Z(2) gauge neural network and its phase structure

We study general phase structures of neural-network models that have Z(2) local gauge symmetry. The Z(2) spin variable _Si=±1$S_i=\pm 1$ on the i$i$-th site describes a neuron state as in the Hopfield model, and the Z(2) gauge variable _Jij=±1$J_{ij}=\pm 1$ describes a state of the synaptic connection between j$j$-th and i$i$-th neurons. The gauge symmetry allows for a self-coupling energy among _Jij$J_{ij}$’s such as _JijJjkJki$J_{ij}{J}_{jk}{J}_{ki}$, which describes reverberation of signals. Explicitly, we consider the three models; (I) an annealed model with full and partial connections of _Jij$J_{ij}$, (II) a quenched model with full connections where _Jij$J_{ij}$ is treated as a slow quenched variable, and (III) a quenched three-dimensional lattice model with the nearest-neighbor connections. By numerical simulations, we examine their phase structures paying attention to the effect of the reverberation term, and compare them with each other and with the annealed 3D lattice model which has been studied beforehand. By noting the dependence of thermodynamic quantities upon the total number of sites and the connectivity among sites, we obtain a coherent interpretation to understand these results. Among other things, we find that the Higgs phase of the annealed model is separated into two stable spin-glass phases in the quenched models (II) and (III).     

On the integrability of symplectic Monge–Ampère equations

Let u$u$ be a function of n$n$ independent variables x¹,…,xⁿ$x^1,\dots ,x^n$, and let U=(_uij)$U=\left(u_{ij}\right)$ be the Hessian matrix of u$u$. The symplectic Monge–Ampère equation is defined as a linear relation among all possible minors of U$U$. Particular examples include the equation detU=1$detU=1$ governing improper affine spheres and the so-called heavenly equation, _u13u24−_u23u14=1$u_{13}{u}_{24}-u_{23}{u}_{14}=1$, describing self-dual Ricci-flat 4$4$-manifolds. In this paper we classify integrable symplectic Monge–Ampère equations in four dimensions (for n=3$n=3$ the integrability of such equations is known to be equivalent to their linearisability). This problem can be reformulated geometrically as the classification of ‘maximally singular’ hyperplane sections of the Plücker embedding of the Lagrangian Grassmannian. We formulate a conjecture that any integrable equation of the form F(_uij)=0$F\left(u_{ij}\right)=0$ in more than three dimensions is necessarily of the symplectic Monge–Ampère type.     

Three PT-symmetric Hamiltonians with completely different spectra

We discuss three Hamiltonians, each with a central-field part H₀$H_0$ and a PT-symmetric perturbation igz$igz$. When H₀$H_0$ is the isotropic Harmonic oscillator the spectrum is real for all g$g$ because H$H$ is isospectral to H₀+g²/2$H_0+g^2/2$. When H₀$H_0$ is the Hydrogen atom then infinitely many eigenvalues are complex for all g$g$. If the potential in H₀$H_0$ is linear in the radial variable r$r$ then the spectrum of H$H$ exhibits real eigenvalues for 0<g<g_c$0<g<g_c$ and a PT phase transition at g_c$g_c$.     

The first Chevalley–Eilenberg Cohomology group of the Lie algebra on the transverse bundle of a decreasing family of foliations

In [L. Lebtahi, Lie algebra on the transverse bundle of a decreasing family of foliations, J. Geom. Phys. 60 (2010), 122–133], we defined the transverse bundle V^k$V^k$ to a decreasing family of k$k$ foliations _Fi$F_i$ on a manifold M$M$. We have shown that there exists a (1,1)$(1,1)$ tensor J$J$ of V^k$V^k$ such that J^k≠0$J^k\ne 0$, J^k+1=0$J^{k+1}=0$ and we defined by _LJ(V^k)$L_J\left(V^k\right)$ the Lie Algebra of vector fields X$X$ on V^k$V^k$ such that, for each vector field Y$Y$ on V^k$V^k$, [X,JY]=J[X,Y]$[X,JY]=J[X,Y]$.     

