Research proof index

For symmetric T and ‖ψ‖ = 1, variance equals ‖Tψ‖ ^ 2 - ⟨T⟩_ψ ^ 2.

For ‖ψ‖ = 1, zero variance iff ψ is an eigenvector with eigenvalue ⟨T⟩_ψ.

If the MCA condition IsMCA holds for a projected generator, then IsMCA holds for the original generator G with the zero-extension defined above.

Let $\mathfrak A$ be an antichain contained in the tile class $\mathfrak C(k,n)$, and let $f$ be measurable with $\lVert f(x)\rVert\le\mathbf 1_F(x)$. On $G\setminus G'$, the Carleson sum over the selected positive tiles in $\mathfrak A$ has an $L^1$ bound with exponential decay in the layer index $n$:

$\int_{G\setminus G'}^+\!\lVert\operatorname{CarlesonSum}f(x)\rVert\,dx\le C(a,q)\,\mu(G)^{1-1/q}\mu(F)^{1/q}\,2^{-((q-1)/(8a^4))n}.$

If the project’s inhomogeneous Lipschitz norm of $\varphi$ on the ball $B(z,R)$ is finite, then $\varphi$ is Lipschitz on that ball. A valid Lipschitz constant is the finite normalized norm divided by $R$.

Illustration that equiv_implies_canonical_eq does not hold for all functions f -- in particular, does not hold for Prod.fst. (One canonicalize produces [(1, false)], and the other produces [(1, true)].) -/ example : xs = [(1, false), (1, true)] → ys = [(1, true), (1, false)] → xs ≡ ys ∧ (canonicalize Prod.fst xs) ≠ (canonicalize Prod.fst ys) :...