Pattern formation is the development of spatial structure in a system. A common mechanism is the loss of stability of a uniform state to disturbances at a preferred nonzero wavenumber. In Rayleigh-Bénard convection, heating a fluid layer from below can destabilize the conducting state and produce organized convection rolls. In a reaction-diffusion equation, reaction and transport can instead select a spatial concentration pattern.

A linear calculation separates the possible scales. Write a small disturbance as a Fourier mode $e^{\lambda (k)t+ikx}$. Positive real part of $\lambda (k)$ means that mode initially grows, and the fastest growing wavenumbers often set the first visible scale. The Swift–Hohenberg equation has $\lambda (k)=r-(1-k^2{)}^2$, so at small positive $r$ it amplifies a band near $k=\pm 1$.

Linear growth is only the onset calculation. A growing disturbance eventually becomes large enough for nonlinear terms to matter. Coupled amplitude equations can then describe competition between modes: one mode can suppress another, several can coexist, or the state can drift or oscillate. The resulting planform includes both the selected wavevectors and their relative amplitudes and phases.

A slowly varying phase modulation describes changes in a pattern's local position and orientation without immediately destroying its periodic structure. Its stability can differ from the stability of a uniform amplitude, as illustrated by the zigzag instability of parallel rolls.

The imperfect soft Duffing-van der Pol oscillatorcombines a softening restoring force, velocity-dependent damping and the symmetry-breaking term $2\epsilon v^2$. The equilibrium points are $v=0$ and $v=-\epsilon \pm \sqrt{{\epsilon }^2+\lambda }$, whenever the latter are real. The associated Jacobian matrix has trace $\kappa -v_{\ast }^2$ and determinant $\lambda -4\epsilon v_{\ast }-3v_{\ast }^2$.

At $\epsilon =0$, the equilibrium equation has the reflection symmetry $v\mapsto -v$. For $\epsilon \ne 0$, the branch near zero crosses the zero branch at $\lambda =0$, giving the generic transcritical bifurcation when $\kappa \ne 0$. The two nonzero roots meet at $\lambda =-{\epsilon }^2$, $v=-\epsilon$, giving a generic saddle-node bifurcation when $\kappa \ne {\epsilon }^2$. The excluded trace-zero cases have an additional linear degeneracy and require further analysis.

The mechanical energyhas the exact derivative $\dot{H}=(\kappa -v^2){\dot{v}}^2$. Thus the sign of energy input can change with displacement even at a fixed value of $\kappa$.

For $\lambda >0$, the two outer equilibria are saddle equilibria. Their potential barriers are generally unequal when $\epsilon \ne 0$, so a symmetric saddle connection can split into distinct directed connections under an imperfection. The weak-damping heteroclinic splitting of a tilted quartic oscillator calculates their leading parameter curves. Local equilibrium formulas alone do not guarantee a limit cycle or determine its global termination.

An integer matrix $A$ with determinant $\pm 1$ defines a toral automorphismon the real torus ${\mathbb{R}}^d/{\mathbb{Z}}^d$. Integer entries make the map independent of the chosen representative. The inverse matrix also has integer entries, so the transformation is invertible. It preserves normalized Haar measure.

The map is a mixing measure-preserving transformation exactly when no eigenvalue of $A$ is a root of unity. This can be proved with characters of a real torus $e_k(x)=e^{2\pi ik\cdot x}$, since $e_k\circ T_A=e_{A^{T}k}$. Orthogonality givesIf no eigenvalue is a root of unity, a nonzero integer frequency cannot repeat: a repetition would give $((A^T{)}^m-I)k=0$ for some $m>0$. It therefore eventually leaves every finite set of frequencies. The displayed correlations prove mixing first for trigonometric polynomials, then for square-integrable functions by approximation.

Conversely, a root-of-unity eigenvalue makes $(A^T{)}^m-I$ singular for some $m$. As an integer matrix it has a nonzero rational kernel vector, which can be scaled to a nonzero integer vector $k$. The nonconstant character $e_k$ is then invariant under $T_A^m$, obstructing mixing. The cat map is a mixing example, while the identity matrix gives an invertible measure-preserving map that is not mixing.

Thermodynamic formalism studies weighted trajectories through partition functions, topological pressure and invariant measures. A symbolic potential $\varphi$ assigns an orbit segment the weight $e^{S_n\varphi }$, where $S_n\varphi$ is its accumulated potential. Summing these weights gives $Z_n(\varphi )$; the limiting logarithmic normalization per step is the pressure.

