Narrow Results

Bayesian belief networks (BBN) are a widely studied graphical model for representing uncertainty and probabilistic interdependence among variables. One of the factors that restricts the model's wide acceptance in practical applications is that the general inference with BBN is NP-hard. This is also true for the maximum a posteriori probability (MAP) problem, which is to find the most probable joint value assignment to all uninstantiated variables, given instantiation of some variables in a BBN . To circumvent the difficulty caused by MAP's computational complexity, we suggest in this paper a neural network approximation approach. With this approach, a BBN is treated as a neural network without any change or transformation of the network structure, and the node activation functions are derived based on an energy function defined over a given BBN. Three methods are developed. They are the hill-climbing style discrete method, the simulated annealing method, and the continuous method based on the mean field theory. All three methods are for BBN of general structures, with the restriction that nodes of BBN are binary variables. In addition, rules for applying these methods to noisy-or networks are also developed, which may lead to more efficient computation in some cases. These methods' convergence is analyzed, and their validity tested through a series of computer experiments with two BBN of moderate size and complexity. Although additional theoretical and empirical work is needed, the analysis and experiments suggest that this approach may lead to effective and accurate approximation for MAP problems.

Random Boolean networks have been used as simple models of gene regulatory networks, enabling the study of the dynamic behavior of complex biological systems. However, analytical treatment has been difficult because of the structural heterogeneity and the vast state space of these networks. Here we used mean field approximations to analyze the dynamics of a class of Boolean networks in which nodes have random degree (connectivity) distributions, characterized by the mean degree k and variance D. To achieve this we generalized the simple cellular automata rule 126 and used it as the Boolean function for all nodes. The equation for the evolution of the density of the network state is presented as a one-dimensional map for various input degree distributions, with k and D as the control parameters. The mean field dynamics is compared with the data obtained from the simulations of the Boolean network. Bifurcation diagrams and Lyapunov exponents for different parameter values were computed for the map, showing period doubling route to chaos with increasing k. Onset of chaos was delayed (occurred at higher k) with the increase in variance D of the connectivity. Thus, the network tends to be less chaotic when the heterogeneity, as measured by the variance of connectivity, was higher.

An interacting-agent model of speculative activity explaining price formation in financial markets is considered in the present paper, which based on the stochastic Ising model and the mean field theory. The model describes the interaction strength among the agents as well as an external field, and the corresponding random logarithmic price return process is investigated. According to the empirical research of the model, the time series formed by this Ising model exhibits the bursting typical of volatility clustering, the fat-tail phenomenon, the power-law distribution tails and the long-time memory. The statistical properties of the returns of Hushen 300 Index, Shanghai Stock Exchange (SSE) Composite Index and Shenzhen Stock Exchange (SZSE) Component Index are also studied for comparison between the real time series and the simulated ones. Further, the multifractal detrended fluctuation analysis is applied to investigate the time series returns simulated by Ising model have the distribution multifractality as well as the correlation multifractality.

The magnetic properties of NiAl${}_x$Fe${}_{2-x}$O₄ (NAFO) spinels were studied. Influence of Al doping on magnetic properties of NiFe₂O₄ spinel were examined. The exchange interactions in NAFO were obtained. The general expression of saturation magnetization and the critical temperature were obtained using mean field theory. The high-temperature series expansions combined with the Padé approximant are given to determine the critical temperature of NAFO. The critical exponent associated with the magnetic susceptibility $\gamma$ was also deduced. The obtained results were in good agreement with magnetic measurement.

Weakly first-order or nearly second-order phase transitions occurring in metal–organic frameworks (MOFs), particularly in DMAKCr and perovskite HyFe, are studied under the mean field model by using the observed data from the literature. In this work, mainly thermal and magnetic properties among various physical properties which have been reported in the literature for those MOFs are studied by the mean field theory. By expanding the free energy in terms of the magnetization (order parameter), the excess heat capacity ($\mathrm{\Delta }$C${}_P$) and entropy ($\mathrm{\Delta }$S), latent heat (L), magnetization (M) and the inverse susceptibility (${\chi }^{-1}$) are calculated as a function of temperature close to the weakly first-order phase transition within the Landau phenomenological model which is fitted to the experimental data from the literature for C${}_P$ (DMAKCr and perovskite HyFe) and for magnetization M (HyFe).

