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The singular-curve adjunction formula in a symplectic manifold of real dimension four gives $c_1(A)=A^2+2-2\delta$ for a simple J-holomorphic sphere, with a nonnegative singularity count $\delta$. Hence the displayed inequality holds. This is the version needed when the sphere need not be embedded.

A sieve $R$ on $A$ is dense when every arrow $f:B\to A$ has a further composite $fh$ in $R$. Equivalently, every pullback sieve $f^{\ast }R$ is nonempty. This condition is stable under pullback and transitive refinement and defines the double-negation topology.

When ${\mu }_Q=a(b-r)$ and $\eta (r{)}^2={\eta }_0^2+{\eta }_1^{2}r$, substituting $P=A(\tau )e^{-B(\tau )r}$ gives $B^{\prime }=1-aB-{\eta }_1^2{B}^2/2$ and $(\log A{)}^{\prime }=-abB+{\eta }_0^2{B}^2/2$, with $B(0)=0$, $A(0)=1$. The Vasicek model uses ${\eta }_1=0$, and the CIR model uses ${\eta }_0=0$.

In a one-factor short-rate model, a derivative $V(t,r)$ can be hedged locally by a nondegenerate traded zero-coupon bond $P(t,r;T_1)$. Match Brownian exposures by holding $V_r/P_r$ bonds and putting the remaining value into the bank account. Substitution into the two pricing equations proves the gain identity. This requires $\eta P_r\ne 0$; completeness at a degenerate boundary cannot be inferred from the dimension count alone.

Removing the first derivative from the truncated photon diffusion equation gives the displayed transformed oscillator. Constant mean free time changes its oscillation frequency only at second order. For a slowly varying mean free time and subhorizon wavelengths, derivative corrections to the phase and prefactor are small; the leading solution is an acoustic sine/cosine multiplied by $\exp [-k^2/k_D^2]$, where $k_D^{-2}=2\int {\tau }_{c}d\tau /15$. Dropping the derivative correction requires slow variation as well as a small mean free time; it is not an exact solution for arbitrary time dependence.

Place a line source a distance $h$ above a plane welded interface and put $q_j=({\beta }_j^{-2}-p^2{)}^{1/2}$, $t_1=h/({\beta }_1^2{q}_1)$ and $r_1=h/({\beta }_1{q}_1)$. The transmitted elastic ray is $x=hp/q_1+{\beta }_2^{2}p(t-t_1)$, $z={\beta }_2^2{q}_2(t-t_1)$. Elastic-interface boundary conditions give $T_{\mathrm{SH}}=2{\mu }_1{q}_1/({\mu }_1{q}_1+{\mu }_2{q}_2)$. Differentiating the ray positions across neighboring values of $p$ and applying SH ray-tube amplitude transport gives the displayed leading amplitude. Both $q_j$ must be positive; critical and evanescent incidence require different asymptotics.

For positive real mass density and shear modulus, real $k$ and $\mathrm{Im}\omega >0$, the displayed derivative of $S_z=-\frac{1}{2}\mathrm{Re}(V^{\ast }T)$ follows from the two first-order shear equations. Every nontrivial solution has strictly decreasing $S_z$. Since $S_z=\frac{1}{2}\mathrm{Re}Z|V{|}^2$, positive-real impedance at a deeper point remains positive at all shallower points; negative-real impedance at a shallower point remains negative deeper down. A zero field is the necessary exception to strictness. At complex frequency this is a spectral flux diagnostic, reducing to a physical cycle average on the real axis.

In a uniform layer, put $q=\mu p_{\beta }$, $\kappa =\omega p_{\beta }$ and $d=z-z_r$. Initial impedance $Z_r$ gives the displayed fractional-linear solution and $V(z)=V_r[\cos \kappa d+i(Z_r/q)\sin \kappa d]$. Expanding into exponentials yields forward and backward shear-horizontal waves. Negative $d$ propagates a bottom load upward. If the denominator vanishes, the nonsingular pair $(V,T)$ must be propagated through the impedance pole.

For distinct simple roots, ${\alpha }_j-{\alpha }_i$ has mixed-sign simple-root coefficients and is not a root. Thus $[f_i,e_j]=0$. In the finite-dimensional adjoint representation of the simple-root SU(2) subalgebra, $e_j$ is a lowest-weight vector of weight $a_{ij}$. Standard SU(2) ladders force $a_{ij}=-n_{ij}$ with $n_{ij}\ge 0$ integral and make $(\mathrm{ad}{e}_i{)}^{n_{ij}+1}{e}_j=0$.

Put $g(n)=\mu (n{)}^2/\phi (n)$. As a Dirichlet convolution, $g=h\ast (n\mapsto 1/n)$, where $h$ is a multiplicative arithmetic function with $h(p)=1/(p(p-1))$, $h(p^2)=-1/(p(p-1))$, and $h(p^j)=0$ for $j\ge 3$. The local sums are one, so ${\sum }_{n}h(n)=1$. The series ${\sum }_n|h(n)|\log (2n)$ converges, since the prime-local errors are $O((\log p)/p^2)$. Consequently $G(z)={\sum }_{d\le z}h(d)H_{\lfloor z/d\rfloor }=\log z+O(1)$ by the harmonic number estimate $H_m=\log m+O(1)$. This identifies the leading constant in the one-dimensional Selberg upper-bound sieve.

