Algebras in higher-dimensional statistical mechanics - the exceptional partition (mean field) algebras

We determine the structure of the partition algebraP _n(Q) (a generalized Temperley-Lieb algebra) for specific values ofQ , focusing on the quotient which gives rise to the partition function ofn siteQ-state Potts models (in the continuousQ formulation) in arbitrarily high lattice dimensions (the mean field case). The algebra is nonsemisimple iffQ is a nonnegative integer less than 2n-1. We determine the dimension of the key irreducible representation in every specialization.Work supported by the Packard Foundation.     

Structure and phase transformation of alkali silicate melts analysed by Raman spectroscopy

Raman spectra of binary alkali silicates were measured at various temperatures from 1300°C to room temperature to investigate the relation between structural change and phase transformation phenomena. Distribution of structural units of Q ⁿ was estimated at each temperature by the deconvolution of spectra based on the equilibrium 2Q³ ? Q² + Q⁴. The Q ⁿ distributions of sodium and potassium silicate systems strongly depend on temperature and the equilibrium shifts to the left-hand side with decreasing temperature, but those of lithium silicate system were less sensitive to the temperature variation. In alkali disilicates (33?mol%?R₂O–67?mol%?SiO₂, where R=Li, Na or K), the Q ⁿ distributions near the melting point were independent of alkali ion species, and they held the relation [Q²]?=?[Q⁴]?≈?[Q³]/4. This means that two of 6Q³ units (six-memberd ring) in crystals are transformed into a pair of Q² and Q⁴ in the melting process. Below the melting point, the Q ⁿ distribution in lithium disilicate melt remained while, in sodium and potassium disilicate melts, [Q³] increased with decreasing temperature. Crystallization of the alkali disilicate melts is discussed considering the configuration entropy of Q ⁿ units. In 25?mol%?Li₂O–75?mol%?SiO₂, which is in the range of the immiscibility dome, the Q ⁿ distribution was maintained even when phase separation occurs in the cooling process.     

High-gradient operators of the unitary matrix-model

A complete classification of all rotationally invariant operators of the two-dimensional unitary matrix model composed of gradients of the fieldQ and their anomalous dimensions are given in one-loop order. Similarly as in the orthogonal case and for then-vector model the leading correction of operators with2n factors Q grows withn(n–1).Work supported in part by the Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 123 Stochastic Mathematical Models     

The N-mode squeezed state with enhanced squeezing

It is known that exp [iλ (Q1P1i/2)] is a unitary single-mode squeezing operator,where Q1,P1 are the coordinate and momentum operators,respectively.In this paper we employ Dirac’s coordinate representation to prove that the exponential operator S n ≡ exp [iλ sum((QiPi+1+Qi+1Pi))) from i=1 to n ],(Qn+1=Q1,Pn+1=P1),is an n-mode squeezing operator which enhances the standard squeezing.By virtue of the technique of integration within an ordered product of operators we derive S n ’s normally ordered expansion and obtain new n-mode squeezed vacuum states,its Wigner function is calculated by using the Weyl ordering invariance under similar transformations.     

Closed Geodesics and Billiards on Quadrics Related to Elliptic KdV Solutions

We consider algebraic geometrical properties of the integrable billiard on a quadric Q with elastic impacts along another quadric confocal to Q. These properties are in sharp contrast with those of the ellipsoidal Birkhoff billiards in ⁿ . Namely, generic complex invariant manifolds are not Abelian varieties, and the billiard map is no more algebraic. A Poncelet-like theorem for such system is known. We give explicit sufficient conditions both for closed geodesics and periodic billiard orbits on Q and discuss their relation with the elliptic KdV solutions and elliptic Calogero system.     

