Vegetation classification should reflect the major environmental and phytogeographical gradients that influence species composition. However, the importance of different gradients depends on the geographical scale of particular studies. Locally defined vegetation units usually reflect local gradients, such as disturbance and soil properties, while regionally defined units reflect macroclimatic patterns and different evolutionary and migration histories of large regions. The classification of Central European oak-hornbeam forests (Carpinion alliance) is an example of a widely accepted, broad-scale classification with geographically delimited associations. However, in some cases it fails to describe adequately local vegetation patterns. In the Czech Republic, six associations, based on a broad-scale regional classification, were traditionally distinguished: (1) Melampyro nemorosiCarpinetum – Hercynian association; (2) Primulo veris-Carpinetum – Pannonian association; (3) Carici pilosae-Carpinetum – Carpathian association; (4) Tilio-Carpinetum – Polonian association; (5) Stellario-Tilietum – local association of southern Bohemia; (6) Tilio-Betuletum – ecologically delimited and local association. The goal of this study was to evaluate, using a cluster analysis of a set of 601 geographically stratified relevés, whether the variation in species composition of oakhornbeam forests in the Czech Republic reflects the traditional geographically based classification. Hercynian, Pannonian and Carpathian types of oak-hornbeam forests were reproduced by cluster analysis. No support was found for the Polonian type and the two local associations, StellarioTilietum and Tilio-Betuletum. Instead, a distinct group confined to wet soils emerged. On the basis of this analysis, we suggest a classification that combines ecological and geographical principles and distinguishes four associations: (1) Hercynian (Melampyro nemorosi-Carpinetum or Galio sylvatici-Carpinetum); (2) Pannonian (Primulo veris-Carpinetum); (3) Carpathian (Tilio cordataeCarpinetum or Carici pilosae-Carpinetum); (4) Stellario holosteae-Carpinetum, which is a geographically delimited association of atlantic northwestern central Europe, defined ecologically by its occurrence on wet soils within the Czech Republic.
Consider the difference equation ∆x(n) +∑m i=1 pi(n)x(τi(n)) = 0, n ≥ 0 [ ∇x(n) − ∑m i=1 pi(n)x(σi(n)) = 0, n ≥ 1 ] , where (pi(n)), 1 6 i 6 m are sequences of nonnegative real numbers, τi(n) [σi(n)], 1 6 i 6 m are general retarded (advanced) arguments and ∆ [∇] denotes the forward (backward) difference operator ∆x(n) = x(n + 1) − x(n) [∇x(n) = x(n) − x(n − 1)]. New oscillation criteria are established when the well-known oscillation conditions lim sup n→∞ ∑m i=1 ∑n j=τ(n) pi(j) > 1 [ lim sup n→∞ ∑m i=1 σ∑ (n) j=n pi(j) > 1 ] and lim inf n→∞ ∑m i=1 n∑−1 j=τi(n) pi(j) > 1⁄e [ lim inf n→∞ ∑m i=1 σ∑i(n) j=n+1 pi(j) > 1⁄e ] are not satisfied. Here τ (n) = max 1≤i≤m τi(n) [σ(n) = min 1≤i≤m σi(n)]. The results obtained essentially improve known results in the literature. Examples illustrating the results are also given.