A fast initial response CUSUM starts a CUSUM with a positive head start below its signalling boundary. Before its first reset, it accumulates a fixed-start log-likelihood ratio like a sequential probability ratio test; after resetting to zero it behaves as an ordinary CUSUM. It gives faster initial detection when the process may already be out of control, at the cost of changed initial false-alarm behavior.

The average run length of a CUSUM is the expectation of its first signalling time under a specified data model and initial chart state. A large in-control average run length discourages false alarms; a small out-of-control value indicates rapid detection. The mean alone does not give a finite-horizon Type I error probability or the complete detection-delay distribution.

For independent Poisson observations, comparing mean ${\lambda }_{0t}$ with $\theta {\lambda }_{0t}$ gives the displayed log-likelihood ratio. Updating a CUSUM with these scores targets the specified multiplicative increase. Its expected score is negative under the null and positive under the alternative when $\theta >1$, so resetting at zero discards earlier evidence against an increase.

For fixed volume $V$ and symmetric spreading, conservation of mass and the cusp flow between touching cylinders equation give width proportional to $t^{-1/8}$ and support length proportional to $t^{3/8}$. The source-type similarity solution has the displayed profile and $z_N=(8Va/\pi {)}^{1/4}(8\gamma t/(21\mu a){)}^{3/8}$. It is a formal outer lubrication solution; its microscopic nose requires a local description when the axial slope becomes large.

For vertical touching cylinders in a bath, the hydrostatic suction at height $z$ is $\rho gz$. Equating it to the meniscus suction $2\gamma a/w^2$ gives the displayed large-height width. Ideal perfect wetting permits an arbitrarily high thin cusp tail, unlike a uniform finite-gap capillary. The prediction assumes the continuum model remains valid as the gap narrows.

The trace of the curvature form of a connection is a global closed differential form of degree two. Under a frame change $e^{\prime }=eg$, the connection matrix and curvature change by $A^{\prime }=g^{-1}Ag+g^{-1}dg$ and ${\Theta }^{\prime }=g^{-1}\Theta g$. Thus trace is independent of frame. The Bianchi identity gives $d\Theta =\Theta \wedge A-A\wedge \Theta$. Taking the trace cancels the two terms: one-forms commute with two-forms in the graded sense, and renaming the matrix indices gives equality of their traces. Locally $\mathrm{tr}\Theta =d\mathrm{tr}A$, since the trace of $A\wedge A$ vanishes by pairing its off-diagonal terms. It is the curvature of the determinant connection.

A Helmholtz coil arrangement uses two identical coaxial circular windings separated by their radius, carrying equal electric currents in the same circulation sense. Their combined magnetic field is especially uniform near the midpoint, making it useful for a current-loop torque experiment. The cancellation of the axial second derivative there follows from the on-axis magnetic field of a circular current loop.

Piezoelectricity is linear coupling between mechanical deformation and electrical response in suitable materials. The direct piezoelectric effect produces electrical polarization under applied stress; the converse piezoelectric effect produces strain under an applied electric field. The two responses are linked by the same electromechanical coupling coefficients when conventions and mechanical constraints are specified consistently.

A piezoelectric stress sensor uses the direct response to turn force or vibration into an electrical signal. A scanning probe uses the converse response to move a tip by very small distances, controlling a tunnel junction's separation. The resulting displacement can depend on drive history through piezoelectric hysteresis and can continue changing after a voltage step through piezoelectric scanner creep. These effects motivate calibration and closed-loop control.

A current dipole idealizes a localized impressed electric current density by the displayed Dirac delta function distribution. Its moment $\mathbf{Q}$ has units of current times length. In a homogeneous volume conductor of conductivity $\sigma$, its primary electric potential is $\Psi (\mathbf{r};{\mathbf{r}}_0)=\mathbf{Q}\cdot (\mathbf{r}-{\mathbf{r}}_0)/(4\pi \sigma |\mathbf{r}-{\mathbf{r}}_0{|}^3)$.

A charge carrier transports electric charge by moving: electrons are one example, and ions are another. For species $s$ with number density $n_s$, electric charge $q_s$ and mean velocity ${\mathbf{v}}_s$, the transport contributions add as $\mathbf{J}={\sum }_s{n}_s{q}_s{\mathbf{v}}_s$. A stationary charged background contributes to charge density but not to this current density. ^[ref].

The wire cross-sectional area is the area of a slice perpendicular to the wire's axis. For a round wire of radius $a$, it is $A=\pi a^2$. It enters both the uniform-flow formula $I=jA$ for current density and the formula $R=L/(\sigma A)$ for the electrical resistance of a uniform Ohmic conductor.

