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Because $⟨B⟩=0$, a periodic primitive $H$ exists. Fix its additive constant by $⟨H⟩=0$. Integrating the conserved-field equation once in space gives $H_T=\sigma (H_X+\delta A^2{)}_X$: the possible integration constant vanishes on taking the spatial mean. Treat the spatial average as the inner product defining functional derivatives. Integration by parts, with no boundary terms under periodic boundary conditions, givesThus the conserved-field real amplitude equation is a gradient flow ascending $V$, or descending $-V$, with positive diagonal mobilities. Along a classical solution,For the mixed term put $P=⟨A^4{⟩}^{1/2}$ and $Q=⟨H_X^2{⟩}^{1/2}$. The Cauchy-Schwarz inequality and a completed square giveConsequently, for $\beta =1-\delta >0$,The negative leading quartic coefficient proves the upper bound. The stronger pointwise square completionis also useful: a superlevel set $V\ge V(0)$ bounds the amplitude in $L^4$, its spatial derivative in $L^2$, and $B=H_X$ in $L^2$, because a quartic polynomial with negative leading coefficient dominates its lower-order terms.

The dynamics dissipates toward the steady-state set rather than sustaining nonconstant periodic motion. Indeed $V$ increases to a finite limit, and the time integral of $⟨A_T^2+H_T^2/(2\delta \sigma )⟩$ is finite. For global regular trajectories with the usual parabolic precompactness, every limit point is stationary, by the LaSalle invariance principle. A nonconstant periodic or recurrent orbit would force a strictly increasing $V$ to return to its original value and is impossible. Boundedness of a monotone functional alone does not prove convergence to one particular equilibrium; isolation of the limiting equilibrium or an additional convergence theorem supplies that stronger claim.

For a stationary finite-wavenumber instability, the relevant Landau–Ginzburg model is the real Ginzburg–Landau equation: its coefficients are real although its pattern amplitude is complex. The complex amplitude records both the strength and the position of the underlying pattern. Assume a simple critical pair of Fourier modes $\pm k_c$, a quadratic nondegenerate maximum of the linear growth rate, and no additional conserved or marginal mode. The last assumption is essential: the conserved field in the previous problem requires an additional evolution equation.

To derive the model, write a real evolution equation schematically as $u_t=L(r)u+N_2(u,u)+N_3(u,u,u)+\cdots$, with symmetric multilinear nonlinearities. Put $r={\epsilon }^2{r}_2$, $X=\epsilon x$, $T={\epsilon }^{2}t$ and use the method of multiple scales expansionHere $L(k_c;0)v=0$. At the next order the noncritical mean and second harmonic are slaved to the critical mode:where $L(0;0)w_0=-2N_2(v,\overline{v})$ and $L(2k_c;0)w_2=-N_2(v,v)$. These inverses exist under the no-other-critical-mode hypothesis. Let $v^{\ast }$ be an adjoint null vector normalized by $⟨v^{\ast },v⟩=1$. At order ${\epsilon }^3$, the Fredholm solvability condition requires the resonant forcing to be orthogonal to $v^{\ast }$. The cubic resonant contribution isThis shows why the cubic coefficient includes feedback from slaved harmonics as well as any original cubic nonlinearity. The resulting amplitude equation, after naming its linear control coefficient $\mu$, isHere $\lambda (k)$ is the critical linear dispersion relation for growth. The condition $g>0$ gives a supercritical bifurcation; $g<0$ calls for higher-order saturation, for example the quintic real Ginzburg-Landau equation.

A concrete check is the cubic Swift–Hohenberg equation $u_t={\epsilon }^2\mu u-(1+{\partial }_x^2{)}^{2}u-g_0{u}^3$. With $u=\epsilon (A{e}^{ix}+\overline{A}{e}^{-ix})+\cdots$, its linear operator contributes $4A_{XX}$ at resonant order, while the cubic resonance is $3|A{|}^{2}A$. Hence $A_T=\mu A+4A_{XX}-3g_0|A{|}^{2}A$. This directly realizes the perturbation-theory coefficients above.

