Local state probabilities for solvable restricted solid-on-solid models:A n,D n,D n (1) , andA n (1)

The local state probabilities (LSPs) are exactly computed for four hierarchies of solvable lattice models. They are restricted solid-on-solid (RSOS) models whose local states and their adjacent conditions are specified by Dinkin diagrams of typesA _n,D _n,D _n ⁽¹⁾ andA _n ⁽¹⁾ . The LSPs are expressed in terms of modular functions characterized by branching identities among the theta functions. Their automorphic properties are used to study the critical behaviors. Some fine structures are found in the spectrum of the critical exponents.     

Spin waves and dual spin waves ofSU(2) chiral models

We study the low coupling (low temperature) limit of the dual versions of chiralSU(2) invariant models. Based on a conjecture of Regge about the asymptotic behaviour of the mean square of 3nj-coefficients forj→∞, we show that the low coupling excitations of the dual chiral models are vibrational modes of a triangular manifold, which is embedded in Euclidean three dimensional space and is formed by classical angular momenta.     

One-dimensional configuration sums in vertex models and affine Lie algebra characters

We study the local state probabilities of the vertex models in the face formulation associated with the simple Lie algebras X _n =A _n, B _n, C _n, D _n. The corner transfer matrix method expresses them in terms of one-dimensional configuration sums. We show that the latter are the string functions of X _n ⁽¹⁾ modules. We also present similar results for the restricted face models of types B _n ⁽¹⁾, C _n ⁽¹⁾, D _n ⁽¹⁾.     

On the configuration of classical Yang-Mills fields with topological charge <Emphasis Type="Italic">k</Emphasis> = 4

The solutions of the nonlinear matrix equation in the Atiyah-Hitchin-Drifeld-Manin (AHDM) construction that determine the Yang-Mills self-dual fields with topological charge k = 4 for symplectic gauge groups are discussed. In the case of Sp(n), n > 2, it is possible to use a procedure that was proposed earlier for generating solutions with k = 3. It is shown that for SU(2) = Sp(1) the AHDM matrix can be generated by using cubic equation solutions with coefficients that depend on 8k — 3 parameters.     

Solvable lattice models related to the vector representation of classical simple Lie algebras

A series of solvable lattice models with face interaction are introduced on the basis of the affine Lie algebraX _n ⁽¹⁾ =A _n ⁽¹⁾ ,B _n ⁽¹⁾ ,C _n ⁽¹⁾ ,D _n ⁽¹⁾ . The local states taken on by the fluctuation variables are the dominant integral weights ofX _n ⁽¹⁾ of a fixed level. Adjacent local states are subject to a condition related to the vector representation ofX _n . The Boltzmann weights are parametrized by elliptic theta functions and solve the star-triangle relation.     

The effect of surface temperature on H2/D2(v = 0, j = 0)–Ni(100) scattering processes

We carry out both four-dimensional (4D×2D) and six-dimensional (6D) quantum dynamics on a parametrically time- and temperature-dependent effective Hamiltonian for H₂/D₂(v = 0,j = 0)–Ni(100) collision process. Such an effective potential was derived within a theoretical framework of mean-field approximation by considering weakly correlated interaction between molecular degrees of freedom, phonon modes and electron– hole pair (elhp) coupling through a Hartree-product-type wave function, where the initial state distribution of the surface modes and elhp coupling were introduced through Bose– Einstein and Fermi– Dirac probability factor, respectively. The temperature-dependent dissociation and state-to-state transition probabilities for H₂/D₂(v = 0,j = 0)–Ni(100) system are depicted as a function of initial kinetic energ of the incoming diatom. Though such effect appears negligibly small for H₂(v = 0,j = 0)–Ni(100) system, it is prominent in the case of D₂(v = 0,j = 0)–Ni(100) collision. It appears that the change of dissociation and transition probabilities of D₂ with the increase of surface temperature is exclusively dictated by the phonon modes directed along Z-axis, but the effect of elhp coupling particularly for transition probabilities is insignificant.     

