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There are many methods that can be used to compute the geographic similarity between regions represented through areas in a two dimensional space. However, it is the rough set based membership function that allows an estimate of the similarity between a subzone formed by the data. The major objective behind this paper is to measure the similarity of GIS subzones on a discrete dataset. This has a varied number of applications not only in GIS and epidemiology but also in clusters analysis and artificial intelligence.

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.

The paper describes a hybrid algorithm of position-trajectory control of a moving object, based on the integration of the wave and ant algorithms. The process of tracing the trajectory is carried out step by step. At each step relative to the current position of the moving object, a zone is formed within which all the obstacles are localized with the help of the radar, after which a separate trajectory section is constructed which is a continuation of the previously constructed section. And the entire trajectory is a collection of individual sections. The time complexity of this algorithm depends on the ant colony lifetime l (the number of iterations), the number of vertices of the graph n, and the number of ants m, and is defined as O(l∙n2∙m).

Euclidean geometry employs Cartesian coordinate system in which reference axes are perpendicular to each other. In this system every point is represented with a unique tuple which is non-ambiguous. In contrast to classical geometry, the fuzzy geometry deals with the objects with hazy boundary and thus can be viewed as a collection of fuzzy points. In fuzzy set theory it is considered that universe is non-fuzzy, the way we perceive any object is fuzzy or imprecise. This hypothesis guided us to establish fuzzy plane geometry in two dimensional space. This chapter starts with the discussion on reference frame considered for developing fuzzy geometry, which act as a yardstick to measure. The concept of fuzzy point and fuzzy line segment are also introduced here.

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

International audience; A prey-predator model with a sexual reproduction in prey population and nonlocal consumption of resources by prey in two spatial dimensions is considered. Patterns produced by the model without nonlocal terms and periodic boundary conditions are studied first. Then, Turing patterns induced by the nonlocal interaction (see Banerjee et al. (2018) [1]) in the two dimensional space are explored along with the effects of the nonlocal interaction range on the resulting patterns under proper parametric restrictions. The Turing bifurcation conditions for the nonlocal model are derived analytically and bifurcation scenario of stationary hotspot pattern generated from the homogeneous steady-state are studied in detail, both analytically and numerically. Also, conversion of periodic and aperiodic solutions exhibited by the local model into stationary Turing pattern as an effect of the nonlocal interaction term is also explored. The resulting patterns are stationary when the range of nonlocal interactions are significantly large.

International audience; This paper deals with the multidimensional stability of time periodic V-shaped traveling fronts in bistable reaction–diffusion equations. It is well known that time periodic V-shaped traveling fronts are asymptotically stable in two dimensional space. In the current study, we further show that such fronts are asymptotically stable under spatially decaying initial perturbations in Rn with n≥3. In particular, we show that the fronts are algebraically stable if the initial perturbations belong to L1 in a certain sense. Furthermore, we prove that there exists a solution oscillating permanently between two time periodic V-shaped traveling fronts, which implies that time periodic V-shaped traveling fronts are not always asymptotically stable under general bounded perturbations. Finally we show that time periodic V-shaped traveling fronts are only time global solutions of the Cauchy problem if the initial perturbations lie between two time periodic V-shaped traveling fronts.

We numerically study the Stokes eigen-modes in two dimensions on isosceles triangles with apex angle θ = π / 3 , π/2, and 2π/3 by using two spectral solvers, i.e., a Lagrangian collocation method with a weak formulation for the primitive variables and a Legendre–Galerkin method for the stream-function. We compute the first 6,400 Stokes eigen-modes. With 72 collocation points in each spatial dimension, the eigen-values λn for n ≤ 400 can be obtained with spectral accuracy and at least ten significant digits. We show the symmetry of the Stokes eigen-modes dictated by the geometry of the bounded flow domain. From the spectrally accurate data of the Stokes eigen-modes, the following features are observed. First, the n-dependence of the spectrum λn obeys the Weyl asymptotic formula in two dimensional space: λ n = C 1 n + C 2 n + o ( n ) . Second, for an isosceles triangle with legs of unit length, the θ-dependence of the spectrum λn can be accurately approximated by λn(θ)/λn(π/2) ≈ 1/(sin θ), as a consequence the volume-dependence of the coefficient C1 in the Weyl asymptotic formula. And third, a linear stream function-vorticity correlation is observed in the interior of the flow domain.

