A sixth-order extrapolation method for special nonlinear fourth-order boundary value problems

A sixth-order convergent finite difference method is developed for the numerical solution of the special nonlinear fourth-order boundary value problem y^(iv)(x) = f(x, y), a < x < b, y(a) = A₀, y″(a) = B₀, y(b) = A₁ y′(b) = B₁, the simple-simple beam problem.The method is based on a second-order convergent method which is used on three grids, sixth-order convergence being obtained by taking a linear combination of the (second-order) numerical results calculated using the three individual grids.Special formulas are proposed for application to points of the discretization adjacent to the boundaries x = a and x= b, the first two terms of the local truncation errors of these formulas being the same as those of the second-order method used at the other points of each grid.Modifications to these two formulas are obtained for problems with boundary conditions of the form y(a) = A₀, y′(a) = C₀, y(b) = A₁, y′(b) = C₁, the clamped-clamped beam problem.The general boundary value problem, for which the differential equation is y^(iv)(x) = f(x, y, y′, y″, y‴), is also considered.     

A numerical method for diffusion-convection equation using high-order difference schemes

In this paper, we propose a simple general form of high-order approximation of O(c²+ch²+h⁴) to solve the two-dimensional parabolic equation αuxx+βuyy=F(x,y,t,u,ux,uy,ut), where α and β are positive constants. We apply the compact form for solving diffusion-convection equation. The results of numerical experiments are presented and compared with analytical solutions to confirm the higher accuracy of the presented scheme.     

Inapproximability of the Tutte polynomial of a planar graph

The Tutte polynomial of a graph G is a two-variable polynomial T(G; x, y) that encodes many interesting properties of the graph. We study the complexity of the following problem, for rationals x and y: given as input a planar graph G, determine T(G; x, y). Vertigan completely mapped the complexity of exactly computing the Tutte polynomial of a planar graph. He showed that the problem can be solved in polynomial time if (x, y) is on the hyperbola H _q given by (x ? 1)(y ? 1) = q for q = 1 or q = 2 or if (x, y) is one of the two special points (x, y) = (?1, ?1) or (x, y) = (1, 1). Otherwise, the problem is #P-hard. In this paper, we consider the problem of approximating T(G; x, y), in the usual sense of “fully polynomial randomized approximation scheme” or FPRAS. Roughly speaking, an FPRAS is required to produce, in polynomial time and with high probability, an answer that has small relative error. Assuming that NP is different from RP, we show that there is no FPRAS for the Tutte polynomial in a large portion of the (x, y) plane. In particular, there is no FPRAS if x > 1, y < ?1 or if y > 1, x < ?1 or if x < 0, y < 0 and q > 5. Also, there is no FPRAS if x < 1, y < 1 and q = 3. For q > 5, our result is intriguing because it shows that there is no FPRAS at (x, y) =?(1 ? q/(1 + ε), ?ε) for any positive ε but it leaves open the limit point ε =?0, which corresponds to approximately counting q-colorings of a planar graph.     

On the entropy function of degree β

Kamiński and Mikusiński (1974) formed a functional equation H(x,y,z) = H(x + y,0,z) + H(x,y,0), x ? 0, y ? 0, z ? 0, xy + yz + zx > 0, and determined its continuous and symmetric solution which is homogeneous of degree 1. This approach simplifies Fadeev's well-known characterization of Shannon's entropy. In this paper we obtain this functional equation from a generalized set of axioms and solve it under continuity and homogeneity of degree β (β ≠ 1, β > 0). This extends Kamiński and Mikusiński's approach of characterizing to degree β the measure of entropy, which has been studied by a number of workers with considerable interest.     

Solutions to the functional equation I(x, y) = I(x, I(x, y)) for three types of fuzzy implications derived from uninorms

This paper investigates an iterative Boolean-like law with fuzzy implications derived from uninorms. More precisely, we characterize the solutions to the functional equation I(x, y) = I(x, I(x, y)) that involve RU-, (U, N)- and QLU-implications generated by the most usual classes of uninorms.     

Torsion problem for elastic cylinder with inserts and holes

An integral equation method to solve the classical torsion problem for an elastic cylinder with inserts and holes is treated. The bounded region outside the inserts and the holes will be termed a matrix. As is well-known the solution depends on finding plane harmonic functions in the matrix and inserts such that (a) on the outer boundary of the matrix and the boundaries of the holes the harmonic function in the matrix takes the values $\text{1}\text{2}\text{(x}{}^2\text{+y}{}^2\text{)+c}{}_{\mathrm{j}}$, and (b) on the interfaces of the matrix and the inserts relations exist between the harmonic functions and between their normal derivatives. Here (x, y) are the coordinates of the point on the boundary and c_j, are unknown constants. The usual methods are cumbersome and lengthy. In this paper a straightforward method is presented which is easily programmable. The numerical solution is obtained by evaluating a few integrals either analytically or numerically and solving a system of linear simultaneous equations. An example of a cylinder with an eccentric insert is given which substantiates the theory developed in this paper and is found to agree with known results. However, the method is general and may be applied to a variety of problems.     

