Null Boundary Conditions Research Articles

Application and Reversibility of Three Dimensional Cellular Automata

In this study, we obtain the characteristic matrices of three-dimensional cellular automata under the null boundary condition. We examine the inverse of characteristic matrices. We obtain a recurrence equation to determine under what conditions the matrix is invertible. Thanks to this equation, we can calculate the inverse of large-dimensional matrices. Finally, we give some applications of cellular automata. We find the minimal polynomial of the characteristic matrix. We find the cycle length and transition length of the characteristic matrix with the help of minimal polynomials. We also find the attractive points of the characteristic matrix. Finally, we draw the State Transition diagram with the results we obtained.

Irreversibility of 2D Linear CA and Garden of Eden

In this paper, we consider a pentagonal lattice and we investigate the rule matrix with null boundary condition for two-dimensional cellular automata with the field [Formula: see text] (the set of integers modulo [Formula: see text]) and analyze their characteristics. Moreover, an algorithm of computing the rank of rule matrix with null boundary condition for von Neumann neighborhood is developed. Finally, necessary and sufficient conditions for the existence of Garden of Eden configurations for two-dimensional cellular automata are obtained.

An Efficient Cellular Automata-Based Classifier with Variance Decision Table

Classification is an important task of machine learning for solving a wide range of problems in conforming patterns. In the literature, machine learning algorithms dealing with non-conforming patterns are rarely proposed. In this regard, a cellular automata-based classifier (CAC) was proposed to deal with non-conforming binary patterns. Unfortunately, its ability to cope with high-dimensional and complicated problems is limited due to its applying a traditional genetic algorithm in rule ordering in CAC. Moreover, it has no mechanism to cope with ambiguous and inconsistent decision tables. Therefore, a novel proposed algorithm, called a cellular automata-based classifier with a variance decision table (CAV), was proposed to address these limitations. Firstly, we apply a novel butterfly optimization, enhanced with a mutualism scheme (m-MBOA), to manage the rule ordering in high dimensional and complicated problems. Secondly, we provide the percent coefficient of variance in creating a variance decision table, and generate a variance coefficient to estimate the best rule matrices. Thirdly, we apply a periodic boundary condition in a cellular automata (CA) boundary scheme in lieu of a null boundary condition to improve the performance of the initialized process. Empirical experiments were carried out on well-known public datasets from the OpenML repository. The experimental results show that the proposed CAV model significantly outperformed the compared CAC model and popular classification methods.

Non-uniform number-conserving elementary cellular automata

In this paper, we investigate non-uniform elementary cellular automata (i.e., one-dimensional cellular automata whose cells can use different Wolfram rules to update their states) in the context of number conservation. As a result, we obtain an exhaustive characterization of such number-conserving cellular automata on all finite grids both with periodic and null boundary conditions. The characterization obtained allows, inter alia, to enumerate all number-conserving non-uniform elementary cellular automata, in particular those that are reversible. Surprisingly, the numbers obtained are closely related to the Fibonacci sequence.

Probabilistic analysis of a cantilever beam subjected to random loads via probability density functions

This paper addresses the probabilistic analysis of the deflection of a cantilever beam by means of a randomization of the classical governing fourth-order differential equation with null boundary conditions. The probabilistic study is based on the calculation of the first probability density function of the solution, which is a stochastic process, as well as the density function of further quantities of interest associated with this engineering problem such as the maximum slope and deflection at the free end of the cantilever beam, that are treated as random variables. In addition, the probability density function of the bending moment and the shear force will also be computed. The study takes extensive advantage of the so called Random Variable Transformation method, also known as Probability Transformation Method, that allows us to fully unify the probabilistic analysis in three relevant cases commonly studied in the deterministic setting. All the theoretical findings are illustrated via detailed numerical examples corresponding to each one of the three scenarios.

