanical efficiency机械原理 theory of machines and mechanisms机械运转不均匀系数 coefficient of speed fluctuation机械无级变速 mechanical stepless speed changes基础机构 fundamental mechanism基本额定寿命 basic rating life基于实例设计 case-based design,CBD基圆 base circle如有你有帮助，请购买下载，谢谢！基圆半径 radius of base circle基圆齿距 base pitch基圆压力角 pressure angle of base circle基圆柱 base cylinder基圆锥 base cone急回机构 quick-return mechanism急回特性 quick-return characteristics急回系数 advance-to return-time ratio急回运动 quick-return motion棘轮 ratchet棘轮机构 ratchet mechanism棘爪 pawl极限位置 extreme (or limiting) position极位夹角 crank angle between extreme (or limiting) positions计算机辅助设计 computer aided design, CAD计算机辅助制造 computer aided manufacturing, CAM计算机集成制造系统 computer integrated manufacturing system, CIMS计算力矩 factored moment; calculation moment计算弯矩 calculated bending moment加权系数 weighting efficient加速度 acceleration加速度分析 acceleration analysis加速度曲线 acceleration diagram尖点 pointing; cusp尖底从动件 knife-edge follower间隙 backlash间歇运动机构 intermittent motion mechanism减速比 reduction ratio减速齿轮、减速装置 reduction gear减速器 speed reducer减摩性 anti-friction quality渐开螺旋面 involute helicoid渐开线 involute渐开线齿廓 involute profile渐开线齿轮 involute gear渐开线发生线 generating line of involute渐开线方程 involute equation渐开线函数 involute function渐开线蜗杆 involute worm渐开线压力角 pressure angle of involute渐开线花键 involute spline简谐运动 simple harmonic motion键 key键槽 keyway交变应力 repeated stress 交变载荷 repeated fluctuating load交叉带传动 cross-belt drive交错轴斜齿轮 crossed helical gears胶合 scoring角加速度 angular acceleration角速度 angular velocity角速比 angular velocity ratio角接触球轴承 angular contact ball bearing角接触推力轴承 angular contact thrust bearing角接触向心轴承 angular contact radial bearing角接触轴承 angular contact bearing铰链、枢纽 hinge校正平面 correcting plane接触应力 contact stress接触式密封 contact seal阶梯轴 multi-diameter shaft结构 structure结构设计 structural design截面 section节点 pitch point节距 circular pitch; pitch of teeth节线 pitch line节圆 pitch circle节圆齿厚 thickness on pitch circle节圆直径 pitch diameter节圆锥 pitch cone节圆锥角 pitch cone angle解析设计 analytical design紧边 tight-side紧固件 fastener径节 diametral pitch径向 radial direction径向当量动载荷 dynamic equivalent radial load径向当量静载荷 static equivalent radial load径向基本额定动载荷 basic dynamic radial load rating 径向基本额定静载荷 basic static radial load tating径向接触轴承 radial contact bearing径向平面 radial plane径向游隙 radial internal clearance径向载荷 radial load径向载荷系数 radial load factor径向间隙 clearance静力 static force静平衡 static balance静载荷 static load静密封 static seal如有你有帮助，请购买下载，谢谢！局部自由度 passive degree of freedom矩阵 matrix矩形螺纹 square threaded form锯齿形螺纹 buttress thread form矩形牙嵌式离合器 square-jaw positive-contact clutch绝对尺寸系数 absolute dimensional factor绝对运动 absolute motion绝对速度 absolute velocity均衡装置 load balancing mechanism抗压强度 compression strength开口传动 open-belt drive开式链 open kinematic chain开链机构 open chain mechanism可靠度 degree of reliability可靠性 reliability可靠性设计 reliability design, RD空气弹簧 air spring空间机构 spatial mechanism空间连杆机构 spatial linkage空间凸轮机构 spatial cam空间运动副 spatial kinematic pair空间运动链 spatial kinematic chain空转 idle宽度系列 width series框图 block diagram雷诺方程Reynolds‘s equation离心力 centrifugal force离心应力 centrifugal stress离合器 clutch离心密封 centrifugal seal理论廓线 pitch curve理论啮合线 theoretical line of action隶属度 membership力 force力多边形 force polygon力封闭型凸轮机构 force-drive (or force-closed) cam mechanism 力矩 moment力平衡 equilibrium力偶 couple力偶矩 moment of couple连杆 connecting rod, coupler连杆机构 linkage连杆曲线 coupler-curve连心线 line of centers链 chain链传动装置 chain gearing 链轮 sprocket ; sprocket-wheel ; sprocket gear ; chain wheel 联组V 带 tight-up V belt联轴器 coupling ; shaft coupling两维凸轮 two-dimensional cam临界转速 critical speed六杆机构 six-bar linkage龙门刨床 double Haas planer轮坯 blank轮系 gear train螺杆 screw螺距 thread pitch螺母 screw nut螺旋锥齿轮 helical bevel gear螺钉 screws螺栓 bolts螺纹导程 lead螺纹效率 screw efficiency螺旋传动 power screw螺旋密封 spiral seal螺纹 thread (of a screw)螺旋副 helical pair螺旋机构 screw mechanism螺旋角 helix angle螺旋线 helix ,helical line绿色设计 green design ; design for environment马耳他机构 Geneva wheel ; Geneva gear马耳他十字 Maltese cross脉动无级变速 pulsating stepless speed changes脉动循环应力 fluctuating circulating stress脉动载荷 fluctuating load铆钉 rivet迷宫密封 labyrinth seal密封 seal密封带 seal belt密封胶 seal gum密封元件 potted component密封装置 sealing arrangement面对面安装 face-to-face arrangement面向产品生命周期设计 design for product`s life cycle, DPLC 名义应力、公称应力 nominal stress模块化设计 modular design, MD模块式传动系统 modular system模幅箱 morphology box模糊集 fuzzy set模糊评价 fuzzy evaluation模数 module如有你有帮助，请购买下载，谢谢！摩擦 friction摩擦角 friction angle摩擦力 friction force摩擦学设计 tribology design, TD摩擦阻力 frictional resistance摩擦力矩 friction moment摩擦系数 coefficient of friction摩擦圆 friction circle磨损 abrasion ;wear; scratching末端执行器 end-effector目标函数 objective function耐腐蚀性 corrosion resistance耐磨性 wear resistance挠性机构 mechanism with flexible elements 挠性转子 flexible rotor内齿轮 internal gear内齿圈 ring gear内力 internal force内圈 inner ring能量 energy能量指示图 viscosity逆时针 counterclockwise (or anticlockwise) 啮出 engaging-out啮合 engagement, mesh, gearing啮合点 contact points啮合角 working pressure angle啮合线 line of action啮合线长度 length of line of action啮入 engaging-in牛头刨床 shaper凝固点 freezing point; solidifying point扭转应力 torsion stress扭矩 moment of torque扭簧 helical torsion spring诺模图 NomogramO 形密封圈密封 O ring seal盘形凸轮 disk cam盘形转子 disk-like rotor抛物线运动 parabolic motion疲劳极限 fatigue limit疲劳强度 fatigue strength偏置式 offset偏( 心) 距 offset distance偏心率 eccentricity ratio偏心质量 eccentric mass偏距圆 offset circle 偏心盘 eccentric偏置滚子从动件 offset roller follower偏置尖底从动件 offset knife-edge follower偏置曲柄滑块机构 offset slider-crank mechanism拼接 matching评价与决策 evaluation and decision频率 frequency平带 flat belt平带传动 flat belt driving平底从动件 flat-face follower平底宽度 face width平分线 bisector平均应力 average stress平均中径 mean screw diameter平均速度 average velocity平衡 balance平衡机 balancing machine平衡品质 balancing quality平衡平面 correcting plane平衡质量 balancing mass平衡重 counterweight平衡转速 balancing speed平面副 planar pair, flat pair平面机构 planar mechanism平面运动副 planar kinematic pair平面连杆机构 planar linkage平面凸轮 planar cam平面凸轮机构 planar cam mechanism平面轴斜齿轮 parallel helical gears普通平键 parallel key其他常用机构 other mechanism in common use起动阶段 starting period启动力矩 starting torque气动机构 pneumatic mechanism奇异位置 singular position起始啮合点 initial contact , beginning of contact气体轴承 gas bearing千斤顶 jack嵌入键 sunk key强迫振动 forced vibration切齿深度 depth of cut曲柄 crank曲柄存在条件 Grashoff`s law曲柄导杆机构 crank shaper (guide-bar) mechanism曲柄滑块机构 slider-crank (or crank-slider) mechanism 曲柄摇杆机构 crank-rocker mechanism如有你有帮助，请购买下载，谢谢！曲齿锥齿轮 spiral bevel gear曲率 curvature曲率半径 radius of curvature曲面从动件 curved-shoe follower曲线拼接 curve matching曲线运动 curvilinear motion曲轴 crank shaft驱动力 driving force驱动力矩 driving moment (torque)全齿高 whole depth权重集 weight sets球 ball球面滚子 convex roller球轴承 ball bearing球面副 spheric pair球面渐开线 spherical involute球面运动 spherical motion球销副 sphere-pin pair球坐标操作器 polar coordinate manipulator燃点 spontaneous ignition热平衡 heat balance; thermal equilibrium人字齿轮 herringbone gear冗余自由度 redundant degree of freedom柔轮 flexspline柔性冲击 flexible impulse; soft shock柔性制造系统 flexible manufacturing system; FMS柔性自动化 flexible automation润滑油膜 lubricant film润滑装置 lubrication device润滑 lubrication润滑剂 lubricant三角形花键 serration spline三角形螺纹 V thread screw三维凸轮 three-dimensional cam三心定理 Kennedy`s theorem砂轮越程槽 grinding wheel groove砂漏 hour-glass少齿差行星传动 planetary drive with small teeth difference 设计方法学 design methodology设计变量 design variable设计约束 design constraints深沟球轴承 deep groove ball bearing生产阻力 productive resistance升程 rise升距 lift实际廓线 cam profile 十字滑块联轴器double slider coupling; Oldham‘s coupling 矢量 vector输出功 output work输出构件 output link输出机构 output mechanism输出力矩 output torque输出轴 output shaft输入构件 input link数学模型 mathematic model实际啮合线 actual line of action双滑块机构 double-slider mechanism, ellipsograph双曲柄机构 double crank mechanism双曲面齿轮 hyperboloid gear双头螺柱 studs双万向联轴节 constant-velocity (or double) universal joint 双摇杆机构 double rocker mechanism双转块机构 Oldham coupling双列轴承 double row bearing双向推力轴承 double-direction thrust bearing松边 slack-side顺时针 clockwise瞬心 instantaneous center死点 dead point四杆机构 four-bar linkage速度 velocity速度不均匀( 波动) 系数 coefficient of speed fluctuation 速度波动 speed fluctuation速度曲线 velocity diagram速度瞬心 instantaneous center of velocity塔轮 step pulley踏板 pedal台钳、虎钳 vice太阳轮 sun gear弹性滑动 elasticity sliding motion弹性联轴器 elastic coupling ; flexible coupling弹性套柱销联轴器 rubber-cushioned sleeve bearing coupling 套筒 sleeve梯形螺纹 acme thread form特殊运动链 special kinematic chain特性 characteristics替代机构 equivalent mechanism调节 modulation, regulation调心滚子轴承 self-aligning roller bearing调心球轴承 self-aligning ball bearing调心轴承 self-aligning bearing调速 speed governing如有你有帮助，请购买下载，谢谢！