Solution of the unanimity rule on exponential, uniform and scalefree networks: A simple model for biodiversity collapse in foodwebs

We solve the Unanimity Rule on networks with exponential, uniform and scalefree degree distributions. In particular we arrive at equations relating the asymptotic number of nodes in one of two states to the initial fraction of nodes in this state. The solutions for exponential and uniform networks are exact, the analytical approximation for the scalefree case is in perfect agreement with simulation results. We use these solutions to provide a theoretical understanding for biodiversity loss in experimental data of foodwebs, which is available for the three network types discussed. The model allows in principle to estimate the critical value of species that have to be removed from the system to induce a complete diversity collapse.     

基于K-阶结构熵的网络异构性研究

结构熵可以考察复杂网络的异构性.为了弥补传统结构熵在综合刻画网络全局以及局部特性能力上的不足,本文依据网络节点在K步内可达的节点总数定义了K-阶结构熵,可从结构熵随K值的变化规律、最大K值下的结构熵以及网络能够达到的最小结构熵三个方面来评价网络的异构性.利用K-阶结构熵对规则网络、随机网络、Watts-Strogatz小世界网络、Barabási_-Albert无标度网络以及星型网络进行了理论研究与仿真实验,结果表明上述网络的异构性依次增强.其中K-阶结构熵能够较好地依据小世界属性来刻画小世界网络的异构性,且对星型网络异构性随其规模演化规律的解释也更为合理.此外, K-阶结构熵认为在规则结构外新增孤立节点的网络的异构性弱于未添加孤立节点的规则结构,但强于同节点数的规则网络.本文利用美国西部电网进一步论证了K-阶结构熵的有效性.     

Closed-form solutions for non-uniform Euler–Bernoulli free–free beams

In this paper, the free vibration of a non-uniform free–free Euler–Bernoulli beam is studied using an inverse problem approach. It is found that the fourth-order governing differential equation for such beams possess a fundamental closed-form solution for certain polynomial variations of the mass and stiffness. An infinite number of non-uniform free–free beams exist, with different mass and stiffness variations, but sharing the same fundamental frequency. A detailed study is conducted for linear, quadratic and cubic variations of mass, and on how to pre-select the internal nodes such that the closed-form solutions exist for the three cases. A special case is also considered where, at the internal nodes, external elastic constraints are present. The derived results are provided as benchmark solutions for the validation of non-uniform free–free beam numerical codes.     

Three-dimensional treatment of convective flow in the earth's mantle

A three-dimensional finite-element method is used to investigate thermal convection in the earth's mantle. The equations of motion are solved implicitly by means of a fast multigrid technique. The computational mesh for the spherical problem is derived from the regular icosahedron. The calculations described use a mesh with 43,554 nodes and 81,920 elements and were run on a Cray X. The earth's mantly is modeled as a thick spherical shell with isothermal, free-slip boundaries. The infinite Prandtl number problem is formulated in terms of pressure, density, absolute temperature, and velocity and assumes an isotropic Newtonian rheology. Solutions are obtained for Rayleigh numbers up to approximately 10⁶ for a variety of modes of heating. Cases initialized with a temperature distribution with warmer temperatures beneath spreading ridges and cooler temperatures beneath present subduction zones yield whole-mantle convection solutions with surface velocities that correlate well with currently observed plate velocities.     

Zum allgemeinen Dispersionsgesetz der linearen Boltzmanngleichung

The conditions for the existence of particular solutions are investigated for the space- and time-dependent linear Boltzmann-equation which describes the neutron field. The regular solutions correspond to a discrete spectrum of parameters. Therefore, such particular states can be isolated asymptotically in suitable experiments. The number of possible asymptotic neutron-field-experiments is determined by the number of possible combinations of discrete parameter values corresponding to regular solutions. In the course of the analysis, two possible types of new experiments were found: the instationary exponential experiment and the neutron-wave experiment with damped excitation. The formulation of the problem has been extended to include the energy dependence of neutron distribution functions.     

Static spherically symmetric solutions of the Einstein-Yang-Mills equations

We study the global behaviour of static, spherically symmetric solutions of the Einstein-Yang-Mills equations with gauge groupSU(2). Our analysis results in three disjoint classes of solutions with a regular origin or a horizon. The 3-spaces (t=const.) of the first, generic class are compact and singular. The second class consists of an infinite family of globally regular, resp. black hole solutions. The third type is an oscillating solution, which although regular is not asymptotically flat.This article was processed by the author using the Springer-Verlag TEX CoMaPhy macro package 1991.     

Small World Properties Generated by a New Algorithm Under Same Degree of All Nodes

Based on the model of the same degree of all nodes we proposed before, a new algorithm, the so-called “spread all over vertices” (SAV) algorithm, is proposed for generating small-world properties from a regular ring lattices. During randomly rewiring connections the SAV is used to keep the unchanged number of links. Comparing the SAV algorithm with the Watts-Strogatz model and the “spread all over boundaries” algorithm, three methods can have the same topological properties of the small world networks. These results offer diverse formation of small world networks. It is helpful to the research of some applications for dynamics of mutual oscillator inside nodes and interacting automata associated with networks.     

