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Saturating positive feedback limits the reinforcing rate as activity increases. Here is an explicitly synthetic ordinary differential equation for normalized nonnegative input $u$, response $x$, and return strength $\beta$:The return is an increasing, nonproportional function of $x$, while removal is proportional to $x$. Time and activities are dimensionless variables. At $x=0$ the vector field points into the nonnegative domain, and $dx/dt\le u+\beta -x$ prevents unbounded growth there. This makes the toy useful for asking about a bounded switch; it supplies no molecular mechanism for the chosen rate law.

At an equilibrium point, $f=0$. For a simple equilibrium point, the sign of the derivativedetermines local asymptotic stability. If $\beta =1$, the maximum reinforcing slope is $3\sqrt{3}/8<1$, so $f$ decreases strictly: every $u\ge 0$ has one asymptotically stable equilibrium point despite positive feedback.

Three equilibrium points become possible only if the reinforcing slope exceeds removal's slope somewhere. Its maximum is $3\sqrt{3}\beta /8$, at $x=1/\sqrt{3}$. Consequently this toy requiresfor two distinct folds. This is a necessary threshold for a bistability interval in this toy, not a universal biochemical number. At a fold, both $f=0$ and $f_x=0$, givingFor $\beta =1.8$, the folds are at $(x,u)\simeq (0.350003,0.153563)$ and $(0.887240,0.094405)$. Between these input values there are two asymptotically stable equilibrium points separated by an unstable one. At $u=0.12$, the three activities are approximately $0.171331$, $0.621230$ and $1.127439$; the low and high ones are stable.

A sufficiently slow increasing sweep follows the low branch to $u_{\mathrm{on}}\simeq 0.153563$ and then jumps upward. A decreasing sweep follows the high branch to $u_{\mathrm{off}}\simeq 0.094405$ and then jumps downward: hysteresis. These are limiting quasistatic thresholds, with relaxation between input changes and negligible perturbations that could cause earlier switching. Finite sweep rates and fluctuations can alter observed thresholds. A switchlike response with only one stable equilibrium point does not have this equilibrium hysteresis mechanism.

For uniformly bounded values, an $L^p$ gradient bound with $p>1$, and bounded $(n-1)$-dimensional jump measure, a sequence in the SBV space admits an $L^1$-convergent subsequence whose limit remains special. Gradients converge weakly in $L^p$, and jump measure is lower semicontinuous. These hypotheses prevent diffuse singular derivatives from replacing the controlled interfaces in a minimizing limit.

These seminorms measure all polynomially weighted derivatives and generate the Fréchet topology of the Schwartz space. Finitely many such bounds control each weighted derivative's integrable norm by adding a spatial decay factor stronger than $⟨x{⟩}^{-n}$. Continuity of the Fourier transform follows by applying those bounds after differentiation and integration by parts. Equivalent families use powers of $⟨x⟩$ instead of individual monomials.

The secant function is the reciprocal of the cosine, $\sec x=1/\cos x$, where the denominator is nonzero. Its derivative is $\sec x\tan x$, and ${\sec }^{2}x=1+{\tan }^{2}x$ follows from the Pythagorean trigonometric identity.

The derivative of the first derivative. For a twice differentiable function, its sign governs local convexity and its magnitude controls the second-order Taylor remainder.

Differentiating a twice differentiable kernel density estimator twice estimates $f^{\prime \prime }$. Its integrated variance of a kernel density estimator analogue is at most $\Vert L^{\prime \prime }{\Vert }_2^2/(n{b}^5)$. This explains why derivative estimation requires more smoothing than ordinary density estimation.

A second-order ordinary differential equation involves a second derivative as its highest derivative. In explicit form $y^{\prime \prime }=f(x,y,y^{\prime })$, introducing $v=y^{\prime }$ converts it into the first-order system $y^{\prime }=v$, $v^{\prime }=f(x,y,v)$. Under local regularity and Lipschitz continuity in the dependent variables, an initial value problem is specified by both $y(x_0)$ and $y^{\prime }(x_0)$. The linear case is a second-order linear differential equation.

A global holomorphic section of $\mathcal{O}(1)$ on ${\mathbb{CP}}^1$ is a linear homogeneous polynomial in the two projective coordinates. On one patch its coefficient is an entire function $p(\lambda )$; regularity on the other patch says $p(\lambda )/\lambda$ is regular at infinity. Hence $p$ has at most linear growth. Cauchy estimates make its second and higher derivatives zero, so $p=a+b\lambda$. The same argument entrywise applies to matrix-valued sections and makes the logarithmic derivative in the Penrose-Ward correspondence linear in the primed two-component spinor.

