A characteristic surface may remain smooth while its projection to the independent-variable plane fails to be locally invertible. At a generic fold, smooth changes of parameter and plane coordinates reduce this projection to $(u,v)\mapsto (u,v^2)$. There are two sheets on one side of the image of $v=0$, no sheet on the other, and one merged sheet on the fold itself. The image is a fold caustic. A characteristic graph can terminate there even though the space curves defining the surface continue smoothly. Vanishing of the Jacobian determinant detects a candidate singular locus; verifying the fold requires a nonzero quadratic term in its null direction, as in $v^2$.

For the first-order quasilinear partial differential equation $P(x,y,z)z_x+Q(x,y,z)z_y=R(x,y,z)$, the displayed ordinary differential equations trace curves tangent to every classical solution surface. Indeed, if $z=u(x,y)$ solves the equation and $x_s=P$, $y_s=Q$, then the chain rule gives $z_s=u_{x}P+u_{y}Q=R$. Conversely a surface swept out by these curves solves the equation wherever its projection to $(x,y)$ is invertible and hence represents a differentiable graph. The coefficients can depend on the unknown $z$, which distinguishes these equations from characteristic equations for a prescribed linear transport equation.

For a first-order partial differential equation with the displayed projected characteristic field, label the initial segment by $r=y(0)\in [0,c]$. The characteristic curves are $x=a-(a-b)e^s$, $y=r{e}^s$. They are the branches of straight lines through $(a,0)$ on the side $x<a$. The inverse coordinates areThey are unique exactly in the displayed sector. The Jacobian determinant is $-(a-b)e^{2s}\ne 0$, so the initial segment is noncharacteristic and its transported data determine a unique classical solution on the sector's interior whenever the characteristic equation for the dependent variable has a unique solution. The bounding rays inherit data from the two endpoints. The point $(a,0)$ is approached only at infinite parameter time, and characteristics on the opposite side cannot reach the initial segment.

A characteristic caustic is a locus where the coordinate map from initial-data and evolution parameters to physical coordinates has zero Jacobian determinant. The inverse function theorem then fails, and different characteristic branches can reach the same point. A classical solution constructed from initial data must be restricted to its appropriate invertible branch before such a fold. In geometrical optics the corresponding ray-envelope phenomenon is a caustic; in nonlinear first-order equations a fold can signal the onset of multiple gradients or shock formation by characteristic intersection.

For a fixed real unit vector $k$, the linear ordinary differential equation $\dot{r}=r-k\times r$ combines dilation with rotation about the axis $\mathbb{R}k$. Its solution from a line $s\ell$, with $|\ell |=1$, isSince $k\cdot r=s{e}^t\kappa$ and $|r|=|s|e^t$, the union of these characteristic curves is given by $|k\cdot r|=|\kappa ||r|$. For $0<|\kappa |<1$ it is a circular cone of semi-angle $\arccos |\kappa |$: varying $t$ supplies every azimuth and varying $s$ supplies every radius on both halves. For $\kappa =0$ it is a plane; for $|\kappa |=1$ it is only the axis. In the last case the line is itself a characteristic curve, so the initial data do not generate a two-dimensional surface. The cone is a smooth surface away from its vertex, and it is a graph of a function only on suitable patches. Invariance of a graph patch under characteristic curves holds only while the curve stays in its domain; global flow invariance requires continuation of that patch.

For a first-order nonlinear partial differential equation $F(x,y,u,p,q)=0$, with $p=u_x,q=u_y$, Charpit's equations extend the method of characteristics to the five-dimensional space of position, value and gradient:The chain rule gives $dF/dt=0$ by direct cancellation, so a trajectory starting on $F=0$ remains there. Initial data $u_0(s)$ on $(x_0(s),y_0(s))$ require both $F(x_0,y_0,u_0,p_0,q_0)=0$ and $p_0{x}_0^{\prime }+q_0{y}_0^{\prime }=u_0^{\prime }$. The non-characteristic condition is $x_0^{\prime }{F}_q-y_0^{\prime }{F}_p\ne 0$. It makes the projection onto $(x,y)$ locally invertible. Compatible initial contact data then propagate as $du=pdx+qdy$, reconstructing a classical solution. Multiple permitted initial normal derivatives can give different local branches; loss of the projection Jacobian determinant marks a characteristic caustic.

For $H_t+H^{n+1}{H}_x=1-2x$ with $H(\xi ,0)=0$, method of characteristics conserves $H^{n+2}/(n+2)-(x-x^2)$. A positive downstream branch therefore requires $0\le \xi <x<1-\xi$. Initial points in the negative-accumulation region cannot create nonnegative ice from a bare bed. Characteristic intersections and missing dry-bed physics prevent using the formal initial/boundary branches as an unrestricted global solution.

A smooth characteristic surface in three-dimensional $(x,y,u)$ space need not remain the graph of a differentiable function after projection to the $(x,y)$ plane. Locally, if its implicit equation is $F(x,y,u)=0$, the implicit function theorem produces a smooth graph wherever $F_u\ne 0$. At a point where $F_u=0$, the projection can fold and two graph branches can meet. A real radical formula may continue to that boundary even though its partial derivatives diverge there. Thus the closed set on which the radical is real can be larger than the open domain of a classical solution.

For example,has two branches $u=(15x+12y\pm 5\sqrt{9x^2+40xy})/16$. On them $F_u=\pm 10\sqrt{9x^2+40xy}$, so the graph projection folds on the nonzero boundary points of $9x^2+40xy\ge 0$. Its gradient in the base variables does not vanish at those points, and differentiation of the radical gives the transverse derivative singularity. The characteristic surface itself can be smooth there: its graph coordinate is the part which fails. Boundary values may still be defined continuously, but they do not create an open classical-solution neighborhood.