Index of a family of Dirac operators on loop space

We use methods of constructive field theory to generalize index theory to an infinite-dimensional setting. We study a family of Dirac operatorsQ on loop space. These operators arise in the context of supersymmetric nonlinear quantum field models with HamiltoniansH=Q ². In these modelsQ is self-adjoint and Fredholm. A natural grading operator Γ exists such that ΓQ+QΓ=0. We studyQ ₊=P _? QP ₊, whereP _±=1/2 (1±Γ) are the orthogonal projections onto the eigenspaces of Γ. We calculate the indexi(Q ₊) for Wess-Zumino models defined by a superpotentialV(ω). HereV is a polynomial of degreen≧2. We establish thati(Q ₊)=n?1=degδV. In particular, the field theory models have unbroken supersymmetry, and (forn≧3) they have degenerate vacua. We believe that this is the first index theorem for a Dirac operator that couples infinitely many degrees of freedom.     

Measurements of total and positronium-formation cross sections for the scattering of positrons by alkali atoms

We present a review of our recent measurements of total cross sections (Q _T's) for the scattering of positrons by Na, K, and Rb, and positronium-formation cross sections (Q _Ps's) for Na and K. For our total cross section measurements, a beam transmission technique has been used. For ourQ _Ps measurements, our approach involves setting upper and lower limits onQ _Ps using a combination of (1) measuring the transmission of the positron beam with the angular discrimination of the apparatus made as poor as possible, and (2) measuring the 511 keV annihilation gamma rays in coincidence produced by the decay of para-Ps formed in the scattering cell. Comparison with theoretical calculations shows that our measuredQ _T's andQ _Ps's for Na and K agree reasonably well with a close coupling approximation (CCA) calculation which takes into account the formation of Ps in then=1 andn=2 states. In the 3–10 eV energy range, this calculation predicts a peak in theQ _T's andQ _Ps's for K which also appears in our measurements. The absence of such a peak in our measuredQ _T's andQ _Ps's (preliminary) for Na in this energy range is also consistent with the same theory. Comparisons with five-state CCA calculations ofQ _T which do not take Ps-formation into account also show good agreement with our positron-Na, K, and RbQ _T measurements for energies above 20 eV, but show dramatic departures from our measurements below 10 eV for K and Rb.     

基于马尔科夫链的射频识别防碰撞测试

分析了自适应Q值算法的防碰撞原理以及射频识别(RFID)通信的时序,定义防碰撞过程的识别效率、识别速度和标签数目及Q值的数据状态(Q,n).在此基础上讨论并建立了多标签的状态转移过程的马尔科夫链模型.通过蒙特卡罗统计方法,对马尔科夫链模型求解,得出识别效率和识别速度.用软件无线电测试方案实现防碰撞测试,有效地实现了RFID防碰撞过程的识别效率和识别速度的量化分析.模型仿真结果和测试数据的一致证明了测试模型的有效性和测试方法的正确性. 关键词： 射频识别 防碰撞测试 马尔科夫链 时隙计数器     

Creep behaviour in a discontinuous SiC–Zn-22% Al composite

Creep experiments were conducted on Zn-22%?Al in which SiC particulates were introduced by variable co-deposition of multi-phase materials (VCM). The objective of the investigation is to determine the effect of SiC particulates on the creep behaviour in region I (the low-stress region) and region II (the intermediate-stress or superplastic region) of the sigmoidal plot between stress and strain rate, which was previously reported for the reinforcement-free Zn-22%?Al. The creep data show that the presence of SiC particulates has no effect on the sigmoidal trend between stress and strain rate; and that in region II, the stress exponent, n, and the activation energy for creep, Q, agree well with those reported for SiC-free grades of Zn-22%?Al; n?=?2.5 and Q?～?Q _gb, where Q _gb is energy for grain boundary diffusion in the alloy. However, the data indicate that the presence of the particulates results in narrowing region II and reducing maximum ductility. An analysis of the creep data reveals the presence of a threshold stress that depends strongly on temperature. The microstructural data inferred from an examination of the crept specimens by the means of transmission electron microscopy (TEM) suggest that the origin of τ ₀ may be related to the interaction between moving dislocations and dispersion particles. These particles are introduced in the material as a result of processing the material by thermal spray and deposition.     

