Analogues of the nine-point circle for orthocentric n-simplexes

This article presents a generalization of some theorems connected with the nine-point circle for n-dimensional Euclidean space.     

A fourth-order difference method for elliptic equations with nonlinear first derivative terms

We present a nine-point fourth-order finite difference method for the nonlinear second-order elliptic differential equation Au_xx + Bu_yy = f(x, y, u, u_x, u_y) on a rectangular region R subject to Dirichlet boundary conditions u(x, y) = g(x, y) on ?R. We establish, under appropriate conditions O(h⁴)-convergence of the finite difference scheme. Numerical examples are given to illustrate the method and its fourth-order convergence.     

A high-order compact difference scheme for 2D Laplace and Poisson equations in non-uniform grid systems

In this study, a high-order compact scheme for 2D Laplace and Poisson equations under a non-uniform grid setting is developed. Based on the optimal difference method, a nine-point compact difference scheme is generated. Difference coefficients at each grid point and source term are derived. This is accomplished through the consideration of compatibility between the partial differential equation and its difference discretization. Theoretically, the proposed scheme has third- to fourth-order accuracy; its fourth-order accuracy is achieved under uniform grid settings. Two examples are provided to examine performance of the proposed scheme. Compared with the traditional five-point difference scheme, the proposed scheme can produce more accurate results with faster convergence. Another reference scheme with the same nine-point grid stencil is derived based on the five-point scheme. The two nine-point schemes have the same coefficients for each grid points; however, their coefficients for the source term are different. The overall accuracy level of the solution resulting from the proposed scheme is higher than that of the nine-point reference scheme. It is also indicated that the smoothness of grids has significant effects on accuracy and convergence of the solutions; efforts in optimizing the grid configuration and allocation can improve solution accuracy and efficiency. Consequently, with the proposed method, solution under the non-uniform grid setting with appropriate grid allocation would be more accurate than that under the uniform-grid manipulation, with the same number of grid points.     

A nine-point scheme with explicit weights for diffusion equations on distorted meshes

In this paper, for the structured quadrilateral mesh we derive a nine-point difference scheme which has five cell-centered unknowns and four vertex unknowns. The vertex unknowns are treated as intermediate ones and are expressed as a linear combination of the neighboring cell-centered unknowns, which reduces the scheme to a cell-centered one with a local stencil involving nine cell-centered unknowns. The coefficients in the linear combination are known as the weights and two types of new weights are proposed. These new weights are neither discontinuity dependent nor mesh topology dependent, have explicit expressions, can reduce to the one-dimensional harmonic-average weights on the nonuniform rectangular meshes, and moreover, are easily extended to the unstructured polygonal meshes and non-matching meshes. Both the derivation of the nine-point scheme and that of new weights satisfy the linearity preserving criterion. Numerical experiments show that, with these new weights, the nine-point difference scheme and its simple extension have a nearly second order accuracy on many highly distorted meshes, including structured quadrilateral meshes, unstructured polygonal meshes and non-matching meshes.     

Reduktionsverfahren für Differenzengleichungen bei Randwertaufgaben II

Summary In a recent paper [11], two of the authors investigated a fast reduction method for solving difference equations which approximate certain boundary value problems for Poisson's equation. In this second paper, we prove the numerical stability of the reduction method, and also report on further developments of the method. For the general case, the provided bounds for the numerical errors behave roughly like the condition numberO(n ²) of the linear system; for more realistic model problems estimates of order less thanO(n) are obtained (n ^–1=h=mesh width). The number of operations required for the reduction method isO(n ² ), for the usual five-point difference formula, as well as for the common nine-point formula with discretization error of orderh ⁴.     

Higher order recurrences for analytical functions of Tchebysheff type

Relation of hyperbolons of volume one to generalized Clifford algebras is described in [1b] and there some applications are listed. In this note which is an extension of [8] we use the one parameter subgroups of the group of hyperbolons of volume one in order to define and investigate generalization of Tchebysheff polynomial system. Parallely functions of roots of polynomials of any degree are studied as possible generalization of symmetric functions considered by Eduard Lucas. It is found how functions of roots of polynomial of any degree are related to this generalization of Tchebysheff polynomials. The relation is explicit. In a primary sense the considered generalization is in passing fromZ ₂ toZ _n group decomposition of the exponential. We end up with an application of the discovered generalization to quite large class of dynamical systems with iteration.     