Perfect quantum state transfer with spinor bosons on weighted graphs

A duality between the properties of many spinor bosons on a regular lattice and those of a single particle on a weighted graph reveals that a quantum particle can traverse an infinite hierarchy of networks with perfect probability in polynomial time, even as the number of nodes increases exponentially. The one-dimensional "quantum wire" and the hypercube are special cases in this construction, where the number of spin degrees of freedom is equal to one and the number of particles, respectively. An implementation of a near-perfect quantum state transfer across a weighted parallelepiped with ultracold atoms in optical lattices is discussed.     

Spatially compact solutions and stabilization in Einstein-Yang-Mills-Higgs theories

New solutions to the static, spherically symmetric Einstein-Yang-Mills-Higgs equations with the Higgs field in the triplet (doublet) representation are presented. They form continuous families parametrized by alpha = M(W)/M(Pl) [M(W) (M(Pl)) denoting the W boson (the Planck) mass]. The corresponding space-times are regular and have spatially compact sections. A particularly interesting class with the Yang-Mills amplitude being nodeless is exhibited and is shown to be linearly stable with respect to spherically symmetric perturbations. For some solutions with nodes of the Yang-Mills amplitude a new stabilization phenomenon is found, according to which their unstable modes disappear as alpha increases (for the triplet) or decreases (for the doublet).     

A kernel-free boundary integral method for elliptic boundary value problems

This paper presents a class of kernel-free boundary integral (KFBI) methods for general elliptic boundary value problems (BVPs). The boundary integral equations reformulated from the BVPs are solved iteratively with the GMRES method. During the iteration, the boundary and volume integrals involving Green’s functions are approximated by structured grid-based numerical solutions, which avoids the need to know the analytical expressions of Green’s functions. The KFBI method assumes that the larger regular domain, which embeds the original complex domain, can be easily partitioned into a hierarchy of structured grids so that fast elliptic solvers such as the fast Fourier transform (FFT) based Poisson/Helmholtz solvers or those based on geometric multigrid iterations are applicable. The structured grid-based solutions are obtained with standard finite difference method (FDM) or finite element method (FEM), where the right hand side of the resulting linear system is appropriately modified at irregular grid nodes to recover the formal accuracy of the underlying numerical scheme. Numerical results demonstrating the efficiency and accuracy of the KFBI methods are presented. It is observed that the number of GMRES iterations used by the method for solving isotropic and moderately anisotropic BVPs is independent of the sizes of the grids that are employed to approximate the boundary and volume integrals. With the standard second-order FEMs and FDMs, the KFBI method shows a second-order convergence rate in accuracy for all of the tested Dirichlet/Neumann BVPs when the anisotropy of the diffusion tensor is not too strong.     

A kernel-free boundary integral method for elliptic boundary value problems

This paper presents a class of kernel-free boundary integral (KFBI) methods for general elliptic boundary value problems (BVPs). The boundary integral equations reformulated from the BVPs are solved iteratively with the GMRES method. During the iteration, the boundary and volume integrals involving Green’s functions are approximated by structured grid-based numerical solutions, which avoids the need to know the analytical expressions of Green’s functions. The KFBI method assumes that the larger regular domain, which embeds the original complex domain, can be easily partitioned into a hierarchy of structured grids so that fast elliptic solvers such as the fast Fourier transform (FFT) based Poisson/Helmholtz solvers or those based on geometric multigrid iterations are applicable. The structured grid-based solutions are obtained with standard finite difference method (FDM) or finite element method (FEM), where the right hand side of the resulting linear system is appropriately modified at irregular grid nodes to recover the formal accuracy of the underlying numerical scheme. Numerical results demonstrating the efficiency and accuracy of the KFBI methods are presented. It is observed that the number of GMRES iterations used by the method for solving isotropic and moderately anisotropic BVPs is independent of the sizes of the grids that are employed to approximate the boundary and volume integrals. With the standard second-order FEMs and FDMs, the KFBI method shows a second-order convergence rate in accuracy for all of the tested Dirichlet/Neumann BVPs when the anisotropy of the diffusion tensor is not too strong.     

Scalar-vector topological soliton

We present classical scalar-vector equations which admit soliton solutions in three space dimensions. Exact spherical solutions are obtained which are everywhere regular and resemble charged particles of finite self-energy. The corresponding 4-current is identically conserved and leads to quantized charges. The scale of the soliton is unique and determined by boundary conditions, which also ensure its topological stability.     

