Commensurate and incommensurate Haldane phases for a spin-1 bilinear-biquadratic model

Commensurate and incommensurate Haldane phases for a spin-1 bilinear-biquadratic model are investigated using an infinite matrix product state algorithm.The bipartite entanglement entropy can detect a transition point between the two phases.In both phases,the entanglement spectrum shows double degeneracy.We calculate the nonlocal order parameter of the bond-centered inversion in both phases,which rapidly approaches a saturation value of-1 as the segment length increases.The nonlocal order parameter of the bond-centered inversion with a saturation value-1 and the nonzero value string order indicate that the Haldane phase is a symmetry-protected topological phase.To distinguish the commensurate and incommensurate Haldane phases,the transversal spin correlation and corresponding momentum distribution of the structure factor are analyzed.As a result,the transversal spin correlations exhibit different decay forms in both phases.

Modeling of Catfish Farm Using Lexicographic Linear Goal Programming

In this paper, a fish farm was modeled using the Lexicographic linear goal programming approach due to incommensurability in objectives. The study considered the fish farming plan with two sizes of catfish from stocking to harvesting at four-month intervals. The multi-objective goals developed are required raw materials feed, water, light (resource utilization), sales revenue, profit realized, labor utilization, production costs, and pond utilization. The developed model was tested using related data collected from the farm records with the use of TORA 2007 software. The compromised solution from the results showed that the developed model is an efficient tool for decision-making process in the fish farm business organization.

Multiscale mechanics of noncovalent interface in graphene oxide layered nanocomposites

Noncovalent interfaces play a vital role in inelastic deformation and toughening mechanisms in layered nanocomposites due to their dynamical recoverability. When interfacial engineering is applied to design layered nanocomposites, shear-lag analysis is usually implemented to evaluate the capability of interfacial loading transfer. Here, we introduce a multiscale shear-lag model that correlates macroscale mechanical properties with the molecular mechanisms to quantify the effects of interfacial configuration in graphene oxide(GO) layered nanocomposites. By investigating the mechanical responses of commensurate and incommensurate interfaces, we identify that the commensurate interface exhibits a pronounced size effect due to the nucleation and propagation of interfacial defects, whereas the incommensurate interface displays uniform deformation. Our predictions are further validated through large-scale molecular dynamics simulations for GO layered nanocomposites. This work demonstrates how size effects and interfacial configurations can be exploited to fabricate layered nanocomposites with superior mechanical properties despite relying on weak noncovalent interfaces.

Incommensurate Antiferromagnetism in the Extended t-J Model

The effect of the extra second neighbor hopping t' on the incommensurate spin correlation in the t-J modelin the underdoped regime is studied within the fermion-spin theory. It is shown that although the extra second neighborhopping t' is systematically accompanied with the increasing of the weight of the incommensurate peaks in the dynamicalspin structure factor, for the physical reasonable small value of t' the qualitative behavior of the incommensurate spincorrelation in the t-t'-J model is the same as in the case of t-J model.

Incommensurate Magnetic Order in Sm_(3)BWO_(9)with Distorted Kagome Lattice

We investigate the magnetic ground state of Sm_(3)BWO_(9)with a distorted kagome lattice.A magnetic phase transition is identified at T_(N)=0.75 K from the temperature dependence of specific heat.From 11B nuclear magnetic resonance measurements,an incommensurate magnetic order is shown by the double-horn type spectra under a c-axis magnetic field,and absence of line splitting is observed for field oriented within the ab-plane,indicating the incommensurate modulation of the internal field strictly along𝑐c-axis.From the spin dynamics,the critical slowing-down behavior is observed in the temperature dependence of 1/T_(1)with𝜇_(μ0)H⊥c-axis,which is completely absent in the case with𝜇_(μ0)H||c-axis.Based on the local symmetry of 11B sites,we analyze the hyperfine coupling tensors and propose two constraints on the possible magnetic structure.The single ion anisotropy should play an important role in determination of contrasting ground states of Sm_(3)BWO_(9) and Pr_(3)BWO_(9).

Misfit-layered compound PbTiS_3 with incommensurate modulation:Transmission electron microscopy analysis and transport properties

The microstructural characteristic of the misfit-layered compound PbTiS3 has been studied with transmission electron microscopy. All the incommensurate modulation-induced satellite spots and main diffraction spots of basic sublattices can be indexed systematically with a superspace group method. Finally, the relationship between the electronic transport properties and the crystal structure is discussed.

Modified adaptive controller for synchronization of incommensurate fractional-order chaotic systems

We investigate the synchronization of a class of incommensurate fractional-order chaotic systems, and propose a modified adaptive controller for fractional-order chaos synchronization based on the Lyapunov stability theory, the fractional order differential inequality, and the adaptive strategy. This synchronization approach is simple, universal, and theoretically rigorous. It enables the synchronization of O fractional-order chaotic systems to be achieved in a systematic way. The simulation results for the fractional-order Qi chaotic system and the four-wing hyperchaotic system are provided to illustrate the effectiveness of the proposed scheme.

