New features of a smectic-A-crystal-B phase transition in a homologous series of liquid crystals

The components L _j of the Lorentz tensor and the polarizability density of molecules G in the smectic-A and crystalline-B phases have been determined for homologues of the series of alkyl-p-(4-alkoxybenzylideneamino-)cinnamates. The quantity L _j (G) in both phases is a linear (quadratic) function of the orientational order parameter of molecules S, which is invariant (noninvariant) with respect to the A-B transition, which is manifested in the form of jumps δL _j and δG and enhancement of the G(S) dependence. An increase in the length of terminal molecular chains and weakening of interlayer correlation of molecules are accompanied by strengthening of the A-B transition of the first order and G(S) dependences in both phases together with an increase in δL _j and δG. Change δG and dependence G(S) in the B phase are related to change in the conformation (flattening) of aromatic molecular cores.     

A spin-3/2 Ising model on a square lattice

The spin-3/2 Ising model described by the most general Hamiltonian with up-down symmetry, −βH=Σ_〈ij〉{JS _i S _j +KS ₂ ⁱ S ₂ ^j +LS ₃ ⁱ S ₃ ^j +M/2(S _i S ₃ ^j +S _j S ₃ ⁱ )}−ΔΣ_i S ₂ ⁱ , is investigated on a square lattice. It is shown that this model is reducible to an eight-vertex model on a surface in the parameter space spanned by the coupling constants J, K, L, and M. It is shown that this model is equivalent to an exactly solvable free fermion model along two lines in the parameter space. Consequently, the critical behavior and, in particular, the critical temperature for the second-order phase transitions of the model is found exactly. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 4, 270–275 (25 February 1996) Published in English in the original Russian journal. Edited by Steve Torstveit.     

On asymptotic behaviour of four spin correlation functions in a linear chain of spins

Second and fourth moments of the four spin frequency correlation functions S^Z_i(t) S^Z_i+1 (t) S^Z_j+1 〉_k,ω and 〈S^x_i(t) S^z_i+1 (t) S^x_jS^z_j+1 〉_kω have been calculated in the high temperature limit for an arbitrary spin with an isotropic Heisenberg Hamiltonian.     

Dual model for dislocation and disclination melting

We show that defect melting involving dislocations and disclinations is dually equivalent to an extension of an XY model with an energy of the type Σ_{i, j}{[cos(?_iu_j + ?_ju_i) + ? cos ?_iω_j] }, where $\text{ω}{}_{\mathrm{t}}\text{=}\text{1}\text{2}\text{?}{}_{\mathrm{tjk}}\text{?}{}_{\mathrm{j}}\text{u}{}_{\mathrm{k}}$ is the local rotation field. The. model clarifies the proper choice of defect core energies arising from nonlinear elasticity. These permit the pile-up of dislocations to disclinations which is essential for the first order of the melting transition.     

Initial perpendicular isothermal susceptibility formulas for the transverse general-spin Ising and Blume-Capel models

Using a Kubo formula and the Suzuki identities, expressions are derived for the initial perpendicular susceptibilities χ_⊥ of the transverse spin-S Ising and spin-S Blume-Capel models on regular and irregular lattices. χ_⊥ is given in terms of the thermal average of a function of the peripheral sumO _i= ε_j J _i,j S _j, where coupling to distant neighbors may be included, as well as arbitrary local parallel magnetic fieldsh _j. For the Ising model on a Bravais lattice, e.g., the susceptibility is given by $$\chi _ \bot = Nm^2 S^{ - 2} \langle B_s (\beta [O_i + h_i ])/[O_i + h_i ]\rangle $$ whereB _s is the Brillouin function. ForS=1/2, the formula of Fisher and the results of Horiguchi and Morita are regained. A connection is made with the general-spin work of Essam and Garelick.     

Effects of random biquadratic couplings in a spin-1 spin-glass model

A spin-1 model, appropriated to study the competition between bilinear (J _ij S _i S _j ) and biquadratic (K _ij S _i ² S _j ²) random interactions, both of them with zero mean, is investigated. The interactions are infinite-ranged and the replica method is employed. Within the replica-symmetric assumption, the system presents two phases, namely, paramagnetic and spin-glass, separated by a continuous transition line. The stability analysis of the replica-symmetric solution yields, besides the usual instability associated with the spin-glass ordering, a new phase due to the random biquadratic couplings between the spins. Received 18 May 1999 and Received in final form 20 October 1999     

An exactly solvable 5-state IRF model

A 5-state IRF model (σ_l = 0, 1, 2, 3, 4) with a hard-core condition 0 ⩽ σ_i + σ_j ⩽ 4 on adjacent spins σ_i and σ_j is exactly solved. Explicit parametrizations of 70 independent Boltzmann weights which satisfy 510 star-triangle equations are presented with an exact expression for the free energy.     

