Research proof index

Completeness theorem for Merkle trees: for any full binary tree with 2 ^ n leaves, and for any index i, the honestly-generated opening proof verifies against the honestly-built root with probability one.

A functional completeness theorem for Merkle proofs built from buildMerkleTree_with_hash.

Let $1 < q \le 2$ and let $q'$ be its Hölder conjugate. In the project’s cancellative metric-space setting, assume the associated nontangential operators satisfy the required uniform $L^2$ bound. If $F$ and $G$ are measurable and $f$ is measurable with $\lVert f(x)\rVert \le \mathbf{1}_F(x)$, then the Carleson operator obeys the restricted estimate

$\int_G^+ \mathcal{C}_K f(x)\,dx \le C(a,q)\,\mu(G)^{1/q'}\mu(F)^{1/q}.$

The two definitions of entropy, in terms of T or β = 1 / T, are equivalent.

The two definitions of entropy, in terms of T or β, are equivalent.

The "definition of temperature from entropy": 1/T = (∂S/∂U), when the derivative is at constant extrinsic d (typically N/V). Here we use β instead of 1/T on the left, and express the right actually as (∂S/∂β)/(∂U/∂β), as all our things are ultimately parameterized by β. This identity requires the denominator ∂U/∂β to be nonzero.