One-dimensional Ising chain with competing interactions: Exact results and connection with other statistical models

We study the ground state properties of a one-dimensional Ising chain with a nearest-neighbor ferromagnetic interactionJ ₁, and akth neighboranti-ferromagnetic interactionJ _k . WhenJ _k/J₁=–1/k, there exists a highly degenerate ground state with a residual entropy per spin. For the finite chain with free boundary conditions, we calculate the degeneracy of this state exactly, and find that it is proportional to the (N+k–1)th term in a generalized Fibonacci sequence defined by,F _N ^(k) =F _N–1 ^(k) +F _N–k ^(k) . In addition, we show that this one-dimensional model is closely related to the following problems: (a) a fully frustrated two-dimensional Ising system with a periodic arrangement of nearest-neighbor ferro- and antiferromagnetic bonds, (b) close-packing of dimers on a ladder, a 2× strip of the square lattice, and (c) directed self-avoiding walks on finite lattice strips.Work partially supported by grants from AFOSR and ARO.     

Thermodynamics on noncommutative geometry in coherent state formalism

The thermodynamics of ideal gas on the noncommutative geometry in the coherent state formalism is investigated. We first evaluate the statistical interparticle potential and see that there are residual “attraction (repulsion) potential” between boson (fermion) in the high temperature limit. The characters could be traced to the fact that, the particle with mass m in noncommutative thermal geometry with noncommutativity θ and temperature T will correspond to that in the commutative background with temperature T(1+kTmθ)⁻¹. Such a correspondence implies that the ideal gas energy will asymptotically approach to a finite limiting value as that on commutative geometry at T_θ=(kmθ)⁻¹. We also investigate the squeezed coherent states and see that they could have arbitrary mean energy. The thermal properties of those systems are calculated and compared to each other. We find that the heat capacity of the squeezed coherent states of boson and fermion on the noncommutative geometry have different values, contrast to that on the commutative geometry.     

Hamilton Spaces of Order k ≥ 1

A suitable dual for the k-acceleration bundle(T ^k M, ^k ,M) is the fiberedbundle (T ^k–1 M× _M T*M). The mentioned bundle carries a canonicalpresymplectic structure and k canonical Poisson structures. By means of thisdual we define the notion of Hamilton spaces of orderk, whose total spaceconsists of points x of the configuration spaceM, accelerations of order 1,...,k – 1, y ⁽¹⁾,...,y ^(k–1), and momenta p. Some remarkable Hamiltonian systemsare pointed out. There exists a Legendre mapping from the Lagrange spaces oforder k to the Hamilton space of order k.     

Eley–Rideal reaction dynamics between O atoms on β-cristobalite (1 0 0) surface: A new interpolated potential energy surface and classical trajectory study

We present a theoretical study of the collisions of atomic oxygen with O-precovered β-cristobalite (1 0 0) surface. We have constructed a multidimensional potential energy surface for the O₂/β-cristobalite (1 0 0) system based mainly on a dense grid of density functional theory points by using the interpolation corrugation-reducing procedure. Classical trajectories have been computed for quasithermal (100–1500 K) and state-specific (e.g., collision energies between 0.01 and 4 eV) conditions of reactants for different O incident angles (θ_v). Atomic sticking and O_2(adsorbed) formation are the main processes, although atomic reflection and Eley–Rideal (ER) reaction (i.e., O₂ gas) are also significant, depending their reaction probabilities on the O incident angle. ER reaction is enhanced by temperature increase, with an activation energy derived from the atomic recombination coefficient (γ_O(θ_v = 0°, T)) equal to 0.24 ± 0.02 eV within the 500–1500 K range, in close agreement with experimental data. Calculated γ_O(θ_v = 0°, T) values compare quite well with available experimental γ_O(T) although a more accurate calculation is proposed. Chemical energy accommodation coefficient β_O(T) is also discussed as a function of ER and other competitive contributions.     

