In [7] a new infinite class $\mathbb G_n$, $n \geq 0$, of near polygons was defined. The near $2n$-gon $\mathbb G_n$ has the property that it contains $\mathbb G_{n-1}$ as a big geodetically closed sub near polygon. In this paper, we determine all near $2n$-gons, $n \geq 4$, having $\mathbb G_{n-1}$ as a big geodetically closed sub near $2(n-1)$-gon under the additional assumption that every two points at distance 2 have at least two common neighbours. We will prove that such a near $2n$-gon is isomorphic to either $\mathbb G_n$, $\mathbb G_{n-1} \otimes \mathbb G_2$, or $\mathbb G_{n-1} \times L$ for some line $L$.

Nous établissons des résultats de non existence de solutions pour des problèmes semilinéaires elliptiques et d'évolution d'ordre arbitraire en temps. Deux types de domaines sont considérés : les produits de cônes et les produits de cônes tronqués. On retrouve, en particulier et d'une manière tout à fait différente, le résultat dû à Ohta et Kaneko concernant l'exposant critique de Fujita dans le cas d'un produit de domaines. Notre approche est fondée sur la méthode des {\em fonctions test} développée par Mitidieri & Pohozaev, Pohozaev & Tesei et Pohozaev & Véron.

Nonexistence of global solutions to semilinear elliptic and "heigher- order evolution" inequalities is studied. Two types of domains are considered: product of cones and product of cone-like domains. We find, in particular and with a different method, the result of Ohta and Kaneko concerning Fujita's exponents on product domains. Our approach is based on the test-function method, developed by Mitidieri & Pohozaev, Pohozaev & Tesei and Pohozaev & Véron.

In this paper, it is proved that if $f:D\rightarrow D^{\prime } $ is a holomorphic homeomorphism between two domains $D$ and $D^{\prime }$ in $\CC^n(n\geq 2)$ which commutes with the Lelong transformation $T,$ then $f$ extends to a holomorphic homeomorphism $\widetilde{f}$ between the corresponding cells of harmonicity ${\cal H}(D)$ and ${\cal H}(D^{\prime }).$ In such way a generalization is given of Jarnicki 's result obtained in the case $n=1.$

Let $\mathbb R_{0,n}$ be the Clifford algebra of the antieuclidean vector space of dimension $n$. The aim is to built a function theory analogous to the one in the $\mathbb C$ case. In the latter case, the product of two holomorphic functions is holomorphic, this fact is, of course, of paramount importance. Then it is necessary to define a product for functions in the Clifford context. But, non-commutativity is inconciliable with product of functions. Here we introduce a product which is commutative and we compute some examples explicitely.

In this paper we prove the basic matrix theorem of Antosik-Swartz under weaker hypotheses than the ones they used. We obtain the converse result for complete normed spaces and generalize Antosik's interchange theorem for double series in a normed space. As a consequence, a number of characterizations on convergence in several spaces of vector sequences are derived. Finally, we obtain a version of the Orlicz-Pettis theorem for Banach spaces with a Schauder basis.

For a nonlinear differential equation $x^{\prime\prime}+a(t)f(x)=0,$ we obtain limit-point criteria by proving first stronger results which guarantee nonexistence of nontrivial bounded (uniformly continuous) $L^{2}$-solutions under milder restrictions on the coefficient $a(t)$ and nonlinearity $f(x)$.

We consider the existence and nonexistence of the positive periodic solutions of the non-autonomous nonlinear Duffing equation with delay and variable coefficients $$ x''(t)-p(t)x'(t)+ q(t)x(t)=\lambda h(t)f(t,x(t-\tau(t)))+r(t), $$ It is shown that the equation has positive periodic solutions under certain conditions and no positive $T-$periodic solution under some other conditions by using a fixed theorem in cones.

The sets we are going to consider here are of the form ${z\in\mathbb C \mid |A(z)|=1}$ (equipotential) and ${z\in\mathbb C \mid IM A(z)=0}$ (harmonic) with $A$ being a polynomial with complex coefficients. There are two themes which we want to focus on and which come out from invariance property of inner products on $\mathbb C[Z]$ related to the aforesaid sets. First, we formalize the construction of integral representation of the inner products in question with respect to matrix measure. Then we show that these inner products when represented in a Sobolev way are precisely those with discrete measures in the higher order terms of the representation. In this way we fill up the case already considered in [3] by extending it from the real line to harmonic sets on the complex plane as well as we describe completely what happens in this matter on equipotential sets. As a kind of smooth introduction to the above we are giving an account of standard integral representations on the complex plane in general and of those supported by these two kinds of real algebraic sets.