In this paper, we have studied the topology of some classical functional spaces. Among these spaces, there are standard spaces, spaces that can be metrizable and others that cannot be metrizable. But they are all topo...In this paper, we have studied the topology of some classical functional spaces. Among these spaces, there are standard spaces, spaces that can be metrizable and others that cannot be metrizable. But they are all topological vector spaces and it is in this context that we have chosen to present this work. We are interested in the topology of its spaces and in the topologies of their dual spaces. The first part, we presented the fundamental topological properties of topological vector spaces. The second part, we studied Frechet spaces and particularly the space S(R<sup>n</sup>) of functions of class C<sup>∞ </sup>on R<sup>n</sup> which are as well as all their rapidly decreasing partial derivatives. We have also studied its dual S'(Rn</sup>) the space of tempered distributions. The last part aims to define a topological structure on an increasing union of Frechet spaces called inductive limit of Frechet spaces. We study in particular the space D(Ω) of functions of class C<sup>∞</sup> with compact supports on Ω as well as its dual D' (Ω) the space distributions over the open set Ω.展开更多
The objectives of this study were to assess distribution patterns of plant species richness,plant diversity and vegetation structure in relation to environmental factors along elevation gradient in Al-Jabal Al-Akhdar,...The objectives of this study were to assess distribution patterns of plant species richness,plant diversity and vegetation structure in relation to environmental factors along elevation gradient in Al-Jabal Al-Akhdar,Libya.For each species,its growth form,chorological affinities,degree of occurrence,endemism status and originality were provided.A total of 534 taxa were generated from 70 families of the flowering plants,and 3 of the non-flowering plants were identified.The native flora of the study area was counted for 80.2%of the total indigenous taxa(465).The hump-shaped pattern of plant species richness was obtained,with the highest species richness at the mid-elevations,and both ends of the gradient have the lowest.The annuals(300 species,56.2%)and non-succulent perennial herbs(165 species,30.9%)were the most dominant growth forms of the total species composition.Along the elevation gradient,Asteraceae,Fabaceae,Poaceae,Lamiaceae and Apiaceae were the dominant families with the highest numbers of species.This investigation recorded 31 endemic taxa,comprising 25 dicots and 6 monocots,primarily of Mediterranean origin,with most belonging to the Asteraceae and Lamiaceae families.Mediterranean chorotype was the dominant,whether pure(mono-),or combined with one(bi-and pluri-regional)or more(pluri-regional).Application of cluster analysis on the vegetation data yielded four cluster groups;each was linked to an elevation level.It emphasized the importance of establishing conservation strategies to minimize human disturbance and safeguard relic habitats of Juniperus phoenicea L.at its southern distribution limits in Africa,underlining the proactive management required for species preservation.The application of Redundancy Analysis revealed that Shannon diversity index(H'),pH and Fe were the determinant soil factor in the mid-elevation levels(L2 and L3)whereas altitude,fine sand,HCO3,OM,and soil contents of Na,SO4 and Cl for the extreme levels(L1 and L4).展开更多
文摘In this paper, we have studied the topology of some classical functional spaces. Among these spaces, there are standard spaces, spaces that can be metrizable and others that cannot be metrizable. But they are all topological vector spaces and it is in this context that we have chosen to present this work. We are interested in the topology of its spaces and in the topologies of their dual spaces. The first part, we presented the fundamental topological properties of topological vector spaces. The second part, we studied Frechet spaces and particularly the space S(R<sup>n</sup>) of functions of class C<sup>∞ </sup>on R<sup>n</sup> which are as well as all their rapidly decreasing partial derivatives. We have also studied its dual S'(Rn</sup>) the space of tempered distributions. The last part aims to define a topological structure on an increasing union of Frechet spaces called inductive limit of Frechet spaces. We study in particular the space D(Ω) of functions of class C<sup>∞</sup> with compact supports on Ω as well as its dual D' (Ω) the space distributions over the open set Ω.
文摘The objectives of this study were to assess distribution patterns of plant species richness,plant diversity and vegetation structure in relation to environmental factors along elevation gradient in Al-Jabal Al-Akhdar,Libya.For each species,its growth form,chorological affinities,degree of occurrence,endemism status and originality were provided.A total of 534 taxa were generated from 70 families of the flowering plants,and 3 of the non-flowering plants were identified.The native flora of the study area was counted for 80.2%of the total indigenous taxa(465).The hump-shaped pattern of plant species richness was obtained,with the highest species richness at the mid-elevations,and both ends of the gradient have the lowest.The annuals(300 species,56.2%)and non-succulent perennial herbs(165 species,30.9%)were the most dominant growth forms of the total species composition.Along the elevation gradient,Asteraceae,Fabaceae,Poaceae,Lamiaceae and Apiaceae were the dominant families with the highest numbers of species.This investigation recorded 31 endemic taxa,comprising 25 dicots and 6 monocots,primarily of Mediterranean origin,with most belonging to the Asteraceae and Lamiaceae families.Mediterranean chorotype was the dominant,whether pure(mono-),or combined with one(bi-and pluri-regional)or more(pluri-regional).Application of cluster analysis on the vegetation data yielded four cluster groups;each was linked to an elevation level.It emphasized the importance of establishing conservation strategies to minimize human disturbance and safeguard relic habitats of Juniperus phoenicea L.at its southern distribution limits in Africa,underlining the proactive management required for species preservation.The application of Redundancy Analysis revealed that Shannon diversity index(H'),pH and Fe were the determinant soil factor in the mid-elevation levels(L2 and L3)whereas altitude,fine sand,HCO3,OM,and soil contents of Na,SO4 and Cl for the extreme levels(L1 and L4).
