Narrow Results

We construct an infinite particle/infinite volume Langevin dynamics on the space of simple configurations in ℝ^d having velocities as marks. The construction is done via a limiting procedure using N-particle dynamics in cubes (-λ, λ]^d with periodic boundary condition. A main step to this result is to derive an (improved) Ruelle bound for the canonical correlation functions of N-particle systems in (-λ, λ]^d with periodic boundary condition. After proving tightness of the laws of the finite particle dynamics, the identification of accumulation points as martingale solutions of the Langevin equation is based on a general study of properties of measures on configuration space fulfilling a uniform Ruelle bound (and their weak limits). Additionally, we prove that the initial/invariant distribution of the constructed dynamics is a tempered grand canonical Gibbs measure. All proofs work for a wide class of repulsive interaction potentials ϕ (including, e.g., the Lennard–Jones potential) and all temperatures, densities and dimensions d ≥ 1.

The effect of the periodic boundary condition (PBC) on the structure and energetics of nanotubes has been investigated by performing molecular-dynamics computer simulation. Calculations have been realized by using an empirical many-body potential energy function for carbon. A single-wall carbon nanotube has been considered in the simulations. It has been found that the periodic boundary condition has no effect at low temperature (1 K), however, it plays an important role even at intermediate temperature (300 K).

The reversibility problem for linear cellular automata with null boundary defined by a rule matrix in the form of a pentadiagonal matrix was studied over the binary field ℤ₂ by Martín del Rey et al. [Appl. Math. Comput.217, 8360 (2011)]. Recently, the reversibility problem of 1D Cellular automata with periodic boundary has been extended to ternary fields and further to finite primitive fields ℤ_p by Cinkir et al. [J. Stat. Phys.143, 807 (2011)]. In this work, we restudy some of the work done in Cinkir et al. [J. Stat. Phys.143, 807 (2011)] by using a different approach which is based on the theory of error correcting codes. While we reestablish some of the theorems already presented in Cinkir et al. [J. Stat. Phys.143, 807 (2011)], we further extend the results to more general cases. Also, a conjecture that is left open in Cinkir et al. [J. Stat. Phys.143, 807 (2011)] is also solved here. We conclude by presenting an application to Error Correcting Codes (ECCs) where reversibility of cellular automata is crucial.

Olympic systems are collections of small ring polymers whose aggregate properties are largely characterized by the extent (or absence) of topological linking in contrast with the topological entanglement arising from physical movement constraints associated with excluded volume contacts or arising from chemical bonds. First, discussed by de Gennes, they have been of interest ever since due to their particular properties and their occurrence in natural organisms, for example, as intermediates in the replication of circular DNA in the mitochondria of malignant cells or in the kinetoplast DNA networks of trypanosomes. Here, we study systems that have an intrinsic one, two, or three-dimensional character and consist of large collections of ring polymers modeled using periodic boundary conditions. We identify and discuss the evolution of the dimensional character of the large scale topological linking as a function of density. We identify the critical densities at which infinite linked subsystems, the onset of percolation, arise in the periodic boundary condition systems. These provide insight into the nature of entanglement occurring in such course grained models. This entanglement is measured using Gauss linking number, a measure well adapted to such models. We show that, with increasing density, the topological entanglement of these systems increases in complexity, dimension, and probability.

We consider the Cauchy problem for the kinetic derivative nonlinear Schrödinger equation on the torus $T=R/2\pi Z:{\partial }_{t}u-i{\partial }_x^{2}u=\alpha {\partial }_x(|u{|}^{2}u)+\beta {\partial }_x[H(|u{|}^2)u]$ for $(t,x)\in [0,T]\times T$, where the constants $\alpha ,\beta$ are such that $\alpha \in R$ and $\beta <0$, and $H$ denotes the Hilbert transform. This equation has dissipative nature, and the energy method is applicable to prove local well-posedness of the Cauchy problem in Sobolev spaces $H^s$ for $s>3/2$. However, the gauge transform technique, which is useful for dealing with the derivative loss in the nonlinearity when $\beta =0$, cannot be directly adapted due to the presence of the Hilbert transform. In particular, there has been no result on local well-posedness in low regularity spaces or global solvability of the Cauchy problem.

In this paper, we shall prove local and global well-posedness of the Cauchy problem for small initial data in $H^s(T)$, $s>1/2$. To this end, we make use of the parabolic-type smoothing effect arising from the resonant part of the nonlocal nonlinear term $\beta {\partial }_x[H(|u{|}^2)u]$, in addition to the usual dispersive-type smoothing effect for nonlinear Schrödinger equations with cubic nonlinearities. As by-products of the proof, we also obtain forward-in-time regularization and backward-in-time ill-posedness results.

This paper focuses on the buckling instabilities of periodic porous elastomers under combined multiaxial loading. A numerical model based on the periodic boundary condition (PBC) for the 2D representative volume element (RVE) is proposed, in which two proportional loading parameters are employed to control the complex stressing state applied to the RVE model. A homogenization-based orthogonal transformation matrix is established by satisfying the equality of the total work rate to realize a proportional multiaxial loading on the RVE. First, the transition behavior of buckling patterns of periodic porous structures is revealed through instability analysis for the RVE consisting of $2\times 2$ primitive cells with circular holes subjected to different proportional loading conditions. Simulation results show that the first-order buckling mode of RVE may change suddenly from a uniaxial shearing buckling pattern to a biaxial rotating buckling pattern at a critical loading proportion. Then the influences of the number of primitive cells in the enlarged RVE on the buckling behavior are discussed. When the number of primitive cells in any enlarging direction is odd, the points of buckling pattern transition of the enlarged RVEs vary significantly with the number of cells in RVE. When the number of primitive cells is even in both enlarging directions, there is no apparent difference for the critical buckling stresses of the enlarged RVEs.

This paper deals with flow field prediction in a blade cascade using the convolution neural network. The convolutional neural network (CNN) predicts density, pressure and velocity fields based on the given geometry. The blade cascade is modeled as a single interblade channel with periodic boundary conditions. In this paper, an algorithm that enforces periodic boundary conditions onto the CNN is presented. The main target of this study is to parametrize the CNN model depending on the Reynolds number. A new parametrization approach based on a so-called hypernetwork is employed for this purpose. The idea of this approach is that when the Reynolds number is modified, the hypernetwork modifies the weights of the CNN in such a way that it produces flow fields corresponding to that particular Reynolds number. The concept of the hypernetwork-based parametrization is tested on the problem of a compressible fluid flow through a blade cascade with variable blade profiles and Reynolds numbers.

In this paper we introduce extended cone metric spaces, prove some fixed point theorems of contractive mappings in such spaces. The results directly improve and generalize some fixed point results in cone metric spaces. In addition, we use our conclusions to obtain the existence and uniqueness of solution for an ordinary differential equation with periodic boundary condition.