Contact Terms and Duality Symmetry in The Critical Dissipative Hofstadter Model

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特种粉末冶金及复合材料制备加工第五届学术会议在安徽合肥隆重召开

第2期郭彦青等：2A I2铝合金粉末与T C4钛合金热等静压粉-固扩散连接• 73 •在进一步促进了 CU的扩散。

在扩散层靠近钛合金的一侧，并未检测出具体的化合物。

(3)利用CU作为中间层的扩散连接接头中间区域相比直接扩散连接的中间区域，硬度较低，为120HV,其剪切强度相比铝合金粉末和钛合金固体的直接扩散连接增加了 64%,达到了 23 MPa。

参考文献：[1] Leyens C, Peters M. Titanium and Titanium Alloys: Fundamentals and Applications[M]. Weinheim： Wiley-VCH, 2005.[2]Heinz A, Haszler A, Keidel C, et al. Recent development in aluminium alloys for aerospace applications[J]. Materials Scienceand Engineering(A), 2000, 280(1): 102.[3] WEI Y, LI J, XIONG J, et al. Joining aluminum to titanium alloy by friction stir lap welding with cutting pin[J]. MaterialsCharacterization, 2012,71(5): 1.[4] LI Y, LIU P, WANG J, et al. XRD and SEM analysis near thediffusion bonding interface of Mg/AI dissimilar materials[J].Vacuum, 2007, 82(1): 15.[5] REN J, LI Y, FENG T. Microstructure characteristics in the interface zone of Ti/Al diffusion bonding[J]. Materials Letters,2002, 56(5): 647.[6]Jiangwei R, Yajiang L. Tao F. Microstructure characteristics inthe interface zone of Ti/Al diffusion bonding[J]. Materials Letters, 2002, 56(5): 647.[7] W Y, A P W, G S Z, et al. Formation process of the bondingjoint in Ti/Al diffusion bonding[J]. Materials Science and Engi-neering(A), 2008, 480(1/2): 456.[8] Prescott R, Graham M J. The formation of aluminum oxidescales on high- temperature alloys[J]. Oxidation of Metals,1992,38(3/4): 233.[9] Cook G O, Sorensen C D. Overview of transient liquid phaseand partial transient liquid phase bonding[J]. Journal of Materials Science, 2011, 46( 16): 5305.[10] Kenevisi M S, Mousavi Khoie S M. An investigation on microstructure and mechanical properties of A17075 to Ti - 6A1 - 4Vtransient liquid phase (TLP) bonded joint[J]. Materials & Design, 2012(38): 19.[11] Alhazaa A, Khan T I, Haq I. Transient liquid phase (TLP) bonding of A17075 to Ti-6A1-4V alloy[J]. Materials Characterization, 2010, 61(3): 312.[12]郎利辉，王刚，布国亮，等.钛合金粉末热等静压数值模拟及性能研究[J].粉末冶金工业,2015, 25(3): 1.[13]喻思，郎利辉，王刚，等.热等静压成形2A12铝合金粉末的数值模拟研究[J].粉末冶金工业,2016, 26(2): 17.[14]喻思，郎利辉，王刚，等.2A12铝合金粉末热等静压成形的性能研究[J].粉末冶金工业，2015, 25(5): 42.[15]郎利辉，王刚，布国亮，等.热等静压工艺参数对2A12粉末铝合金性能的影响研究粉末冶金工业，2014, 24(5): 19.[16] Geng J, Oelhafen P. Photoelectron spectroscopy study of Al-Cuinterfaces[J], Surface Science, 2000,452(1 )： 161.•行业劲特种粉末冶金及复合材料制备/加工第五届学术会议在安徽合肥隆重召开2020年12月24-26日，“特种粉末冶金及复合材料制备/加工第五届学术会议”在安徽省合肥市世纪金源大饭店召开。

Integrability and PT-symmetry of $N$-Body Systems with Spin-coupling $delta$-Interactions

0+
1 0 c 1
ϕ ϕ′
,
0−
(1)
where c ∈ R, ϕ(x) is the scalar wave function of two spinless particles with relative coordinate x. (1) also describes two particles with spin s but without any spin coupling between the particles when they meet (i.e. x = 0), in this case ϕ represents any one of the components of the wave function. It is easily veriﬁed that the boundary condition (1) is both self-adjoint and PT-symmetric. Now consider two particles with spin s and have both δ-interactions and spin couplings when they meet. For a particle with spin s, the wave function has n = 2s + 1 components. Therefore two particles with δ-interactions have a general boundary condition, described in the center of mass coordinate system: ψ ψ′ =
interactions are studied [13]. In this article we study the PT-symmetric boundary conditions for “spin”-related δ-interactions and the corresponding integrability for many-body case. The family of the usual point interactions for the one dimensional Schr¨ odinger operator −

八次对称二维准晶的有限摩擦接触问题

punch. As a special case, the results m the paper can be degenerated into the corresponding conclusions m 2D decag
onal quasicrystals. K eyw ords：two-dimensional(2D) octagonal quasicrystals；frictional contact problem ；singularity；Riemann-Hilbert boundary problem ；complex variable method
in 1D and 2D qu asicrystals h alf-sp ac e，and the in trinsic Unks between the penetration depth and the indentation force w a s presented[10 11]. B y using complex variable method，the frictional and the adhesive contact problems w ere studied in 1D orth orh o m b ic，2D dodecagonal and 3D c o s a h e d ra l quasicrystals， and the expressions of contact stress and contact displacement were obtained under the action of a single rigid flat punch[12 14]. T h e smooth contact problem with cracks was discussed in elastic hal--plane of one-dimensional hexagonal quasicrystals. By means of reasonable stress function decomposition， the distribution of contact stress under the punch and the stress intensity factor at the crack ttp were obtained[15].

金兹堡朗道理论

Ginzburg–Landau theoryFrom Wikipedia, the free encyclopediaIn physics, Ginzburg–Landau theory, named after Vitaly Lazarevich Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomenological model which could describe type-I superconductors without examining their microscopic properties. Later, a version of Ginzburg–Landau theory was derived from the Bardeen-Cooper-Schrieffer microscopic theory by Lev Gor'kov, thus showing that it also appears in some limit of microscopic theory and giving microscopic interpretation of all its parameters.Contents•1Introduction•2Simple interpretation•3Coherence length and penetration depth•4Fluctuations in the Ginzburg–Landau model•5Classification of superconductors based on Ginzburg–Landau theory•6Landau–Ginzburg theories in string theory•7See also•8References•8.1PapersIntroduction[edit]Based on Landau's previously-established theory of second-order phase transitions, Ginzburg and Landau argued that the free energy, F, of a superconductor near the superconducting transition can be expressed in terms ofa complex order parameter field, ψ, which is nonzero below a phase transition into a superconducting state and isrelated to the density of the superconducting component, although no direct interpretation of this parameter was given in the original paper. Assuming smallness of |ψ| and smallness of its gradients, the free energy has the form ofa field theory.where F n is the free energy in the normal phase, α and β in the initial argument were treated as phenomenologicalparameters, m is an effective mass, e is the charge of an electron, A is the magnetic vector potential, and is the magnetic field. By minimizing the free energy with respect to variations in the order parameter and the vector potential, one arrives at the Ginzburg–Landau equationswhere j denotes the dissipation-less electric current density and Re the real part. The first equation — which bears some similarities to the time-independent Schrödinger equation, but is principally different due to a nonlinear term —determines the order parameter, ψ. The second equation then provides the superconducting current.Simple interpretation[edit]Consider a homogeneous superconductor where there is no superconducting current and the equation for ψ simplifies to:This equation has a trivial solution: ψ = 0. This corresponds to the normal state of the superconductor, that is for temperatures above the superconducting transition temperature, T>T c.Below the superconducting transition temperature, the above equation is expected to have a non-trivial solution (that is ψ ≠ 0). Under this assumption the equation above can be rearranged into:When the right hand side of this equation is positive, there is a nonzero solution for ψ (remember that the magnitude of a complex number can be positive or zero). This can be achieved by assuming the following temperature dependence of α: α(T) = α0 (T - T c) with α0/ β > 0:•Above the superconducting transition temperature, T > T c, the expression α(T) / β is positive and the right hand side of the equation above is negative. The magnitude of a complex number must be a non-negative number, so only ψ = 0 solves the Ginzburg–Landau equation.•Below the superconducting transition temperature, T < T c, the right hand side of the equation above is positive and there is a non-trivial solution for ψ. Furthermorethat is ψ approaches zero as T gets closer to T c from below. Such a behaviour is typical for a second order phase transition.In Ginzburg–Landau theory the electrons that contribute to superconductivity were proposed to forma superfluid.[1] In this interpretation, |ψ|2 indicates the fraction of electrons that have condensed into a superfluid.[1] Coherence length and penetration depth[edit]The Ginzburg–Landau equations predicted two new characteristic lengths in a superconductor which wastermed coherence length, ξ. For T > T c (normal phase), it is given bywhile for T < T c (superconducting phase), where it is more relevant, it is given byIt sets the exponential law according to which small perturbations of density of superconducting electrons recover their equilibrium value ψ0. Thus this theory characterized all superconductors by two length scales. The second one is the penetration depth, λ. It was previously introduced by the London brothers in their London theory. Expressed in terms of the parameters of Ginzburg-Landau model it iswhere ψ0 is the equilibrium value of the order parameter in the absence of an electromagnetic field. The penetration depth sets the exponential law according to which an external magnetic field decays inside the superconductor. The original idea on the parameter "k" belongs to Landau. The ratio κ = λ/ξ is presently known asthe Ginzburg–Landau parameter. It has been proposed by Landau that Type I superconductors are those with 0 < κ< 1/√2, and Type II superconductors those with κ> 1/√2.The exponential decay of the magnetic field is equivalent with the Higgs mechanism in high-energy physics. Fluctuations in the Ginzburg–Landau model[edit]Taking into account fluctuations. For Type II superconductors, the phase transition from the normal state is of second order, as demonstrated by Dasgupta and Halperin. While for Type I superconductors it is of first order as demonstrated by Halperin, Lubensky and Ma.Classification of superconductors based on Ginzburg–Landau theory[edit]In the original paper Ginzburg and Landau observed the existence of two types of superconductors depending on the energy of the interface between the normal and superconducting states.The Meissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. In Type I superconductors, superconductivity is abruptly destroyed when the strength of the applied field rises above a critical value H c. Depending on the geometry of the sample, one may obtain an intermediate state[2] consisting of a baroque pattern[3] of regions of normal material carrying a magnetic field mixed with regions of superconducting material containing no field. In Type II superconductors, raising the applied field past a critical value H c1 leads to a mixed state (also known as the vortex state) in which an increasing amount of magnetic flux penetrates the material, but there remains no resistance to the flow of electric current as long as the current is not too large. At a second critical field strength H c2, superconductivity is destroyed. The mixed state is actually caused by vortices in the electronic superfluid, sometimes called fluxons because the flux carried by these vortices is quantized. Most pure elemental superconductors, except niobium and carbon nanotubes, are Type I, while almost all impure and compound superconductors are Type II.The most important finding from Ginzburg–Landau theory was made by Alexei Abrikosov in 1957. He used Ginzburg–Landau theory to explain experiments on superconducting alloys and thin films. He found that in a type-II superconductor in a high magnetic field, the field penetrates in a triangular lattice of quantized tubes offlux vortices.[citation needed]Landau–Ginzburg theories in string theory[edit]In particle physics, any quantum field theory with a unique classical vacuum state and a potential energy witha degenerate critical point is called a Landau–Ginzburg theory. The generalization to N=(2,2) supersymmetric theories in 2 spacetime dimensions was proposed by Cumrun Vafa and Nicholas Warner in the November 1988 article Catastrophes and the Classification of Conformal Theories, in this generalization one imposes thatthe superpotential possess a degenerate critical point. The same month, together with Brian Greene they argued that these theories are related by a renormalization group flow to sigma models on Calabi–Yau manifolds in thepaper Calabi–Yau Manifolds and Renormalization Group Flows. In his 1993 paper Phases of N=2 theories intwo-dimensions, Edward Witten argued that Landau–Ginzburg theories and sigma models on Calabi–Yau manifolds are different phases of the same theory. A construction of such a duality was given by relating the Gromov-Witten theory of Calabi-Yau orbifolds to FJRW theory an analogous Landau-Ginzburg "FJRW" theory in The Witten Equation, Mirror Symmetry and Quantum Singularity Theory. Witten's sigma models were later used to describe the low energy dynamics of 4-dimensional gauge theories with monopoles as well as brane constructions. Gaiotto, Gukov & Seiberg (2013)See also[edit]•Domain wall (magnetism)•Flux pinning•Gross–Pitaevskii equation•Husimi Q representation•Landau theory•Magnetic domain•Magnetic flux quantum•Reaction–diffusion systems•Quantum vortex•Topological defectReferences[edit]1.^ Jump up to:a b Ginzburg VL (July 2004). "On superconductivity and superfluidity (what I have and havenot managed to do), as well as on the 'physical minimum' at the beginning of the 21 st century". Chemphyschem.5 (7): 930–945. doi:10.1002/cphc.200400182. PMID15298379.2.Jump up^ Lev D. Landau; Evgeny M. Lifschitz (1984). Electrodynamics of Continuous Media. Course ofTheoretical Physics8. Oxford: Butterworth-Heinemann. ISBN0-7506-2634-8.3.Jump up^ David J. E. Callaway (1990). "On the remarkable structure of the superconductingintermediate state". Nuclear Physics B344 (3): 627–645. Bibcode:1990NuPhB.344..627C.doi:10.1016/0550-3213(90)90672-Z.Papers[edit]•V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz.20, 1064 (1950). English translation in: L. D. Landau, Collected papers (Oxford: Pergamon Press, 1965) p. 546• A.A. Abrikosov, Zh. Eksp. Teor. Fiz.32, 1442 (1957) (English translation: Sov. Phys. JETP5 1174 (1957)].) Abrikosov's original paper on vortex structure of Type-II superconductors derived as a solution of G–L equations for κ > 1/√2•L.P. Gor'kov, Sov. Phys. JETP36, 1364 (1959)• A.A. Abrikosov's 2003 Nobel lecture: pdf file or video•V.L. Ginzburg's 2003 Nobel Lecture: pdf file or video•Gaiotto, David; Gukov, Sergei; Seiberg, Nathan (2013), "Surface Defects and Resolvents" (PDF), Journal of High Energy Physics。

