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For independent event times with survivor function $S(t)=\mathbb{P}(T>t)$, an exact event at $t$ contributes mass $S(t-)-S(t)$, a right-censored observation at $c$ contributes $S(c)$, and an interval-censored observation $l<T\le u$ contributes $S(l)-S(u)$. Multiplying these factors gives the event-time likelihood when the observation mechanism is noninformative. Maximizing over nonincreasing right-continuous survivor functions gives a nonparametric maximum-likelihood estimator rather than requiring a parametric density.

If $u\in H^1(V)$ and $\Delta u=f\in L^2(V)$ weakly, then $u\in H_{\mathrm{loc}}^2(V)$. Test the weak equation with $-{\delta }_{-h}({\eta }^2{\delta }_{h}u)$, move the first difference quotient onto $Du$, and bound the remaining quotient on the forcing side by the first derivative of ${\eta }^2{\delta }_{h}u$. Young inequality absorbs the localized second derivatives. Uniform bounds in $h$, together with the Sobolev characterization by bounded difference quotients, give the assertion without differentiating $f$. The constant depends only on dimension and the positive distance from $U$ to the boundary of $V$.

// Target: geometry-and-topology.bigb

Choose the holomorphic square root equal to one at zero on the unit disc. The resulting elliptic integral maps that disc conformally onto the square with vertices $\pm B,\pm iB$, where $B={\int }_0^{1}dt/\sqrt{1-t^4}$. For the inverse Cayley transform between the half-plane and disk $h(z)=i(1+z)/(1-z)$, the square of the derivative of $F_1\circ h$, with $F_1^{\prime }(w)=1/\sqrt{w(1-w^2)}$, is $2i/(1-z^4)$. Thus the two square maps differ by an affine change of coordinate.

For an ellipse with semiaxes $1,x$, the perimeter integral has integrand $\sqrt{1-t{\cos }^2\theta }$. The uniformly convergent binomial series near $t=0$ permits termwise integration. Integrals of ${\cos }^2\theta$ and ${\cos }^4\theta$ over a full period give the displayed first corrections. The same coefficients follow from the Frobenius method for the transformed perimeter equation.

The semi-major and semi-minor axes are the larger and smaller half-axis lengths of an ellipse. In principal-axis coordinates the ellipse has equation $x^2/a^2+y^2/b^2=1$ with $a\ge b>0$. The same terminology is used for a hyperbola $x^2/a^2-y^2/b^2=1$, where $a$ is half its transverse axis and $b$ half its conjugate axis. Orbital semiaxes apply these geometric quantities to conic sections.

In the space of infinite subsets of the natural numbers, the families $C_n=\{X:n\notin X\}$ are clopen sets in the ordinary topology on infinite subsets and thus in the finer Ellentuck topology. Their union is $[\mathbb{N}{]}^{\omega }\setminus \{\mathbb{N}\}$. Every basic Ellentuck topology neighborhood of $\mathbb{N}$ contains $\mathbb{N}\setminus \{n\}$ for some sufficiently large $n$, so this union is not a closed set.

For an infinite coinfinite set $A\subseteq \mathbb{N}$, let $E_A=[A{]}^{\omega }$ in the space of infinite subsets of the natural numbers. This cone is a nonempty clopen set in the Ellentuck topology, because it is the basic neighbourhood $[\varnothing ,A]$, and any $X\notin E_A$ has a finite initial segment containing a point outside $A$ and hence a basic neighbourhood disjoint from $E_A$.

The cone is a closed set with empty interior in the ordinary topology on infinite subsets: membership of a point outside $A$ witnesses its complement as an open set, while every ordinary finite-stem neighbourhood can be extended by an arbitrarily large point outside $A$. Thus $E_A$ is an ordinary nowhere dense set, but it is neither ordinary open nor Ellentuck nowhere dense. It provides both distinctions between these two topologies.

