Research proof index

The quantized wavenumbers form a subset of the BrillouinZone.

In a conditionally complete linear order, suppose the values $f(i)$ for $i\in s$ are bounded above. If one of those values is exactly $a$, then $a$ is at most the supremum $\sup_{i\in s} f(i)$.

If $X, Y$ are $G$-valued random variables, and $Z$ is another random variable defined on the same sample space as $X$, then $D_{KL}((X|Z)\Vert Y) = D_{KL}(X\Vert Y) + \bbH[X] - \bbH[X|Z].$

Conditional multidistance is unchanged when both the random variables and their conditioning variables are reindexed along an equality $m'=m$. As with ordinary multidistance, the value does not depend on the chosen equal presentation of the finite index type.

A lower bound forced by $\tau$-minimality. If $(X_1,X_2)$ minimizes the source's $\tau$ functional, then for measurable $X_1',X_2'$ and conditioning variables $Z,W$, $d[X_1'\mid Z;X_2'\mid W]\ge d[X_1;X_2]-\eta\bigl(d[X_1^0;X_1'\mid Z]-d[X_1^0;X_1]\bigr)-\eta\bigl(d[X_2^0;X_2'\mid W]-d[X_2^0;X_2]\bigr).$

Conjugation is an involution: conjugating twice returns the original tensor, up to the bar_involution recolouring (the identity permutation permT).