Abstract:
We consider the semilinear elliptic equation $-L u = f(u)$ in a general smooth bounded domain $\Omega \subset R^{n}$ with zero Dirichlet boundary condition, where $L$ is a uniformly elliptic operator and $f$ is a $C^{2}$ positive, nondecreasing and convex function in $[0,\infty)$ such that $\frac{f(t)}{t}\rightarrow\infty$ as $t\rightarrow\infty$. We prove that if $u$ is a positive semistable solution then for every $0\leq\beta<1$ we have $$f(u)\int_{0}^{u}f(t)f"(t)~e^{2\beta\int_{0}^{t}\sqrt{\frac{f"(s)}{f(s)}}ds}~dt\in L^{1}(\Omega),$$ by a constant independent of $u$. As we shall see, a large number of results in the literature concerning a priori bounds are immediate consequences of this estimate. In particular, among other results, we establish a priori $L^{\infty}$ bound in dimensions $n\leq 9$, under the extra assumption that $\limsup_{t\rightarrow\infty} \frac{f(t)f"(t)}{f'(t)^{2}} < \frac{2}{9-2\sqrt{14}}\cong 1.318$. Also, we establish a priori $L^{\infty}$ bound when $n\leq 5$ under the very weak assumption that, for some $\epsilon>0$, $\liminf_{t\rightarrow\infty} \frac{(tf(t))^{2-\epsilon}}{f'(t)} > 0$ or $\liminf_{t\rightarrow\infty} \frac{t^{2}f(t)f"(t)}{f'(t)^{\frac{3}{2}+\epsilon}} > 0$.

Abstract:
We consider the semilinear elliptic equation $-\Delta u =\lambda f(u)$ in a smooth bounded domain $\Omega$ of $R^{n}$ with Dirichielt boundary condition, where $f$ is a $C^{1}$ positive and nondeccreasing function in $[0,\infty)$ such that $\frac{f(t)}{t}\rightarrow\infty$ as $t\rightarrow\infty$. When $\Omega$ is an arbitrary domain and $f$ is not necessarily convex, the boundedness of the extremal solution $u^{*}$ is known only for $n= 2$, established by X. Cabr\'{e} \cite{C1}. In this paper, we prove this for higher dimensions depending on the nonlinearity $f$. In particular, we prove that if $$\frac{1}{2}<\beta_{-}:=\liminf_{t\rightarrow\infty} \frac{f'(t)F(t)}{f(t)^{2}}\leq \beta_{+}:=\limsup_{t\rightarrow\infty} \frac{f'(t)F(t)}{f(t)^{2}}<\infty$$ where $F(t)=\int_{0}^{t}f(s)ds$, then $u^{*}\in L^{\infty}(\Omega)$, for $n\leq 6$. Also, if $\beta_{-}=\beta_{+}>\frac{1}{2}$ or $\frac{1}{2}<\beta_{-}\leq\beta_{+}<\frac{7}{10}$, then $u^{*}\in L^{\infty}(\Omega)$, for $n\leq 9$. Moreover, if $\beta_{-}>\frac{1}{2}$ then $u^{*}\in H^{1}_{0}(\Omega)$ for $n\geq 2$.

Abstract:
Cotton is one of the most important crops in Iran, and is cultivated in different regions of the country. Gossypium herbaceum is one of the A-genome cottons, which is a potentially important genetic resource for cotton breeding programs. Collecting native cultivars of this species growing in different regions is a vital step in broadening variability of the gene pool. The G. herbaceum is one of the two cultivated species under cultivation in Iran, which is specifically adapted to a given environment and includes more than 40 ecotypes, named as landrace cottons. The present paper reports the intragenomic characteristics analysis of 42 G. herbaceum cultivars in the cotton genebank using cytological methods. The karyological studies showed variations within the species in the size of chromosome, chromosome volume and karyotype formulae. All cultivars possessed 2n=26 chromosome, but varied with regard to number of SAT-chromosomes (ranging from 1 to 3) and the chromosomes carrying secondary constructions. Karyotypes were of symmetrical type, having small chromosomes. Analysis of variance revealed significant differences between the cultivars as well as the chromosomes. Cluster analysis could group the cultivars in four distinct clusters. The present study indicates genomic differences among diploid G. herbaceum cultivars, which can be used in cotton hybridization programs in Iran or other countries.

