Enumeration of Rosenberg-type hypercompositional structures defined by binary relations

Boolean matrices, i.e., matrices over a Boolean algebra, do not satisfy many of the fundamental properties of matrices over the real or complex field. Algebraic aspects of the Boolean pseudoinverse matrix concept are investigated. Maps operating on Boolean vector spaces by analogy with those on linear spaces are defined. Some properties of them are obtained. Most of these properties remain the same in the case of linear spaces and they are essential for solving linear systems. In particular orthogonal projections in linear spaces receive their algebraic analogue in Boolean vector spaces. The maps defined are useful for solving equations in the Boolean algebra, if no pseudoinverse matrix to a given matrix exists. We also share our experience in using computer algebra systems for teaching discrete mathematics and linear algebra. Some examples for computations with binary relations and Boolean matrices using computer algebra systems are given.

Binary relations, matrices and inference developments

Factorizing Boolean matrices using formal concepts and iterative usage of essential entries

The development of the symbolic inverse of a manipulator Jacobian and its use in controlling the manipulator is discussed. The ease of symbolically inverting a matrix is directly related to the simplicity of the expressions in the matrix. Therefore, a general method for obtaining a simplified form of the Jacobian of a manipulator in any of the intermediate link frames is developed and presented using an example. Also, a general method is presented for the verification of a manipulator Jacobian using the forward kinematics. A hand-waving discussion of the inversion of the Kraft slave Jacobian using symbolic manipulation software is given and the final form of the inverse is presented. An approximate solution to the inverse Jacobian problem for the Kraft slave using symbolic inverses is presented. This solution is used in dealing with the only reachable singularity of the Kraft slave. Finally, a teleoperational control system using the inverse Jacobian to calculate the inverse kinematics is discussed.

We study properties determined by idempotents in the following families of matrix semigroups over a semiring [Formula: see text]: the full matrix semigroup [Formula: see text], the semigroup [Formula: see text] consisting of upper triangular matrices, and the semigroup [Formula: see text] consisting of all unitriangular matrices. Il’in has shown that (for [Formula: see text]) the semigroup [Formula: see text] is regular if and only if [Formula: see text] is a regular ring. We show that [Formula: see text] is regular if and only if [Formula: see text] and the multiplicative semigroup of [Formula: see text] is regular. The notions of being abundant or Fountain (formerly, weakly abundant) are weaker than being regular but are also defined in terms of idempotents, namely, every class of certain equivalence relations must contain an idempotent. Each of [Formula: see text], [Formula: see text] and [Formula: see text] admits a natural anti-isomorphism allowing us to characterise abundance and Fountainicity in terms of the left action of idempotent matrices upon column spaces. In the case where the semiring is exact, we show that [Formula: see text] is abundant if and only if it is regular. Our main interest is in the case where [Formula: see text] is an idempotent semifield, our motivating example being that of the tropical semiring [Formula: see text]. We prove that certain subsemigroups of [Formula: see text], including several generalisations of well-studied monoids of binary relations (Hall relations, reflexive relations, unitriangular Boolean matrices), are Fountain. We also consider the subsemigroups [Formula: see text] and [Formula: see text] consisting of those matrices of [Formula: see text] and [Formula: see text] having all elements on and above the leading diagonal non-zero. We prove the idempotent generated subsemigroup of [Formula: see text] is [Formula: see text]. Further, [Formula: see text] and [Formula: see text] are families of Fountain semigroups with interesting and unusual properties. In particular, every [Formula: see text]-class and [Formula: see text]-class contains a unique idempotent, where [Formula: see text] and [Formula: see text] are the relations used to define Fountainicity, but yet the idempotents do not form a semilattice.

Characteristic numbers and approximation operators in generalized rough approximation system

We describe the use of a symbolic manipulation package, written in REDUCE 3.2, for the construction of supergravity actions. As illustrative examples, some calculations in D=10, N=1 supergravity are performed.

In this paper we attempt to revive interest in the reversion method for solving non-linear differential equations by combining the reversion technique with the Cauchy integral method for solving differential equations by direct quadratures. The resultant synthesis is easy to implement on any standard symbolic manipulation package and various standard problems are solved showing the superiority of the current method over previous attempts to employ the reversion technique using, for example, Laplace transforms.

There have been many approaches to solve ordinary differential equations numerically. Even though many numerical methods can provide very good approximate solutions they need considerable calculation effort, often through iterations. The advance of symbolic manipulation packages such as Maple gives the opportunity for new approaches to this type of problems. This paper will discuss an improvement to one of these new approaches enabled by the availability of these packages, to obtain a numerical solution of an initial‐value problem governed by a high‐order non‐linear ordinary differential equation. The method we propose here is based on Zhu and Phan's earlier paper (Commun. Numer. Meth. Engng. 2003; 19:601–614) but we shall show an improved order of accuracy for the case of the highest‐order derivative being of an even order. In any case it provides the ability to further increase the order of accuracy under certain conditions of differentiability. Like Zhu and Phan's (Commun. Numer. Meth. Engng. 2003; 19:601–614) original method, an important feature of the new method is that no iterations are needed except in the preparatory phase for some cases. Copyright © 2006 John Wiley & Sons, Ltd.

On enumeration of hypergroups of order 3

The analysis of prismatic members made of reinforced concrete under inclined bending, especially the computation of ultimate loads, is a pronounced non-linear problem which is frequently solved by discretizing the stress distribution in the cross-section using interpolation functions. In the approach described in the present contribution the exact analytical stress distribution is used instead. The obtained expressions are integrated by means of a symbolic manipulation package and automatically converted to optimized Fortran code. The direct problem-computation of ultimate internal forces given the position of the neutral axis-is first described. Subsequently, two kinds of inverse problem are treated: the computation of rupture envelops and the dimensioning of reinforcement, given design internal forces. An iterative Newton-Raphson procedure is used. Examples are presented.

A new class of conformal invariants for a given spacetime M is introduced exploiting the conformal geometry of any light ray [Formula: see text]. Each congruence of light rays passing through a given point p defines the sky S(p) of such point. The new conformal invariants are defined on the bundle of skies of the spacetime M, being called sky invariants accordingly. The natural conformal covariant derivative defined on a light ray and its associated covariant calculus allows us to show the existence of a natural conformal invariant differential of arc that, together with the restriction of the curvature of the conformal covariant derivative, can be used to construct a sky invariant that will be called the sky curvature. An algorithm, that can be implemented on any symbolic manipulation software system, to compute the sky curvature will be discussed and the main ideas and the explicit computation of the sky curvature are illustrated in Schwarzschild spacetime.

An improved Vietoris sine inequality

Curvilinear-axis electron optical focusing and aberration theory of wide electron beams in immersion magnetic lenses, which can be used for projection electron beam lithography, are put forward, and second-order aberration formulas have been derived using the symbolic manipulation software MATHEMATICA®. A program was written to calculate numerically the wide beam Gaussian focusing and aberration properties of these lenses by using the curvilinear-axis theory. The magnetic structures that have convergent force lines have been studied. As an example, lenses with an analytical axial magnetic field distribution are calculated. The results show that wide beam projection images without crossovers can be formed with demagnified lateral magnification, and with a submicron and nanometer lateral resolution over a larger field size.

Computer generation of geometrical error equations applicable for improvement of robots' positioning accuracy