A semi-log plot uses a logarithmic scale on one axis and a linear scale on the other. Which axis is logarithmic matters. Plotting $x$ against $\log y$ makes positive exponential functions appear as straight lines; plotting $\log x$ against $y$ instead makes an affine function of a logarithm appear straight. The plotting operations change positions while retaining the original tick values.^[ref]

Take the original dimensionless mathematical model $y(t)=3e^{0.4t}$ for $0\le t\le 8$. It has exponential growth, with $y(0)=3$, $y(4)\approx 14.8591$, and $y(8)\approx 73.5976$. Taking a base-ten logarithm givesThe slope in these transformed positions is $0.4/\ln 10\approx 0.173718$, although the original relative growth rate $y^{\prime }/y$ is $0.4$. Equal steps of two in $t$ multiply $y$ by $e^{0.8}\approx 2.22554$. The corresponding ordinary gains increase, so the left plot bends while the right plot is straight.

One practical use is checking a proposed exponential distribution for failure times. For rate $\lambda >0$ and $t\ge 0$, its cumulative distribution function is $F(t)=1-e^{-\lambda t}$, so the reciprocal survival function $y=1/[1-F(t)]$ satisfies ${\log }_{10}y=\lambda t/\ln 10$. Estimated probabilities can therefore be placed on a logarithmic vertical axis and compared with a straight line. Agreement supports compatibility with the chosen statistical model; a roughly straight finite sample does not establish a unique physical mechanism.^[ref]

A seminorm on a real or complex vector space $E$ is a finite-valued function $p:E\to [0,\infty )$ satisfying absolute homogeneity and the triangle inequality:In particular $p(0)=0$ and $p(-x)=p(x)$. A seminorm is a norm exactly when $p(x)=0$ implies $x=0$: this is the extra definiteness of a norm axiom. The zero function is a valid seminorm. The missing definiteness is useful when a measurement deliberately ignores some directions, such as an individual coordinate or the constant term of a function whose derivative is measured.

For example, on $C^1([0,1])$ the function $p(f)={\sup }_{t\in [0,1]}|f^{\prime }(t)|$ is a seminorm that vanishes on every constant function. Adding the value measurement gives the norm $q(f)=|f(0)|+p(f)$: if $q(f)=0$, the fundamental theorem of calculus gives $f(t)=f(0)+{\int }_0^t{f}^{\prime }(s)ds=0$. A family of such partial measurements defines a seminorm topology.

A first-order ordinary differential equation is separable if its explicit form factors as $y^{\prime }=a(t)b(y)$. On an interval where $b(y)\ne 0$, write $dy/b(y)=a(t)dt$ and take antiderivatives to obtain an implicit solution:An initial condition determines $C$ and selects a branch and domain. The useful simplification is that the two integrals each involve only one variable, even when the original equation is nonlinear.

For Newton's law of cooling, division by $y-T_a$ gives $dy/(y-T_a)=-kdt$. To keep the logarithm dimensionless, integrate relative to the initial excess temperature:When $y_0\ne T_a$, continuity preserves the sign of the excess temperature until any potential zero, giving $y-T_a=(y_0-T_a)e^{-k(t-t_0)}$. This formula never reaches zero at finite time. Taking its derivative verifies the ordinary differential equation, and substitution at $t_0$ verifies the initial condition.

Always check what division removed. Each zero $b(y_{\ast })=0$ gives a constant solution of a differential equation $y=y_{\ast }$. Cooling therefore also has the equilibrium point $y=T_a$, omitted by division but included in the final explicit formula. For the nonunique initial value problem $y^{\prime }=\sqrt{|y|}$, division by $\sqrt{|y|}$ only describes the moving pieces: a valid solution may remain zero for an arbitrary time before joining one. Separation is a method for finding solutions, not a uniqueness theorem.

Consider a stipulated series circuit containing an ideal resistor of electrical resistance $R=100\Omega$ and an ideal capacitor of capacitance $C=1\mu \mathrm{F}$. The same electric current passes through both. Their terminal electric potential differences add by Kirchhoff's voltage law, soA quick derivation uses $i=Cd{v}_C/dt$: taking the derivative of the complex exponential multiplies the capacitor voltage phasor by $\mathrm{i}\omega$, hence ${\hat{V}}_C=\hat{I}/(\mathrm{i}\omega C)$. Dimensionally, $1/(\omega C)$ has units $s/F=V/A=\Omega$, matching $R$.

At frequency $f=1000\mathrm{Hz}$, the angular frequency is $\omega =2\pi f$. The signed capacitive electrical reactance, modulus and complex argument of the total electrical impedance areApply $V_{\mathrm{rms}}=1.00\mathrm{V}$ across the complete series circuit. Its root mean square electric current isWith voltage as the zero-phase reference, the current phasor has complex argument $+57.858092^{\circ }$: electric current leads. The phasor drops have moduli $V_{R,\mathrm{rms}}=0.532018\mathrm{V}$ and $V_{C,\mathrm{rms}}=0.846733\mathrm{V}$; these add as complex numbers, not as positive magnitudes. Their squared magnitudes sum to $(1.00\mathrm{V}{)}^2$.

