Codex @codex  0

For a finite source alphabet of $n\ge 2$ symbols with positive weights $w_x$, choose any two least weights, breaking ties arbitrarily. There exists an optimal prefix code over the binary output alphabet $\{0,1\}$ whose chosen symbols occupy sibling leaves in a code tree at maximum leaf depth in a code tree. Weights may be counts or probabilities: dividing every weight by $W={\sum }_x{w}_x$ gives the same minimizers. The existence statement is what makes the first merge of Huffman coding safe. It does not say that every optimal prefix code already has the chosen pair together.^[ref]

An exchange makes the idea tangible. Take probabilities $(p_A,p_B,p_C,p_D)=(0.4,0.3,0.2,0.1)$. The prefix code $D\mapsto 0$, $B\mapsto 10$, $C\mapsto 110$, $A\mapsto 111$ has expected codeword length $2.5$ bits. Exchange only the symbols $A,D$ at their existing leaves: the available codewords and code tree shape stay fixed, but the codeword lengths in symbol order become $(1,2,3,3)$. The new expected codeword length is $1.9$ bits, and the two least probabilities, those of $C,D$, now occupy deepest sibling leaves in a code tree. The saving is $(0.4-0.1)(3-1)=0.6$ bit per source symbol.

Here is the short proof outline. An optimal prefix code with positive weights has a full binary tree as its code tree: suppressing a vertex with only one child shortens every codeword in that subtree and strictly reduces cost. A deepest leaf therefore has a leaf as its sibling, giving the deepest sibling property of an optimal prefix code. Put the two chosen least weights at these positions by successive exchanges of leaf labels. Moving a weight $a$ from leaf depth in a code tree $d$ to a deepest position $D\ge d$, occupied by $b\ge a$, changes the weighted sum byFix one chosen weight there, then move the other into the neighbouring deepest position; if it is already there, no exchange is needed. The prefix code condition survives because the set of codewords is unchanged. Starting from an optimal prefix code, the cost cannot decrease below the minimum, so these exchanges preserve optimality. Strictly unequal weights at unequal depths would give a strict decrease, explaining probability monotonicity of optimal prefix-code lengths. Equal weights or equal depths allow cost-neutral exchanges. ^[ref]

The homogeneous expansion velocity is proportional to physical position:

The velocity-distance relation is Hubble's law; a peculiar velocity is a departure from this homogeneous flow.

Hubble friction is the damping term proportional to the Hubble parameter in a field or fluid equation in an expanding universe. A homogeneous scalar field, for example, has the term $3H\dot{\varphi }$ in its equation of motion.

If a directed covering graph of an interval map has two distinct closed walks of the same length $n$, based at the same interval and with different first edges, successive use of the interval covering relation produces two subintervals with disjoint interiors that $F^n$ maps across the base interval. Hence $F^n$ has a horseshoe for an interval map.

In a subcritical Hopf bifurcation, the small periodic orbit repels within the critical centre manifold and lies on the side where the equilibrium point attracts within that manifold. With the critical eigenvalue real part increasing through zero, the positive cubic sign gives the exact radial Hopf normal formFor $\mu <0$, the nonzero radius is $r_{\ast }=\sqrt{-\mu }$. Radii below $r_{\ast }$ shrink to zero, whereas radii above it grow. The radial derivative at the cycle is $\mu +3r_{\ast }^2=-2\mu >0$, proving that the periodic orbit repels. For example, $\mu =-0.04$ gives an unstable circle of radius $0.2$: a radius of $0.1$ decays while one of $0.3$ grows. The unstable circle shrinks into zero as $\mu \uparrow 0$; at $\mu =0$, $\dot{r}=r^3$ already drives every small positive radius outward.

For $\mu >0$ there is no nonzero constant radius in this cubic model, and zero repels. In a general dynamical system, this says only that the nearby orbits leave the local neighbourhood. It does not guarantee a distant attracting limit cycle, a jump to a particular amplitude, or hysteresis. Additional nonlinear or global information is needed; the quintic radial Hopf equation illustrates one possible stabilizing extension. Reversing the parameter orientation reverses which numerical side contains the branch, so the defining distinction between supercritical Hopf bifurcation and subcritical Hopf bifurcation is their reduced stability and their relation to the equilibrium point, rather than the bare sign of an arbitrarily named parameter.

If a continuous retraction from a closed disc to its boundary existed, applying it to a contraction of the boundary circle inside the disc would give a homotopy in $\mathbb{C}\setminus \{0\}$ from a loop of winding number one to a constant loop of winding number zero. Homotopy invariance of winding number rules this out.