We prove uniform optimal-order error estimates for the characteristics-mixed finite element for advection-diffusion equations.In the proof, the interpolation operators are used instead of the mixed elliptic projections.The generic constants of the estimates do not explicitly depend on e, but depend linearly on Sobolev norms of the true solution. We prove uniform error estimates for the characteristics-mixed finite element method for two-dimensional transient convection-dominated diffusion equations, in the sense that the generic constants in the error estimates depend linearly on certain Sobolev norms of the true solution but not on the scaling diffusion parameter e. Numerical experiments are presented to confirm our theoretical findings.

This work focuses on the construction of rigid formation from non-rigid ones in the two-dimensional space. Analogously to operations of Henneberg sequence aiming to guarantee the minimal rigidity of formation, two new operations are introduced, allowing one to sequentially build any rigid graph by connecting non-rigid ones. A systematic construction sequence is developed based on proposed operations, and is shown to be able to restore rigidity by introducing minimum number of new edges during the construction process. Further applications of the proposed operations are also presented, one of which is successfully employed in the problem of persistence analysis of directed graphs, and can verify the persistence of a given graph with a speed two times faster comparing with existing solution.

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.

The static structural and energetic properties of thin crystalline films (∼two dimensional bilayers) of silica, SiO2, are modelled. Two potential models are considered in which the key interactions are described by purely harmonic terms and more complex electrostatic terms, respectively. The relative energetic stability of two potential crystalline forms, which represent alternative ways of tiling two dimensional space, is discussed. Coherent and incoherent distortions are introduced to the simulated crystals and their effects considered in terms of the ring structure formed by the Si atoms. The spatial relationship between distorted rings is analysed. An experimentally-observed single crystalline configuration is considered for comparison throughout.

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.

Two new rank properties for time difference of arrival (TDOA) measurements in two-dimensional space are reported in this paper. On the basis of these rank properties, we propose a class of algorithms to refine TDOAs from their observations. Since only the singular value decomposition (SVD) technique is used, these proposed algorithms are very simple. Simulative experiments show that the accuracy of TDOA estimations is significantly improved using the proposed refining algorithms. Moreover, their ability to improve TDOA-based joint source and sensor localization is also proven by simulative experiments.

In this paper, we study the large time behavior of solutions to the Cauchy problem for the anisotropic conservation laws in two dimensional space. Without any smallness assumption on the initial data, the decay rates of solutions in L 2 space and homogeneous Sobolev space H ˙ γ are obtained by using the method of time-frequency decomposition and the classical energy method.

Conventional identification algorithms of space- frequency block codes (SFBC) only utilize the space- domain redundancy between any two receive antennas. In this paper, a novel two-dimensional space-frequency domain redundancy based SFBC identification algorithm for frequency selective fading is proposed in which the detection probability varies with the number of subcarriers. In particular, space-domain redundancy is utilized to construct the cross-correlation function of the estimator while frequency-domain redundancy is incorporated in the hypothesis test statistic. Simulation results verify the viability of the proposed algorithm and its superior performance for short observation periods with comparable computational complexity to the conventional algorithms.

A numerical method for a kind of time-dependent two-dimensional two-sided space fractional diffusion equations is developed in this paper. The proposed method combines a time scheme based on direction splitting approaches and a spectral method for the spatial discretization. The direction splitting approach renders the underlying two-dimensional equation into a set of one-dimensional space fractional diffusion equations at each time step. Then these one-dimensional equations are solved by using the spectral method based on weak formulations. A time error estimate is derived for the semi-discrete solution, and the unconditional stability of the fully discretized scheme is proved. Some numerical examples are presented to validate the proposed method.