调速电动机 adjustable speed motors调速系统 speed control system调压调速 variable voltage control调速器 regulator, governor铁磁流体密封 ferrofluid seal停车阶段 stopping phase停歇 dwell同步带 synchronous belt同步带传动 synchronous belt drive凸的，凸面体 convex凸轮 cam凸轮倒置机构 inverse cam mechanism凸轮机构 cam , cam mechanism凸轮廓线 cam profile凸轮廓线绘制 layout of cam profile凸轮理论廓线 pitch curve凸缘联轴器 flange coupling图册、图谱 atlas图解法 graphical method推程 rise推力球轴承 thrust ball bearing推力轴承 thrust bearing退刀槽 tool withdrawal groove退火 anneal陀螺仪 gyroscopeV 带 V belt外力 external force外圈 outer ring外形尺寸 boundary dimension万向联轴器 Hooks coupling ; universal coupling 外齿轮 external gear弯曲应力 beading stress弯矩 bending moment腕部 wrist往复移动 reciprocating motion往复式密封 reciprocating seal网上设计 on-net design, OND微动螺旋机构 differential screw mechanism位移 displacement位移曲线 displacement diagram位姿 pose , position and orientation稳定运转阶段 steady motion period稳健设计 robust design蜗杆 worm蜗杆传动机构 worm gearing蜗杆头数 number of threads 蜗杆直径系数 diametral quotient蜗杆蜗轮机构 worm and worm gear蜗杆形凸轮步进机构 worm cam interval mechanism蜗杆旋向 hands of worm蜗轮 worm gear涡圈形盘簧 power spring无级变速装置 stepless speed changes devices无穷大 infinite系杆 crank arm, planet carrier现场平衡 field balancing向心轴承 radial bearing向心力 centrifugal force相对速度 relative velocity相对运动 relative motion相对间隙 relative gap象限 quadrant橡皮泥 plasticine细牙螺纹 fine threads销 pin消耗 consumption小齿轮 pinion小径 minor diameter橡胶弹簧 balata spring修正梯形加速度运动规律 modified trapezoidal acceleration motion 修正正弦加速度运动规律 modified sine acceleration motion斜齿圆柱齿轮 helical gear斜键、钩头楔键 taper key泄漏 leakage谐波齿轮 harmonic gear谐波传动 harmonic driving谐波发生器 harmonic generator斜齿轮的当量直齿轮 equivalent spur gear of the helical gear心轴 spindle行程速度变化系数 coefficient of travel speed variation行程速比系数 advance-to return-time ratio行星齿轮装置 planetary transmission行星轮 planet gear行星轮变速装置 planetary speed changing devices行星轮系 planetary gear train形封闭凸轮机构 positive-drive (or form-closed) cam mechanism虚拟现实 virtual reality虚拟现实技术 virtual reality technology, VRT虚拟现实设计 virtual reality design, VRD虚约束 redundant (or passive) constraint许用不平衡量 allowable amount of unbalance许用压力角 allowable pressure angle如有你有帮助，请购买下载，谢谢！许用应力 allowable stress; permissible stress悬臂结构 cantilever structure悬臂梁 cantilever beam循环功率流 circulating power load旋转力矩 running torque旋转式密封 rotating seal旋转运动 rotary motion选型 type selection压力 pressure压力中心 center of pressure压缩机 compressor压应力 compressive stress压力角 pressure angle牙嵌式联轴器 jaw (teeth) positive-contact coupling雅可比矩阵 Jacobi matrix摇杆 rocker液力传动 hydrodynamic drive液力耦合器 hydraulic couplers液体弹簧 liquid spring液压无级变速 hydraulic stepless speed changes液压机构 hydraulic mechanism一般化运动链 generalized kinematic chain移动从动件 reciprocating follower移动副 prismatic pair, sliding pair移动关节 prismatic joint移动凸轮 wedge cam盈亏功 increment or decrement work应力幅 stress amplitude应力集中 stress concentration应力集中系数 factor of stress concentration应力图 stress diagram应力—应变图 stress-strain diagram优化设计 optimal design油杯 oil bottle油壶 oil can油沟密封 oily ditch seal有害阻力 useless resistance有益阻力 useful resistance有效拉力 effective tension有效圆周力 effective circle force有害阻力 detrimental resistance余弦加速度运动 cosine acceleration (or simple harmonic) motion 预紧力 preload原动机 primer mover圆带 round belt圆带传动 round belt drive 圆弧齿厚 circular thickness圆弧圆柱蜗杆 hollow flank worm圆角半径 fillet radius圆盘摩擦离合器 disc friction clutch圆盘制动器 disc brake原动机 prime mover原始机构 original mechanism圆形齿轮 circular gear圆柱滚子 cylindrical roller圆柱滚子轴承 cylindrical roller bearing圆柱副 cylindric pair圆柱式凸轮步进运动机构 barrel (cylindric) cam圆柱螺旋拉伸弹簧 cylindroid helical-coil extension spring圆柱螺旋扭转弹簧 cylindroid helical-coil torsion spring圆柱螺旋压缩弹簧 cylindroid helical-coil compression spring 圆柱凸轮 cylindrical cam圆柱蜗杆 cylindrical worm圆柱坐标操作器 cylindrical coordinate manipulator圆锥螺旋扭转弹簧 conoid helical-coil compression spring圆锥滚子 tapered roller圆锥滚子轴承 tapered roller bearing圆锥齿轮机构 bevel gears圆锥角 cone angle原动件 driving link约束 constraint约束条件 constraint condition约束反力 constraining force跃度 jerk跃度曲线 jerk diagram运动倒置 kinematic inversion运动方案设计 kinematic precept design运动分析 kinematic analysis运动副 kinematic pair运动构件 moving link运动简图 kinematic sketch运动链 kinematic chain运动失真 undercutting运动设计 kinematic design运动周期 cycle of motion运动综合 kinematic synthesis运转不均匀系数 coefficient of velocity fluctuation运动粘度 kenematic viscosity载荷 load载荷—变形曲线 load—deformation curve载荷—变形图 load—deformation diagram窄V 带 narrow V belt如有你有帮助，请购买下载，谢谢！毡圈密封 felt ring seal展成法 generating张紧力 tension张紧轮 tension pulley振动 vibration振动力矩 shaking couple振动频率 frequency of vibration振幅 amplitude of vibration正切机构 tangent mechanism正向运动学 direct (forward) kinematics正弦机构 sine generator, scotch yoke织布机 loom正应力、法向应力 normal stress制动器 brake直齿圆柱齿轮 spur gear直齿锥齿轮 straight bevel gear直角三角形 right triangle直角坐标操作器 Cartesian coordinate manipulator 直径系数 diametral quotient直径系列 diameter series直廓环面蜗杆 hindley worm直线运动 linear motion直轴 straight shaft质量 mass质心 center of mass执行构件 executive link; working link质径积 mass-radius product智能化设计 intelligent design, ID中间平面 mid-plane中心距 center distance中心距变动 center distance change中心轮 central gear中径 mean diameter终止啮合点 final contact, end of contact周节 pitch周期性速度波动 periodic speed fluctuation周转轮系 epicyclic gear train肘形机构 toggle mechanism轴 shaft轴承盖 bearing cup轴承合金 bearing alloy轴承座 bearing block轴承高度 bearing height轴承宽度 bearing width轴承内径 bearing bore diameter轴承寿命 bearing life 轴承套圈 bearing ring轴承外径 bearing outside diameter轴颈 journal轴瓦、轴承衬 bearing bush轴端挡圈 shaft end ring轴环 shaft collar轴肩 shaft shoulder轴角 shaft angle轴向 axial direction轴向齿廓 axial tooth profile轴向当量动载荷 dynamic equivalent axial load轴向当量静载荷 static equivalent axial load轴向基本额定动载荷 basic dynamic axial load rating轴向基本额定静载荷 basic static axial load rating轴向接触轴承 axial contact bearing轴向平面 axial plane轴向游隙 axial internal clearance轴向载荷 axial load轴向载荷系数 axial load factor轴向分力 axial thrust load主动件 driving link主动齿轮 driving gear主动带轮 driving pulley转动导杆机构 whitworth mechanism转动副 revolute (turning) pair转速 swiveling speed ; rotating speed转动关节 revolute joint转轴 revolving shaft转子 rotor转子平衡 balance of rotor装配条件 assembly condition锥齿轮 bevel gear锥顶 common apex of cone锥距 cone distance锥轮 bevel pulley; bevel wheel锥齿轮的当量直齿轮 equivalent spur gear of the bevel gear 锥面包络圆柱蜗杆 milled helicoids worm准双曲面齿轮 hypoid gear子程序 subroutine子机构 sub-mechanism自动化 automation自锁 self-locking自锁条件 condition of self-locking自由度 degree of freedom, mobility总重合度 total contact ratio总反力 resultant force如有你有帮助，请购买下载，谢谢！总效率 combined efficiency; overall efficiency组成原理 theory of constitution组合齿形 composite tooth form组合安装 stack mounting组合机构 combined mechanism阻抗力 resistance最大盈亏功 maximum difference work between plus and minus work纵向重合度 overlap contact ratio纵坐标 ordinate组合机构 combined mechanism最少齿数 minimum teeth number最小向径 minimum radius作用力 applied force坐标系 coordinate frame。

