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The method of numerical Fourier analysis is used to investigate the fractal properties of 2D objects with micrometer–centimeter sizes. This numerical method simulates the small-angle light scattering experiment. Different geometric 2D-regular fractals, such as the Sierpinski carpet, Sierpinski triangle, Koch snowflake, and Vishek snowflake, are studied. We can divide 2D fractals, by analogy with 3D fractals, into “plane” and “boundary” fractals with fractal dimensions lying in the intervals from 1 to 2 and from 2 to 3, respectively. For an object with a smooth boundary, i.e., a circle, the model small-angle scattering curve decreases according to the power law q–3, where q is the momentum transferred.

A simple kinematic approach to the description of the interaction between solitons is developed. It is applicable to both integrable and non-integrable two-dimensional models, including those commonly used for studying the surface and internal oceanic waves. This approach allows obtaining some important characteristics of the interaction between solitary waves propagating at an angle to each other. The developed theory is validated by comparison with the exact solutions of the Kadomtsev-Petviashvili equation and then applied to the observed interaction of solitary internal waves in a two-layer fluid within the two-dimensional Benjamin-Ono model.

A general ejection theory of polymer is developed in a two- and three-dimensional space.A polymer is confined initially in a cavity and ejects spontaneously to the outer space through a nanopore channel without the help of any external stimulus. A reflective wall boundary is set at the pore entrance to prevent the falling of the head monomer of chain into the cavity. Three stages are distinguished in a process: (1) the entering stage, in which the head monomer enters the pore to search for a way to traverse the pore channel, (2) the main ejection stage, in which the chain body is transported from the cavity to the outer space, (3) the leaving stage, in which the tail monomer passes through and leaves the pore channel. Depending on the number of the monomers remaining in the cavity, the main ejection stage can be divided into the confined and the non-confined stages. The non-confined stage can be further split into the thermal escape and the entropic pulling stages. The Onsager's variational principle is used to derive the kinetics equation of ejection. The escape time is calculated from the corresponding Kramers' escape problem.Extensive molecular dynamics simulations are then performed in a quasi two-dimensional space to verify the theory. The variation of the ejection speed is carefully examined in a process. The decreasing behavior of the number of monomers in the cavity is studied in details. The scaling properties of the spending time at each processing stage are investigated systematically by varying the chain length, the cavity diameter, and the initial volume fraction of chain. The results of simulation support firmly the predictions of the theory, cross-checked in the studies of various topics. Together with the previous investigations in the three-dimensional space, the generalized theory is very robust able to explain the two seemly different phenomena, polymer ejection and polymer translocation, under the same theoretical framework in the two space dimensions.

In this paper, a fourth-order difference scheme (FODS) is proposed for solving the two-dimensional Riesz space-fractional diffusion equations with homogeneous Dirichlet boundary conditions. It is proved that the FODS is uniquely solvable, unconditionally stable, and convergent with order O ( τ 2 + h x 4 + h y 4 ) in the discrete L∞- norm, where τ is the time step size, and hx, hy are the space grid sizes in the x direction and the y direction, respectively. Based on the special structure and symmetric positive definiteness of the coefficient matrix, a fast method is developed for the implementation of the FODS. The fast method reduces the storage requirement of O(N2) and computational cost of O(N3) down to O ( M + J ) and O(Nlog N), where N = M J , M and J are the numbers of the spatial grid points in the x direction and the y direction, respectively. Finally, several numerical results are shown to verify the theoretical results and the efficiency of the fast method.

In this article, a simple expression for the center of mass of a system of material points in a two-dimensional surface of Gaussian constant negative curvature is given. By using the basic techniques of geometry, we obtained an expression in intrinsic coordinates, and we showed how this extends the definition for the Euclidean case. The argument is constructive and serves to define the center of mass of a system of particles on the one-dimensional hyperbolic sphere LR1.