On thecd-variation polynomials of André and simsun permutations

We prove a conjecture of Stanley on thecd-index of the semisuspension of the face poset of a simplicial shelling component. We give a new signed generalization of André permutations, together with a new notion ofcd-variation for signed permutations. This generalization not only allows us to compute thecd-index of the face poset of a cube, but also occurs as a natural set of orbit representatives for a signed generalization of the Foata-Strehl commutative group action on the symmetric group. From the induction techniques used, it becomes clear that there is more than one way to define classes of permutations andcd-variation such that they allow us to compute thecd-index of the same poset. This research was supported by the UQAM Foundation.     

Direct constructions of additive codes

A code is q^m‐ary q‐linear if its alphabet forms an m‐dimensional vector space over ??_q and the code is linear over ??_q. These additive codes form a natural generalization of linear codes. Our main results are direct constructions of certain families of additive codes. These comprise the additive generalization of the Kasami codes, an additive generalization of the Bose‐Bush construction of orthogonal arrays of strength 2 as well as a class of additive codes which are being used for deep space communication. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 207–216, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.20000     

Zero commutativity of nilpotent elements skewed by ring endomorphisms

The reversible property is an important role in noncommutative ring theory. Recently, the study of the reversible ring property on nilpotent elements is established by Abdul-Jabbar et al., introducing the concept of commutativity of nilpotent elements at zero (simply, a CNZ ring) as a generalization of reversible rings. We here study this property skewed by a ring endomorphism α, and such ring is called a right α-skew CNZ ring which is an extension of CNZ rings as well as a generalization of right α-skew reversible rings, and then investigate the structure of right α-skew CNZ rings and their related properties. Consequently, several known results are obtained as corollaries of our results.     

A note on Kazdan-Warner-type identities

In this article we deal with f-conformal Killing vector fields, a generalization of conformal Killing vector fields involving a function f and two real parameters. Under certain conditions on f and the parameters, some non-existence results for such vector fields are proven. Moreover we derive a generalization of the Kazdan-Warner and Bourguignon-Ezin identities to the case of f-conformal Killing vector fields. Based on these, finally we present an application to the f-conformal solitons, a generalization of the Yamabe solitons.     

On approximate solutions of the Pexider equation

Summary LetX be an abelian (topological) group andY a normed space. In this paper the following functional inequality is considered: {ie143-1} This inequality is a similar generalization of the Pexider equation as J. Tabor's generalization of the Cauchy equation (cf. [3], [4]). The solutions of our inequality have similar properties as the solutions of the Pexider equation. Continuity and related properties of the solutions are investigated as well.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

Guided discovery of the nine-point circle theorem and its proof

The nine-point circle theorem is one of the most beautiful and surprising theorems in Euclidean geometry. It establishes an existence of a circle passing through nine points, all of which are related to a single triangle. This paper describes a set of instructional activities that can help students discover the nine-point circle theorem through investigation in a dynamic geometry environment, and consequently prove it using a method of guided discovery. The paper concludes with a variety of suggestions for the ways in which the whole set of activities can be implemented in geometry classrooms.     

L-fuzzy ⋎ and ⋏ hyperoperations and the associated l-fuzzy hyperalgebras

In this paper we study two fuzzy hyperoperations, denoted by ⋎ (which can be seen as a generalization of ∨) and ⋏ (which can be seen as a generalization of ∧). ⋎ is obtained from a family of crisp ∨; _p hyperoperations and ⋏ is obtained from a family of crisp ∧ _p hyperoperations. The hyperstructure (X, ⋎, ∧) resembles ahyperlattice and the hyperstructure (X, ∨, ⋏) resembles adual hyperlattice     

Fully Coprime Comodules and Fully Coprime Corings

Prime objects were defined as generalization of simple objects in the categories of rings (modules). In this paper we introduce and investigate what turns out to be a suitable generalization of simple corings (simple comodules), namely fully coprime corings (fully coprime comodules). Moreover, we consider several primeness notions in the category of comodules of a given coring and investigate their relations with the fully coprimeness and the simplicity of these comodules. These notions are applied then to study primeness and coprimeness properties of a given coring, considered as an object in its category of right (left) comodules. Supported by King Fahd University of Petroleum and Minerals, Research Project # INT/296.     

