For an abelian variety and an integer $N$ invertible in the base field, a torsion line bundle $M\in \hat{A}[N]$ has trivial pullback by $[N]$. Choose a trivialization. The descent action of the kernel $A[N]$ on that trivial line bundle acts through a character ${\chi }_M:A[N]\to {\mu }_N$. Define $e_N(P,M)={\chi }_M(P)$. This is independent of rescaling the trivialization and bilinear under translation and tensor product. A trivial character descends the trivial bundle, so $M$ is trivial. Since both groups have order $N^{2\dim A}$, the character map is an isomorphism and the Weil pairing for an abelian variety is perfect. The theta autoduality of a Jacobian turns it into a pairing on two copies of the Jacobian variety's torsion group.

For initial spot $s>b$ in the Black-Scholes model, put $\nu =(\rho -{\sigma }^2/2)/\sigma$ and $\kappa =(s/b{)}^2$. A down-and-in claim with terminal payoff $f$ has the same initial price as the terminal claim $g(x)=f(x)1_{x\le b}+{\kappa }^{-\nu /\sigma }f(x/\kappa )1_{x>\kappa b}$. Reflection of the log-price endpoint above the lower barrier changes its normal probability density to $e^{2\nu \ell }{\varphi }_T(y-2\ell -\nu T)$, where $\ell =\log (b/s)/\sigma$. Substitution $z=y-2\ell$ gives the payoff identity. This is price equality, not pathwise equality of payments.

For strike $c>b$, write $C(s,c,\tau )$ for the ordinary European call option value. Before activation the down-and-in claim is $I(s,\tau )=(b/s{)}^{\alpha }C(b^2/s,c,\tau )$, where $\alpha =2\rho /{\sigma }^2-1$. Consequently its left-limit option delta at the barrier is $-\alpha C(b,c,\tau )/b-C_s(b,c,\tau )$, whereas after activation it is $C_s(b,c,\tau )$. Their difference is strictly positive for $\tau >0$: differentiating the killed normal transition density at its boundary gives the integral of a strictly positive function against the positive call payoff. The value is continuous, so changing the bank holding finances the stock rebalance.

This doubly stochastic matrix replaces two values $a,b$ by $ta+(1-t)b$ and $(1-t)a+tb$, preserving their sum. It fixes all other coordinates when embedded in a larger matrix. A finite product of such averaging matrices and permutation matrices is again doubly stochastic.

Let $T_i,S_i$ be the temperature and salinity of the original supplied liquid, and $T_{\infty },S_{\infty }$ those of the surroundings. For reservoir values $T,S$, define the dimensionless cooling and freshening measureswith positive reference differences. Here $\alpha$ is the thermal expansion coefficient, $\beta$ the haline contraction coefficient, $A$ the interface area and $R_{\rho }=\beta (S-S_{\infty })/[\alpha (T-T_{\infty })]$ the density ratio. Increasing $\theta$ describes cooling toward the surroundings; increasing $\eta$ describes freshening. The flux law below is an assumed double-diffusive convection closure.

With a constant buoyancy-flux ratio and the preceding steady initial state, the heat and salt balance equations give $d\eta /d\theta =q$ and ${\eta }_s=q{\theta }_s$, yielding $\eta =q\theta$ after shutdown.

After inflow and overflow cease, a fixed-volume reservoir initially in the preceding steady state obeys $\eta =q\theta$ and $\dot{\theta }=k(1-\theta {)}^{10/3}/(1-q\theta {)}^2$, with $k=Ab[\alpha (T_i-T_{\infty }){]}^{1/3}/(V{R}_0^2)$. Integrating gives $k(t-t_0)=G(\theta )-G({\theta }_s)$, where $G=3(1-q{)}^2(1-\theta {)}^{-7/3}/7+3q(1-q)(1-\theta {)}^{-4/3}/2+3q^2(1-\theta {)}^{-1/3}$. For $q<1$, heat contrast tends to zero only at infinite time, while a nonzero salt contrast remains.