Our predictions of the excess heat capacity ($\mathrm{\Delta }$C${}_P$) and entropy ($\mathrm{\Delta }$S) agree below T${}_c$ with the observed data within the temperature intervals studied for DMAKCr and perovskite HyFe. From our predictions, we find that magnetization decreases continuously whereas the inverse susceptibility decreases linearly with increasing temperature toward the transition temperature in those MOFs as expected for a weakly first-order transition from the mean field model.

Controlling the virus spreading in complex networks is very important to the research fields ranging from social science to technology. However, recent researches only consider information transmission and virus spreading as independent and uncorrelated, which neglect their interactive process in real systems. To study the interactive process between information transmission and virus spreading, in this paper, we propose a new virus information interactive spreading model on multiplex network. The multiplex networks can be divided into information transmission layer and virus spreading layer. The interactive spreading model considers the inherent characteristics of information by introducing the information forgetting and information loss mechanisms. Specially, by utilizing the mean field theory, the information transmission and virus process are unified to deduce the critical threshold of virus breakout. It shows that the critical threshold is not only related to the spreading properties of virus, but also influenced by the information density of individuals. We also find that the critical threshold is independent of the information generation probability, and the loss of the information will lead to a decrease. Finally, to verify the theoretical derivations, we apply it on both artificial network and real social network. The experimental results show that the simulations perfectly coincide with theoretical results, which verified the effectiveness of our model.

The dynamic magnetic behaviors of the spin-1 Blume–Capel Ising bilayer system (BCIS) are studied in an oscillating external magnetic field on a two-layer square lattice by utilizing the mean field theory (MFT) based on Glauber-type stochastic dynamics [dynamic mean field theory (DMFT)]. The dynamic equations describing the time-dependencies of the average magnetizations are obtained with the Master equation. The dynamic phases in this system are found by solving these dynamic equations. The temperature dependence of the dynamic order parameters is examined to characterize the nature (continuous or discontinuous) of the phase transitions and to obtain the dynamic phase transition (DPT) points. The dynamic phase diagrams (DPDs) are shown for ferromagnetic/ferromagnetic, antiferromagnetic/ferromagnetic, antiferromagnetic/antiferromagnetic interactions in the plane of the reduced temperature versus magnetic field amplitude and they display dynamic tricritical and re-entrant behavior as well as the dynamic triple point (tp).

The understanding of vortex structures in 3D turbulent fluids is a basic problem. One of the questions is whether some large scale structure can emerge as the macroscopic result of the self-organization of small scale vortex filaments, similarly to the 2D case of point vortices. This paper gives a first step in this direction: a mean field result is proved for a dense collection of vortex filaments. The filaments considered here are described by stochastic processes, including Brownian motion. Under a special rescaling of the energy, a mean field result is proved for a model of 3D vortex filaments described by stochastic processes, including Brownian motion, Brownian bridge, fractional Brownian motion and other semimartingales. Propagation of chaos, variational characterization of the limit Gibbs density h and an equation for h are proved.

We use the self-consistent mean-field theory to discuss the ground state and decay properties of Λ hypernuclei. We first discuss the deformation of Λ hypernuclei using the relativistic mean-field (RMF) approach. We show that, although most of the hypernuclei have a similar deformation parameter to the core nucleus, the shape of ²⁸Si is drastically altered, from oblately deformed to spherical, if a Λ particle is added to this nucleus. We then discuss the pionic weak decay of neutron-rich Λ hypernuclei using the Skyrme Hartree-Fock + BCS method. We show that, for a given isotope chain, the decay rate increases as a function of mass number, due to the strong neutron-proton interaction.

We have performed a systematic study for the nuclear structure of superheavy nuclei with a special emphasis on the nuclei with possible central depletion of proton and neutron density in the mass region $\mathrm{\text{A}}\sim 300$ using the Relativistic Hartree–Bogoliubov (RHB) framework. It has been observed that in the case of neutron density distribution, the occurrence of central depletion is related to the occupancy of 4s orbital and it is found to decrease with increasing occupancy of the 4s orbital. On the other hand, in the case of proton density distribution, the central density depletion is mainly due to the lowering of weakly bound p-orbital states close to the continuum as it is energetically favored to lower the Coulomb repulsion in the case of superheavy nuclei. Also, occupation probability of the lower angular momentum states (p-orbitals) lying near the Fermi level is strongly suppressed due to the weak centrifugal barrier and strong Coulomb repulsion in comparison to large angular momentum states (contributing to surface region mainly), resulting in central density depletion. Among the considered cases in the present work, the maximum depletion is observed for ${}^{292}120$ and for ${}^{294}$Og under spherically symmetric and axially deformed cases, respectively.