For a variation of a unit-speed geodesic, let $T$ be its tangent, $J$ its variation vector field and $A={\nabla }_{s}J$ its variation acceleration. The second variation of Riemannian arc length has the displayed endpoint term in addition to the Riemannian index form of $J^{\perp }=J-⟨J,T⟩T$. Differentiating the speed twice gives $|D_{t}J{|}^2-⟨D_{t}J,T{⟩}^2+⟨D_s{D}_{t}J,T⟩$. Commuting derivatives and integrating $D_{t}A$ by parts gives the formula. The term cannot be discarded unless the endpoints are fixed or an appropriate endpoint condition makes it zero.

For $x^{\prime \prime }+[1+ϵ\cos t+f{ϵ}^2]x=0$, let $T={ϵ}^{2}t$ and $x_0=a(T)\cos t+b(T)\sin t$. First-order forcing produces a constant and second harmonics, not a resonant harmonic. Their feedback at second order gives the displayed solvability condition in the method of multiple scales. The slow squared growth rate is $(f+1/12)(5/12-f)/4$. Consequently the leading instability interval is $-1/12<f<5/12$; outside its closure the slow amplitudes oscillate. The endpoints are degenerate and require the fourth-order edges of the second Mathieu instability tongue for exact small-parameter stability.

A correlogram plots the estimated autocorrelation function against lag. The plot helps distinguish the finite autocorrelation cutoff of a moving-average model from the decaying dependence of a causal autoregressive model. Estimated cutoffs are diagnostics, not exact finite-sample identities.

With string tension $T=1/(2\pi {\alpha }^{\prime })$, the free-ended open string $X^0=a\tau$, $X^1+i{X}^2=a\cos \sigma e^{i\tau }$, $0\le \sigma \le \pi$, is straight at every target time. Its Nambu–Goto equations of motion and Virasoro constraints hold, and its ends move at light speed. Integrating the target-space Noether charges of a string gives $M=\pi Ta$ and $J=\pi T{a}^2/2$. Thus $J={\alpha }^{\prime }{M}^2$. Extending the same map to $0\le \sigma \le 2\pi$ describes a double-covered folded closed string, with twice the mass and twice the angular momentum, hence $J={\alpha }^{\prime }{M}^2/2$. The different slopes record the different boundary topology.

The divisibility is between generators of characteristic ideals. Auxiliary-prime derivative operators applied to a root-of-unity Euler system produce Kummer classes with controlled valuations. The congruence axiom identifies the valuation at a new auxiliary prime with a residue symbol of the preceding class. The Chebotarev density theorem chooses prime representatives of desired ideal classes, and iterating bounds their elementary divisors by the index of the cyclotomic unit. Passing to the norm tower gives the displayed divisibility, with finite errors removed at height-one primes. The reverse divisibility requires the global index and norm-defect comparison.

For a cyclic group of order $m$ with generator $\sigma$, let $N={\sum }_{i=0}^{m-1}{\sigma }^i$. Subtracting adjacent coefficients and using ${\sigma }^m=1$ proves the displayed group-ring identity. If $p^r\mid m$, then $(\sigma -1)D\equiv -N(\mathrm{mod}{p}^r)$. Together with the norm relations of a root-of-unity Euler system, this makes derivative classes invariant modulo p-power powers and permits Kummer descent, with roots-of-unity corrections where necessary. The congruence axiom controls the associated residue and valuation maps.

Let $q\nmid m$ be outside the excluded primes, let $\zeta$ have order $m$, and let $\rho$ be a primitive qth root of unity. Galois equivariance identifies the conjugates with $\Phi ({\rho }^i\zeta )$ for $1\le i\le q-1$. The distribution axiom includes the missing $i=0$ factor, so the field norm is $\Phi ({\zeta }^q)/\Phi (\zeta )$. Arithmetic Frobenius sends $\zeta$ to ${\zeta }^q$, giving the displayed factor. At higher powers of an already ramified prime all q translates occur instead. These are different norm relations; norm compatibility in just a p-power tower does not supply the auxiliary-prime relation by itself.

At one matrix entry $a$, let $\rho ,\mu ,\nu ,\lambda$ be the shapes of the northwest, northeast, southwest and southeast matrix prefixes under the RSK correspondence. The displayed row-length recurrence expresses merged insertion paths: the new entry supplies the first-row carry, and the overlap of the two incoming row extensions beyond the northwest shape is bumped into the next row. Zero boundary shapes determine the whole diagram. Transposing the matrix exchanges $\mu ,\nu$, leaving the rule invariant.

For a type $\alpha$, each successive difference is a rim hook of size ${\alpha }_i$; zero parts give empty differences of height zero. Label its cells by $i$. Such fillings are weakly increasing along rows and columns, and the height is the sum of the occupied strips' numbers of rows minus one. Empty parts are omitted from the associated product of positive power sums; the finite-variable degree-zero power sum is not inserted.

For a connected Riemannian manifold of dimension $n\ge 2$, assume geodesic completeness and $\mathrm{Ric}\ge (n-1)\kappa g$ for a constant $\kappa >0$. The Riemannian distance then has diameter at most $\pi /\sqrt{\kappa }$; the manifold is compact and its fundamental group is finite. The dimension restriction excludes the vacuous one-dimensional Ricci bound.