The extrinsic and intrinsic curvature of a two-dimensional waveguide influences wave propagation therein. While this can already be apprehended from a geometric point of view in terms of geodesics generalizing straight lines as the shortest distance between any two points, in wave optics interference phenomena strongly govern the field evolution, too. Radii of curvature in the order of the wavelength of light modify the local effective refractive index by altering the mode profile. Macroscopic radii only influence light propagation for non-vanishing intrinsic (or Gaussian) curvature. A positive Gaussian curvature leads to refocusing and thus an imaging behavior, whereas negative Gaussian curvature forces the field profile to diverge exponentially. These effects can be explained by an effective transverse potential acting on the electromagnetic field distribution’s envelope. This can also be extended to nonlinear beam propagation. In this review paper we give a thorough introduction to differential geometry in two-dimensional manifolds and its incorporation with Maxwell’s equations. We report on first fundamental experiments in this newly emerging field, which may lead to applications in integrated optical circuits. The close conceptual analogy to phenomena in four-dimensional spacetime with constant curvature as well as toy-models of the Schwarzschild metric and a wormhole topology are also discussed.

In this paper we present a high-accuracy charge conservation scheme of current assignment for particle-in-cell (PIC) simulations in two dimensional space. The current continuity integral equation is applied to set up a linear equation set about the assigned currents on edges of the cells through which a particle moves during a time step. The currents on the edges can be very accurately assigned by solving the linear equation set. This scheme can be used in Cartesian and cylindrical geometry, and is not affected by the form-factor of quasi-particle. It has been applied to the PIC simulation of Z-pinch of the current-carrying rarefied deuterium plasma shell without resorting to the electric field correction by solving Poisson equation . The simulated results reveal that the relative and absolute errors in satisfying the current continuity integral equation and Poisson equation are very tiny.

The two-dimensional space with a topological defect is a transverse section of the three-dimensional space with an Abrikosov–Nielsen–Olesen vortex, i.e. a gauge-flux-carrying tube which is impenetrable for quantum matter. Charged spinor matter field is quantized in this section with the most general mathematically admissible boundary condition at the edge of the defect. We show that a current and a magnetic field are induced in the vacuum. The dependence of results on the boundary conditions is studied, and we find that the requirement of finiteness of the total induced vacuum magnetic flux removes an ambiguity in the choice of boundary conditions. The differences between the cases of massive and massless spinor matters are discussed., Двовимiрний простiр з топологiчним дефектом є поперечним зрiзом тривимiрного простору з вихором Абрикосова–Нiльсена–Олесена, який являє собою непроникливу для квантованої матерiї трубку з потоком калiбрувального поля. Заряджене поле спiнорної матерiї квантується в цьому зрiзi, задовiльняючи найбiльш загальним математично допустимим граничним умовам. Показано, що струм та магнiтне поле iндукуються у вакуумi. Вивчається залежнiсть результатiв вiд граничних умов i встановлено, що вимога скiнченностi повного iндукованого вакуумного магнiтного потоку усуває неоднозначнiсть у виборi граничних умов. Обговорюються вiдмiнностi мiж випадками масивної та безмасової спiнорної матерiї.

A new photonic platform allows implementing 2D Quantum Walks in the space of transverse wavevector components of a single light beam. Detection of an anomalous velocity demonstrates that this system simulates a Quantum Hall Insulator.

We investigate the linearized hydrodynamic equations of interacting self-propelled particles in two dimensional space. It is found that the small perturbations of density and polarization fields satisfy the hyperbolic partial differential equations---that admit analytical propagating wave solutions. These solutions uncover the questionable traveling band formation in the flocking state of self-propelled particles. Below the critical noise strength, an unstable disordered state (random motion) undergoes a transient vortex and evolves to an ordered state (flocking motion) as unidirectional traveling waves. There appear two possible longitudinal wave patterns depending on the noise strength, including single band in stable state and multiplebands in unstable state. A comparison of theoretical and experimental studies is presented., Comment: 6 pages, Accepted for publication in Physical Review E

In this paper, we intend to develop an effective numerical method to solve a class of two-dimensional space-fractional advection-diffusion-reaction equations. After spatially discretizing this equation using the fractional centered difference formula, it leads to a system of nonlinear ordinary differential equations. The compact implicit integration factor method is applied to solve the resulting system to achieve good stability and robustness. Linear stability analysis and numerical experiments are given to verify that the compact implicit integration factor method has excellent efficiency and stability properties.

