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For $\zeta >1$, let $r_{+}>r_{-}$ be the real characteristic roots. The response with the specified initial conditions isThe generic late response is governed by $r_{+}$, the root nearer zero. For very large damping ratio, $r_{+}\sim -{\omega }_0/(2\zeta )=-k/b$: an increasingly strong brake delays the return to equilibrium. This approximation comes from balancing the spring and viscous damping terms after the fast transient. Choosing $v_0=r_{-}{x}_0$ makes $C_{+}=0$ and leaves only the faster root; that exception requires different initial velocities as $b$ varies.

For $0<\zeta <1$, define the damped angular frequency ${\omega }_{\mathrm{d}}={\omega }_0\sqrt{1-{\zeta }^2}$. The initial value problem has solutionEvery nonzero underdamped response crosses equilibrium repeatedly. Its oscillation period is $T_{\mathrm{d}}=2\pi /{\omega }_{\mathrm{d}}$, not $2\pi /{\omega }_0$. The amplitude envelope decays as $e^{-\zeta {\omega }_{0}t}$, but the actual displacement also changes sign. The envelope coefficient need not equal $|x_0|$: initial velocity fixes the oscillation phase as well as that coefficient.

Viscous damping means a force proportional to, and opposite to, velocity: $F_{\mathrm{d}}=-b\dot{x}$ with $b\ge 0$. Its coefficient has units $\mathrm{kg}{\mathrm{s}}^{-1}$, equivalently $\mathrm{N}\mathrm{s}{\mathrm{m}}^{-1}$. The name describes a force law; eddy-current damping can approximate the same law without a viscous fluid. Constant $m$, $b$ and $k$ produce a linear ordinary differential equation, allowing superposition of responses. Kinetic friction with approximately constant magnitude and quadratic drag obey different laws, so their decay need not have the exponential decay or critical threshold derived here.

For the cyclotomic field $\mathbb{Q}({\zeta }_n)$, arithmetic Frobenius automorphisms at primes $\ell \nmid n$ send ${\zeta }_n$ to ${\zeta }_n^{\ell }$: reduction distinguishes the nth roots of unity, and Frobenius raises residues to their $\ell$th powers. Multiplicativity therefore sends a fractional ideal with positive rational generator $a$, prime to $n$, to the exponent $a\mathrm{mod}n$. Its kernel consists exactly of positive generators congruent to one modulo $n$. This proves Artin reciprocity for the ray class group of modulus $n\infty$ without assuming the general reciprocity theorem.

A vector is separating for an operator algebra when evaluation at that vector is injective. A cyclic vector for the commutant of an operator algebra is separating for the original algebra: if $a\Omega =0$, then $a$ vanishes on its dense commutant orbit.

If a nonempty subset $A$ of ${\mathbb{Z}}_p$ is invariant under translation by a nonzero $t$, then repeated translation of any member runs through all $p$ residues, so $A={\mathbb{Z}}_p$. Thus all $p$ translates of a nontrivial subset are distinct. Partitioning the $k$-element subsets into translation classes of size $p$ proves $p\mid \left(\genfrac{}{}{0ex}{}{p}{k}\right)$ for $0<k<p$.

// Target: analysis.bigb

A nontrivial subgroup $H$ of $⟨g⟩$ is generated by $g^d$, where $d$ is the least positive exponent with $g^d\in H$. Divide any exponent $k$ with $g^k\in H$ by $d$; its remainder also gives an element in $H$ and must be zero. A quotient group by a normal subgroup is generated by the coset of $g$, because every coset is a power of that coset. The trivial subgroup and quotient are also cyclic.

This is the sum of content monomials of colorings up to the group action. It equals the cycle index specialized at $t_r=p_r(x)$. One may first use finitely many colors; stabilization defines the corresponding homogeneous symmetric function. Calling the specialized cycle index a cycle indicator makes $Z_G=F_G$ an equality in the same variables.

An induced cycle is a cycle in a graph whose vertices induce exactly its cycle edges, with no edge between nonconsecutive vertices. A triangle is an induced cycle of length three. A chordal graph forbids induced cycles of length at least four.

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.