On the Geometry of the 7-Sphere

A sphere of dimension 4n+3 admits three Sasakian structures and it is natural to ask if a submanifold can be an integral submanifold for more than one of the contact structures. In the 7-sphere it is possible to have curves which are Legendre curves for all three contact structures and there are 2 and 3-dimensional submanifolds which are integral submanifolds of two of the contact structures. One of the results here is that if a 3-dimensional submanifold is an integral submanifold of one of the Sasakian structures and invariant with respect to another, it is an integral submanifold of the remaining structure and is a principal circle bundle over a holmophic Legendre curve in complex projective 3-space.     

On the Number of Conserved Quantities for the Two-Layer Shallow-Water Equations

The shallow-water equations for two-layer inviscid flow with a free surface overlying a rigid horizontal bottom subject to gravitational forcing only are examined to determine the possible forms of conservation laws that the equations permit. In the case of a single layer with flow in only one horizontal direction, it is known that there are an infinite number of associated equations in conservation form, where the conserved quantity is a multinomial in the layer variables. The method used to determine this result is generalized to show that in the two-layer case, the result does not generalize, and it is discovered that only a finite number of conservation equations exist when the density difference between the layers is nonzero. The subsequent conservation equations are given explicitly, and a systematic method for deriving conservation laws from an arbitrary first-order system is described. For the case when the flow is in both horizontal dimensions, the method of analysis is straightforward in the one-layer case, and the finite number of conservation equations are derived. The two-layer case is similar, and the finite number of generalized conserved quantities are stated, although the question of whether or not there are only a finite number is posed as an open question.     

On the Description of the Content-Equality by a Relation

In the geometry of polyhedra we understand by an elementary content-functional a real valued, non-negative, finite additive measure on the set of polyhedra which is invariant under isometries. There are close relations between the content-measurement and the relation of equidecomposability. Two polyhedra are called equidecomposable if they are decomposed into pairwise congruent pieces. For an example we consider the set of all polygons in the euclidean plane. It is well known that planar polygons have the same area if and only if they are equidecomposable. In the three-dimensional euclidean space one also can describe the content-equality of polyhedra by a relation. Two polyhedra have the same volume if they are equidecomposable with respect to equiaffine mappings (see [3]). In [4] the concept of an invariant content of polyhedra in a topological Klein space is introduced. Each regular closed quasicompact set ot the space is called polyhedron. Under this supposition two polyhedra have equal contents if they are equivalent by decomposition. The relation “equivalent by decomposition” is closely related to the relation “equidecomposable”.     

Equations for the continuous estimation of the perturbations of dynamical systems

A dynamical system acted upon by external perturbations is considered. It is assumed that the phase state of the system (or a part of it) is observed with certain errors. The problem is to construct differential equations for estimating (reconstructing) the perturbations using measurement data. Unlike in papers in which cases of discrete instants of the observations are analysed, the continuous case is considered for which differential equations of an auxiliary system are derived, the controls in which are approximations of an unknown input. The general constructions are illustrated by means of an example.     

On the recoverable robust traveling salesman problem

We consider an uncertain traveling salesman problem, where distances between nodes are not known exactly, but may stem from an uncertainty set of possible scenarios. This uncertainty set is given as intervals with an additional bound on the number of distances that may deviate from their expected, nominal values. A recoverable robust model is proposed, that allows a tour to change a bounded number of edges once a scenario becomes known. As the model contains an exponential number of constraints and variables, an iterative algorithm is proposed, in which tours and scenarios are computed alternately. While this approach is able to find a provably optimal solution to the robust model, it also needs to solve increasingly complex subproblems. Therefore, we also consider heuristic solution procedures based on local search moves using a heuristic estimate of the actual objective function. In computational experiments, these approaches are compared.     

