An air track supports a cart on a cushion of air to reduce contact friction. It provides a nearly one-dimensional platform for a damped harmonic oscillator; residual drag remains part of the baseline damping.

For positive locally constant $A,C$, supplement damped first-order resonant complex eccentricity by $\dot{a}/a=-2(p+1)Ce\sin \varphi -4A/5$. Balancing the two dissipative rates gives $e_f^2=2/[5(p+1)]$, $\sin {\varphi }_f=-A{e}_f/C$ and $B{x}_f=-(C/e_f)\cos {\varphi }_f$. Two fixed points exist when $A<C\sqrt{5(p+1)/2}$. The negative-cosine branch has $x_f>0$ and is stable in this leading model: a phase increase raises the positive resonant torque, raises semi-major axis, lowers mean motion, and reverses the phase drift. The positive-cosine branch has the opposite feedback.

For a coupled verification put $z=u+iv$. The characteristic polynomial at either fixed point is ${\lambda }^3+2A{\lambda }^2+[A^2+B^2{x}_f^2-2B(p+1)C{u}_f]\lambda -4AB(p+1)C{u}_f$. For $u_f<0$ its positive coefficients satisfy the cubic Routh-Hurwitz stability criterion, since the remaining inequality reduces to $2A(A^2+B^2{x}_f^2)>0$. For $u_f>0$ its negative constant term forces a positive real root. At the equality threshold the equilibria merge and an eigenvalue is zero. This is local stability for the frozen smooth-coefficient approximation, not a conclusion for arbitrary rapidly varying dissipation laws.

Half exponents mean the inverse of two in the odd-order root-of-unity group. For $p\nmid ab$, the ratio is a real global unit and is compatible under the norms in the p-power tower. Its Coleman power series is $(1+T{)}^{(b-a)/2}(1-(1+T{)}^a)/(1-(1+T{)}^b)$, with constant term $a/b$. Its higher logarithmic derivatives vanish at odd indices and equal $(a^k-b^k)B_k/k$ at even indices, using $B_1=-1/2$.

For a real cyclotomic field of odd prime-power conductor, the index of its cyclotomic-unit group has the same p-part as its ideal class number. The analytic class number formula compares the regulator of all units with the explicit cyclotomic-unit regulator. The equality here is stated at the p-part level, so harmless prime-to-p normalization factors do not affect it.

For $1\le k<p$, choose $l$ with $kl\equiv 1(\mathrm{mod}p)$. The geometric sum $1+{\zeta }_p^k+\cdots +{\zeta }_p^{k(l-1)}$ is the inverse of $u_k$. Hence these are units, and their residues cover ${\mathbb{F}}_p^{\times }$. In the cyclotomic Milnor square, this makes the $K_1$ map onto the common field surjective and eliminates its connecting obstruction to $K_0$.

Here $a\ge 0$ and $p\nmid m$. The prime-to-$p$ roots of unity generate an unramified extension of degree equal to the multiplicative order of $p$ modulo $m$; for $m=1$ this degree is one. Indeed, their distinct residues lift uniquely by Hensel lemma, and the Frobenius automorphism acts by raising them to $p$. The $p$-power roots of unity generate a totally ramified extension, because the translated cyclotomic polynomial ${\Phi }_{p^a}(1+T)$ is Eisenstein. The two fields have trivial intersection and their compositum has the same residue degree as the prime-to-$p$ field, by unramified base change preserves total ramification. Thus total ramification is equivalent to $p\equiv 1(\mathrm{mod}m)$.

Over a field of characteristic different from two and three, let $D=\{y^{2}z=x^3\}\subset {\mathbb{P}}^2$. Its homogeneous Hessian matrix has determinant $24x{y}^2$. Its common zeros with $D$ are the cusp $[0:0:1]$ and the smooth point $p=[0:1:0]$. The Hessian criterion for a flex therefore leaves exactly one smooth flex.

At $p$, the projective tangent line is $L=\{z=0\}$. Restricting the cubic equation to $L$ gives $x^3=0$, so its intersection multiplicity at $p$ is three. Consequently a triple projective line and a cuspidal cubic meeting only at a smooth point are projectively equivalent to $(3L,D)$ under these field hypotheses.

The characteristic restriction matters. In characteristic three, every smooth point can be written $[a:1:a^3]$ and has projective tangent line $z=a^{3}y$. Substitution gives $(x-ay{)}^3=0$, so every smooth point has triple contact. The unique-flex conclusion then fails.

The new subspace is the orthogonal complement of the old subspace of cusp forms for the Petersson inner product. A form belongs to it precisely when the adjoints of all lower-level oldform by argument dilation maps annihilate it.

The old subspace is spanned by $h(tz)$ for cusp forms $h$ of level $M<N$, where $M\mid N$ and $t\mid N/M$. It is stable under the Hecke operators, but need not have a basis of eigenvectors for every bad-prime operator. Its Petersson inner product orthogonal complement is the new subspace of cusp forms.

