Splitting Balanced Incomplete Block Designs with Block Size 2×k

This paper investigated the existence of splitting balanced incomplete block designs with blck size 2× k .The necessary conditions for such a design are λ(v-1) ≡0(mod k ),and λv(v- 1)≡0(mod 2 k 2).It will show that the above necessary conditions are also sufficient for k =3 with the definite exception( v,λ )=(10,1) and with several possible exceptions.

Neighbour effects on plant biomass and its allocation for forbs growing in heterogeneous soils

Focal plants are considerably affected by their neighbouring plants,especially when growing in heterogeneous soils.A previous study on grasses demonstrated that soil heterogeneity and species composition affected plant biomass and above-and belowground allocation patterns.We now tested whether these findings were similar for forbs.Three forb species(i.e.Spartina anglica,Limonium bicolor and Suaeda glauca)were grown in pots with three levels of soil heterogeneity,created by alternatively filling resource-rich and resource-poor substrates using small,medium or large patch sizes.Species compositions were created by growing these forbs either in monocultures or in mixtures.Results showed that patch size×species composition significantly impacted shoot biomass,root biomass and total biomass of forbs at different scales.Specifically,at the pot scale,shoot biomass,root biomass and total biomass increased with increasing patch size.At the substrate scale,shoot biomass and total biomass were higher at the large patch size than at the medium patch size,both in resource-rich and resource-poor substrates.Finally,at the community scale,monocultures had more shoot biomass,root biomass and total biomass than those in the two-or three-species mixtures.These results differ from earlier findings on the responses of grasses,where shoot biomass and total biomass decreased with patch size,and more shoot biomass and total biomass were found in resource-rich than resource-poor substrates.To further elucidate the effects of soil heterogeneity on the interactions between neighbour plants,we advise to conduct longer-term experiments featuring a variety of functional groups.

The lower and upper bounds of the degree distribution of a graded algebra

For a graded algebra,the minimal projective resolution often reveals amounts of information.All generated degrees of modules in the minimal resolution of the trivial module form a sequence,which can be called the degree distribution of the algebra.We try to find lower and upper bounds of the degree distribution,introduce the notion of(s,t)-(homogeneous) determined algebras and construct such algebras with the aid of algebras with pure resolutions.In some cases,the Ext-algebra of an(s,t)-(homogeneous) determined algebra is finitely generated.