For the seismic moment tensor $M=e_1{e}_2^{\mathsf{T}}+e_2{e}_1^{\mathsf{T}}$, source differentiation of the outgoing elastodynamic Green tensor gives $A_P=n^{\mathsf{T}}Mn$ and ${\mathbf{A}}_S=(I-n{n}^{\mathsf{T}})Mn$. In the $x_1{x}_2$ plane the signed radial P wave and tangential S wave factors are $\sin 2\varphi$ and $\cos 2\varphi$. The full S wave magnitude squared is $n_1^2+n_2^2-4n_1^2{n}_2^2$.

Suppose a chordal SLE trace stays in the open complex upper half-plane at every positive time almost surely. For every deterministic rational time $s$, the domain Markov property of a chordal Loewner chain makes its mapped and centered future a fresh SLE, so its positive-time future stays in the open half-plane as well. The inverse mapping-out function then places that future outside the entire old hull. Intersecting these probability-one events over rational $s$ rules out every self-intersection: if two times $u<v$ shared a trace point, choose rational $s$ strictly between them. This proves simplicity using the allowed boundary-avoidance assertion for $0<\kappa \le 4$.

On an oriented Riemannian manifold with boundary, the divergence of a Riemannian vector field obeys the displayed integral identity for smooth compactly supported vector fields $X$, or for arbitrary smooth $X$ if the manifold is compact. Here $N$ is the outward unit normal vector and the boundary has the outward-normal-first boundary orientation. Its induced Riemannian volume form is $j^{\ast }{\iota }_N{\omega }_g$. Splitting $X$ into its normal and tangential components gives $j^{\ast }{\iota }_X{\omega }_g=g(X,N){\omega }_{\tilde{g}}$, since a top form vanishes on $n$ boundary-tangent arguments. Apply the Stokes theorem to $d{\iota }_X{\omega }_g=(\mathrm{div}X){\omega }_g$.

Without a support or decay condition, flux at infinity can invalidate the result even if the integrals converge. On $[0,\infty )$ with Riemannian metric $d{x}^2$, $X=(\arctan x){\partial }_x$ has boundary flux zero but integral of divergence $\pi /2$.

For $X\in C^1({\mathbb{R}}^n;{\mathbb{R}}^n)$, integration over $B_R$ gives $|\kappa ||B_R|\le \Vert X{\Vert }_{\infty }|\partial B_R|$, so $|\kappa |\le n\Vert X{\Vert }_{\infty }/R$. Let $R\to \infty$. In particular, the bounded flux $Du/\sqrt{1+|Du{|}^2}$ of an entire constant-mean-curvature graph forces its divergence constant to be zero, independently of any bound on $Du$.

For a bounded sufficiently regular region, the divergence theorem applied to the vector field $x_i{e}_j$ gives the displayed identity. Contracting with constant vectors gives ${\int }_{\partial \Omega }(b\cdot x)(c\cdot n)dS=|\Omega |b\cdot c$. Contracting with the Kronecker delta or the Levi-Civita symbol respectively gives $\int x\cdot ndS=3|\Omega |$ and $\int x\times ndS=0$ in three dimensions.

For a continuously differentiable scalar-valued function $f$ on a bounded sufficiently regular region, apply the divergence theorem to $fk$, where $k$ is any constant vector. Since $\nabla \cdot (fk)=k\cdot \nabla f$, the dot products of the two displayed vector integrals with every $k$ agree. The vector integrals are therefore equal. This is the scalar-gradient analogue of the usual flux identity.

For a continuously differentiable scalar $\Omega$ on a bounded region $V$ with outward boundary normal, ${\int }_V\nabla \Omega dV={\int }_{\partial V}\Omega \mathbf{n}dS$. Apply the divergence theorem to $\mathbf{c}\Omega$ for every constant vector $\mathbf{c}$; equality of all scalar projections gives the vector identity. The theorem applies to piecewise smooth boundaries, with singular points removed by a limiting argument when necessary.