Rayleigh waves in thermoelasticity: Triple porous media in local thermal non-equilibrium

Assuming the hypothesis of local thermal non-equilibrium, this work investigates the propagation of Rayleigh surface waves in a thermoelastic half-space, isotropic, homogeneous and structured with a triple level of porosity. Its surface is supposed to be stress-free, thermally insulated and characterized by null pressure boundary conditions. A class of wave solutions is highlighted for the differential system of the model, each solution satisfying suitable asymptotic conditions in the depth of the considered half-space. Then, the Rayleigh wave solution is sought as a linear combination of the elements of this class, and, moreover, by means of the selected boundary conditions, the associated secular equation is found. By solving the secular equation, the characteristics of the wave solution are determined: the propagation speed as well as the damping in time. With the purpose of clearly highlighting the characteristics of the model, the secular equation is solved numerically and significant graphical representations are provided, using the software packages Mathematica and MATLAB. This suggests an increase of the Rayleigh wave propagation speed, corroborated with the appearance of the damping in time of its amplitude.

Reversibility of One-Dimensional Cellular Automata

Bu makalede bir boyutlu sonlu lineer hücresel dönüşümler üzerinde çalışıyoruz. Sıfır Sınır Şartı altında yerel kurallar yardımıyla temsili matrisi elde ettik. Elde edilen bu matrislerin sonlu cisimler üzerinde hangi şartlar altında tersinin olup olmadığını veren bir algoritma tanımladık. Bu aynı zamanda bize hücresel dönüşüm ailesinin terslenebilirliği hakkında fikir veriyor. Son olarak hücresel dönüşümlerin bu ailelerine bazı örnekler veriyoruz.

Efficient methods with polynomial complexity to determine the reversibility of general 1D linear cellular automata over [formula omitted

The property of reversibility is quite meaningful for the classic theoreticabl computer science model, cellular automata. This paper focuses on the reversibility of general one-dimensional (1D) linear cellular automata (LCA), under null boundary conditions over the finite field Zp. Although the existing approaches have split the reversibility challenge into two sub-problems: calculate the period of reversibility first, then verify the reversibility in a period, they are still exponential in the size of the CA’s neighborhood. In this paper, we use two efficient algorithms with polynomial complexity to tackle these two challenges, making it possible to solve large-scale reversible LCA, which substantially enlarge its applicability. Finally, we provide an interesting perspective to inversely generate a 1D LCA from a given period of reversibility.

Reversibility of non-saturated linear cellular automata on finite triangular grids.

Discrete dynamical systems such as cellular automata are of increasing interest to scientists in a variety of disciplines since they are simple models of computation capable of simulating complex phenomena. For this reason, the problem of reversibility of such systems is very important and, therefore, recurrently taken up by researchers. Unfortunately, the study of reversibility is remarkably hard, especially in the case of two- or higher-dimensional cellular automata. In this paper, we propose a novel and simple method that allows us to completely resolve the reversibility problem of a wide class of linear cellular automata on finite triangular grids with null boundary conditions.

Recurrent Misconceptions in the Study of CA Reversibility on Triangular Grids

We present counterexamples illustrating that the characterization of the reversibility of linear cellular automata on finite triangular grids given by Uguz et al. [2017] and Uguz et al. [2019] is not valid, neither in the case of null boundary conditions nor in the case of periodic boundary conditions.

Eigenvalues with respect to a weight for general boundary value problems on networks

In this work we analyze self-adjoint boundary value problems on networks for Schrödinger operators, in which a part of the boundary with a Neumann condition is always considered. We first characterize when the energy is positive semi-definite on the space of functions satisfying the null boundary conditions. To do this, the fundamental tools are the Doob transform and the discrete version of the trace function. Then, we raise eigenvalue problems with respect to a weight for general boundary value problems and we prove the discrete version of the Mercer Theorem. Finally, we apply the obtained results to a Dirichlet-Robin boundary value problem on a star-like network.