Limiting case of modified electroweak model for contracted gauge group

The modification of the Electroweak Model with 3-dimensional spherical geometry in the matter fields space is suggested. The Lagrangian of this model is given by the sum of the free (without any potential term) matter fields Lagrangian and the standard gauge fields Lagrangian. The vector boson masses are generated by transformation of this Lagrangian from Cartesian coordinates to coordinates on the sphere S ³. The limiting case of the bosonic part of the modified model, which corresponds to the contracted gauge group SU(2; j) × U(1) is discussed. Within framework of the limit model Z boson and electromagnetic fields can be regarded as external ones with respect to W-boson fields in the sence that W-boson fields do not effect on these external fields. The masses of all particles of the Electroweak Model remain the same, but field interactions in contracted model are more simple as compared with the standard Electroweak Model.     

Non-abelian T-duality,Ramond fields and coset geometries

We extend the recently constructed double field theory formulation of the low-energy theory of the closed bosonic string to the heterotic string. The action can be written in terms of a generalized metric that is a covariant tensor under O(D, D + n), where n denotes the number of gauge vectors, and n additional coordinates are introduced together with a covariant constraint that locally removes these new coordinates. For the abelian subsector, the action takes the same structural form as for the bosonic string, but based on the enlarged generalized metric, thereby featuring a global O(D, D + n) symmetry. After turning on non-abelian gauge couplings, this global symmetry is broken, but the action can still be written in a fully O(D, D + n) covariant fashion, in analogy to similar constructions in gauged supergravities.

     

Meron-antimeron solution in Minkowski space

An elementary derivation, using Witten's Ansatz, is given of the elliptic meron-antimeron solution of the (Minkowski) SU(2) gauge theory in the W ₀=0 gauge.     

SU(2) q in a Hilbert space of analytic functions

The algebraSU(2) _q is realized in a Hilbert spaceH _q ² of analytic functions; the starting point is the differential realization of operators that satisfyq-algebra in a Hilbert spaceH _q. The Weyl realization ofSU(2) _q is constructed exhibiting the reproducing kernel and the principal vectors; the noncommutativity of the matrix elements of a 2×2 linear representation ofSU(2) _q is obtained as consistency conditions for couplingj1=j2=1/2 toj=0, 1; the derivation of Clebsch-Gordan coefficients is sketched and theq-generalization of the rotation matrices is included. The unitary correspondence ofH _q with a Hilbert space of complex functions of a real variable is also studied. The study presented in this paper follows Bargmann's formalism for the rotation group as closely as possible.     

Many-atom interactions in the theory of higher order elastic moduli: A general theory

The total potential energy of a crystal U({r _ik }) as a function of the vectors r _ik connecting centers of equilibrium positions of the ith and kth atoms is assumed to be represented as a sum of irreducible interaction energies in clusters containing pairs, triples, and quadruples of atoms located in sites of the crystal lattice A2: U({r _ik }) ≡ Σ _N=1 ⁴ E _N ({r _ik }). The curly brackets denote the “entire set.” A complete set of invariants {I _j ({r _ik })} _N , which determine the energy of each individual cluster as a function of the vectors connecting centers of equilibrium positions of atoms in the cluster E _N ({r _ik }) ≡ E _N ({I _j ({r _ik })} _N ), is obtained from symmetry considerations. The vectors r _ik are represented in the form of an expansion in the basis of the Bravais lattice. This makes it possible to represent the invariants {I _j ({r _ik })} _N in the form of polynomials of integers multiplied by τ ₂ ^m . Here, τ₂ is one-half of the edge of the unit cell in the A2 structure and m is a constant determined by the model of interaction energy in pairs, triples, and quadruples of atoms. The model interaction potential between atoms in the form of a sum of the Lennard-Jones interaction potential and similarly constructed interaction potentials of triples and quadruples of atoms is considered as an example. Within this model, analytical expressions for second-order and third-order elastic moduli of crystals with the A2 structure are obtained.     