Symmetry independently constrains the form. A fast spatial translation rotates $A$ by a constant phase, so every nondifferential monomial must transform like $A$. The leading allowed nonlinear term is $|A{|}^{2}A$, and a quadratic term is forbidden. A spatial reflection maps $A(X)$ to $\overline{A}(-X)$; it makes the leading coefficients real and rules out a drift term in a reflection-symmetric rest frame. The vanishing first derivative of the linear growth rate at $k_c$ is consistent with the absence of a leading first spatial derivative. These arguments constrain terms but do not calculate $\xi ,g$ or replace the solvability calculation. Breaking reflection or studying an oscillatory instability generally leads to the complex Ginzburg–Landau equation with genuinely complex coefficients.

The principal steady solutions are the zero state and the Ginzburg-Landau plane wavesTheir physical carrier wavenumber is $k_c+\epsilon q$. For $q=0$ the constant phase is arbitrary, giving a circle of uniform-amplitude states. The zero state has Fourier growth rate $\mu -\xi p^2$ and is stable for $\mu <0$. For $g>0$, each existing plane wave is stable to spatially homogeneous amplitude perturbations, with radial eigenvalue $-2g{R}^2$, and neutral to a constant phase shift.

Steady modulated states are also possible. Writing $A=a(X)e^{i\varphi (X)}$ gives the conserved spatial phase current $j=a^2{\varphi }_X$ andIts first integral is $\xi a_X^2/2+\mu a^2/2-g{a}^4/4+\xi j^2/(2a^2)=E$. Boundary conditions select nonuniform periodic amplitudes, fronts or defects from this spatial dynamics. In the invariant real subspace, for example,is a steady kink joining opposite real states. It is not stable in the unrestricted complex equation: the imaginary perturbation $v=\mathrm{sech}(\sqrt{\mu /(2\xi )}X)$ has eigenvalue $\mu /2>0$ under $v_T=\xi v_{XX}+(\mu -g{A}^2)v$. This illustrates why a real-amplitude calculation can miss a phase instability. With real coefficients the Lyapunov functionalsatisfies $A_T=-\delta \mathcal{F}/\delta \overline{A}$ and $d\mathcal{F}/dT=-2\int |A_T{|}^{2}dX$ under compatible boundary conditions. It supplies a variational structure absent from a generic complex-coefficient model.

To test full linear stability of a plane wave, set $A=e^{iqX}(R+u+iv)$ with real perturbations $u,v$. Let $b=g{R}^2=\mu -\xi q^2>0$. The linear equations areFor perturbation wavenumber $p$, the sideband spectrum of a real Ginzburg-Landau plane wave isThe phase branch at small $p$ isTherefore the robust stable band on the infinite line is $q^2<\mu /(3\xi )$, narrower than the existence band $q^2<\mu /\xi$. Crossing its boundary gives the longitudinal Eckhaus instability. At the boundary the quadratic phase-diffusion coefficient vanishes and the leading nonzero term is negative of order $p^4$. More exactly, a nonzero sideband grows only when $p^2<6q^2-2\mu /\xi$; a finite periodic domain for the wave envelope may exclude that interval. Constant-amplitude stability is understood modulo its neutral global phase, and the allowed perturbation wavenumbers must always be specified.

In a second spatial dimension, an anisotropic medium that selects one roll orientation permits an equation with ${\xi }_x{A}_{XX}+{\xi }_y{A}_{YY}$. For positive coefficients its plane-wave analysis is similar, with the second-dimensional contribution adding damping. But simply replacing $A_{XX}$ by an isotropic Laplacian is not a faithful extension for rotationally invariant stationary rolls. There the critical wavevectors form a circle. Near $(k_c,0)$,This forces the anisotropic transverse scaling of an isotropic roll envelope: $X=\epsilon x$, $Y={\epsilon }^{1/2}y$, $T={\epsilon }^{2}t$. For the positive carrier convention used here the Newell–Whitehead–Segel equation isThe opposite fast-carrier convention reverses the accompanying imaginary sign and detuning convention. For a transverse phase perturbation of $A=R{e}^{iqX}$ our convention gives $\sigma =-\xi q{p}^2/k_c-\xi p^4/(4k_c^2)$. Thus $q<0$ permits the transverse zigzag instability, in addition to the longitudinal Eckhaus instability. The fourth transverse derivative and mixed derivative in the square encode the critical-circle geometry.