Commonly encountered shapes in nuclei are spherical, prolate and oblate. The shape of a nucleus is decided by the residual interactions among its nucleons. In our current investigation, we incorporate both axial and triaxial degrees of freedom to study how the shape of thorium nuclei changes within the boundaries of the drip lines. We employed the relativistic mean-field theory and utilized density-dependent meson-exchange as the functional throughout our calculations. We calculated the total binding energy as it varies with the deformation parameter ${\beta }_2$ and subsequently depicted the resulting binding energy curves. Furthermore, we examined the self-consistent energy surface for thorium isotopes with triaxial quadrupole deformations. The emergence of potential energy minima can be attributed to the presence of gaps or regions with reduced density of single-particle energy levels around the Fermi surface, as the nucleus undergoes deformation. The nuclear shape evolution of thorium isotopes ranging from ${}^{204}\mathrm{\text{Th}}$ to ${}^{280}\mathrm{\text{Th}}$ was investigated, ${}^{216}\mathrm{\text{Th}}$ and ${}^{274}\mathrm{\text{Th}}$ are spherical in shape and the nearby isotopes are less deformed and nearly spherical. Isotopes ranging from ${}^{220}\mathrm{\text{Th}}$ to ${}^{264}\mathrm{\text{Th}}$ exhibit a higher level of deformation. The most deformed nucleus is ${}^{240}\mathrm{\text{Th}}$ and is prolate. The weak triaxial behavior is observed within the region associated with shape phase transition.

We analyze a science collaboration network, i.e. a network whose nodes are scientists with edges connecting them for each paper published together. Furthermore, we develop a model for the simulation of discontiguous small-world networks that shows good coherence with empirical data.

The aggregation behavior of superparamagnetic iron oxide nanoparticles in an aqueous medium has been studied using SQUID magnetometry and XRD analysis. Iron oxide nanoparticles were synthesized by the coprecipitation method using iron salt and sodium hydroxide as precursors. The reactions were initially carried out at high temperature with different concentrations of iron salt. The synthesized nanoparticles were then collected for subsequent bio-functionalization. Structural characterizations were done using XRD and TEM measurements and show that the particles are magnetite (cubic spinel structure) with an average size of 9.5 nm. Magnetic measurements done using SQUID predict a magnetic particle size of 7.75 nm, which suggests a "magnetically dead layer" is present on the surface.

Understanding the thermodynamic driving forces underlying any chemical process requires a description of the underlying free energy surface. However, computation of free energies is difficult, often requiring advanced sampling techniques. Moreover, these computations can be further complicated by the evaluation of any long-ranged interactions in the system of interest, such as Coulomb interactions in charged and polar media. Local molecular field theory is a promising approach to avoid many of the conceptual and computational difficulties associated with long-ranged interactions. We present frameworks for performing alchemical free energy calculations and non-Boltzmann sampling with local molecular field theory. We demonstrate that local molecular field theory can be used to perform these free energy calculations with accuracy comparable to traditional methodologies while eliminating the need for explicit treatment of long-ranged interactions in simulations.

A Hamiltonian $H$, with locally smeared Ising-type s-d exchange between s-electrons and magnetic impurities, in a dilute magnetic alloy, is investigated. The Feynman-Kac theorem, Laplace expansion and Bogolyubov inequality are applied to obtain a lower and upper bound (lb and ub) on the system’s free energy per conducting electron $f(H,\beta )$. The two bounds differ, in the infinite-volume limit by a term $\mathrm{\text{lt}}>0$, linear in impurity concentration: lb$=f(h,\beta )-\mathrm{\text{lt}}$, ub$=f(h,\beta )$, $h$ denoting the Hamiltonian of the approximating mean-field s-d system. $h$ represents randomly positioned impurities interacting with a mean field implemented by the gas of conduction s-electrons, the latter interacting with the field of barriers and wells (according to the s-electron’s spin orientation) localized at the impurity sites. The inequality $f(h,\beta )-\mathrm{\text{lt}}\le f(H,\beta )\le f(h,\beta )$ demonstrates increasing accuracy of the mean-field $h$-theory, with decreasing impurity concentration.

