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A tractrix is a plane curve whose tangent segment from the point of contact to a fixed axis has constant length. For segment length one and a positive acute tangent angle $\theta$, its height is $r=\sin \theta$. The slope relation $dr/dh=\tan \theta$ gives $dh/d\theta ={\cos }^2\theta /\sin \theta$. Integrating yields the displayed axial coordinate up to an additive constant. Conversely these coordinates have the required slope, and their tangent intercept is at distance $r/\sin \theta =1$. Rotating this profile about its axis gives a tractoid.

A transported planar potential dipole follows the surrounding regular velocity, so its position satisfies the displayed equation after its own singular term is removed. The conjugation comes from the convention $w^{\prime }=v_x-i{v}_y$. Reflected image parameters are held fixed when taking the spatial derivative, then evaluated at the instantaneous dipole position.

For a unit circular wall, the Milne-Thomson circle theorem gives image potential $\bar{m}z/(1-\bar{a}z)$ and therefore $\dot{\bar{a}}=\bar{m}/(1-|a{|}^2{)}^2$. For a horizontal straight wall the image is $\bar{m}/(z-\bar{a})$, giving $\dot{\bar{a}}=-\bar{m}/(a-\bar{a}{)}^2$. For the line $y=-x$, reflected conjugation gives image $-i\bar{m}/(z+i\bar{a})$ and hence $\dot{\bar{a}}=i\bar{m}/(a+i\bar{a}{)}^2$. In each case the image lies outside the fluid region and makes the wall a streamline; differentiation of the regular image proves the motion formula. Several compatible straight-wall reflections can be summed before the same operation is applied.

For the autonomous differential equation $y^{\prime }=-|y-y_{\ast }{|}^p$, the displayed family stays at $y_{\ast }$ until time $c$ and then decreases below it. Here $[z{]}_{+}=\max (z,0)$. Direct differentiation verifies the equation on each side; since $1/(1-p)>1$, the derivative tends to zero at departure, making the joined function continuously differentiable. Distinct departure times give different solutions from the same equilibrium initial condition. The ratio $|f(y_{\ast }+\epsilon )-f(y_{\ast })|/\epsilon ={\epsilon }^{p-1}$ is unbounded, so the local Lipschitz continuity hypothesis of the Picard-Lindelöf theorem fails.

A solution starting above $y_{\ast }$ reaches it in finite time. It can then use any of these continuations. For $p=1/2$, $y_{\ast }=-1$, and initial value $y(0)=0$, arrival occurs at time two; departure at $c\ge 2$ gives $y=-1-(t-c{)}^2/4$ afterwards. Waiting-time solutions of a sublinear Bernoulli equation give a related mechanism for departures in the opposite direction.

Suppose $f$ is continuous and Lipschitz in its state variable on a closed cylinder $|t-a|\le h$, $\Vert y-b\Vert \le k$, with bounds $\Vert f\Vert \le M$ and Lipschitz constant $L_f$. On a smaller interval of radius $\eta$, the displayed Picard operator maps continuous curves in that closed ball into itself when $\eta M\le k$, and it is a contraction mapping in the supremum norm when $\eta L_f<1$. This function space is complete because it is a closed subset of a Banach space of continuous vector-valued functions. The contraction mapping theorem gives a unique fixed point, and the fundamental theorem of calculus makes it a solution of the ordinary differential equation. Choosing $\eta M<k$ also prevents any solution with the same initial data from first exiting the cylinder during that interval.

A coefficient that vanishes at the initial time can hide failure of the uniform Lipschitz condition required for an initial value problem. For $f(x,y)=x|y{|}^{2/3}$, the map $y\mapsto f(0,y)$ is identically zero. But every rectangle of positive widths around $(0,0)$ contains some $x\ne 0$, andNo Lipschitz constant works uniformly on the rectangle. Actual uniqueness also fails: $y=0$ and $y=x^6/216$ are distinct differentiable solutions with initial value zero. Substitution gives $y^{\prime }=x^5/36=x|y{|}^{2/3}$ for the second solution. A condition at the single initial time cannot replace control on a neighbourhood.

For the ordinary differential equation $y^{\prime }=3x^2/(4y^3)$ through $y(0)=b>0$, consider a symmetric existence rectangle $|x|\le h$, $|y-b|\le k<b$. Its maximum slope bound is $M=3h^2/[4(b-k{)}^3]$. The condition $Mh\le k$ becomes $h^3\le 4k(b-k{)}^3/3$. Differentiating $k(b-k{)}^3$ shows its maximum occurs at $k=b/4$, givingThis is the largest interval guaranteed by this particular centered-rectangle criterion. Direct separation of variables instead gives $y(x)=(b^4+x^3{)}^{1/4}$ on the maximal real interval $x>-b^{4/3}$. The comparison shows the difference between a sufficient local existence theorem and the actual continuation interval. The endpoint obstruction is the zero of the denominator at $y=0$.

The source of the medicinal seed called Calabar bean.

Explanation of bulk assimilate transport from source to sink through hydrostatic pressure differences associated with solute loading and unloading.

A periodic phase response function describes the first-order phase shift produced by a perturbation along a limit cycle. With period-one orbit $U$, choose the periodic adjoint solution normalized by ${\int }_0^{1}Y\cdot U^{\prime }ds=1$. The Fredholm solvability condition then projects forcing $p$ to ${\int }_0^{1}Y\cdot pds$. For voltage-difference coupling, the averaged interaction is $h(\varphi )={\int }_0^1{Y}^1(s)U^1(s+\varphi )ds$.