本稿はセンサーと呼ばれる中途打ち切りデータが存在するケースにおける2次元データ上の生存関数の推定問題について考える。特にパス従属推定量について研究する。それは2次元空間上に引かれた経路に観測値を射影した後1次元データに置き換えて、ノンパラメトリックな推定量を求める手法である。センサーメカニズムが全順序であるとき、パス従属推定量は有効な推定を行えることがすでにわかっているが、ここでは全順序でないときの適用可能性や、パスの選び方について考察する。さらに従来のパス従属推定量とは若干異なる新しい推定量を提案する。, This paper focuses on the estimation problem of a survival function on two-dimensional data when the data points are subject to censoring. In particular, a path-dependent estimator is studied. In the methods, observations are projected onto the path drawn on twodimensionalspace, and replaced by one dimensional data, then a nonparametric estimatoris given. When the censoring mechanism is totally ordered, a path-dependent estimatorgive valid inference. This paper discusses the applicability of the estimator when censoring mechanism is not totally ordered and proposes a new path-dependent estimator. It is slightly different from former one.

We study the two-dimensional generalized Lans alpha magnetohydrodynamics system. We show that the solution pairs of velocity and magnetic fields to this system preserve their initial regularity in two cases: dissipation logarithmically weaker than a full Laplacian and zero diffusion, zero dissipation and diffusion logarithmically weaker than a full Laplacian.

The backward Euler and Crank-Nicolson-Galerkin fully-discrete approximate schemes for two-dimensional space-fractional advection-dispersion equations are established. Firstly, we prove that the corresponding variational problem has a unique solution, and the proposed fully-discrete schemes are unconditionally stable, whose solutions are all unique. Secondly, the optimal error estimates are derived by use of properties of projection operator and fractional derivatives. Finally, numerical examples demonstrate effectiveness of numerical schemes and confirm the theoretical analysis.

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.

Several examples of classical superintegrable systems in two-dimensional spac are shown to possess hidden symmetries leading to their linearization. They are those determined 50 years ago in [Phys. Lett. 13, 354 (1965)], and the more recent Tremblay-Turbiner-Winternitz system [J. Phys. A: Math. Theor. 42, 242001 (2009)]. We conjecture that all classical superintegrable systems in two-dimensional space have hidden symmetries that make them linearizable., Comment: 17 pages

In present work, the exact analytical bound-state solutions of modified Schrödinger equation (MSE) with modified extended Cornel potential (MECP) have been presented using both Bopp’s shift method and standard perturbation theory in the noncommutative two dimensional real space and phase (NC-2D: RSP), we have also constructed the corresponding noncommutative Hamiltonian operator which containing two new terms, the first one is modified Zeeman effect and the second is spin-orbital interaction. The theoretical results show that the automatically appearance for both spin-orbital interaction and modified Zeeman effect leads to the degenerate to energy levels to 2(2l +1) sub states.

Address-Event Representation is a flowering technology that can change the visual perception of the computer vision world. This paper proposes a methodology to associate the input data from this kind of sensors. A new descriptor computed using an extended LBP operator seeks to characterize the connectivity of the asynchronous incoming events in a two dimensional space. Those features can be organized on histograms and combined with others descriptors, as histograms of oriented events. They can be the input of traditional classifiers to detect or recognize objects from the scene.

In this paper, we study a compressible Primitive Equations (CPEs) of the ocean in the two dimensional space, with horizontal periodic and vertical mixed boundary conditions. Thanks to an effective change of variables, we obtain a new CPEs model, which is similar as viscous shallow water equation. Using a new entropy estimate, we prove the stability of weak solutions for this new two dimensional CPEs model.

