A spatial group element combined with a time shift may preserve a periodic orbit even if the spatial element alone does not fix it at each instant. The temporal circle acts complex linearly on Hopf amplitudes. Complex axial isotropy subgroups identify the primary branches guaranteed by the equivariant Hopf theorem.

// Target: dynamical-systems.bigb

Spatially complex coordinates on two real copies need not both have the same temporal phase weight. If their time-circle action is $(z_1,z_2)\mapsto (e^{-i\varphi }{z}_1,e^{i\varphi }{z}_2)$, conjugating the second coordinate gives a common Hopf phase. The linear terms then have the same imaginary frequency in $q$ coordinates but opposite imaginary frequencies in the original $z$ coordinates. Failing to make this conjugation produces amplitude equations inconsistent with the specified symmetries.

// Target: dynamical-systems.bigb

In Hopf coordinates where the rotation acts as $(i{q}_1,-i{q}_2)$, reflection swaps components, and temporal phase multiplies both equally, cubic equivariance permits precisely the three displayed monomials in the first component. Swapping indices gives the second equation, with the same complex coefficients. The rotating, axial standing, and diagonal standing branches are $q_2=0$, $q_2=q_1$, and $q_2=i{q}_1$. Their squared component amplitudes are $\mu /\mathrm{Re}a$, $\mu /\mathrm{Re}(a+b+c)$, and $\mu /\mathrm{Re}(a+b-c)$, respectively, when positive.

// Target: dynamical-systems.bigb

A primary periodic branch retaining a spatial reflection at every time. In the natural plane representation, each real-copy component oscillates along a fixed reflection axis. The fourfold dihedral case has distinct axial and diagonal standing-wave types.

// Target: dynamical-systems.bigb

A primary periodic branch whose phase advance is compensated by a spatial rotation. For the natural plane representation, the spatial pattern rotates rather than remaining on a fixed reflection axis. Opposite chiralities are exchanged by reflection.

// Target: dynamical-systems.bigb

For the natural plane action of the dihedral group, rotations constrain monomial weights and reflection makes coefficients real. The reflection axes support the bifurcating equilibria. For $n=3$, the quadratic anisotropic term gives three transcritical-like branches that are saddles in the full plane. For $n=4$, cubic isotropic and anisotropic terms determine two reflection-isotropy classes. For $n=5$, the cubic isotropic term sets the leading radial amplitude while quartic anisotropy separates stable and saddle angular classes.

// Target: dynamical-systems.bigb

The normalizer of an axial isotropy subgroup preserves its one-dimensional fixed-point subspace of a group action. If its quotient by the isotropy contains an element acting as minus one, the reduced scalar vector field is odd, forcing a pitchfork bifurcation rather than a generic quadratic branch. For odd dihedral groups the reflection normalizer has no such quotient; an even-degree anisotropic term can distinguish the two half-axes at higher order.

// Target: algebra.bigb

For real continuous $2\pi$-periodic $f$, a degree-at-most-$n$ real trigonometric polynomial $p$ is its unique best uniform approximation exactly when $f-p$ has $2n+2$ alternating extrema of magnitude $\Vert f-p{\Vert }_{\infty }$ at distinct ordered points in one period. The closing interval around the circle also counts. A strictly improving approximant would force its difference from $p$ to have at least $2n+2$ zeros, whereas a nonzero degree-$n$ trigonometric polynomial has at most $2n$.

For real $f\in C^{n+1}[0,1]$ with positive $(n+1)$st derivative, equality of consecutive best uniform approximation errors would make a best degree-at-most-$n$ polynomial also best at degree $n+1$. The Chebyshev alternation theorem then supplies $n+3$ alternating extrema of its nonzero error, hence $n+2$ distinct zeros. Repeated Rolle's theorem forces a zero of the $(n+1)$st derivative of that error, contradicting the assumed sign. The same conclusion holds for a strictly negative derivative by replacing $f$ with $-f$.

A three-dimensional hyperbolic equilibrium point is a saddle-focus when its eigenvalues include a nonreal pair and a real eigenvalue with the opposite sign of real part. In the displayed orientation it has a one-dimensional unstable manifold and a two-dimensional spiralling stable manifold; reversing time interchanges the stable and unstable dimensions. The ratio $-{\lambda }_s/{\lambda }_u$ controls local transverse contraction in a Shilnikov return map.

A hyperbolic equilibrium point is a sink when all its eigenvalues have negative real parts. Nearby trajectories converge exponentially to it. In a planar system this includes both stable nodes and stable foci. The condition is stronger than neutral stability: a centre equilibrium is not a sink.

A hyperbolic equilibrium point is a source when all its eigenvalues have positive real parts. It is a sink equilibrium for the reversed flow. Every sufficiently close nonstationary trajectory leaves its neighborhood in forward time. This is distinct from an external forcing source.

The Yanai wave is the $n=0$ branch of the equatorial shallow-water dispersion relation with positive frequency:Its group velocity is $d\omega /dk=c{\omega }^2/({\omega }^2+\beta c)>0$. At low frequency it has westward phase velocity but transports energy eastward; at high frequency its phase and group velocities approach $c$.

