How to use acts properly in a sentence? Find typical usage patterns (collocations)/phrases/context for "acts properly" and check conjugation/comparative form for "acts properly". Mastering all the usages of "acts properly" from sentence examples published by news publications.
Sioux Nation, 448 U.S. at 408. While reaffirming earlier decisions that Congress has "paramount authority over the property of the Indians," the Court concluded that Congress acts properly only if it "makes a good faith effort to give the Indians the full value of the land," which here it had failed to do.U.S. v. Sioux Nation, 448 U.S. at 409.
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In geometric group theory, groups are studied by their actions on metric spaces. A principle that generalizes the bilipschitz invariance of word metrics says that any finitely generated word metric on G is quasi-isometric to any proper, geodesic metric space on which G acts, properly discontinuously and cocompactly. Metric spaces on which G acts in this manner are called model spaces for G. It follows in turn that any quasi-isometrically invariant property satisfied by the word metric of G or by any model space of G is an isomorphism invariant of G. Modern geometric group theory is in large part the study of quasi-isometry invariants.
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In mathematics, a Clifford–Klein form is a double coset space :Γ\G/H, where G is a reductive Lie group, H a closed subgroup of G, and Γ a discrete subgroup of G that acts properly discontinuously on the homogeneous space G/H. A suitable discrete subgroup Γ may or may not exist, for a given G and H. If Γ exists, there is the question of whether Γ\G/H can be taken to be a compact space, called a compact Clifford–Klein form. When H is itself compact, classical results show that a compact Clifford–Klein form exists. Otherwise it may not, and there are a number of negative results.
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A linear fractional transformation defined by a matrix from PSL(2,C) will preserve the Riemann sphere P1(C) = C ∪ ∞, but will send the upper-half plane H to some open disk Δ. Conjugating by such a transformation will send a discrete subgroup of PSL(2,R) to a discrete subgroup of PSL(2,C) preserving Δ. This motivates the following definition of a Fuchsian group. Let Γ ⊂ PSL(2,C) act invariantly on a proper, open disk Δ ⊂ C ∪ ∞, that is, Γ(Δ) = Δ. Then Γ is Fuchsian if and only if any of the following three equivalent properties hold: # Γ is a discrete group (with respect to the standard topology on PSL(2,C)). # Γ acts properly discontinuously at each point z ∈ Δ. # The set Δ is a subset of the region of discontinuity Ω(Γ) of Γ. That is, any one of these three can serve as a definition of a Fuchsian group, the others following as theorems. The notion of an invariant proper subset Δ is important; the so-called Picard group PSL(2,Z[i]) is discrete but does not preserve any disk in the Riemann sphere.
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