The estimation of the strength of the heat source in the heat conduction problems

A general method is proposed to determine the strength of the heat source in the Fourier and non-Fourier heat conduction problems. A finite difference method, the concept of the future time and a modified Newton–Raphson method are adopted in the problem. The undetermined heat source at each time step is formulated as an unknown variable in a set of equations from the measured temperature and the calculated temperature. Then, an iterative process is used to solve the set of equations. No selected function is needed to represent the undetermined function in advance. Three examples are used to demonstrate the characteristics of the proposed method. The validity of the proposed method is confirmed by the numerical results. The results show that the proposed method is an accurate and stable method to determine the strength of the heat source in the inverse hyperbolic heat conduction problems. Furthermore, the result shows that more future times are needed in the hyperbolic equation than that of parabolic equation. Moreover, the robustness and the accuracy of the estimated results in the non-Fourier problem are not as well as those of the Fourier problem.     

可列无限等可能概型概率场的讨论

假设有某试验,对每次试验而言有可列无限个试验结果,由于某种对称性,每个结果的出现又有一定的均匀性.对这种概型本文称之谓可列无限等可能慨型,这种概型的问题其概率场是什么?这是个颇有趣的问题,本文将针对这一问题进行一定的探讨.本文将针对从全体正整数中随机取数的问题的一种特殊的子集类构成事件域,并在其上合理地定义概率,从而建立概率场.     

Solution of problems on the anti-plane deformation of elastic bodies with corner points by the method of integral equations

The problem of the antiplane deformation of an elastic cylinder with a multiconnected finite or infinite section, bounded by a system of closed curves that can have corner points, is examined. Forces or displacements are given on the whole boundary of the body. The problem is reduced to an integral equation whose kernel has strong stationary singularities at the corner points. Results of an investigation of the solvability of this equation and the smoothness of its solution are presented. A procedure for the numerical solution of the integral equation is described. A space with a prismatic hole of rectangular section or a rigid inclusion subjected to a uniform tangential force at infinity is considered as an example. The generalized stress intensity factors are calculated.     

On the numerical approach of the enthalpy method for the Stefan problem

In this article an error bound is derived for a piecewise linear finite element approximation of an enthalpy formulation of the Stefan problem; we have analyzed a semidiscrete Galerkin approximation and completely discrete scheme based on the backward Euler method and a linearized scheme is given and its convergence is also proved. A second‐order error estimates are derived for the Crank‐Nicolson Galerkin method. In the second part, a new class of finite difference schemes is proposed. Our approach is to introduce a new variable and transform the given equation into an equivalent system of equations. Then, we prove that the difference scheme is second order convergent. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

On the Solution of the Inverse Eigenvalue Complementarity Problem

In this paper, we discuss the solution of an Inverse Eigenvalue Complementarity Problem. Two nonlinear formulations are presented for this problem. A necessary and sufficient condition for a stationary point of the first of these formulations to be a solution of the problem is established. On the other hand, to assure global convergence to a solution of this problem when it exists, an enumerative algorithm is designed by exploiting the structure of the second formulation. The use of additional implied constraints for enhancing the efficiency of the algorithm is also discussed. Computational results are provided to highlight the performance of the algorithm.     

An analytical approach to the unified solution of kinetic equations in the rarefied gas dynamics. II. Heat transfer problems

The ADO method, an analytical version of the discrete-ordinates method, is used here to solve a heat-transfer problem in a rarefied gas confined in a channel, as well as to solve a half-space problem in order to evaluate the temperature jump at the wall. This work is an extension of a previous work, devoted to flow problems, where the complete development of the solution, which is analytical in terms of the spatial variable, is presented in a way, such that, a wide class of kinetic models are considered, in an unified approach. A series of numerical results are showed and different simulations are used in order to establish a general comparative analysis based on this consistent set of results provided by the same methodology. In particular, numerical results for heat-flow profile, temperature and density perturbations are obtained for channels (walls), defined by different materials, on which different temperatures are imposed.     

