Research proof index

Zero-extension of a compactly supported local W₀^{1,p} witness to all of ℝ^d.

Δ transforms under S as: Δ(-1/z) = z¹² · Δ(z) -/ lemma Δ_S_transform (z : ℍ) : Δ (ModularGroup.S • z) = z ^ (12 : ℕ) * Δ z := by have h := Discriminant_S_invariant simp only [funext_iff] at h specialize h z rw [SL_slash_apply] at h simp only [ModularGroup.denom_S, zpow_neg] at h field_simp [ne_zero z] at h rw [h, mul_comm] theorem Delta_boundedfactor : Te...

If every tile in $P$ lies at the fixed generation $k$, then the first global density of $P$ is at most the generation-specific density $\mathrm{dens}'(k,P)$.

This is the dual of Dense.comap_val_nhdsWithin_Ioi_neBot.

This is an auxillary lemma used to prove Dense.monotone_of_isRightContinuous. It is saying that if D is a dense set and a, b are two points such that a < b, then the comap of 𝓝[Set.Ioi a] a under the inclusion D → α is nontrivial. Note that a < b is necessary as this is clearly not true if a is a top element.

If f is monotone on a dense set D and is right continuous, then f is monotone. We prove under the assumption that α has a top element ⊤ and ⊤ ∈ D, which is a necessary assumption because otherwise it is possible that ⊤ is an isolated point. This theorem should be also true when α satisfies NoTopOrder α.