接触氧化法英文

接触氧化法英文Introduction:The contact oxidation method is an important wastewater treatment technology that has been widely used in various industries such as chemical engineering, textile, food processing and so on. In this article, we will discuss the principle, advantages and disadvantages, as well as the application scope of contact oxidation method.Principle:Contact oxidation method refers to the use of aerobic microorganisms to degrade organic pollutants in wastewater. The process mainly consists of two stages: the first stage is the adsorption and decomposition of organic pollutants on the surface of the carrier, in which the organic matter is converted into CO2 and H2O under the action of oxygen and bacteria; the second stage is the final purification and release of wastewater, which is achieved through the settling of activated sludge.Advantages and Disadvantages:Advantages:1. High treatment efficiency: contact oxidation method has a higher treatment efficiency compared to other biological treatment technologies, which can effectively remove the biochemical oxygen demand (BOD) and chemical oxygen demand (COD) in wastewater.2. Low investment and operating costs: the equipment required for contact oxidation method is simple and easy to maintain, and the energy consumption is low, which can save a lot ofinvestment and operating costs.3. Wide adaptability: contact oxidation method can adapt to various types of wastewater, including high concentration, low temperature, and low pH wastewater.Disadvantages:1. Limited treatment capacity: the treatment capacity of contact oxidation method is relatively low, which is not suitable for large-scale wastewater treatment.2. High sensitivity to shock: the contact oxidation method is sensitive to the shock of organic pollutants, and the sudden increase of organic pollutants may cause a decrease in the treatment efficiency.3. Impact on the environment: the sludge generated during contact oxidation may cause secondary pollution to the environment if not treated properly.Application Scope:Contact oxidation method is widely used in various industries such as chemical engineering, textile, food processing, and municipal wastewater treatment. It is particularly suitable for the treatment of small and medium-sized enterprises' wastewater, as well as the purification of urban sewage and the reconstruction of old sewage treatment plants.Conclusion:The contact oxidation method is an important technology for wastewater treatment, which has the advantages of high treatment efficiency, low investment and operating costs, and wide adaptability. However, it also has its weaknesses, such as limited treatment capacity and high sensitivity to shock. In practice, it should be used in combination with other wastewater treatment methods to achieve better results.。

CONTSEM-L2-DefiningContact

Solving Contact Problems with New Capabilities in Abaqus
Defining Contact Pairs
L2.4
Defining Contact Pairs
• Features of contact pairs:
• Contact between pairs of surfaces is explicitly defined. The user defines:
L2.15
Defining Contact Pairs
• Nonoverlapping surfaces
• Can include or filter out
• Distance is within search tolerance
• But surface normals do not intersect the opposing surface (nonoverlapping)
TOPPLATE
Surface
• Automatic free surface generation on bumper elements:
*SURFACE,NAME=BUMPER-EXT BUMPER,
SHAFT
Surface
• Discrete rigid surfaces:
*RIGID BODY, ELSET=BOTDIE, REF NODE=BOTRP *SURFACE,NAME=BOTPLATE BOTDIE, SPOS *RIGID BODY, ELSET=TOPDIE, REF NODE=TOPRP *SURFACE, NAME=TOPPLATE TOPDIE, SPOS *RIGID BODY, ELSET=SHAFTDIE, REF NODE=SHAFTRP *SURFACE, NAME=SHAFT SHAFTDIE, SPOS

半导体物理与器件——Terms汉译英

半导体物理与器件——Terms(术语)U1 Terms:Semiconductor physics and devices半导体物理与器件,Space lattice空间晶格, unit cell晶胞, primitive cell原胞,basic crystal structures 基本晶格结构(five), Miller indices密勒指数, atomic bonding原子价键U2 Terms:quantum mechanics量子力学,energy quanta能量子, wave-particle duality波粒二象性,the uncertainty principle测不准原理/海森堡不确定原理Schrodinger's wave equation薛定谔波动方程, eletrons in free Space自由空间中的电子the infinite potential well无限深势阱, the step potential function 阶跃势函数, the potential barrier势垒.U3 Terms:Pauli exclusion principle泡利不相容原理, quantum state量子态. allowed energy band允带, forbidden energy band禁带.conduction band导带, valence band价带,hole空穴, electron 电子.effective mass有效质量.density of states function状态密度函数,the Fermi-Dirac probability function费米-狄拉克概率函数,the Boltzmann approximation波尔兹曼近似,the Fermi energy费米能级.U4 Terms:charge carriers载流子, effective density of states function有效状态密度函数,intrinsic本征的,the intrinsic carrier concentration本征载流子浓度, the intrinsic Fermi level本征费米能级.charge neutrality电中性状态, compensated semiconductor补偿半导体, degenerate简并的,non-degenerate非简并的, position of E F费米能级的位置U5 Terms:drift current漂移电流, diffusion current 扩散电流,mobility迁移率, lattice scattering晶格散射, ionized impurity scattering 电离杂质散射, velocity saturation饱和速度,conductivity电导率,resistivity电阻率.graded impurity distribution杂质梯度分布,the induced electric field感生电场, the Einstein relation爱因斯坦关系, the hall effect霍尔效应U6 Terms:nonequilibrium excess carriers非平衡过剩载流子,carrier generation and recombination载流子的产生与复合,excess minority carrier过剩少子,lifetime寿命,low-level injection小注入,ambipolar transport双极输运, quasi-Fermi energy准费米能级.U7 Terms:the space charge region空间电荷区,the built-in potential内建电势, the built-in potential barrier内建电势差,the space charge width空间电荷区宽度, zero applied bias零偏压, reverse applied bias反偏, onesided junction单边突变结.U8 Terms:the PN junction diode PN结二极管, minority carrier distribution少数载流子分布, the ideal-diode equation理想二极管方程, the reverse saturation current density反向饱和电流密度.a short diode短二极管,generation-recombination current产生-复合电流,the Zener effect齐纳效应, the avalanche effect雪崩效应, breakdown击穿.U9 Terms:Schottky barrier diode (SBD)肖特基势垒二极管,Schottky barrier height肖特基势垒高度.Ohomic contact欧姆接触,heterojunction异质结, homojunction单质结,turn-on voltage开启电压,narrow-bandgap窄带隙, wide-bandgap宽带隙,2-D electron gas二维电子气U10 Terms:bipolar transistor双极晶体管,base基极, emitter发射极, collector集电极.forward active region正向有源区, inverse active region反向有源区, cut-off截止, saturation饱和,current gain电流增益,common-base共基, common-emitter共射.base width modulation基区宽度调制效应, Early effect厄利效应, Early voltage厄利电压U11 Terms:Gate栅极, source源极, drain漏极, substrate基底.work function difference功函数差threshold voltage阈值电压, flat-band voltage平带电压enhancement mode增强型, depletion mode耗尽型strong inversion强反型, weak inversion弱反型,transconductance跨导, I-V relationship电流-电压关系。

The-Tyger英文赏析

The Tyger and The lamb:In The Tyger Blake points to the contrast between these two animals: the tiger is fierce, active, predatory, while The Lamb is meek, vulnerable and harmless. The reference to the lamb in the penultimate stanza reminds the reader that a tiger and a lamb have been created by the same God, and raises questions about the implications of this. It also invites a contrast between the perspectives of "experience" and "innocence" represented here and in the poem "The Lamb." "The Tyger" consists entirely of unanswered questions, and the poet leaves us to awe at the complexity of creation, the sheer magnitude of God's power, and the inscrutability of divine will. The perspective of experience in this poem involves a sophisticated acknowledgment of what is unexplainable in the universe, presenting evil as the prime example of something that cannot be denied, but will not withstand facile explanation, either. The open awe of "The Tyger" contrasts with the easy confidence, in "The Lamb," of a child's innocent faith in a benevolent universe.Theme:The poem is more about the creator of the tiger than it is about the tiger itself. The poet was at a loss to explain how the same God who made the lamb could make the tiger. So, the theme is : humans are incapable of fully understanding the mind of God and the mystery of his handiwork.Symbolism:Black writing his poems in plain an direct language. He presents his view in visual images rather that abstract ideas. Symbolism in wide range is a distinctive feature of his poetry. The Tyger, included in Songs of Experience, is one of Blake's best-known poems. It seemingly praises the great power of tiger, but what the tiger symbolizes remains disputable: the power of man? Or the revolutionary force? Or the evil? The poem is highly symbolic with a touch of mysticism and it is open to various interpretations. The tiger initially appears as a strikingly sensuous image. However, as the poem progresses, it takes on a symbolic character, and comes to embody the spiritual and moral problem the poem explores: perfectly beautiful and yet perfectly destructive, Blake's tiger becomes the symbolic center for an investigation into the presence of evil in the world. Since the tiger's remarkable nature exists both in physical and moral terms, the speaker's questions about its origin must also encompass both physical and moral dimensions. The poem's series of questions repeatedly ask what sort of physical creative capacity the "fearful symmetry"of the tiger bespeaks; assumedly only a very strong and powerful being could be capable of such a creation.Background:"The Tyger" just might be William Blake’s most famous poem. Published in a collection of poems :Songs of Experience in 1794, Blake wrote "The Tyger" during his more radical period. He wrote most of his major works during this time, often railing against oppressive institutions like the church or themonarchy, or any and all cultural traditions – sexist, racist, or classist –which stifled imagination or passion. The French revolution is a revolution against the feudalism, it has profound effects on the Britain. It brings the thoughts of “liberty”, “equality”, “fraternity”to the English. After the industrial revolution, the contradictions of the British social class becomes more serious. People found that the industry and technology just brought them with pain instead of happiness. So more and more people became disappointed about the society. That’s why William Blake has changed his writing style during this time.Blake published an earlier collection of poetry: the Songs of Innocence in 1789. Once Songs of Experience came out five years later, the two were always published together. In general, Songs of Innocence contains idyllic poems, many of which deal with childhood and innocence. Idyllic poems have pretty specific qualities: they’re usually positive, sometimes extremely happy or optimistic and innocent. They also often take place in pastoral settings :think countryside; springtime; harmless, cute wildlife; sunsets; babbling brooks; wandering bards; fair maidens, and many times praise one or more of these things as subjects. William Blake published the Songs of Experience in 1794, often railing against oppressive institutions like the church or the monarchy, or any and all cultural traditions –sexist, racist, or classist –which stifled imagination or passion. The Songs of Innocence was published in 1789. In general, Songs of Innocence contains idyllic poems, many of which deal with childhood and innocence. Idyllic poems have pretty specific qualities: they’re usually positive, sometimes extremely happy or optimistic and innocent. They also often take place in pastoral settings :think countryside; springtime; harmless, cute wildlife; sunsets; babbling brooks; wandering bards; fair maidens, and many times praise one or more of these things as subjects. The themes of the two collections are extremely different.The first and last stanzas are identical except the word "could" becomes "dare" in the second iteration. Kazin says to begin to wonder about the tiger, and its nature, can only lead to a daring to wonder about it. Blake achieves great power through the use of alliteration ("frame" and "fearful") combined with imagery, (burning, fire, eyes), and he structures the poem to ring with incessant repetitive questioning, demanding of the creature, "Who made thee?". In the second stanza the focus moves from the tiger, the creation, to the creator – of whom Blakes wonders "What dread hand? & what dread feet?" . "The Tyger" is six stanzas in length, each stanza four lines long. Much of the poem follows the metrical pattern of its first line and can be scanned as trochaic tetrameter catalectic. A number of lines, however—such as line four in thefirst stanza—fall into iambic tetrameter.The first and last stanzas are identical except the word "could" becomes "dare" in the second iteration. Kazin says to begin to wonder about the tiger, and its nature, can only lead to a daring to wonder about it. Blake achieves great power through the use of alliteration ("frame" and "fearful") combined with imagery, (burning, fire, eyes), and he structures the poem to ring with incessant repetitive questioning, demanding of the creature, "Who made thee?". In the second stanza the focus moves from the tiger, the creation, to the creator – of whom Blakes wonders "What dread hand? & what dread feet?".[1] "The Tyger" is six stanzas in length, each stanza four lines long. Much of the poem follows the metrical pattern of its first line and can be scanned as trochaic tetrameter catalectic. A number of lines, however—such as line four in thefirst stanza—fall into iambic tetrameter.The Tyger" is the sister poem to "The Lamb" (from "Songs of Innocence"), a reflection of similar ideas from a different perspective (Blake's concept of "contraries"), with "The Lamb" bringing attention to innocence. "The Tyger" presents a duality between aesthetic beauty and primal ferocity, and Blake believes that to see one, the hand that created "The Lamb", one must also see the other, the hand that created "The Tyger”. The Songs of Experience were written as a contrary to the "Songs of Innocence" – a central tenet in Blake's philosophy, and central theme in his work The struggle of humanity is based on the concept of the contrary nature of things, Blake believed, and thus, to achieve truth one must see the contraries in innocence and experience. Experience is not the face of evil but rather another facet of that which created us. Kazin says of Blake that, "Never is he more heretical than ... where he glories in the hammer and fire out of which are struck ... the Tyger".[1] Rather than believing in war between good and evil or heaven and hell Blake thought each man must first see and then resolve the contraries of existence and life; in the "The Tyger" he presents a poem of "triumphant human awareness", and "a hymn to pure being", according to Kazin.。

