The iterated defect correction methods for singular two-point boundary value problems

In this paper, the iterated defect correction (IDeC) techniques based on the centered Euler method for the equivalent first order system of the singular two-point boundary value problem in linear case (x ^α y′(x))′ = f(x), y(0) = a,y(1) = b, where 0 < α < 1 are considered. By using the asymptotic expansion of the global error, it is analyzed that the IDeC methods improved the approximate results by means of IDeC steps and the degree of the interpolating polynomials used. Some numerical examples from the literature are given in illustration of this theory.     

Positive solutions to superlinear semipositone periodic boundary value problems with repulsive weak singular forces

This paper is devoted to study the existence of positive solutions to the second-order semipositone periodic boundary value problem x″ + a(t)x = f(t,x), x(0) = x(1), x′(0) = x′(1). Here, f (t, x) may be singular at x = 0 and may be superlinear at x = +∞. Our analysis relies on a fixed-point theorem in cones.     

Positive solutions of singular Dirichlet and periodic boundary value problems

Let A and T be positive numbers. The singular differential equation (r(x)x′)′ = μq(t)f(t, x) is considered. Here r > 0 on (0, A] may be singular at x = 0, and f(t, x) ≤ 0 may be singular at x = 0 and x = A. Effective sufficient conditions imposed on r, μ, q, and f are given for the existence of a solution x to the above equation satisfying either the Dirichlet conditions x(0) = x(T) = 0 or the periodic conditions x(0) = x(T), x′(0) = x′(T), and, in addition, 0 < x < A on (0, T).     

Exact solution of mixed problems for variable coefficient one-dimensional diffusion equation

This paper deals with diffusion problems modeled by the equation a(t)u_xx = u_t, x > 0, t > 0, u(x, 0) = c(x) together with the boundary condition u(0, t) = b(t) or u_x(0, t) = b(t). By using Fourier transforms, existence conditions and exact solutions of the above mixed problems are given.     

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.     

Parametric quintic spline solution of third-order boundary value problems

In this paper, we develop parametric quintic spline function to approximate the solution of third-order boundary value problems of the form u″′=f(x, u), a≤x≤b, subject to the boundary conditions u(a)=k ₁, u′(a)=k ₂ and u(b)=k ₃. The class of methods are second-, fourth- and sixth-order accurate. End equations of the splines are derived and truncation error is given. Three numerical examples are presented to illustrate the practical use of our methods as well as their accuracies when compared with some existing spline function methods. It is shown that the new methods give approximations, which are better than those produced by other methods.     

Approximate Solutions of Polynomial Equations

In this paper, we introduce “approximate solutions" to solve the following problem: given a polynomial F(x, y) over Q, where x represents an n -tuple of variables, can we find all the polynomials G(x) such that F(x, G(x)) is identically equal to a constant c in Q ? We have the following: let F(x, y) be a polynomial over Q and the degree of y in F(x, y) be n. Either there is a unique polynomial g(x)  ∈ Q [ x ], with its constant term equal to 0, such that F(x, y)  = ∑_j = 0ⁿc_j(y − g(x))^jfor some rational numbers c_j, hence, F(x, g(x)  + a)  ∈ Q for all a ∈ Q, or there are at most t distinct polynomials g₁(x),⋯ , g_t(x), t ≤ n, such that F(x, g_i(x))  ∈ Q for 1  ≤ i ≤ t. Suppose that F(x, y) is a polynomial of two variables. The polynomial g(x) for the first case, or g₁(x),⋯ , g_t(x) for the second case, are approximate solutions of F(x, y), respectively. There is also a polynomial time algorithm to find all of these approximate solutions. We then use Kronecker’s substitution to solve the case of F(x, y).     

The interval-merging problem

A closed interval is an ordered pair of real numbers [x, y], with x ? y. The interval [x, y] represents the set {i ∈ R∣x ? i ? y}. Given a set of closed intervals I={[a₁,b₁],[a₂,b₂],…,[a_k,b_k]}, the Interval-Merging Problem is to find a minimum-cardinality set of intervals M(I)={[x₁,y₁],[x₂,y₂],…,[x_j,y_j]}, j ? k, such that the real numbers represented by equal those represented by . In this paper, we show the problem can be solved in O(d log d) sequential time, and in O(log d) parallel time using O(d) processors on an EREW PRAM, where d is the number of the endpoints of I. Moreover, if the input is given as a set of sorted endpoints, then the problem can be solved in O(d) sequential time, and in O(log d) parallel time using O(d/log d) processors on an EREW PRAM.     

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.     

Weighted aggregation operators based on minimization

Minimization based aggregation operators A_g,D are discussed. Special attention is paid to weighting function g based cases related to some fixed dissimilarity function D. When D₂(x,y)=(x-y)², we recognize mixture operators and we recall some sufficient conditions for g ensuring the monotonicity of A_g,D. For D₁(x,y)=|x-y| and non-decreasing (non-increasing) g, A_g,D is shown to be the upper (lower) median whenever A_g,D is an aggregation operator.     

Asymptotic equivalence between two difference systems

In this paper, we study the asymptotic equivalence between the linear system Δx(n) = A(n)x(n) and its perturbation Δy(n) = A(n)y(n)+g(n, y(n)) by using the comparison principle and supplementary projections. Furthermore, we establish some asymptotic properties for the nonlinear system Δx(n) = f(n, x(n)).     