One complex amplitude still describes only a narrow cone of nearby roll orientations. Competing well-separated wavevectors require coupled amplitude equations; resonant triples can admit quadratic interactions and produce the hexagonal convection amplitude equations when the underlying sign symmetry allows them. Strong disorder, defect cores with large gradients, far-from-onset patterns, boundaries, mean flows and conserved fields may all lie outside the single-amplitude approximation. The method of multiple scales gives controlled leading behavior for small amplitudes and slow modulations, rather than a universal model of all two-dimensional pattern selection.

The degree-$n$ Bernstein basis isFor $0\le t\le 1$, every basis function is nonnegative, and the binomial theorem givesTherefore $P(t)={\sum }_i{P}_i{b}_i^n(t)$ is a convex combination of its control points. By the definition of their convex hull,This includes the endpoints because $P(0)=P_0$ and $P(1)=P_n$. It depends on the specified parameter interval: outside it the weights need not be nonnegative.

For $n\ge 1$, differentiating the basis and using the elementary binomial identities giveswhere out-of-range basis indices are zero. ThusHence the Bézier derivative control polygon has points $n(P_{i+1}-P_i)$, andIn particular the endpoint tangent vectors are $n(P_1-P_0)$ and $n(P_n-P_{n-1})$. For $n=0$ the curve is constant and its derivative is zero.

A parametric curve interrogation interface first supplies the parameter domain and the point $C(t)$. At regular parameters, it also supplies $C^{\prime }(t)$ and, when needed, higher derivatives. The unit tangent vector, speed and curvature are then obtained fromin three dimensions. The formulas require $C^{\prime }\ne 0$; a Frenet normal additionally requires nonzero curvature. A practical interface can restrict the curve to a parameter interval, provide endpoints, and give a certified bounding volume or a derivative-range bound. Arc length, closest-point and intersection queries are higher-level operations built from these elementary enquiries rather than consequences of point evaluation alone.

For a plane $n\cdot x=d$ with $n\ne 0$, defineThis reduces certified curve-plane intersection to scalar root isolation. A subdivision curve can expose the parametric enquiries by evaluating its limit at a parameter address: refine the corresponding piece until its certified positional enclosure reaches the requested accuracy, or use its exact local spline representation. Derivative evaluation similarly needs the derivative refinement rule and its error bound. A finite control vertex is not automatically a point of the limit curve.

One reliable algorithm is range-based isolation followed by safeguarded Newton iteration:

For a transverse root, the derivative is nonzero nearby, so sufficiently fine intervals have monotone $F$ and the sign bracket isolates it. Bisection provides a global error bound; near a simple root Newton iteration gives quadratic convergence. Good enclosures discard most of the curve before expensive limit evaluation, and the method does not require a specific subdivision mask.

The weaknesses are equally important. Near tangency $F^{\prime }$ is small and Newton steps are ill-conditioned; same-sign endpoints may conceal a tangent root or two crossings. An arc lying in the plane has infinitely many intersections and should be returned as an interval, not a point. Finite-precision evaluations require error enclosures before a sign is certified. For an arbitrary smooth point-evaluation oracle, no finite sampling scheme can certify all intersections: a smooth bump between sampled parameters can create extra crossings without changing any sampled values or derivatives. With finitely many isolated transverse intersections and convergent certified enclosures, the algorithm terminates to any prescribed tolerance. For tangencies it either uses stronger representation tests or reports unresolved possible-contact intervals, rather than silently omitting them.

A subdivision curve interrogation interface provides its control/topology data, a parameter interval or address, and a refinement operation returning restricted child pieces with their parameter correspondence. It also provides a certified enclosing bounding volume, endpoints or limit-point enclosures, and an error estimate for replacing a sufficiently small piece by its chord. Positional and derivative limit queries may be implemented by refinement or by conversion to the appropriate spline. Nonnegative masks often imply a control-point convex hull bound; a generic scheme with negative weights needs a separate bound.

These enquiries can be applied to a parametric curve by using interval restriction as the subdivision operation and constructing its enclosure from its representation. For example, a Bézier curve is split by De Casteljau's algorithm, and each child's control-point convex hull encloses exactly that child arc. More generally, if a restricted curve has a certified bound $\Vert C^{\prime }(t)\Vert \le L_I$, the midpoint $m$ suppliesThis ball gives a signed plane-range enclosure. Such an upper derivative bound must be supplied or proved; taking the largest derivative seen in a few samples is not a certificate.

