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Independent sign changes of three amplitudes and a cyclic coordinate permutation force each cubic component to be its own amplitude times a cyclic permutation of one quadratic polynomial. One convenient parametrization is $\dot{x}=x(\mu -aX-c{y}^2+e{z}^2)$ with cyclic companions and $X=x^2+y^2+z^2$. The axial and diagonal isotropy types give primary branches; two-coordinate branches can emerge in secondary pitchfork bifurcations when cross-couplings change sign.

The Newell–Whitehead–Segel equation is an amplitude equation for slowly modulated stationary rolls in an isotropic pattern-forming medium. With the displayed sign convention, a roll $A=R{e}^{iqX}$ has $R^2=\mu -q^2$. A transverse phase modulation has linear growth rate $q{k}^2-k^4/4$, so long-wave transverse instability occurs for $q>0$. Reversing the carrier-wave convention reverses the sign attached to its detuning $q$; one must derive the stability sign from the operator actually used.

Two counterpropagating modes transform as $(A,B)\mapsto (e^{ik\delta }A,e^{-ik\delta }B)$ under spatial translation. Forcing at twice their natural frequency breaks continuous temporal phase symmetry but preserves simultaneous sign reversal. Hence $\overline{B}$ transforms like $A$ and $\overline{A}$ like $B$, allowing linear parametric coupling. An equal-amplitude phase-locked standing wave has $|A{|}^2=|B{|}^2=z>0$ with $|\beta +\gamma {|}^2{z}^2-2\mu \mathrm{Re}(\beta +\gamma )z+{\mu }^2-{\nu }^2=0$. A nonnegative discriminant is only an algebraic compatibility condition: a physical nonzero wave also requires a positive root.

A forcing at the carrier wavenumber breaks continuous spatial translation symmetry and adds a constant to its complex amplitude equation. In a frame following a moving forcing pattern, its phase drift gives $i\Lambda A$. A nonzero forcing coefficient is normalized to one by scaling amplitude by its cube root and time by its two-thirds power. A steady intensity $s=|A{|}^2$ satisfies $s[(\mu -s{)}^2+{\Lambda }^2]=1$, and its phase is uniquely determined by the steady complex equation.

Assume $\sigma >0$. With periodic boundary conditions, the spatial mean of $B$ is a conserved quantity. For zero mean write $B=H_X$ with periodic zero-mean $H$. The functional $V=⟨\mu A^2/2+\alpha A^3/3-A^4/4-A_X^2/2-A^2{H}_X/2-H_X^2/(4\delta )⟩$ generates $A_T=\delta V/\delta A$, $H_T=2\delta \sigma \delta V/\delta H$. Thus $V_T=⟨A_T^2+H_T^2/(2\delta \sigma )⟩\ge 0$. Completing the square in $H_X$ bounds $V$ above by a coercive quartic polynomial when $0<\delta <1$, ruling out nonconstant recurrent motion on regular precompact trajectories.

The partial differential equation $T_t=-\mu \Delta T-{\Delta }^{2}T+\nabla \cdot (|\nabla T{|}^2\nabla T)$ on a $2\pi$-periodic square conserves the spatial mean. On a fixed-mean subspace its first instability is at $\mu =1$ and has four real critical amplitudes. With $\mu =1+{\epsilon }^2\nu$ and $T=\epsilon (A{e}^{ix}+B{e}^{iy}+\text{complex conjugate})+\cdots$, projection onto the critical Fourier modes gives the displayed equation and its $A,B$ interchange. For $\nu >0$, squares have $|A{|}^2=|B{|}^2=\nu /5$, amplitude eigenvalues $-2\nu ,-2\nu /5$, and neutral translation phases. Rolls have a positive transverse growth rate $\nu /3$.

A frog is an amphibian of the tailless adult group. Many species develop from aquatic tadpoles, while others have different reproductive modes. The adult’s jumping body plan and the larva’s feeding apparatus belong to distinct developmental stages.

