Volume 57 , issue 1$-$2 ( 2005 ) | back |

Algorithms for triangulating polyhedra into a small number of tethraedra | 1$-$9 |

**Abstract**

Two algorithms for triangulating polyhedra, which give the number of tetrahedra depending linearly on the number of vertices, are discussed. Since the smallest possible number of tetrahedra necessary to triangulate given polyhedra is of interest, for the first--``Greedy peeling" algorithm, we give a better estimation of the greatest number of tetrahedra ($3n-20$ instead of $3n-11$), while for the second one--``cone triangulation", we discuss cases when it is possible to improve it in such a way as to obtain a smaller number of tetrahedra.

**Keywords:** Triangulation of polyhedra, minimal triangulation.

**MSC:** 52C17, 52B05, 05B40

Inequality of Poincaré-Friedrich's type on $L^p$ spaces | 11$-$14 |

**Abstract**

In this paper it is demonstrated that the inequality $$ \biggl(\int_G|u|^p\,dx\biggr)^{1/p}\leq A_p\biggl(\int_D|\nabla u|^p\,dx \biggr)^{1/p},\quad u|_{\partial D}=0,1\leq p\leq\infty $$ holds, where $G\subset D\subset R^2$, $D$ is a convex domain and constant $A_p$ is expressed in terms of areas of $G$ and~$D$.

**Keywords:** Poincaré-Friedrich's inequality, $L^p$-space.

**MSC:** 26D10, 35P15

A necessary condition for the existence of periodical solutions of a differential system | 15$-$17 |

**Abstract**

The problem of necessary conditions for the existence of periodical solutions of the system $X'=\f(t)P(X)+Q(t)$ is considered, where $\f(t)$, $Q_i(t)$ ($i=1,\dots,n$) are continuous functions belonging to certain classes.

**Keywords:** Periodical solution, differential system.

**MSC:** 34C25

On a system of linear thermoelasticity with the Bessel operator | 19$-$26 |

**Abstract**

In this paper, we study an initial value problem for a one-dimensional system of thermoelasticity. Using an a priori bound and a density argument, we prove the existence and uniqueness of a generalized solution.

**Keywords:** Coupled system, thermoelasticiy, a priori bound.

**MSC:** 35K22, 58D25, 73B30

Numerical stability of a class (of systems) of nonlinear equations | 27$-$33 |

**Abstract**

In this article we consider stability of nonlinear equations which have the following form: $$ Ax+F(x)=b, \tag1 $$ where $F$ is any function, $A$ is a linear operator, $b$ is given and $x$ is an unknown vector. We give (under some assumptions about function $F$ and operator $A$) a generalization of inequality: $$ \frac{\|X_{1}-X_{2}\|}{\|X_{1}\|}\leq \|A\|\|A^{-1}\|\frac{\|b_{1}-b_{2}\|}{\|b_{1}\|} \tag2 $$ (equation (2) estimates the relative error of the solution when the linear equation $Ax=b_{1}$ becomes the equation $Ax=b_{2}$) and a generalization of inequality: $$ \frac{\|X_{1}-X_{2}\|}{\|X_{1}\|}\leq \|A_{1}^{-1}\|\|A_{1}\|\left(\frac{\|b_{1}-b_{2}\|}{\|b_{1}\|}+ \|A_{1}\|\|A_{2}^{-1}\|\frac{\|b_{2}\|}{\|b_{1}\|}\cdot \frac{\|A_{1}-A_{2}\|}{\|A_{1}\|}\right) \tag3 $$ (equation (3) estimates the relative error of the solution when the linear equation $A_1x=b_{1}$ becomes the equation $A_2x=b_{2}$).

**Keywords:** Numerical stability, nonlinear equations.

**MSC:** 65J15

A mapping theorem on $\aleph$-spaces | 35$-$40 |

**Abstract**

In this paper we give a mapping theorems on $\aleph$-spaces by means of strong compact-covering mappings and $\sigma$-mappings. As an application, we get characterizations of quotient (pseudo-open) $\sigma$-images of metric spaces.

**Keywords:** $\aleph$-spaces; strong compact-covering mappings;
$\sigma$-mappings; $k$-spaces; Fréchet spaces.

**MSC:** 54E99, 54C10

The Banach algebra $B(X)$, where $X$ is a BK space and applications | 41$-$60 |

**Abstract**

In this paper we give some properties of Banach algebras of bounded operators $B(X)$, when $X$ is a BK space. We then study the solvability of the equation $Ax=b$ for $b\in\{s_{\alpha },s_{\alpha}^{{{}^{\circ}}},s_{\alpha }^{( c)},l_{p}( \alpha)\}$ with $\alpha\in U^{+}$ and $1\leq p<\infty$. We then deal with the equation $T_{a}x=b$, where $b\in\chi(\Delta ^{k})$ for $k\geq 1$ integer, $\chi\in\{s_{\alpha },s_{\alpha }^{{{}^{\circ}}},s_{\alpha}^{(c)},l_{p}(\alpha)\}$, $1\leq p<\infty$ and $T_{a}$ is a Toeplitz triangle matrix. Finally we apply the previous results to infinite tridiagonal matrices and explicitly calculate the inverse of an infinite tridiagonal matrix. These results generalize those given in [4,~9].

**Keywords:** Infinite linear system, sequence space, BK space,
Banach algebra, bounded operator.

**MSC:** 40C05, 46A45