Linearizing weakly coupled locked phases gives $0=\xi -\Omega 1-kL\theta$, where $L$ is the directed graph Laplacian. Multiplying by a normalized left null vector ${\pi }^{\mathsf{T}}$ eliminates coupling and gives the displayed weighted frequency. In a balanced network, including symmetric reciprocal coupling, $\pi =1/N$ and the frequency is the arithmetic mean. A zero diagonal in the coupling matrix alone does not imply balance. For two directed weights $p,q>0$, the weights are $(q,p)/(p+q)$.

The Perron's formula kernel $y^{s+1}/(s(s+1))$ integrates to $(y-1{)}_{+}$, giving the smoothed sum as an integral of $x^{s+1}/(\zeta (s)s(s+1))$. For each fixed height $T$, a zero-free rectangle permits shifting the central contour to a real part below one; its vertical contribution is $o(x^2)$, and its horizontal contributions are $O_T(x^2/\log x)$ when the original real part is $1+1/\log x$. A bound $1/\zeta (\sigma +it)=O({\log }^7(2+|t|))$ for $\sigma \ge 1$ bounds the remaining tails by $C{x}^2{\int }_T^{\infty }{\log }^7(2+t)t^{-2}dt$. Letting $x$ grow first and $T$ afterward proves the displayed cancellation, without uniform estimates to the left of one.

With $e_n(s)=(2\pi {)}^{-1/2}{e}^{ins}$, periodic $L^2$ functions have the displayed convolution identity for every real $t$. Apply the polarized Parseval identity to $f$ and $\overline{g(t-\cdot )}$; a change of variable gives the latter function's pairing with $e_n$. The Cauchy-Schwarz inequality bounds the sum of absolute coefficient products by $\Vert f{\Vert }_2\Vert g{\Vert }_2$, so the resulting series converges absolutely and uniformly. This proves continuity of the convolution even though neither original Fourier series need converge pointwise.

A rank-two period lattice $\Lambda \subset \mathbb{C}$ stable under multiplication by $i$ is, up to nonzero complex scaling, $\mathbb{Z}+i\mathbb{Z}$. Choose a shortest nonzero period $\omega$. Every complex number lies within $1/\sqrt{2}<1$ of a Gaussian integer, so for any $\eta \in \Lambda$, subtraction of a suitable $(m+ni)\omega$ leaves a shorter period and hence zero. This proves $\Lambda =\omega \mathbb{Z}[i]$.

Similarly, stability under a primitive cube root $\zeta$ gives $\Lambda =\omega (\mathbb{Z}+\zeta \mathbb{Z})$. The triangular lattice has covering radius $1/\sqrt{3}<1$, so the same shortest-period argument applies. Stability under a primitive sixth root is equivalent, since $e^{\pi i/3}=1+e^{2\pi i/3}$. These facts identify complex tori with automorphisms fixing their origin of orders four and three or six, respectively.

The outer envelope of a flower, composed of sepals.

The petals of a flower, collectively.

A part of the perianth when its members are not differentiated into sepals and petals.

For disjoint closed sets $A,B$ in a subspace $Y$, their ambient closures $C,D$ satisfy $(C\cap D)\cap Y=\varnothing$. In a perfectly normal space choose continuous functions $f,g$ with exact zero sets $C,D$. The function $f/(f+g)$ is continuous on $X\setminus (C\cap D)$, which contains $Y$, and separates $A,B$. Disjoint inverse images of endpoint neighborhoods give normality of $Y$.

For $h>0$, the third forward difference $L_{h}u=-u(0)+3u(h)-3u(2h)+u(3h)$ annihilates polynomials of degree at most two. Its Peano kernel is $K_h(t)=\frac{1}{2}[3(h-t{)}_{+}^2-3(2h-t{)}_{+}^2+(3h-t{)}_{+}^2]$ for $0\le t\le 3h$, extended by zero outside that interval. It is $h^2$ times the quadratic cardinal B-spline evaluated at $t/h$, is positive on $(0,3h)$, and has integral $h^3$. The Peano kernel theorem therefore represents $L_{h}u$ as a positive weighted average of the third derivative: $L_{h}u=h^3{u}^{(3)}(\xi )$ for some interior $\xi$ when that derivative is continuous. The kernel is the smooth error weight hidden inside a discrete difference.

The Peano kernel theorem writes a derivative-based error functional as $L(u)=\int K(t)u^{(r)}(t)dt$, so its bound in the displayed derivative seminorm is at most the L1 norm of the kernel. This is the sharp constant: continuous derivatives can approximate the sign of $K$ in the weighted integral, and integrating them $r$ times produces admissible test functions. If $K$ has one sign, a constant $r$th derivative already attains equality; in particular $\int K=L[x^r/r!]$. This provides a direct exact check on proposed error constants before estimating individual polynomial pieces.

An infinite path in a transition system represented by a finite prefix followed by infinite repetition of a nonempty finite loop. If stored positions are $q_0,\dots ,q_k$ and $q_k\to q_{\ell }$, the represented sequence is $q_0\cdots q_{\ell -1}(q_{\ell }\cdots q_k{)}^{\omega }$. The linear temporal logic condition $GFp$ holds exactly when $p$ occurs on the repeated part, and $FGp$ holds exactly when $p$ holds everywhere on that part.