拉雷尔电子有限公司产品说明书

LAUREL ELECTRONICS, INC.4-20 mA & Serial Output Transmitter for Resistance Input in OhmsFeatures•4-20 mA, 0-20 mA, 0-10V or -10V to +10V transmitter output, 16 bits, isolated•RS232 or RS485 serial data output, Modbus or Laurel ASCII protocol, isolated•Dual 120 mA solid state relays for alarm or control, isolated•Five precalibrated resistance input ranges from 20.000 Ω to 200.00 kΩ•Fixed 2.0000 ohm, 2.0000 MΩ and 20.000 MΩ range available as a factory specia• 1 mΩ resolution on 20 Ω scale•Custom curve linearization for changing resistance transducers•2, 3 or 4-wire connection with lead resistance compensation•Analog output resolution 0.0015% of span (16 bits), accuracy ±0.02% of span•DIN rail mount housing only 22.5 mm wide, detachable screw-clamp connectors•Universal 85-264 Vac / 90-300 Vdc or 10-48 Vdc / 12-32 Vac powerDescriptionThe Laureate Resistance Transmitter is factory calibrated forfive jumper selectable resistance ranges from 20 Ω to 200 kΩ.Fixed factory-special ranges of 2.000 Ω, 2.0000 MΩ and 20.000MΩ are also available. Accuracy is an exceptional 0.01% of fullscale ± 2 counts. Resolution is one part in 20,000. In the 20 Ωrange, resolution is 1 mΩ, making the transmitter suitable forcontact resistance and conductance measurements.Transmitter connections can be via 2, 3 or 4 wires. With 4-wirehookup, 2 wires are used for excitation and two separate wiresare used to sense the voltage across the resistance to bemeasured, thereby eliminating any lead resistance effects. With3-wire hookup, the transmitter senses the combined voltage dropacross the RTD plus two excitation leads. It also senses thevoltage drop across one excitation lead, and then subtracts twicethis voltage from the combined total. This technique effectivelysubtracts the lead resistance if the excitation leads are the same.All resistance ranges are digitally calibrated at the factory, withcalibration factors stored in EEPROM on the signal conditionerboard. This allows ranges and signal conditioner boards to bechanged in the field without recalibrating the transmitter. Ifdesired, the transmitter can easily be calibrated using externalstandards plus scale and offset in software.Fast read rate at up to 50 or 60 conversions per second whileintegrating the signal over a full power line cycle is provided byConcurrent Slope (Pat 5,262,780) analog-to-digital conversion.High read rate is ideal for peak or valley capture and for real-timecomputer interface and control.Open sensor indication is standard and may be set up to indi-cate either upscale or downscale. Excitation is provided by thetransmitter.Custom curve linearization, available with the Extendedversion, makes this transmitter ideal for use with transducerswhose output is a changing resistance.Standard features of Laureate transmitters include:•4-20 mA, 0-10V or -10V to +10V analog transmitter output,isolated, jumper-selectable and user scalable. All selectionsprovide 16-bit (0.0015%) resolution of output span and 0.02%output accuracy of a reading from -99,999 to +99,999 countsthat is also transmitted digitally. Output isolation from signaland power grounds eliminates potential ground loop problems.•Serial communications output, isolated. User selectableRS232 or RS485, half or full duplex. Three protocols are userselectable: Modbus RTU, Modbus ASCII, or Laurel ASCII.Modbus operation is fully compliant with Modbus Over SerialLine Specification V1.0 (2002). The Laurel ASCII protocolallows up to 31 Laureate devices to be addressed on thesame RS485 data line. It is simpler than the Modbus protocoland is recommended when all devices are Laureates.•Dual solid state relays, isolated. Available for local alarm orcontrol. Rated 120 mA at 130 Vac or 170 Vdc.•Universal 85-264 Vac power. Low-voltage 10-48 Vdc or 12-32 Vac power is optional.Easy Transmitter programming is via Laurel's InstrumentSetup Software, which runs on a PC under MS Windows. Thissoftware can be downloaded from this website at no charge. Therequired transmitter-to-PC interface cable is available from Laurel(P/N CBL04).SpecificationsRange Resolution Accuracy Excitation Current0-2.0000 Ω 0-20.000 Ω 0-200.00 Ω 0-2000.0 Ω 0-20000 Ω 0-200.00 kΩ 0-2.0000 MΩ 0-20.000 MΩ 0.1 mΩ1 mΩ10 mΩ100 mΩ1 Ω10 Ω100 Ω1 kΩ±0.01% of range± 2 counts5 mA5 mA500 µA50 µA5 µA500 nA500 nA75 nASignal InputInput Resolution Input Accuracy Update Rate, Max 16 bits (65,536 steps)±0.01% of full scale ± 2 counts 50/sec at 50 Hz, 60/sec at 60 HzAnalog Output (standard)Output Levels Compliance at 20 mA Compliance at 10V Output Resolution Output Accuracy Output Isolation 4-20 mA, 0-20 mA, 0-10 Vdc, -10 to +10Vdc (user selectable) 10V (0-500Ω load)2 mA (5 kΩ load or higher)16 bits (65,536 steps)0.02% of output span plus conversion accuracy250V rms working, 2.3 kV rms per 1 minute testSerial Communications (standard)Signal TypesData RatesOutput Isolation Serial Protocols Modbus Modes Modbus Compliance Digital Addressing RS232 or RS485 (half or full duplex)300, 600, 1200, 2400, 4800, 9600, 19200 baud250V rms working, 2.3 kV rms per 1 min testModbus RTU, Modbus ASCII, Laurel ASCIIRTU or ASCIIModbus over Serial Line Specification V1.0 (2002)247 Modbus addresses. Up to 32 devices on an RS485 line w/o a repeater.Dual Relay Output (standard)Relay Type Load Rating Two solid state relays, SPST, normally open, Form A 120 mA at 140 Vac or 180 VdcPower InputStandard Power Low Power Option Power Frequency Power Isolation Power Consumption 85-264 Vac or 90-300 Vdc10-48 Vdc or 12-32 VacDC or 47-63 Hz250V rms working, 2.3 kV rms per 1 min test 2W typical, 3W with max excitation outputMechanicalDimensions MountingElectrical Connections 129 x 104 x 22.5 mm case35 mm rail per DIN EN 50022 Plug-in screw-clamp connectorsEnvironmentalOperating Temperature Storage Temperature Relative Humidity Cooling Required 0°C to 55°C-40°C to 85°C95% at 40°C, non-condensingMount transmitters with ventilation holes at top and bottom. Leave 6 mm (1/4") between transmitters, or force air with a fan.PinoutMechanicalQA Application with Relays in Passband ModeA deviation limit (50 mΩ in this example) is set uparound both sides of a setpoint. The relay closes (oropens) when the reading falls within the deviationband, and opens (or closes) when the reading fallsoutside of this band. This mode sets up a passbandaround the setpoint and can be used for contactresistance testing in a production environment.RTD HookupIn 4-wire hookup, different pairs of leads are used to apply the excitation current and sense the voltage drop across the unknown resistance, so that the IR drop across the excitation leads is not a factor.In 3-wire hookup, the transmitter senses the combined voltage drop across the unknown resistance plus two excitation leads. It also senses the voltage drop across one excitation lead, and then subtracts twice this voltage from the combined total. This technique effectively subtracts all lead resistance and compen-sates for ambient temperature changes if the two excitation leads are identical.In 2-wire hookup, the transmitter senses the combined voltage drop across the unknown resistance and both lead wires. The voltage drop across the lead wires can be measured by shorting out the resistance during transmitter setup, and this voltage is then automatically subtracted from the combined total. However, changing resistance of the lead wires due to ambient tempera-ture changes will not be compensated.Ordering GuideCreate a model a model number in this format: LT20R1Transmitter Type LT Laureate 4-20 mA & RS232/RS485 output transmitter Main Board 2 Standard Main BoardPower0 Isolated 85-264 Vac or 90-300 Vdc 1 Isolated 10-48 Vdc or 12-32 VacResistance RangeR0 0-20 ohms (factory special fixed range) R1 0-20 ohms R2 0-200 ohms R3 0-2 kohms R4 0-20 kohms R5 0-200 kohmsR6 0-2 Mohms (factory special fixed range) R7 0-20 Mohms (factory special fixed range)Note: The same signal conditioner board can be used for resistance and RTD temperature measurement.AccessoriesCBL04 RS232 cable, 7ft. Connects RS232 screw terminals of LT transmitter to DB9port of PC.CBL02 USB to RS232 adapter cable. Combination of CBL02 and CBL04 connectstransmitter RS232 terminals to PC USB port.。

Finding community structure in networks using the eigenvectors of matrices

Finding community structure in networks using the eigenvectors of matrices
M. E. J. Newman
Department of Physics and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109–1040
We consider the problem of detecting communities or modules in networks, groups of vertices with a higher-than-average density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the beneﬁt function known as “modularity” over possible divisions of a network. Here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix, which plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations. This result leads us to a number of possible algorithms for detecting community structure, as well as several other results, including a spectral measure of bipartite structure in neteasure that identiﬁes those vertices that occupy central positions within the communities to which they belong. The algorithms and measures proposed are illustrated with applications to a variety of real-world complex networks.

液晶聚合物：从结构到应用说明书

v
vi
Preface
Chapters 2 and 3 describe methods of characterizing the mesophases. In the former a comprehensive guide to textures is given. Chapter 4 describes the dielectric properties of LCPs. Chapters 5 to 8 inclusive deal with comb and longitudinal LCPs, lyotropic and thermotropic. In these chapters are discussed the syntheses, characterization, structures and properties of these materials. Chapter 9 deals with some applications of LCPs.
Division of Applied Physics, School of Science,
Sheffield Hallam University,
UK
ELSEVIER APPLIED SCIENCE
LONDON and NEW YORK
ELSEVIER SCIENCE PUBLISHERS LTD Crown House, Linton Road, Barking, Essex IGll 8JU, England
A.A. COLLYER
Sheffield Hal/am University, UK
Springer-sbm Archive Dordrecht
LIQUID CRYSTAL POLYMERS:

材料科学相关英文术语

仪器分析方法：差热分析：Differential thermal analysis，简称DTA差示扫描量热法：Differential scanning calorimetry，简称DSC热重法：Thermogravimetry，简称TGCharacterization, property, crystalline, inorganic, nanoscale, emphasis, solids, periodic, crystal, bonding, structure, diffraction, classical, quantum models, electronic, applications, semiconductors, chemical, structure-property relationships, synthesis methods, optical, electronic, junction, nanomaterials, real space lattices, reciprocal lattice and diffraction, diffraction/phase diagrams, band theory(能带理论), photonic, amorphous, synthetic biomaterials, infrared, conducting, polymers, solar cells, plasmonics, dielectrics, classifications, natural, artificial, ceramics, alloy, ferroelectrics, superconductors, magnetic, optical, microelectronic, surface engineering, superalloys, structural, composites, composition, biological, measure, characterize, remake, experimentalist, theoretician, materials by design, characterization techniques, crystallography, X-ray/electron scattering, microscopy, spectroscopy, scanning probes, electrical, magnetic, materials modeling, advanced, experimental, optimization, integration, certification, manufacturing, initiative, acceleration, liquid, resistance, planes, chains, diode(二极管), transistors(晶体管), zeolites(沸石), catalysis, separation, purification, hexagon(六边形), pentagon(五边形), spherical, fullerene(富勒烯), allotrope(同素异形体), graphene, dimethyl(二甲基), size-dependent property, solution, discrete(分裂的，不连续的), bandgap(能带隙), periodicity, quartz, formation, kinetics, favorable kinetics, periodic, array, atom, ion, molecules, lattice, symmetry(对称性), Bravais lattices, orientation, vector(矢量), constant(常数), zincblende, miller indices, crystallography, planes, directions, row, d-spacing, individual, set of, equivalent, axis, intercept(截距), specify, lattice directions, quasicrystals(准晶体), diagonal(对角线), interpenetrate(互相渗透), hexagonal close packed(hcp, 密方六排结构), crystallography晶体学, integer 整数, inverse intercept 截距的倒数, axes 轴的复数, reciprocal倒数, equivalent, symmetry 对称性, perpendicular 垂直的, parallelepiped 平行六面体, translational invariance 平移不变性, point symmetry 点对称, aperiodic 非周期性的(quasi-periodic), binary 二元的, ternary 三元的, intermetallic 金属间的, geology地质学, spectroscopy光谱学, pyrite黄铁矿, galena 方铅矿, coordination number(CN, 配位数), surface plasmon resonance(SPR, 表面等离子共振), quantum dots 量子点, coprecipitation 共沉淀, high-temperature decomposition 高温分解法, micro-emulsion 微乳液法, gel-sol 溶胶凝胶法, sonochemistry 超声化学法, laser pyrolysis 激光分解法, biotinylated 生物素化, real-time 实时, longitudinal relaxation 纵向弛豫, transverse relaxation 横向弛豫, superparamagnetic 超顺磁性, boundary effects 边界效应, intergration by parts 分步积分, curie temperature 居里温度, ferromagnetic 铁磁性, aperiodic 非周期性的, stereograms 立体图, pyrite 黄铁矿, galena 方铅矿, cosine 余弦, sine 正弦, tangent 正切, stoichiometry 化学计量学, thermochemistry 热化学, enthalpy 焓, electron spin 电子旋转, Pauli exclusion principle保利不相容原理, electron affinity 电子亲和性, electronegativity 电负性, energetics动力学, covalent bonding 共价键, molecular geometry 分子几何学, bonding theories 键合理论, chemical kinetics 化学动力学, classical thermodynamics 经典热力学, entropy 熵, Gibbs free energy 吉布斯自由能, crystal field theory, 晶体场理论, mineralogy 矿物学, metallurgy 冶金学, array 排列, hollow 空隙, stack 堆垛, repulsion 排斥力, coordination number 配位数, packing efficiency 致密度, density 晶体密度, tetrahedrons 四面体, pentagons 五角形, constituent 构成的/成分, spatial dimensions 三维空间, arbitrary 随意的, planar 平面的, enantiomorphic 对映体, parallelepiped 平行六面体, coplanar 共面的,perpendicular 垂直的, adjacent 邻近的, interplanar spacing 平面间距, parallel 平行的, fractional 分数的, ionization energy 电离能, cation 阳离子, anion 阴离子, electrostatic 静电的, covalency 共价, intermolecular force 分子力, dipole-dipole force 偶极作用, rock salt 岩盐, alkali 碱halide 卤化物, hydride 氢键, octahedra 八面体, polyhedral 多面体的, arsenide 砷化物, fluorite 萤石, calcium 钙, hole 空隙, antifluorite 反萤石, wurtzite 纤维锌矿, van der Waals attraction 范德瓦尔斯力, rutile(TiO2) 金红石, spinel 尖晶石, perovskite 钙钛矿, Ilmenite 钛铁矿, radii(radius的名词复数), Lanthanide contraction 镧系收缩, proportional 比例的, interlocking 联锁的, valence 原子价, brittle 易碎的, silica 硅石,二氧化硅, large coefficient of expansion 扩散系数, Band model 能带模型, silicates 硅酸盐, polymorphic 多晶形的, quartz 石英, ionization energy 离子能, electron affinity 电子亲和势, lattice energy 晶格能, magnitudes 数级, permittivity 介电常数, permittivity of vacuum 真空介电常数, equation 方程式, polarizable 可极化的, molten state 熔融状态, solubility 溶解度, solvent 溶剂, polar solvent 极性溶剂,句子：1.Materials Chemistry is the foundation for the field of Nanoscience and technology.材料化学是纳米科学技术领域的基石。

