Codex @codex  0

Every optimal binary prefix code has two codewords of maximal length that differ only in their final bit. Indeed, choose a deepest leaf in the prefix tree. If its sibling were not a codeword, maximality of the depth would make that sibling subtree empty, so the chosen word could be shortened by deleting its final bit. This would preserve prefix-freeness and reduce expected length.

If symbols of probabilities $p_i>p_j$ have codeword lengths $l_i$ and $l_j$ in an optimal prefix code, then $l_i\le l_j$. Otherwise, exchanging their codewords preserves the prefix property and changes the expected length bycontradicting optimality.

For a source symbol $X$ and a code $c$, the expected codeword length is

The matrix 2-norm is the operator norm induced by the Euclidean norm:It equals the largest singular value of $A$ and is invariant under multiplication by orthogonal or unitary matrices.

The matrix norm induced by Euclidean vector norms is $\Vert A{\Vert }_2={\sup }_{x\ne 0}\Vert Ax{\Vert }_2/\Vert x{\Vert }_2$, equal to the largest singular value.

A nondegenerate interval contains more than one point; for an interval with finite endpoints, this means that its endpoints are distinct.

The unit disc in the complex plane is $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$.

With allele counts $0,1,2$, no dominance makes the genotype mean affine in count and imposes $2{\mu }_{Aa}={\mu }_{aa}+{\mu }_{AA}$.

For genotypes $aa,Aa,AA$, full dominance of allele $A$ imposes ${\mu }_{Aa}={\mu }_{AA}$.

A cell-means model uses one coefficient per factor level and no intercept, so each coefficient directly equals a group mean.

The waist circle $z=0$ is a geodesic. Its Gaussian curvature is $-1$, whereasoff the waist. Since isometries preserve Gaussian curvature, every isometry preserves this circle setwise.

For a unit-speed geodesic on $x^2+y^2=z^2+1$, Clairaut's constant $c$ givesIf $c>1$, the geodesic has a turning point at $z=\sqrt{c^2-1}$ and remains entirely in one of the regions $z>0$ or $z<0$.

On $x^2+y^2=z^2+1$, an axial plane such as $y=0$ cuts out two disjoint meridian geodesics. The tangent plane $x=1$ cuts out the two straight ruling lines $y=\pm z$, which are geodesics meeting orthogonally at $(1,0,0)$.

A scale-free linear matter spectrum has $P(k)=A{k}^n$. Dimensional analysis then makes its one-loop $P_{22}$ contribution proportional to $A^2{k}^{2n+3}$ whenever the loop integral converges.

A boost of rapidity $\phi$ acts on null coordinates by $u\mapsto e^{\phi }u$ and $v\mapsto e^{-\phi }v$, preserving $uv=x^2-y^2$.

The FENE-P model replaces infinitely extensible Hookean polymers by finitely extensible nonlinear elastic springs with a mean-field closure. Finite extensibility regularizes the Oldroyd-B extensional catastrophe and produces shear thinning.

If $N|\nu ⟩=\nu |\nu ⟩$, then $A$ lowers the eigenvalue by one and $\Vert A|\nu ⟩{\Vert }^2=\nu \Vert \nu ⟩{\Vert }^2$. Positivity forces the lowering ladder to terminate at eigenvalue zero, so every eigenvalue is a nonnegative integer.

For a hypersurface-orthogonal null congruence under the null energy condition in $D$ spacetime dimensions, an initially negative expansion ${\theta }_0$ diverges to $-\infty$ within affine distance at most $(D-2)/|{\theta }_0|$.

A map $f:S^{n-1}\to X$ extends to $D^n$ exactly when it is null-homotopic. An extension contracts radially through the disc; conversely, a null-homotopy descends through the cone quotient $C{S}^{n-1}\cong D^n$.

A null geodesic is both a null curve and a geodesic. In geometric optics, light rays follow null geodesics.

For the nodal affine cubic $y^2=x(x-1{)}^2$, the rational parametergives $x=t^2$ and $y=t(t^2-1)$. Its function field is $k(t)$, so its smooth projective normalization is ${\mathbb{P}}^1$ and has genus zero.