Locomotion and dispersal are important processes that affect the distribution and abundance of organisms in aquatic environment. In this study we observed the movement of a group of Daphnia magna called an ensemble. In laboratory conditions, the distribution of fifty daphnids (in triplicate) at the release from the point source in two-dimensions was examined. In experiments, animals were placed in a square plastic container with thin layer of culture medium and the video of their movements was recorded. Video processing and measuring of swimming behaviour was carried out by the TrackTox software. Mathematical and statistical analyses were performed using the functions and packages of the R software. The diffusion motion equation used allowed to obtain the value of the diffusion coefficient, which in our case was 0.051±0.009 cm2 s–1. The approach used can be used to model the migration and spatial distribution of these microscopic crustaceans. Moreover, given the fact that certain parameters of swimming behavior are already used to toxicity assessment, the diffusion parameters of a Daphnia ensemble can also be proposed as a characteristic in ecotoxicological studies.

In this paper, we present a curvature based modification to the existing beam element formulation to model cables which undergoes large displacements. The proposed approximation is numerically tested against the P-delta formulation and true nonlinear formulation of cables. As a result, a limiting value for the curvature based stiffness is reported.

The transformations of the partial fractional derivatives under spatial rotation inR2are derived for the Riemann-Liouville and Caputo definitions. These transformation properties link the observation of physical quantities, expressed through fractional derivatives, with respect to different coordinate systems (observers). It is the hope that such understanding could shed light on the physical interpretation of fractional derivatives. Also it is necessary to be able to construct interaction terms that are invariant with respect to equivalent observers.

Exploring the concept of non-Hermitian Hamiltonians respecting parity-time symmetry with classical wave systems is of great interest as it enables the experimental investigation of parity-time-symmetric systems through the quantum-classical analogue. Here, we demonstrate unidirectional wave vector manipulation in two-dimensional space, with an all passive acoustic parity-time-symmetric metamaterials crystal. The metamaterials crystal is constructed through interleaving groove- and holey-structured acoustic metamaterials to provide an intrinsic parity-time-symmetric potential that is two-dimensionally extended and curved, which allows the flexible manipulation of unpaired wave vectors. At the transition point from the unbroken to broken parity-time symmetry phase, the unidirectional sound focusing effect (along with reflectionless acoustic transparency in the opposite direction) is experimentally realized over the spectrum. This demonstration confirms the capability of passive acoustic systems to carry the experimental studies on general parity-time symmetry physics and further reveals the unique functionalities enabled by the judiciously tailored unidirectional wave vectors in space.

Geometric phase metasurface (GPM) elements are two dimensional space variant gradient structures, which enable exotic light manipulation. Such structures consist of a dense assembly of resonant optical nanoantennas, the size parameters and orientation of which dictate local light-matter interactions. The GPM elements have been extensively studied, showing that they can control of the phase, amplitude, polarization and orbital angular momentum of light beams [1-4]. The GPM elements have been used as flat optical elements with unique features, as polarization control elements, and as spectro-polarimetric devices.

A variant of the well known Vicsek model of the collective motion of a group of agents has been studied where the range of interactions are spatially quenched and non-overlapping. To define such interactions, the underlying two dimensional space is discretized and is divided into the primitive cells of an imaginary square lattice. At any arbitrary time instant, all agents within one cell mutually interact with one another. Therefore, when an agent crosses the boundary of a cell, and moves to a neighboring cell, only then its influence is spread to the adjacent cell. Tuning the strength of the scalar noise $\eta$ it has been observed that the system makes a discontinuous transition from a random diffusive phase to an ordered phase through a critical noise strength $\eta_c$ where directed bands with high agent densities appear. Unlike the original Vicsek model here a host of different types of bands has been observed with different angles of orientation and different wrapping numbers. More interestingly, two mutually crossed independent sets of simultaneously moving bands are also observed. A prescription for the detailed characterization of different types of bands have been formulated., Comment: 13 figures

Bats use frequency-modulated echolocation to identify and capture moving objects in real three-dimensional space. Experimental evidence indicates that bats are capable of locating static objects with a range accuracy of less than 1 μs. A previously introduced model estimates ranges of multiple, static objects using linear frequency modulation (LFM) sound and Gaussian chirplets with a carrier frequency compatible with bat emission sweep rates. The delay time for a single object was estimated with an accuracy of about 1.3 μs by measuring the echo at a low signal-to-noise ratio (SNR). The range accuracy was dependent not only on the SNR but also the Doppler shift, which was dependent on the movements. However, it was unclear whether this model could estimate the moving object range at each timepoint. In this study, echoes were measured from the rotating pole at two receiving points by intermittently emitting LFM sounds. The model was shown to localize moving objects in two-dimensional space by accurately estimating the object's range at each timepoint.

