Null Boundary Conditions Research Articles

Reversibility Problem of Multidimensional Finite Cellular Automata

While the reversibility of multidimensional cellular automata is undecidable and there exists a criterion for determining if a multidimensional linear cellular automaton is reversible, there are only a few results about the reversibility problem of multidimensional linear cellular automata under boundary conditions. This work proposes a criterion for testing the reversibility of a multidimensional linear cellular automaton under null boundary condition and an algorithm for the computation of its reverse, if it exists. The investigation of the dynamical behavior of a multidimensional linear cellular automaton under null boundary condition is equivalent to elucidating the properties of block Toeplitz matrix. The proposed criterion significantly reduce the computational cost whenever the number of cells or the dimension is large; the discussion can also apply to cellular automata under periodic boundary condition with a minor modification.

Fredholm theory connected with a Douglis–Nirenberg system of differential equations over an exterior region in

We consider a boundary problem over an exterior subregion of for a Douglis–Nirenberg system of differential operators under limited smoothness asumptions and under the assumption of parameter‐ellipticity in a closed sector in the complex plane with vertex at the origin. We pose the problem in an Sobolev–Bessel potential space setting, , and denote by the operator induced in this setting by the boundary problem under null boundary conditions. We then derive results pertaining to the Fredholm theory for for values of the spectral parameter λ lying in as well as results pertaining to the invariance of the Fredholm domain of with p.

On the irreversibility of Moore cellular automata over the ternary field and image application

Cellular automata have rich computational properties and provide many models in mathematical and physical processes. In this paper, one of the most commonly used neighborhood types of two dimensional (2D) cellular automata which is called the Moore neighborhood in two dimensional integer lattice is considered. We study the characterization of 2D linear cellular automata defined by the Moore neighborhood with periodic and null boundary conditions over ternary fields. Furthermore, we analyze some results of 2D cellular automata defined by the rule number 9840NB and finally also present applications to error correcting codes and image processing field.

Elliptic problems for a Douglis–Nirenberg system over ℝn and over an exterior subregion

We consider both a problem over ℝn and a boundary problem over an exterior subregion for a Douglis–Nirenberg system of differential operators under limited smoothness assumptions as well as under the assumption of parameter ellipticity in a closed sector Ꮭ in the complex plane with vertex at the origin. We pose each problem on an Lp Sobolev space setting, 1 < p < ∞, and denote by the operator induced in this setting by the problem over ℝn and by Ap the operator induced in this setting by the boundary problem under null boundary conditions. We then derive results pertaining to the Fredholm theory for each of these operators for values of the spectral parameter λ lying in Ꮭ as well as results for these values of λ pertaining to the invariance of their Fredholm domains with p.

Granular vortices: Identification, characterization and conditions for the localization of deformation

We relate the micromechanics of vortex evolution to that of force chain buckling and, on this basis, formulate the conditions for strain localization in a continuum model of dense granular media. Using the traditional bifurcation analysis of shear bands, we show that kinematic vortex fields are in fact solutions to the boundary value problem satisfying null boundary conditions. To establish an empirical basis for our study, we first develop a method to identify the location of the core and boundary of each vortex from a given displacement field in two dimensions. We then employ this method to characterize the residual deformation field (i.e., the deviation of particle motions from the continuum deformation) in a physical experiment and a discrete element simulation of dense granular samples submitted to biaxial compression. Vortices in the failure regime are essentially confined to the shear band. Primary vortices, the clear majority, rotate in the same direction as the shear band; secondary vortices, the so-called wakes, rotate in the opposite direction. Primary vortices align in spatial succession along the central axis of the band; wakes form next to the band boundaries, in between and beside two adjacent primary vortices. Force chain buckling, the governing mechanism for shear bands, is responsible for vortex formation in the failure regime. Vortex dynamics are consistent with stick-slip dynamics. From quiescent conditions of jamming or stick, vortical motions arise from force chain buckling and associated relative particle rotations and sliding; these in turn precipitate intermittent periods of unjamming or slip, evident in the attendant drops in stress ratio and bursts in both kinetic energy and local nonaffine deformation. A kinematic vortex field inside shear bands is proposed that is consistent with the equations of continuum mechanics and the underlying instability of force chain buckling: such a field is periodic with a repeating unit cell comprising a primary vortex at the center of the band, with two trailing wakes close next to the band boundaries.

Null warped AdS in higher spin gravity

We equip three-dimensional spin-3 gravity in the principal embedding with a new set of boundary conditions that we call "asymptotically null warped AdS". We find a chiral copy of the Polyakov-Bershadsky algebra as asymptotic symmetry algebra, reminiscent of the situation in topologically massive gravity with strict null warped AdS boundary conditions. We prove the invertibility of the map between zuvielbein and metric variables and construct a global gauge transformation to half of AdS spin-3 gravity in the diagonal embedding. This explains why the theory is chiral and why the Polyakov-Bershadsky algebra arises. We then introduce chemical potentials, derive the entropy, free energy, and the holographic response functions, and conclude with a discussion.