Sodium perchlorate is the ionic salt $\mathrm{NaCl}{\mathrm{O}}_4$, containing sodium cations and perchlorate anions. Its aqueous solution supplies dissolved ions whose interactions with water can be investigated spectroscopically.

In spectroscopic measurement of hydration dynamics, sodium perchlorate supplies the salt environment for a selectively observed water response. An observed relaxation concerns the chosen spectroscopic probe and solution conditions, rather than a permanently fixed hydration cage.

For the linear dispersive Stokes equation, $\omega (\nu )=\omega (k)$ factors as $(\nu -k)({\nu }^2+k\nu +k^2-1)=0$. In $D_{+}=\{\mathrm{Im}k>0,\mathrm{Re}\omega (k)<0\}$ the two displayed other roots lie in the lower half-plane, so both can be used in the global relation. To check the root locations, at $k=ib$ they have imaginary part $-b/2$. A root could reach the real axis only where $\mathrm{Re}\omega (k)=0$, so continuity preserves their signs throughout this connected domain. The branch points $\pm 2/\sqrt{3}$ lie outside its closure. The weights $A_1=(k-{\nu }_2)/({\nu }_1-{\nu }_2)$ and $A_2=({\nu }_1-k)/({\nu }_1-{\nu }_2)$ obey $A_1+A_2=1$ and ${\nu }_j^{\prime }=-A_j$.

For the backward-sign Airy half-line global relation, suppose exactly one trace of derivative order $j\in \{0,1,2\}$ is unknown. Its spectral multiplier is a constant times $k^r$, $r=2-j$. Use the rotated null contours for half-range Fourier inversion with the displayed coefficients. The unknown contribution becomes an integral over the boundary of the middle upper sector $\pi /3<\arg k<2\pi /3$. There $\mathrm{Im}{k}^3<0$, sois bounded, while $e^{ikx}$ decays exponentially. The Cauchy integral theorem therefore removes that entire unknown contribution. The resulting solution depends on the initial transform and the two prescribed boundary transforms only.

// Target: analysis.bigb

A jump occurs when both finite one-sided limits of a function exist but differ. Unlike an isolated change in value, a jump changes limiting values on whole one-sided neighbourhoods. One-sided Fourier spectrum excludes jumps explains an obstruction to jumps for functions with only nonnegative Fourier coefficients.

A removable discontinuity has an existing finite limit that differs from the assigned point value, or a missing point value. Changing the point value removes it. Integrals and Fourier coefficients do not detect changes on a set of measure zero, so purely integral hypotheses cannot prohibit this kind of discontinuity for an arbitrary representative.

A connected complete Riemannian manifold that is disconnected at infinity contains a line in a Riemannian manifold. Choose points escaping in two components outside a fixed compact set and join them by minimizing geodesics. Each segment crosses the compact set. Parametrize from a crossing point; both parameter endpoints tend to infinity in opposite directions. Compactness of the unit tangent vectors over the crossing set gives a subsequence converging to an entire geodesic. Every finite subsegment is minimizing by continuity of distance, so the limit is a line.

For a locally Lipschitz function that is convex near $x$, increasing secant slopes have a finite right limit. Positive rescaling of $t$ gives positive homogeneity in $y$. The convexity bound $f(x+t(y+z))\le [f(x+2ty)+f(x+2tz)]/2$ gives subadditivity after subtracting $f(x)$, dividing by $t$ and taking the limit. Thus $p_x$ is a finite sublinear functional, although it need not be a linear functional.

Define $f(x,y)=x^3/(x^2+y^2)$ away from the origin and zero at the origin. Every directional derivative exists. Along a unit vector $n$, the derivative at the origin is $n_1^3$, which is not linear in $n$. Thus these directional derivatives cannot be the restrictions of one Fréchet derivative, and the function is not Fréchet differentiable at that point.