Delamination of Composites along the Interface as Buckling Failure of the Stressed Layer

A refined model for free vibrations of a Griffith crack in layered composites is investigated. The delamination of the composites along an interface is studied in the context of growth of instability of the stressed layer. This is analogous to the bending of a beam on an elastic foundation. The second-order gradient terms are taken into account in the power-series expansion of deflections. These terms are responsible for nonlinear phenomena in the interaction zone. The zero-order frequency of free vibrations of the crack along the interface and the time of instability growth are determined.     

Concertina-like movements of the error curve in the alternation theorem

If a continuous function f is approximated by elements of a Haar space in the maximum norm on an interval, the error curve of the best approximation has well known alternation properties. It is shown that if f is adjoined to the Haar space all zeros of the error function are monotonously increasing functions of the endpoints, and that under an additional hypothesis, the entire graph of the error curve is shifted to the left or right when the endpoints are moved accordingly.     

On the convergence of operator splitting for the Rosenau–Burgers equation

We present convergence analysis of operator splitting methods applied to the nonlinear Rosenau–Burgers equation. The equation is first splitted into an unbounded linear part and a bounded nonlinear part and then operator splitting methods of Lie‐Trotter and Strang type are applied to the equation. The local error bounds are obtained by using an approach based on the differential theory of operators in Banach space and error terms of one and two‐dimensional numerical quadratures via Lie commutator bounds. The global error estimates are obtained via a Lady Windermere's fan argument. Lastly, a numerical example is studied to confirm the expected convergence order.     

On the Stone and Bishop appoximation theorems

Generalizations of the Stone-Weierstrass and Bishop approximation theorems are presented. Given an algebra, a subspace in a continuous function space coinciding with the closure of this algebra is constructed. Analogs of these results are obtained in the case where the set of functions under consideration is not an algebra, but its closure is related to some algebra.     

New proof of the cobordism invariance of the index

We give a simple proof of the cobordism invariance of the index of an elliptic operator. The proof is based on a study of a Witten-type deformation of an extension of the operator to a complete Riemannian manifold. One of the advantages of our approach is that it allows us to treat directly general elliptic operators which are not of Dirac type.

     

On the structure of the teaching-learning interactive process

This article is a follow-up to an earlier paper byMarchi andMiguel [1974] in which a mathematical structure for the teaching-learning process has been developed. Now, possible paths for teaching in a given knowledge set and the possible choices open to different teachers and groups of students are investigated. Furthermore, considering that teachers and students are in an interactive situation, an optimal global procedure corresponds exactly to the concept of equilibrium point. The existence proof gives rise to a constructive algorithm which would allow using the results for practical applications.     

Algebraic transition matrices in the Conley index theory

We introduce the concept of an algebraic transition matrix. These are degree zero isomorphisms which are upper triangular with respect to a partial order. It is shown that all connection matrices of a Morse decomposition for which the partial order is a series-parallel admissible order are related via a conjugation with one of these transition matrices. This result is then restated in the form of an existence theorem for global bifurcations. Simple examples of how these results can be applied are also presented.

     

On the Weierstrass Preparation Theorem

An analytic function of several variables is considered. It is assumed that the function vanishes at some point. According to the Weierstrass preparation theorem, in the neighborhood of this point the function can be represented as a product of a nonvanishing analytic function and a polynomial in one of the variables. The coefficients of the polynomial are analytic functions of the remaining variables. In this paper we construct a method for finding the nonvanishing function and the coefficients of the polynomial in the form of Taylor series whose coefficients are found from an explicit recursive procedure using the derivatives of the initial function. As an application, an explicit formula describing a bifurcation diagram locally up to second-order terms is derived for the case of a double root.     

Some features of the deceleration of an inhomogeneous ball in the air

A model of the deceleration of an inhomogeneous ball acted upon by the drag of the air is discussed, taking into account the interaction of the translational and rotational motions. The problem is reduced to analysing a non-linear second-order dynamical system. The steady motions of the ball, including self-oscillatory and self-rotation motions, are obtained. The bifurcation values of the parameters defining these motions are determined. The corresponding phase portraits are constructed and an interesting interpretation of them is given.