An obstruction to small time local null controllability for a viscous Burgers' equation

In this work, we are interested in the small time local null controllability for the viscous Burgers' equation $y_t - y_{xx} + y y_x = u(t)$ on the line segment $[0,1]$, with null boundary conditions. The second-hand side is a scalar control playing a role similar to that of a pressure. In this setting, the classical Lie bracket necessary condition $[f_1,[f_1,f_0]]$ introduced by Sussmann fails to conclude. However, using a quadratic expansion of our system, we exhibit a second order obstruction to small time local null controllability. This obstruction holds although the information propagation speed is infinite for the Burgers equation. Our obstruction involves the weak $H^{-5/4}$ norm of the control $u$. The proof requires the careful derivation of an integral kernel operator and the estimation of residues by means of weakly singular integral operator estimates.

Trend of the water content profile in a homogeneous soil layer from analytical solutions

Summary The linearized one-dimensional Richards equation is solved analytically in a horizontal, homogeneous soil layer of finite thickness. The obtained solution is the soil water content at any required time and depth in the layer. Any discrete soil water content profile (e.g. experimentally measured) can be assumed as initial condition; the boundary conditions are two arbitrary functions representing the time evolution of the soil volumetric water content. Both initial and boundary conditions are approximated by means of a suitable number of step functions; therefore the solution presented will be hereafter called Step Function Solution (SFS). Making use of the variables separation method and of the superposition principle, the general solution is obtained by the sum of two solutions: one is derived for null boundary conditions and an arbitrary initial condition; the other is derived for a null initial condition and two arbitrary boundary conditions. The instantaneous fluxes at the top and at the bottom of the layer are calculated. From the time integration of these instantaneous fluxes the cumulative ones and the water gained by the soil layer in a specified time interval are obtained. These hydrological fields are relevant parameters for many studies. The stationary solution and the value of the corresponding flux are also calculated. Finally, the SFS is compared with two analytical solutions and two experimental sets of soil volumetric water content data. The finite thickness domain has been studied to represent a more realistic scheme of the surface soil with respect to the half space domain (Menziani et al., 2007). The comparison of the two sets of solutions (finite-thickness layer and half space schemes) with experimental data can help to decide when the depth of a layer can be assumed as infinite.

Reversibility of general 1D linear cellular automata over the binary field [formula omitted] under null boundary conditions

The reversibility of cellular automata(CA) has long been studied but there are still many problems untouched. This paper tackles the reversibility problem of the general case concerning 1D CA under null boundary conditions defined by linear rules over the binary field Z2. Although transition matrix has been widely used for some special linear CA(LCA) rules, it has severe limitations: its complexity depends on the number of cells, and determining its reversibility becomes another unsolved tough problem for general rules. By constructing deterministic finite automata(DFA) of the de Bruijn graph presentation of CA, we conclude the reversibility problem of all 1D linear rules over Z2 under null boundary conditions. It turns out that any 1D LCA, except for some unilateral rules, over Z2 under null boundary conditions can be reversible once having a proper number of cells, and DFA provides an efficient way to find those numbers. Moreover, the complexity of the DFA solution is independent of the number of cells. In addition, the relative low complexity of DFA can be further significantly reduced by using only a small part of it.

PERIODICITIES OF SOME ADDITIVE CELLULAR AUTOMATA

Abstract. We investigate periodicities of some additive cellular automatawith null boundary condition. 1. IntroductionCellular automata have been demonstrated by many researchers to be agood computational model for physical systems simulation since the concept ofcellular automata rst introduced by John Von Neumann in the 1950’s. Andresearchers have studied on cellular automata con gured with rules 51, 60, 102,153, 195 or 204 [1-8].In this note, we will investigate periodicities of some additive cellular au-tomata with null boundary condition.2. PreliminariesA cellular automaton (CA) is an array of sites (cells) where each site is inany one of the permissible states. At each discrete time step (clock cycle) theevolution of a site value depends on some rule (the combinational logic) which isa function of the present state of its k neighbors for a k-neighborhood CA. Fora 2-state 3-neighborhood CA, the evolution of the i th cell can be representedas a function of the present states of (i 1)

PERIODICITIES OF SOME HYBRID CELLULAR AUTOMATA WITH RULES 102 AND 60

Abstract. We investigate periodicities of some hybrid cellular au-tomata con gured with rule 102 and 60 and null boundary condi-tion. 1. IntroductionCellular automata have been demonstrated by many researchers tobe a good computational model for physical systems simulation sincethe concept of cellular automata rst introduced by John Von Neumannin the 1950’s. And researchers have studied on cellular automata con- gured with rules 51, 60, 102, 153, 195 or 204 [1-7].In this note, we will investigate periodicities of some hybrid cellularautomata con gured with rule 102 and 60 and null boundary condition.2. PreliminariesA cellular automaton (CA) is an array of sites (cells) where eachsite is in any one of the permissible states. At each discrete time step(clock cycle) the evolution of a site value depends on some rule (thecombinational logic) which is a function of the present states of its kneighbors for a k-neighborhood CA. For a 2-state 3-neighborhood CA,the evolution of the (i)th cell can be represented as a function of thepresent states of (i 1)th, (i)th, and (i + 1)th cells as: x

Some remarks on the asymptotic profiles of solutions for strongly damped wave equations on the 1-D half space

We consider a mixed problem with the Dirichlet null boundary conditions on the 1-dimensional half space (0,+∞). We will derive an asymptotic profiles of the corresponding solutions in the case when the initial data belong to a weighted L1,2(0,∞) space by employing a method recently introduced in [7] or [8].

