Human activities have more than doubled inputs of nitrogen (N) to terrestrial ecosystems worldwide (1). Until recently, most of this increase was focused in the developed regions of the world. With fertilizer consumption (2), fossil-fuel emissions (3), and biomass burning (4) rapidly increasing in less developed regions, sources and distribution of anthropogenic N now occur globally (5), as doe...

Long-term day-to-day monitoring of physical activity (PA) has not been undertaken in adolescents despite PA declines rapidly during adolescence. This study monitored the school year-round pedometer-determined PA of pupils attending high school in the Czech Republic. We assessed their PA levels; appraised the school year-round variability of their PA; and, assessed the associations between their...

V = U ⊕ U ′, for some other subspace U ′ ⊂ V , by Theorem 3.21. Consider T restricted to U ′. This is a map T : U ′ →W , which is injective, since U ′∩ker(T ) = {0}. And note that injective maps, in general, preserve linear independence: if we have a linearly independent set {~x1, . . . , ~xn} ∈ X in a vector space X, a linear map of vector spaces A : X → Y , and a linear combination 0 = λ1A~x1...

Proof. Suppose that U1, . . . , Um are invariant subspaces of T ∈ L(V ). We wish to show that U1 + · · ·+ Um is also an invariant subspace. Consider any ~v ∈ U1 + · · ·+ Um. By definition, we can write ~v = ~u1 + · · ·+ ~um, for some ~u1 ∈ U1, . . . , ~um ∈ Um. Since U1, . . . , Um are invariant subspaces, by definition, we know T~u1 ∈ U1 ⊂ U1 + · · ·+ Um, ..., T~um ∈ Um ⊂ U1 + · · ·+ Um. Hence...

Water having entered a stream channel from the surrounding catchment may continue to have connections with the catchment. In the stream’s hyporheic zone, water “in the channel” exchanges with “groundwater” in the bed of the stream. Hyporheic exchange flows typically occur at scales small relative to the length and volumetric transport characteristics of the stream. Nevertheless, it is well docu...