Computing monodromy via continuation methods on random Riemann surfaces

We consider a Riemann surface X defined by a polynomial f(x,y) of degree d, whose coefficients are chosen randomly. Hence, we can suppose that X is smooth, that the discriminant δ(x) of f has d(d−1) simple roots, Δ, and that δ(0)≠0, i.e. the corresponding fiber has d distinct points {y₁,…,y_d}. When we lift a loop 0∈γ⊂C−Δ by a continuation method, we get d paths in X connecting {y₁,…,y_d}, hence defining a permutation of that set. This is called monodromy.Here we present experimentations in Maple to get statistics on the distribution of transpositions corresponding to loops around each point of Δ. Multiplying families of “neighbor” transpositions, we construct permutations and the subgroups of the symmetric group they generate. This allows us to establish and study experimentally two conjectures on the distribution of these transpositions and on transitivity of the generated subgroups.Assuming that these two conjectures are true, we develop tools allowing fast probabilistic algorithms for absolute multivariate polynomial factorization, under the hypothesis that the factors behave like random polynomials whose coefficients follow uniform distributions.     

Filter theory of BL algebras

In this paper we consider fundamental properties of some types of filters (Boolean, positive implicative, implicative and fantastic filters) of BL algebras defined in Haveshki et al. (Soft Comput 10:657–664, 2006) and Turunen (Arch Math Logic 40:467–473, 2001). It is proved in Haveshki et al. (2006) that if F is a maximal and (positive) implicative filter then it is a Boolean filter. In that paper there is an open problem Under what condition are Boolean filters positive implicative filters? One of our results gives an answer to the problem, that is, we need no more conditions. Moreover, we give simple characterizations of those filters by an identity form ? x, y(t(x, y) ∈ F), where t(x, y) is a term containing x, y.      

TRIANGULATOR: Excel spreadsheets for converting relative bearings to XYZ coordinates, with applications to scaling photographs and orienting surfaces

TRIANGULATOR comprises two Microsoft Excel spreadsheets for processing relative bearing data from an electronic total station. Program XYZ converts bearings to positions. It determines x, y, z coordinates of points from relative bearings from two base stations of known relative position. Program LINES uses the output of XYZ. It converts the x, y, z coordinates of three points into the equation of a plane, yielding strike and dip of that plane. Then it uses bearings from one base station to points on a cliff face, and calculates their x′, z′ coordinates for either direct measurement of features, or determination of the scale of a photograph. An example demonstrates the application of the program to scale the photograph of an exfoliation fracture.     

Efficient computation of the cylindrical Bessel functions of complex argument

An algorithm that generates the cylindrical Bessel function very accurately for a wide range of complex arguments has been developed by Mason. The Mason algorithm consists of four different methods that apply to different portions of the complex plane. Experience with the Floating Point Systems FPS-364 minisupercomputer indicates several ways by which these methods can be made more efficient. Specific improvements relate to: 1) the method for determination of the point where backward recursion is initiated for the Bessel functions of the first kind; 2) the way that the Bessel functions of the first and second kind are normalized when |y| < 5 and |x| ⩽ 20; and 3) the extent that asymptotic expansions are used when |x| > 20 and |y| < 5. The first and third modifications will result in increased efficiency for all architectures. The second modification will be of value for many, but probably not all, architectures.     

On a family of copulas constructed from the diagonal section

We characterize the class of copulas that can be constructed from the diagonal section by means of the functional equation C(x,y)+|x−y|=C(x∨y,x∨y), for all (x,y) in the unit square such that C(x,y)>0. Some statistical properties of this class are given.     

Embedding hamiltonian paths in hypercubes with a required vertex in a fixed position

Assume that n is a positive integer with n?2. It is proved that between any two different vertices x and y of Qn there exists a path Pl(x,y) of length l for any l with h(x,y)?l?n²−1 and 2|(l−h(x,y)). We expect such path Pl(x,y) can be further extended by including the vertices not in Pl(x,y) into a hamiltonian path from x to a fixed vertex z or a hamiltonian cycle. In this paper, we prove that for any two vertices x and z from different partite set of n-dimensional hypercube Qn, for any vertex y∈V(Qn)−{x,z}, and for any integer l with h(x,y)?l?n²−1−h(y,z) and 2|(l−h(x,y)), there exists a hamiltonian path R(x,y,z;l) from x to z such that dR(x,y,z;l)(x,y)=l. Moreover, for any two distinct vertices x and y of Qn and for any integer l with h(x,y)?l?²n−1 and 2|(l−h(x,y)), there exists a hamiltonian cycle S(x,y;l) such that dS(x,y;l)(x,y)=l.     