Parabolic Equations of Infinite Order withL1Data

We prove an existence result of a nonlinear parabolic equation under Dirichlet null boundary conditions in Sobolev spaces of infinite order, where the second member belongs toL1QT.

A Cellular Automata based Optimal Edge Detection Technique using Twenty-Five Neighborhood Model

Cellular Automata (CA) are common and most simple models of parallel computations. Edge detection is one of the crucial task in image processing, especially in processing biological and medical images. CA can be successfully applied in image processing. This paper presents a new method for edge detection of binary images based on two dimensional twenty five neighborhood cellular automata. The method considers only linear rules of CA for extraction of edges under null boundary condition. The performance of this approach is compared with some existing edge detection techniques. This comparison shows that the proposed method to be very promising for edge detection of binary images. All the algorithms and results used in this paper are prepared in MATLAB.

Generalized eigenvalue problems of nonhomogeneous elliptic operators and their application

We consider the equation div.a.x;jruj/ru/D juj p 2 u (whose special case a.x; t/D t p 2 is the p-Laplace equation) on a bounded domainR N with C 2 boundary, with null boundary condition. We prove that there are 2 R for which the equation has a nontrivial solution. As an applica- tion, by variational methods, we present the existence of a positive solution to div.a.x;jruj/ru/D f.x; u/ in , where f is asymptotically . p 1/- linear near zero and1, considering the nonresonant, resonant, and dou- bly resonant cases. We show that, generally, the spectrum of the operator div.a.x;jruj/ru/ on W 1; p 0 ./ is not discrete.

The infected frontier in an SEIR epidemic model with infinite delay

An SEIR epidemic model with infinite delay and the Neumannboundary condition is investigated, as well as the correspondingfree boundary problem, in which the free boundary exactly describesthe spreading frontier of the disease. For the problem in a fixed domain with null Neumann boundary condition, thetransmission dynamics of the disease is determined by the basicreproduction number $R_0$. More specifically, whether the diseasewill die out or not depends on $R_01$; while for the free boundary problem, we show thatunder certain conditions the disease will die out even $R_0>1$. Our results indicate that besides the basic reproductionnumber, the initial size of the infected domain and the diffusivityof the disease in a new region also produce a non-negligibleinfluence to the disease transmission, and it seems more reasonableand acceptable.

The inverse problem of seismic fault determination using part time measurements

This paper shows how a part time measurement T of the acceleration on the ground can be done to avoid the consideration of reflections of seismic waves on the Moho discontinuity. A bounded hemispherical solid is considered with radius greater than c pT , but smaller than the Moho depth, in order to get the null boundary conditions on the hemispherical surface. The inverse problem to determine the fault plane and the time-dependent fault geometry is well-defined and solved in closed form by the reciprocity gap functional method. Two formulae are established for the components of the fault slip, which involve time and spatial Fourier transforms of some quantities related to the reciprocity gap functional. A forward initial boundary value problem in elastodynamics enables us to get the internal wave field, the shear stress as well as the normal stress on the fault. This full solution would be useful for understanding the friction mechanism on the fault during a real strike, whenever the initial uniform tectonic stresses are known from geophysical considerations.

On the invertible cellular automata 150 over [formula omitted

In this work the reversibility problem for cellular automata with rule number 150 over the finite field Fp is tackled. It is shown that when null boundary conditions are stated, the reversibility is independent of the state set Fp and it appears when the number of cells of the cellular space satisfies some conditions. Furthermore, the explicit expressions of the inverse cellular automata are computed.

On the resolvent arising in a parameter‐elliptic multi‐order boundary problem

AbstractIn this paper we consider a boundary problem for a parameter‐elliptic, multi‐order system of differential equations defined over a bounded region in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^n$\end{document} and under limited smoothness assumptions as well as under boundary conditions which include those of Dirichlet. Information is then derived concerning the asymptotic behaviour of the trace of a power of the resolvent of the Hilbert space operator, in general non‐selfadjoint, induced by the boundary problem under null boundary conditions. This information will then be used in a subsequent work to derive various results pertaining to the asymptotic behaviour of the eigenvalues of this operator.

Eigenvalue Asymptotics for the Non-Selfadjoint Operator Induced by a Parameter-Elliptic Multi-Order Boundary Problem

In this paper we consider a boundary problem for a parameter-elliptic, multi-order system of differential equations defined over a bounded region in \({\mathbb{R}^n}\) and under Dirichlet boundary conditions. In addition, the problem is considered under limited smoothness assumptions. Information is then derived concerning the asymptotic behaviour of the eigenvalues of the Hilbert space operator, in general non-selfadjoint, induced by the boundary problem under null boundary conditions.