Piezoelectric resonators with mechanical damping and resistance in current conduction

A novel design method for high Q piezoelectric resonators was presented and proposed using the 3-D equations of linear piezoelectricity with quasi-electrostatic approximation which include losses attributed to mechanical damping in solid and resistance in current conduction. There is currently no finite element sofware for estimating the Q of a resonator without apriori assumptions of the resonator impedance or damping. There is a necessity for better and more realistic modeling of resonators and filters due to miniaturization and the rapid advances in frequency ranges in telecommunication. We presented new three-dimensional finite element models of quartz and barium titanate resonators with mechanical damping and resistance in current conduction. Lee, Liu and Ballato’s 3-D equations of linear piezoelectricity with quasi-electrostatic approximation which include losses attributed to mechanical damping in solid and resistance in current conduction were formulated in a weak form and implemented in COMSOL. The resulting finite element model could predict the Q and other electrical parameters for any piezoelectric resonator without apriori assumptions of damping or resistance. Forced and free vibration analyses were performed and the results for the Q and other electrical parameters were obtained. Comparisons of the Q and other electrical parameters obtained from the free vibration analysis with their corresponding values from the forced vibration analysis were found to be in excellent agreement. Hence, the frequency spectra obtained from the free vibration analysis could be used for designing high Q resonators. Results for quartz thickness shear AT-cut and SC-cut resonators and thickness stretch poled barium titanate resonators were presented. An unexpected benefit of the model was the prediction of resonator Q with energy losses via the mounting supports.     

Supersymmetric Quantum Mechanics and Super-Lichnerowicz Algebras

We present supersymmetric, curved space, quantum mechanical models based on deformations of a parabolic subalgebra of osp(2p+2|Q). The dynamics are governed by a spinning particle action whose internal coordinates are Lorentz vectors labeled by the fundamental representation of osp(2p|Q). The states of the theory are tensors or spinor-tensors on the curved background while conserved charges correspond to the various differential geometry operators acting on these. The Hamiltonian generalizes Lichnerowicz’s wave/Laplace operator. It is central, and the models are supersymmetric whenever the background is a symmetric space, although there is an osp(2p|Q) superalgebra for any curved background. The lowest purely bosonic example (2p, Q) = (2, 0) corresponds to a deformed Jacobi group and describes Lichnerowicz’s original algebra of constant curvature, differential geometric operators acting on symmetric tensors. The case (2p, Q) = (0, 1) is simply the superparticle whose supercharge amounts to the Dirac operator acting on spinors. The (2p, Q) = (0, 2) model is the supersymmetric quantum mechanics corresponding to differential forms. (This latter pair of models are supersymmetric on any Riemannian background.) When Q is odd, the models apply to spinor-tensors. The (2p, Q) = (2, 1) model is distinguished by admitting a central Lichnerowicz-Dirac operator when the background is constant curvature. The new supersymmetric models are novel in that the Hamiltonian is not just a square of super charges, but rather a sum of commutators of supercharges and commutators of bosonic charges. These models and superalgebras are a very useful tool for any study involving high rank tensors and spinors on manifolds. Dedicated to the memory of Tom Branson     

Chaotic properties of multipoint correlation functions of an Ising model with long-range interactions on the Sierpiński—Gasket lattice

A class of multispin correlation functions of an Ising model with ferromagnetic nearest neighbor interactionsK and constant (distance-independent) long-range interactionsQ ₁=Q,l=1,2,..., on the Sierpiski-gasket lattice is considered. Using an exact method for calculating thermodynamic functions of hierarchically constructed Ising systems, it is shown that, for a set of values ofQ and for almost all values ofK, someM _k-spin correlation functions, whereM _k=3 ^k +3 withk=1,2,...,n andn=1,2,... being the order of lattice construction, change chaotically asn, k, and therebyM _k increase to infinity. Accordingly, in the thermodynamic limit, these correlation functions prove to be nonanalytic for appropriate values ofQ andK. SinceM _k-point correlation functions withk being finite, i.e., correlation functions involving finite numbers of spins, remain analytic asn tends to infinity, there is a smooth crossover between analytic properties of correlation functions of the two types.     