In this paper, we propose a tailored finite point method for linearized incompressible flow (Oseen equations) in two dimensions based on the equation decomposition technique. Unlike the usual vorticity-stream function formulation, the velocities are decomposed to irrotational and rotational parts. We only need to solve a system of two elliptic equations which are decoupled in the interior domain. They are only coupled in boundary conditions. When the domain is unbounded, we use the artificial boundary method to reduce the original problem to a problem on a bounded computational domain. Our finite point method has been tailored to some particular properties of the problem. Therefore, our scheme satisfies the discrete maximum principle in the interior domain automatically. We also give some remarks on more generally linearized incompressible flow, and it can be considered as the first step to solve the incompressible Navier–Stokes problem. Finally, several numerical examples show the efficiency and feasibility of our method whatever the Reynolds number is small or large.

This paper concerns the Cauchy problem of the barotropic compressible Navier–Stokes equations on the whole two-dimensional space with vacuum as far field density. In particular, the initial density can have compact support. When the shear and the bulk viscosities are a positive constant and a power function of the density respectively, it is proved that the two-dimensional Cauchy problem of the compressible Navier–Stokes equations admits a unique local strong solution provided the initial density decays not too slow at infinity. Moreover, if the initial data satisfy some additional regularity and compatibility conditions, the strong solution becomes a classical one.

Twenty-eight sex- and age-matched participants, half dextrals and half sinstrals, were instructed to move a pen-sized planometer three inches (7.6 cm) while blindfolded. Under separate trials, movements were made at four angles, towards and away from the body, and at two distances from the body (30 cm, 53 cm). Half were made with the right hand and half with the left hand. Line estimates increased in length across blocks of trials in a linear fashion and progressively overestimated the three-inch imagined criterion. Lines made moving towards the body were longer than those made moving away from the body, implying an egocentric frame of reference in making the estimates. Line estimates made at an oblique angle differed significantly from estimates made at other angles. No influences of sex, handedness, or the hand used in making the estimates were observed. The findings suggest that motoric estimates of line lengths made without visual cues—a unique measure of an implicit cognitive concept—are significantly altered by temporal and spatial factors, but not by sex or hemispheric laterality.

By means of the Hirota bilinear method, explicit representations of general rogue waves for the Mel’nikov equation are explored in terms of determinants. As applications, it is found that this system admits bright- and dark-types rogue waves localized in two dimensional space. Furthermore, the superposition of such bright rogue waves are investigated graphically by different choices of the free parameters.

Curve or contour correspondence has been extensively explored. Previous work was mainly concentrated on the rigid correspondence or alignment. This paper presents a spectral analysis method to resolve the problem of curves correspondence with non-rigid deformation. Using the embedding of original affinity matrix to the spectral domain, we can build the point correspondence of no-rigid deformation shapes.

This work presents a systematic study of population games of the Prisoner's Dilemma, Hawk–Dove, and Stag Hunt types in two-dimensional Euclidean space under two-person, one-shot game-theoretic interactions, and in the presence of agent random mobility. The goal is to investigate whether cooperation can evolve and be stable when agents can move randomly in continuous space. When the agents all have the same constant velocity cooperation may evolve if the agents update their strategies imitating the most successful neighbor. If a fitness difference proportional is used instead, cooperation does not improve with respect to the static random geometric graph case. When viscosity effects set-in and agent velocity becomes a quickly decreasing function of the number of neighbors they have, one observes the formation of monomorphic stable clusters of cooperators or defectors in the Prisoner's Dilemma. However, cooperation does not spread in the population as in the constant velocity case.

To modify the power factor and balance the three-phase currents simultaneously, this paper proposes the instantaneous compensator to calculate the compensation current. The instantaneous compensator utilizes two-dimensional instantaneous space vector and setting the active power as a constant for each cycle which can improve power quality effectively. Moreover, the instantaneous compensator requires an independent power source, whose capacity can be reduce by using a static var compensator (SVC). An SVC does not interfere with the capability of the instantaneous compensator. Field measurement data were analyzed. Simulation results confirmed the feasibility of correcting the power factor and balancing load currents simultaneously using the proposed method.