Oriented matroids and Ky Fan’s theorem

L. Lovász (Matroids and Sperner’s Lemma, Europ. J. Comb. 1 (1980), 65–66) has shown that Sperner’s combinatorial lemma admits a generalization involving a matroid defined on the set of vertices of the associated triangulation. We prove that Ky Fan’s theorem admits an oriented matroid generalization of similar nature. Classical Ky Fan’s theorem is obtained as a corollary if the underlying oriented matroid is chosen to be the alternating matroid C ^m,r .     

A note on Baskakov-Kantorovich type operators preserving e−x

In this paper, we give a generalization of the Baskakov-Kantorovich type operators that reproduce functions e₀ and e^−x. We discuss uniform convergence of this generalization by means of the modulus of continuity and establish quantitive asymptotic formula.     

Monadic Bounded Commutative Residuated ℓ-monoids

Bounded commutative residuated ℓ-monoids are a generalization of algebras of propositional logics such as BL-algebras, i.e. algebraic counterparts of the basic fuzzy logic (and hence consequently MV-algebras, i.e. algebras of the Łukasiewicz infinite valued logic) and Heyting algebras, i.e. algebras of the intuitionistic logic. Monadic MV-algebras are an algebraic model of the predicate calculus of the Łukasiewicz infinite valued logic in which only a single individual variable occurs. We introduce and study monadic residuated ℓ-monoids as a generalization of monadic MV-algebras. Jiří Rachůnek was supported by the Council of Czech Goverment MSM 6198959214.     

Derivation of the energy–momentum and Klein–Gordon equations from El Naschie’s complex time

In a previous note, we have provided a formal derivation of the transverse Doppler shift of special relativity from the generalization of El Naschie’s complex time. Here, we show that the relativistic energy–momentum equation, and hence the Klein–Gordon equation, are also natural consequences of the complex time generalization.     

A generalization of the Looman-Menchoff theorem

In this paper we give a generalization of the classical Looman-Menchoff theorem:If f is a complex-valued continuous function of a complex variable in a domain G, f has partial derivatives f _x and f _y everywhere in G and the Cauchy Riemann equations f _x +if _y = 0are satisfied almost everywhere, then f is holomorphic in G. From our generalization of this theorem, we deduce a theroem of Sindalovskii [9] as a corollary and also answer some of the questions raised in [9]. We note in this context that, as far as we know, Sindalovskii’s result is the best published to date in this area.     

Historical origins of the nine-point conic. The contribution of Eugenio Beltrami

In this paper, we examine the evolution of a specific mathematical problem, i.e. the nine-point conic, a generalisation of the nine-point circle due to Steiner. We will follow this evolution from Steiner to the Neapolitan school (Trudi and Battaglini) and finally to the contribution of Beltrami that closed this journey, at least from a mathematical point of view (scholars of elementary geometry, in fact, will continue to resume the problem from the second half of the 19th to the beginning of the 20th century). We believe that such evolution may indicate the steady development of the mathematical methods from Euclidean metric to projective, and finally, with Beltrami, with the use of quadratic transformations. In this sense, the work of Beltrami appears similar to the recent (after the anticipations of Magnus and Steiner) results of Schiaparelli and Cremona. Moreover, Beltrami's methods are closely related to the study of birational transformations, which in the same period were becoming one of the main topics of algebraic geometry. Finally, our work emphasises the role played by the nine-point conic problem in the studies of young Beltrami who, under Cremona's guidance, was then developing his mathematical skills. To this end, we make considerable use of the unedited correspondence Beltrami – Cremona, preserved in the Istituto Mazziniano, Genoa.