A disordered ising model with internal degrees of freedom

A solvable Ising-system withnn-interactions as a basic system is considered. This system is assumed to undergo a phase transition so that the thermal quantityq=Σ _i> ₀ ² /N, which is not necessarily the order parameter of the basic system, is different from zero (q≠0) forT _c. The whole system consists of the basis {S _i} and an equal number of additional spins η _i which are coupled linearly with the basic spinsS _i through an exchange constantJ _i which is distributed at random. The pairs of spinsS _i ?η _i can be interpreted as clusters with an internal degree of freedom. The specific heat Δc and the susceptibility Δ_χ of the internal degree of freedom can be separated exactly and show typical spin glass properties: the specific heat Δc is an analytic function over the whole temperature range and the susceptibility Δ_χ shows a cusp at the transition temperatureT _c of the basic system.     

Spectral methods and cluster structure in correlation-based networks

We investigate how in complex systems the eigenpairs of the matrices derived from the correlations of multichannel observations reflect the cluster structure of the underlying networks. For this we use daily return data from the NYSE and focus specifically on the spectral properties of weight W_ij=|C|_ij−δ_ij and diffusion matrices D_ij=W_ij/s_j−δ_ij, where C_ij is the correlation matrix and s_i=∑_jW_ij the strength of node j. The eigenvalues (and corresponding eigenvectors) of the weight matrix are ranked in descending order. As in the earlier observations, the first eigenvector stands for a measure of the market correlations. Its components are, to first approximation, equal to the strengths of the nodes and there is a second order, roughly linear, correction. The high ranking eigenvectors, excluding the highest ranking one, are usually assigned to market sectors and industrial branches. Our study shows that both for weight and diffusion matrices the eigenpair analysis is not capable of easily deducing the cluster structure of the network without a priori knowledge. In addition we have studied the clustering of stocks using the asset graph approach with and without spectrum based noise filtering. It turns out that asset graphs are quite insensitive to noise and there is no sharp percolation transition as a function of the ratio of bonds included, thus no natural threshold value for that ratio seems to exist. We suggest that these observations can be of use for other correlation based networks as well.     

An exact result for the 1D random Ising model in a transverse field

It is shown exactly that for an N-site cyclic chain with hamiltonian H = ?Σ^N_i=1(γ_iS^x_i + J_iS^z_iS^z_iS^z_i+1), the gap in the excitation spectrum goes to zero when N → ∞ at the “critical point” given by the relation Π^N_i=1Γ_i = Π^N_i=1J_i.     

Robust output synchronization of second-order systems

A technique to synchronize a network of dynamical systems described by second-order ordinary differential equations is presented. Each system can be driven by a coupling control signal, which is synthesized such that, at steady-state, the outputs of two given systems, say y _i and y _j , i ≠ j, satisfy a specified ratio, that is, y _i /y _j = α _i /α _j , α _i ≠ 0 ≠ α _j . Among others, this includes the cases where the outputs are synchronized in-phase or anti-phase. The proposed synchronization technique is robust; this means that a small synchronization error is preserved at steady-state, even if the systems were perturbed by external disturbances. Some level of parameter uncertainty can also be tolerated. The coupling control signals are synthesized based on a classical controller and a robust observer that estimates the generalized velocities and provides an estimation of the perturbation terms. Some experimental results, showing the performance of the proposed synchronization technique, are included.     

The electromagnetic energy and momentum of finite charged bodies moving at constant velocity

It is shown that a purely electromagnetic, divergence-free tensorS ^ij , can be defined for any electrically charged body which is held in equilibrium by some cohesive force and moving at some constant velocity. This tensor appears to represent the electromagnetic energy-momentum of the body; the integral (1/c) ∝S ^ij dS _j (dS _j is the differential element of any spacelike hypersurface) iscM ₀ μ ⁱ the electromagnetic four-momentum of the system (M ₀ is the electromagnetic rest mass of the system,U _i is the four-velocity). The divergence-free property ofS ^ij depends only on Maxwell's equation and the condition of uniform motion. It is suggested that whatever the nature of the cohesive forces within such a system the total stress-energy tensor will, in effect, break up into two parts which are separately divergence-free: the purely electromagnetic tensor,S ^ij , and a tensor representing the energy-momentum of the cohesive forces. Just as it makes sense to speak of the electromagnetic mass of a system at rest without regard to the cohesive forces, it makes sense to talk about the electromagnetic momentum of the system, when it is moving at constant velocity, without reference to the cohesive forces.     

A sequence of two ferrotoroidal phase transitions in nickel-bromine boracite Ni3B7O13Br

The phenomenological theory of a sequence of two second-order phase transitions in Ni-Br boracite is presented. Two different components of the toroidal moment vector T _i are the order parameters of these transitions. Expressions are derived for the temperature dependences of the spontaneous values of T _i , polarization P _i , and magnetization M _i and the dielectric χ_ij=dP _i /dE _j , magnetic k _ij =dM _i /H _j , and magnetoelectric α _ij=dP _i /dH _j =dM _j /dE _i susceptibilities. Some of these susceptibilities display sharp temperature peaks in the vicinity of phase transitions.     