Rexnord Omega Elastomer Couplings Installation Ins

Installation Instructions• Rexnord® Omega® ElastomerCouplings(Page 1 of 6) Type E and ES•Sizes 2-140 This is the Original Document in English LanguageT he designation ATEX (A tmosphere E xplosibles) has established itself for the new guidelines. ATEX 100a controls allregulations for the condition of explosion-proof equipment.Model No. _____________________Category ______________________Reference _____________________Mfg Year _______________________Max Temperature _______________1. General Information1.1. Omega Couplings are designed to provide a mechanical connection between the rotating shafts of mechanical equipment, using atorsionally soft flexible element to accommodate inherent misalignment while transmitting the power and torque between the shafts.1.2. These instructions are intended to help you to install and maintain your Omega coupling. Please read these instructions prior toinstalling the coupling, and prior to maintenance of the coupling and connected equipment. Keep these instructions near the coupling installation and available for review by maintenance personnel.1.3. Rexnord Industries, LLC owns the copyright of this material. These Installation and Maintenance instructions may not bereproduced in whole or in part for competitive purposes.1.4. Symbol descriptions:Danger of injury to persons.Damages on the machine possible.Pointing to important items.2. Safety and Advice HintsDANGER!2.1. Safety should be a primary concern in all aspects of coupling installation, operation, and maintenance.2.2. All rotating power transmission products are potentially dangerous and can cause serious injury. They must be guarded incompliance with OSHA, ANSI, ATEX and any other local standard for the applications they are used. It is the responsibility ofthe user to provide proper guarding.2.3. Failure to secure capscrews properly could cause coupling component(s) to dislodge during operation and result in personalinjury. See table 3 for proper tightening torques.2.4. Do not use on turbine drives if the coupling cannot be protected from steam leakage or overspeed situations beyond thecouplings published speed rating.2.5. Before installing this coupling on systems involving sleeve bearings, herringbone gearsets or other axially sensitive devices,consult Rexnord.2.6. Elastomeric couplings can hold a static electric charge that may discharge and ignite in an explosive environment Both shafts ofthe connected equipment must have a path to ground.Installation Instructions • Rexnord ® Omega ® Elastomer Couplings (Page 2 of 6) Type E and ES • Sizes 2-1403. Rexnord Omega Coupling Design and Part NumbersType EType ES4.Drive AlignmentDANGER!Stop the motor and lock it out to prevent start-up during installation of coupling.A TTENTION! Improper alignment of the equipment or hubs may result in hub contact and sparking.⑥⑤①④⑥⑦③②Table 1 - Omega part numbersSizeElastomer ElementHubsElementcapscrews METRIC ⑥High speedrings⑦SleeveextensionE①ES ②Rough bore③BSW UNFQD hub⑤Taper Bush hub ④Taper Bush hub ④27300005M 7300075M 7300215M ---7301410-37300010M 7300080M 7300240M 7300795M 7300730M-73014207369574M 47300015M 7300085M 7300270M 7300800M 7300740M 7300860M 73014207369575M 57300020M 7300090M 7300305M 7300805M 7300745M 7300865M 73014207369576M 107300025M 7300095M 7300340M 7300810M 7300750M 7300870M 73014507369577M207300030M 7300100M 7300650M 7300815M 7300755M 7300875M 7393101 7301100M 7369578M 307300035M 7300105M 7300660M 7300820M 7300760M 7300880M 7393101 7301105M 7369579M 407300040M 7300110M 7300670M 7300825M 7300765M 7300885M 7393105 7301110M 7369580M 507300045M 7300115M 7300680M 7300830M 7300770M 7300890M 7393105 7301115M 7369581M 607300050M 7300120M 7300690M 7300835M 7300775M 7300895M 7393109 7301120M 7369582M 707300055M 7300125M 7300700M 7300840M 7300780M 7300900M 7393109 7301125M 7369583M 807300060M 7300130M 7300710M 7300845M 7300785M 7300905M 7393109 7301130M 7369584M1007300065M -7300720M 7300850M 7300850M -7301530-7369834M 1207300070M -7300725M 7300855M 7300855M -7301540-7369835M1407300071M-7300727M7300858M7300857M-7301545--5. Rexnord Omega Coupling InstallationSTEp 15.1. Clean dirt and burrs from shafts and hub bores.5.2. Be sure the keys fit shafts properly .5.3. Position both hubs on the shaft without tighteningthe setscrews.5.4. Use a half element to set proper hub spacing. 5.5. When the hubs are properly spaced, tighten the setscrews.5.6.When using tapered bushings, follow bushing manufacturers instructions.STEp 25.7. Mount first half element to the hubs using cap screws provided.5.8. Rotate the shaft 180 degrees and secure second half element.5.9.If shaft cannot be rotated, mount half elements at 90 degrees.STEp 35.10. Tighten all cap screws to the torques specified in Table 3. 5.11. Align equipment.5.12. Install proper guarding prior to equipment start up.A TTENTION! When installing the element, first seat all the cap screws with a light torque, then tighten all cap screws to proper torque using a torque wrench.Type E Type ESType EType ESType EType ESTable 2 - Drive Alignment23451020304050607080100120140(b-a)mm 3456767911881091214Δ Kr mm222222222333555Installation Instructions • Rexnord ® Omega ® Elastomer Couplings (Page 4 of 6) Type E and ES • Sizes 2-1406. Rexnord Omega Hub Mounting Options6.1.Hubs can be installed:• flush with the shaft end (D)• extended beyond the end of the shaft (E) • recessed behind the shaft end (F)A TTENTION! Shaft engagement length should be >0,8 times shaft diameter , bushed hubs must engage 100%.7. Cap Screw Torque7.1.Do not lubricate cap screws threads.7.2.Cap screws must have a thread-locking adhesive applied.7.3. Tighten cap screws by using torque wrench.A TTENTION! Do not lubricate cap screw threads 8. Rexnord Omega “Type E” Mounting OptionsArrangementAArrangementBCE Table 3 - Cap Screw Torque Couplingsize Quantity Torque - DRYMetricIn. Lbs Nm Part number Steel Part number Stainless steel Thread size Wrench size28+82042373014107301417M61038+87301420730142748+87301420730142758+8730142073014271012+87301450730145720124685373931017393102M101630127393101739310240167393105739310650167393105739310660168169273931097393110M12187016739310973931108016739310973931101002032403707301530-M203012024730154073015471403270808007301545-M24Table 4 - Type E Mounting Options23451020304050607080100120140A 368888131281181817445776B 4127273434394142515364837091102C4646465959656975919710914995124127Installation Instructions • Rexnord® Omega® ElastomerCouplings(Page 5 of 6) Type E and ES•Sizes 2-140 9.Rexnord Omega “Type ES” Mounting Optionsoutward One hub mounted inwardinwardTable 5 - Spacer coupling (ES) Hub mounting options for industry shaft gaps* Hub mounted inboardInstallation Instructions • Rexnord ® Omega ® Elastomer Couplings (Page 6 of 6) Type E and ES • Sizes 2-14010. Preventative MaintenanceDANGER!Do not make contact with the coupling when it is rotating and/or in operation10.1. Periodic visual inspection is necessary to evaluate the condition of the flex element. Inspection can be done during the operationusing a strobe light.10.2. When inspecting the element look for:• Fatigue cracks at element splits • Discoloration• Surface cracking in body of element.A TTENTION! Replace Element if necessary.11. Element ReplacementDANGER!Stop the motor and lock it out to prevent start-up during installation of coupling.11.1.Always replace both half elements.11.2. Install both half elements from the same box.11.3. Follow installation instructions (see Section 5, Rexnord Omega Coupling Installation).11.4.Tighten element cap screws to proper torque (see Table 3).。