Every Borel set for the Ellentuck topology is completely Ramsey. First, an acceptance-and-rejection fusion makes every open set homogeneous after thinning the infinite reservoir while keeping its finite stem. Every nowhere-dense set is consequently completely Ramsey-null, and a fusion handling all subsets of each finite selected prefix shows that countable unions of such null sets remain null. Finally, sets differing from an open set by a meagre set form a sigma-algebra containing the open sets. Thus every Borel set has the required stem-preserving homogeneous refinement.

For zonal averages, $u^{\prime }=-{\psi }_y^{\prime }$, $v^{\prime }={\psi }_x^{\prime }$, and $Q^{\prime }={\nabla }_H^2{\psi }^{\prime }+{\partial }_z(S{\psi }_z^{\prime })$, with $S=f_0^2/N^2$ depending only on height, define $F=-\overline{u^{\prime }{v}^{\prime }}$ and $G=S\overline{{\psi }_x^{\prime }{\psi }_z^{\prime }}$. Integrating zonal derivatives by parts gives $\overline{v^{\prime }{Q}^{\prime }}={\partial }_y\overline{{\psi }_x^{\prime }{\psi }_y^{\prime }}+{\partial }_z(S\overline{{\psi }_x^{\prime }{\psi }_z^{\prime }})$. This identity is exact within quasi-geostrophic potential vorticity, without assuming small disturbance amplitudes. Periodicity or appropriate vanishing zonal boundary terms is essential.

In the $(A^1,A^2,A^3,B)$ basis with gauge transformations $U=e^{i\alpha Q}$ and $Q=T_3+Y$, the non-Abelian connection transforms by conjugation. The Lie brackets $[T_3,T_1]=i{T}_2$ and $[T_3,T_2]=-i{T}_1$ give $\delta A^1=\alpha A^2$, $\delta A^2=-\alpha A^1$, with neutral $A^3,B$. Thus $\delta A=i\alpha Q_{\mathrm{gauge}}A$ gives the displayed Hermitian matrix. The fields $W^{\pm }=(A^1\mp i{A}^2)/\sqrt{2}$ transform as $e^{\pm i\alpha }{W}^{\pm }$, while both neutral combinations are uncharged. This separates the Adjoint representation on the gauge fields from the doublet charge matrix $\mathrm{diag}(1,0)$ on the Higgs field.

With $k_{\parallel }/k_{\perp }\sim \delta B/B_0\ll 1$, define $b=\delta B_{\parallel }/B_0$ and $\delta {\mathbf{B}}_{\perp }/B_0=\hat{\mathbf{z}}\times {\nabla }_{\perp }\Psi /v_A$ to leading order. Let ${\nabla }_{\parallel }={\partial }_z+\{\Psi ,\cdot \}/v_A$, where $\{f,g\}=f_x{g}_y-f_y{g}_x$. The leading electron velocities are ${\mathbf{u}}_{e\perp }=v_A{d}_i\hat{\mathbf{z}}\times {\nabla }_{\perp }b$ and $u_{e\parallel }=-d_i{\nabla }_{\perp }^2\Psi$. Substitution into frozen-in induction yieldsLinearization gives ${\omega }^2=v_A^2{d}_i^2{k}_{\parallel }^2{k}_{\perp }^2$, agreeing with the full EMHD relation to leading anisotropic order. If $b$ varies along the guide field, an order-smaller perpendicular gradient correction restores exact solenoidality; the two-field representation is not an exact decomposition at arbitrary anisotropy.

In the inertia-free, constant-density electron magnetohydrodynamics model with uniform guide field $B_0\hat{\mathbf{z}}$, a transverse magnetic plane wave obeys $\omega \mathbf{b}=i{v}_A{d}_i{k}_{\parallel }\mathbf{k}\times \mathbf{b}$. The helicity decomposition of a transverse Fourier mode gives the displayed dispersive branches with opposite circular polarizations. This electron-current mode is called a whistler wave; see Lyutikov's analysis of electron magnetohydrodynamics. Its pressure-free induction equation should not be identified with a general kinetic Alfvén model.