INCOMMENSURATE AND COMMENSURATE PHASE TRANSITIONS IN Ti-Ni-Fe ALLOY

Measurements by using the ultrasonic technique together for electric resistance verified that the first sound velocity valley on cooling is mainly due to the soft-mode of driving the com- mensurate phase transition,and the second one due to the soft-mode of driving the martensitic transformation.The starting point,R_s,and finishing point,R_f,of the commensurate phase transition of the alloy have been determined.

Full Euclidean Algorithm by Means of a Steady Walk

Let x and y be two positive real numbers with x < y. Consider a traveler, on the interval [0, y/2], departing from 0 and taking steps of length equal to x. Every time a step reaches an endpoint of the interval, the traveler rebounds off the endpoint in order to complete the step length. We show that the footprints of the traveler are the output of a full Euclidean algorithm for x and y, whenever y/x is a rational number. In the case that y/x is irrational, the algorithm is, theoretically, not finite;however, it is a new tool for the study of its irrationality.

Reversion of Incommensurate Modulation in Cubic Lazurite: Example of Reversible Forced Equilibrium?

The sample of cubic lazurite, collected in the Baikal region, with incommensurately 3D modulated (ITM) structure has been studied by the method of high-temperature X-ray powder diffraction. At short time of annealing in high-temperature diffraction experiment the modulation recovery proceeds during cooling down the sample to room temperature. The identity of the period of both initial and recovered modulation demonstrates that the system has a structural memory. The acquired results are interpreted through comparison of thermal behavior of lazurite, sodalite and quartz structures. It is supposed that two kinetically different and thermally activated processes proceed under heating: 1) reversible framework expansion due to Si-O-Al angle increase, and 2) equalizing of periodic local distortions via the diffusion-controlled transfer of cage ions between adjacent subcells. The second process seems to be much slower than the first one, especially at lower temperatures. With increasing temperature, both processes are activated. However, the framework expands more rapidly than the cage clusters migrate, and the periodic distortions of the framework are aligned. Under lower temperatures, the framework shrinks and again accommodates to the configuration of cage cations (clusters), which may be changed at high temperature and sufficient time or may not at lower temperature, short time, unfavorable SO2 fugacity values. In the first case the modulation disappears entirely, while in the second case it arises again. The probable reason for ITM formation is the balance of counteracting energetic terms: the elastic strain energy of structure deformation and the energy of cluster ordering providing the state of forced equilibrium. The excess free energy due to structure distortion is compensated by the increment associated with the cluster ordering process. However, no significant variations in sulphur anion speciation for different degrees of modulation retention were observed by XPS S 2p. This may be due to the ordering of Na- and Ca-containing clusters rather than the clusters with different sulphur species. ITM reversion is considered as an example of reversible forced equilibrium with completely reproducible forcing factor.

Crystallography in Spaces E², E³, E⁴, E⁵...N⁰I Isomorphism Classes: Properties and Applications to the Study of Incommensurate Phase Structures, Molecular Symmetry Groups and Crystal Families of Space E<sup>5</sup

This paper mainly consists of the classification of all crystallographic point groups of n-dimensional space with n ≤ 6 into different isomorphism classes. An isomorphism class is defined by a type of finite mathematic group;for instance, the different types of mathematic groups have been well defined and studied by Coxeter. This classification may be used in the investigation of several domains of crystallography such as the study of the incommensurate phases, the quasi crystals … Indeed, each mathematic substitution group characterizes an isomorphism class of crystallographic point groups (spaces E2 or E3), of point groups of super crystals (spaces E4 or E5), and of molecular symmetry groups (spaces E2 or E3). This mathematic group gives interesting information about: 1) the incommensurate phase structures and their phase transitions according to the Landau’s theory in their super spaces E4, E5, E6, ···;2) the molecular symmetry group of chemisorbed molecules in space E2 (paragraph 2) or of the molecular crystal or solution in view of studying the molecule structure or its rotations or vibrationsin space E3;3) the geometric polyhedron symmetry groups as the regular rhombohedron in space E3, the rhombotope in space E4 or the rhombotope in space E5. Then, thanks to the isomorphism classes, we shall give properties of some crystal families that we have not published up to now. This formalism may be used to study crystal families in n-dimensional space with n > 6.

Lyapunov Stability Analysis for Incommensurate Nabla Fractional Order Systems

This paper investigates the problem of stability analysis for a class of incommensurate nabla fractional order systems.In particular,both Caputo definition and Riemann-Liouville definition are under consideration.With the convex assumption,several elementary fractional difference inequalities on Lyapunov functions are developed.According to the essential features of nabla fractional calculus,the sufficient conditions are given first to guarantee the asymptotic stability for the incommensurate system by using the direct Lyapunov method.To substantiate the efficacy and effectiveness of the theoretical results,four examples are elaborated.