The lie algebra of invariant group of the KdV,MKdV, or Burgers equation

Let (E): u _t=H(u) denote the KdV, MKdV or Burgers equation, and U(s)=(D^j _s)/u _j, where D=d/dx, u _i=Dⁱ _u, s=s(u, u ₁, ..., u _n) is a polynomial of u _i with constant coefficients, be the generator of invariant group of equation (E). We prove in this paper that all such generators form a commutative Lie algebra, from which it follows that for any symmetry s(u, ..., u _n) of (E), the evolution equation u _t=s(u, ..., u _n) possesses an infinite number of symmetries (or conservation laws in the case of KdV and MKdV equations).     

Virial series around non-null density: density dependence of virial coefficients

It is shown that the equation of state of fluid systems can be expanded around non-null densities if the well known virial series is generalized by considering its coefficients as density dependent. This in turn leads to a hierarchy of differential equations that describe the coefficients b_j (ρ). Starting from the already known equation of state for hard bodies in d = 0,1,2,3 dimensions this hierarchy is analysed and the behaviour of both the reducible b_j (ρ) and irreducible β～ _j (ρ) cluster integrals is discussed. New virial coefficients b_j (ρ) have been introduced with a simpler density dependence. Their asymptotic (j → ∞) behaviour is discussed.     

Field Equations in Quantum Electrodynamics

A formulation of quantum electrodynamics is presented, based on finite local field equations. These Dirac and Maxwell equations have the usual form except that the current operators f(x) and j^μ (x) are explicitly expressed as local limits of sums of non-local field products and suitable subtraction terms. These limits are shown to exist and to yield finite operators in the sense that the iterative solutions to the field equations are equivalent to conventional renormalized perturbation theory. The various invariance properties of the theory, including Lorentz invariance, gauge invariance, charge conjugation invariance, and renormalization invariance, are discussed and related directly to the field equations and current definitions. Initially only the general forms of the currents, based on dimensional arguments, are given. The electric current, for example, contains the (suitably defined) term :A³(x) :.The corresponding field equations are used to derive renormalized Dyson-Schwinger-type integral equations for the renormalized proper part functions ∑, II^μν, Λ^μ, and X^αβγδ (the four-photon vertex function), etc. Application of the boundary conditions ∑(p̀ = m) = ∑′(p̀ = m) = II(O) = II′(O) = II″(O) = Λ(p̀ = m, o) = X(O, O, O, O) = O completely specifies the current operators. Consistency is established by deriving the same equations from rigorous renormalization theory so that their iterative solutions are proved to reproduce the correct renormalized perturbation expansion. The electric current operator is exhibited in a manifestly gauge invariant form and in a form which is manifestly negative under charge conjugation. It is shown, in fact, that much of j^μ (x) can be determined directly from the requirements of gauge invariance and charge conjugation covariance, without recourse to the integral equations. It is suggested that equal time commutation relations can serve to similarly specify the rest of the current.     

The application of a new approach based on organic homo‐rank compounds and homologous compounds to the structure–property relationship study of mono‐substituted alkanes

The two conceptual systems of organic homologous compounds and homo‐rank compounds give insight into the influence of structures on the properties of mono‐substituted alkanes X_i–(CH₂)_j–H from the transverse (change of repeating unit number j of CH₂) and longitudinal (change of functional group X_i) perspectives, respectively. This paper aims to combine the organic homo‐rank compounds approach together with the homologous compounds approach to explore the property change rules of mono‐substituted alkanes involving various substituents. Firstly, based on the concept of organic homologous compounds, the properties of mono‐substituted straight‐chain alkane homologues were linearly correlated to the two‐thirds power of the number of carbon atoms (N^2/3) in alkyl, and regression equations such as Q = A + BN^2/3 were obtained. The regression coefficients A and B vary with different substituents X_i, so coefficients A and B were employed to characterize the structural information of substituent X_i. The structural features of alkyls (–(CH₂)_j–H, that is, –C_jH_2j+1) were described by the polarizability effect index (PEI_(R)) and vertex degree–distance index (VDI). Then based on four parameters A, B, PEI_(R), and VDI, quantitative structure–property relationship models were built for the boiling points (Bp) and refractive indexes (n_D) of each mono‐substituted alkane homo‐rank series, where j = 3–10 and the substituents X_i involve F, Cl, Br, I, NO₂, CN, NH₂, COOH, CHO, OH, SH, and NC. Good results indicate that the combination of an organic homo‐rank compounds method and a homologous compounds method has exhibited obvious advantages over traditional methods in the quantitative structure–property relationship study of mono‐substituted alkanes concerning various substituents. Copyright © 2015 John Wiley & Sons, Ltd.     