Minimum-wear characteristics for boundary friction between solids

A new parameter is proposed for characterizing boundary friction between solids—the probability Q that some contact spot will be converted to a wear particle as the real contact area shifts by one mean contact-spot diameter. A method is proposed for the phenomenological detection of optimum regimes of boundary friction, which are characterized by the minimum wear intensity of the sample material, and for measuring the corresponding quantity Q=Q ^opt. For babbitt, one of the most frictionless materials, the value of Q ^opt is found to be ≈2×10⁻¹⁰. When data on the linear wear intensity I _h , the contact pressure p _n , and the hardness H are available, Q can be calculated for the given test conditions. Deviations of Q from the value Q ^opt (for a given material) can serve as a criterion for how closely a prepared surface structure approaches the optimal. Zh. Tekh. Fiz. 68, 48–52 (April 1998)     

Photon,quantum and collective,effects from rydberg atoms in cavities

I review some theories of the interaction ofN Rydberg atoms interacting collectively with radiation in microwave cavities. The radiation may be incoherent (black body) radiation or it may be coherent. In the former case theories of the steady state inversion and of the superradiance from initially inverted atoms in low-Q cavities agree well with experimental observations. In the latter case in low-Q cavities ‘phase transitions’ of both first and second order types are predicted and should be observable by monitoring the output of an atomic beam by an atomic ionisation detector. The first order transition which occurs at opposite detunings of the cavity and atoms from the frequency of the coherent driving field is of “optically” bistable type but hysteresis is suppressed by quantum fluctuations which can be large in the cavity field close to the transition. I also review a theory of the spectra from single atoms in cavities ofarbitrary Q containing a few microwave photons. A transition from a single peaked Lorentzian spectrum at low-Q to a double-peaked spectrum forQ≃10⁶ is predicted and peaks representing one or more photon transitions of the Jaynes-Cummings model are also expected to be observable at these or largerQ values. The collective theories are all based onN atom Dicke type models driven by the coherent or incoherent field. Substantial squeezing of the fluorescent radiation field from these Dicke models is also predicted and may be observable with Rydberg atoms.     

Investigation of the Mesostructure of Transition-Metal Monogermanides Synthesized under Pressure

The mesostructure of transition-metal monogermanides Mn_{1 – x}Co _x Ge is studied by small-angle neutron scattering in a wide range of concentrations x = 0–0.95. These compounds were synthesized under high pressure and are metastable under normal conditions. The experimental dependences I(Q) obtained for the whole series of samples in the range of transferred momenta (6 × 10^–2 nm^–1 < Q< 2.5 nm^–1) are described by the power dependence Q^–n with an exponent n = 2.99 ± 0.02, uniquely related to the fractal properties of the system under study. The dependence obtained indicates that the superatomic structure of the samples is characterized by the presence of defects with a spatial organization described by a fractal model with a logarithmic dependence of the correlation function of the defect density. It is interesting to note that such defects are absent in the isostructural FeGe compound, i.e., the experimental dependences of the intensity are described well by the expression Q^–n with an exponent n = 4.1 ± 0.1, which demonstrates the presence of crystallites with a uniform density distribution inside and a sharp boundary characterizing the surface.

     

Metastable decay rates,asymptotic expansions,and analytic continuation of thermodynamic functions

The grand potentialP(z)/kT of the cluster model at fugacityz, neglecting interactions between clusters, is defined by a power series _n Q _n z ⁿ , whereQ _n , which depends on the temperatureT, is the partition function of a cluster of sizen. At low temperatures this series has a finite radius of convergencez _s . Some theorems are proved showing that ifQ _n , considered as a function ofn, is the Laplace transform of a function with suitable properties, thenP(z) can be analytically continued into the complexz plane cut along the real axis fromz _s to + and that (a) the imaginary part ofP(z) on the cut is (apart from a relatively unimportant prefactor) equal to the rate of nucleation of the corresponding metastable state, as given by Becker-Döring theory, and (b) the real part ofP(z) on the cut is approximately equal to the metastable grand potential as calculated by truncating the divergent power series at its smallest term.     