Nonlinear neural networks. II. Information processing

Information processing in nonlinear neural networks with a finite numberq of stored patterns is studied. Each network is characterized completely by its synaptic kernelQ. At low temperatures, the nonlinearity typically results in 2^q–2–q metastable, pure states in addition to theq retrieval states that are associated with theq stored patterns. These spurious states start appearing at a temperature , which depends onq. We give sufficient conditions to guarantee that the retrieval states bifurcate first at a critical temperatureT _c and that /T _c 0 asq. Hence, there is a large temperature range whereonly the retrieval states and certain symmetric mixtures thereof exist. The latter are unstable, as they appear atT _c . For clipped synapses, the bifurcation and stability structure is analyzed in detail and shown to approach that of the (linear) Hopfield model asq. We also investigate memories that forget and indicate how forgetfulness can be explained in terms of the eigenvalue spectrum of the synaptic kernelQ.     

Clustering in a Self-Gravitating One-Dimensional Gas at Zero Temperature

We study a system of gravitationally interacting sticky particles. At the initial time, we have n particles, each with mass 1/n and momentum 0, independently spread on [0, 1] according to the uniform law. Due to the confining of the system, all particles merge into a single cluster after a finite time. We give the asymptotic laws of the time of the last collision and of the time of the kth collision, when n. We prove also that clusters of size k appear at time n ^{–1/2(k–1)}. We then investigate the system at a fixed time t<1. We show that the biggest cluster has size of order logn, whereas a typical cluster is of finite size.     

Convergence to Equilibrium of Random Ising Models in the Griffiths Phase

We consider Glauber-type dynamics for disordered Ising spin systems with nearest neighbor pair interactions in the Griffiths phase. We prove that in a nontrivial portion of the Griffiths phase the system has exponentially decaying correlations of distant functions with probability exponentially close to 1. This condition has, in turn, been shown elsewhere to imply that the convergence to equilibrium is faster than any stretched exponential, and that the average over the disorder of the time-autocorrelation function goes to equilibrium faster than exp[–k(log t) ^d/(d–1)]. We then show that for the diluted Ising model these upper bounds are optimal.     

Effective action in higher derivative (Super)gravities

'The one-loop effective action (EA) with an accuracy up to linear curvature terms ind=4R ²-gravity, conformal gravity, andN=1,d=4 conformal supergravity on the backgroundR ₄×T_4–k,k=1, 2, 3 is calculated. (Here,R _k is thek-dimensional curved space, T_n is then-dimensional torus). The one-loop EA in multidimensionalR ²-gravity and ind=10 conformal supergravity on the backgroundR ₄ ×T _d–4 is also obtained. The mechanism of inducing the Einstein gravity from the EA of considered theories of higher derivative (super)gravity is presented.We are grateful to I. L. Bukhbinder for the numerous discussions of considered questions.     

Dislocation retardation due to the destruction of short-range order in the L12 superstructure

The magnitude and temperature dependence of the dislocation retardation due to the destruction of short-range order are calculated in the quasichemical approximation. The atomic interaction in two coordination spheres and dislocation slip in cubic and octahedral planes are taken into account. The retardation stress for the first superdislocation is essentially the same in these planes for T < T _k (where T _k is the phase-transition temperature), while the retardation stress for subsequent superdislocations is much lower, so large planar accumulations of superdislocations may arise. The retardation stress is maximal at T = T _k , and for T > T _k the first dislocations should move in pairs. Comparison of the experimental cleavage stresses with calculated values shows that, by itself, superdislocation retardation due to correlation destruction cannot explain the behavior of the yield point.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 78–85, April, 1971.     

Painlevé test of the McKean and Carleman models

The Painlevé test of the system of nonlinear partial differential first-order equations u₁+u_k=k₁v²+k₂u²+k₃uv, v₁–v_x=–k₁v²–k₂u²–k₃uv is performed. The system includes the Carleman and McKean models which are caricatures of the Boltzmann equation. For k ₁=k ₂=0 the system describes the interaction of two waves u and v. The results of the Painlevé test are discussed in connection with whether or not the system is integrable. We also study in detail the constraint on (whose vanishing defines a noncharacteristic hypersurface S) which arises at the resonance.     