Thermodynamics of the phase equilibrium of multicomponent solid solutions containing nano-sized precipitates of the second phase

Based on the theory of regular solutions, the interphase boundary model for the case of an arbitrary number of components is developed. General relations for the calculation of the phase composition of multicomponent solutions containing nano-sized precipitates of the second phase are derived. By the example of binary and ternary solutions, it is shown that a considerable mutual enrichment of the conjugated phases by atoms of components dissolved in them occurs in the case if precipitates of the second phase are characterized by the sizes of the order of several nanometers.     

Singularities of regular black holes and the monodromy method for asymptotic quasinormal modes

We use the monodromy method to investigate the asymptotic quasinormal modes of regular black holes based on the explicit Stokes portraits. We find that, for regular black holes with spherical symmetry and a single shape function, the analytical forms of the asymptotic frequency spectrum are not universal and do not depend on the multipole number but on the presence of complex singularities and the trajectory of asymptotic solutions along the Stokes lines.     

On the Road Map of Vogel’s Plane

We define “population” of Vogel’s plane as points for which universal character of adjoint representation is regular in the finite plane of its argument. It is shown that they are given exactly by all solutions of seven Diophantine equations of third order on three variables. We find all their solutions: classical series of simple Lie algebras (including an “odd symplectic” one), \({D_{2,1,\lambda}}\) superalgebra, the line of sl(2) algebras, and a number of isolated solutions, including exceptional simple Lie algebras. One of these Diophantine equations, namely \({knm=4k+4n+2m+12,}\) contains all simple Lie algebras, except so\({(2N+1).}\) Among isolated solutions are, besides exceptional simple Lie algebras, so called \({\mathfrak{e}_{7\frac{1}{2}}}\) algebra and also two other similar unidentified objects with positive dimensions. In addition, there are 47 isolated solutions in “unphysical semiplane” with negative dimensions. Isolated solutions mainly belong to the few lines in Vogel plane, including some rows of Freudenthal magic square. Universal dimension formulae have an integer values on all these solutions at least for first three symmetric powers of adjoint representation.     

Evolution of the exact spatiotemporal periodic wave and soliton solutions of the (3 + 1)-dimensional generalized nonlinear Schrödinger equation with distributed coefficients

In this paper the spatiotemporal evolution of the periodic wave is investigated analytically when the laser passes through the inhomogeneous nonlinear medium. Firstly, the (3 + 1)-dimensional generalized nonlinear Schrödinger equation with distributed coefficients is solved analytically by an improved homogeneous balance principle and F-expansion technique. A number of exact periodic traveling wave and spatiotemporal soliton solutions are obtained. Then, their propagation characteristics are analyzed in detail. It is found that the evolutions of propagation of spatiotemporal soliton and periodic wave solutions are regular when the diffraction and dispersion coefficients are the identical distributed coefficients, but the evolutions of propagation of these solutions are irregular with other coefficients.     

Thick brane in 7D and 8D spacetimes

We consider a thick brane model using two interacting scalar fields in 7D and 8D gravity. Using a special choice of potential energy, we obtain numerically regular asymptotically flat vacuum solutions. The possibility of obtaining the similar solutions for an arbitrary number of extra spatial dimensions is being estimated.     

Game of life on phyllosilicates: Gliders,oscillators and still life

A phyllosilicate is a sheet of silicate tetrahedra bound by basal oxygens. A phyllosilicate automaton is a regular network of finite state machines — silicon nodes and oxygen nodes — which mimics structure of the phyllosilicate. A node takes states 0 and 1. Each node updates its state in discrete time depending on a sum of states of its three (silicon) or six (oxygen) neighbours. Phyllosilicate automata exhibit localisations attributed to Conway?s Game of Life: gliders, oscillators, still lifes, and a glider gun. Configurations and behaviour of typical localisations, and interactions between the localisations are illustrated.     

Global monopole in general relativity

We consider the gravitational properties of a global monopole on the basis of the simplest Higgs scalar triplet model in general relativity. We begin with establishing some common features of hedgehog-type solutions with a regular center, independent of the choice of the symmetry-breaking potential. There are six types of qualitative behaviors of the solutions; we show, in particular, that the metric can contain at most one simple horizon. For the standard Mexican hat potential, the previously known properties of the solutions are confirmed and some new results are obtained. Thus, we show analytically that solutions with the monotonically growing Higgs field and finite energy in the static region exist only in the interval 1 < λ < 3, where λ is the squared energy of spontaneous symmetry breaking in Planck units. The cosmological properties of these globally regular solutions apparently favor the idea that the standard Big Bang might be replaced with a nonsingular static core and a horizon appearing as a result of some symmetry-breaking phase transition at the Planck energy scale. In addition to the monotonic solutions, we present and analyze a sequence of families of new solutions with the oscillating Higgs field. These families are parametrized by n, the number of knots of the Higgs field, and exist for λ < γ_n=6/[(2n + 1)(n + 2)]; all such solutions possess a horizon and a singularity beyond it.