The Entropy and Reversibility of Cellular Automata on Cayley Tree

In this paper, we study linear cellular automata (CAs) on Cayley tree of order [Formula: see text] over the field [Formula: see text] (the set of prime numbers modulo [Formula: see text]). After revealing the rule matrix corresponding to cellular automata on Cayley tree with the null boundary condition, we analyze the reversibility problem of these cellular automata for some given values of [Formula: see text] and the levels [Formula: see text] of Cayley tree. The necessary and sufficient conditions for determining whether a CA is reversible or not are demonstrated. Furthermore, we compute the measure-theoretical entropy of the cellular automata which we define on Cayley tree. We show that for CAs on Cayley tree, the measure entropy with respect to uniform Bernoulli measure is infinity.

Method of Images for a Charge inside a Three-Layer Medium and Implementation of a Dirichlet Border Condition

A process to apply the method of images for a charge located in a three-layer medium is presented. The images are found according to the boundary conditions between the layers for the electric field. The characteristics of the electric potential are also considered, thus the number of unknown variables becomesa guide to set the image charges needed to solve the problem. The results are compared with finite element simulations through the use of the software FEMM 4.2, showing good agreement. The found limitations of the process are also noted, mainly in regards to the dependence of the images on the coordinates where the field is to be calculated. The model obtained was applied to different cases, where it was seen that it was not limited to three material media only. Finally, the null potential boundary condition was applied, showing how the method of images could be applied to this type of problems.

Reversibility of Symmetric Linear Cellular Automata with Radius r = 3

The aim of this work is to completely solve the reversibility problem for symmetric linear cellular automata with radius r = 3 and null boundary conditions. The main result obtained is the explicit computation of the local transition functions of the inverse cellular automata. This allows introduction of possible and interesting applications in digital image encryption.

Trajectory generation via FIR filters: A procedure for time-optimization under kinematic and frequency constraints

This paper starts from the results reported in the article “FIR Filters for Online Trajectory Planning with Time- and Frequency-Domain Specifications”, where the use of a cascade of FIR (Finite Impulse Response) filters for planning minimum-time multi-segment polynomial trajectories, i.e. trajectories composed of several polynomial segments, under constraints of velocity, acceleration, etc. is proposed. In particular, in that paper the relationship between the limits acting on the trajectory derivatives (i.e. velocity, acceleration, jerk, etc.), and the parameters of the filters is deduced, along with a set of constraints among these parameters that guarantees the time-optimality of the trajectory in the rest-to-rest case, that is with null boundary conditions on the trajectory derivatives. However, the choice of the parameters, when these conditions are not satisfied, was still an open problem, at least for high order trajectories. In this paper, we show that in case the conditions are not met by the filters parameters, the optimality of the trajectory under the given kinematic bounds can be assured in any case. An algorithm for the selection of the optimal parameters for a generic nth order trajectory planner subject to n kinematic limits is provided. Additionally, the optimal combination of kinematic and frequency constraints is considered. In fact, the compliance with these two types of constraints may lead to a planner composed by a redundant number of filters and, therefore, a procedure for the selection of the minimum number of FIR filters is devised. The effectiveness of the time-optimal trajectory planner is proved by means of numerical simulations and experimental tests.

Maximally mathcal{N} -extended super-BMS3 algebras and generalized 3D gravity solutions

We consider the maximal mathcal{N} -extended supergravity theory in 3 dimensions with fermionic generators transforming under real but non necessarily irreducible representations of the internal algebra. We obtain the symmetry algebra at null infinity preserving boundary conditions of asymptotically flat solutions, i.e. the maximal mathcal{N} -extended super-BMS3 algebra, which possesses non-linear correction in the anti-commutators of supercharges. We present the supersymmetric energy bound and derive the explicit form of the asymptotic Killing spinors. We also find the most generic circular symmetric ground state of the theory, which corresponds to a non-supersymmetric cosmological solutions and derive their entropy.

Existence and Hyers\u2013Ulam stability for three-point boundary value problems with Riemann\u2013Liouville fractional derivatives and integrals

This paper is concerned with a class of two-term fractional differential equations. Three-point boundary value problems with mixed Riemann–Liouville fractional differential and integral boundary conditions are discussed. The Green’s function is investigated and the existence results are obtained based on some fixed point theorems. The Hyers–Ulam stability is also studied for null boundary conditions. As an auxiliary result, a Gronwall type inequality of fractional order integral is obtained.