In this paper, we consider solutions to a Cauchy problem for a parabolic-elliptic system in two dimensional space. This system is a simplified version of a chemotaxis model, and is also a model of self-interacting particles. The behavior of solutions to the problem closely depends on the $L^1$-norm of the solutions. If the quantity is larger than $8\pi$, the solution blows up in finite time. If the quantity is smaller than the critical mass, the solution exists globally in time. In the critical case, infinite blowup solutions were found. In the present paper, we direct our attention to radial solutions to the problem whose $L^1$-norm is equal to $8\pi$ and find bounded and unbounded oscillating solutions.

The problem of existence of bounded relativistic fermion states in a spherically symmetric well of finite depth in a two-dimensional space is investigated. The well depth critical for the appearance of standard states with energies E = m, 0, and –m is determined; moreover, cases with zero and nonzero fermion momenta are considered. Approximate analytical expressions for the critical depths of narrow and wide wells are derived which are in good agreement with the results of numerical calculations. Approximate energies of levels located on the boundaries of the upper and lower continuums and determined analytically are in good agreement with the results of numerical calculations.

The self-consistent field theory (SCFT) is applied to the microphase separation of concentrated solutions of weakly charged polyelectrolytes. The generalized Poisson–Boltzmann equation describing t...

A paradox concerning the flight of insects in two-dimensional space is identified: insects maintaining their bodies in a particular position (hovering) cannot, on average, generate hydrodynamic force if the induced flow is temporally periodic and converges to rest at infinity. This paradox is derived by using the far-field representation of periodic flow and the generalized Blasius formula, an exact formula for a force that acts on a moving body, based on the incompressible Navier–Stokes equations. Using this formula, the time-averaged force can be calculated solely in terms of the time-averaged far-field flow. A straightforward calculation represents the averaged force acting on an insect under a uniform flow, −〈V〉, determined by the balance between the hydrodynamic force and an external force such as gravity. The averaged force converges to zero in the limit 〈V〉 → 0, which implies that insects in two-dimensional space cannot hover under any finite external force if the direction of the uniform flow has a component parallel to the external force. This paradox provides insight into the effect of the singular behaviour of the flow around hovering insects: the far-field wake covers the whole space. On the basis of these assumptions, the relationship between this paradox and real insects that actually achieve hovering is discussed.

The basic theory on relativistic positioning systems in a two-dimensional space-time and the analysis of the possibility of making relativistic gravimetry with these systems have been presented elsewhere [Phys. Rev. D 73 , 084017 (2006); Phys. Rev. D 74 , 104003 (2006)]. Here we summarize these results and we outline new issues on the relativistic positioning systems in Minkowski plane. We point out that the accelerations of the emitters and of the user along their trajectories are determined by the sole knowledge of the emitter positioning data and of the acceleration of only one of the emitters and only during a light echo interval.

The vacuum structure of N = 2 linear supersymmetry (LSUSY) invariant QED, which is equivalent to N = 2 nonlinear supersymmetry (NLSUSY) model, is studied explicitly in two-dimensional space–time ( d = 2 ). Two different isometries SO ( 1 , 3 ) and SO ( 3 , 1 ) appear for the vacuum field configuration corresponding to the various parameter regions. Two different field configurations of SO ( 3 , 1 ) isometry describe the two different physical vacua, i.e. one breaks spontaneously both U ( 1 ) and SUSY and the other breaks spontaneously SUSY alone.