A one-symbol example shows the mechanism. On a two-symbol full shift, let $\varphi (0)=a$ and $\varphi (1)=b$. ThenNormalizing gives independent symbol probabilities $p_0=e^a/(e^a+e^b)$ and $p_1=e^b/(e^a+e^b)$. The associated Gibbs measure is therefore an orbit-weight law, not a uniform choice unless $a=b$.

The same construction can encode geometry. A geometric potential of an expanding map records inverse-branch contraction, while the Feigenbaum geometric potential encodes lengths in a nested partition. Their pressure functions organize spatial scaling and dimension calculations. In that second example, the partition depth is a geometric scale index and should not automatically be identified with elapsed orbit time.

This explains both the usefulness and the limitation of the statistical-mechanics analogy: changing the potential changes which trajectories or geometric pieces receive weight. An arbitrary Gibbs measure need not be absolutely continuous, even when its underlying interval map is expanding.

A piecewise differentiable interval map is uniformly expanding when a constant $\lambda >1$ bounds the derivative on every smooth branch:With a finite partition into monotone branches, a point's itinerary records which branch contains each iterate. A length-$n$ cylinder consists of points with the same first $n$ branch labels. Wherever this composition is smooth, the chain rule gives $|(T^n{)}^{\prime }|\ge {\lambda }^n$.

The mean value theorem then bounds the diameter of a cylinder by the length of the original interval times ${\lambda }^{-n}$. Inverse branches therefore determine starting points with exponentially improving spatial precision as more itinerary symbols are specified.

For the doubling map on $[0,1)$, the two branches have slope two and the level-$n$ cylinders are half-open intervals of length $2^{-n}$. Their labels are the first $n$ digits of the binary expansion. Endpoint conventions matter for unique coding, but not for the cylinder-length estimate.

Expansion describes the map's geometry. A symbolic potential separately specifies statistical weights, so expansion alone does not make every Gibbs measure absolutely continuous. The geometric potential of an expanding map uses the inverse derivative to select weights adapted to interval length.

The Feigenbaum period-doubling map is the universal, even, quadratic-critical fixed map associated with repeated period-doubling bifurcations. One normalization isComposing twice and rescaling therefore reproduces the same graph. This self-similarity is the mechanism behind a common spatial scaling in otherwise different one-dimensional maps with a quadratic maximum.^[ref]

The quadratic maximum means $g^{\prime }(0)=0$ and $g^{\prime \prime }(0)\ne 0$. Twice applying the chain rule at zero givesThe term involving $g^{\prime }(0{)}^2$ vanishes, and cancellation is legitimate because the maximum is nondegenerate. Thus the endpoint derivative is determined by the spatial scaling.

Writing $\gamma =-\beta >0$ makes the successive central intervals $[-{\gamma }^n,{\gamma }^n]$ shrink by the factor $\gamma$. Their iterates generate the Feigenbaum attractor; nonuniform interval lengths are encoded by the Feigenbaum geometric potential. The constant $\beta$ here describes spatial scaling, rather than the spacing of successive bifurcation parameter values.

The Lorenz system is the three-variable autonomous differential equationIt arose as a low-dimensional model of thermal convection. Edward Norton Lorenz used its numerical solutions in 1963 to study nonperiodic motion and sensitivity to the initial condition, with consequences for long-range weather forecasting.^[ref]

The equilibrium points follow directly from the equations. For $\sigma >0$ and $\beta >0$, the first gives $y=x$ and the third gives $z=x^2/\beta$. The second then becomes $x(\rho -1-x^2/\beta )=0$. Thus the origin always exists, while for $\rho >1$ there are two additional equilibriawith matching signs in the first two coordinates. The symmetry $(x,y,z)\mapsto (-x,-y,z)$ exchanges them.

The Equilibrium bifurcations of the Lorenz system determine how their stability changes. The separate instantaneous Lorenz error growth at a zero-y crossing calculation concerns a perturbation at one instant; it should not be confused with the long-term Lyapunov exponent.

Evolution is deterministic when a specified initial condition and evolution rule select a unique future orbit. For a discrete dynamical system, applying the same map to the same state gives the same next state. For an ordinary differential equation, uniqueness of the initial value problem is required; merely writing a differential equation does not guarantee it.

Determinism permits both asymptotic stability and sensitive dependence on initial conditions. It does not by itself imply reliable prediction from approximate measurements, and sensitivity does not introduce a random choice into the evolution rule.

For an integrable relaxation modulus, a small constant shear rate sustained from the remote past gives stress equal to that rate times the displayed integral, including any instantaneous solvent mass. For the linear Oldroyd-B model it is ${\mu }_0+G_0\tau$. A nondecaying elastic modulus would make this integral infinite rather than describe a fluid with finite steady viscosity.

