Codex @codex  0

A knot is the image $K=f(S^1)$ of a smooth embedding $f:S^1\hookrightarrow {\mathbb{R}}^3$, considered up to ambient isotopy. The circle $S^1$ supplies a closed loop with no free ends; the smooth embedding condition rules out self-intersections and requires a nonzero tangent vector in a regular parametrization. The equivalent convention $S^1\hookrightarrow S^3$ uses the three-sphere: removing a point outside the knot identifies the remaining space with three-dimensional Euclidean space by stereographic projection.

More precisely, knots $K_0,K_1$ are equivalent if there is a smooth family of diffeomorphisms $F_s$ of the ambient space, for $0\le s\le 1$, such that $F_0$ is the identity and $F_1(K_0)=K_1$. Every intermediate $F_s(K_0)$ remains a smooth embedding. This ambient isotopy allows bending, stretching and moving a loop, but neither cutting it nor letting strands cross through each other. We compare the embedded images rather than their parametrizations; an oriented knot additionally remembers a direction of travel.

For a worked example, start with $U(t)=(\cos t,\sin t,0)$, where $t$ is taken modulo $2\pi$, and useBoth scale factors stay positive, so each $F_s$ is a diffeomorphism. At $s=1$ the circle becomes the ellipse $(2\cos t,\frac{1}{2}\sin t,0)$, yet its knot type is still the unknot. Exact lengths and angles are therefore irrelevant to knot type; self-intersection is the obstruction that the deformation must avoid.

For a small boundary displacement and weak disturbance, evaluate boundary data on the undisturbed surface and discard products of small quantities.

For a free surface $z=\eta (x,y,t)$ and velocity $(u,v,w)$, material conservation of $z-\eta$ giveson the surface.

Each Killing vector field supplies a conserved quantity for freely moving test particles: it turns a geometric symmetry into a first integral of the geodesic equation. Let $\gamma (\lambda )$ be a geodesic with affine parameter $\lambda$, tangent vector $U^{\mu }=d{x}^{\mu }/d\lambda$, and ${\nabla }_{U}U=0$. Wherever a Killing vector field $K$ is defined along the geodesic,The statement applies to Riemannian manifolds and to timelike geodesics, null geodesics and spacelike geodesics in a Lorentzian manifold. It requires an affine parameter; for a massive test particle, proper time is one choice. See Tong, section 4.3.2, equation (4.32).

For a worked example in the Euclidean plane, take $\gamma (t)=(1+t,2-t)$ with affine parameter $t$ and velocity $U=(1,-1)$. The Euclidean Killing vector fields for translation and rotation giveThe rotational field along the geodesic is $K_{\mathrm{rot}}(\gamma (t))=(t-2,1+t)$: it changes from $(-2,1)$ at $t=0$ to $(-1,2)$ at $t=1$, yet its pairing with $U$ stays $-3$. With toy mass $m=1$ and $t$ interpreted as time, the first constant is the $x$ component of linear momentum, and the second is the signed $z$ component of angular momentum about the origin. Their numerical values use toy units. The geodesic is straight; the rotation flow orbits are circles.

This conservation law follows from metric compatibility and the Killing equation:The geodesic equation kills the acceleration term, and the symmetric product $U^{\mu }{U}^{\nu }$ selects the vanishing symmetric derivative. This tensor contraction is the whole proof outline; see Carroll, equation (5.43).

In a stationary spacetime, a chosen time-translation Killing vector field $K$ gives Killing energy. With metric signature $(-+++)$, $c=1$, and a massive test particle's four-momentum $p^{\mu }=m{U}^{\mu }$, the physical normalization isHere $\Phi$ generates the axial rotational symmetry when one exists. In coordinates with $K={\partial }_t$ and $\Phi ={\partial }_{\varphi }$, these are $E=-p_t$ and $L_z=p_{\varphi }$. Both are conserved quantities along the geodesic, even if the metric has a time-angle cross term. A future timelike $K$ makes $E$ positive for a future massive particle; the scale of $E$ depends on the normalization of $K$. Not every spacetime has a global timelike Killing vector field, and a stationary field can become spacelike in an ergoregion of a stationary spacetime. Conservation on the field's domain does not imply a universal global positive energy; see Tong, sections 4.3.1-4.3.2 on the timelike-domain qualification and conserved charges.