Curve generation of implicit functions by incremental computers

The conventional numerical solution of an implicit function f(x, y) = 0 is substantially complicated for calculating by any computer. We propose a new method representing the argument of the implicit function as a unary function of a parameter, t, if the continuous and unique solution of f(x, y) = 0 exists. The total differential $\text{d}\text{f}\text{d}\text{t}$ constitutes simultaneous differential equations of which the solution about x and y is unique. The Newton-Raphson method must be used to calculate the values near singular points of an implicit function and then the sign of dt has to be decided according to four special cases. Incremental computers are suitable for curve generation of implicit functions by the new method, because the incremental computer can perform more complex algorithms than the analog computer and can calculate faster than the digital computer. This method is easily applicable to curve generation in three-dimensional space.     

Substitutions into propositional tautologies

We prove that there is a polynomial time substitution (y₁,…,yn):=g(x₁,…,xk) with k?n such that whenever the substitution instance A(g(x₁,…,xk)) of a 3DNF formula A(y₁,…,yn) has a short resolution proof it follows that A(y₁,…,yn) is a tautology. The qualification “short” depends on the parameters k and n.     

An explicit two-step method exact for the scalar test equation y′ = λy

An explicit two-step method exact for the scalar test equation y′ = λy, Re(λ) < 0 is presented in this paper. It is exponentially fitted, L-stable (thus, A-stable), and of order 2. With a new set of vector computations, we also extend directly the method to systems of ordinary differential equations. The numerical experiments demonstrate that this explicit two-step method is suitable for stiff systems.     

Comparing two independent samples using the risk difference under inverse sampling

Tang and Tian (2009) derived formulas for the development of confidence bounds for the risk difference under inverse sampling. Unfortunately, the last formula in the appendix for the inverse of the expected Fisher Information matrix I⁻¹ contains an error which leads-if used-to incorrect confidence bounds: I⁻¹ does not remain invariant if (x₀,y₀) is exchanged with (x₁,y₁) and, simultaneously, Δ is exchanged with −Δ. In the following I will present an idea that corrects this error and that thereby leads to a considerable simplification of the formulas needed for the calculation of the confidence bounds. Furthermore I point out an existing alternative for the calculation of the zeros of a cubic equation that may be of interest.     

Computing Bessel functions of the second kind in extreme parameter regimes

A useful method of computing the integral order Bessel functions of the second kind Y_n(x+iy) when either, the absolute value of the real part, or the imaginary part of the argument z=x+iy is small, is described. This method is based on computing the Bessel functions for extreme parameter regimes when x∼0 (or y∼0) and is useful because a number existing algorithms and methods fail to give correct results for small x or small y. The approximating equations are derived by expanding the Bessel function in Taylor series, are tested and discussed. The present work is a continuation of the previous one conducted in regard to the Bessel function of the first kind. The results of our formalism are compared to the available existing numerical methods used in Mathematica, IMSL, MATLAB, and the Amos library. Our numerical method is easy to implement, efficient, and produces reliable results. In addition, this method reduces the computation of the Bessel functions of the second complex argument to that of real argument which simplify the computation considerably.     

Fast error estimates for indirect measurements: applications to pavement engineering

In many real-life situations, we have the following problem: we want to know the value of some characteristicy that is difficult to measure directly (e.g., lifetime of a pavement, efficiency of an engine, etc.). To estimatey, we must know the relationship betweeny and some directly measurable physical quantitiesx ₁,...,x _n . From this relationship, we extract an algorithmf that allows us, givenx _i , to computey: y=f(x ₁, ...,x _n ). So, we measurex _i , apply an algorithmf, and get the desired estimate. Existing algorithms for error estimate (interval mathematics, Monte-Carlo methods, numerical differentiation, etc.) require computation time that is several times larger than the time necessary to computey=f(x ₁, ...,x _n ). So, if an algorithmf is already time-consuming, error estimates will take too long. In many cases, this algorithmf consists of two parts: first, we usex _i to determine the parametersz _k of a model that describes the measured object, and second, we use these parameters to estimatey. The most time-consuming part is findingz _k ; this is done by solving a system of non-linear equations; usually least squares method is used. We show that for suchf, one can estimate errors repeating this time-consuming part off only once. So, we can compute bothy and an error estimate fory with practically no increase in total computation time. As an example of this methodology, we give pavement lifetime estimates.     

Geodesic methods in quantitative image analysis

Let X be a part of an image to be analysed. Given two arbitrary points x and y of X, let us define the number d_x(x, y) as follows: d_x(x, y) is the lower bound of the lengths of the arcs in X ending at points x and y, if such arcs exist, and + α if not. The function d_x is an X-intrinsic distance function, called ‘geodesic distance’. Note that if x and y belong to two disjoint connected components of X, d_x(x, y) = + α. In other words, d_x seems to be an appropriate distance function to deal with connectivity problems.In the metric space (X, d_x), all the classical morphological transformations (dilation, erosion, skeletonization, etc.) can be defined. The geodesic distance d_x also provides rigorous definitions of topological transformations, which can be performed by automatic image analysers with the help of parallel iterative algorithms.All these notions are illustrated by several examples (definition of the length of a fibre and of an effective length factor; automatic detection of cells having at least one nucleus or having one single nucleus; definitions of the geodesic center and of the ends of an object without a hole; etc.). The corresponding algorithms are described.