The scaling function at strong coupling from the quantum string Bethe equations

a r X i v :h e p -t h /0703131v 1 14 M a r 2007Preprint typeset in JHEP style -HYPER VERSIONMatteo Beccaria Dipartimento di Fisica,Universita’del Salento,Via Arnesano,I-73100Lecce INFN,Sezione di Lecce E-mail:matteo.beccaria@le.infn.it Gian Fabrizio De Angelis Dipartimento di Fisica,Universita’del Salento,Via Arnesano,I-73100Lecce INFN,Sezione di Lecce E-mail:deangelis@le.infn.it Valentina Forini Dipartimento di Fisica,Universita’di Perugia,Via A.Pascoli,I-06123Perugia INFN,Sezione di Perugia and Humboldt-Universit¨a t zu Berlin,Institut f¨ur Physik,Newtonstraße 15,D-12489Berlin E-mail:forini@pg.infn.it,forini@physik.hu-berlin.de A BSTRACT :We study at strong coupling the scaling function describing the large spin anomalous dimension of twist two operators in super Yang-Mills theory.In the spirit of AdS/CFT duality,it is possible to extract it from the string Bethe Ansatz equa-tions in the slsector of the superstring.To this aim,we present a detailed analysis of the Bethe equations by numerical and analytical methods.We recover several short string semiclassical results as a check.In the more difﬁcult case of the long string limit providing the scaling function,we analyze the strong coupling version of the Eden-Staudacher equation,including the Arutyunov-Frolov-Staudacher phase.We prove that it admits a unique solution,at least in perturbation theory,leading to the correct prediction consistent with semiclassical string calculations.K EYWORDS :integrable quantum ﬁeld theory,integrable spin chains (vertex models),quantum integrability (Bethe ansatz).1.IntroductionThe-gluon maximally helicity violating(MHV)amplitudes in planar SYM obey very remarkable iterative relations[1,2,3]suggesting solvability or even integrability of the maximally supersymmetric gauge theory.The main ingredient of the construction is the so-called scaling function deﬁned in terms of the large spin anomalous dimension of leading twist operators in the gauge theory[4].The scaling function can be obtained by considering operators in the sl sector of the formTr permutations(1.1) The classical dimension is,so is the twist,with minimal value.The minimal anomalous dimension in this sector is predicted to scale at large spin as(1.2) where the planar’t Hooft coupling is deﬁned as usual byThe one and two loops explicit perturbative calculation of is described in[5,6,7] and[8,9].Based on the QCD calculation[10],the three-loop calculation is per-formed in[11,12]by exploiting the so-called trascendentality principle(KLOV).In principle,one would like to evaluate the scaling function,possibly at all loop order by Bethe Ansatz methods exploiting the conjectured integrability of SYM.This strategy has been started in[13].In that paper,an integral equation providing is proposed by taking the large spin limit of the Bethe equations[14].Its weak coupling expansion disagrees with the four loop contribution.The reason of this discrepancy is well under-stood.The Bethe Ansatz equations contain a scalar phase,the dressing factor,which is not constrained by the superconformal symmetry of the model.Its effects at weak-coupling show up precisely at the fourth loop order.A major advance was done by Beisert,Eden and Staudacher(BES)in[15].In the spirit of AdS/CFT duality,they considered the dressing factor at strong coupling.In that regime,it has been conjectured a complete asymptotic series for the dressing phase[16]. This has been achieved by combining the tight constrains from integrability,explicit1-loop-model calculations[17,18,19,20]and crossing symmetry[21].By an impressive insight,BES proposed a weak-coupling all-order continuation of the dressing.Including it in the ES integral equation they obtained a new(BES)equation with a rather complicated kernel.The predicted analytic four-loop result agrees with the KLOV principle.Very re-markably,an explicit and independent perturbative4-loop calculation of the scaling func-tion appeared in[22].In theﬁnal stage,the4-loop contribution is evaluated numerically with full agreement with the BES prediction.This important result is one of the main checks of AdS/CFT duality.Indeed,a non trivial perturbative quantity is evaluated in the gauge theory by using in an essential way input data taken from the string side.As a further check,one would like to recover at strong coupling the asymptotic behav-ior of the scaling function,as predicted by the usual semiclassical expansions of spinning string solutions[23,24,25].Actually,the BES equation passes this check,partly numer-ically[26]and partly by analytical means[27].One could say that this is a check that nothing goes wrong if one performs the analytic continuation of the dressing phase from strong to weak coupling.From a different perspective,one would like to close this logical circle and check that the same result is obtained in the framework of the quantum string Bethe equations pro-posed originally in[17].Indeed,it would be very nice to show that these equations repro-duce the scaling function in the suitable long string limit.Also,one expects toﬁnd some simpliﬁcations due to the fact that only theﬁrst terms in the strong coupling expression of the dressing must be dealt with.On the other side,the BES equation certainly requires all the weak-coupling terms if it has to be extrapolated at large coupling.In this paper,we pursue this approach.As aﬁrst step,we study numerically the quantum Bethe Ansatz equations in the sl sector and check various results not directly related to the scaling function.Then,we work out the long string limit which is relevant to the calculation of.From our encouraging numerical results,we move to an analytical study of a new version of the BES equation suitable for the string coupling region.Thisequation has beenﬁrst derived by Eden and Staudacher in[13]as a minor result.Indeed,it has been left over because the main interest was focusing on matching the weak-coupling 4-loop prediction.However,we believe that it is a quite comfortable tool if the purpose is that of reproducing the strong coupling behavior of the scaling function.We indeed prove that the solution described in[27]is the unique solution of the strong coupling ES equation.The plan of the paper is the following.In Sec.(2)we recall the Bethe Ansatz equations valid in the sl sector of SYM without and with dressing corrections.In Sec.(3)we present various limits obtained in the semiclassical treatment of the superstring. We present our results for short and long string conﬁgurations.In Sec.(4)we analyze the strong coupling ES equation building explicitly its solution and checking that it agrees with the result of[27].We also investigate numerically the equation without making any strong coupling expansion to show that the equation is well-deﬁned.Sec.(5)is devoted to a summary of the presented results.2.Gauge Bethe Ansatz predictions for the scaling function without dressing In the seminal paper[13],Eden and Staudacher(ES)proposed to study the scaling func-tion in the framework of the Bethe Ansatz for the sl sector of SYM.The states to be considered in this rank-1perturbatively closed sector take the general formTr(2.1) They are associated to the states of an integrable spin chain.The anomalous dimension is related to the chain energies by(2.2) The all-loop conjectured Bethe Ansatz equations valid for up to wrapping terms are fully described in[28,14].Some explicit tests are can be found in[29,30].The Bethe Ansatz equations for the roots are(2.3)where we have deﬁned the maps(2.4) The solutions of Eq.(2.3)must obey the following constraint to properly represent single trace operatorsThe quantum part of the anomalous dimension,i.e.the chain spectrum,is obtained from(2.6) Taking the large limit of the Bethe Ansatz equations,ES obtain the following represen-tation of the scaling function(2.9) Notice that given we can simply write[31].These equations are independent on the twist which drops in the large limit.This is important since the scaling function is expected to be universal[32,13]and therefore can be computed at large twist.Unfortunately,the perturbative expansion of disagrees at4loops with the ex-plicit calculation in the gauge theory.This is well known to be due to the missing contri-bution of the dressing phase.2.1Input from string theory:Dressing correctionsThe effect of dressing is discussed in[15]to which we defer the reader for general dis-cussions about its origin and necessity.The Bethe equations are corrected by a universal dressing phase according to1This general formula holds unchanged in various deformations of the SYM theory[35,36],see for ex.[37].Theﬁrst non trivial constant is.Indeed,consistently with the3-loops agreement with explicit perturbation theory.The proposed coefﬁcients for the all-order weak-coupling expansion of the dressing phase[16]are given in[15](see also[38]and[39]).They read(2.15) with(2.17) The modiﬁed integral equation can be exploited to compute the perturbative expansion of.Now,there is agreement with the4loop explicit calculation.As we stressed in the Introduction,it is very remarkable that this weak coupling agreement is found with various inputs from string theory.In this sense,this is a powerful check of AdS/CFT duality.3.Strong coupling regime and the string Bethe equationsAs we explained,the BES equation is obtained by including in the ES equation an all-order weak-coupling expansion of the dressing phase.This expansion comes from a clever combination of string theory inputs and constraints from integrability.In our opinion,this is the essence of the integrability approach to AdS/CFT duality.As a consistency check, one would like to recover from the BES equation the known semiclassical predictions valid in the superstring at large coupling.There are indeed several limits that can be computed.The semiclassical limit is eval-uated in terms of the BMN-like[40]scaled variables which are keptﬁxed as(3.1) where are the semiclassical energy of a string rotating in with angular momen-tum and spinning in with spin.The classical solution,and theﬁrst quantum corrections as well,are described in[23,24,25].The simplest limits that can be considered are those describing short strings that do not probe regions with large curvature.We call themshort-GKP(3.2)short-BMNﬁxed(3.3) The scaling function is instead reproduced in the simplest long string limit which readslong string(3.4) In this limit,one can read the strong coupling behavior of the scaling function which iswhich at means.One can ask if it is possible to explore numerically the Bethe equations in the gauge theory up to and with to extract the scaling function.Actually,this is a hard task.At the regime is not perturbative.The complete dressing should be resummed and it is not easy to do that,although some very interesting results have been presented in[15].An alternative hybrid approach would be that of taking the string Bethe equations with leading dressing.This should be enough to study the leading terms of at strong coupling.Of course,the problem is now that must be large and then must be unre-alistically large to deal with the numerical solution.However,not much is known about the properties of the strong coupling dressing expansion.It is divergent,but possibly asymptotic.Therefore,it would be difﬁcult to estimate its accuracy at.In the next Section,we illustrate the detailed exploration of the above three limits.3.1The String Bethe equations and their numerical solutionFor the subsequent analysis it is convenient to pass to the variables deﬁning(3.7)(3.8) The loop corrections to the energy are now(3.9) where are obtained by solving the Bethe equations with dressing Eq.(2.10).At strong coupling,we use the leading dressing phase and write the string Bethe equations in loga-rithmic form as(3.10)(3.11) where and we have utilized the symmetry of the solution for the ground state as well as its known mode numbers[13].The numerical solution of the String Bethe equations is perfectly feasible.The tech-niques have already been illustrated in the two compact rank-1subsectors su and su as discussed in[41,42].First,we solve the equations at.This is the one-loop contribution.It is known exactly at.The Bethe roots are obtained,for an even spin,as the roots of the resolvent polynomial[43,13]At,we use the solution at as the starting point for the numerical root ﬁnder.Then,we increase.At each step,we use a linear extrapolation of the previous solutions to improve the guess of the new solution.This procedure is quite stable and allows to explore a wide range of values.Notice that changing the twist is trivial since the complexity of the equations does not change.3.2Short string in the GKP limitAs aﬁrst numerical experiment weﬁx and and increase up to large values where the equations are reliable.This is the short string limit where the geometry is approx-imatelyﬂat.One expects to recover the Gubser-Klebanov-Polyakov law[44]. Solving the equations,we indeed verify that the Bethe momenta have the asymptotic scaling.This is clearly illustrated in Fig.(1).Despite the non trivial distribution of the Bethe roots,it is straightforward to compute the anomalous dimension at large.To do that,weﬁrst write the Bethe equations in the form(3.14)At large we can write(3.15)The Bethe momenta can be divided into a set with and a symmetric set obtained byﬂipping.The expansion of the energy is(3.16)(3.17) From the Bethe equations we have(3.18) where we exploited.Each Bethe momentum with can be written as with.Hence we can write(3.19) At large(3.21)The next-to-leading terms are(3.22)(3.24) The subleading term cancels the classical contribution leaving a pure behavior.This can be compared with theﬂat limit in the semiclassical approximation that reads for(3.25) with full agreement.3.3Short string in the BMN limitIf we keepﬁxed and increase withﬁxed,we can reach the BMN limit[40]. This is numerically very easy because enters trivially the equations.Fig.(2)shows the convergence to the BMN limit when is increased from10to100and isﬁxed at. The various curves clearly approach a limiting one.This is very nice since it is an explicit show of how the BMN regimes sets up.Fig.(3)shows the limiting curves forat very large.The three curves are perfectlyﬁt by the expected law(3.26)3.4Long string limit and the scaling functionThe previous pair of tests in the(easy)short string limits is a clear illustration that the numerical solution of the Bethe equations is reliable.The slow string limit is much more difﬁcult.We begin with a plot of the energy at ﬁxed twist and increasing spin from10to60.It is shown in Fig.(4).Each curve bends downward as increases,since it ultimately must obey the law.However,at ﬁxed,when increases the energy increases slowly eventually following the law. We attempted an extrapolation at at each.In Fig.(5)we show our estimate for the derivative of the scaling function,byﬁtting the data with.We also show the analytical prediction.It seems to be roughly reproduced as soon as .The above”dirty”numerical procedure shows that it is reasonable to expect that the quantum string Bethe equations are able to capture the correct strong coupling behavior of the scaling function.However,the above extrapolation has a high degree of arbitrariness, especially concerning theﬁtting function employed to estimate the limit.Also, one would like to go to quite larger requiring a huge number of Bethe roots,equalindeed to the spin.In practice,as it stands,the numerical investigation could hardly be signiﬁcantly improved.For all these reasons,in theﬁnal part of this paper we explore an equation analogous to the BES equation,but derived for the string Bethe equation,at least with the leading order dressing phase.4.The strong coupling ES equationThe inclusion of the AFS phase[17]in the ES equation(4.1)(4.3)(4.4) where the AFS kernel is(4.6) For the AFS kernel we have2(4.8) We change variables to put the equation in a somewhat simpler form and deﬁne(4.9)(4.7) The result Eq.(4.8)follows immediately from the expansions reported in the appendices of[15].The strong coupling ES equation for is then(4.12) Taking the terms with the leading power of weﬁnd that satisﬁes the remarkably simple equation(4.15) and now the equation reads(4.16) Now,the following question arises:Is this equation a constraint or does it determine a unique solution?As aﬁrst step,we prove that the solution of[27]is indeed a solution to the above equation.This solution reads(4.17) Now,the detailed values of the matrix elements areand(4.19) The linear equations Eq.(4.16)are then(stands for even Bessel)(4.20) whereThese equations are indeed satisﬁed by the solution deﬁned through Eq.(4.17).This can be checked by evaluating the inﬁnite sum in closed form by the Sommerfeld-Watson trans-formation methods.For instance theﬁve terms in Eq.(4.21)read at(4.22)(4.23)(4.24) However,this is not the unique solution of Eq.(4.13).As is quite usual,the straightfor-ward strong coupling limit of Bethe equations does not determine completely the solution which isﬁxed by the tower of subleading corrections.A similar difﬁculty is explained in details in[27].For instance,a second solution of Eq.(4.16)is(4.25) Indeed,this produces the remarkably simple solution(4.26) Notice that,before summing the series,this second solution is precisely of the general class Eq.(4.15).As a check,we have indeedfor(4.27) In practice,we still need the equal-weight condition on the even/odd Bessel functions contained in the solution[27].Toﬁnd a unique solution,we must examine the next orders in the strong coupling expansion.Indeed,the next orders provide both equations for the various subleading corrections to the solution and constraints on the previous contributions.This is due to the fact that the AFS kernel is a function of expressed as a Neumann series of purely even Bessel functions.The odd Bessel functions provide the above mentioned constraints as we now illustrate.4.2NLO order at strong couplingLet us work out the constraints from the subleading correction.If we take into account the next terms in the expansion and write(4.28)(4.29) weﬁnd the following equation for(4.30)(4.34) Hence,the equation can be written(4.35) where,explicitly,the constants are given by(4.36)(4.38)otherwiseandEquating to zero the coefﬁcients of the odd Bessel functions we obtain the con-straint on(4.40) where(4.42) So this constraint adds nothing new and,in particular,is satisﬁed by the Alday’s solu-tion[27].Looking at the even Bessel functions,we obtain the homogeneous equation(4.44)(4.47)and(4.48) One obtains immediately the crucial relationThis relation permits to write all odd coefﬁcients in terms of the even ones.Substi-tuting this relation in the truncated versions of the basic conditions(4.50) one obtains a well-posed problem converging rapidly to the solution[27]without any a priori condition on the solution.For instance,by truncating the problem dropping all with,weﬁnd the table Tab.(1)of values for.A simple polynomial extrapolation to provides the correct limit.Hence,the strong coupling expansion is well deﬁned and the leading solution is unique.Of course,it is the one described in[27].1020304050607080Table1:Coefﬁcient from the truncated full rank linear problem.4.4Numerical integration of the strong coupling ES equationTo summarize,we have shown that the strong coupling ES equation is consistent with the results of[27].In that paper,it was crucial toﬁx the relative weights of the even/odd Bessel functions appearing in the general solution.These weights were shown to be more than an Ansatz.They are encoded in the full equation before expanding at strong cou-pling.Alternatively,they can be derived by analyzing the next-to-leading and next-to-leading corrections.As aﬁnal calculation and check,we provide the results from a numerical investigation without any strong coupling expansion to see how the correct strong coupling solution arises.This can be done along the lines illustrated in[26,27].We start again from the Neumann expansion5.ConclusionsIn this paper,we have considered several properties of the quantum string Bethe equa-tions in the sl sector with the leading strong coupling dressing,i.e.the AFS phase.We have performed a numerical investigation of the equations showing that their analysis is quite feasible.As an interesting result,we have repeated the calculation of the GKP limit of the anomalous dimensions as for the highest excited states in the compact rank-1 subsectors su and su.Also,we have been able to observe the setting of the BMN scaling regime by reproducing the plane wave energy formula atﬁxed spin and large twist.In the case of the long string regime,we have been able to provide numerical ev-idences for a scaling function exhibiting an early strong coupling behavior as expected from the numerical solution of the BES equation.Motivated by these results,we have analyzed analytically and perturbatively at strong coupling an almost trivially modiﬁed version of the BES equation with the very simple strong coupling dressing[17].In particular,we have proved that this equation admits,as it should,a unique solution for the asymptotic Bethe root(Fourier transformed)density in full agreement with existing results.While this work was under completion,the paper[45]presented an analysis partially overlapping with our results.That paper derives an integral equation for the Bethe root density taking into account the dressing at strong coupling and is based on a novel integral representation of the dressing kernel.We hope that the two alternative approaches will turn out to be useful in computing the one-loop string correction to the large scaling function.Indeed,this interesting contribution has been checked numerically in[26]but it still evades an analytical conﬁrmation.Hopefully,these various efforts might give insight on the general structure of the dressing phase as well as on the role of the asymptotic Bethe equations in an exact de-scription of the planar spectrum[46].Signiﬁcative studies aboutﬁnite size effects[47] and corrections that arise in aﬁnite volume to the magnon dispersion relation at strong coupling[48],see also[49],as well as the recent observation[50]that the dressing phase could originate from the elimination of”novel”Bethe roots,strongly demand a deeper understanding.AcknowledgmentsWe thank D.Serban and M.Staudacher for very useful discussions and comments.M.B.also thanks G.Marchesini for conversations about the properties of twist-2anomalous dimensions atﬁnite spin.The work of V.F.is supported in part by the PRIN project 2005-24045”Symmetries of the Universe and of the Fundamental Interactions”and by DFG 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Coupling Effect of Ion Channel Clusters on Calcium Signalling

