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

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

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

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

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

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

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