Let (X,+)(X,+) be a topological abelian group. We discuss regularity of solutions f:X®Rf:X®R of Hlawka’s functional inequality f(x+y)+f(y+z)+f(x+z)£f(x+y+z)+f(x)+f(y)+f(z),f(x+y)+f(y+z)+f(x+z)≤f(x+y+z)+f(x)+f(y)+f(z), postulated for all x,y,z∈Xx,y,z∈X. We study the lower and upper hull of ff. Moreover, we provide conditions which imply continuity of ff. We prove, in particular, that if XX is generated by any neighborhood of zero, f is continuous at zero, and f(0)=0f(0)=0, then f is continuous on X.

At the Fourty-forth International Symposium on Functional Equations in 2006, Louisville, Kentucky, USA, T.M.K. Davison presented a talk under the title "D’Alembert’s functional equation on compact groups" in which he offered a solution method for this equation on any compact group. In this paper we offer another approach based on some very recent results on spectral synthesis, which method seems to be useful in the case of other equations.

In this paper, we study the stability of the system of functional equations ¦(xy) + ¦(xy-1) = 2¦(x) + ¦(y) + ¦(y-1) and ¦(yx) + ¦(y-1x) = 2¦(x) + ¦(y) + ¦(y-1) on groups. Here f is a real-valued function that takes values on a group. Among others we proved the following results: 1) the system, in general, is not stable on an arbitrary group; 2) the system is stable on Heisenberg group UT(3,K), where K is a commutative field with characteristic different from two; 3) the system is stable on certain class of n-Abelian groups; 4) any group can be embedded into a group where this system is stable.

Aim Plant functional groups are categories of species that exhibit similar responses to environmental parameters. The current research was carried out to evaluate factors that describe the distribution of plant functional groups in mountainous rangelands of Chaharbagh, Golestan. Methods Three transects of 300 meters were considered in the representative area of each plant type. Along each transect, 10 plots were considered in dimensions of 2 x 2 meters. The names of species, the percentage of vegetation, and the number of species were determined. Six soil samples were collected for each plant type. The biological form of each plant species was determined. Plants were classified based on edibility, vegetative form, biological form, and life span. To evaluate the relationship between environmental factors and plant functional groups, Redundancy Detrended Analysis (RDA) was applied. Findings The results showed that the occurrence of phanerophytes was affected by electrical conductivity and lime. With an increase in potassium, the frequency of camophytes increased. Palatability values of the first class were affected by direction, saturated moisture content, and organic carbon. Values of tree-type growth were affected by electrical conductivity and lime. Forb vegetation increased in correlation with clay percentage. Grass and pseudo-grass vegetative forms were affected by direction factors, soil saturated moisture content, and organic carbon. Conclusion Overall, a precise study of differences in plant functional groups can be an important approach for monitoring changes in rangeland under management strategies which require further attention in future research.