(Received 28 November 2007)
Based on a modiﬁed intracellular Ca model involving diﬀusive coupling of two calcium ion channel clusters, the eﬀects of coupling on calcium signalling are numerically investigated. The simulation results indicate that the diﬀusive coupling of clusters together with internal noise determine the calcium dynamics of single cluster, and for either homogeneous or heterogeneous coupled clusters, the synchronization of clusters, which is important to calcium signalling, is enhanced by the coupling eﬀect.
by the National Natural Science Foundation of China under Grant Nos 10575041 and 10747005. whom correspondence should be addressed. Email: jiay@ c 2008 Chinese Physical Society and IOP Publishing Ltd
)1 , YU Guang(
)1
Department of Physics and Institute of Biophysicsan 430079 2 College of Science, China University of Mining and technology, Xuzhou 221008

Entropy of the Randall-Sundrum brane world with the generalized uncertainty principle

gr-qc/0605084 SOGANG-HEP 317/06
Entropy of the Randall-Sundrum brane world with the generalized uncertainty principle
Wontae Kim1,a , Yong-Wan Kim2,b , and Young-Jai Park1,c
(5) 16πGN
d4 x
dy −g(5) R(5) + 12k 2 −g(−) λ(−) , (1)
−
d4 x
−g(+) λ(+) +
where λ(+) and λ(−) are tensions of the branes at y = 0 and y = πrc , respectively, while 12k 2 is a cosmological constant. We assume that orbifold S 1 /Z2 possesses a periodicity in the extra coordinate y , and identify −y with y . Two singular points on the orbifold are located at y = 0 and y = πrc . Two 3-branes are placed at these points. Note that the metric on each brane is (+) (5) µ (−) (5) µ deﬁned as gµν ≡ gµν (x , y = 0) and gµν ≡ gµν (x , y = πrc ). Since we are interested in black holes, let us assume the bulk metric as

Top Physics

a Another
useful reference on this topic is ref.8
Table 1. Measured11 mt and σtt from CDF and DØ.
experiment CDF
channel dilepton lepton + jets all jets combined dilepton lepton + jets all jets combined combined
DØ
Tevatron
will also describe the increases in measurement precision anticipated at Run II and future accelerators and discuss what we hope to learn. 1.1 Mass
TOP PHYSICS∗
ELIZABETH H. SIMMONS Department of Physics, Boston University 590 Commonwealth Avenue, Boston, MA 02215, USA and Radcliﬀe Institute for Advanced Study, Harvard University 34 Concord Avenue, Cambridge, MA 02138 e-mail: sim 11.4 175.9 ± 7.1 186.0 ± 11.5 176.0 ± 6.5 168.4 ± 12.8 173.3 ± 7.8 172.1 ± 7.1 174.3 ± 5.1
σtt (pb) .5 8.4+4 −3.5 5.1 ± 1.5 9.2 ± 4.3 (SVX b-tag) (soft lepton tag) .5 7.6+3 −2.7 +1.7 6.5−1.4 (mt = 175) 4.1 ± 2.1 8.3 ± 3.6 (topological) (soft lepton tag) 7.1 ± 3.2 5.9 ± 1.7 (mt = 172)

Muon anomalous magnetic moment and the heavy photon in a little Higgs model

hep-ph 0306112
KIAS P03039
Muon anomalous magnetic moment and the heavy photon in a little Higgs model
Seong Chan Park∗ Korea Institute for Advanced Study (KIAS) 207-43 Cheongryangri-dong Dongdaemun-gu, Seoul 130-012, Korea
scale and remains weakly coupled at TeV scale. The model predicts the presence of new heavy gauge bosons (WH , ZH and AH ) and a new heavy top-like vector quark T and their couplings. The minimality of the Littlest Higgs model would leave characteristic signatures at the present and future collider experiments. Since the tree level corrections of the Littlest Higgs model to electroweak precision data constrain the heavy particles as massive as a few TeV, a 500 GeV linear collider (LC) has not been expected to eﬃciently test the model. In literatures, Large Hadronic Collider(LHC) of the CERN is shown to have a potential to detect the new particles [16, 17, 18]. In the Littlest Higgs model, however, we ﬁnd that the global symmetry structure SU (5)/SO (5) yields substantially light AH , light enough to be produced on-shell at a 500 GeV LC. Moreover, as shall been shown below, the AH becomes lighter in the parameter space where the corrections to the electroweak precision measurements are minimized. The presence of a few hundreds GeV heavy photon can be dangerous to other low energy observables. We study its one-loop contributions to a well measured observable, the muon anomalous magnetic moment. Another issue here is the collider signatures of the AH . The process of e+ e− → µ+ µ− is to be discussed, which is one of the most eﬀective processes to probe the model, as the branching ratios (BR) of the heavy photon suggests. The paper is organized as follows. In Sec. II, we brieﬂy review the Littlest Higgs model. We point out the preferred parameter space by considering some tree level relations of the SM gauge boson masses and couplings. In Sec. III, physical properties of the heavy photon is studied, focused on its mass and decay patterns. In Sec. IV, the one-loop corrections of the new gauge bosons to the muon anomalous magnetic moment are calculated. Numerical value is to be compared with the latest experimental data. In Sec. V, we study the eﬀects of the Littlest Higgs model on the process e+ e− → µ+ µ− , of which the dominant contribution is from the heavy photon. We summarize our results in Sec. VI.

