Matter fields in triangle-hinge models

The worldvolume theory of membrane is mathematically equivalent to three-dimensional quantum gravity coupled to matter fields corresponding to the target space coordinates of embedded membrane. In a recent paper [arXiv:1503.08812] a new class of models are introduced that generate three-dimensional random volumes, where the Boltzmann weight of each configuration is given by the product of values assigned to the triangles and the hinges. These triangle-hinge models describe three-dimensional pure gravity and are characterized by semisimple associative algebras. In this paper, we introduce matter degrees of freedom to the models by coloring simplices in a way that they have local interactions. This is achieved simply by extending the associative algebras of the original triangle-hinge models, and the profile of matter field is specified by the set of colors and the form of interactions. The dynamics of a membrane in $D$-dimensional spacetime can then be described by taking the set of colors to be $\mathbb{R}^D$. By taking another set of colors, we can also realize three-dimensional quantum gravity coupled to the Ising model, the $q$-state Potts models or the RSOS models. One can actually assign colors to simplices of any dimensions (tetrahedra, triangles, edges and vertices), and three-dimensional colored tensor models can be realized as triangle-hinge models by coloring tetrahedra, triangles and edges at a time.Comment: 21 pages, 14 figures. v2: discussions in section 4 improved. v3: title changed, introduction enlarge

Melnikov integral formula for beam sea roll motion utilizing a non-Hamiltonian exact heteroclinic orbit

Chaos appearing in a ship roll equation in beam seas, known as the escape equation, has been intensively investigated so far because it is closely related to capsizing accident. In particular, many applications of Melnikov integral formula have been reported in the existing literature. However, in all the analytical works concerning with the escape equation, Melnikov integral is formulated utilizing a separatrix for Hamiltonian part or a numerically obtained heteroclinic orbit for non-Hamiltonian part, of the original escape equation. To overcome such limitations, this paper attempts to utilise an analytical expression of the non-Hamiltonian part. As a result, an analytical procedure making use of a heteroclinic orbit of non-Hamiltonian part within the framework of Melnikov integral formula is provided

Melnikov Integral Formula for Beam Sea Roll Motion Utilizing a Non-Hamiltonian Exact Heteroclinic Orbit (Part II)

In the research filed of nonlinear dynamical system theory it is well known that a homoclinic/heteroclinic point leads to unpredictable motions, such as chaos. Melnikov’s method enables us to judge whether the system has a homoclinic/heteroclinic orbit. Therefore, in order to assess a vessels safety against capsizing, Melnikov’s method has been applied for the investigations of chaos that appears in beam sea rolling. This is because chaos is closely related to capsizing incidents. In a previous paper 1), the formula to predict the capsizing boundary by applying Melnikov’s method to analytically obtain the non-Hamiltonian heteroclinic orbit, was proposed. However, in that paper, limited numerical investigation had been carried out. Therefore further comparative research between the analytical and numerical results is conducted, with the result being that the formula is validated

Emergence of AdS geometry in the simulated tempering algorithm

In our previous work [1], we introduced to an arbitrary Markov chain Monte Carlo algorithm a distance between configurations. This measures the difficulty of transition from one configuration to the other, and enables us to investigate the relaxation of probability distribution from a geometrical point of view. In this paper, we investigate the geometry of stochastic systems whose equilibrium distributions are highly multimodal with a large number of degenerate vacua. Implementing the simulated tempering algorithm to such a system, we show that an asymptotically Euclidean anti-de Sitter geometry emerges with a horizon in the extended configuration space when the tempering parameter is optimized such that distances get minimized.Comment: 19 pages, 5 figures. v2: typos corrected, some discussions improve

Analytical methods to predict the surf-riding threshold and the wave-blocking threshold

For the safe design and operation of high-speed craft it is important to predict their behaviour in waves. There still exists a concern, however, in the framework of the International Maritime Organization (IMO) with regards to the stability criteria. In particular, for high-speed craft, the higher limit of operational speed resulting in wave blocking as well as the lower limit known as the surf-riding threshold are important features. Therefore, by applying the polynomial approximation to wave induced surge force including the nonlinear surge equation, an analytical formula in order to predict the wave blocking and surf-riding thresholds is proposed. Comparative results of the surf-riding threshold and wave blocking threshold utilizing the proposed formula and the numerical bifurcation analysis indicate fairly good agreement. In addition, previously proposed analytical formulae are inclusively examined. It is concluded that the analytical formulae based on a continuous piecewise linear approximation and Melnikov’s method agrees well with the wave blocking threshold and the surf-riding threshold obtained by the numerical bifurcation analysis and the free-running model experiment. As a result, it is considered that these two calculation methods could be recommended for the early design stage tool for avoiding broaching and bow-diving

Analytical methods to predict the surf-riding threshold and the wave-blocking threshold

For the safe design and operation of high-speed craft it is important to predict their behaviour in waves. There still exists a concern, however, in the framework of the International Maritime Organization (IMO) with regards to the stability criteria. In particular, for high-speed craft, the higher limit of operational speed resulting in wave blocking as well as the lower limit known as the surf-riding threshold are important features. Therefore, by applying the polynomial approximation to wave induced surge force including the nonlinear surge equation, an analytical formula in order to predict the wave blocking and surf-riding thresholds is proposed. Comparative results of the surf-riding threshold and wave blocking threshold utilizing the proposed formula and the numerical bifurcation analysis indicate fairly good agreement. In addition, previously proposed analytical formulae are inclusively examined. It is concluded that the analytical formulae based on a continuous piecewise linear approximation and Melnikov’s method agrees well with the wave blocking threshold and the surf-riding threshold obtained by the numerical bifurcation analysis and the free-running model experiment. As a result, it is considered that these two calculation methods could be recommended for the early design stage tool for avoiding broaching and bow-diving

Triangle–hinge models for unoriented membranes

Triangle–hinge models [M. Fukuma, S. Sugishita, and N. Umeda, J. High Energy Phys. 1507, 088 (2015)] are introduced to describe worldvolume dynamics of membranes. The Feynman diagrams consist of triangles glued together along hinges and can be restricted to tetrahedral decompositions in a large-NN limit. In this paper, after clarifying that all the tetrahedra resulting in the original models are orientable, we define a version of triangle–hinge models that can describe the dynamics of unoriented membranes. By regarding each triangle as representing a propagation of an open membrane of disk topology, we introduce a local worldvolume parity transformation which inverts the orientation of a triangle, and define unoriented triangle–hinge models by gauging the transformation. Unlike two-dimensional cases, this local transformation generally relates a manifold to a nonmanifold, but still is a well-defined manipulation among tetrahedral decompositions. We further show that matter fields can be introduced in the same way as in the original oriented models. In particular, the models will describe unoriented membranes in a target spacetime by taking matter fields to be the target space coordinates

Nonlinear dynamics of ship capsizing at sea

Capsizing is one of the worst scenarios in oceangoing vessels. It could lead to a high number of fatalities. A considerable number of studies have been conducted until the 1980s, and one of the discoveries is the weather criterion established by the International Maritime Organization (IMO). In the past, one of the biggest difficulties in revealing the behavior of ship-roll motion was the nonlinearity of the governing equation. On the other hand, after the mid-1980s, the complexity of the capsizing problem was uncovered with the aid of computers. In this study, we present the theoretical backgrounds of the capsizing problem from the viewpoint of nonlinear dynamics. Then, we discuss the theoretical conditions and mechanisms of the bifurcations of periodic solutions and numerical attempts for the bifurcations and capsizing