Fit values of N subshell fluorescence yields and Coster-Kronig yields for elements with 38?Z?103

Most probable values of Coster-Kronig (CK) transition probabilities (S₁₂, S₁₃, S₁₄, S₁₅, S_1,67, S₂₃, S₂₄, S_2,67, S₃₄, S₃₅, S₃₅, S_3,67, S₄₅ and S_4,67) and N_i subshell fluorescence yields (N₁, N₂, N₃, N₄, N₅ and N₆₇) have been determined by using least-squares fitted to obtain the polynomials, which is plotted by using the McGuire's values (Phys Rev A 9(5) (1974) 1840), representing them as a function of atomic number. The S_ij represents the N subshell vacancy transitions from ith subshell to jth subshell. The obtained values of N_i subshell fluorescence yields—N₁, N₂, N₃, N₄, N₅, N₆₇, and CK transition probabilities—S₁₂, S₁₃, S₁₄, S₁₅, S_1,67, S₂₃, S₂₄, S_2,67, S₃₄, S₃₅, S₃₅, S_3,67, S₄₅ and S₄₆ are presented.     

Dominant decays of scalar leptoquarks and scalar gluons in the minimal four-color symmetry model

Fermionic and weak decays of the scalar leptoquarks S = S ₁ ⁽⁺⁾ , S ₁ ^(?) , and S _m and the scalar gluons F = F ₁ and F ₂ predicted by the minimal model involving four-color symmetry and the Higgs mechanism of quark-and lepton-mass splitting are considered. The widths and the branching ratios are calculated for these decays, and the results are analyzed versus the couplings and masses of decaying particles. It is shown that, at relatively small mass splittings Δm within scalar doublets (Δm < m _W), the fermionic decays S ₁ ⁽⁺⁾ → tl _j ⁺ , S ₁ ^(?) → v _i \(\tilde b\), S _m → t \(\tilde \nu \) _j, F ₁ → t \(\tilde b\), and F ₂ → t \(\tilde t\), which are characterized by few-GeV widths for m _S, m _F < 1 TeV and decay branching ratios close to unity, are dominant, but that, for Δm > m _W, the weak decays S → S′W and F → F’W compete with the above fermionic decays. In the case of m _S < m _t, the processes S ₁ ⁽⁺⁾ → cl _j ⁺ , S ₁ ^(?) → v _i \(\tilde b\), S _m → bl _j ⁺ , and S _m → c \(\tilde \nu \) _j, whose total branching ratios are Br(S ₁ ⁽⁺⁾ → cl ⁺) ≈ Br(S ₁ ^(?) → v \(\tilde b\)) ≈ 1, Br(S _m → bl ⁺) ≈ 0.9, and Br(S _m → c \(\tilde \nu \)) ≈ 0.1, appear to be dominant decays of scalar leptoquarks. Searches for these decays at LHC and the Tevatron are of interest.     

A phenomenological theory of possible sequences of ferrotoroidal phase transitions in boracites

A phenomenological theory of the sequence of two second-order phase transitions with close temperatures is considered; such transitions occur in the Ni-Br boracite. The thermodynamic potential is written as a function of polarization P _i, magnetization M _i, and toroidal moment T _i vectors and fields E _i and H _i; T _i is treated as an order parameter. It is assumed that only one coefficient of T _i ² passes through zero as T decreases. The possibility of a sequence of two proper ferrotoroidal phase transitions along the T ₁ and T ₂ components is demonstrated. Spontaneous T _i, P _i, and M _i vector values and equations for susceptibility tensors (dielectric χ _ij = dP _i/dE _j, magnetic k _ij = dM _i/dH _j, and magnetoelectric α_ij = dP _i/dH _j = dM _j/dE _i) were obtained for three phases. Some of these values have well-defined anomalies in the vicinity of transitions. All possible sequences of ferrotoroidal phase transitions in boracites are considered. Depending on two potential coefficient values, these sequences may consist of one, two, or three such transitions.     

Functional properties of quantum logics

A quantum logic is defined as a setL of functions from the set of all statesS into [0, 1] satisfying the orthogonality postulate: for any sequencea ₁,a ₂, ... of members ofL satisfyinga _i+a _j≤1 fori≠j there isb∈L such thatb+a ₁+a ₂+...=1. Every logicL is in a natural way an orthomodular σ-orthocomplemented partially ordered set (L, ≤, ′) with members ofS inducing a full set of measures onL. It is shown that a logicL is quite full if and only if (L,≤,′) is isomorphic to an orthocomplemented set lattice of subsets ofS. Sufficient conditions are given in order that a quite full logic be representable in the set of projection quadratic formsf(u)=(Pu, u) on a complex Hilbert space, or in the set of trace functionsf(A)=Trace (AP) generated by projectionsP, where the domain off is the set of non-negative self-adjoint trace operators of trace 1 in a complex Hilbert space.     

Effect of damage in neural networks

The effect of damage on the pattern recognition in the Hopfield-model of neural networks is studied. It is assumed that in a damaged network the synaptic efficaciesJ _i,j=J_j,i, between pairs of neuronsS _i andS _j follow the Hebb rule with probability (1–p) and are equal to zero with probabilityp. Numerical simulations are performed for a net consisting of 400 neurons. It is investigated in detail how the retrieval of noisy patterns and the storage capacity of the net depends, for varying initial noise, on the concentrationp of the damaged synaptic efficacies.