A Workshop of MIT-RTG

Here are several broad goals of the workshop we could keep in mind:1)Understand equivalence between constructible sheaves and Fukaya category of cotangent bundle.Consider applications such as homological characterizations of com-pact exact branes,mirror symmetry for toric varieties,and Springer theory.In the ﬁrst two cases,sheaves help us understand branes;in the third case,branes help us understand sheaves.All of the material here is available in the literature.2)Understand what the above equivalence should imply about quantizations of more general exact symplectic targets arising in representation theory.For example,we should be able to see the relation between quantizations of Slodowy slices in the form of the Fukaya category and in the form of modules over W-algebras.To my knowledge, this is not completely mapped out by the literature.3)Discuss directions for further investigation of relations between Fukaya categories and categories in representation theory.Starting point:Fukaya categories of cotangent bundles toﬂag varieties,D-modules on moduli spaces of bundles.Here the literature points to many open questions.1.Singularities and constructible sheaves.1.1.Tame geometry.Subanalytic geometry.Deﬁning functions.Whitney stratiﬁca-tions and triangulations.Thom isotopy lemmas.Example of real line.Build up the notion of subanalytic subset of a real analytic manifold by starting with the real line and then considering standard operations(with an emphasis on the special role of the image of a map).Explain relation between closed subanalytic subsets and zeros of subanalytic functions.Discuss axioms of Whitney stratiﬁcations and results about stratifying and triangulating subanalytic sets.Discuss Thom iso-topy lemmas,in particular the assertion:if f:M→N is a proper stratiﬁed map, then stratum-preserving homeomorphisms of N(smooth along each stratum)lift to stratum-preserving homeomorphisms of M(smooth along each stratum).Describe local structure of Whitney stratiﬁcations as iterated cone bundles along strata.This lecture should contain many simple counterexamples.For example,to illustrate the Whitney conditions,one could discuss the Whitney umbrella and cusp.Refs:[BM88],[VM96]1.2.Homotopical categories.Diﬀerential graded and A∞-categories.Functors and modules.Linear structure:shifts and cones.Localization with respect to collection of morphisms.Homological perturbation theory.This lecture should approach categories as multi-pointed versions of algebras.In fact,we should have in mind the case where the number of points isﬁnite so that in the end we could think in terms of algebras.Introduce chain complexes and basic notions:tensor and hom,shift,sum and sum-mand,cone,quasi-equivalence,...Introduce strong notion of algebra(diﬀerential graded12algebra)and weak notion of algebra(A∞-algebra).Draw operadic pictures for A∞-categories and functors between them.Describe equivalence of diﬀerential graded cate-gories and A∞-categories via homological perturbation theory.Explain what is gained (and perhaps lost)in the stabilization(Morita theory)of A∞-categories by passing to perfect modules:idempotent-completion of complexes of representable functors to chain complexes.Reminder that triangulated categories arise in nature as the under-lying discrete categories of stable A∞-categories.Describe localization of a category (basic example:passing from modules over an algebra to modules over a localization of the algebra).Refs:[Ke06],[S],[L]1.3.Constructible sheaves.Diﬀerential graded category of sheaves.Functoriality under maps.Standard triangles and bases.Relation to constructible functions.This lecture can be in the more traditional language of triangulated categories as long as it is understood that all of the constructions and results can be lifted to the diﬀerential graded setting.Fix Whitney stratiﬁcation S of real analytic manifold X.Introduce diﬀerential graded category of S-constructible complexes.Discuss case where S consists of one stratum X itself(local systems and complexes with locally constant cohomology).In-troduce Grothendieck’s6operations(f∗,f∗),(f!,f!),⊗,H om and Verdier duality.Con-struct standard triangles associated to pair of an open U⊂X and closed V=X\U. Calculate morphisms between standard extensions of constant sheaves on strata of S. Possibly include:informal discussion of exit-path simplicial category of a Whitney stratiﬁcation,and constructible sheaves asﬁnitely-generated modules over the exit-path category.Explain how stalk Euler characteristic identiﬁes Grothendieck group of constructible sheaves with constructible functions.Refs:[KS84],[GM83]1.4.Examples.Constructible sheaves on R stratiﬁed with a single marked point.Con-structible sheaves on S1stratiﬁed with a single marked point.Constructible sheaves on A1stratiﬁed with a single marked point.Constructible sheaves on P1stratiﬁed with a single marked point.The aim here is to give quiver presentations of categories of constructible sheaves in some simple examples.By choosing enough functionals,we can describe a category with the description depending on the functionals.For the above examples,choose various functionals and describe resulting quivers.Describe objects representing the functionals considered.For example,ﬁrst construct quiver arising from considering generic stalk and stalk at marked point,then construct quiver using generic stalk and vanishing cycles at marked point.Further example:a three stratum space such as A2with a marked singular curve.2.Microlocal geometry of sheaves2.1.Cotangent bundles.Exact symplectic structure.Geodesicﬂow.Examples of Lagrangians:conormals,graphs and generalizations.Conormals to stratiﬁ-grangian correspondences.3 Summary of basic structures in exact symplectic geometry with emphasis on the case of cotangent bundles,including Liouvilleﬂow,contact hypersurfaces,compatible almost complex structures,exact Lagrangians,...Explain meaning of basic objects in terms of classical mechanics.Describe graph Lagrangians and conormal Lagrangians and their hybrids.Construct Lagrangian correspondences of cotangent bundles from maps of base manifolds,emphasizing case of projection and inclusion.Refs:[A],[KS84]2.2.Characteristic cycles.From constructible sheaves to conical Lagrangian cycles. Functoriality under maps.Introduce group of conical Lagrangian cycles.Construct characteristic cycle of con-structible sheaf on a manifold.Calculate everything in case when manifold is real line or complex line.Explain functoriality for Grothendieck’s6operations(f∗,f∗),(f!,f!),⊗,H om and Verdier duality.Show characteristic cycle construction descends to iso-morphism between group of constructible functions and group of conical Lagrangian cycles.Refs:[KS84],[SV96]2.3.Intersection of Lagrangian cycles.Perturbations near inﬁnity.Intersections of characteristic cycles:compatibility with ext-pairing of constructible sheaves and corresponding pairing of constructible functions.Index theorems.Describe framework of perturbing conical Lagrangians by normalized geodesicﬂow near inﬁnity.Discuss Z/2-grading on intersections of conical Lagrangian cycles.Show characteristic cycle takes pairing on constructible functions to intersection of conical Lagrangian cycles.Dubson-Kashiwara index formula(generalization of Poincar´e-Hopf index formula):calculate global Euler characteristic of constructible sheaf as intersec-tion with zero section.Construct automorphisms of group of conical Lagrangian cycles via motions of pieces of support.Example of Dehn twist on conical Lagrangian cycles in T∗S1.This topic is logically independent of the preceding but is reasonable to discuss at this juncture.Refs:[GrM97],[NZ09]2.4.Riemann-Hilbert correspondence.Diﬀerential operators as quantization of functions on cotangent bundle.Algebraic model of constructible sheaves:regular holo-nomic D-modules.Explain Riemann-Hilbert correspondence between regular holonomic D-modules and constructible sheaves.Discuss the failure of an abelian version and the resulting notion of a perverse sheaf.Introduce the singular support of a D-module and its relation to characteristic cycles.Illustrate everything with the case of A1stratiﬁed by a single marked point.Refs:[Be],[Kap]3.Exact Lagrangians in cotangent bundles3.1.Morse category of submanifolds.Gradient tree A∞-category of submanifolds with local systems.Equivalence with constructible sheaves.4This lecture should explain how the diﬀerential graded category of constructiblesheaves on a manifold can be reformulated in terms of a Morse A∞-category whoseobjects are locally closed submanifolds equipped with local systems.Basic case:explain equivalence of de Rham algebra of compact manifold with MorseA∞-algebra.This will provide opportunity to interpret A∞-operad in terms of trivalentgraphs.Show how Morse theory provides geometric ingredients to apply homologicalperturbation theory.(For bonus points:mention other sources of parallel geometricingredients such as Hodge theory.)Main topic:interpret constructible sheaves in termsof Morse theory.Explain how Thom’s isotopy lemma allows one to replace locally closedsubmanifolds with singular boundary with open submanifolds with smooth boundary.Draw vectorﬁelds for constructible sheaves and calculate morphisms,for example forR stratiﬁed with a single marked point,and A1stratiﬁed with a single marked point. Possible further topic:explain some of Grothendieck’s6operations in terms of MorseA∞-category.Refs:[HL01],[KS01],[NZ09]3.2.Exact Floer-Fukaya theory.Fukaya category of compact exact Lagrangians inexact symplectic target.Brane structures.Moduli spaces of anization intoA∞-category.This lecture should be an introduction to Fukaya categories of exact targets.Seidel’sbook provides the foundations and the speaker should choose appropriate highlights.Itis likely more worthwhile that we understand the broad picture than the analytic details.We should know what brane structures are and why that is what they are(gradingsof intersections and orientations of moduli spaces).We should hear enough about thebehavior of moduli of disks to see the A∞-structure(most prominently,there shouldbe a discussion of the A∞-equations coming from the boundary of moduli).We shouldhear enough about continuation maps to believe that everything is well-deﬁned.Thelecture can restrict to compact Lagrangians as we will be hearing about non-compactones soon enough.Should discuss the Piunikhin-Salamon-Schwarz(PSS)calculationof endomorphisms of compact branes.Resolutions of du Val singularities and theirdeformations would be a good example to illustrate the theory(and will appear inlater talks).Refs:[S]3.3.Inﬁnitesimal Fukaya category of cotangent bundle.Noncompact branes:perturbations,tameness,bounds on parisons with directed and wrappedFukaya categories.Equivalence of subcategory of standard branes with Morse categoryof submanifolds.This talk should consist of roughly two halves:general theory of Fukaya categorieswith non-compact branes and example of the cotangent bundle.First half.Survey general techniques for dealing with disks along noncompact branes:energy bounds,tameness,diameter estimates.For exact target withﬁxed energy func-tion,introduce the inﬁnitesimal Fukaya category where small Hamiltonian perturba-tions of branes are used near inﬁparisons could be made with directed Fukaya-Seidel categories of Lefschetzﬁbrations,and also wrapped Fukaya categories where theHamiltonian perturbations are not small but rather linear near inﬁnity.5 Second half.Explain why the Morse A∞-version of constructible sheaves embeds in the inﬁnitesimal Fukaya category of the cotangent bundle.Here the main ingredient is Fukaya-Oh’s analytic equivalence between gradient trees and pseudo-holomorphic disks (or alternatively,hybrid moduli spaces interpolating between them).Refs:[S],[Sik94],[FO97],[NZ09],[Nspr]3.4.Equivalence of sheaves and branes.Formalism of Yoneda lemma and bimod-ules.Beilinson’s argument.Decomposition of diagonal.Noncharacteristic motions.The aim of this talk is to prove that the inﬁnitesimal Fukaya category of the cotangent bundle is equivalent to constructible sheaves.At this point,what is left to prove is that the standard branes coming from standard sheaves indeed generate.Begin with general discussion of the Yoneda lemma,functors and bimodules,and the formalism of generators.Introduce Beilinson’s construction of generators for coherent sheaves on projective space as guiding example.Bulk of talk should be devoted to applying this argument to the inﬁnitesimal Fukaya category of the cotangent bundle. Here the main ingredient is the notion of non-characteristic propagation.Thom’s iso-topy lemma should be reinterpreted in the language of non-characteristic maps.An analogous lemma for continuation maps of branes should be formulated.Finally,we should see at least a sketch of Beilinson’s argument in the setting of the inﬁnitesimal Fukaya category.Application:homological characterization of compact exact branes in cotangent bundle.Refs:[B78],[N09],[Nspr]4.Some examples and applicationsThis day’s talks are more independent of each other and the speciﬁc material covered can be determined by the speaker’s taste.4.1.Mirror symmetry for toric varieties.Fukaya category of cotangent bundle of torus.Consider torus(S1)n=R n/Z n.Introduce alternative viewpoints on symplectic geometry of(C×)n T∗(S1)n via two projections T∗(S1)n→(S1)n and T∗(S1)n→(R∨)n.Describe branes arising from considering a toric compactiﬁcation of(C×)n. Explain how to think about them in terms of constructible sheaves.Discuss mirror symmetry and dual description of coherent sheaves in terms of constructible sheaves. Extra credit:equivariant generalization.If we understand nothing else,we should at least understand mirror symmetry be-tween A-model of T∗S1and B-model of P1.Refs:[FLTZ]and related papers.4.2.Springer theory.Fukaya category of cotangent bundle of Lie algebra.Fourier transform from Floer perspective.Describe basic diagram of Springer theory arriving at Springer brane in T∗g.Intro-duce Fourier dual perspective and Fourier transform for branes.Deduce consequences for Springer brane.Interpret preceding in classical language of constructible sheaves.Refs:[BoM81],[Nspr]64.3.Microlocalization and Hamiltonian reduction.Formalism of microlocaliza-tion and Hamiltonian reduction.Introduction to crepant resolutions,their deformations and quantizations.This talk and the one that follows could be planned in tandem.One approach would be to have theﬁrst talk cover theory,and the second cover examples.In any event,the speakers should strategize together.We should see that many important examples of exact symplectic manifolds(sym-plectic resolutions with C×-action)arising in representation theory can be constructed from conical open subsets of cotangent bundles via Hamiltonian reduction.We should learn how to think about categories(Fukaya,modules over deformation quantization) associated to such targets and their deformations can be arrived at from categories (Fukaya,microlocal constructible sheaves and D-modules)associated to conical open subsets of cotangent bundles via Hamiltonian reductions.4.4.W-algebras from topological viewpoint.Fukaya category beyond compact branes in Slodowy slices.This talk could map out the relation between branes in Slodowy slices and their deformations,sheaves onﬂag manifolds(or equivalently,regular holonomic D-modules onﬂag manifolds),and modules over W-algebras.The speciﬁc example of du Val resolutions could be discussed concretely.Refs:[KhS],[SS],[Ma],[Lo]among many related papers.5.Further directions5.1.Gauge theory setting.Hitchin integrable system.Relation to talks of previous day.Challenge of quantization ofﬁbers.Refs:[BD],[KW],[Kap]5.2.Where to go from here.References[A]Arnold,V.I.“Mathematical methods of classical mechanics.”Translated from the Russian byK.Vogtmann and A.Weinstein.Graduate Texts in Mathematics,60.Springer-Verlag,New York-Heidelberg,1978.x+462pp.ISBN:0-387-90314-3[B78]A.A.Be˘ılinson,“Coherent sheaves on P n and problems in linear algebra,”(Russian)Funktsional.Anal.i Prilozhen.12(1978),no.3,68–69;English translation:Functional Anal.Appl.12(1978), no.3,214–216(1979).[BD]A.Beilinson and V.Drinfeld,“Quantization of Hitchin Hamiltonians and Hecke Eigensheaves,”preprint.[Be]J.Bernstein,“Algebraic theory of D-modules.”/~mitya/langlands/Bernstein/Bernstein-dmod.ps[BM88]E.Bierstone and man,“Semianalytic and subanalytic sets,”Inst.Hautes´Etudes Sci.Publ.Math.67(1988),5–42.[FLTZ]Bohan Fang,Chiu-Chu Melissa Liu,David Treumann,Eric Zaslow.“T-Duality and Homolog-ical Mirror Symmetry of Toric Varieties”,arXiv:0811.1228[BoM81]W.Borho and R.MacPherson,“Repr´e sentations des groupes de Weyl et homologie d’intersection pour les vari´e t´e s nilpotentes,”C.R.Acad.Sci.Paris S´e r.I Math.292(1981),no.15,707–710.7 [FO97]K.Fukaya and Y.-G.Oh,“Zero-loop open strings in the cotangent bundle and Morse homo-topy,”Asian.J.Math.1(1997)96–180.[FSS]Kenji Fukaya,Paul Seidel,Ivan Smith,“The symplectic geometry of cotangent bundles from a categorical viewpoint”.arXiv:0705.3450.[GM83]M.Goresky and R.MacPherson.“Intersection homology.II.”Invent.Math.72(1983),no.1, 77–129.[GrM97]M.Grinberg and R.MacPherson.“Euler characteristics and Lagrangian intersections.”Sym-plectic geometry and topology(Park City,UT,1997),265–293,IAS/Park City Math.Ser.,7, Amer.Math.Soc.,Providence,RI,1999.[HL01]F.R.Harvey and wson,Jr.,“Finite Volume Flows and Morse Theory,”Annals of Math.vol.153,no.1(2001),1–25.[Kap]A.Kapustin,“A-branes and noncommutative geometry,”arXiv:hep-th/0502212.[KW]A.Kapustin and E.Witten,“Electric-Magnetic Duality And The Geometric Langlands Pro-gram,”arXiv:hep-th/0604151.[KS84]M.Kashiwara and P.Schapira,Sheaves on manifolds.Grundlehren der Mathematischen Wis-senschaften292,Springer-Verlag(1994).[Ke06]B.Keller,On diﬀerential graded categories.arXiv:math.AG/0601185.International Congress of Mathematicians.Vol.II,151–190,Eur.Math.Soc.,Z¨u rich,2006.[KhS]Mikhail Khovanov,Paul Seidel“Quivers,Floer cohomology,and braid group actions”, arXiv:math/0006056.[KS01]M.Kontsevich and Y.Soibelman,“Homological Mirror Symmetry and Torus Fibrations,”Sym-plectic geometry and mirror symmetry(Seoul,2000),203–263,World Sci.Publ.,River Edge,NJ, 2001.[Lo]I.Losev,“Finite W-algebras”,arXiv:1003.5811.[L]J.Lurie,“Stable Inﬁnity Categories,”arXiv:math/0608228.[Ltft]J.Lurie,“On the Classiﬁcation of Topological Field Theories,”arXiv:0905.0465[Ma]Ciprian Manolescu“Link homology theories from symplectic geometry”,arXiv:math/0601629. [N09]D.Nadler,“Microlocal Branes are Constructible Sheaves,”Selecta Math.15(2009),no.4,563–619.[Nspr]D.Nadler,“Springer theory via the Hitchinﬁbration”,arXiv:0806.4566.[NZ09]D.Nadler,E.Zaslow,“Constructible sheaves and the Fukaya category.”J.Amer.Math.Soc.22(2009),no.1,233–286.[SV96]W.Schmid and K.Vilonen,“Characteristic cycles of constructible sheaves,”Invent.Math.124 (1996),451–502.[S]P.Seidel,Fukaya Categories and Picard-Lefschetz Theory.[SS]Paul Seidel,Ivan Smith“A link invariant from the symplectic geometry of nilpotent slices”, arXiv:math/0405089.[STZ]N.Sibilla,D.Treumann,E.Zaslow,“Ribbon Graphs and Mirror Symmetry I”,arXiv:1103.2462. [Sik94]J.-C.Sikorav,“Some properties of holomorphic curves in almost complex manifolds,”in Holo-morphic Curves in Symplectic Geometry,Birkh¨a user(1994),165–189.[Sl80]P.Slodowy,Simple singularities and simple algebraic groups.Lecture Notes in Mathematics, 815.Springer,Berlin,1980.[VM96]L.van den Dries and ler,“Geometric categories and o-minimal structures,”Duke Math.J.84,no.2(1996),497–539.。