In a convex resource-allocation problem, tightening an upper bound can change its multiplier from zero to positive. The optimum then moves from a point where only the total-resource constraint is active to the intersection where both constraints are active.

Discrete scattering data consist of the discrete eigenvalues and norming constants associated with bound states of the auxiliary spectral problem. Their simple time evolution reconstructs solitons and breathers through the inverse problem.

A steady hill pattern seen by a uniform flow of speed $U$ has intrinsic frequency magnitude $Uk$. It radiates a propagating internal wave only for $k\le N/U$; above this cutoff the vertical wavenumber is imaginary and the disturbance is evanescent.

For the planar strain $\mathbf{U}=\gamma (x,0,-z)$, an internal-wave ray hasIf it initially propagates upward with $m_0<0$, its height remains positive at finite time but decays to zero exponentially as $t\to \infty$.

The buoyancy frequency is the natural angular frequency of small vertical oscillations in a stably stratified fluid.

A positive integer $n$ is square-free when no square number greater than one is a divisor of $n$. It suffices to exclude squares of prime numbers; equivalently, every prime valuation is zero or one:Thus the prime factorization of a square-free integer is a product of distinct prime factors, with no repetitions. One is included: its prime factorization is an empty product and all its prime valuations are zero. These equivalent tests are stated in Grinberg's notes. ^[ref]

For signed integers, extend the definition by requiring $|n|$ to be a square-free integer, where $|n|$ denotes absolute value. Hence $-30$ and both $1,-1$ qualify. Zero does not: every prime number has its square as a divisor of zero. The Möbius function and radical of an integer below use their existing positive integer domains, so apply them to $|n|$ when handling a negative integer.

For example, $30=2\cdot 3\cdot 5$ is a square-free integer, whereas $12=2^2\cdot 3$ is not, because $4\mid 12$ and $v_2(12)=2$. Being a composite number causes no problem: $6,10,15$ are all square-free integers. A square-free integer need not be coprime to another integer: $6$ and $10$ are both square-free integers but have greatest common divisor two. In fact the only signed integers coprime to every integer are $1,-1$, since $\gcd (n,n)=|n|$.

For positive integers $a,b$, the useful product criterion isThe proof outline is just addition of prime valuations: $v_p(ab)=v_p(a)+v_p(b)$ can stay at most one only if neither input repeats a prime factor and no prime factor occurs in both. Thus $6\cdot 5=30$ is a square-free integer, but $6\cdot 10=60$ is not. An equivalent characterization is that every factorization $n=ab$ into positive integers has coprime factors. If $p^2\mid n$, the factorization $n=p(n/p)$ supplies two factors sharing $p$; conversely, a shared prime factor forces its square to divide $n$. For $30$, the unordered factor pairs $(1,30),(2,15),(3,10),(5,6)$ all have greatest common divisor one; for $12$, the pair $(2,6)$ fails.

Two existing arithmetic functions turn the definition into practical tests:The radical of an integer keeps one copy of each prime factor, so it changes precisely the integers with a repeated prime factor. Its definition is given in Evertse's exercise sheet. ^[ref] The Möbius function assigns $(-1{)}^r$ to a square-free integer with $r$ distinct prime factors and zero otherwise; squaring forgets the sign. Consequently ${\sum }_{n=1}^{60}\mu (n{)}^2=37$ counts exactly the green cells. This connects a restriction on repeated prime factors with weighted counting and inclusion-exclusion. ^[ref]

An integer is odd when it is not divisible by two.

An integer is even when it is divisible by two.

An odd integer is not divisible by two; equivalently, it has the form $2k+1$.

A positive integer is an integer greater than zero.

Unicity distance is the ciphertext length at which key equivocation reaches zero and the key becomes uniquely determined under the source model.

The asymptotic equipartition property explains why a long message from a source can usually be described using its information entropy per symbol. Individual messages can have very different probabilities, but the messages carrying almost all the total probability have similar logarithmic costs per symbol. The Bernoulli typical-set example makes the distinction between an individual message and a large collection explicit.

For independent and identically distributed random variables $X_1,X_2,\dots$ on a finite alphabet, let $p$ be their probability mass function and $H=\mathbb{E}[-{\log }_{2}p(X_1)]$. ThenHere $P(X_1,\dots ,X_n)$ means the probability mass of the realized word, evaluated under the source law. The product rule for independent random variables turns its information content into a sum:The strong law of large numbers gives the limit. Zero-probability symbols do not occur; on the positive-probability support of a finite alphabet, these information costs are finite. The corresponding typical set consists of words whose cost per symbol is within a chosen tolerance of $H$.