On the local structure of the phase separation line in the two-dimensional Ising system

We investigate the structure of the phase separation line between the pure phases in the two-dimensional Ising model, the liquid and vapor phase in lattice gas language, at low temperatures. The fluctuations in the location of this line are known to diverge in the thermodynamic limit, something which is also believed to happen to the continuum liquid-vapor interface in three dimensions (in the absence of the gravitational field). We show that despite this global divergence it is possible to define precisely the local structure of the phase separation line. This has a finite, exponentially small, width at low temperatures which is related by a central limit theorem⁽¹⁾ to the width of the global fluctuations on the appropriate (divergent) length scale. The latter has been computed explicitly⁽²⁾ for all temperatures below the critical temperatureT _c, where it diverges as (T _c –T)^–1/2. We also prove a Gibbs formula for the surface tension at low temperature, which relates it to the local structure of the phase separation line.Supported in part by NSF grant No. M^rPHY 78-15920 and MCS78-01885.On leave from: Departement de Physique Théorique, Université de Louvain, Belgium.     

Dependency of control parameters on a rate coefficient giving the potential curvature in a reaction cascade model

Many chemical reactions in vivo are self-controlled by fluxes of chemical energy and matter through biological systems, so the induction of such reactions can be governed by changes in the control parameters of the rate equation. A potential of a system is assumed to be given by Gibbs' functionG(T, P, x), which is continuously differentiable, and the rate equation can be derived from the differential (–G/x) of Taylor's expansion ofG _(T,P)(x) for the order parameterx, which corresponds to the product number, at around the critical pointC(T _C, P_C). The equation is described bydx/dt=(x)–k₁x–k₂x³, andk ₂>0. In this equation,k ₁ andk ₂ are functions of the control parameters, temperatureT and pressureP, andk ₁ is allowed to have a positive or negative values as (T, P). Thenk ₁ is an important factor that decides the induction conditions of the reactions with a phase transition in the steady statex=0. Because bothk ₁ (the transition parameter) andG are the quantity of state, they are given by the total differential, and functions that decideG andk ₁ are related to a mutual inverse function. From the above relation, the rate of change ink ₁ by G, which corresponds to the reaction energy of the system, is uniquely determined by a function ofk ₁, [f(k ₁ ^± )] andf(k ₁ ^± ) is described approximately by ±₁ k ₁ ^± in the transient process thatk ₁ approaches zero, where ₁ implies 1/RT. These results indicate that internal driving forces caused by a stimulus in a system are proportional tok ₁ ^± and that the system is regulated by competition of the forces. an approximate function fork ₁ in the transient process is described by tanh (G/RT) and Arrhenius' law is elucidated from this theory.Decreased January 19, 1992     

Karhunen–Loève decomposition of peripheral blood flow signal

The Karhunen–Loève expansion is applied to scalar signals and the effect of window length (t_w), time lag (τ) and embedding dimension (d) is analysed for periodic signals and for signals modeled by the Lorenz equations. For τ≠k/2f_i (f_i are characteristic frequencies of the signal, k is positive integer), we obtain 2m modes from an m-periodic signal. For a large set of parameters a finite number of modes was not obtained from the Lorenz system. It is further shown that, on the time scale of a minute, the peripheral blood flow signal contains oscillatory modes that occur in pairs thereby confirming that the blood flow through the cardiovascular system is oscillatory. Some of the difficulties of applying Karhunen–Loève expansion to scalar signals are pointed out.     

Reaction of the muonium substituted cyclohexadienyl radical with 2,3-dimethyl-1,3,-butadiene

The rate constant for the title reaction is represented by log (k _M/M^–1s^–1)=6.9(4)–700(350)K/T between 279 and 343K, whereas the H analogous radical, C₆H₇, reacts at room temperature withk _H 12 M^–1s^–1 with dimethylbutadiene. The title reaction is proposed to be transfer of the light hydrogen isotope, Mu, and the large kinetic isotope effect is discussed.Support by the Swiss National Foundation for Scientific Research and by the Swiss Institute for Nuclear Research (SIN) are gratefully acknowledged.     