The quark gluon plasma (QGP) at zero temperature and high baryon number is a strongly interacting system that may exist in the core of dense stars. Using an equation of state based on a mean-field approximation of QCD to describe this cold QGP, we study the structure of compact stars and obtain masses which are compatible with recent astrophysical data.

We report on the effect of Carbon insertion on the microstructure and magnetic properties of nanocrystalline Pr₂Co₇C_x($x\le 1$). The Pr₂Co₇C_x were characterized by X-ray diffraction (XRD), transmission electron microscopy (TEM) and physical property measurement system (PPMS9) Quantum Design. Mean field theory was utilized to depict the temperature dependence of magnetization and deduce the exchange interactions. The approach to saturation magnetization was as well used. The results were interpreted in the framework of random magnetic anisotropy model. From such analysis, some fundamental parameters were extracted.

In this work, we model the salient magnetic properties of the alloy lamellar ferrimagnetic nanostructures ${[{\mathrm{\text{Co}}}_{1-c}{\mathrm{\text{Gd}}}_c]}_{{\ell }^{\prime }}{[\mathrm{\text{Co}}]}_{\ell }{[{\mathrm{\text{Co}}}_{1-c}{\mathrm{\text{Gd}}}_c]}_{{\ell }^{\prime }}$ between $\mathrm{\text{Co}}$ semiinfinite leads. We have employed the Ising spin effective field theory (EFT) to compute the reliable magnetic exchange constants for the pure cobalt $J_{\mathrm{\text{Co–Co}}}$ and gadolinium $J_{\mathrm{\text{Gd–Gd}}}$ materials in complete agreement with their experimental data. The sublattice magnetizations of the $\mathrm{\text{Co}}$ and $\mathrm{\text{Gd}}$ sites on the individual hcp atomic (0001) planes of the Co–Gd layered nanostructures are computed for each plane and corresponding sites by using the combined EFT and mean field theory (MFT) spin methods. The sublattice magnetizations, effective site magnetic moments, and ferrimagnetic compensation characteristics for the individual hcp atomic planes of the embedded nanostructures, are computed as a function of temperature, and for various stable eutectic concentrations in the range $c\le 0.5$. The theoretical results for the sublattice magnetizations and the local magnetic variables of these ultrathin ferrimagnetic lamellar nanostructured systems, between cobalt leads, are necessary for the study of their magnonic transport properties, and eventually their spintronic dynamic computations. The method developed in this work is general and can be applied to comparable magnetic systems nanostructured with other materials.

Double-stranded DNA (dsDNA) is one of the most used model polymers in studying polymer dynamics. Some recent studies with the experimental data via fluorescence correlation spectroscopy (FCS) proposed that the end-monomer dynamics of dsDNA in dilute solution should be fallen into Rouse-type at intermediate times. This viewpoint is inconsistent with the classical polymer dynamics, therefore arousing controversy. To have a further looking clearly at what else meaning could be revealed from the original data of the FCS measurements, we made a re-calculation by two methods: one is based on Lodge and Wu's model (LWM) modified from the classical bead-spring model, in which parameters used in modeling are needed to be adjusted; and another is a mean field theory (MFT) for semiflexible chain with no parameter fitting needed. In LWM, we find not so weak hydrodynamic interactions (HI) which is not expected in Rouse theory, and the scaling of mean square displacement (MSD) is between Rouse-type and Zimm-type. MFT can reproduce experimental data well at larger time scales, whereas also gives rather different picture in intermediate regime — a dynamical scaling between Rouse-like and Zimm-like rather than Rouse-like scaling is found, indicating there may be sample problems or limitations in the setup for the experiment.

We use the self-consistent mean-field theory to discuss the ground state and decay properties of Λ hypernuclei. We first discuss the deformation of Λ hypernuclei using the relativistic mean-field (RMF) approach. We show that, although most of the hypernuclei have a similar deformation parameter to the core nucleus, the shape of ²⁸Si is drastically altered, from oblately deformed to spherical, if a Λ particle is added to this nucleus. We then discuss the pionic weak decay of neutron-rich Λ hypernuclei using the Skyrme Hartree-Fock + BCS method. We show that, for a given isotope chain, the decay rate increases as a function of mass number, due to the strong neutron-proton interaction.