How much information is needed to be the majority during the binary-state opinion formation?

In this paper, we try to propose a toy model, which follows the majority rule with the Fermi function, to uncover the role of the heterogeneous interaction between individuals in opinion formation. In order to do this, we define the impact factor I_Fi$I{F}_i$, says individual i$i$, as the exponential function of its connectivity _ki$k_i$ with the tunable parameter β$\beta$. β$\beta$ also shows the public information that can be collected by individuals in the system. We realize our model in scale-free networks with mean connectivity 〈k〉$〈k〉$. We find that much more public information (β>_β2$\beta >{\beta }_2$) and less public information (β<_β1$\beta <{\beta }_1$) cannot let either of the two opinions be the majority during the opinion formation. Furthermore, _β1${\beta }_1$ is a constant and equal to −0.76(±0.04)$-0.76(\pm 0.04)$, and _β2${\beta }_2$ decreases as a power-law function of the mean connectivity 〈k〉$〈k〉$ of the network. Our work can provide some perspectives and tools to understand the diversity of opinion in social networks.     

Is space-time symmetry a suitable generalization of parity-time symmetry?

We discuss space-time symmetric Hamiltonian operators of the form H=H₀+igH^′$H=H_0+ig{H}^{\prime }$, where H₀$H_0$ is Hermitian and g$g$ real. H₀$H_0$ is invariant under the unitary operations of a point group G$G$ while H^′$H^{\prime }$ is invariant under transformation by elements of a subgroup G^′$G^{\prime }$ of G$G$. If G$G$ exhibits irreducible representations of dimension greater than unity, then it is possible that H$H$ has complex eigenvalues for sufficiently small nonzero values of g$g$. In the particular case that H$H$ is parity-time symmetric then it appears to exhibit real eigenvalues for all 0<g<g_c$0<g<g_c$, where g_c$g_c$ is the exceptional point closest to the origin. Point-group symmetry and perturbation theory enable one to predict whether H$H$ may exhibit real or complex eigenvalues for g>0$g>0$. We illustrate the main theoretical results and conclusions of this paper by means of two- and three-dimensional Hamiltonians exhibiting a variety of different point-group symmetries.     

Diffusion-limited aggregates grown on nonuniform substrates

In the present paper, patterns of diffusion-limited aggregation (DLA) grown on nonuniform substrates are investigated by means of Monte Carlo simulations. We consider a nonuniform substrate as the largest percolation cluster of dropped particles with different structures and forms that occupy more than a single site on the lattice. The aggregates are grown on such clusters, in the range the concentration, p$p$, from the percolation threshold, _pc$p_c$ up to the jamming coverage, _pj$p_j$. At the percolation threshold, the aggregates are asymmetrical and the branches are relatively few. However, for larger values of p$p$, the patterns change gradually to a pure DLA. Tiny qualitative differences in this behavior are observed for different k$k$ sizes. Correspondingly, the fractal dimension of the aggregates increases as p$p$ raises in the same range _pc≤p≤_pj$p_c\le p\le p_j$. This behavior is analyzed and discussed in the framework of the existing theoretical approaches.     

