A decision procedure for the order of regular events

A regular event R is said to be of finite order if R^k=R^k+1 for some nonnegative integer k. It is shown that there exists an algorithm for deciding whether an arbitrary regular events is of finite order or not.     

Finite difference discretization of the extended Fisher-Kolmogorov equation in two dimensions

Approximate solutions are considered for the extended Fisher-Kolmogorov (EFK) equation in two space dimension with Dirichlet boundary conditions by a Crank-Nicolson type finite difference scheme. A priori bounds are proved using Lyapunov functional. Further, existence, uniqueness and convergence of difference solutions with order O(h²+k²) in the L^∞-norm are proved. Numerical results are also given in order to check the properties of analytical solutions.     

The construction of a priori bounds for the solution of a two point boundary value problem with finite elements I

Let û be the solution of a boundary value problem for an ordinary differential equation of the second order. Function boundsv andw are constructed to û such thatv ≦ û ≦w. From this other bounds are derived for the derivatives û′ and û″. To this end a collocation method with finite elements is used. The inclusion property is proven with the aid of theorems on differential inequalities. Leth be the maximal step size and letk be an arbitrary natural number. Then the accuracy can be made to have arbitrarily high order such thatw?v=C(h ^2k ).     

On some subclasses of the class of generable languages

A [k]-machine is a quadruple M = (W, V, k, δ), where W ≠ Ø is a finite set, V ? W, k∈ {1, 2,?}, δ: V^k → (2W?{Ø}). A [k]-machine is of the type α (β) if δ: V^k → V (δ: V^k → (2V?{Ø})). The language of M is the set of all words (sequences of elements of V) “generated” by M. In the paper the languages of [k]-machines of type α and β are studied and their relations to the sets of computations of machines as mentioned in references are discussed. For a fixed finite set A the number of languages E is given for which E ? A^∞ (of sequences of elements of A) generated by [k]-machines of type α and β.     

Divergence-Free HDG Methods for the Vorticity-Velocity Formulation of the Stokes Problem

We study a hybridizable discontinuous Galerkin method for solving the vorticity-velocity formulation of the Stokes equations in three-space dimensions. We show how to hybridize the method to avoid the construction of the divergence-free approximate velocity spaces, recover an approximation for the pressure and implement the method efficiently. We prove that, when all the unknowns use polynomials of degree k??0, the L ² norm of the errors in the approximate vorticity and pressure converge with order k+1/2 and the error in the approximate velocity converges with order k+1. We achieve this by letting the normal stabilization function go to infinity in the error estimates previously obtained for a hybridizable discontinuous Galerkin method.     

Pressure projection stabilized finite element method for Navier–Stokes equations with nonlinear slip boundary conditions

In this paper, we consider the pressure projection stabilized finite element method for the Navier–Stokes equation with nonlinear slip boundary conditions whose variational formulation is the variational inequality problem of the second kind with Navier–Stokes operator. The H ¹ and L ² error estimates for the velocity and the L ² error estimate for the pressure are obtained. Finally, the numerical results are displayed to verify the theoretical analysis.     

A New Nonsymmetric Discontinuous Galerkin Method for Time Dependent Convection Diffusion Equations

We propose a discontinuous Galerkin finite element method for convection diffusion equations that involves a new methodology handling the diffusion term. Test function derivative numerical flux term is introduced in the scheme formulation to balance the solution derivative numerical flux term. The scheme has a nonsymmetric structure. For general nonlinear diffusion equations, nonlinear stability of the numerical solution is obtained. Optimal kth order error estimate under energy norm is proved for linear diffusion problems with piecewise P ^k polynomial approximations. Numerical examples under one-dimensional and two-dimensional settings are carried out. Optimal (k+1)th order of accuracy with P ^k polynomial approximations is obtained on uniform and nonuniform meshes. Compared to the Baumann-Oden method and the NIPG method, the optimal convergence is recovered for even order P ^k polynomial approximations.     