A supersymmetricSU(5)×U(1) model with natural gauge hierarchy

We present a supersymmetricSU(5)×U(1) model. This model has the following features. The gauge hierarchy is naturally generated by the quadratically divergent nature of the Fayet-IliopoulosD term. TheSU(5)×U(1) gauge symmetry breaks uniquely intoSU(3) _W ×SU(2) _c ×U(1) _y at an energy scale of 10^17–18GeV. The non-vanishing vacuum expectation value of an auxiliary field component ofU(1) gauge vector multiplet induces the breaking ofSU(2) _W ×U(1) _y . It gives a mass of 10^2–3GeV to scalar quarks and scalar leptons at the tree level. The renormalization group analysis shows that the color fine structure constant α _C (M _W ) becomes somewhat small and the Weinberg angle sin²θ _W (M _W ) somewhat too large in a simple version of the model.     

Complete determination of the transition amplitudes from a comprehensive data analysis of the fusion reactions D(d,p)3H and D(d,n)3He for Ed<500 keV

All relevant low-energy transition amplitudes for the D(d,n)³He and D(d,p)³H reactions were determined from a fit to Legendre expansion coefficients of the available experimental data. A simple barrier penetrability model was used. Quintet S-wave transitions are found to contribute strongly thus obliterating the idea of neutron-lean “polarized” fusion energy production. The D+D interaction radius was determined with good accuracy for both reactions individually. The astrophysical S functions show a small S-wave enhancement and P-wave suppression of the D(d,p)³H branch.     

Precision measurements of optical excitation functions for zinc atom

The excitation of zinc atoms by ultramonoenergetic electrons is experimentally studied. The optical excitation functions for 19 atomic spectral lines that originate from the n ¹ S ₀, 4¹ P ₁, n ¹ D ₂, n ³ S ₁, 4³ P ₁, 6³ P ₂, and n ³ D _j levels are studied in detail. In the excitation functions measured from the excitation threshold to 19 eV for the spectral lines originating from the n ¹ S ₀, n ³ S ₁, n ¹ D ², and n ³ D _j levels, specific features caused by postcollision interactions of emitted and scattered electrons are observed for the first time in the energy region of 10.9–17.0 eV near the thresholds of autoionization states.     

Capacity of Multivariate Channels with Multiplicative Noise: Random Matrix Techniques and Large-N Expansions (2)

We study memoryless, discrete time, matrix channels with additive white Gaussian noise and input power constraints of the form Y i = ∑ _j H _ij X _j + Z _i , where Y _i , X _j and Z _i are complex, i = 1… m, j = 1… n, and H is a complex m× n matrix with some degree of randomness in its entries. The additive Gaussian noise vector is assumed to have uncorrelated entries. Let H be a full matrix (non-sparse) with pairwise correlations between matrix entries of the form E[H ik H ^* jl] = 1/n C ij D kl, where C, D are positive definite Hermitian matrices. Simplicities arise in the limit of large matrix sizes (the so called large-n limit) which allow us to obtain several exact expressions relating to the channel capacity. We study the probability distribution of the quantity f(H) = log (1+PH ^† SH) . S is non-negative definite and hermitian, with TrS = n and P being the signal power per input channel. Note that the expectation E[f(H)], maximised over S, gives the capacity of the above channel with an input power constraint in the case H is known at the receiver but not at the transmitter. For arbitrary C, D exact expressions are obtained for the expectation and variance of f(H) in the large matrix size limit. For C = D = I, where I is the identity matrix, expressions are in addition obtained for the full moment generating function for arbitrary (finite) matrix size in the large signal to noise limit. Finally, we obtain the channel capacity where the channel matrix is partly known and partly unknown and of the form α; I+ β H, α,β being known constants and entries of H i.i.d. Gaussian with variance 1/n. Channels of the form described above are of interest for wireless transmission with multiple antennae and receivers.