Abstract:
We study the existence and multiplicity of solutions to the elliptic system $$displaylines{ -hbox{div}(| abla u|^{p-2} abla u)+m_1(x)|u|^{p-2}u =lambda g(x,u) quad xin Omega,cr -hbox{div}(| abla v|^{p-2} abla v)+m_2(x)|v|^{p-2}v=mu h(x,v) quad xin Omega,cr | abla u|^{p-2}frac{partial u}{partial n}=f_u(x,u,v),quad | abla v|^{p-2}frac{partial v}{partial n}=f_v(x,u,v), }$$ where $Omegasubset mathbb{{R}}^N$ is a bounded and smooth domain. Using fibering maps and extracting Palais-Smale sequences in the Nehari manifold, we prove the existence of at least two distinct nontrivial nonnegative solutions.

Abstract:
An efficient and eco-friendly method for the synthesis of polyhydroquinoline derivatives using Ce(SO4)2.4H2O as mild and heterogeneous Lewis acid catalyst via the Hantzsch reaction in very short reaction time is reported. A mixture of an appropriate aldehyde, dimedone, ethyl acetoacetate and malononitrile in the presence of the Ce(SO4)2.4H2O at reflux conditions in water based media resulted in good to excellent yields of the corresponding products. The catalyst can be used as selective for some aromatic aldehydes in the reaction conditions.DOI: http://dx.doi.org/10.4314/bcse.v26i3.16

Abstract:
We study generalized projectively symmetric spaces. We first study some geometric properties of projectively symmetric spaces and prove that any such space is projectively homogeneous and under certain conditions the projective curvature tensor vanishes. Then we prove that given any regular projective s-space ( , ), there exists a projectively related connection , such that ( , ) is an affine s-manifold. 1. Introduction Affine and Riemannian s-manifolds were first defined in [1] following the introduction of generalized Riemannian symmetric spaces in [2]. They form a more general class than the symmetric spaces of E. Cartan. More details about generalized symmetric spaces can be found in the monograph [3]. Let be a connected manifold with an affine connection , and let be the Lie transformation group of all affine transformation of . An affine transformation will be called an affine symmetry at a point if is an isolated fixed point of . An affine manifold will be called an affine s-manifold if there is a differentiable mapping , such that for each , is an affine symmetry at . In [4] Podestà introduced the notion of a projectively symmetric space in the following sense. Let be a connected manifold with an affine torsion free connection on its tangent bundle; is said to be projectively symmetric if for every point of there is an involutive projective transformation of fixing and whose differential at is . The assignment of a symmetry at each point of can be viewed as a map , and can be topologised, so that it is a Lie transformation group. In the above definition, however, no further assumption on is made; even continuity is not assumed. In this paper we define and state prerequisite results on projective structures and define projective symmetric spaces due toPodestà. Then we generalize them to define projective s-manifolds as manifolds together with more general symmetries and consider the cases where they are essential or inessential. A projective s-manifold is called inessential if it is projectively equivalent to an affine s-manifold and essential otherwise. We prove that these spaces are naturally homogeneous, and moreover under certain conditions the projective curvature tensor vanishes. Later we define regular projective s-manifolds and prove that they are inessential. 2. Preliminaries Let be a connected real manifold whose tangent bundle is endowed with an affine torsion free connection . We recall that a diffeomorphism of is said to be projective transformation if maps geodesics into geodesics when the parametrization is disregarded [5];

Abstract:
We define a symmetry for a Finsler space with Chern connection and investigate its implementation and properties and find a relation between them and flag curvature.

Abstract:
The Baer theorem states that for a group $G$ finiteness of $G/Z_i(G)$ implies finiteness of $\gamma_{i+1}(G)$. In this paper we show that if $G/Z(G)$ is finitely generated then the converse is true.

Abstract:
In this paper a new approach for obtaining an approximation global optimum solution of zero-one nonlinear programming (0-1 NP) problem which we call it Parametric Linearization Approach (P.L.A) is proposed. By using this approach the problem is transformed to a sequence of linear programming problems. The approximately solution of the original 0-1 NP problem is obtained based on the optimum values of the objective functions of this sequence of linear programming problems defined by (P.L.A).