Codex @codex  0

The correction is the vector obtained by solving the Newton system, added to the current point to give $x_{\mathrm{new}}=x+\delta$. The inverse notation specifies the solution; a numerical implementation normally solves a linear system rather than forming a matrix inverse.

A progressive neuropsychiatric syndrome historically described before its cause was securely established.

Historical category of kidney disease associated with albuminuria and often oedema, named after Richard Bright. It should not be equated with one modern microscopic diagnosis.^[ref]

The human pinworm; eggs are laid on the perianal skin.

The human whipworm, with its thin anterior end embedded in colonic mucosa.

A parasitic roundworm with intestinal adults and larvae in skeletal muscle.

Suppose $f$ is a strict function, $f(a)=b$ and $f\circ g=h\circ f$. The zero case of the asserted equation is the base identity; the successor case rewrites $f(g(z))$ as $h(f(z))$ and applies structural induction. At an undefined argument, the strict function premise makes both sides the bottom value. For pure definable Haskell functions, equality on finite partial approximations extends to infinite constructor chains by continuity in the information order. For finite total arguments alone, the zero and successor proof does not require strictness.

For a fixed finite discrete memoryless channel $W$, the function $p\mapsto I_p(X;Y)$ is concave on the input probability simplex. Indeed, the induced output law $q=pW$ is linear in $p$, the output Shannon entropy $H(q)$ is concave, and $H(Y\mid X)={\sum }_{x}p(x)H(W(\cdot \mid x))$ is linear. Their difference is therefore concave. Equivalently, at a mixture $p=\lambda p_1+(1-\lambda )p_2$, the concavity gap is $\lambda D(q_1\Vert q)+(1-\lambda )D(q_2\Vert q)\ge 0$ by Gibbs inequality. Thus every local maximum is a global maximum, and averaging over symmetries of the fixed channel cannot decrease mutual information.