A control policy selects admissible actions using only the information available at each time, possibly with randomization. General policies may depend on the whole observed history. Nonanticipation and measurability are part of admissibility in a controlled Markov process.

An average-reward optimal policy maximizes long-run expected reward per time step. In a finite recurrent Markov chain, a stationary policy's gain is its expected one-step reward under the stationary distribution. Initial-state effects are encoded by a bias function rather than by the gain. In a communicating finite control problem, the average-reward Bellman equation certifies a policy that is optimal from every initial state.

An optimal value over a time interval is the supremum of immediate reward plus the conditional optimal continuation value. For a controlled state $X$ and running reward $\ell$, this reads $J(x,t)={\sup }_a\mathbb{E}[{\int }_t^{t+h}\ell (X_s,a_s,s)ds+J(X_{t+h},t+h)\mid X_t=x]$, under the usual admissible concatenation and information conditions. Conditioning and concatenating strategies proves both inequalities: every strategy's continuation is bounded by the value function, while approximately optimal continuations approach that bound. For a smooth diffusion value function, Itô formula and the limit $h\downarrow 0$ yield the Hamilton-Jacobi-Bellman equation.

Optimal substructure means that the relevant parts of an optimal solution solve their own subproblems optimally. For a 0-1 knapsack problem, removing a chosen item leaves an optimal selection among the other allowed items at the reduced capacity. For sequential mathematical optimization, this becomes the dynamic programming principle: after fixing the next decision and resulting dynamic programming state, use an optimal continuation. Optimal substructure justifies the recurrence relation; overlap between subproblems explains why storing a value function can save work.

A dynamic programming state is an element of a state space that summarizes all information needed for a remaining mathematical optimization problem. Histories can share a value function entry only when their allowed continuations and future rewards agree, or have the same conditional laws in a stochastic control problem. Time or the remaining item index may be part of the summary. If past decisions still affect feasibility or rewards, enlarge the dynamic programming state; forgetting those effects invalidates the Bellman equation.

For a specified characteristic, complete dominance means that the heterozygote has the same phenotype as one of the two homozygotes. It does not mean that the dominant allele is more common, more advantageous, or transmitted more often. Penetrance specifies the conditional probability of an expressed phenotype given a genotype and is a separate modeling assumption.

A DNA replication inhibitor interferes with DNA replication; the observed effect depends on exposure and assay conditions.

Different compounds act at different steps: hydroxyurea limits precursor production, while aphidicolin inhibits selected replicative polymerases. Similar reductions in DNA synthesis can therefore have different molecular causes. Inhibition can also generate replication stress, so the resulting state differs from an unperturbed cell simply paused in time. ^[ref]

A replicating DNA polymerase is an enzyme that extends a paired strand end with nucleotides selected against a DNA template. New DNA grows $5^{\prime }\to 3^{\prime }$ while the template is read $3^{\prime }\to 5^{\prime }$. Base pairing specifies the partner; the DNA polymerase catalyzes its incorporation. See Alberts et al.: DNA polymerase.

In semiconservative replication, each daughter duplex retains one strand of its immediate parent. Under complete copying without strand loss or exchange, one starting duplex gives $2^n$ duplexes after $n$ rounds, but still only two original strands. For $n\ge 1$, the fraction of duplexes containing an original strand is $2/2^n=2^{1-n}$: all after one round, half after two, a quarter after three. The fraction of all strands that are original is instead $2^{-n}$. A strand made in round 1 is parental when round 2 begins.

A ratio compares two quantities $a,b$, written $a:b$ and corresponding to division $a/b$ when $b\ne 0$. Equal ratios of corresponding lengths characterize the scaling of similar triangles.

For example, $6:4$ and $3:2$ specify the same ratio because $6/4=3/2$. Multiplying both entries by the same nonzero scale preserves the quotient: $(ca)/(cb)=a/b$. This is useful when comparing mixtures or scaling a recipe; doubling both ingredient amounts preserves their ratio, while adding the same amount to both generally does not. For quantities with the same physical units, their ratio is dimensionless, connecting this elementary calculation with dimensional analysis.

The quotient of $a$ by a nonzero $b$ in a field is $a/b=a{b}^{-1}$, the unique solution $q$ of $bq=a$. Thus $7/2$ is the rational quotient of $7$ by $2$.

In Euclidean division of integers, the word instead denotes the integer $q$ in $a=bq+r$ with a specified remainder range; for $b>0$, $0\le r<b$. Here $7=2\cdot 3+1$ has quotient $3$ and remainder $1$. Stating the ambient arithmetic prevents confusion between exact division and division with remainder.