At the critical impact parameter $b=3\sqrt{3}M$, the null orbit equation factors as $(u^{\prime }{)}^2=(u-1/3{)}^2(2u+1/3)$ for $u=M/r$. One oriented family has the displayed solution. As $\varphi \to -\infty$, positive $A$ approaches the photon sphere from larger radii and negative $A$ from smaller radii; $A=0$ is the circular orbit. The physical domain must retain $r>0$ and the positive square root in its implicit parametrization.

Quintessence models dark energy through an evolving scalar field. For a homogeneous canonical field in natural units,A positive potential dominating the kinetic term gives negative pressure. The Friedmann acceleration equation requires $w=p_{\varphi }/{\rho }_{\varphi }<-1/3$ for scalar-dominated accelerated expansion. A nonnegative canonical kinetic term prevents $w<-1$ when ${\rho }_{\varphi }>0$.

Unlike a strict cosmological constant, the field can evolve, so $w$ need not remain $-1$. This is a model class, not an identified substance.^[ref]

In A Mathematical Theory of Communication by Claude Shannon, Claude Shannon compares block uncertainty per symbol, denoted $G_N$, with conditional uncertainty given the preceding symbols, denoted $F_N$. Theorems 5 and 6 connect both decreasing approximations to the source's limiting uncertainty. He uses symbol statistics at increasing block lengths to expose structure missed by single-letter counts.^[ref] The fair marginal bits of the binary Markov source provide a small exact example of that distinction.

Consider a binary Markov source with fair initial bit and flip probability $q=1/8$. Its exact $n$-symbol information entropy is $H_n=1+(n-1)h_2(1/8)$. At $n=16$,This includes the uncertainty of the initial bit; the entropy rate itself is $h_2(1/8)\simeq 0.54356444$ bits per symbol. The finite-block difference is $(1-h_2(1/8))/n$.

For any finite block distribution, the entropy lower bound for prefix codes gives expected codeword length $L_n\ge H_n$, while Shannon coding achieves $L_n<H_n+1$. These bounds apply to each block as one source symbol, without requiring consecutive blocks to be independent. Here Huffman coding on all $2^{16}$ possible 16-symbol words gives the independently calculated value $L_{16}\simeq 9.19511361$ bits per block, or $0.57469460$ bits per symbol. The noninteger value is an expected value; each actual codeword has an integer length. As the block length grows, the prefix code bounds squeeze $L_n/n$ towards the entropy rate when an optimal code is used.

An entropy rate below one does not allow every word to fit into a shorter fixed binary word with zero error. For $0<q<1$, all $2^n$ binary words are possible, so an injective fixed-length code needs at least $n$ bits. Compression instead exploits variable codeword lengths, or the vanishing-error convention in reliable source encoding at a rate. These are different guarantees.

For a strictly stationary process on a finite alphabet, put $a_1=H(X_1)$ and $a_n=H(X_n\mid X_1,\dots ,X_{n-1})$ for $n\ge 2$. Each $a_n$ is the average remaining uncertainty in the next symbol after its history is known. More history can improve prediction, so this sequence is nonincreasing:The inequality uses conditioning reduces entropy; the last equality uses strict stationarity. Nonnegative $a_n$ therefore have a limit. By the chain rule for information entropy,The Cesàro mean converges to the same limit, proving the equality with the entropy rate. This is a statement about average information entropy. A constant almost-sure limit of the realized information content requires the further hypotheses in the Shannon-McMillan-Breiman theorem; stationarity without ergodicity in an information source explains the distinction.

Why does a gas spread through an available container? One part of the answer is counting: a nearly uniform macrostate can be realized by vastly more microstates than a macrostate with every particle crowded into one corner. Boltzmann's entropy formula turns that multiplicity into entropy. The useful insight is that a large entropy measures how many microscopic realizations fit a specified macroscopic description, rather than how untidy a picture looks.

For a specified macrostate $M$ containing $\Omega (M)$ accessible, equally likely microstates, writeHere $k_B$ is the Boltzmann constant, and $\ln$ is the natural logarithm, also written $\log$ in this article's title. The count $\Omega (M)$ is dimensionless; entropy has units of energy divided by temperature. Always specify the macrostate, the constraints and the model's microscopic resolution before counting. A microcanonical ensemble provides a familiar equilibrium setting for the equal-probability assumption. ^[ref]

The natural logarithm compresses an enormous count into a manageable number and makes products of counts into sums of entropies; see additivity of Boltzmann entropy. With this counting convention, one allowed microstate has zero Boltzmann entropy. This zero refers to the chosen model and resolution; it does not by itself assign an absolute entropy to a real material.

For unforced homogeneous incompressible flow, the vorticity equation gives $\dot{Z}=⟨{\omega }_i{\omega }_j{S}_{ij}⟩-\nu ⟨|\nabla \times \mathbf{\omega }{|}^2⟩$. Curl and full-gradient squared averages agree because the vorticity is divergence free and boundary divergences average to zero. In established high-Reynolds-number turbulence, slow integral-time evolution makes production and destruction nearly balance.