This paper investigates the characteristics of evanescent waves in the generalized nonstandard finite difference time domain (NS-FDTD) method and derives the dispersion equation and stability condition for two-dimensional space. The validity of the dispersion equation is confirmed by comparison with results of numerical simulations. The NS-FDTD method is applied to the analysis of an optical switching device using a photon tunneling effect. It is shown that the NS-FDTD method has highly accurate characteristics not only for propagating waves, but also for evanescent waves compared with the standard FDTD method., application/pdf

We clarify the relation between two kinds of statistics for particle excitations in planar systems: the braid statistics of anyons and the Haldane exclusion statistics(HES). It is shown non-perturbatively that the HES exists for incompressible anyon liquid in the presence of a Hall response. We also study the statistical properties of a specific quantum anomalous Hall model with Chern-Simons term by perturbation in both compressible and incompressible regimes, where the crucial role of edge states to the HES is shown.

The nature of nonstatistical fluctuations in the density distribution of singly charged particles produced in 32S–Ag/Br interactions at an incident momentum of 200 A GeV/c has been investigated. With the help of the nuclear photographic emulsion technique, the data were collected, and using different statistical methods of multiplicity moments these data have been analysed in terms of intermittency and fractal properties of one- and two-dimensional phase space distributions. Wherever possible, experimental results have been compared with the results obtained in other similar heavy-ion-induced experiments, as well as with those obtained from simulated events of the Lund Monte Carlo code FRITIOF. In some cases events simulated by generating (pseudo)random numbers have also been used for the purpose of comparison. The investigation shows the presence of weak intermittency in one dimension (1D) that is consistent with self-similarity in the density distribution. In two dimensions (2D), anisotropy in the distributions along longitudinal and transverse phase space variables indicates that self-affinity, rather than self-similarity in the distributions, is responsible for the observed behaviour. From the analysis of factorial correlators and oscillatory multiplicity moments, the presence of a few-particle short-range correlation has also been established. In almost all cases, the simulated interactions fail to replicate the experimental results.

The equation of the geodesics on a hot plate with a radially symmetric temperature profile is derived using the Lagrangian approach. Numerical solutions are presented with an eye towards (a) teaching two-dimensional curved space and the metric used to determine the geodesics, (b) revealing some characteristics of two-dimensional curved spacetime and (c) providing insight into understanding the curved space which emerges in teaching relativity. In order to provide a deeper insight, we also present the analytical solutions and show that they represent circles whose characteristics depend on curvature of the space, conductivity and the coefficient of thermal expansion.

The analytical solution of gradual change media in two-dimensional free space is studied. Using separation of variables, the solution of electromagnetic wave in time-varying media, which is an exponential function of time, is derived in two-dimensional space. The rationality of the solution is verified indirectly. According to the solution, the figures of the wave are depicted. Based on these figures, the character of the wave in time-varying media is obtained, which shows frequency shift and changes of phase velocity and amplitude.

Making use of the adiabatic hyperspherical approach, we report a calculation for the energy spectrum of the ground and low-excited states of a two-dimensional helium in a magnetic field. The results show that the ground and low-excited states of helium in low-dimensional space are more stable than those in three-dimensional space and there may exist more bound states.

We discuss the solitary wave solutions of a particular two-component scalar field model in two-dimensional Minkowski space. These solitary waves involve one, two or four lumps of energy. The adiabatic motion of these composite non-linear non-dispersive waves points to variations in shape., Comment: 21 pages, 15 figures. To appear in Physica D: Nonlinear Phenomena

Making use of hyperspherical coordinates, we investigate the qualitative features of the ground and low-lying states of a positronium molecule confined in a two-dimensional (2D) space under a magnetic field. We find that a positronium molecule has more bound states in 2D than in 3D. With the increase of the magnetic field, the second bound state experiences a transition in angular momentum. The result shows that symmetry plays an essential role in the energy spectrum of low-lying states.

Making use of the adiabatic hyperspherical approach, a feature of the low-lying spectrum of a positronium negative ion in a two-dimensional (2D) space under a magnetic field is investigated. The results show that there may exist more bound states for a positronium negative ion in the 2D case.