For $b_l{l}_{\parallel }/l_{\perp }\ll 1$, the weak wave cascade estimate gives ${\tau }_{\mathrm{cas}}\sim l_{\perp }^3/(v_A{d}_i{b}_l^2{l}_{\parallel })$. Constant energy flux $ϵ\sim v_A^2{b}_l^2/{\tau }_{\mathrm{cas}}$ yields the displayed amplitude. An additionally isotropic local cascade has $b_l\propto l^{1/2}$ and shell-integrated magnetic energy spectrum $E_B(k)\propto k^{-2}$. These statements are phenomenological consequences of the chosen locality and decorrelation assumptions.

Critical balance sets ${\tau }_{\mathrm{nl}}\sim {\tau }_w$, hence $l_{\parallel }\sim l_{\perp }/b_l$. A local strong cascade with constant energy flux has $ϵ\sim v_A^3{d}_i{b}_l^3/l_{\perp }^2$, giving the displayed exponents and perpendicular magnetic energy spectrum $E_B(k_{\perp })\propto k_{\perp }^{-7/3}$. The anisotropy becomes stronger towards small perpendicular scales.

An electrochemical gradient combines spatial changes in chemical potential and electric potential for a charged species. Passive transport proceeds downhill in electrochemical potential when an appropriate route is available; a concentration difference alone does not determine the direction for an ion.

For an ohmic element with finite nonzero electrical resistance $R$, its electrical conductance is $G=1/R$: an electric potential difference $\Delta V$ drives electric current $I=G\Delta V$. Greater positive electrical conductance means a greater electric current for the same positive electric potential difference. Electrical conductivity is instead a bulk material coefficient relating current density to electric field.

The motivating observation is a pair of sinusoids on an oscilloscope: changing frequency changes both their relative amplitudes and their phase difference. Gary Tuttle's EE 201 laboratory uses resistor voltage as the current-phase reference, compares predictions with readings at several frequencies, and explicitly asks students to measure component values and account for coil electrical resistance. The following bench arrangement applies that method to a series RLC circuit; its component values and illustrated traces are our own toy example. ^[ref].

Connect a low-voltage sine generator to a series chain ending in a known sensing resistor $R_s$ at the common return. Use a resistor whose unwanted inductance is negligible over the chosen frequency range. An oscilloscope displays voltage against time: channel A reads the loaded voltage across the whole chain, and channel B reads the voltage $v_s$ across $R_s$. Both channel ground connections and the generator return meet at that same node. Ordinary bench oscilloscope grounds are electrically common; attaching one to an intermediate circuit node would change the circuit. A separate floating component voltage needs a suitable differential measurement, rather than moving a ground clip. ^[ref].

After the free transient decays, record frequency $f$, period $T=1/f$, both voltage amplitudes, and matched rising zero-crossing times $t_A,t_B$. Check that the traces are sinusoidal. Define $\Delta t=t_B-t_A$ on the branch $|\Delta t|<T/2$. With negligible probe loading and negligible sensing-resistor electrical reactance, $v_s=R_{s}i$, so channel B is the electric current reference, and in the reactance phase conventionA later current crossing gives $\Delta t>0$, hence positive electrical reactance; an earlier crossing gives negative electrical reactance. Use the same amplitude convention for both channels. An amplitude-only ratio recovers $|Z|$, so it cannot distinguish the two signs: timing supplies the missing information. The formulas follow from $Z=|Z|e^{\mathrm{i}\varphi }$ and the chosen terminal orientation.

Begin with a known resistor replacing the reactive chain: its expected phase difference is approximately zero, testing channel polarity and relative delay. Measure the same signal with both channels to establish the instrumental delay baseline, then exchange probes to check that a phase offset does not belong to one channel. Next test an inductor and a capacitor separately with $R_s$: after subtracting $R_s$ from the inferred real part, expect approximately $X/f=2\pi L$ and $fX=-1/(2\pi C)$, respectively. Finally combine $L,C,R,R_s$ and sweep through the predicted resonance. These are proposed controls and consequences of the model, not reported observations.

