Volume 55 , issue 3$-$4 ( 2003 ) | back |

On gnomons | 59$-$64 |

**Abstract**

A gnomon is a shape which, when added to a figure, yields a figure that is similar to the original one. Gazalé~[4] conjectured that if the gnomon is a regular polygon and the figure to which it is added is a finite polygon then the gnomon is a triangle or a square. We prove this conjecture to be correct.

**Keywords:** Gnomon, polygon, tesselation, spiral, discontinuous group action.

**MSC:** 51M99

Some remarks on the category SET(L), Part II | 65$-$82 |

**Abstract**

This paper considers some intrinsic properties of the category SET(L) of L-subsets of sets with a fixed basis L and is a continuation of our previous work [4]. Here we study properties of some abstract functors when applied to the category SET(L) as well as some special objects related to them. In the last section we consider two standard constructions, namely, inverse and direct systems in this category.

**Keywords:** Category theory, L-fuzzy set,
functor, special morphism, special object, standard construction.

**MSC:** 03E72, 04A72, 18B99

A class of univalent functions defined by using Hadamard product | 83$-$96 |

**Abstract**

In this paper we introduce the class $L_{\alpha}^*(\lambda,\beta)$ of functions defined by $f*S_{\alpha}(z)$ of $f(z)$ and $S_{\alpha}=\dfrac z{(1-z)^{2(1-\alpha)}}$. We determine coefficient estimates, closure theorems, distortion theorems and radii of close-to-convexity, starlikeness and convexity. Also we find integral operators and some results for Hadamard products of functions in the class $L_{\alpha}^*(\lambda,\beta)$. Finally, in terms of the operators of fractional calculus, we derive several sharp results depicting the growth and distortion properties of functions belonging to the class $L_{\alpha}^*(\lambda,\beta)$.

**Keywords:** Univalent functions, Hadamard product, fractional calculus.

**MSC:** 30C45

Spectral problems for parabolic equations with conditions of conjugation and dynamical boundary conditions | 97$-$106 |

**Abstract**

The initial boundary value problems for the heat equation with discontinuous heat flow and concentrated heat capacity in the interior points or at the boundary are considered. The corresponding spectral problems, where the eigenvalues appear in the boundary or interface conditions, are derived and studied.

**Keywords:** Parabolic equations, conjugation
conditions, dynamical boundary conditions, spectral problems.

**MSC:** 35K20, 35P99

Sufficient conditions for elliptic problem of optimal cnotrol in $R^n$, where $n>2$ | 107$-$119 |

**Abstract**

This paper is concerned with the local minimization problem for a variety of non Frechet-differentiable G\^ateaux functional $J(f)\equiv ınt_{Q}v(x,u,f)\,dx$ in the Sobolev space $(W^{1,2}_0(Q),\|\cdot\|_p)$, where $u$ is the solution of the Dirichlet problem for a linear uniformly elliptic operator with nonhomogenous term $f$ and $\|\cdot\|_{p}$ is the norm generated by the metric space $L^p(Q)$, $(p>1)$. We use a recent extension of Frechet-differentiability (approach of Taylor mappings, see [5]), and we give various assumptions on $v$ to guarantee a critical point to be a strict local minimum. Finally, we give an example of a control problem where classical Frechet differentiability cannot be used and their approach of Taylor mappings works.

**Keywords:** Elliptic problem, optimal control, local minimization, Dirichlet
problem, Frechet-differentiability.

**MSC:** 49K20