Volume 53 , issue 3$-$4 ( 2001 ) | back |

Recherche des ensembles minimum pour une cascade de files d'attente avec impatiences | 61$-$70 |

**Abstract**

Des conditions de récurrence et de récurrence positives sont obtenues sur un syst\`eme de files d'attente dans lequel les clients impatients quittent la chaine. Les temps des inter-arrivés, les temps de service et les temps d'impatience sont indépendants et indépendantes entre eux.

**Keywords:** Chaines de Markov, récurrence, récurrence positive,
file d'attente, tandem.

**MSC:** 60K25

Approximate solution of a boundary problem for a linear complex differential equation of the second order | 71$-$76 |

**Abstract**

The paper considers the problem $$ D^{(2)}w+a(z,\bar z)Dw+b(z,\bar z)w=f(z,\bar z), $$ with boundary conditions $$ \aligned c_1(z)\a_{g(z)}w+c_2\a_{g(z)}Dw&=c_3(z),\\ d_1(z)\a_{h(z)}w+d_2\a_{h(z)}Dw&=d_3(z). \endaligned $$ The problem is solved approximately, by using the formulas $$ \align 2\df{z^2}{h^2}(w_{i+1}-2w_i+w_{i-1})+a_i\df zh(w_{i+1}-w_{i- 1})+b_iw_i&=f_i,\quad i=1,\dots,n-1,\\ c_1(z)w_0+c_2(z)\df zh(-w_2+4w_1-3w_0)&=c_3(z),\\ d_1(z)w_n+d_2(z)\df zh(3w_n-4w_{n-1}+w_{n-2})&=d_3(z). \endalign $$

**Keywords:** Complex $\psi$ differences, boundary problem.

**MSC:** 34M20, 65L10

A generalization of Darboux theorem | 77$-$78 |

**Abstract**

We show a generalization of the fundamental Darboux theorem that states intermediate property for the derivative function of a real differentiable function. We extend this result for pairs of differentiable functions, i.e., for flat differentiable arcs.

**Keywords:** Mean value theorems, analysis.

**MSC:** 26A24

On a nonlocal singular mixed evolution problem | 79$-$89 |

**Abstract**

In the present paper, the existence and uniqueness of the strong solution of a mixed problem for a second order plurihyperbolic equation with an integral condition is proved. The proof is essentially based on an a priori bound and on the density of the range of the operator generated by the considered problem. In spite of the apparant simplicity of the problem, the solution requires a delicate set of techniques. It seems very difficult to extend these technics to the considered equation in more than one dimension without imposing complementary conditions.

**Keywords:** Strong solution, a priori bound, plurihyperbolic equation.

**MSC:** 35L20, 35L67

Some new properties of sequence spaces and application to the continued fractions | 91$-$102 |

**Abstract**

We give two methods of approximation of a solution of an infinite linear system. First, we will construct a sequence of finite matrices which approaches a solution, this one being defined by an infinite sequence. Then, we will apply these results to the continued fractions.

**Keywords:** Sequence spaces, continued fractions

**MSC:** 46A15, 40A15

Some theorems on cosymplectic hypersurfaces of six-dimensional Hermitian submanifolds of Cayley algebras | 103$-$110 |

**Abstract**

Diverse properties of cosymplectic hypersurfaces in six-dimensional Hermitian submanifolds of Cayley algebra are considered.

**Keywords:** Hermitian manifold, almost contact
metric structure, ruled manifold, minimal submanifols, cosymplectic
structure, $g$-cosymplectic hypersurfaces axiom, type number.

**MSC:** 53C40

Extremal properties of the chromatic polynomials of connected 3-chromatic graphs | 111$-$116 |

**Abstract**

In this paper the greatest $\lceil n/2\rceil$ values of $P(G;3)$ in the class of connected 3-chromatic graphs $G$ of order $n$ are found, where $P(G;\lambda)$ denotes the chromatic polynomial of~$G$.

**Keywords:** Chromatic polynomial, connected
3-chromatic graph, 3-color partition, skeleton of a graph.

**MSC:** 05C15

Bilinear expansions of the kernels of some nonselfadjoint integral operators | 117$-$123 |

**Abstract**

Let $H$ and $S$ be integral operators on $L^2(0,1)$ with continuous kernels. Suppose that $H>0$ and let $A=H(I+S)$. It is shown that if the (nonselfadjoint) operator $S$ is small in a certain sense with respect to $H$, then the corressponding Fourier series of functions from $R(A)$ (or $R(A^*)$) converges uniformly on $[0,1]$.

**Keywords:** Nonselfadjoint integral operators, bilinear expansion.

**MSC:** 47G10, 45P05