For independent $N(\mu ,{\mu }^2)$ observations with $\mu >0$, the sufficient statistics are their sum and squared norm. Condition on $a=\sqrt{n}\Vert y\Vert /{\sum }_i{y}_i>0$ and write $v=\hat{\mu }=\Vert y\Vert /(q\sqrt{n})$, where $q=(\sqrt{1+4a^2}+1)/(2a)$. Polar coordinates supply radial measure $\Vert y{\Vert }^{n-1}d\Vert y\Vert$, while the conditional angular factor does not depend on $\mu$. This proves the displayed shape. Its normalization depends on $a,n$ but not $\mu$ after the substitution $v=\mu z$, and agrees with the normalized P-star approximation.

If a regular plane curve has positive curvature $\kappa$ and a particle moves with speed $v=1/\kappa$ in the chosen time units, its elapsed time is $\int \kappa ds$, the total curvature traversed. On a logarithmic spiral $r=a{e}^{b\theta }$, both $ds/d\theta$ and $1/\kappa$ equal $r\sqrt{1+b^2}$. Hence $dt=d\theta$ and every complete turn takes time $2\pi$, regardless of its radius.

Where a regular curve has positive curvature $\kappa$, its radius of curvature is the radius $\rho =1/\kappa$ of its osculating circle. A sharper bend has a smaller radius. At zero curvature there is no finite radius; writing $\rho =\infty$ is a limiting convention rather than an ordinary real reciprocal of zero.

At a point of a twice continuously differentiable regular curve with positive curvature $\kappa$, its osculating circle has radius $1/\kappa$ and center $c=r+N_p/\kappa$, where $N_p$ is the principal normal vector. It matches the point, tangent line and curvature. For a space curve, it lies in the plane spanned by the unit tangent vector and principal normal vector. The parabola example under curvature of a plane curve gives center $(0,1/2)$ and radius $1/2$ at the origin.

In the exterior orthonormal coframe $E^0=\sqrt{V}dt$, $E^1=dr/\sqrt{V}$, $E^2=rd\theta$, $E^3=r\sin \theta d\varphi$, use ${\Omega }^a{}_b=\frac{1}{2}{R}^a{}_{bcd}{E}^c\wedge E^d$. The independent mixed-index curvature 2-forms are ${\Omega }^0{}_1=-V^{\prime \prime }{E}^0\wedge E^1/2$, ${\Omega }^0{}_A=-V^{\prime }{E}^0\wedge E^A/(2r)$, ${\Omega }^1{}_A=-V^{\prime }{E}^1\wedge E^A/(2r)$ for $A=2,3$, and ${\Omega }^2{}_3=(1-V)E^2\wedge E^3/r^2$. The orthonormal Ricci tensor is diagonal with entries $V^{\prime \prime }/2+V^{\prime }/r$, $-V^{\prime \prime }/2-V^{\prime }/r$, and twice $(1-V-r{V}^{\prime })/r^2$. The Einstein field equations with negative cosmological constant then give $V=1-2M/r+r^2/l^2$.

// Target: general-relativity.bigb

A torsion dynamometer compares an applied torque with the restoring torque of a twisted elastic element. Returning the lever to its zero position allows a current-loop torque experiment to hold the test orientation fixed while reading the compensating torque.

A radial current dipole has moment parallel to its displacement from the sphere centre. The Sarvas formula gives identically zero exterior magnetic field for such a source in a spherical volume conductor model, although its electric potential need not vanish. This is a nonuniqueness mechanism for the inverse problem in magnetoencephalography.

For a nonnegative continuous event time with survival function S, hazard function h and H the cumulative hazard, split the expectation at c. The censored contribution is S(c)H(c). The event contribution is ${\int }_0^{c}H(t)f(t)dt=1-S(c)-S(c)H(c)$ by integration by parts, since $dH=hdt$ and $f=hS$. Adding gives the displayed identity. Linearity extends it to the expected observed event count in a cohort, without requiring independence between subjects.

Let $X=\min (T,C)$, $V=1_{\{T\le C\}}$, and suppose $T$ and $C$ are independent, with continuous failure density $f=hS$. Then the Tonelli theorem givesThus $V-H(X)$ has mean zero. The identity also holds conditionally on covariates when censoring is independent conditionally on them. For an estimated hazard, the mean error is controlled by $\mathbb{E}|\hat{H}(X)-H(X)|$, not by pointwise fit alone.

// Target: survival-analysis.bigb

For independent and identically distributed random variables with a moment-generating function finite on an open interval containing zero, the empirical mean satisfies a large deviation principle with Rate function given by the Legendre transform of a cumulant-generating function. The exponential-moment hypothesis is part of this theorem; mere integrability does not guarantee this full large-deviation principle.

For independent identically distributed continuous random variables, let $K(t)=\log \mathbb{E}{e}^{tY}$ be finite near zero. At a mean value $x$ reached by $K^{\prime }(\hat{t})=x$ in the interior of that domain, the displayed leading approximation describes the density of the sum $S_n$. Exponential tilting by $\hat{t}$ puts the sum at its tilted mean; a local normal approximation gives the denominator, and undoing the tilt gives the exponential factor. Under suitable smooth nonlattice regularity, at fixed interior $x$ the exact density is the displayed quantity times $1+O(n^{-1})$. This statistical approximation is an application of the saddle-point approximation to a transform inversion integral. It requires more than the existence of a few ordinary moments.