材料领域英文术语

metal 金属ceramic 陶瓷polymer 聚合物Composites 复合材料Semiconductors 半导体Biomaterials 生物材料Processing 加工过程Structure 组织结构Properties 性质Performance 使用性能Mechanical properties力学性能Electrical properties 电性能Thermal behavior 热性能Magnetic properties 磁性能Optical properties 光性能Deteriorative characteristics 老化特性Atomic mass unit (amu) 原子质量单位Atomic number 原子数Atomic weight原子量Bohr atomic model 波尔原子模型Bonding energy 键能Coulombic force 库仑力Covalent bond 共价键Dipole (electric) 偶极子electronic configuration 电子构型electron state 电位Electronegative 负电的Electropositive 正电的Ground state 基态Hydrogen bond 氢键Ionic bond 离子键Isotope 同位素Metallic bond 金属键Mole 摩尔Molecule 分子Pauli exclusion principle 泡利不相容原理Periodic table 元素周期表Polar molecule 极性分子Primary bonding 强键Quantum mechanics 量子力学Quantum number 量子数Secondary bonding 弱键valence electron 价电子van der waals bond 范德华键Wave-mechanical model 波粒二象性模型Allotropy 同素异形现象Amorphous 无定形Anion 阴离子Anisotropy 各向异性atomic packing factor(APF) 原子堆积因数body-centered cubic (BCC) 体心立方结构Bragg’s law 布拉格定律Cation 阳离子coordination number 配位数crystal structure 晶体结构crystal system 晶系crystalline 晶体的diffraction 衍射face-centered cubic (FCC) 面心立方结构grain 晶粒grain boundary 晶界hexagonal close-packed (HCP) 六方密堆积isotropic 各向同性的lattice 晶格lattice parameters 晶格参数miller indices 密勒指数noncrystalline 非结晶的octahedral position 八面体配位polycrystalline 多晶的polymorphism 多晶形single crystal 单晶tetrahedral position 四面体配位unit cell 晶胞covalent bonds: 共价键hydrogen bonds: 氢键van der Waals bonds: 范德华键（分子键）Hydrocarbons: 烃: 由碳和氢组成的物质methane (CH4) 甲烷Ethylene C2H4 乙烯acetylene, C2H2 乙炔Unsaturated Hydrocarbons: 不饱和烃Saturated Hydrocarbons: 饱和烃Paraffin: 石蜡Macromolecule: 高分子Isomerism: 同质异构Mer: 链节:组成聚合物链重复单位的原子集合。

Duality symmetry, strong coupling expansion and universal critical amplitudes in two-dimens