Non-Abelian fusion rules from an Abelian system

We demonstrate the emergence of non-Abelian fusion rules for excitations of a two dimensional lattice model built out of Abelian degrees of freedom. It can be considered as an extension of the usual toric code model on a two dimensional lattice augmented with matter fields. It consists of the usual C(Z_p)$\mathbb{C}\left({\mathbb{Z}}_p\right)$ gauge degrees of freedom living on the links together with matter degrees of freedom living on the vertices. The matter part is described by a n$n$ dimensional vector space which we call H_n$H_n$. The Z_p${\mathbb{Z}}_p$ gauge particles act on the vertex particles and thus H_n$H_n$ can be thought of as a C(Z_p)$\mathbb{C}\left({\mathbb{Z}}_p\right)$ module. An exactly solvable model is built with operators acting in this Hilbert space. The vertex excitations for this model are studied and shown to obey non-Abelian fusion rules. We will show this for specific values of n$n$ and p$p$, though we believe this feature holds for all n>p$n>p$. We will see that non-Abelian anyons of the quantum double of C(S₃)$\mathbb{C}\left(S_3\right)$ are obtained as part of the vertex excitations of the model with n=6$n=6$ and p=3$p=3$. Ising anyons are obtained in the model with n=4$n=4$ and p=2$p=2$. The n=3$n=3$ and p=2$p=2$ case is also worked out as this is the simplest model exhibiting non-Abelian fusion rules. Another common feature shared by these models is that the ground states have a higher symmetry than Z_p${\mathbb{Z}}_p$. This makes them possible candidates for realizing quantum computation.     

Entropy and stability of phase synchronisation of oscillators on networks

I examine the role of entropy in the transition from incoherence to phase synchronisation in the Kuramoto model of N$N$ coupled phase oscillators on a general undirected network. In a Hamiltonian ‘action-angle’ formulation, auxiliary variables J_i$J_i$ combine with the phases θ_i${\theta }_i$ to determine a conserved system with a 2N$2N$ dimensional phase space. In the vicinity of the fixed point for phase synchronisation, θ_i≈θ_j${\theta }_i\approx {\theta }_j$, which is known to be stable, the auxiliary variables J_i$J_i$ exhibit instability  . This manifests Liouville’s Theorem in the phase synchronised regime in that contraction in the θ_i${\theta }_i$ parts of phase space are compensated for by expansion in the auxiliary dimensions. I formulate an entropy rate based on the projection of the J_i$J_i$ onto eigenvectors of the graph Laplacian that satisfies Pesin’s Theorem. This leads to the insight that the evolution to phase synchronisation of the Kuramoto model is equivalent to the approach to a state of monotonically increasing entropy. Indeed, for unequal intrinsic frequencies on the nodes, the networks that achieve the closest to exact phase synchronisation are those which enjoy the highest entropy production. I compare numerical results for a range of networks.     

Complex scale-free networks with tunable power-law exponent and clustering

We introduce a network evolution process motivated by the network of citations in the scientific literature. In each iteration of the process a node is born and directed links are created from the new node to a set of target nodes already in the network. This set includes m$m$ “ambassador” nodes and l$l$ of each ambassador’s descendants where m$m$ and l$l$ are random variables selected from any choice of distributions _pl$p_l$ and _qm$q_m$. The process mimics the tendency of authors to cite varying numbers of papers included in the bibliographies of the other papers they cite. We show that the degree distributions of the networks generated after a large number of iterations are scale-free and derive an expression for the power-law exponent. In a particular case of the model where the number of ambassadors is always the constant m$m$ and the number of selected descendants from each ambassador is the constant l$l$, the power-law exponent is (2l+1)/l$(2l+1)/l$. For this example we derive expressions for the degree distribution and clustering coefficient in terms of l$l$ and m$m$. We conclude that the proposed model can be tuned to have the same power law exponent and clustering coefficient of a broad range of the scale-free distributions that have been studied empirically.     

On the parallel transport of the Ricci curvatures

Geometrical characterizations are given for the tensor R⋅S$R\cdot S$, where S$S$ is the Ricci tensor   of a (semi-)Riemannian manifold (M,g)$(M,g)$ and R$R$ denotes the curvature operator   acting on S$S$ as a derivation, and of the Ricci Tachibana tensor  _∧g⋅S${\wedge }_g\cdot S$, where the natural metrical operator  _∧g${\wedge }_g$ also acts as a derivation on S$S$. As a combination, the Ricci curvatures   associated with directions on M$M$, of which the isotropy determines that M$M$ is Einstein, are extended to the Ricci curvatures of Deszcz   associated with directions and planes on M$M$, and of which the isotropy determines that M$M$ is Ricci pseudo-symmetric in the sense of Deszcz.     