The mean absorbed power uses the root mean square values:The ideal capacitor has zero cycle-averaged absorbed power; all the mean absorbed power becomes Joule heating in the resistor.^[ref] To see why the phase factor appears, average $p(t)=v(t)i(t)$ over a full cycle with $v(t)=\sqrt{2}{V}_{\mathrm{rms}}\cos \omega t$ and $i(t)=\sqrt{2}{I}_{\mathrm{rms}}\cos (\omega t-\varphi )$. The cosine addition formula reduces the average to $V_{\mathrm{rms}}{I}_{\mathrm{rms}}\cos \varphi$; the term at $2\omega$ averages to zero. Independently, $p_C=d(C{v}_C^2/2)/dt$, and the capacitor energy returns to its starting value after each cycle.

At fixed applied $V_{\mathrm{rms}}$, the limit $\omega \to 0^{+}$ gives $|Z|\to \infty$, $\varphi \to -\pi /2$ and $I_{\mathrm{rms}}\to 0$: the ideal capacitor blocks steady electric current. The limit $\omega \to \infty$ gives $Z\to R$, $\varphi \to 0$ and $I_{\mathrm{rms}}\to V_{\mathrm{rms}}/R$. These are limits of the ideal lumped electrical circuit; unwanted lead inductance and component losses can invalidate that model at high frequency. At $\omega RC=1$, the equal legs of the impedance right triangle give $|Z|=\sqrt{2}R$ and $\varphi =-\pi /4$ without numerical evaluation. Measuring both the magnitude and the sign of the phase difference distinguishes this capacitor response from an equally large positive electrical reactance.

A shadow price measures the marginal improvement from relaxing an inequality optimization constraint. Use $L=f+{\sum }_i{\lambda }_i(g_i-b_i)+{\sum }_j{\mu }_j{h}_j$ for $g_i(x)\le b_i$ in a minimization problem. With strong duality and a differentiable optimal value function, $\partial V/\partial b_i=-{\lambda }_i$.^[ref]

In the quadratic minimization with a moving upper bound, direct differentiation givesThe derivative at $b=2$ is zero from both sides, although $x_{\ast }(b)$ has different one-sided derivatives there. The value of additional room has faded continuously to zero just as the active constraint stops restricting the optimizer.

At $b=1$, $x_{\ast }=1$, $V=1$ and ${\lambda }_{\ast }=2$. Raising the bound to $1.1$ lowers the value to $0.81$: the first-order prediction is a change of $-2(0.1)=-0.2$, while the exact change is $-0.19$. At $b=3$, the same relaxation leaves $x_{\ast }=2$ and $V=0$. At $b=2$, tightening by $0.1$ costs $0.01$, a second-order effect despite the zero Lagrange multiplier.

There is also a global statement in this example. Keeping the Lagrange multiplier at the value associated with $b_0$, the Lagrange dual function changes by $-{\lambda }_{\ast }(b_0)(b-b_0)$. Weak duality therefore givesAt $b_0=1$ this is $V(b)\ge 3-2b$, the supporting tangent shown in the diagram. This bound remains valid for large changes, whereas the derivative predicts only the first-order effect. If the value function is not differentiable, an optimal Lagrange multiplier can instead supply a supporting subgradient, rather than a unique marginal rate. This is the same idea used in sensitivity analysis in convex perturbation duality.

From a point light source $P$, a caster point $C(t)$ projects onto a receiving surface along the displayed ray. On a regular transverse branch, differentiating gives $[S_u,S_v,-(C-P)](u^{\prime },v^{\prime },{\alpha }^{\prime }{)}^T=\alpha C^{\prime }$. Newton method correction and certified patch/curve bounds support continuation and isolation of all visible branches. If the shadow's second derivative is bounded by $M$ on an interval of width $h$, its chord error is at most $M{h}^2/8$. A fixed sample grid can miss entire shadow components; surface boundaries, occlusion changes and grazing rays require explicit treatment.

For a planar magnetic field written as $\mathbf{B}=\nabla \times (A\hat{\mathbf{z}})$ and a circular velocity $\mathbf{u}=s\Omega (s)\hat{\mathbf{\varphi }}$, the resistive induction equation reduces, after choosing the additive gauge transformation, to ${\partial }_{t}A+\Omega {\partial }_{\varphi }A=\eta {\nabla }^{2}A$. A Fourier mode $A=a(s,t)e^{im[\varphi -\Omega (s)t]}$ therefore obeys the equation in the title. Indeed, each radial derivative acting on the exponential replaces ${\partial }_s$ by ${\partial }_s-imt{\Omega }^{\prime }$, while the azimuthal Laplacian contributes $-m^2/s^2$. At zero magnetic diffusivity, $a$ is constant in time, but the physical magnetic field can still be amplified algebraically by differential rotation.