(2012) used an

arcsin transformation if percent coverage data sets were not normal.

[[??].sup.ref.sub.k] =

arcsin ([angle][[PHI].sub.c]].sub.k,k]/4[pi]d/[lambda]), k = 1, ..., K.

Data were transformed to

arcsin [square root of (x/100)] for statistical analysis (Taylor, 1984).

Y = sgn(y) [square root of [x.sup.2] + [y.sup.2]]/[square root of 2[pi]] ([pi]/2 -

arcsin [x.sup.2] - [y.sup.2]/[x.sup.2] + [y.sup.2]).

[alpha] [less than or equal to] 90[degrees] -

arcsin ([n.sub.2]/[n.sub.1]) = 90[degrees] -

arcsin (1.46/1.49) = 90[degrees] - 78[degrees] = 12[degrees] (11)

[alpha] = [2/[pi]]

arcsin ([7/128] [[square root of 207] + [square root of 15]]) (21)

Dendrogram was performed on the basis of

arcsin transformed percentage relative frequency (%O) and consumed biomass (%B) data of 10 main food types (same food types as listed above, except pheasant and other birds were merged).

[alpha] =

arcsin h/R =

arcsin 2gh/[g.sup.2] + [h.sup.2].

(1.3) [q.summation over (j=1)] ([[alpha].sub.k+1] - [[beta].sub.j]) < 4/[sigma]

arcsin [square root of ([delta]/2)],

Therefore, we also transformed the original data using

arcsin, a process typical for percentage data (Sokal and Rolf, 1981), and analyzed them, using a paired t-test in addition to a related-samples Wilcoxon signed-rank test.

c) If L is quasismooth (in the sense of Lavrentiev), that is, for every pair [z.sub.1], [z.sub.2] [member of] L, if s([z.sub.1], [z.sub.2]) represents the smallest of the lengths of the arcs joining [z.sub.1] to [z.sub.2] on L, there exists a constant c > 1 such that s([z.sub.1], [z.sub.2]) [less than or equal to] c [absolute value of ([z.sub.1] - [z.sub.2])], then [PHI] [member of] Lip [alpha] for [alpha] = [1/2][(1 - [1/[pi]]

arcsin [1/c]).sup.-1] and [PSI] [member of] Lip [beta] for [beta] = [2/[(1 + c).sup.2]] [26], [27].

Analysis of variance was used for all statistical comparisons (PROC GLM, SAS Institute, 2013) and proportions were

arcsin, square root transformed prior to analysis.