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...

Visual-vestibular and postural interactions can act as cues that trigger motion sickness and can also have a role in some anxiety disorders. We explore a method to detect individual sensitivity to visual-vestibular unusual patterns, which can signal a vulnerability to develop motion sickness and possibly anxiety disorders such as a fear of heights and panic. 65 undergraduate students were recru...

In the last few years in Malaysia, dengue fever has increased dramatically and has caused huge public health concerns. The present study aimed to establish a spatial distribution of dengue cases in the city of Kuala Lumpur using a combination of Geographic Information System (GIS) and spatial statistical tools. Collation of data from 1,618 dengue cases in 2009 was obtained from Kuala Lumpur Cit...

The creation of a surface breaks the translational symmetry of the bulk with the electrons redistributing to lower their energy, resulting in a new lattice structure. Surface relaxation and reconstruction have been studied for years, and it is generally assumed that theory and experiment are in fairly good agreement for the ground state properties at a metal surface. There is a conceptual pictu...

Four novel oxapenem compounds (i.e., AM-112, AM-113, AM-114, and AM-115) were investigated for their beta-lactamase inhibitory activity against a panel of isolated class A, C, and D enzymes, which included expanded-spectrum beta-lactamase enzymes (ESBLs). The oxapenems were potent beta-lactamase inhibitors. Activity varied within the group, with AM-113 and AM-114 proving to be the most active c...