In vacuum magnetostatics, the Ampère-Maxwell equation reduces to $\nabla \times \mathbf{B}={\mu }_0\mathbf{J}$. Applying Stokes theorem gives ${\oint }_C\mathbf{B}\cdot d\mathbf{\ell }={\mu }_0{I}_{\mathrm{enc}}$, where $I_{\mathrm{enc}}$ is the electric current crossing an oriented spanning surface. Time-dependent electric fields require the additional displacement current term in the Ampère-Maxwell equation.

Cysteine is an amino acid whose amino acid side chain contains sulfur, represented by Cys or C for an incorporated amino acid residue. Two cysteine positions can be joined through a protein disulfide bond. See IUPAC Blue Book, P-103.1: cysteine's structure and symbols and Cooper: cysteine crosslinks in proteins. The C in a sequence means cysteine, whereas the C-end label in the toy diagram means C-terminus.

Phenylalanine is an amino acid, represented by Phe or F for an incorporated amino acid residue. It was identified at the other insulin N-terminus in Sanger's insulin sequencing experiment. See IUPAC Blue Book, P-103.1: amino-acid names and symbols.

Glycine is an amino acid, represented by Gly or G for an incorporated amino acid residue. Its amino acid side chain is a hydrogen chemical atom. It was identified at one insulin N-terminus in Sanger's insulin sequencing experiment. See IUPAC Blue Book, P-103.1: amino-acid names, symbols and structures.

An alpha-amino acid has amino functionality on the carbon next to its carboxyl carbon. The skeleton in amino acid shows this adjacency explicitly. Numbering the carboxyl carbon as C1 makes the adjacent position C2, also called the alpha position. ^[ref]

Alanine is an amino acid represented by Ala or A in protein sequence notation. GCU is one of its codons in the standard genetic code, and supplies the second residue in the worked gene expression example. ^[ref]

Methionine is an amino acid represented by Met or M in protein sequence notation. In the standard genetic code, AUG encodes it, as in the worked gene expression example. ^[ref]

In a discrete-time binomial market with $B_t=(1+r{)}^t$ and one-step stock factors $1\pm \epsilon$, use the risk-neutral probability $p=(1+r/\epsilon )/2$, where $0\le r<\epsilon <1$. For the exercise payoff $S_t^2$, the discounted one-step reward multiplier has meanThe Snell envelope of a multiplicative process gives the value $V_t=S_t^2{\lambda }^{T-t}$. Continuation strictly exceeds exercise before maturity, so the optimal exercise time is $T$. The replicating portfolio in a binomial market holds $2S_t{\lambda }^{T-t-1}$ shares on the next step, found by subtracting the two next-state values and dividing by the stock-price difference. At time zero the price is $S_0^2{\lambda }^T$.

An American call option permits buying the underlying at strike price $X$ at any chosen stopping time from the current date $t$ through maturity $T$. Under standard complete-market pricing and integrability hypotheses,The conditional expectation uses information available at time $t$, and the optimization allows only future exercise times. A European call option restricts exercise to $T$. The no early exercise of a call without dividends result applies with nonnegative interest under its stated assumptions; dividends or negative rates can change that conclusion. The discounted value is the Snell envelope of discounted exercise rewards, as in American option.

For a countable discrete group, the Reiter condition requires a sequence of probability measures in ${\ell }^1(G)$ asymptotically invariant under every left translation. It is equivalent to the existence of an invariant mean and a Følner sequence. Separation by bounded functions shows that amenability gives approximate invariance simultaneously on each finite set of translations; countability makes the successive finite tests into a sequence.

An inclinometer measures tilt relative to the vertical established by gravity, usually reporting an angle from the perpendicular horizontal plane. A weighted-string inclinometer compares a sighting tube with a freely hanging reference; its scale must be oriented so a level sight gives zero altitude. A raw protractor may instead indicate the complementary angle, so check the markings rather than assuming that every instrument has the same scale. ^[ref].

A finite alphabet is an alphabet whose symbols form a finite set. The binary alphabet $\{0,1\}$ and a finite vocabulary of possible messages are examples.

A group variety is an algebraic variety whose multiplication and inversion are morphisms.