New explicit filters and smoothers for diffusions with nonlinear drift and measurements

The optimal least-squares filtering of a diffusion x(t) from its noisy measurements {y(τ); 0 τ t} is given by the conditional mean E[x(t)|y(τ); 0 τ t]. When x(t) satisfies the stochastic diffusion equation dx(t) = f(x(t)) dt + dw(t) and y(t) = ∫₀^tx(s) ds + b(t), where f(·) is a global solution of the Riccati equation /xf(x) + f(x)² = f(x)² = αx² + βx + γ, for some , and w(·), b(·) are independent Brownian motions, Benes gave an explicit formula for computing the conditional mean. This paper extends Benes results to measurements y(t) = ∫₀^tx(s) ds + ∫₀^t dx(s) + b(t) (and its multidimensional version) without imposing additional conditions on f(·). Analogous results are also derived for the optimal least-squares smoothed estimate E[x(s)|y(τ); 0 τ t], s < t. The methodology relies on Girsanov's measure transformations, gauge transformations, function space integrations, Lie algebras, and the Duncan-Mortensen-Zakai equation.     

Numerische Lösung von Anfangswertaufgaben gewöhnlicher Differentialgleichungen n-ter Ordnung

In the present paper a new method is given for the numerical treatment of the initial problemsy ⁽ⁿ⁾=f(x,y,y′, ...,y ^(n?1),y ⁽ⁱ⁾ (x _o )=y _o ⁽ⁱ⁾ , i=0, 1, ...,n?1. This method is an one-step process of order four. For a class of linear differential equations the exact solution is obtained. Moreover some numerical results are presented.     

Sufficient conditions for the existence and asymptotic behaviour of solution to a quasi-linear elliptic problem

In this article, we combine the existing regularity theory, perturbation method and the lower and upper solutions method to study the existence and asymptotic behaviour of positive solution to a boundary value problem for the p-Laplacian operator. More exactly, we study the existence and asymptotic behaviour of the positive solution to a quasi-linear elliptic problem of the form?Δ _p u=λ a(x)g(u) in D′(Ω), u>0 in Ω, lim _{x→? Ω} u(x)=0. Under some conditions on a and g, we show that there is a non-negative number Λ₀ such that for all λ∈(0, Λ₀], the problem has a solution u _λ in the sense of distribution, which is bounded from above by some positive numbers μ(λ). Such estimates and the asymptotic behaviour are important in computer programs when we know an algorithm for determining the solution.     

Interpolation that leads to the narrowest intervals and its application to expert systems and intelligent control

In many real-life situations, we want to reconstruct the dependencyy=f(x ₁,…, x_n) from the known experimental resultsx _i ^(k) , y^(k). In other words, we want tointerpolate the functionf from its known valuesy ^(k)=f(x ₁ ^(k) ,…, x _n ^(k) ) in finitely many points $\bar x^{(k)} = (x_1^{(k)} , \ldots ,x_n^{(k)} )$ , 1≤k≤N There are many functions that go through given points. How to choose one of them? The main goal of findingf is to be able to predicty based onx _i. If we getx _i from measurements, then usually, we only getintervals that containx _i. As a result of applyingf, we get an interval y of possible values ofy. It is reasonable to choosef for which the resulting interval is the narrowest possible. In this paper, we formulate this choice problem in mathematical terms, solve the corresponding problem for several simple cases, and describe the application of these solutions to intelligent control.     

Truncations of infinite matrices and algebraic series associated with some of grammars

We consider an algebraic system over $\text{R}$[x] of the form X = a₀(x)X^k+ a_k1(x)X+a_k(x), where a₀(x) and a_k(x) are in x$\text{R}$[x] and a_k?1(x) is in x$\text{R}$. Let A be the infinite incidence matrix associated with the algebraic system. Then we prove that the eigenvalues of northwest corner truncations of A are dense in some algebraic curves.Using this we get a result on positive algebraic series. We consider the case that the coefficients of a₁(x)(i = 0,…,k?1, k) are positive. The algebraic series generated by the algebraic system may be viewed as a function in the complex variable x. Then by the above fact we prove that the radius of convergence of the function equals the least positive zero of the modified discriminant of the system.As an application to context free languages we show a procedure for calculating the entropy of some one counter languages. Other applications to Dyck languages and the Lukasiewicz language are also described.     

Approximation algorithms for covering/packing integer programs

Given matrices A and B and vectors a, b, c and d, all with non-negative entries, we consider the problem of computing . We give a bicriteria-approximation algorithm that, given ε∈(0,1], finds a solution of cost O(ln(m)/ε²) times optimal, meeting the covering constraints (Ax?a) and multiplicity constraints (x?d), and satisfying Bx?(1+ε)b+β, where β is the vector of row sums β_i=∑_jB_ij. Here m denotes the number of rows of A.This gives an O(lnm)-approximation algorithm for CIP—minimum-cost covering integer programs with multiplicity constraints, i.e., the special case when there are no packing constraints Bx?b. The previous best approximation ratio has been O(ln(max_j∑_iA_ij)) since 1982. CIP contains the set cover problem as a special case, so O(lnm)-approximation is the best possible unless P=NP.     

Two point hermite approximations for the solution of linear initial value and boundary value problems

Application of an idea originally due to Ch. Hermite allows the derivation of an approximate formula for expressing the integral ∫xixi?1y(x)dx as a linear combination of y(xi?1), y(xi), and their derivatives y(v)(xi?1) up to order v = α and y(v)(xi) up to order v = β. In addition to this integro-differential form a purely differential form of the 2-point Hermite approximation will be derived. Both types will be denoted by Hαβ-approximation. It will be shown that the well-known Obreschkoff-formulas contain no new elements compared to the much older Hαβ-method.The Hαβ-approximation will be applied to the solution of systems of ordinary differential equations of the type y'(x) = M(x)y(x) + q(x), and both initial value and boundary value problems will be treated. Function values at intermediate points x? (xi?1, xi) are obtained by the use of an interpolation formula given in this paper.An advantage of the Hαβ-method is the fact that high orders of approximation (α, β) allow an increase in step size hi. This will be demonstrated by the results of several test calculations.