The method of characteristics from $s\ne 0$ are $(x,y,z)=(s{e}^t,s{e}^{2t},s^2(2+e^{3t})/3)$. Their projections are half-parabolas $y=x^2/s$ with $x/s>0$. The inverse parameters $s=x^2/y$, $t=\log (y/x)$ exist exactly on the first and third quadrants, giving the unique solution $z=2x^4/(3y^2)+xy/3$ there. The projection Jacobian $s{e}^{3t}$ is nonzero, so every allowed initial point is noncharacteristic.

The equations of the method of characteristics give $(x,y,u)=(s+t^2/2,t,s{e}^t)$. Their parameter map to the plane has Jacobian determinant one, so it is globally invertible and $u=(x-y^2/2)e^y$ is uniquely determined exactly where $1\le x-y^2/2\le 2$. Both positive and negative $y$ occur. Outside this closed parabolic strip, $u=e^{y}F(x-y^2/2)$ allows arbitrary smooth changes of $F$ outside the initial parameter interval. The boundary characteristic curves carry the data from its endpoints.

For the partial differential equation $u_x+u{u}_y=x$ with initial profile $u(0,\eta )=1+\sin \eta$, the method of characteristics gives the parametrized surfaceThe projection to the $(x,y)$ plane has Jacobian determinant $1+x\cos \eta$. It is positive for $|x|<1$, so the inverse function theorem makes the surface a unique local solution graph there. For $\eta \ge 0$, strict monotonicity and divergence to infinity give a global inverse onto $y\ge x+x^3/6$ at each such $x$.

The derivative $u_y=\cos \eta /(1+x\cos \eta )$ becomes singular when that projection first degenerates, although $u$ itself stays finite. At $x=1$ the degenerate labels are odd multiples of $\pi$, and at $x=-1$ they are even multiples. At either boundary the limiting map remains strictly increasing, with isolated zero derivatives, so fixed image values determine convergent labels. Consequently the gradient diverges at fixed $y=(2k+1)\pi +7/6$ as $x\uparrow 1$, and at fixed $y=2k\pi -7/6$ as $x\downarrow -1$, using $k\ge 0$ and $k\ge 1$, respectively, to stay in the half-line initial domain. At these first singularities $y_{\eta \eta }=0$ and $y_{\eta \eta \eta }\ne 0$; this is the onset of loss of invertibility, rather than an ordinary nondegenerate fold at the initial singular time.

A process with finite positive capacity whose output sequence is a prefix of its input sequence and whose input-output length discrepancy never exceeds that capacity. While active, it offers input whenever non-full and output whenever nonempty; upstream closure drains retained data, while downstream closure cancels delivery.

Mercury(II) sulfate is the salt represented by $\mathrm{HgS}{\mathrm{O}}_4$, combining mercury in oxidation state +2 with sulfate. The alternative name mercuric sulfate refers to the same composition.^[ref]

Charge balance distinguishes it from a mercury(I) sulfate: the latter contains the dimeric mercury(I) ion, $\mathrm{H}{\mathrm{g}}_2^{2+}$, and has two mercury atoms per sulfate group. A formula alone does not establish which dissolved complexes or hydrolysis products occur in a particular solution; those also depend on chemical equilibrium and conditions such as pH.

A mercury barometer with its lower end in a reservoir exposed to atmospheric pressure. A fixed-cistern scale must allow for the changing reservoir level, while an adjustable cistern can restore a fixed lower reference before each reading.

A measured mercury column represents pressure through its actual mass density and local gravitational acceleration. An equivalent reference column therefore obeys ${\rho }_0{g}_s{h}_s=\rho (T)gh$. If the indicated scale length expands as $1+{\alpha }_s\Delta T$ and mercury volume as $1+\beta \Delta T$, the first-order thermal factor multiplying the scale reading is $1+({\alpha }_s-\beta )\Delta T$. These factors describe a pressure-unit reduction, not a physical change in local gravity.^[ref]

For the five-point Dirichlet Laplacian on an $n\times n$ square grid, a block-Jacobi splitting that solves each horizontal line exactly has blocks $\mathrm{tridiag}(-1,4,-1)$. Its tensor-product sine modes have angles ${\theta }_r=r\pi /(n+1)$ and the displayed iteration matrix eigenvalues. With $c=\cos (\pi /(n+1))$, its spectral radius is $c/(2-c)<1$, so it converges on every fixed grid. However the mode $r=1,s=n$ has multiplier $-c/(2-c)\to -1$. It is highly oscillatory between lines and is barely damped. The undamped method is therefore not a uniformly effective smoother for the multigrid method, even though it converges. Convergence and smoothing test different properties of the error spectrum.

A real symmetric matrix whose entries are the moments $H_{ij}={\int }_0^1{x}^{i-1}{x}^{j-1}dx$. Thus it is the Gram matrix of the monomials in the real L2 inner product on $[0,1]$. For a nonzero real coefficient vector $v$, $v^{T}Hv={\int }_0^1({\sum }_i{v}_i{x}^{i-1}{)}^{2}dx>0$, so every finite Hilbert matrix is positive-definite. The moment description connects these matrices with polynomial approximation and with the shifted Hilbert quadratic-form bound.

A rectangular array whose entries are nondecreasing in each coordinate. Every subarray obtained by restricting to a row interval and a column interval has the same property. Comparing a target with the top-right entry discards a whole row or column, giving staircase matrix search in time linear in the sum of the dimensions.