Quantitative Raman spectroscopy: speciation of cesium silicate glasses

The silicate speciation forms an important aspect of the structure of silicate melts, a subject of interest to both the earth‐ and materials science communities. In this study, the Qⁿ speciation of binary cesium silicate glasses was studied by Raman spectroscopy. A method to extract the equilibrium constant from a set of Raman spectra is presented, and the least‐squares optimization algorithm is given (in Supporting Information). Log(K), the equilibrium constant of the speciation reaction, 2Q³ = Q⁴ + Q², equals −2.72 ± 0.11 at the glass transition. This extends the previously established correlation between log(K) and the inverse of the ionic radius of the network modifier to cesium. Copyright © 2009 John Wiley & Sons, Ltd.     

The properties of some derivative autocorrelation functions computed with the atom-atom potential

High-resolution quasi-elastic neutron-scattering measurements have been made on two nematogens: DMBCA with a nematic range 108 to 119°C, and 5CB and a tail-deuteriated sample (D5CB), having a nematic range 22·6 to 35·1°C.

Results on 5CB in the crystal phase at ～18°C showed no significant quasielastic broadening, which means that any random motions of the alkyl chain are slower than about 5 × 10⁹ rad s^-1. Measurements were made at a single temperature in the nematic phases on specimens aligned in a magnetic field of 0·25T; for DMBCA with scattering vector Q⊥n (n is the nematic director) and for 5CB and D5CB with Q⊥n and Q∥n and also on the isotropic liquid phase of D5CB at 45°C. Analysis of the coherent scattering from nematic D5CB at Q = 1·2 Å^-1 and 25°C gave an order parameter <P ₂>=0·55, close to the simple mean field value for this temperature. The coherent scattering from DMBCA is too weak to allow this experiment to be performed.

The most remarkable qualitative feature of the results is the close similarity of the scattering law S(Q, ω) for D5CB (and 5CB) with Q⊥n and Q∥n. Analysis of the results in all cases was made using values for the translational diffusion constants measured previously. Corrections for multiple scattering are shown to be important and a single simple model has been devised which fits the line shapes of all the results for D5CB in nematic (Q⊥n and Q∥n) and isotropic liquid phases and DMBCA. The model involves uniaxial rotational diffusion about the long molecular axis m coupled to a displacement along the rotation axis giving a net rotation in a plane whose normal makes an angle ∝ relative to the direction m. Values for the rotational diffusion constant D _rd ns^-1 are as follows: D5CB, 25°C, 6 (∝ ～ 50°); 45°C, 10. DMBCA, 112°C, 16, (all ±10–15 per cent).

The results for D5CB and 5CB are so similar that no additional detailed model fitting was attempted for the fully hydrogenous sample and it is concluded that while the motion of the alkyl tails is freer, the time scale of the motions is not more than about a factor of 2 faster than that of the molecular cores.     

Bifurcation of a torus from a stationary state in the presence of symmetry

We investigate the formation of a torus from a stationary state in an autonomous dynamical system with order-parameter dimensionn=4 and symmetry groupG=QC ₂ (Q =quaternion group). The quasi-periodic temporal states bifurcating from the origin due to a symmetry-induced hard-mode instability are characterized by two-independent frequencies.Dedicated to Professor Harry Thomas on the occasion of his 60th birthdayWork supported by the Sonderforschungsbereich 185 Frankfurt-Darmstadt     

Modular conformal scaling group evidenced in lepton-quark mass

Lepton-quark mass may reflect a correspondence in spacetime structure described by a modular conformal scaling group. Stemming in part from a spacetime line element correspondenceds → (expλ _n)ds in which the eight quantitiesλ ₀,λ ₁, ...,λ ₇ constitute a closed set under a modular addition, the associated formula for lepton-quark mass (yielding values at the 1 GeV scale for the leptons and lighter quarks and at the physical pole for the top) is conjectured to bem=m _fQ²(exp −λ _n), wherem _f=10.245 TeV is the progenitor fermion mass,Q is the charge number of the lepton or quark, and the modular group parameterλ _n is indexed by a fermion principal quantum numbern that depends on three mutually independent projection operators.