a r X iv:c ond-ma t/99868v1[c ond-m at.stat-m ech]4Au g1999Duality symmetry,strong coupling expansion and universal critical amplitudes in two-dimensional Φ4ﬁeld models Giancarlo Jug INFM –UdR Milano,and Dipartimento di Scienze,Universit`a dell’Insubria Via Lucini 3,22100Como (Italy)∗and Max-Planck-Institut f¨u r Physik komplexer Systeme,N¨o thnitzer Str.38D-01187Dresden (Germany)Boris N.Shalaev Fachbereich Physik,Universit¨a t-Gesamthochschule Essen,D-45117Essen (Germany)∗∗Abstract We show that the exact beta-function β(g )in the continuous 2D g Φ4model possesses the Kramers-Wannier duality symmetry.The duality symmetry transformation ˜g =d (g )such that β(d (g ))=d ′(g )β(g )is constructed and the approximate values of g ∗computed from the duality equation d (g ∗)=g ∗are shown to agree with the available numerical results.The calculation of the beta-function β(g )for the 2D scalar g Φ4ﬁeld theory based on the strong coupling expansion is developed and the expansion of β(g )in powers of g −1is obtained up to order g −8.The numerical values calculated for the renormalized coupling constant g ∗+are in reasonable good agreement with the best modern estimates recently obtained from the high-temperature series expansion and with those known from the perturbative four-loop renormalization-group calculations.The application of Cardy’s theorem for calculating the renormalized isothermal coupling constant g c of the 2D Ising model and the related univer-sal critical amplitudes is also discussed.PACS numbers:05.50.+q,03.70.+k,64.60.-i,75.10.HkTypeset using REVT E XI.INTRODUCTIONIn this paper we study mainly the symmetry properties of the beta-functionβ(g)for the 2D gΦ4theory,regarded as a continuum limit of the exactly solvable2D Ising model.In contrast to the latter,the2D gΦ4theory is not an integrable quantumﬁeld theory.This means,in particular,that the theory does not possess the factorized scattering matrix,and therefore that the thermodynamic Bethe ansatz method cannot be applied at all.Thus,despite the fact that the2D Ising model at h=0can be solved by many diﬀerent methods(see[1]for an excellent review),the beta-functionβ(g)of its continuum limit is to date known only in the four-loop approximation within the framework of conventional perturbation theory atﬁxed dimension d=2[2–4].Calculations of beta-functions are of great interest in statistical mechanics and quantumﬁeld theory.The beta-function contains the essential information on the renormalized coupling constant g∗,this being important for constructing the equation of state of the2D Ising model–for example–which remains still a challenging problem,rich in applications.This and other considerations do not allow us to regard the2D Ising model as having fully been solved.The2D Ising model and some other lattice spin models are known to possess the remark-able Kramers-Wannier(KW)duality symmetry,playing an important role both in statistical mechanics and in quantumﬁeld theory[5–7].The self-duality of the isotropic2D Ising model means that there exists an exact mapping between the high-T and low-T expansions of the partition function[7].In the transfer-matrix language this implies that the transfer-matrix of the model under discussion is covariant under the duality transformation.If we assume that the critical point is unique,the KW self-duality would yield the exact Curie tempera-ture of the model.This holds for a large set of lattice spin models including systems with quenched disorder(for a review see[7,8]).Over twenty years ago the KW self-duality was shown to be equivalent to a Fourier tranformation in target space[9].Also,it has been recognised long ago that self-duality combined with some special algebraic properties of a model leads to the existence of an inﬁnite set of conserved charges[10].Duality is thus known to impose some important constraints on the exact beta-function[11,12].The other main purpose of this paper is to develop a strong coupling expansion for the calculation of the beta-function of the2D scalar gΦ4theory as an alternative approach to standard perturbation theory.It will then be of interest to match this expansion with the results of a four-loop approximation(where possible)by constructing a smooth interpolation with respect to g.It is in fact well known from quantumﬁeld theory and statistical mechanics that any strong coupling expansion is closely connected with a suitable high-temperature (HT)series expansion for a lattice model[1,7].From theﬁeld-theoretical point of view the HT series are nothing but strong coupling expansions forﬁeld models,the lattice being considered as a technical device to deﬁne cutoﬀ-regularisedﬁeld theories.Recently,the high-temperature(HT)series expansions and perturbative calculations for the gΦ4ﬁeld theory atﬁxed dimensions d<4have been a topic of intense studies(for references see below).Computing critical exponents and various critical amplitude ratios from series expansion data has a long history going back to the early1960s.Nowadays there are a good number of papers containing a large body of information for the N-vector model deﬁned on diﬀerent lattices for d=2,3,4,5and arbitrary N[13–15].It is remarkable thatthe HT series data for the zeroﬁeld susceptibilityχand the second correlation momentµ2 of the N-component classical Heisenberg ferromagnet have been extended up to the order K21(K=J/T),the data for the secondﬁeld derivative of the susceptibility(χ4)being available through to the order K17.Having been equiped with this information,one may try to employ diﬀerent techniques of resummation of the existing HT series expansions, like Pad´e approximants or more subtle approaches,for computing critical exponents and universal critical amplitude ratios[13,16–21].It is worth noting that the strong-coupling behavior of the gΦ4theory has recently been treated within the framework of a variational perturbative approach[22].The paper is organized as follows.In Sect.II we set up basic notations and deﬁne the duality symmetry transformation˜g=d(g).Then it is proved thatβ(d(g))=d′(g)β(g).An approximate expression for d(g)providing good estimates for g∗+(the renormalisedﬁxed-point coupling constant along the isochore line)is found.In Sect.III the HT series expansion data are used to obtain the strong coupling expansion ofβ(g)for the2D0(N)-symmetric gΦ4 theory in powers of1/g up to the order g−8.Some numerical estimates for the renormalized coupling constant g∗+above T c are obtained.We then compare theﬁxed point values found to those already known from the four-loop renormalization-group(RG)calculations and from the HT series expansions.In Sect.IV we also discuss the application of Cardy’s formula both for the exact calculation of the renormalized isothermal coupling constant g∗c at T c and,for some universal critical amplitudes,along the isothermal critical line.Sect.V ﬁnally contains some concluding remarks.The Appendix presents a simple derivation of the correlation lengthξand of the exact beta-functionβIsing(T)for the lattice2D Ising model, where the temperature T plays the role of an eﬀective coupling constant,and we discuss some of their properties.II.DUALITY SYMMETRY OF THE BETA-FUNCTION We begin by considering the classical Hamiltonian of the2D Ising model(in the absence of an external magneticﬁeld),deﬁned on a square lattice with periodic boundary conditions; as usual:H=−J <i,j>σiσj(2.1)where<i,j>indicates that the summation is over all nearest-neighboring sites;σi=±1are spin variables and J is a spin coupling.The standard deﬁnition of the spin-pair correlation function reads:G(R)=<σRσ0>(2.2) where<...>stands for a thermal average.The correlation length may be deﬁned in many diﬀerent ways,all deﬁnitions being equiv-alent to each other in the close vicinity of the critical point[13].This,in fact,reﬂects the arbitrariness somewhat inherent in any renormalization scheme.The statistical mechanics deﬁnition of the correlation length is given by[23]ξ2=d ln G(p)2dµ0(2.5)where d is the spatial dimension(in our case d=2).It should be mentioned that the above deﬁnition ofξdiﬀers from the one used in other related approaches,e.g.[11].The2D Ising model near T c is known to be equivalent to the gΦ4theory with a one-component real order parameter.In order to extend the KW duality symmetry to the continuousﬁeld theory we have need for a”lattice”model deﬁnition of the coupling constant g,equivalent to the conventional one exploited in the RG approach.The renormalization coupling constant g of the gΦ4theory is closely related to the fourth derivative of the ”Helmholtz free energy”,namely∂4F(T,m)/∂m4,with respect to the order parameter m= Φ .It may be deﬁned as follows(see[13,14,24]and references therein)g(T,h)=−(∂2χ/∂h2)χ3ξd(2.6)whereχis the homogeneous magnetic susceptibilityχ= d2xG(x)(2.7) It is in fact easy to show that g(T,h)in Eq.(2.6)is merely the standard four-spin correlation function taken at zero external momenta.The renormalized coupling constant of the critical theory is deﬁned by the double limitg∗=limh→0limT→T cg(T,h)(2.8)and it is well known that these limits do not commute with each other.As a result,g∗is a path-dependent quantity in the thermodynamic(T,h)plane[13].Here we are mainly concerned with the coupling constant on the isochore line g(T> T c,h=0)in the disordered phase and with its critical valueg∗+=limT→T+c g(T,h=0)=−∂2χ/∂h2The”lattice”coupling constant g∗+deﬁned in Eq.(2.9)is in a given correspondence with the temperature T c.We shall see that it will be more convenient to deal with a new variable s=exp(2K)tanh(K),where K=J/T.The standard KW duality tranformation is known to be as follows[6,7]sinh(2˜K)=1(1−s)2is a self-dual quantity.Now,on the one hand,we have the formal relationξds(g)dgβ(g)(2.11)where s(g)is deﬁned as the inverse function of g(s),i.e.g(s(g))=g and the beta-function is given,as usual,byξdgdξ=2s(1−s)(1+s(g))(ds(g)/dg)(2.14) Let us deﬁne the dual coupling constant˜g and the duality transformation function d(g)ass(˜g)=1s(g))(2.15)where s−1(x)stands for the inverse function of x=s(g).It is easy to check that a further application of the duality map d(g)gives back the original coupling constant,i.e.d(d(g))= g,as it should be.Notice also that the deﬁnition of the duality transformation given by Eq.(2.15)has a form similiar to the standard KW duality equation,Eq.(2.10).It is easy to prove that d′(g∗)=±1.The maps we are looking for have d′(g∗)=−1,since the opposite sign leads to the trivial solution d(g)≡g.This is also shown in the Appendix.Consider now the symmetry properties ofβ(g).We shall see that the KW duality sym-metry property,Eq.(2.10),results in the beta-function being covariant under the operation g→d(g):β(d(g))=d′(g)β(g)(2.16)To prove it let us evaluateβ(d(g)).Then Eq.(2.14)yieldsβ(d(g))=2s(˜g)(1−s(˜g))s(g)(1+s(g))(ds(˜g)/d˜g)(2.18)The derivative in the r.h.s.of Eq.(2.18)should be rewritten in terms of s(g)and d(g).It may be easily done by applying Eq.(2.15):ds(˜g)d˜g 1s2(g)1g +24g1g−12(2.20)Combining this Pad´e-approximant with the deﬁnition of d(g),Eq.(2.15),one is led tod(g)=43g−35the appropriate expansion parameter.Having been equipped with these formulas,one may easily calculate the beta-functionβ(g)as a power series in g−1.Inserting Eq.(A.10)into Eq.(2.14)and performing simple but somewhat cumbersome calculations,we are led to the desired asymptotic expansion forβ(g)β(g)=−2g+32−64/g+512/g2+512/g3−30720/g4−172032/g5+32768/g6−172032/g7+32768/g8+0(g−9)(3.1)¿From Eq.(3.1)it follows that in the large-g limitβ(g)→−2g+32,whilst in the weak coupling regime one has for g→0:β(g)→+2g[2–4,23,24].It implies that the continuous functionβ(g)changes sign at least once at someﬁxed point g∗.Let us get some numerical estimates for g∗+now,from Eq.(3.1),and compare these results with those found from the HT series expansions and those of the four-loop RG calculations. In the standard perturbative approach to quantumﬁeld theory atﬁxed dimension one must apply some resummation technique to the expansions ofβ(g)and other RG-functions.It is interesting that at least in low orders of perturbation theory the1/g-expansion,Eq.(3.1), does not require the application of a resummation technique.The most reliable numerical estimates of g∗+were obtained by means of the straightforward solution of the equation β(g∗)=0,from Eq.(3.1),taken within the g−6-approximation(without the last two terms in g−7and g−8).Theﬁve(and best)subsequent approximation are as followsg∗(1) +=16;g∗(2)+=13.6568;g∗(3)+=15.0044;g∗(4) +=15.0784;g∗(5)+=14.7632(3.2)Here the index k in g∗(k)+indicates that k+1terms are retained in theﬁxed point equationunder discussion.These estimates exhibit a regular behavior,the last value being in very good agreement with the most recent estimate g∗+=14.700±0.017obtained for the square lattice[14,15].The estimates obtained after taking into account the g−7and g−8terms diﬀer signiﬁcantly from the above values.This is apparently an indication that1/g-series also require the application of some resummation technique.Another approach to obtain a numerical estimate for g∗+is a straightforward solution of Eq.(A.10),given in the Appendix,after setting s=1.In contrast to theﬁxed point equation,β(g)=0,it yields a rather poor value of the renormalized coupling constant, g∗+=12.533,compared to the value reported in[3,14,15].It is interesting to compare our results with those obtained from the beta-function of the 2D Ising model and computed in the four-loop approximation,known to provide more or less satisfactory results for the critical indices[2–4]:β(v)=2v−2v2+1.432346v3−1.861533v4+3.1647764v5+0(v6)(3.3) To obtain the beta-function in our normalization we have to change variables[24]g=8π3β(v)(3.4)The analysis based on the Pad´e-Borel method of resummation of asymptotic series yields g∗+=15.08±2.5[24],which slightly exceeds the best values obtained from the HT seriescalculations:g ∗+=14.70±0.017[14,15];g ∗+=14.67±0.04[24](it is tempting to conjecturethat g ∗+=14π3[23].IV.ISOTHERMAL COUPLING CONSTANT AND CRITICAL AMPLITUDES The two preceeding Sections were devoted to computing the approximate value of therenormalized coupling constant g ∗+at h =0in the isochore limit.Here we remark thatin two dimensions there is a possibility of calculating the exact value of the renormalizedcoupling constant g ∗c in the isothermal limit,i.e.at the Curie point in an applied magneticﬁeld,namelyg ∗c =lim h →0g (T =T c ,h )(4.1)by virtue of Cardy’formula [26].It is in fact essential to stress that,in contrast to otherisothermal critical amplitudes,g ∗c is ﬁxed by this formula,which reads [26]c =3πh 2(2−η4−η;χ(h )=C c h 2η−4(4−η)2C c (f c 1)2(4.5)On the other hand,from Eq.s (2.6)and (4.3)it is seen that the correlation length ξdropsout of the product g ∗c c :cg ∗c =3π(4−η)2h 2 −∂2χ/∂h 2χ2(4.6)Inserting Eq.(4.4)into the r.h.s.of Eq.(4.6),one obtains the renormalized coupling constant value at the end point of the isothermal lineg∗c=6π3π(4−η)2(4.8)¿From Eq.(4.8)it follows that what we actually found,by virtue of Cardy’s formula,is only the product C c(f c1)2.To compute these quantities separately one needs more powerful techniques.In some seminal papers[27–29]it was shown how to compute the isothermal amplitudes by making use of the Thermodynamic Bethe Ansatz and within the framework of the form-factor approach.In particular,in his paper Fateev obtained the following remarkable result [27]f c1=Γ(2/3)Γ(8/15)4π2Γ(3/4)Γ2(13/16)]4/15=0.2270194675(4.9)¿From Eq.s(4.8)and(4.9)it follows thatC c=0.0731998414(4.10) All these results allow one to compute exactly the two following universal combinations[13] (see also[23]and[30])Q1=C cδC c ( f c1The exact values found provide a good opportunity to test the numerical results ob-tained from the HT series expansions and from Monte Carlo simulations.The fair esti-mates obtained from the analysis of HT series in the2D Ising model on the square lattice yield f c1=0.233and C c=0.0706([13]),whilst the exact results are given by Eq.s(4.9) and(4.10).As for the universal combinations Q1,2,the series expansion analysis yields Q1=0.88023,Q2=2.88[13].Notice in conclusion that Eq.s(4.7)and(4.8)hold good also for the general case of the 2D gΦ40(N)-symmetric model for−2<N<2,in particular for the minimal models of con-formalﬁeld theory corresponding to the discrete values of N:N=2cos(πall the corrections to the scaling laws in the2D Ising model are analytical.For instance, the susceptibility near T c is given byχ=C+τ−7/4+C+1τ−3/4+...[31].On the other hand, corrections to scaling are known to be powers ofτων[23].All this would lead toω=1, in obvious contradiction to conformalﬁeld theory.Moreover,the spectrum of conformal dimensions of the2D Ising model consists of just three numbers,these being(0,1)are as follows[15]Tχ=1+4K+12K2+104K3/3+92K4+3608K5/15+3056K6/5+484528K7/315+400012K8/105(A.1)µ2=4K+32K2+488K3/3+2048K4/3+38168K5/15+394624K6/45+8994736K7/315+28064768K8/315(A.2)χ′′hh=−2−32K−264K2−4864K3/3−8232K4−553024K5/15−2259616K6/15−180969728K7/315−217858792K8/105(A.3) withχ′′hh being the second derivative of the homogenous susceptibility with respect to a magneticﬁeld h;χandµ2were deﬁned in Sect.II.The standard RG equation for the eﬀective temperature T is given byξdT2dχ(A.5)The key observation for computingξis to make use of the new variable s= exp(2K)tanh(K).One has to substitute Eq.s(A.1)and(A.2)into Eq.(A.5)and then to rewrite the expression obtained in terms of s.At this order of approximation the proce-dure gives the resultξ2=s+2s2+3s3+4s4+5s5+6s6+7s7+8s8+0(s9)=sτ;τ=T−T cln(√4ln(√d lnξ=βIsing(s)=2s(1−s)1d lnξ=βIsing(T)=T21−2exp(−2/T)−2exp(−4/T)+2exp(−6/T)+exp(−8/T)ds|s=1=1(A.12)(iii)the beta-function is covariant under the duality transformation,namelyβIsing(s)=−s2βIsing(12sinh2T(A.14)does not satisfy either properties(ii)and(iv)[35].To end with,we prove some useful relations concerning the duality transformation˜g= d(g)introduced in Sect.II.i)Let us show that d′(g∗)=±1.First of all,from the deﬁnitiond(g)≡s−1(1d(d(g))=g;d(g∗)=g∗d′(d(g))d′(g)=1;g=g∗d′(g∗)d′(g∗)=1(A.16) showing that indeed d′(g∗)=±1.iii)Diﬀerentiating Eq.(A.16)(second from top)with respect to g one obtainsd′[d(g)]d′(g)=1;d′′[d(g)]d′(g)2+d′[d(g)]d′′(g)=0(A.17) and at theﬁxed point we arrive at(d(g∗)=g∗):d′′(g∗)(d(g∗))2+d′(g∗)d′′(g∗)=0(A.18) If d′(g∗)=−1we have an identity d′′(g∗)=d′′(g∗).In the opposite case from d′(g∗)=+1 it follows that d′′(g∗)=0.Proceeding in the same way,it is easy to see that all higher derivatives vanish identicatically at theﬁxed point.This,alone,does not imply that d(g)≡g,since it could be that d(g)=g+f(g)where f(g)is some nonanalytic function having vanishing derivatives at g∗.Let us then assume that f(g)is theﬁrst term of an asymptotic expansion of d(g)around theﬁxed point,so that f(g)→0as g→g∗.Then remembering that d(d(g))=g,g=d(d(g))=d(g+f(g))=g+f(g)+f(g+f(g))≃g+2f(g)(A.19) and we do obtain f(g)≡0.REFERENCES∗(permanent address)∗∗On leave of absence from:A.F.Ioﬀe Physical&Technical Institute,Russian Academy of Sciences,Polytechnicheskaya str.26,194021St.Petersburg(Russia)(permanent ad-dress)[1]McCoy B.M.,HEPTH/9403084[2]Nickel B.G.,Meiron D.I.and Baker G.A.Jr.,Compilation of2-pt and4-pt graphs forcontinuous spin models,University of Guelph report(1977)[3]Baker G.A.Jr.,Nickel B.G.and Meiron D.I.,Phys.Rev.B17,1365(1978)[4]Le Guillou J.C.and Zinn-Justin J.,Phys.Rev.B21,3976(1980)[5]Kramers H.A.and Wannier G.H.,Phys.Rev.60,252(1941)[6]Savit R.,Rev.Mod.Phys.52,453(1980)[7]Kogut J.B.,Rev.Mod.Phys.51,659(1979)[8]Shalaev B.N.,Phys.Rep.237,129(1994)[9]Dotsenko V.S.,Sov.Phys-JETP75,1083(1978)[10]Dolan L.and Grady M.,Phys.Rev.D25,1587(1982)[11]Damgaard P.H.and Haagensen P.E.,J.Phys.A:Math.Gen.30,4681(1997)[12]Correia J.D.,hep-th/9806229[13]Tarko H.B.and Fisher M.E.,Phys.Rev.B11,1217(1975)[14]Shun-Yong Zinn,Sheng-Nan Lai and Fisher M.E.,Phys.Rev.E54,1176(1996)[15]Butera P.and Comi M.,Phys.Rev.B54,15828(1996)[16]Butera P.and Comi M.,Phys.Rev.E55,6391(1997)[17]Reisz T.,Phys.Lett.B360,77(1995)[18]Guida R.and Zinn-Justin J.,Nucl.Phys.B489,626(1997)[19]Sokolov A.I.,Phys Solid State40,N7(1998)[20]Sokolov A.I.,Orlov E.V.and Ul’kov V.A.,Phys.Lett.A227,255(1997)[21]Shun-Yong Zinn and Fisher M.E.,Physica A226,168(1996)[22]Kleinert H.,Phys.Rev.D57,2264(1998);cond-mat/9801167[23]Zinn-Justin J.,Quantum Field Theory and Critical Phenomena,3rd ed.,(ClarendonPress,Oxford)(1996)[24]Baker G.A.Jr.,Phys.Rev.B15,1552(1977)[25]Wegner F.,Phys.Rev.B5,4529(1972);Phys.Rev.B6,1891(1972)[26]Cardy J.L.,Phys.Rev.Lett.26,2709(1988)[27]Fateev V.A.,Phys.Lett.B324,45(1994)[28]Mussardo G.in Integrable Quantum Field Theories NATO ASI Series B:Physics310173(1993)(Plenum Press,New York)(1993)[29]Delﬁno G.and G.Mussardo,Nucl.Phys.B455,724(1995)[30]Privman V.,P.C.Hohenberg and A.Aharony,Phase Transitions and Critical Phenom-ena,Vol.14,p.4,C.Domb and J.Lebowitz eds.,(Academic Press,New York)(1991)[31]Barouch E.,B.M.McCoy and T.T.Wu,Phys.Rev.Lett.31,1409(1973)[32]Pelissetto A.and E.Vicari Nucl.Phys.B540[FS],639(1999)[33]Wegner F.J.,Nucl.Phys.B316,663(1989)[34]Baxter R.J.,Exactly Solved Models in Statistical Mechanics(Academic Press,New York)(1982),p.120[35]Migdal A.A.,Sov.Phys.JETP42,413(1975)。

数字1到11的英语

数字1到11的英语From the tiniest speck to the vast expanse of the universe, numbers have always played a crucial role in our lives. They are the building blocks of mathematics, the foundation of science, and the language of measurement. However, numbers are not just about calculations and measurements; they are also deeply rooted in our culture, traditions, and languages. In this article, we will explore the numbers from 1 to 11 in English and Chinese, delving into their significance, history, and usage in both languages.**1. One (一)**In English, "one" is the fundamental building block of all numbers. It represents unity, individuality, and the beginning of all things. In Chinese, "一" (yī) alsocarries a profound meaning, symbolizing harmony, balance, and the integration of all elements. Both languages use "one" as a starting point for counting and measurement.**2. Two (二)**"Two" in English represents duality, partnership, and balance. It is often associated with concepts like duality, symmetry, and harmony. In Chinese, "二" (èr) also embodies these concepts, but it is also used to indicate sequence or order, as in "the second floor" or "the second son."**3. Three (三)**"Three" in English is often considered a magical number, associated with concepts like creativity, communication,and diversity. It is also significant in many religions and cultures. In Chinese, "三" (sān) holds a similar significance, representing the triad principle of harmony, change, and balance. It is also used to symbolize the past, present, and future or the Trinity in some religions.**4. Four (四)**In English, "four" is often associated with stability, solidity, and completeness. It is the number of directions (north, south, east, west) and the number of elements (earth, air, fire, water). However, in Chinese culture, "四" (sì) is often avoided in certain contexts due to its association with death (as in the phrase "four deaths," which refers to birth, aging, sickness, and death).Nevertheless, it is also used to symbolize the four seasons, the four cardinal directions, and the four virtues of Confucianism (ren, yi, li, zhi).**5. Five (五)**"Five" in English is often considered a幸运数字due to its association with the five senses, the five fingers ofthe hand, and the five points of the star. It is also significant in geometry and music. In Chinese, "五" (wǔ)is also auspicious, symbolizing the five elements (wood, fire, earth, metal, water) and the five grains (rice, wheat, beans, corn, and millet). It is often used to represent diversity and abundance.**6. Six (六)**"Six" in English is often associated with harmony and balance, as it is the sum of the first three natural numbers (1+2+3=6). It is also considered a lucky number in many cultures. In Chinese, "六" (liù) is also considered auspicious, often used to represent luck and prosperity. It is also the number of directions in hexagrams of the IChing (Book of Changes).**7. Seven (七)**"Seven" in English is often considered a mysterious and sacred number, associated with concepts like completion, perfection, and divinity. It is also significant in religions like Christianity (seven days of creation, seven deadly sins, etc.). In Chinese, "七" (qī) is also considered auspicious in some contexts, but it is also associated with melancholy and sorrow due to its use in the phrase "the seven sorrows of life" (rensheng qijian).**8. Eight (八)**"Eight" in English is often associated with infinity, abundance, and prosperity. It is considered a lucky number in many cultures due to its shape, which resembles a figure eight on its side, symbolizing continuous flow and growth. In Chinese, "八" (bā) is also considered extremely auspicious, often used to symbolize prosperity, wealth, and good fortune. It is often seen in dates and prices to attract good luck and positive energy.**9. Nine (九)**"Nine" in English is often associated with concepts like completion, perfection, and divinity. It is the highest single-digit number and represents the culmination of all things. In Chinese, "九" (jiǔ) is also considered auspicious, often used to symbolize the highest level or state of something. It is also the。