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

In this work, a new modified homotopy perturbation (NHPM) is applied to solve a special kind of nonlinear Volterra integral equations in two-dimensional space. The two most important steps in the application of the new Homotopy perturbation method are to construct a suitable homotopy equation and to choose a suitable initial guess. In the present paper, convergence of the new approach is proved. Comparison of our solution with the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM) shows that the NHPM is effective and accurate in solving these kind of problems.

The exponential synchronization problem of complex networks is investigated. The complex network model considered is a moving agent network in a two dimensional space and the coupling of between nodes is nonlinear links. In order to achieve the objective of exponential synchronizaiton, linearizing error system is presented. Base on the Lyapunov stability theory, a new criterion is proposed for exponetial synchronization of moving agnets networks under the condition of fast switching. In addition, an numerical example of the Lorenz chaotic system are analyzed, and numerical simulations results show the effectiveness of proposed exponential synchronization criterion.

This newly developed program gives predictions of a climber's fall in two dimensional space. It numerically calculates all forces acting on the climber, the belayer, the rope and various fix points. Also it shows the movement of both the climber and the belayer, taking air drag, friction, deformation of the climber, real rope and belay device characteristics into account. By using a 3-parameter viscoelastic rope model the different behaviour of the rope for dynamic and static loading is correctly simulated. The rope parameters are calibrated using actual rope values which are commonly available from product data sheets of rope manufacturers. A new concept is introduced to describe the rope forces due to the friction at the 1st protection. The climber is modelled with 2-masses to describe the body deformation. An energy analysis shows in detail the significance of all present mechanisms to absorb the fall energy compared to all others.

In this paper, we discuss the space-time Caputo-Riesz fractional diffusion equation with variable coefficients on a finite domain. The finite difference schemes for this equation are provided. We theoretically prove and numerically verify that the implicit finite difference scheme is unconditionally stable (the explicit scheme is conditionally stable with the stability condition @t^@c(@Dx)^@a+@t^@c(@Dy)^@b

In this study, we consider an acyclic rigid formation with a group of mobile autonomous agents moving in a two-dimensional space. The formation is generated via a Henneberg sequence construction in which there is one global leader that does not follow any other agents, one first-follower that only follows the global leader, and each of other agents has two leaders, which is added by a vertex addition or an edge splitting operation. The entire formation moves with the leadership of the global leader. Every follower agent tries to maintain distances towards its leaders. Under the constraint of the acceleration for the global leader, the distributed formation control laws are proposed for the followers that only use the locally relative distance measurement. The control law of the first-follower is proposed, which needs to know the velocity of the global leader and the relative distance between the global leader and itself. The global asymptotic stability of the expected formation is proved via a Lyapunov-based technique for the considered multi-agent system. Moreover, the stable rigidity problem of a formation is investigated for the proposed distributed relative position-only formation control law. Necessary and sufficient conditions are provided that must be satisfied by the architecture of the underlying graph. Simulation results illustrate the effectiveness of the proposed formation control approach.

Many disciplines are faced with the problem of handling time-series data. This study introduces an innovative visual representation for time series, namely the continuous triangular model. In the continuous triangular model, all subintervals of a time series can be represented in a two-dimensional continuous field, where every point represents a subinterval of the time series, and the value at the point is derived through a certain function (e.g. average or summation) of the time series within the subinterval. The continuous triangular model thus provides an explicit overview of time series at all different scales. In addition to time series, the continuous triangular model can be applied to a broader sense of linear data, such as traffic along a road. This study shows how the continuous triangular model can facilitate the visual analysis of different types of linear data. We also show how the coordinate interval space in the continuous triangular model can support the analysis of multiple time series through spatial analysis methods, including map algebra and cartographic modelling. Real-world datasets and scenarios are employed to demonstrate the usefulness of this approach.

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.