The largest number of positive-degree cohomology classes whose cup product is nonzero, with a specified coefficient ring $R$; it can be infinite. It is zero if there is no nonzero positive-degree class. The vanishing cup product from an open cover argument implies that Lusternik–Schnirelmann category is at least cup length plus one. For Complex projective space ${\mathbb{CP}}^r$, its integral cup length is $r$.

The cross product takes classes on two factors to a class on their product. Its multiplication rule has the graded tensor product sign: $(a\times b)(c\times d)=(-1{)}^{|b||c|}(ac)\times (bd)$. The Künneth theorem makes these products an additive basis when the integral cohomology of the finite cell complexes is torsion-free.

For a cohomology class $a$, its rth cup power is its r-fold product in the cohomology ring. Naturality gives $f^{\ast }(a^r)=(f^{\ast }a{)}^r$. A nilpotent class has some positive cup power zero; the nilpotence order is preserved by ring isomorphisms and can obstruct a homotopy equivalence or a nonzero mapping degree.

A cubic surface with empty singular locus. Over an algebraically closed field, it is irreducible: distinct positive-degree hypersurface components would meet, and every point of their intersection would be singular in their product equation.

The bipartite cubic graph with vertices $A_i,B_i$, $i\in \mathbb{Z}/7\mathbb{Z}$, and edges $A_i{B}_i,A_i{B}_{i+1},A_i{B}_{i-2}$. The three offsets give a Tait coloring.

Near $\arg z=\pi /3$, set $\lambda =-z^3$ on the branch with argument near zero and truncate the endpoint expansion at $N$ terms. Its remaining factorial series, indexed by $r=N+p$, is ${\lambda }^{-1/3}/[3\Gamma (2/3)]$ times the Borel summation of ${\sum }_{p\ge 0}\Gamma (N+2/3+p)e^{\lambda }/{\lambda }^{N+2/3+p}$. The upper-pole contour fixes the stated remainder and its continuation from the sector below the switching line. With $N+2/3\sim |z{|}^3$, the Gaussian pole transition for a large gamma parameter smooths the algebraic coefficient from $C$ to $C{e}^{2\pi i/3}$.

For binary linear codes $C\subseteq C^{\prime }$, the coset superpositions of the CSS code have $X$ checks labelled by $C$ and $Z$ checks by $(C^{\prime }{)}^{\perp }$. A bit-error pattern $a$ gives the classical syndrome for $C^{\prime }$, and a phase-error pattern $b$ gives the syndrome for the dual code $C^{\perp }$. Errors with $2t_X<{\min }_{a\in C^{\prime }\setminus C}\mathrm{wt}(a)$ and $2t_Z<{\min }_{b\in C^{\perp }\setminus (C^{\prime }{)}^{\perp }}\mathrm{wt}(b)$ are jointly correctable. Every pairwise Pauli-error product either anticommutes with a check, giving zero code compression, or its low-weight components belong to the stabilizer and give scalar compression. Thus the Knill--Laflamme condition covers coherent superpositions and quantum channels as well as definite bit and phase errors.

Here a loopless multigraph has $v$ vertices, $e$ edges, and at most $\mu$ parallel edges per pair. In a minimum-crossing drawing, crossings between incident edges can be removed by exchanging their initial segments, so every crossing involves four distinct vertices. For each parallel class choose an edge with probability $1/\mu$ per edge, with the remaining probability choosing none. Retain vertices independently with probability $p$. The sampled simple graph has expected edge count $p^{2}e/\mu$, expected vertex count $pv$, and expected crossing count $p^4\mathrm{cr}(G)/{\mu }^2$. Deleting one edge per crossing and using the planar graph edge bound gives $p^4\mathrm{cr}(G)/{\mu }^2\ge p^{2}e/\mu -3pv$. Choose $p=4\mu v/e$.

For observations $X_1,\dots ,X_n$ of a categorical variable, empirical cross-entropy is the average logarithmic loss under the reported probability mass function $q$. Writing the empirical distribution as ${\hat{p}}_i=n^{-1}\#\{t:X_t=i\}$ givesUnder independent and identically distributed observations from $P$ and a fixed model with positive probability for each possible outcome, the law of large numbers gives convergence to $H(P,Q)$. A finite empirical distribution is an estimate of $P$, rather than the true source law; unobserved outcomes need not be impossible. If the same data are used to choose $Q$, evaluating the resulting empirical cross-entropy on that data need not estimate its prediction performance on fresh observations without bias.