Weak mirror symmetry of complex symplectic Lie algebras

A complex symplectic structure on a Lie algebra h$\mathfrak{h}$ is an integrable complex structure J$J$ with a closed non-degenerate (2,0)$(2,0)$-form. It is determined by J$J$ and the real part Ω$\Omega$ of the (2,0)$(2,0)$-form. Suppose that h$\mathfrak{h}$ is a semi-direct product g?V$\mathfrak{g}?V$, and both g$\mathfrak{g}$ and V$V$ are Lagrangian with respect to Ω$\Omega$ and totally real with respect to J$J$. This note shows that g?V$\mathfrak{g}?V$ is its own weak mirror image in the sense that the associated differential Gerstenhaber algebras controlling the extended deformations of Ω$\Omega$ and J$J$ are isomorphic.     

Thermodynamics properties of an anisotropic Ising antiferromagnet in both external longitudinal and transverse fields

We have studied the anisotropic two-dimensional nearest-neighbor Ising model with competitive interactions in both uniform longitudinal field H$H$ and transverse magnetic field Ω$\Omega$. Using the effective-field theory (EFT) with correlation in cluster with N=1$N=1$ spin we calculate the thermodynamic properties as a function of temperature with values H$H$ and Ω$\Omega$ fixed. The model consists of ferromagnetic interaction _Jx$J_x$ in the x$x$ direction and antiferromagnetic interaction _Jy$J_y$ in the y$y$ direction, and it is found that for H/_Jy∈[0,2]$H/J_y\in [0,2]$ the system exhibits a second-order phase transition. The thermodynamic properties are obtained for the particular case of λ=_Jx/_Jy=1$\lambda =J_x/J_y=1$ (isotropic square lattice).     

On m-accretive Schrödinger-type operators with singular potentials on Riemannian manifolds

We consider a Schrödinger-type differential expression _HV=∇^∗∇+V$H_V={\nabla }^{\ast }\nabla +V$, where ∇$\nabla$ is a Hermitian connection on a Hermitian vector bundle E$E$ over a complete Riemannian manifold (M,g)$(M,g)$ with metric g$g$ and positive smooth measure dμ$d\mu$, and V$V$ is a locally integrable section of the bundle of endomorphisms of E$E$. We give a sufficient condition for m$m$-accretivity of a realization of _HV$H_V$ in L²(E)$L^2\left(E\right)$.     

The Bethe lattice treatment of sound attenuation for a spin- 3/2 Ising model

The sound attenuation phenomena is investigated for a spin- 3/2 Ising model on the Bethe lattice in terms of the recursion relations by using the Onsager theory of irreversible thermodynamics. The dependencies of sound attenuation on the temperature (T$T$), frequency (w$w$), Onsager coefficient (γ$\gamma$) and external magnetic field (H$H$) near the second-order (_Tc)$\left(T_c\right)$ and first-order (_Tt)$\left(T_t\right)$ phase transition temperatures are examined for given coordination numbers q$q$ on the Bethe lattice. It is assumed that the sound wave couples to the order-parameter fluctuations which decay mainly via the order-parameter relaxation process, thus two relaxation times are obtained and which are used to obtain an expression for the sound attenuation coefficient (α)$\left(\alpha \right)$. Our investigations revealed that only one peak is obtained near _Tt$T_t$ and three peaks are found near _Tc$T_c$ when the Onsager coefficient is varied at a given constant frequency for q=3$q=3$. Fixing the Onsager coefficient and varying the frequency always leads to two peaks for q=3,4$q=3,4$ and 6 near _Tc$T_c$. The sound attenuation peaks are observed near _Tt$T_t$ at lower values of external magnetic field, but as it increases the sound attenuation peaks decrease and eventually disappear.