Cellular automata-based efficient method for the removal of high-density impulsive noise from digital images

Cellular automata (CA) are simple dynamical systems used for solving simple as well as complex problems. It uses a discrete space, like the space used in spatial domain for image processing, to store the values of the input data. Then, the input data can be processed according to the transition rules, functions, or, programmes, for the required output. This research work presents an efficient algorithm based on two dimensional cellular automata (2D CA), with hybrid rules under null and periodic boundary conditions, for filtering high-density impulsive noise from corrupted digital images. The most important advantage of the proposed method is that, it can be applied on all types of digital images (binary, greyscale, or, true color). The paper is organized as follows: Sect. 1 gives a brief introduction towards the problem of noise in digital images with emphasis on its solution. Further, this section presents a brief review of existing standard noise filtering methods based on general image processing and CA techniques. Section 2 discusses the basic concept of CA with special emphasis on 2D CA and related concepts. Section 3 presents the proposed algorithm for the removal of high-density impulsive noise from corrupted digital images. Section 4 discusses, in detail, the experimental results using both mathematical and subjective analysis; and, the extensive experimentation reveals that the proposed 2D CA based algorithm yields better results than the standard noise filtering algorithms. And, then finally Sect. 5 presents the conclusions and future scope.

A Time-Domain Propagator Numerical Method for Computational Electromagnetics

A new time-domain numerical method for full-wave electromagnetic scattering and propagation is presented. The propagator method can be described as a numerical integral operator technique for moving the electromagnetic field through numerical space at successive increments in time. The numerical equations are obtained by discretizing electric and magnetic field propagator integrals. A primary feature of the propagator method is that all six electromagnetic-field components are calculated at each numerical grid point and all components are in time synchronization. Constant spatial and time increments are maintained throughout an inhomogeneous numerical space by employing an extrapolation procedure that ensures causality. A simple and effective first-order absorbing boundary condition, described as the null boundary condition, is introduced. Examples, provided in one-, two-, and three-dimensions, include plane wave reflection from and transmission through a planar boundary, and scattering and radar cross sections for canonical dielectric objects.

Simulation of trace metals and PAH atmospheric pollution over Greater Paris: Concentrations and deposition on urban surfaces

Urban areas can be subject not only to poor air quality, but also to contamination of other environmental media by air pollutants. Here, we address the potential transfer of selected air pollutants (two metals and three PAH) to urban surfaces. To that end, we simulate meteorology and air pollution from Europe to a Paris suburban neighborhood, using a four-level one-way nesting approach. The meteorological and air quality simulations use urban canopy sub-models in order to better represent the effect of the urban morphology on the air flow, atmospheric dispersion, and deposition of air pollutants to urban surfaces. This modeling approach allows us to distinguish air pollutant deposition among various urban surfaces (roofs, roads, and walls). Meteorological model performance is satisfactory, showing improved results compared to earlier simulations, although precipitation amounts are underestimated. Concentration simulation results are also satisfactory for both metals, with a fractional bias <0.5. Concentrations of benzo[a]pyrene are overestimated, probably because continental emissions may be overestimated. Concentrations of benzo[b]fluoranthene and indeno[1,2,3,cd]pyrene are underestimated, in part because of null boundary conditions. PAH deposition fluxes are consistent with earlier measurements obtained in the Greater Paris region. The model simulation results suggest that both wet and dry deposition processes need to be considered when estimating the transfer of air pollutants to other environmental media. Dry deposition fluxes to various urban surfaces are mostly uniform for PAH, which are entirely present in fine particles. However, there is significantly less wall deposition compared to deposition to roofs and roads for trace metals, due to their coarse fraction. Meteorology, particle size distribution, and urban morphology are all important factors affecting air pollutant deposition. Future work should focus on the collection of data suitable to evaluate the performance of atmospheric models for both wet and dry deposition with fine spatial resolution.