The statistics of polymer‐chain conformations in the intrinsic coordinate system, the directions of whose axes are determined by the anisotropy of a polymer coil at certain instantaneous distribution of the conformations, is considered. It is shown that even for an absolutely flexible polymer chain the shape of the coil in the intrinsic coordinate system is anisotropic. Full distribution functions of the chain links in space are constructed for both the absolutely flexible linear polymer and the linear polymer chain which possesses finite flexural rigidity, and degrees of anisotropy of these macromolecules are calculated.

Empirical and analytical procedures are developed to determine the morphological properties of galaxy clusters. The apparent orientations and shapes are obtainted in two dimensional space while the direction towards the cluster pole is found in three dimensional space. These properties were determined for three Abell clusters and found to be strongly related.

Very few AdS6 × M4 supersymmetric solutions are known: one in massive IIA, and two IIB solutions dual to it. The IIA solution is known to be unique; in this paper, we use the pure spinor approach to give a classification for IIB supergravity. We reduce the problem to two PDEs on a two-dimensional space Σ. M4 is then a fibration of S2 over Σ; the metric and fluxes are completely determined in terms of the solution to the PDEs. The results seem likely to accommodate near-horizon limits of (p, q)-fivebrane webs studied in the literature as a source of CFT5’s. We also show that there are no AdS6 solutions in eleven-dimensional supergravity., SCOPUS: ar.j, info:eu-repo/semantics/published

The relation between a nonlinear supersymmetic (NLSUSY) theory and a SUSY Yang-Mills (SYM) theory is studied for N = 3 SUSY in two-dimensional space-time. We explicitly show the NL/L SUSY relation for the (pure) SYM theory by means of cancellations among Nambu-Goldstone fermion self-interaction terms., Comment: 9 pages

We study the sine-Gordon model in two dimensional space time in two different domains. For beta > 8 pi and weak coupling, we introduce an ultraviolet cutoff and study the infrared behavior. A renormalization group analysis shows that the model is asymptotically free in the infrared. For beta < 8 pi and weak coupling, we introduce an infrared cutoff and study the ultraviolet behavior. A renormalization group analysis shows that the model is asymptotically free in the ultraviolet., 43 pages, Latex 2.09

Using variational procedure in infinite two dimensional space, the existence of one dimer and one trimer (spin-1/2) bound state, with binding energies below −0.014 mK and −0.0057 mK (respectively), is definitively found. In a holding harmonic potential the strongest binding is achieved for the width of about 3 A. For dimer at 3.2 A it is −8.39 mK and for trimer at 3A it is −5.5 mK.

International audience; The topological and metric properties of a few natural 2D random cellular structures, namely an armadillo shell structure and young soap froths, which are formed from two classes of cells, large and small, have been characterized. The topological properties of a model generated from a Kagome tiling, which mimics such random binary structures, have also been exactly calculated. The distribution of the number of cell sides is bimodal for the structures investigated. In contrast to the classical Aboav-Weaire law for homogeneous 2D random cellular structures, nm(n), the mean total number of edges of neighbouring cells of cells with n sides does not vary linearly with n. Only the nm(i, n) (i = 1, 2) determined separately for every class of cells are linear in n for all investigated structures. Topological properties and correlations between metric and topological properties are finally compared with the predictions of various literature models.

Many elementary concepts of crystallography can be presented in two dimensions, where they are relatively easy to visualize. A computer simulation program for two-dimensional crystallography has been developed for use in teaching at an introductory level. The program will generate and display direct and reciprocal lattices, based on arbitrary user input, and is able to apply the symmetry operations of any of the 17 two-dimensional plane groups to a user-defined motif. Examples using the program are presented and discussed.

We investigate lattices of interacting integrate-and-fire neurons. Excitatory short-range interactions lead to coherent firing patterns such as moving stripes on a two-dimensional surface. They have been suggested as possible explanations of hallucinatory phenomena. Other known formations include rotating spirals and expanding concentric rings. We obtain all of them using a novel two-variable description of integrate-and-fire neurons that allows for a continuum formulation of neural fields. Within our model we can formulate a topological constraint that explains why all the moving coherent patterns have a dimension lower by one unit from that of the manifold. In the presence of strong inhomogeneity, neural continuity breaks down, and the field approximation no longer holds. In that case, one obtains incoherent firing patterns driven by the excitatory interactions.