英语作文讲数学题

英语作文讲数学题Mathematics, often perceived as a complex and abstract subject, is fundamentally about patterns, relationships, and problem-solving. It is a language that transcends cultural and linguistic barriers, enabling us to describe and understand the world around us with precision and clarity. In this essay, we will delve into a mathematical problem, exploring its intricacies and the beauty of the logic that underpins it.Consider the problem of finding the sum of an arithmetic series. An arithmetic series is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the preceding term. For example, in the series 2, 5, 8, 11, ..., the difference is 3. To find the sum of the first n terms of such a series, one could laboriously add each term, or one could employ a more elegant approach.The formula for the sum of the first n terms of an arithmetic series is given by:$$ S_n = \frac{n}{2} (a_1 + a_n) $$。

Density Correlation Functions in Calogero Sutherland Models

k n i zi ,
giving (6)
f=
k
f {n1 , . . . nk }
i=1
Wni ≡
k
f {n1 , . . . nk }|n1 , . . . nk
where f {ni } are expansion coeﬃcients. Since here we will be interested only in chiral states, all ni can be chosen positive. This is the “string” picture, where the state |{ni } is interpreted as a collection of strings with winding numbers ni . Deﬁning string creation and annihilation operators |n ≡ a+ n |0 and n|0 >≡ an |n which obey the commutation relations [an , a+ m ] = nδm,n and interpreting the action of (5) on (6) in terms of an and a+ n , we get[12,13] H= 1 mL′2
CERN-TH.7243/94 USC-94/008 hep-th/9404192 April 1994
Density Correlation Functions in Calogero-Sutherland Models
arXiv:hep-th/9404192v3 4 May 1994
Joseph A. Minahan

Symmetry-Curvature Duality

COMPUTER VISION,GRAPHICS,AND IMAGE PROCESSING,38,327-341(1987)327Symmetry-Curvature DualityMichael LeytonCenter for Discrete Mathematics&Theoretical Computer Science(DIMACS), Busch Campus,Rutgers University,New Brunswick,NJ08904,USA.mleyton@Received December12,1985AbstractSeveral studies have shown the importance of two very different descriptors for shape:symmetry structure and curvature extrema.The main theorem proved bythis paper,i.e.the Symmetry-Curvature Duality Theorem,states that there is animportant relationship between symmetry and curvature extrema:If we say thatcurvature extrema are of two opposite types,either maxima or minima,then thetheorem states:Any segment of a smooth planar curve,bounded by two consecutivecurvature extrema of the same type,has a unique symmetry axis,and the axisterminates at the curvature extremum of the opposite type.The theorem is initiallyproved using Brady’s SLS as the symmetry analysis.However,the theorem is thengeneralized for any differential symmetry analysis.In order to prove the theorem,anumber of results are established concerning the symmetry structure of Hoffman’sand Richards’codons.All results are obtainedﬁrst by observing that any codon isa string of two,three,or four spirals,and then by reducing the theory of codons tothat of spirals.We show that the SLS of a codon is either(1)an SAT,which is amore restricted symmetry analysis that was introduced by Blum,or(2)an ESAT,which is a symmetry analysis that is introduced in the present paper and is dual toBlum’s SAT.c 1987Academic Press,Inc.Perceptual studies in both psychology and computer vision have revealed the im-portance of two very different structural characteristics for shape:(1)Symmetry.Since the German gestalt schoolﬁrst showed that symmetry is a crucial organizing principle of shape(Wertheimer[24];Goldmeier[8]),many studies have corroborated and extended their results.For example,Psotka[18]presented subjects withﬁgures,such as the outline of a man,and asked the subjects to place a dot at theﬁrst place that came to mind.Pooling the results,he found that the dots were distributed along the local symmetry axes.Leyton[12,13,15,16]gave subjects a set of twenty-two complex abstract and natural shapes;e.g.of animals,birds and plants,and asked them to place lines along the directions of greatest shapeﬂexibility.The subjects consistently chose local symmetry axes.Richards&Kaufman[21]placed cutouts of ﬁgures against a TV screen exhibiting random"snow"and found that subjects saw328MICHAEL LEYTONFigure1:An illustration of the SAT construction for two curves c1and c2.The SAT symmetry axis is the locus of the circle centers O.ﬂow patterns that accorded with the symmetry structure of theﬁgures.Leyton[14, 15,16]showed that human subjects successively prototypify objects by the successive introduction of global symmetries.Furthermore,Leyton[11]demonstrated that the orientation and form multistability phenomenon(e.g.Goldmeier[8];Rock[22])is due to the perceptual deﬁnition of shape as a symmetry structure.Again,several researchers have offered geometric theories of the encoding of shape in terms of what have been called generalized cones,cylinders,ribbons,etc.Despite the differences between these proposals,all of them crucially determine an axis of symmetry for each shape region encoded.One example is that of Blum[3],Blum&Nagel[4],called the Symmetric Axis Transform(SAT).Fig1shows the SAT for two curves c1and c2.The SAT traces a trajectory of maximal inscribed discs between c1and c2.The locus of disc centers,O, is then regarded as the symmetry axis(which might be curved).Other constructions, besides the SAT,include those by Binford[2],Brady[5],Brooks[7],and Leyton[17]. Various relationships between a number of these proposals have been elaborated by Rosenfeld[23].(2)Curvature extrema.A structurally different type of shape description is curvature extrema.The pioneering paper by Attneave[1]showed that the information associated with a contour is not evenly distributed along it but is concentrated at points of curvature extrema.(For example,subjects,asked to represent a contour with a limited number of dots,tend to use the dots such that they correspond to points of maximal curvature on the contour.)Hoffman&Richards[10]and Richards&Hoffman[20]have examined curvature extrema with very different concerns.They have argued that a contour is perceptually partitioned at points of negative curvature extrema.They reasoned as follows:(1)Using the differential-topological notion of transversality,two smooth surfaces in3-space tend to intersect at a contour of concave discontinuity of their tangent planes.(2)In particular,this can be said of two intersecting parts of a shape.(3)The contour of intersection can be considered to be a locus of greatest negative curvature.(4) The two-dimensional equivalent on a plane-curve is an extremum of negative curvature.(5)Thus the perceptual parts of a two-dimensional contour tend to be the segments bounded by points of greatest negative curvature.In support of this claim,Hoffman& Richards[10]present several compelling examples,including Rubin’s classical face-vase illusion.SYMMETRY-CURV ATURE DUALITY329 The above brief review clearly shows that two different descriptors,symmetry struc-ture and curvature extrema,are fundamental to shape perception.In fact,it is important to notice just how different symmetry and curvature are,as structural characteristics. Firstly,symmetry is an algebraic concept;i.e.it is determined by a transitive group action.Furthermore,symmetry,as considered above,is reﬂectional symmetry,and is thus determined by a discrete group.In contrast,curvature is a continuity property determined by an asymmetric limiting process.Thus:Symmetry algebraically relates topologically separable points,i.e.,points that can be isolated from each other in distinct neighborhoods of the contour.Curvature,on the other hand,is deﬁned with respect to the converging non-separable neighborhood structure at a single point.These considerations are reﬂected in the two different types of perceptual theories for which these two structural characteristics have been used.Theories of perceptual symmetry have concentrated on the perceptual binding together of quite separate parts of the contour.Theories involving curvature have concentrated on points of special signiﬁcance,i.e.points of informational richness,or curve break-points.Given these fundamental structural differences between symmetry and curvature, the theorem to be proved in this paper is somewhat surprising.Intuitively,it states:The symmetry axes of a curve tend to terminate at points of curvature extrema.In fact,the differential geometry is captured more by saying that the symmetry axis is forced into a point of maximal curvature.It turns out that the theorem relates symmetry and curvature extrema even more intimately than this.If we let curvature extrema be of two opposite types,either maxima or minima1,then the theorem states:SYMMETRY-CURV ATURE DUALITY THEOREM:Any segment of a smooth planar curve,bounded by two consecutive curvature extrema of the same type,has a unique symmetry axis,and the axis terminates at the curvature extremum of the opposite type.One term in this theorem statement needs to be deﬁned;it is the term symmetry axis.There are a number of alternative mathematical schemes for the description of symmetry.The scheme which we will use for most of the paper is one established by Brady[5]called SLS.However,we shall then generalize our argument to other symmetry schemes,quite easily,because we shall prove that,on segments of the type given in the theorem,a number of different symmetry schemes coincide.Again,amongst the curve segments themselves,it will become apparent that we only need to consider a subset of such segments.These are the segments that correspond to what Hoffman and Richards[20]call codons.Thus we shallﬁrst prove an apparently restricted statement of the above theorem in terms of Brady’s SLS and Hoffman’s and Richards’codons.An 1In fact,we will adopt the convention that curvature is a map from an entire curve to a single1-dimensional continuum.That is,we will not assume two continua,one of positive curvature and the other of negative curvature.Note that the single continuum description arises from a speciﬁcation of a single traversing direction to a curve330MICHAEL LEYTONFigure2:(a)The SLS construction for two curves c1and c2.The SLS symmetry axis is the locus of chord points P.(b)The complete SAT for an ellipse.(c)The complete SLS for an ellipse.(d)A contour having two SLS cross sections at some point A. extensive generalization will then be extracted from the proof of the restricted version. We begin now by brieﬂy describing the SLS and codon analyses,respectively.(1)Brady’s SLS.Michael Brady[5]has provided an analysis of symmetry in shape that can be understood by considering Fig.2a.The purpose is toﬁnd a line of symmetry for the two curved contours c1and c2.In order to do this,points A and B are found on the contour such that the line AB subtends the same angleαwith the normal at A and that at B.The cross-section AB is then divided by two;and the half-way point P is deﬁned to be the symmetry point.The locus of dots in Fig.2represents the line of symmetry points established in this way.The curved symmetry axis is called the Smoothed Local Symmetry(SLS).The importance of Brady’s analysis is that it handles a wider variety of situations than some of the other analyses.For example,it is more exhaustive than the SAT(as deﬁned by Blum).This is illustrated in Fig2b which shows the SAT for a ellipse, in contrast to Fig2c,which shows the ellipse’s SLS.In this example,the SAT fails to produce the minor axis,which is captured by the SLS.The twoﬁgures illustrate also another generality:Every SAT induces an SLS(see Rosenfeld[23],for further discussion).This can be seen by looking back at Fig1.The chord of an SAT disc subtends the same angle with the two radii at the tangent points A and B.These radii are also normal to the curves at A and B.Note that the SAT center,O,is not necessarily the SLS center,which is the point,P,that bisects the chord AB.This fact will beSYMMETRY-CURV ATURE DUALITY331Figure3:The codon classiﬁcation.Slashes represent curvature minima,and dots represent inﬂection points.crucial later when we prove that symmetry axes"are forced"to terminate at points of maximal curvature.Observeﬁnally that the exhaustiveness,of the SLS,results in the non-uniqueness of the latter.For example,as Brady&Asada[6]point out,a situation like Fig2d yields two SLS cross-sections at A;i.e.the line-segments AB and AC.(2)Hoffman’s and Richards’codon analysis.As our brief review of some curvature literature indicated,Hoffman&Richards[10]and Richards&Hoffman[20]have argued that a smooth planar curve is perceptually partitioned into segments at the extrema of negative curvature.Any segment whose ends are two consecutive curvature minima is called a codon.Any curve can be segmented as a string of codons.In fact,Hoffman and Richards establish that there are only six possible types of codon;those given by the six possible orderings of curvature singularities as shown in Fig.3.Theﬁrst one given in Fig.3is trivial,and will be ignored throughout this article.On any codon shown,the direction of traversing the curve is in the direction of the arrow shown,and the curvature minima(indicated by slashes)are the points of greatest clockwise rotation of the arrow.The dots represent points of zero curvature.In this paper,we are going to make the harmless assumption that,except at the points of curvature extrema,the curvature of the codon changes strictly monotonically.We now prove that there is a particularly close relationship between the above two very different kinds of structural descriptors;i.e.between the SLS and the codon-analysis:SLS-CODON THEOREM:The SLS of a codon is unique,and terminates at the point of maximal curvature on the codon.332MICHAEL LEYTONTable1:A Redescription of the Codon Classiﬁcation in Terms of Spirals.Codon type Spiral Sequence0+[+ve incr]·[+ve decr]0-[-ve incr]·[-ve decr]1+[+ve incr]·[+ve decr]·[-ve decr]1-[-ve incr]·[+ve incr]·[+ve decr]2[-ve incr]·[+ve incr]·[+ve decr]·[-ve decr]Note:Each bracket-pair corresponds to a spiral.The symbols within each bracket giveﬁrst,the curvature sign and second,the direction of change of curvature along the single1-dimensional curvature continuum.The proof of this theorem occupies most of the rest of the paper and reveals some surprisingly interesting facts about the relation between symmetry and curvature ex-trema.After the completion of the proof,we return to the Symmetry-Curvature Duality Theorem.Proof.To prove this theorem,one observesﬁrst that a codon is itself built from a number of examples of only one type of primitive subpart.This can be seen as follows.To get to the subparts,we needﬁrst to break the codon at the point M of maximal curvature.One obtains two curves l1and l2,which are respectively the curve before and the curve after M,in traversing the complete codon.We shall call the curves l1and l2,the limbs of the codon.Now observe that each limb is a curve of monotonically changing curvature.A curve with monotonic curvature of one sign,is called a spiral.If a codon limb has no point of zero curvature,it must be a spiral.If the limb has a point of zero curvature then it consists of two spirals joined at the zero point.Thus each codon consists of a string of examples of only one type of subpart,a spiral.Codons0+and0-consist of two spirals,codons1+and1-of three spirals,and codon2of four spirals.If we take the direction of curve transversal into account,this leads to a redescription of the codon classiﬁcation.Recallﬁrst(Footnote1)that we will use the convention that curvature is given by a single one-dimensional continuum.Let us deﬁne a positive increasing spiral to be one which has positive curvature that increases as the spiral is traversed;and let us deﬁne a negative increasing spiral to be one with negative curvature where the curvature increases(moves in the positive direction along the curvature continuum)as the spiral is traversed.Correspondingly,we can deﬁne positive decreasing and negative decreasing spirals.Then the redescription of the codon classiﬁcation is shown in Table 1.We now prove a lemma that will be crucial to the proof of our theorem.SYMMETRY-CURV ATURE DUALITY333 LEMMA1:An SLS cannot be constructed on a spiral.2Proof.Let the curve be parameterized by arc-length s.Let A and B be any pair of SLS-cross-section points.Deﬁne a Cartesian(x,y)plane at point A,such that the x-axis is along AB and the curve AB is in the negative half-plane of y,as shown in Fig4. Deﬁneθ(s)to be the angle of the tangent at any point s to the x-axis.Now observe that θ(B)θ(A)sinθdθ=cosθ(A)−cosθ(B)=cos(π2−α)−cos(π2−α)=0(1) But observe:θ(B)θ(A)sinθdθ=BAdydx·dθds·ds=BAdydx·κ·ds=y(B)κ(B)−y(A)κ(A)−BAy(s)·dκ(s)=−BAy(s)·dκ(s)=0(2)But Eq(2)contradicts Eq(1).Therefore the SLS cannot be constructed3.The lemma in particular shows that an SLS cannot be constructed on any spiral subpart of a codon.However,since a codon consists only of spirals,this means that if an SLS can be constructed,the SLS cross-section points A and B must each come from different spirals.Theﬁrst such case we consider is of two adjacent spirals which are linked by a point of zero curvature;i.e.by a dot as shown on codons1+,1-,and2,in Fig.3.Any such spiral-pair will be called a bi-spiral.It is easy to see that an SLS cannot be constructed on bi-spiral,because points A and B must come from opposite sides of the dot;and the cross-section must therefore cross over the contour.2We will assume that normals cannot change sides on a curve.3This proof is a variant of Ostrowski’s proof of V ogt’s theorem(see Guggenheimer[9])334MICHAEL LEYTONFigure4:Theﬁgure required for Lemma1.Thus we are left with a crucial conclusion that no codon limb can have an SLS, because a codon limb consists either of a spiral or a bi-spiral.In other words,we have proved:LEMMA2:The SLS cross-section points of a smooth planar curve must be separated by a curvature turning-point.In other words,the SLS cross-section points of a codon must lie on different limbs.What is necessary now to prove is(1)that an SLS does in fact exist and that it does have the property of terminating at the maximal curvature point M;and(2)that there is no other possible SLS for a codon.ExistenceWeﬁrst divide theﬁve non-trivial codons into two classes:(1)the codons which have positive maximal curvature,and(2)the codons which have negative maximal curvature. The former will be called the positive codons and the latter the negative codons.Observe that four of the codons are positive;the only negative codon being0-.Observe also, from Table1,that the maximal curvature point M,for each codon,isﬂanked by two spirals of the same sign.These two spirals will be called the s-region of the codon.We prove existenceﬁrst for positive codons.Recall that any SAT has an associated SLS.We shall call such an SLS,an SAT-induced SLS.We will now show that any positive codon has an SAT-induced SLS.To start the construction of an SAT,choose any point A on the codon(where A=M and M is the point of maximal curvature).Without loss of generality,we will assume that A∈l1.Put a circle at A,tangential to the codon at A and inside the codon(to theleft of the traversal direction),and ensure that the circle is small enough not to touch or intersect any other point on the codon.Now simply increase the radius of the circle tillSYMMETRY-CURV ATURE DUALITY335it touches the codon at one other point.Two conditions are satisﬁed,which we prove in the two subsequent lemmas:LEMMA3:A maximal inscribed circle cannot be tangential to a positive codon at more than two separate points.Proof.Suppose the maximal circle is tangential to the codon at three points A,B,and C.Suppose also that neither A,B,nor C is M(the point of maximal curvature).Then two of the three points,say A and C,must lie on the same limb,say l1.However,the line AC is a chord of the circle.Furthermore,because angles<OAC and<OCA must be equal(where O is the center of the circle),the chord AC is an SLS cross-section; which contradicts Lemma2,because A and C are on the same limb.Now suppose that one of the three points,say C,is M.Then by Lemma1,points A and B must lie on different limbs,say l1and l2respectively.Consider now the curve l1∪M.This is a spiral or bi-spiral.But this means that points A and C lie on the same spiral or bi-spiral,which contradicts Lemma1and Lemma2.The next lemma requires an understanding of a codon’s evolute.The evolute of a curve is the locus of its centers of curvature.The following are standard facts about evolutes(e.g.see Guggenheimer[9]):(1)The evolute of a convex curve is convex4;(2) the normal line,at any point on the curve,is tangent to the curve’s evolute;and(3)the evolute at a point of maximal curvature is a cusp.Putting these facts together it is easy to show that the evolute of the s-region5of a codon is of the form shown in Fig5;that is,two convex curves that form a cusp that is tangential to the normal at M and points towards M.Fig5puts the codon on a Cartesian frame with x-axis along the normal at M.Fig5also shows an arbitrary point A with center of curvature E(A)on the evolute and normal that is tangent to the evolute at E(A).It is now easy to prove:LEMMA4:The center,O,of a maximal inscribed circle of a positive codon,must lie in between the two evolutes of the two codon limbs.Proof.Let the maximal circle be tangential to the evolute at A and B.Let us dealﬁrst with the s-region of the codon.By Lemma1,the points A and B must lie on opposite spirals s1and s2respectively.But s1is separated from its evolute by the evolute of s2,and vice versa.Therefore the normal lines must intersect in between the evolutes. However,the center of the inscribed circle is the intersection of normal lines;which means that the center is in between the evolutes of the two spirals.When one now includes the non s-region,one merely adds evolutes that lie outside the codon.So the result remains.4A curve AB is convex if the region bounded by the curve and the chord AB is a topologically convex set.5We deﬁned the s-region to be the two spiralsﬂanking the maximal curvature point.336MICHAEL LEYTONFigure5:A positive codon with its evolute.Observe now that Lemma4provides the following ordering constraint.Considerthe normal line at A,and three points on that line:point A,the evolute point E(A),andthe point O which is the center of the maximal circle.Lemma4tells us that the orderof these points along the normal must be A,O and E(A).This order must be preservedunder projection onto the x-axis in Fig5.Now observe that,as point A moves along the codon to M,point E(A)moves alongthe evolute to the cusp point.The ordering constraint forces O to reach the cusp pointbefore E(A)does.However,the tangent at the cusp is the normal at M.Thus whenO reaches the cusp point it is the center of a circle tangential to M.Observe however that at this point the circle cannot touch the codon also at another point on l2,becausel2∪M is a spiral or bi-spiral.Thus point B must have reached point M at the same time as point A.Which means that the SLS symmetry point,which lies on the chord AB,must have reached M.This completes the proof that a positive codon has a unique SAT-induced SLS,andthat this SLS terminates at the point of maximal curvature.Observe that the above shows why,earlier,we had said intuitively that a symmetryaxis is forced into the point of maximal curvature.We can now see that this refers to thefact that the SAT center O is forced in between the evolutes and is eventually pushedby the ordering constraints into the cusp point.Let us now prove the existence of an SLS for negative codons.All negative codonsare of the singularity form shown as codon type0-in Fig. 3.Above,we proved theexistence of an SLS for positive codons by showing that those codons have an SAT.However,we cannot apply the above proofs to negative codons because weﬁnd thefollowing:SYMMETRY-CURV ATURE DUALITY337Figure6:A negative codon with its evolute.LEMMA5:An SAT cannot be constructed for a negative codon.Material in the proof will be required later.Proof.Fig6shows a negative codon with its evolute,such that the evolute cusp point is tangential to the x-axis of the Cartesian plane.In this case,the evolute of each spiral is on the same side of the x-axis as its spiral.Consider points A and B on different spirals.Their normals must intersect at some point O.However this is further along the normals from A and B than the centers of curvature E(A)and E(B).However,any inscribed circle that is tangential to the codon at two points must have a radius smaller or equal to the radii of curvature at both points.Thus an inscribed circle cannot exist, and an SAT is impossible.It might appear atﬁrst that Lemma5precludes the use of any of the types of techniques used above in the existence proof for positive codons.However,the situation is rescued when one realizes that negative codons can have exscribed circles that will have the properties that we require.We propose a new kind of symmetry analysis that is dual to the SAT.Instead of using circles that are(1)maximal and(2)inscribed,the new analysis uses circles that are(1) minimal and(2)exscribed.To distinguish these two types of analyses,let us call Blum’s analysis,the Inscribed SAT(or just ISAT),and the new analysis,the Exscribed SAT (or just ESAT).One of the things that our proof has so far shown is that positive codons yield an ISAT.We will now go on to show that negative codons yield an ESAT.To illustrate this, in advance,consider Fig.7a.It shows the ESAT of a negative codon.The codon is the bold curve,and the circles are the minimal exscribed ones of the ESAT.The dotted line shows the locus of symmetry points.Now consider Fig7b.It shows an ellipse;and an ellipse turns out to be the conjunction of two negative codons,bounded by the slashes338MICHAEL LEYTONFigure7:(a)A negative codon(thickened curve)with its ESAT.(b)An ellipse interpreted as a string of two negative codons.(c)An ellipse interpreted as a string of two positive codons.shown(see Richards&Hoffman[20]for fuller discussion).However,looking at Fig 7a,it is easy to see that the ESAT of an ellipse creates the short axis.Let us examine this situation a little more closely.Richards&Hoffman[20]show that the ellipse has in fact two interpretations depending on the direction of curve traver-sal6The interpretations are(1)the conjunction of two positive codons,as shown in Fig 7c;and(2)the conjunction of two negative codons,as shown in Fig7b.We have seen that positive codons yield an SAT,which we now call an ISAT.This produces the long axis of an ellipse.We shall also see that negative codons yield an ESAT.This produces the short axis of an ellipse.Putting these together,we obtain the complete SLS of the ellipse;i.e.the two axes,as shown in Fig2c.Let us now return to the general SLS existence-proof for negative codons.It is easy to prove the duals of Lemmas3and4given earlier:LEMMA3 :A minimal exscribed circle cannot be tangential to a negative codon at more than two separate points.LEMMA4 :The center,O,of a minimal exscribed circle of a negative codon must lie in between the two evolutes of the two codon limbs.Proofs.The proofs of these lemmas are obtained by writing out the proofs of Lemmas 3and4,and substituting the dual constructs;i.e.substituting"minimal"for"maximal", and"exscribed"for"inscribed",etc.6As Richards and Hoffman[20]point out,the two directions also lead to two alternativeﬁgure-ground relationships because the interior of aﬁgure is to the left of the transversal direction.Therefore,in Fig7b, the ellipse is a hole,and in Fig7c the ellipse is a solid shape.SYMMETRY-CURV ATURE DUALITY339Figure8:AB and AC are SLS chords–a situation that we prove does not exist.Now observe that different ordering constraints emerge from Lemma4 than fromLemma4.Going back to Fig6,we see that,for negative codons,the order of points A,O,and E(A)on the normal line at A must in fact be A,E(A),O;which reverses the order of E(A)and O for the positive codons.However,this new order is just what weneed to prove our result for negative codons;as we can now see.As point A travels topoint M,the point E(A)travels to the evolute cusp-point.However,the new orderingconstraint pushes O to the cusp point ahead of E(A),ensuring that O reaches the cusppoint.But the cusp point is the center of the circle of curvature at M.Thus,as before,the SLS center reaches M,and the existence result follows.UniquenessObserve now that,although we have just proved that a codon does have an SLS–thatwhich is induced either by the codon’s unique ISAT,or unique ESAT-the codon mighthave another SLS,because there are many SLSs that are not induced by an ISAT orESAT.It remains therefore to show that no such alternative SLS is possible for a codon.We prove uniquenessﬁrst for positive codons.The situation we are going to describeis illustrated in Fig8.Let A and B be cross-section points determined by the ISAT-induced SLS described above;i.e.there is a maximal inscribed circle tangential to thecodon at A and B.If the SLS is not unique,then,without loss of generality,for some A,there is a point C on the same limb of B(say l2)such that A and C are SLS cross-sectionpoints to the codon.Crucial to our analysis is the fact that the latter SLS also induces acircle(not a maximal inscribed one)that is tangential to the codon at points A and C.(This is easy to see:By the SLS construction,angleβ,between the normal at A andAC,must be the same as the angle between the normal at C and AC.Furthermore,the inward normals must lie on the same side of AC,and thus must intersect at some point O.This means that triangle OAC is isoseles and OA is the same length as OC.Thus a circle S,with center at O,can be drawn tangentially to the codon at A and C.)We shall call this new circle,the SLS-induced circle.All pairs of SLS-cross-section。