For the toy series RLC circuit, the prediction at $500\mathrm{Hz}$ is $X=-286.894\Omega$, $|Z|=307.259\Omega$, $\varphi =-69.02^{\circ }$, and $V_{s,\mathrm{rms}}=32.55\mathrm{mV}$. At $5000\mathrm{Hz}$, $X=+282.328\Omega$ and $\varphi =+68.71^{\circ }$. Near $1591.55\mathrm{Hz}$ the model predicts zero phase difference and $V_{s,\mathrm{rms}}=90.91\mathrm{mV}$. Keep actual loaded $V_{A,\mathrm{rms}}$ fixed or normalize by it at each frequency; the generator's display alone does not establish that voltage. The $10\Omega$ sensing resistor is part of the $110\Omega$ total and therefore changes the predicted peak electric current.

Real components add unintended electrical impedance: coil wire has electrical resistance, nearby turns add capacitance, and capacitor leads add inductance. These effects can produce a component's own resonance, beyond which the ideal $\omega L$ or $-1/(\omega C)$ approximation fails. Work in a range where that approximation is adequate, measure coil wire electrical resistance, and compare the inferred $R_{\mathrm{chain}}$ with the sensing resistor, intended resistor, and component losses. A measured nonzero real part is evidence of loss in the assembly, not negative or positive electrical reactance becoming heat. ^[ref]; ^[ref].

The probes also add electrical resistance and capacitance to the measured nodes; their loading becomes more important as frequency increases. Use compensated probes with known input electrical impedance and short ground connections, and repeat with a smaller drive voltage to test whether the inferred $X$ depends on amplitude. ^[ref]. Record actual voltage and phase readings separately from calculated predictions, with component tolerances and timing uncertainty. A phase error $\delta \varphi$ alone produces approximately $\delta X=|Z|\cos \varphi \delta \varphi$, so near zero electrical reactance, a small delay can imitate a small residual $X$. Agreement within stated uncertainties tests the model; the smooth toy curves establish no experimental accuracy.

Use complex amplitudes only for signals at one common positive angular frequency $\omega$. Define the real terminal voltage and electric current byThe complex amplitudes here use peak magnitudes; both can instead use root mean square magnitudes without changing their ratio. Choose positive electric current entering the terminal assigned positive voltage, so $p=vi$ is absorbed power. For an ideal linear inductor and capacitor, their terminal laws $v_L=Ldi/dt$ and $i_C=Cd{v}_C/dt$ giveThe derivative multiplies a complex amplitude by $\mathrm{i}\omega$: this single algebraic step explains both signs. Taking $i(t)=I_0\cos \omega t$ as a reference gives $v_L(t)=-\omega L{I}_0\sin \omega t$ and $v_C(t)=I_0\sin \omega t/(\omega C)$ for the sinusoidal response. Thus inductor voltage leads electric current by $\pi /2$, while capacitor voltage lags electric current by $\pi /2$. A constant offset or a free transient is outside this single-frequency representation. ^[ref].

With the alternative $e^{-\mathrm{i}\omega t}$ convention, the same real signals use conjugated complex amplitudes and $Z_L=-\mathrm{i}\omega L$, $Z_C=+\mathrm{i}/(\omega C)$. Their physical leads and lags stay the same; the sign of $\mathrm{Im}Z$ changes. Also, some textbooks call the positive magnitude $1/(\omega C)$ the capacitive reactance and write the series expression as $X_L-|X_C|$. This article consistently uses signed $X_C=-1/(\omega C)$ and hence $X=X_L+X_C$. ^[ref].

A Rayleigh wave propagates along a stress-free boundary of an isotropic elastic half-space and decays into its interior. It combines an evanescent longitudinal wave with an evanescent vertically polarized shear wave. If its speed is $0<c<c_S<c_P$, the decay factors are $\alpha =\sqrt{1-c^2/c_P^2}$ and $\beta =\sqrt{1-c^2/c_S^2}$. Vanishing shear and normal traction gives the displayed determinant condition.

A Love wave is a horizontally polarized shear wave trapped in a slower elastic surface layer above a faster elastic half-space. Zero shear traction at the free surface, displacement continuity and traction continuity at the interface produce its dispersion relation. A trapped mode has phase speed between the two shear speeds and decays exponentially into the half-space.