Solving BVPs using two-point Taylor formula by a symbolic software

For the numerical solution of boundary value problems a global method, based on two-point Taylor formula is proposed. A Mathematica package to compute approximate solution of BVPs is presented.

Numerical examples provide favorable comparisons with other existing methods, especially with respect to accuracy.     

The structure of finite groups with trait of non-normal subgroups

In this paper, we investigate the finite groups all of whose non-normal nilpotent subgroups are cyclic. We show that such groups are solvable with cyclic centers. If G is a non-supersolvable group, then G has only one non-cyclic Sylow subgroup which is either isomorphic to Q₈ or is of type (q, q).     

Estimating a reduced rank regression model for non-normal variables

A method using third order moments for estimating the regression coefficients as well as the latent state scores of the reduced-rank regression model when the latent variable(s) are non-normally distributed is presented in this paper. It is shown that the factor analysis type indeterminacy of the regression coefficient matrices is eliminated. A real life example of the proposed method is presented. Differences of this solution with the reduced-rank regression eigen solution are discussed.     

Derived tame local and two-point algebras

We determine derived representation type of complete finitely generated local and two-point algebras over an algebraically closed field.     

A uniqueness theorem for two-point boundary value problems

We prove a uniqueness theorem for solutions of two-point boundary value problems, which says that when the nonlinearity is sublinear and Lipschitzian, and one of the boundary values is fixed, then for sufficiently large values of the other boundary value there is a unique solution of the problem. The proof is based on the Banach's fixed-point theorem, and the argument used to show that the relevant operator is a contraction makes use of a generalization of the Riemann-Lebesgue lemma     

A selection theorem for a new class of set-valued maps

A new class of set-valued maps that includes all upper and lower semicontinuous set-valued maps is introduced. For this class, a selection theorem having applications in the theory of differential inclusions is presented. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 503–507, October, 1999.     

The dual geometry of Hermitian two-point codes

In this paper, we study the algebraic geometry of any two-point code on the Hermitian curve and reveal the purely geometric nature of their dual minimum distance. We describe the minimum-weight codewords of many of their dual codes through an explicit geometric characterization of their supports. In particular, we show that they appear as sets of collinear points in many cases.     

Cubic spline for a class of singular two-point boundary value problems

In this paper we have presented a method based on cubic splines for solving a class of singular two-point boundary value problems. The original differential equation is modified at the singular point then the boundary value problem is treated by using cubic spline approximation. The tridiagonal system resulting from the spline approximation is efficiently solved by Thomas algorithm. Some model problems are solved, and the numerical results are compared with exact solution.     

About a numerical method of successive interpolations for two-point boundary value problems with deviating argument

A new numerical method for two-point boundary value problems with deviating argument is obtained. The method uses the fixed point technique, the trapezoidal quadrature rule, and a Birkhoff interpolation procedure. The convergence of the method is proved without smoothness conditions, the kernel function being only Lipschitzian in each argument. The interpolation procedure is used only on the points where the argument is modified. A stopping criterion of the algorithm is obtained and the accuracy of the method is illustrated on four numerical examples of pantograph type.     

A time-decomposition algorithm for a stiff linear two-point boundary-value problem and its application to a nonlinear optimal control problem

An algorithm is proposed to solve a stiff linear two-point boundary-value problem (TPBVP). In a stiff problem, since some particular solutions of the system equation increase and others decrease rapidly as the independent variable changes, the integration of the system equation suffers from numerical errors. In the proposed algorithm, first, the overall interval of integration is divided into several subintervals; then, in each subinterval a sub-TPBVP with arbitrarily chosen boundary values is solved. Second, the exact boundary values which guarantee the continuity of the solution are determined algebraically. Owing to the division of the integration interval, the numerical error is effectively reduced in spite of the stiffness of the system equation. It is also shown that the algorithm is successfully imbedded into an interaction-coordination algorithm for solving a nonlinear optimal control problem.The authors would like to thank Mr. T. Sera and Mr. H. Miyake for their help with the calculations.     

Arbitrary groups as two-point stabilisers of symmetric groups acting on partitions

We give a short, direct proof that given any finite group G there exist positive integers k and l and partitions α₁and α₂ of {1, …, kl } into l subsets of size k such that (S _kl ) _{α 1}, _{α 2}≅ G.The method used will also show that given any finite group G there exists a regular bipartite graph whose automorphism group is isomorphic to G     

Solutions of two-point boundary value problems for even-order differential equations

The existence of solutions of the two-point boundary value problems consisting of the even-order differential equations

     

Back-and-forth shooting method for solving two-point boundary-value problems

The paper proposes a special iterative method for a nonlinear TPBVP of the form (t)=f(t, x(t),p(t)), (t)=g(t, x(t),p(t)), subject toh(x(0),p(0))=0,e(x(T),p(T))=0. Certain stability properties of the above differential equations are taken into consideration in the method, so that the integration directions associated with these equations respectively are opposite to each other, in contrast with the conventional shooting methods. Via an embedding and a Riccati-type transformation, the TPBVP is reduced to consecutive initial-value problems of ordinary differential equations. A preliminary numerical test is given by a simple example originating in an optimal control problem.     

Estimate of boundary layer thickness for linear,singularly perturbed two-point boundary-value problems

In this note, the thickness of the boundary layers for linear, homogeneous, singularly perturbed first-order ordinary differential systems is considered. Inequalities concerning this thickness are derived. It is also remarked that the present analysis can be carried over to the nonhomogeneous case.The authors wish to express their sincere thanks to Dr. S. M. Roberts for his comments and valuable suggestions.     

Strict positive definiteness on products of compact two-point homogeneous spaces

We present an explicit characterization for the real, continuous, isotropic and strictly positive definite kernels on a product of compact two-point homogeneous spaces, covering almost all possible choices for the spaces. The result complements similar characterizations previously obtained for products of high-dimensional spheres.     

Existence of solutions to a class of nonlinear second order two-point boundary value problems

In this paper, the existence and multiplicity results of solutions are obtained for the second order two-point boundary value problem −u^″(t)=f(t,u(t)) for all t∈[0,1] subject to u(0)=u^′(1)=0, where f is continuous. The monotone operator theory and critical point theory are employed to discuss this problem, respectively. In argument, quadratic root operator and its properties play an important role.     

Metrically generated theories

Many examples are known of natural functors describing the transition from categories of generalized metric spaces to the ``metrizable" objects in some given topological construct . If preserves initial morphisms and if is initially dense in , then we say that is -metrically generated. Our main theorem proves that is -metrically generated if and only if can be isomorphically described as a concretely coreflective subconstruct of a model category with objects sets structured by collections of generalized metrics in and natural morphisms. This theorem allows for a unifying treatment of many well-known and varied theories. Moreover, via suitable comparison functors, the various relationships between these theories are studied.

     

A selection theorem for quasi-lower semicontinuous mappings in hyperconvex spaces

We prove a continuous selection theorem for quasi-lower semicontinuous mappings with values that are closed sub-admissible subsets of a hyperconvex metric space and apply this result to obtain fixed point theorems in these spaces.