Evolution at small x_bj The Color Glass Condensate

arXiv:hep-ph/0501087v1 11 Jan 2005
When probed at very high energies or small Bjorken xbj , QCD degrees of freedom manifest themselves as a medium of dense gluon matter called the Color Glass Condensate. Its key property is the presence of a density induced correlation length or inverse saturation scale Rs = 1/Qs . Energy dependence of observables in this regime is calculable through evolution equations, the JIMWLK equations, and characterized by scaling behavior in terms of Qs . These evolution equations share strong parallels with speciﬁc counterparts in jet physics. Experimental relevance ranges from lepton proton and lepton nucleus collisions to heavy ion collisions and cross correlates physics at virtually all modern collider experiments.
49 51 53 55 57 61 62 66 67 69 69 71 72 74 74 80 82 83 84 85 87 88

coupling例句

CouplingIntroductionIn computer science, coupling refers to the degree of interdependence between software modules or components. It measures how closely one module relies on another module. Coupling is an important concept in software design and can have a significant impact on the maintainability, reusability, and testability of a system. In this article, we will explore the different types of coupling, their implications, and strategies to minimize coupling in software systems.Types of Coupling1. Content CouplingContent coupling occurs when one module directly accesses or modifiesthe contents of another module. This is the strongest form of coupling and should be avoided whenever possible. It creates a tight dependency between modules, making it difficult to modify or replace one module without affecting the others. Content coupling can lead to code duplication, increased complexity, and poor encapsulation.2. Common CouplingCommon coupling happens when multiple modules share the same global data or state. Changes to the shared data can impact multiple modules,leading to unintended side effects. Common coupling can make it challenging to understand and reason about the behavior of a system. It also reduces the potential for parallelism and can introduce subtle bugs due to race conditions or inconsistent state.3. External CouplingExternal coupling occurs when a module depends on external entities such as libraries, frameworks, or databases. While some level of externalcoupling is inevitable, excessive reliance on external entities can make a system less flexible and portable. Changes in the external entities may require modifications to multiple modules. It is essential to carefully manage external dependencies and minimize their impact on the system.4. Control CouplingControl coupling happens when one module controls the behavior ofanother module by passing control information, such as function parameters or flags. This type of coupling can lead to complex control flows and make the system more difficult to understand and maintain. It is generally recommended to minimize control coupling and instead relyon well-defined interfaces and message passing mechanisms.Implications of CouplingHigh coupling in a software system can have several negative consequences:1.Reduced Maintainability: When modules are tightly coupled, makingchanges to one module can have a cascading effect on other modules.This makes it harder to understand the impact of changes andincreases the risk of introducing bugs. Low coupling, on the other hand, allows for more modular and independent development, makingmaintenance easier.2.Decreased Reusability: Coupled modules are less reusable sincethey are tightly bound to specific contexts or dependencies.Modules with low coupling can be easily extracted and reused indifferent projects or scenarios.3.Limited Testability: Modules with high coupling are difficult totest in isolation. Changes to one module may require modifications to multiple modules, making it challenging to write focused andreliable tests. Low coupling promotes testability by allowingindividual modules to be tested independently.4.Increased Complexity: Coupling adds complexity to a system byintroducing interdependencies between modules. This complexity canmake it harder to understand the system’s behavior, leading tolonger development cycles and increased maintenance costs.Strategies to Minimize CouplingTo reduce coupling in software systems, the following strategies can be employed:1. EncapsulationEncapsulation is a fundamental principle of object-oriented programming that promotes low coupling. By encapsulating the internal details of a module and exposing only a well-defined interface, dependencies between modules can be minimized. Encapsulation allows modules to be modified or replaced without affecting other parts of the system.2. Dependency InjectionDependency injection is a design pattern that helps manage dependencies between modules. Instead of creating dependencies directly within a module, they are provided from the outside. This allows for loose coupling and easier testing, as dependencies can be easily mocked or replaced.3. Interface DesignWell-defined interfaces help decouple modules by providing a clear contract for communication. By relying on interfaces instead of concrete implementations, modules can be easily swapped or extended without affecting other parts of the system. Careful interface design is crucial for minimizing coupling.4. ModularityBreaking a system down into smaller, independent modules promotes low coupling. Each module should have a clear responsibility and minimal dependencies on other modules. Modularity allows for easier understanding, maintenance, and reuse of code.5. Loose Coupling with MessagingReplacing direct method calls with messaging systems, such as message queues or event-driven architectures, can reduce coupling. Modules communicate with each other through messages, enabling asynchronous and decoupled communication. This approach allows for greater flexibility and scalability.ConclusionCoupling is an important concept in software design that measures the interdependence between modules. High coupling can lead to reduced maintainability, decreased reusability, limited testability, and increased complexity. By employing strategies such as encapsulation, dependency injection, interface design, modularity, and loose coupling with messaging, software developers can minimize coupling and build more flexible and maintainable systems.。

Extended

Extended renovation theory and limit theorems for stochastic ordered graphs∗
S ERGUEI F OSS TAKIS KONSTANTOPOULOS
Abstract We extend Borovkov’s renovation theory to obtain criteria for coupling-convergence of stochastic processes that do not necessarily obey stochastic recursions. The results are applied to an “inﬁnite bin model”, a particular system that is an abstraction of a stochastic ordered graph, i.e., a graph on the integers that has (i, j ), i < j , as an edge, with probability p, independently from edge to edge. A question of interest is an estimate of the length Ln of a longest path between two vertices at distance n. We give sharp bounds on C = limn→∞ (Ln /n). This is done by ﬁrst constructing the unique stationary version of the inﬁnite bin model, using extended renovation theory. We also prove a functional law of large numbers and a functional central limit theorem for the inﬁnite bin model. Finally, we discuss perfect simulation, in connection to extended renovation theory, and as a means for simulating the particular stochastic models considered in this paper. Keywords and phrases. S TATIONARY

Coupling of mixed finite elements and boundary elements

COUPLING OF MIXED FINITE ELEMENTS AND BOUNDARY ELEMENTS
CARSTEN CARSTENSEN AND STEFAN A. FUNKEN
Abstract. The symmetric coupling of mixed nite element and boundary element methods is analysed for a model interface problem with the Laplacian. The coupling involves a further continuous ansatz function on the interface to link the discontinuous displacement eld to the necessarily continuous boundary ansatz function. Quasi-optimal a priori error estimates and sharp a posteriori error estimates are established which justify adaptive mesh-re ning algorithms. Numerical experiments prove the adaptive coupling as an e cient tool for the numerical treatment of transmission problems.
1991 Mathematics Subject Classi cation. 65 N 38, 65 N 15, 65 R 20, 45 L 10. Key words and phrases. coupling of nite elements and boundary elements, mixed nite elements, adaptive algorithms, a posteriori error estimates.