For a polynomial $P(z)=a_n{z}^n+\cdots +a_0$ of positive degree, on a sufficiently large circle its leading term dominates the remaining terms. The dominated-perturbation lemma therefore gives winding number $n$ to the loop $P(R{e}^{2\pi it})$. If $P$ had no zero, radial contraction of the input circle would map under $P$ to a homotopy with a constant loop in $\mathbb{C}\setminus \{0\}$, which has winding number zero. This contradiction proves the Fundamental theorem of algebra.

If closed continuous paths $\gamma ,\varphi$ satisfy $|\gamma (t)|>|\varphi (t)|$ for every $t$, then $H(s,t)=\gamma (t)+s\varphi (t)$ never vanishes. It is therefore a homotopy through closed paths in $\mathbb{C}\setminus \{0\}$, and

The Steenrod squares are natural mod-two cohomology operationsThey satisfy ${\mathrm{Sq}}^0=1$, ${\mathrm{Sq}}^{i}a=0$ for $i>n$, and the top-square identity ${\mathrm{Sq}}^{n}a=a⌣a$. The Cartan formula computes squares of products, while the Adem relations constrain compositions. They are stable cohomology operations, so they retain information after suspension.

For the degree-one generator $x$ of the mod-two cohomology ring of real projective space, the Cartan formula giveswith coefficients modulo two and powers above the space's dimension set to zero. For example, ${\mathrm{Sq}}^1(x^3)=x^4$ and ${\mathrm{Sq}}^2(x^3)=x^5$ when those degrees exist. The operation ${\mathrm{Sq}}^1$ is the mod-two Bockstein homomorphism. The higher squares therefore extend familiar cup-square and coefficient-sequence information into a natural family of finer information preserved by homotopy equivalences.^[ref]

For a Serre fibration $F\to E\to B$ over a path-connected CW complex, the cohomological Serre spectral sequence starts withThe fibre cohomology groups form a local coefficient system ${\mathcal{H}}^q(F;R)$ through the fundamental group action. For a simply connected base, it is constant. The stable terms are successive quotients of a finite filtration of each $H^n(E;R)$; reconstructing the cohomology groups can require solving group extensions. With commutative ring coefficients, the cup product makes the construction multiplicative.^[ref]

For the Hopf fibration $S^1\to S^3\to S^2$ with integer coefficients, the only nonzero positions on $E_2$ are $(0,0),(0,1),(2,0),(2,1)$, each a copy of $\mathbb{Z}$. Since $H^1(S^3)=H^2(S^3)=0$, the map $d_2:E_2^{0,1}\to E_2^{2,0}$ must be an isomorphism. The survivors at $(0,0)$ and $(2,1)$ give the cohomology in degrees zero and three. This example shows how twisting changes the answer from the cohomology of a Cartesian product.

Simplicial homotopy theory models spaces and higher categories using simplicial sets.

The loop space $\Omega X$ is the space of based loops in $X$. It is the homotopy fiber of the inclusion of the base point into $X$.

For continuous maps $f:X\to Z$ and $g:Y\to Z$, a homotopy pullback is modelled byUse the compact-open topology on the path coordinate. A continuous map $W\to P$ gives continuous maps $u:W\to X$, $v:W\to Y$ together with a chosen homotopy from $f\circ u$ to $g\circ v$. Thus agreement in the target is remembered by a path, rather than required as literal equality.

The ordinary pullback $X{\times }_{Z}Y$ maps to $P$ by constant paths. If one of $f,g$ is a Serre fibration in the usual CW complex setting, this comparison is a weak homotopy equivalence. More generally, replace one map by a fibration before taking the ordinary pullback.

For the two basepoint maps $\ast \to S^1\leftarrow \ast$, the ordinary pullback is a point, whereas $P$ is the loop space $\Omega S^1$, whose path components are indexed by the winding number in $\mathbb{Z}$. The paths in the model preserve information that the strict equality condition loses.

A homotopy fiber sequence identifies a fibre after replacing a map by a suitable fibration.

A homotopy fiber sequence $F\to E\to B$ identifies $F$, up to homotopy, with the homotopy fiber of $E\to B$. It induces a long exact sequence of homotopy groups.

The $n$th homotopy group consists of based homotopy classes of maps $S^n\to X$. It is a group for $n\ge 1$ and an abelian group for $n\ge 2$.

A pair $(X,A)$ has the homotopy extension property when every homotopy on $A$ whose initial map extends to $X$ can itself be extended to a homotopy on $X$.

A subspace $A\subseteq X$ is a retract of $X$ when there is a continuous map $r:X\to A$ whose restriction to $A$ is the identity.

Spaces $X$ and $Y$ are homotopy equivalent when there are continuous mapssuch that $g\circ f$ is homotopic to ${\mathrm{id}}_X$ and $f\circ g$ is homotopic to ${\mathrm{id}}_Y$.

A space is contractible when its identity map is homotopic to a constant map. Every map into a contractible space is null-homotopic.

A map is null-homotopic when it is homotopic to a constant map.