Phononic origin of structural lubrication

Atomistic mechanisms of frictional energy dissipation have attracted significant attention.However,the dynamics of phonon excitation and dissipation remain elusive for many friction processes.Through systematic fast Fourier transform(FFT)analyses of the frictional signals as a silicon tip sliding over a graphite surface at different angles and velocities,we experimentally demonstrate that friction mainly excites non-equilibrium phonons at the washboard frequency and its harmonics.Using molecular dynamics(MD)simulations,we further disclose the phononic origin of structural lubrication,i.e.,the drastic reduction of friction force as the contact angle between two commensurate surfaces changes.In commensurate contacting states,friction excites a large amount of phonons at the washboard frequency and many orders of its harmonics that perfectly match each other in the sliding tip and substrate,while for incommensurate cases,only limited phonons are generated at mismatched washboard frequencies and few low order harmonics in the tip and substrate.

THE CONFIGURATION OF QUASICRYSTAL UNIT CELL AND DEDUCTION OF QUASILATTICE

From the crystal chemistry and icosahedral phase, two kinds of coordinational polyhedron with 8-fold rotational symmetry——hexakaicahedra and bicapped antiprism were possibly suggested and their one- and two-dimensional quasilattices were deducted. According to the principle of Bravais in conventional crystallography, four kinds of two-dimensional unit cell have been defined in 5, 8, 10, 12-fold rotational symmetry quasicrystal. The authors considered that quasicrystal is a kind of crystal which possesses an incommensurable translational period. This kind of translation is carried out by inflation or deflation symmetry operation.

Incommensurate modulations in Bi-2212 high-T_c superconductor revealed by single-crystal X-ray analysis using direct methods

The incommensurate modulated structure of the Bi2Sr2Ca1Cu2Oy high-Tc superocnductor has been studied by direct methods with X-ray diffraction data from a single crystal. The super-space group is N111Bbmb with three-dimensional unit cell parameters a = 0.54222 nm, b=0.5437 nm, c=3.0537 run, α=β=γ=90° andthemodula-tion wave vector q=(0, 0.22, 1). 543 main reflections, 867 first-order and 469 second-order satellite reflections have been observed. Phases of main reflections are derived by a conventional direct-method program SAPI91. The phases of satellite reflections are derived by the program DIMS running with default control based on the known phases of main reflections. The atomic parameters of the basic structure and of the modulation so obtained consist with the results from other authors. For the first time the structure details of the incommensurate modulation in the Bi-2212 phase are observed directly without relying on any assumed model of modulation.

A Theoretical Analysis of Torque and Superlubric Motion in Bilayer Graphene Disks

To identify the relation between torque and superlubric motion, we investigate the interlayer sliding behavior of two graphene disks with numerical computation methods. The potential energy, lateral force and torque between the top and bottom graphene disks, which are associated with misfit angle, translational displacement and interlayer distance, are analyzed. The results show that the rotation of the top disk is feeble for commensurate state, but it is difficult to realize superlubricity due to the lateral force fluctuating remarkably. For incommensurate state, the flake exhibits vanishing torque approaching to zero only for partial sliding directions. The superlubricity between the top and bottom disks will be eliminated due to torque-induced reorientation along other sliding directions. Whether for commensurate or incommensurate contact, the amplitudes of the lateral force （516 pN and 13 pN, respectively） are in qualitative agreement with experimental observation （typically 250 pN and 50 pN, respectively）. It shows that the interlayer torque is insensitive to the top disk size with incommensurate contact. The results suggest that the superlubric motion of graphene disk can be controlled by adjusting the torque.

Complex Modified Projective Synchronization for Fractional-order Chaotic Complex Systems

The aim of this paper is to study complex modified projective synchronization（CMPS） between fractional-order chaotic nonlinear systems with incommensurate orders. Based on the stability theory of incommensurate fractional-order systems and active control method, control laws are derived to achieve CMPS in three situations including fractional-order complex Lorenz system driving fractional-order complex Chen system, fractional-order real Rssler system driving fractional-order complex Chen system, and fractionalorder complex Lorenz system driving fractional-order real Lü system. Numerical simulations confirm the validity and feasibility of the analytical method.

Phonon dispersion calculations using the Vaks model in antiferroelectric lead zirconate

The calculations of low-energy phonon dispersion relations in lead zirconate(PZ)have been carried out using the Vaks model.This model takes into account only five low-energy phonon modes and their interactions.It has been shown that this model allows to achieve good agreement not only between calculated and experimental transverse acoustic phonon dispersion but also between their temperature dependences.It has been shown 5-mode system,which describes low-energy phonon dispersion in PZ,is close to unstable state.Even a little change of interaction parameter can lead to lattice instability with nonzero wave vector qcrit.Wave vector qcrit could have any value,including irrational one.In this case,system will undergo an incommensurate phase transition.