核磁共振中自旋裂分或J偶合

核磁共振中自旋裂分或J偶合Spin-spin splitting or J couplingCoupling in 1H NMR spectraWe have discussed how the chemical shift of an NMR absorption is affected by the magnetic field B e produced by the circulation of neighboring electrons. Now we wish to examine how the magnetic field produced by neighboring nuclei B n affects the appearance of the 1H NMR absorption. The effect occurs through the interaction of nuclear spins with bonding electron spins rather than through space. Let's first consider the absorption of a hydrogen nucleus labeled A with only one neighboring hydrogen nucleus in a vicinal position labeled X. Let's also assume that H A and H X have significantly different chemical shifts.H X will have approximately equal probability of existing in either the low energy alpha state or high energy beta state. Again because of the small energy difference between the low and high energy states, the high energy state is easily populated from thermal energy. For those molecules in which H X exists in the low energy state, about half the molecules in the sample, its magnetic field B n will subtract from the magnetic field B o-B e and for those molecules in which H X exists in the higher energy state, again about half the molecules, its magnetic field B n will add to B o-B e.Note: whether B n for a particular spin state adds to or subtracts from B o is a function of the number of intervening bonds; this phenomenon doesn't usually affect the appearance of the signal and will not be explained here but results from the mechanism of coupling involving interaction of nuclear spins with electron spins. For the example of vicinal coupling (3 intervening bonds), the B n field is negative for H X in the alpha spin state; for geminal coupling B n is positive for H X in the alpha spin state. Geminal coupling occurs between protons of different chemical shift bonded to the same carbon (2 intervening bonds); it will be discussed later.As a consequence of the B n field in a vicinal system, at fixed external magnetic field B o, a lower frequency will be required to achieve resonance for those molecules which have H X in the state than for those molecules which have H X in the state. The NMR signal for H A will appearas a two line pattern as shown in Figure 16. We say the H X splits the absorption H A into a doublet and the two protons are coupled to each other. The intensity of the two lines will be equal since the probability of H X existing in the or states is approximately equal. The chemical shift, which is defined as the position of resonance in the absence of coupling, is the center of the doublet. Just as H X splits the signal of H A into a doublet, H A splits the signal of H X into a doublet. The overall splitting pattern consisting of two doublets is call an A X pattern. The splitting of H A by H X is diagramed in Figure 16.When the molecule bears two equivalent vicinal protons, four possibilities exist for their combined magnetic fields: both are in spin states, one is in the spin state and one in the spin state, and vice versa, or both in the spin state. These four possibilities have about equal probability, and the appearance of the NMR signal is a 3-line pattern, a triplet(Figure 17), with intensities 1:2:1 because the effect of and are the same. With one adjacent proton in the spin state andthe other in the spin state, the effect of the Bn field becomes zero,and the center line of the triplet is the position of the chemical shift. The two H X protons split the H A signal into a triplet and the H A proton splits the two H X protons into a doublet. The overall splitting pattern consisting of a triplet and a doublet is called an A X2 pattern.Three chemical shift equivalent vicinal protons H X split the absorption of H A into a quartet with intensity pattern 1:3:3:1 as shown in Figure 10. The chemical shift is the center of the quartet. The three H X protons split the H A signal into a quartet and the H A proton splits the signal for the three H X protons into a doublet. The overall splitting pattern consisting of a quartet and a doublet is called an A X3 pattern.The spacing between the lines of a doublet, triplet or quartet is called the coupling constant. It is given the symbol J and is measured in units of Hertz (cycles per second). The magnitude of the coupling constant can be calculated by multiplying the separation of the lines in units (ppm) by the resonance frequency of the spectrometer in megaHertz.J Hz = ppm x MHz (typically 300, 400, or 500 MHz)In general, N neighboring protons with the same coupling constant J will split the absorbance of a proton or set of equivalent protons into N+1lines. Note that the splitting pattern observed for a particular proton or set of equivalent protons is not due to anything inherent to that nucleus but due to the influence of the neighboring protons. The relative intensity ratios are given by Pascal's triangle as shown in Figure 18.Because of the mechanism of J coupling, the magnitude is field independent: coupling constants in Hertz will be the same whether the spectrum is measured at 300 MHz or 500 MHz. Coupling constants range in magnitude from 0 to 20 Hz. Observable coupling will generally occur between hydrogen nuclei that are separated by no more than three sigma bonds.H-C-H, two sigma bonds or geminal couplingH-C-C-H, three sigma bonds or vicinal couplingCoupling is never observed between chemical shift equivalent nuclei, be it from symmetry or by accident, not because the B n field disappears but because spin transitions that would reveal the coupling are forbidden by symmetry. The role of symmetry in forbidding spectral transitions is of general importance in spectroscopy but is beyond the scope of this discussion. The magnitude of the coupling constant also provides structural information; for example, trans-alkenes show larger vicinal coupling than cis-alkenes. Sometimes, coupling is not observed between protons on heteroatoms such as the OH proton of an alcohol and adjacent protons on carbon. In this case the absence of coupling results from rapid exchange of the OH protons via an acid base mechanism; because of rapidexchange the identity of the spin state, or , of the acidic proton is lost. Examples of coupling constants J are shown in Figure 12.The example of geminal coupling of protons on a saturated carbon requires a structure in which the protons have different chemical shifts. This commonly occurs in a chiral molecule with a tetrahedral stereocenter adjacent to the methylene group as shown in the following compounds with stereocenters labeled with an asterisk. The geminal protons are labeled H A and H B rather than H A and H X because they have similar chemical shifts (A and B are close in the alphabet). Coupling between the geminal protons is independent of optical activity and rotation about single bonds. The hydrogens H A and H B are said to be diastereotopic hydrogens because if alternately each one is replaced with a deuterium atom, the resulting two structures are diastereomers (stereoisomers that aren't mirror images).Now let's examine the 1H NMR spectrum of methyl propanoate (methyl propionate). Notice that hydrogen atoms of the methyl group bonded to oxygen appear as a singlet at 3.6 ppm. They are chemical shift equivalent and hence, do not couple with each other. The chemical shift results from the deshielding effect of the strongly electronegative oxygen atom. Theresonance for the methylene protons appear as a quartet at 2.3 ppm. The splitting is caused by the three chemical shift equivalent protons on the adjacent methyl group. The methylene protons do not split each other since they are also chemical shift equivalent. The methyl protons appear at 1.1 ppm and are split into a triplet by the adjacent methylene protons.The coupling constant for the methyl triplet and the methylene quartet is 7 Hz. The overall splitting pattern consisting of a three-proton triplet and a two-proton quartet is called an A3X2 pattern.next section: Spin-spin splitting and coupling - More complex 1H NMR splitting© U niversity of Colorado, Boulder, Chemistry and Biochemistry Department, 2003 Spin-spin splitting or J couplingMore complex splitting patterns1H NMR patterns are more complex than predicted by the N+1 coupling rule when coupling of one proton or set of equivalent protons occurs to two different sets of protons with different size coupling constants or when coupling occurs between protons with similar but not identical chemical shifts. The former situation can still be analyzed in terms of overlappingN+1 patterns using stick diagrams. This is shown for the spectrum of phenyloxirane which has three oxirane protons of different chemical shift all coupled to each other. The protons are labeled H A, H M, and H X to reflect that they are not close to each other in chemical shift. Each resonance appears as a doublet of doublets, and the overall pattern of three doublets of doublets is called an A M X pattern.The situation of protons with close chemical shifts coupled to each other is more complex. If only two protons are coupled to each other, the pattern still appears as two doublets but the intensities are no longer 1:1 and the chemical shifts are not the centers of the doublets; the separationbetween the lines of each doublet is still the coupling constant J. The chemical shifts are closer to the larger peaks of each doublet and can be calculated using a simple equation as shown below.If more than two protons of close chemical shift are coupled to each other, more complex patterns, often described as complex multiplets, are observed. Multiplets still provide useful structural information because they indicate the presence of coupled protons of similar chemical shift. The AB pattern and complex multiplet patterns result from what is called second order effects. Second order effects occur when the ratio of the chemical shift separation in Hz to the coupling constant is less than approximately 10 or /J < 10. Even when this ratio is greater than 10,slight intensity perturbation is evident in first order patterns as shown by the spectrum for 2-butanone. In fact, if we draw an arrow over the pattern showing the slight tilt (blue arrows in Figure 25), the arrows point toward each other. So we say the patterns for coupled protons point towards each other.Spin-spin splitting and couplingCoupling in 13C NMR spectraBecause the 13C isotope is present at only 1.1% natural abundance, the probability of finding two adjacent 13C carbons in the same molecule of a compound is very low. As a result spin-spin splitting between adjacent non-equivalent carbons is not observed. However, splitting of the carbon signal by directly bonded protons is observed, and the coupling constants are large, ranging from 125 to 250 Hz. Methyl groups appear as quartets, methylenes as triplets, methines as doublets, and unprotonated carbons as singlets. Commonly, splitting of the signal by protons is eliminated by a decoupling technique which involves simultaneous irradiation of the proton resonances at 300 MHz while observing the carbon resonances at 75 MHz. The decoupling is accomplished with a second broad band, continuous, oscillating magnetic field B2(as opposed to the pulsed B1field), and the decoupling is continued during data collection. The B2field causes rapid proton spin transitions such that the 13C nuclei lose track of the spin states of the protons. Figure 26 shows a proton decoupled 13C spectrum of ethyl acetate. The purpose of proton decoupling is to eliminate overlapping signal patterns and to increase the signal to noise ratio. Decoupling of the protons increases the signal to noise ratio by causing the collapse of quartets, triplets, and doublets to singlets and by causing a favorable increase in the number of carbons in the -spin state relative to the -spin state. The latter effect is called the NuclearOverhauser Effect (NOE); how it causes this change in spin state populations will not be discussed here.Integration of 1H NMR spectraThe area under each pattern is obtained from integration of the signal (or better the function that defines the signal) and is proportional to the number of hydrogen nuclei whose resonance is giving rise to the pattern. The integration is sometimes shown as a step function on top of the peak with the height of the step function proportional to the area. The integration of the patterns at 1.1, 2.4, and 3.7 ppm for methyl propanoate is approximately 3:2:3 (see figure 22). Note, the error in integration can be as high as 10% and depends upon instrument optimization. The integration of an 1H NMR spectrum gives a measure of the proton count adjusted for the molecular symmetry. Methyl propanoate has no relevant molecular symmetry and so, the integration gives the actual proton count: 3+2+3=8 protons. In contrast diethyl ether (Et-OEt) has a plane of symmetry which makes the two ethyl groups equivalent, and so, only two signal are observed, a triplet and a quartet, with integration 3:2.The areas represented by the integration step function is usually integrated by the instrument and displayed as numerical values under the scale. For instance, the normalized integration values for 2-butanoneare shown in Figure 27. Note that these values are not exact integers and need to be rounded to the nearest integer to obtain the proper value.Integration of 13C NMR SpectraIn a 1H NMR spectrum, the area under the signals is proportional to the number of hydrogens giving rise to the signal. As a result the integration of the spectrum is a measure of the proton count. In a 13C NMR spectrum the area under the signal is not simply proportional to the number of carbons giving rise to the signal because the NOE from proton decoupling is not equal for all the carbons. In particular, unprotonated carbons receive very little NOE, and their signals are always weak, only about 10% as strong as signals from protonated carbons.Because the resolution in 13C NMR is excellent, the number of peaks in the spectrum is a measure of the carbon count adjusted for the symmetry of the molecule. For example, hexane gives three peaks: the two methyls are equivalent as are two sets of methylenes. Several examples are analyzed as follows; the chemical shifts shown are not the observed values but calculated values from empirical rules:Hexane shows three peaks, two methyls and two sets of methylenes.Acetone shows two peaks, one for the methyls and one for the carbonyl carbon.Ethyl benzoate shows 7 peaks; the benzene ring shows only 4 peaks because of two sets of equivalent carbons.Ethyl 3-chlorobenzoate, however, shows 9 peaks, a separate signal for each carbon because it has no symmetry.Cis-1,2-dimethylcyclohexane shows 4 peaks; because of rapid chair-chair interconversion, we can analyze the NMR spectrum in terms of a flat structure; hence, the methyls are equivalent, as are the methines, and there are two sets of equivalent methylenes.Solvents for NMR spectroscopyA common solvent for dissolving compounds for 1H and 13C NMR spectroscopy is deuteriochloroform, DCCl3. In 1H NMR spectra, the impurity of HCCl3 in DCCl3gives a small signal at 7.2 ppm (see spectrum of methyl propanoate). In 13C spectroscopy 1.1% of the deuteriochloroform has a 13C isotope and it is bonded to a deuterium atom. The nucleus of the deuterium atom, the deuteron, has a more complicated nuclear spin than does the proton, and it has a gyromagnetic ratio () 1/6 as large. This more complicated nuclear spin gives rise to three spin states instead of the two spin states forthe proton, and the deuteron undergoes resonance at a different frequency than either the proton or 13C nucleus. These spin states are approximately all equally populated. Because the spin-spin coupling between the 13C and the deuterium is not eliminated during proton decoupling, the DCCl3shows three equal peaks of low to moderate intensity at about 77 ppm (see Figure 13). The separation is the carbon-deuterium coupling constant JCD. The intensity is low to moderate because the 13C receives no Nuclear Overhauser Enhancement from the proton decoupling.。

Coupling and Cohesion - CSECUET耦合与内聚csecuet

Stamp Coupling
• Definition: Component passes a data structure to another component that does not have access to the entire structure.
• Requires second component to know how to manipulate the data structure (e.g., needs to know about implementation)
Common Coupling (cont)
• Have to look at many modules to determine the current state of a variable
• Side effects require looking at all the code in a function to see if there are any global effects
– module p modifies a statement of module q – Module p refers to local data of module q
(in terms of a numerical displacement) – Module p branches to a local label of
• Internal glue with which component is constructed
• All elements of a component are directed toward and essential for performing the same task