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1. INTRODUCTION The dissipative Wannier–Azbel–Hofstadter model is a quantum-mechanical description of a particle moving in two dimensions, subject to a doubly periodic potential, a transverse magnetic ﬁeld and a dissipative force. In the absence of friction, the energy spectrum, as a function of ﬂux quanta per unit cell, exhibits a remarkable fractal structure [2,3,4]. As the friction/unit cell, α, and ﬂux/unit cell, β , are varied, the system undergoes a series of phase transitions between localized and delocalized behavior. In the limit as the friction goes to zero, the phase diagram becomes fractal, and throughout the entire β -α plane it exhibits an SL(2, Z ) symmetry [5]. In addition to having an unusual phase diagram, this system is also of interest because it is related to several other models. It is equivalent to a generalized, neutral Coulomb gas restricted to one dimension. If the gas is allowed to have a total net charge, then the correlation functions of this theory can be used to ﬁnd the boundary state in open string theory [6]. As a consequence, if the critical theories also satisfy a set of Ward identities reﬂecting the reparametrization independence of the boundary state, then they represent new solutions of open string theory in a non-trivial background of tachyons and gauge ﬁelds [7]. One can also directly use these correlation functions to calculate the scattering of ﬁelds oﬀ a boundary with background tachyon and gauge ﬁelds. In earlier papers [5] [8], we have shown that there is a critical circle when α/ α2 + β 2 = point α = 1, β = 0, the theory fermionizes and the large-time behavior of the system can be solved exactly. In this theory, all coordinate correlation functions (other than two-point functions) are zero, except for contact terms. However, because the fermionization is valid only for large-time behavior, it does not appear to give the correct form for the contact terms that satisfy the Ward identities. Because of the SL(2, R) invariance of the phase diagram, these contact terms are related by a duality transformation to correlation functions with nonzero magnetic ﬁeld. In Ref. [8] we have shown that these correlation functions with β = 0 are ﬁnite at large times. This result suggests that the contact terms are physical and that the symmetries of the theory should determine them. However, this line of reasoning is not entirely reliable because the duality transformation has not been proven with the regulator that satisﬁes the Ward identity. In this paper, we address these issues and the evaluation of the contact terms by giving a detailed study of the theories on the critical circle α/ α2 + β 2 = 1. Using methods motivated by the derivation of fermionization and the duality symmetry, we derive several 1 1, and that the points on this circle where β/α ∈ Z Z are multi-critical points. At hep-th/9304006v1 2 Apr 1993
Abstract The dissipative Hofstadter model describes the quantum mechanics of a charged particle in two dimensions subject to a periodic potential, uniform magnetic ﬁeld, and dissipative force. Its phase diagram exhibits an SL(2,Z) duality symmetry and has an inﬁnite number of critical circles in the dissipation/magnetic ﬁeld plane. In addition, multi-critical points on a particular critical circle correspond to non-trivial solutions of open string theory. The duality symmetry is expected to provide relations between correlation functions at diﬀerent multi-critical points. Many of these correlators are contact terms. However we expect them to have physical signiﬁcance because under duality they transform into functions that are non-zero for large separations of the operators. Motivated by the search for exact, regulator independent solutions for these contact terms, in this paper we derive many properties and symmetries of the coordinate correlation functions at the special multi-critical points. In particular, we prove that the correlation functions are homogeneous, piecewise-linear functions of the momenta, and we prove a weaker version of the anticipated duality transformation. Consequently, the possible forms of the correlation functions are limited to lie in a ﬁnite dimensional linear space. We treat the potential perturbatively and these results are valid to all orders in perturbation theory.