Connected sum of digital closed surfaces

During the past 20 years the research of digital surfaces has proceeded to find their properties in the digital space Zⁿ, such as a topological number, a simple k-point, the 3D-Jordan theorem, a k-separating set, a boundary detecting algorithm and so on. Actually, unlike surfaces in a continuous space, the features of digital surfaces have different characteristics. The aim of this paper is to introduce the notion of a digital closed k-surface in Zⁿ, n ? 3, with the general k-adjacency relations as a generalization of Malgouyres’ and Morgenthaler’s k-surfaces in Z³, to establish some minimal simple closed k-surfaces in Z³ and to find their digital topological properties in relation with the k-fundamental group and k-contractibility. Moreover, a connected sum of two digital closed surfaces is introduced and its digital topological properties are investigated.     

A linear-time algorithm to construct a rectilinear Steiner minimal tree fork-extremal point sets

Ak-extremal point set is a point set on the boundary of ak-sided rectilinear convex hull. Given ak-extremal point set of sizen, we present an algorithm that computes a rectilinear Steiner minimal tree in timeO(k ⁴ n). For constantk, this algorithm runs inO(n) time and is asymptotically optimal and, for arbitraryk, the algorithm is the fastest known for this problem.     

Erzeugende Funktionen bei Spline-Interpolation mit äquidistanten Knoten

Using generating functions we obtain in the case ofn+1 equidistant data points a method for the calculation of the interpolating spline functions(x) of degree 2k+1 with boundary conditionss ^(κ) (x₀)=y ₀ ^(κ) ,s ^(κ) (x _n )=y _n ^(κ) , κ=1(1)k, which only needs the inversion of a matrix of orderk. The applicability of our method in the case of general boundary conditions is also mentioned.     

Tension spline approach for the numerical solution of nonlinear Klein-Gordon equation

The nonlinear Klein-Gordon equation describes a variety of physical phenomena such as dislocations, ferroelectric and ferromagnetic domain walls, DNA dynamics, and Josephson junctions. We derive approximate expressions for the dispersion relation of the nonlinear Klein-Gordon equation in the case of strong nonlinearities using a method based on the tension spline function and finite difference approximations. The resulting spline difference schemes are analyzed for local truncation error, stability and convergence. It has been shown that by suitably choosing the parameters, we can obtain two schemes of O(k²+k²h²+h²) and O(k²+k²h²+h⁴). In the end, some numerical examples are provided to demonstrate the effectiveness of the proposed schemes.     

On linear interpolation under interval data

Some results related to the problem of interpolation of n vertical segments (x_k, Y_k), k = 1,…,n, in the plane with generalized polynomial functions that are linear combinations of m basic functions are presented. It is proved that the set of interpolating functions (if not empty) is bounded in every subinterval (x_k, x_k+1) by two unique such functions η_k⁻ and η_k⁺. An algorithm with result verification for the determination of the boundary functions η_k⁻, η_k⁺ and for their effective tabulation is reported and some examples are discussed.     

An adaptive method for the determination of the order of element for acquiring the desired accuracy in 2D elastostatic FEM analysis

An adaptive method for the determination of the order of element (or element order) was developed for the finite element analysis of 2D elastostatic problems. Here, the order of element means the order of the polynomial function that interpolates the displacement distribution in the element. This method was based on acquiring the desired accuracy for each finite element. From the numerical experiments, the relationship ξ=k(1/p)^β was deduced, where ξ is the error of the result of the finite element analysis relative to the exact value, p is the order of element, and k and β are constants. Applying this relationship to the two results of computations with different orders of element, the order of element for the third analysis was deduced. A computer program using this adaptive determination method for the order of element was developed and applied to several 2D elastostatic problems of various shapes. The usefulness of the method was illustrated by these application results.     

拟（h,k）存贮有限自动机的可逆性

主要研究拟（h,k）阶存贮有限自动机的延迟k步与k＋1步弱可逆性,以及它的弱逆,得到了拟（h,k）阶存贮有限自动机的延迟k步与k＋1步弱可逆的充分必要条件,并且通过所得结果可以比较简便地构造出延迟k步与k＋1步弱可逆拟（h,k）阶存贮有限自动机的延迟k步与k＋1步弱逆。     

Infinite cube-connected cycles

An infinite network for parallel computation is presented which can for every k become partitioned in cube-connected cycles-networks of size 2^2k2^k [1]. This construction extends a result from [2], where finite such networks are constructed. This infinite network is useful for simplifying the structure and improving the efficiency of the general purpose parallel computer shown in [3].     