Annealing effect on the superconductivity of In2O3–ZnO thin films

We have investigated the relation among ρ–T characteristics, superconductivity, annealing conditions and the crystallinity of polycrystalline (In₂O₃)_1−x–(ZnO)_x films. We annealed as-grown amorphous films in air by changing annealing temperature and time. It is found that the films annealed at 200 °C or 300 °C for a time over 0.5 h shows the superconductivity. Transition temperature T_c and the carrier density n are T_c < 3.3 K and n ≈ 10²⁵–10²⁶ m⁻³, respectively. Investigations for films with x = 0.01 annealed at 200 °C have revealed that the T_c, n and crystallinity depend systematically on annealing time. Further, we consider that there is a suitable annealing time for sharp resistive transition because the transition width becomes wider with longer annealing times. We studied the upper critical magnetic field H_c2(T) for the film with different annealing time. From the slope of dH_c2/dT for all films, we have obtained the resistivity ρ dependence of the coherence length ξ(0) at T = 0 K.     

Classification of degraded cartilage through multiparametric MRI analysis

MRI analysis of cartilage matrix may play an important role in early detection and development of therapeutic protocols for degenerative joint disease. Correlations between MRI parameters and matrix integrity have been established in many studies, but the substantial overlap in values observed for normal and for degraded cartilage greatly limits the specificity of these analyses. We implemented established multiparametric analysis methods to define data clusters corresponding to control and degraded bovine nasal cartilage in two-, three-, and four-dimensional parameter spaces, and applied these results to discriminant analysis of a validation data set. Analyses were performed using the parameters (T₁, T₂, k_m, ADC), where k_m is the magnetization transfer rate and ADC is the apparent diffusion coefficient. Results were compared to univariate analyses. Multiparametric k-means clustering led to no improvement over univariate analyses, with a maximum sensitivity and specificity in the range of 60–70% for the detection of degradation using T₁, and in the range of 80% sensitivity but only 36% specificity using the parameter pair (T₁, k_m). In contrast, model-based analysis using more general Gaussian clusters resulted in markedly improved classification, with sensitivity and specificity reaching levels of 80–90% using the pair (T₁, k_m). Finally, a fuzzy clustering technique was implemented which may be still more appropriate to the continuum of degradation seen in degenerative cartilage disease. In view of its success in identifying mild cartilage degradation, the formal multiparametric approach implemented here may be applicable to the nondestructive evaluation of other biomaterials using MRI.     

The q-boson realizations of the quantum groups U q(B n)

We give explicit realization for the quantum enveloping algebras U _q(B _n). In these formulae the generators of the algebra are expressed by means of 2n–1 canonical q-boson pairs and one auxiliary representation of U _q(B _n–1)     

Elastic double diffractive production of axial-vector mesons and the Landau–Yang theorem

We discuss exclusive elastic double diffractive axial-vector χ_c(1⁺) meson production in proton–antiproton collisions at the Tevatron. The amplitude for the process is derived within the k_t-factorization approach with unintegrated gluon distribution functions (UGDFs). We show that the famous Landau–Yang theorem is not applicable in the case of off-shell gluons. Differential cross sections for different UGDFs are calculated. We compare exclusive production of χ_c(1⁺) and χ_c(0⁺). The contribution of χ_c(1⁺) to the J/Ψ+γ channel is smaller than that of the χ_c(0⁺) decay, but not negligible and can be measured. The numerical value of the ratio of the both contributions is much less dependent on the UGDFs modeling than the cross sections themselves.     

Lift order problems for ordinary differential equations on manifolds

Fork≥0, let ττ_k:T ^k+1(M)=T(T ^k(M))→T ^k(M) denote the (k+1)th iterated tangent bundle in relation to a base manifoldT ⁰(M)=M. LetV represent a possibly nonstationary vector field overT ^k(M), and letQ be a subset/submanifold inT ^k(M). Sufficient conditions (and, whenV is completely integrable inQ, necessary and sufficient conditions) are established to ensure that all solutionsg toy′=V(t, y) lying entirely inQ have the formG=f ^[k], wheref ^[k] is thekth-order differential lift of a curvef lying inM. The relevance of the issue for higher order dynamical systems (especially in mechanics) is discussed. Higher order involutions and complete vector field lifts are examined from the viewpoint of the differential equations they present. Collateral results on the general solvability of initial value problems are obtained and numerous examples are discussed in detail. To the memory of my teacher and friend M. Kuga (1928–1990).