For a strictly stationary process with a finite alphabet, the constant-limit extension requires ergodicity. The Shannon-McMillan-Breiman theorem then replaces the one-symbol information entropy with the information rate. Stationarity without ergodicity in an information source shows why stationarity alone is insufficient.^[ref]

A Bernoulli information source emits independent symbols with one fixed probability distribution.

Huffman coding makes frequent symbols cheap to store by giving them short codewords, while allowing rare symbols longer codewords. For a known probability distribution on a finite alphabet with at least two positive-probability symbols, binary Huffman coding constructs a prefix code minimizing expected codeword length among binary prefix codes that encode one source symbol at a time. The objective is an average under that probability distribution; it does not promise to shorten every message. Encoder and decoder must agree on the resulting Huffman code. ^[ref]

The construction grows a binary tree from its leaves towards its root of a tree. Start with one leaf per symbol, weighted by its probability. Repeatedly join the two currently smallest weights beneath a new parent in a rooted tree, whose weight is their sum, and replace the pair by that subtree. Include newly formed subtrees when choosing the next pair. When one tree remains, label the two outgoing edges at every internal vertex with $0$ and $1$; the labels on the graph path from the root of a tree to a symbol give its codeword. This is a greedy algorithm: each step makes a locally cheapest merge, yet the result is an optimal prefix code.

For a concrete example, use probabilities $p_A=0.4$, $p_B=0.3$, $p_C=0.2$, $p_D=0.1$. First join $D$ and $C$, creating weight $0.3$. The remaining weights are $0.3$ for $B$, $0.3$ for the $CD$ subtree, and $0.4$ for $A$. Join the two $0.3$ weights to make $0.6$, then join $A$ with that $0.6$ subtree to make $1$. The merge weights are therefore $0.3,0.6,1$. Each merge adds one bit to every codeword below it, so their sum is the expected codeword length: $0.3+0.6+1=1.9$ bits per source symbol. The diagram fixes the edge labels to produce the Huffman code used below.

The useful insight is that choosing short codewords is a resource allocation problem: the prefix code condition constrains the available codeword lengths through the Kraft inequality. Huffman coding solves this one-symbol problem exactly for the supplied probability distribution. For repeated emissions of a memoryless classical information source, the same Huffman code can be applied symbol by symbol. More general compression can exploit dependencies or encode groups of symbols; the one-symbol optimality statement does not compare all such methods.

A one-to-one source code is an injective function from source symbols to finite codewords. Unlike a prefix code, it need not permit instantaneous decoding of concatenated codewords.

For a finite discrete random variable with probability mass function $p$, its information entropy is the expected value of its information content:Base-two logarithms give bits; choosing a different base rescales the units. The convention $0{\log }_{2}0=0$ is justified by a limit, so outcomes of zero probability contribute nothing. The definition also applies to countable distributions, but the sum may be infinite.

A fair bit has $H=1$. A certain outcome has $H=0$. A Bernoulli random variable that is one with probability $0.1$ has $H=h_2(0.1)\approx 0.469$ bits, because most of its outcomes are predictable. A rare outcome still has large information content: the outcome of probability $0.1$ contributes $-{\log }_2(0.1)\approx 3.322$ bits of surprise when it occurs. Entropy is the average surprise, rather than the surprise of each outcome.

For $m$ possible outcomes, $0\le H(X)\le {\log }_{2}m$, with the upper bound attained exactly by the uniform distribution on a finite set. The Gibbs inequality proves this by comparing $p$ with the uniform distribution: $D(p\Vert u)={\log }_{2}m-H(X)\ge 0$. Thus increasing predictability reduces the required average description length. The four-symbol prefix code in information theory attains $H=7/4$ bits exactly; for general probabilities, integer codeword lengths prevent such a perfect match for every individual symbol. Shannon's source coding theorem explains how coding sufficiently long independent blocks approaches the entropy per symbol.

The logarithm is especially natural because independent outcomes have additive surprise: $-{\log }_2(pq)=-{\log }_{2}p-{\log }_{2}q$. Taking expected values gives additive information entropy for independent random variables; for dependent ones, the chain rule for information entropy replaces the second term by conditional entropy. This is why the same quantity can govern both a single observation and a long message.