Analysis of Agglomerative Clustering

The diameter k-clustering problem is the problem of partitioning a finite subset of ? ^d into k subsets called clusters such that the maximum diameter of the clusters is minimized. One early clustering algorithm that computes a hierarchy of approximate solutions to this problem (for all values of k) is the agglomerative clustering algorithm with the complete linkage strategy. For decades, this algorithm has been widely used by practitioners. However, it is not well studied theoretically. In this paper, we analyze the agglomerative complete linkage clustering algorithm. Assuming that the dimension d is a constant, we show that for any k the solution computed by this algorithm is an O(logk)-approximation to the diameter k-clustering problem. Our analysis does not only hold for the Euclidean distance but for any metric that is based on a norm. Furthermore, we analyze the closely related k-center and discrete k-center problem. For the corresponding agglomerative algorithms, we deduce an approximation factor of O(logk) as well.     

Anelastic Stokes Eigenmodes in Infinite Channel

The anelastic Stokes eigenmodes are computed for a fluid confined, in presence of gravity, between two horizontally infinite plates. These eigenmodes are described by one horizontal wave number k. The eigenvalues λ(k ²) are proved to be all negative. They depend monotonically upon k, behaving like k ² for very large k. Two particular values of k are considered, i.e., k=2?π and k=0, and the stratification parameter of the equilibrium state is taken between 0 (incompressible approximation) and 10 (upper limit of the anelastic configuration). The k=2?π eigenvalue problem is solved numerically while the k=0 is solved both numerically and analytically. Two physical configurations are analyzed, one with no-slip boundary conditions imposed on both horizontal walls, and one with no-stress, while imposing no flow through these boundaries in both cases. The main results are: (i) the smaller the stratification, the larger the decay rate, (ii) the eigenmodes are localized in the lower part of the channel, their vertical extension increasing with the eigenmode spatial frequency, (iii) the Neumann eigenmode decay rates are smaller than their Dirichlet counterparts, except for k=0, where it is just the reverse, (iv) a general trend seems to emerge from the present study, regarding the way the numerical eigenvalues of an elliptic operator compare with the analytical ones, viz., the numerical spectrum overestimates (in absolute value) the analytical spectrum, slightly in the low frequency part of the spectrum and more and more strongly in the upper part.     

A delay optimal coterie on the k-dimensional folded Petersen graph

Coteries are an effective tool for enforcing mutual exclusion in distributed systems. Communication delay is an important metric to measure the performance of a coterie. The topology of the interconnection network in a distributed system also has an impact on its performance. The k-dimensional folded Petersen graph, a graph with 10^k nodes and diameter 2k, qualifies as a good network topology for large distributed systems. In this paper, we present a delay optimal coterie on the k-dimensional folded Petersen graph, FP_k. For any positive integer k, the coterie has message complexity 4^k and delay k. Moreover, this coterie is not vote-assignable.     

A numerical computation of a 3-D stokes problem in the description of intestinal peristaltic waves

The mathematical aspects of a physical model of intestinal peristaltic waves involve numerical methods for calculating and simulating the solution. A finite element method is applied to Stokes' equations in R³ in order to calculate the velocity-pressure couple at each vertex of the pentahedron elements of the geometrical model domain, for viscous and non-compressible fluid. By using special computational programs, the domain may be stitched to create a grid domain in R³. The finite reference element of the grid is a pentahedron. The resultant linear system may be solved by applying Gauss' direct method to obtain velocities and pressures at each grid point. Graphical representations of velocity profiles in R³, show positive and negative zones for the output, with positions that vary from one geometrical model to another. The calculated velocity and pressure values are shown to change according to the applied vector efforts. The numerical results obtained are in good agreement with experimental data. This suggests that the finite element method is useful for solving Stokes' equations to describe intestinal peristalsis waves.