The recursive intersection algorithm is as follows. Start with the whole domain. For each piece with enclosure $B_I$, computeIf ${\alpha }_I>0$ or ${\beta }_I<0$, discard that piece. Otherwise split it and process the children, unless a root has already been uniquely isolated and can be polished as in Question 3. For a convex hull bound, the extrema of this affine plane function are attained among its control points, so only dot products are required. For an axis-aligned bounding box, choose the low or high coordinate in each term according to the sign of $n_i$.

At a small flat piece, the chord-plane parameter iswhen the denominator is nonzero. It supplies an initial estimate, which is corrected on the actual curve. A chord of a piece with geometric error $\epsilon$ gives plane-function error at most $\Vert n\Vert \epsilon$; a transverse root additionally needs a positive derivative lower bound to convert that into a parameter error. A chord hit is not by itself a proof of the exact crossing count. Keep unresolved tangent contacts, merge shared-endpoint hits, and return any certified coplanar parameter intervals separately.

Good speed comes from a bounding volume hierarchy, conservative cheap rejection before accurate evaluation, caching shared subdivision work, local rather than uniform refinement, and fast root polishing only after isolation. Convex-hull or box extrema cost far less than solving every potential intersection. Tight bounds shrink rapidly with subdivision and reduce the active search to neighborhoods of actual crossings. Derivative information can prove monotonicity, accelerate with the Newton root-finding iteration, or tighten range bounds using interval arithmetic. There is no universal fixed speed guarantee when the curve is nearly coplanar or has arbitrarily many intersections: those are genuinely difficult inputs. The representation must provide meaningful bounds and restriction operations; point/derivative enquiries alone do not imply such capabilities.

In a convenient cardinal parameter normalization, take the unit $m$-cube and project by the sum of its coordinates. Its pushforward density is the unit-direction univariate box splineEquivalently, with $B_1=1_{[0,1)}$ at its discontinuity points,Here there are $m$ convolution factors; some conventions instead index the spline by its degree $m-1$. The half-open convention for $m=1$ assigns knot values consistently and does not change the pushforward measure.

Support and positivity. A sum of $m$ numbers in $[0,1]$ lies in $[0,m]$, so the closed support is finite. For $0<x<m$, the interval of $u\in (0,1)$ with $0<x-u<m-1$ has positive length. Induction in the convolution recurrence therefore gives $B_m(x)>0$ in the interior. For $m\ge 2$, $B_m(0)=B_m(m)=0$: strict positivity includes the open support interval, not its endpoints.

Partition of unity. The translates of $B_1$ cover the line exactly once under the half-open convention. Nonnegativity permits interchange of sum and integral, and induction givesThus $\boxed{\sum _j{B}_m(x-j)=1}$, a locally finite sum. Each $B_m$ also has integral one, although partition of unity does not mean its maximum value is one.

For $m\ge 2$, repeated integration gives the explicit truncated-power formOn each unit knot interval this is a polynomial of degree at most $m-1$, and on $(0,1)$ it is $x^{m-1}/(m-1)!$, so the degree is attained. Each truncated power has $m-2$ continuous derivatives; the derivative of order $m-1$ jumps at the first knot. Therefore the dependence on the projected box dimension isFor $m=1$ the pieces are constants and the function is discontinuous at the knots. If geometric width is measured under orthogonal projection onto the unit diagonal instead, the projected coordinate is $x/\sqrt{m}$, so the physical support width is $\sqrt{m}$; its unit-mass density is $\sqrt{m}{B}_m(\sqrt{m}x)$. These are different scalings of the same cardinal spline, not different continuity or degree claims.

For a bivariate box spline, choose a spanning direction matrix $\Xi =({\xi }_1,\dots ,{\xi }_m)$ in the plane and project the unit cube by $u\mapsto \Xi u$. Define its normalization byThe support is the zonotope ${\sum }_r[0,1]{\xi }_r$, and the polynomial pieces generally have total degree $m-2$. The grid comes from the chosen direction families and their lattice translates. Two independent directions, with repetitions, give tensor-product B-splines on a parallelogram grid. Three families such as $e_1,e_2,e_1+e_2$ give a three-direction triangular grid; an affine change of basis can make it the usual equilateral triangular grid. The four families $e_1,e_2,e_1+e_2,e_1-e_2$ give the criss-cross square grid and the quadratic four-direction box spline. Changing the direction matrix and repeating directions therefore changes the grid, support and smoothness while retaining the box-projection construction.