a3g__金属材料术语

Metal Terminology (F)金属术语F2006-6-16 11:28:00FACE CENTERED(Concerning cubic sp ace lattices) - Having equivalent points at the corn ers of th e unit cell and at the cent ers of its six faces. A face-centered cubi c space lattice is characteristic of one o f the close-pack ed arrangements of equal hard spheres.FATIGUEThe ph eno menon leading to fracture und er repeated or fluctuating stress. Fatigue fractu res are prog ressive b eginning a s minute crack s and grow under the action of fluctuating stress.FERRITIC STAINLESS STEELHas a body centered cubi c (BCC) structure. Th ese alloy s are th e chro miu m stainless steels containing low carbon lev els. They are h ard enable primarily b y cold work ing, although some will hard en slightly b y heat treating. Ferritic stainless steels work harden much slower than austentitic stainless steels. FERROALLOYAn alloy o f iron with a sufficient amount of so me element or elements such as manganes e, ch romiu m or v anadiu m for us e as a means in adding thes e elements into molten steel.FERRO-MANGANESEAn allo y of iron and manganes e (80% mangan ese) us ed in mak ing additions of manganes e to steel or cast-iron.FERROUSRelated to iron (derived fro m the Latin ferru m.) Ferrous allo y s are, therefore, iron bas e allo y s.FIBER OR FIBREDirection in which metals have b een caus ed to flow, as b y rolling, with microscopic evidence in the fo rm o f fibrous app earan ce in the direction of flow. FIBER STRE SSUnit stress which exists at an y given point in a structural element sub jected to load; given as load p er unit area.FILED EDGES Finished edges, the fin al contours of which are produced b y drawing the strip over a series of s mall steel files. This is the usual and accepted method of dressing the edges of annealed spring steel strip after slitting in cases wh ere edgewise slitting crack s are ob jectionable or slitting burr is to be remov ed.FINISHED STEEL Steel that is read y for the mark et without further work or treatment. Blooms, billets, slabs, sheet bars and w ire rods are termed “semi-finished”.FINISHES The surface appearan ce o f the various metals after final treat ment such as rolling, etc. Ov er the years the following finishes have b eco me recognized as standard in their respective fields.ALUMINUM SHEET(A) Co mmercially Bright.(B) Bright one side.(A) Bright both sides(D) Embossed Sheets (Produ ced b y using embossed rolls.)BLACK PLATE(A) Dull finish without luster produced b y us e of rough ened rolls.(B) Bright finish - a luster finish produced b y use of rolls having a mod eratel y smooth surface.COLD ROLLED STEEL SHEETS(A) Co mmercial Finish. A dull satin surface texture produ ced b y rough ened rolls.(B) Commercial Bright Finish. Bright in appearance with a texture between luster and a very fine matte finish.(C) Luster Finish. Produced b y use of g round and polished rolls. (Note: This is not a nu mber 3 finish.)COLD ROLLED STRIP STEELSNo. 1 Finish - A dull finish produced without luster by rolling on roughened rolls.No. 2 Finish - A regular bright finish produ ced b y rolling on mod eratel y bright rolls.No. 3 Finish - Best Bright Finish. A lustrous or high gloss finish produced by rolling on highly polished rolls. Also referred to as “Mirror Finish”.COPPER BASE ALLOYSAcid Dipp ed - Dry rolled finished. Produced b y dry cold rolling bi-chro mate dipp ed allo y with polished rolls, resulting in a burnished appearance and retaining the color obt ained b y dipping (Tru e Metal Color).Bright Dipped Finish - Fin ish resulting from an acid dip.Buffed or Polished Surface - A finish obtained b y buffing, resulting in a high gloss or polished finish.Cold Rolled Finish - A relatively s mooth finish obtained b y cold rolling plain pick led strip with a lubricant.Dry Rolled Finish - A burnished finish resulting from d ry cold rolling b y use of polished rolls without an y metal lubricant.Hot Rolled Finish - A dark relatively rough oxidized finish resulting fro m rolling the metal while hot. May subsequ ently b e pick led or bright dipped but the rough surface remains.Stretched Brushed Finish (Satin Finish) Obtained by mechanically b rushing with wire brushes o r b y buffing.FLAT WIRENo. 2 Finish - A regular bright finish.No. 3 Finish - Best Bright High Gloss fin ish produced b y use of polished rolls. Or b y sp ecial bu ffing - this is a negotiated finish.STAINLESS COLD ROLLED SHEET and STRIP Nos. 1, 2B & 2D. No. 1 Finish - C. R. Annealed and pick led appearan ce varies from dull gray matte finish to a fairly reflective surface.No. 2B Finish - Same as No.1 Finish followed by a final light cold rolled p ass generall y on highly polished rolls.No. 2D Finish - A dull cold rolled finish produced b y cold rolling on dull rolls.STAINLESS C.R. SHEET - Polished FinishesNo. 3 Finish - This is an intermediat e polished finish.No. 4 Finish - Ground and Polished finish.No. 6 Finish - Ground, Polished and Tampico Brush ed.No. 7 Finish - Ground and High Luster Polished.No. 8 Finish - Ground and Polished to Mirror Finish.TEMPERED and UNTEMPERED COLD ROLLED CARBON SPRINGSTEEL STRIPClassified b y d escription as follows:(A) Black Oil Temp ered.(B) Scaleless Temp ered.(A) Bright Tempered.(D) Temp ered and Polished.(E) Temp ered, Polished and Colored (Blue or Straw).TIN PLATE(A) Bright Hot Dipped Finish.(B) Electro M atte Dull Finish.(C) Electro Bright Reflow Finish - produced b y the in-the-line thermal treat ment following electrodeposition.FINISHING TEM PERATURE Temp erature of final hot-work ing of a metal.FLAME ANNEALING A pro cess of so ftening a metal b y the application of h eat fro m a high temp erature flame.FLAME HARDENING A process of hardening a ferrous allo y b y heating it above th e transfo rmation rang e b y means o f a high-temp eratu re fl ame, and then cooling as required. FLAPPER VALVE STEEL An extremel y flat, v ery s mooth, very accu rate to g age, polished, harden ed and temp ered sp ring steel produced from approxi matel y 1.15% carbon. The name is derived from its co mmon and p rinciple usag e.FLATTENING(See Roller and Stretcher Lev eling)FLAT LATCH NEEDLE STEEL Supplied cold rolled and ann ealed. Carbon content .85. Supplied both in coil and flat length. Us ed to mak e flat latch needles which are us ed in the manu facture o f k nitted goods.FLAT WIRE A flat Cold Rolled, prepared edge section up to ?#148; wide, rectangular in shape. Generall y produ ced from hot rolled rods or speciall y prep ared round wire b y one or more cold rolling operations, primaril y for th e purpose o f obtaining the size and section desired. May also b e produced b y slitting cold rolled flat metal to desired width followed b y edge d ressing.FLOWLINESAlway s visible to a great er or less d egree wh en a longitudinal section has been subject ed to Macro etching, indicating the direction of wo rk or rolling.FLOW STRESSThe shear stress required to cause plastic d eformation of solid metals.FLUTINGKink ing or break age due to curving of met al strip on a radius so small, with relation to thick ness, as to stretch the outer su rface above its elastic limit. Not to be confused with the specific p roduct, Fluted Tubes.FOILMetal in an y width but no more than about 0.005” thick.FOLDSDefects caused in met al by continued fabrication of overl apping surfaces.FRACTURESurface app earance of met als when brok en.FRACTURE TESTNicking and break ing a bar b y means of sudden i mpact, to enable macros copic study o f the fracture.FRICTION GOUGES OR SCRATCHESA series of rel atively sho rt surface s crat ches variable in fo rm and sev erity. (See Galling)FULL ANNEALINGUsed p rincipall y on iron and steel, means h eating the metal to about 100癋. above th e critical temp erature range, followed b y “so ak ing” at this point and slow cooling below the critical temp erature.FULL FINISH PLATESteel sheet or strip reduced either hot or cold, cleaned, annealed, and then cold-rolled to a bright finish.FULL HARD TEMPER(A) No. 1 Temp er. In low carbon sheet or strip steel, stiff and spring y, not suitable fo r bending in an y dire ction. It is the hard est temp er obtain able b y hard cold rolling. (B) In Stainless Steel Strip, temp ers are b ased on minimu m tensile or y ield strength. For Chromiu m-Nick el grad es Full Hard temper is 185,000 TS, 140,000 YS Min. Term also used in connection with copper b ase alloy s and considered s y non y mous with Hard Temp er.。

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2000 Mathematics Subject Classiﬁcation. 81Q10, 47A10, 35P05. Key words and phrases. Quantum waveguides, quantum graphs, spectral theory, Kirchhoﬀ rule, Dirichlet boundary conditions, collapsing mesoscopic systems. The authors thank Pavel Exner and Rodolfo Figari for useful comments and discussions. This work was supported by the EU-Project “Quantum Probability with Applications to Physics, Information Theory and Biology”, by the Collaborative Research Center (SFB) 611 “Singular Phenomena and Scaling in Mathematical Models” and by the Deutsche Forschungsgemeinschaft (DFG).
1. Introduction In many microscopic systems a quantum particle is constrained by a conﬁning potential to a region with transversal dimensions small with respect to the longitudinal ones. For example in organic molecules the atoms make strong bonds and organize themselves on a regular structure, then the π -electrons move in correspondence of the bonds under the action of a strong conﬁning potential. Since the early 50s one dimensional models were used to describe the dynamics of π -electrons in such molecules (see, e.g., [26]). In more recent times a growing interest in the quantum dynamics of particles in quasi one dimensional structures has been driven by the possibility to realize devices with transversal dimensions on the scale of length of hundreds of nanometers, such as nanotubes or quantum wires. The possibility of a “nanotechnology” was already envisaged by R. Feynman in 1959 (see [12]), but the turning point can பைடு நூலகம்e ﬁxed in 1981 when G. Binnig and H. Rohrer, of IBM’s Z¨ urich Lab, invented the scanning tunneling microscope, making it possible to inspect and manipulate matter on the atomic scale. Quantum-graphs represent an excellent model for many quasi one dimensional structures like organic molecules, nanotubes and systems of quantum wires. In mathematical terms a quantum-graph is realized by a graph (i.e. a set of points, the vertices, and a set of ﬁnite or inﬁnite segments that connect the vertices, the edges), together with a quantum dynamics for a particle on the graph generated by self-adjoint diﬀerential or pseudo-diﬀerential operators on the graph (see [16], [17] and [18]). From the point of view of mathematical physics it is an open question to understand in which sense the one dimensional dynamics on a quantum graph approximates the dynamics of a particle constrained on a region with small transversal dimensions. Essentially one can isolate two problems: to determine which one dimensional, diﬀerential or pseudo-diﬀerential operators are most suitable in order to describe the dynamics on the edges and which couplings in the vertices among the edges can be physically feasible.
COUPLING IN THE SINGULAR LIMIT OF THIN QUANTUM WAVEGUIDES
SERGIO ALBEVERIO, CLAUDIO CACCIAPUOTI, AND DOMENICO FINCO
arXiv:math-ph/0611059v2 23 Feb 2007
Abstract. We analyze the problem of approximating a smooth quantum waveguide with a quantum graph. We consider a planar curve with compactly supported curvature and a strip of constant width around the curve. We rescale the curvature and the width in such a way that the strip can be approximated by a singular limit curve, consisting of one vertex and two inﬁnite, straight edges, i.e. a broken line. We discuss the convergence of the Laplacian, with Dirichlet boundary conditions on the strip, in a suitable sense and we obtain two possible limits: the Laplacian on the line with Dirichlet boundary conditions in the origin and a non trivial family of point perturbations of the Laplacian on the line. The ﬁrst case generically occurs and corresponds to the decoupling of the two components of the limit curve, while in the second case a coupling takes place. We present also two families of curves which give rise to coupling.
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SERGIO ALBEVERIO, CLAUDIO CACCIAPUOTI, AND DOMENICO FINCO
A strategy to approach both these open problems, in the case of diﬀerential operators, consists in studying the limit of the operator minus the Laplacian deﬁned on two or three dimensional domains with a graph-like topology but ﬁnite width, when the width goes to zero. This paper deals with the problem of the coupling in the vertices. For this reason we want to consider the simplest possible limit dynamics on the edges. We take a planar domain of constant width and which is straight outside a compact reg