A share paying a continuous dividend rate ${\delta }_t$ produces dollar gains $d{S}_t+{\delta }_{t}dt$. If its price follows a geometric Brownian motion with drift ${\mu }_{\mathrm{price}}$ and its dividend yield ${\delta }_t/S_t$ is constant, its drift in an investment-consumption problem is their sum. The self-financing portfolio equation uses this total return, even if the agent consumes the dividends rather than reinvesting them. Omitting the dividend yield changes both the optimal stock fraction and the equilibrium interest rate.

For an integrable discrete trading strategy under an equivalent martingale measure, discounted portfolio wealth plus accumulated discounted strategy payments is a martingale. Conditioning its terminal value gives the displayed identity. A formula involving future payments alone requires terminal liquidation, $V_n=0$, or inclusion of that terminal value in the final payment. For an attainable claim paid only at maturity, liquidation yields its usual conditional discounted expected value.

In the semistrip Laplace spectral global relation, reality makes $W(\pm \kappa )$ real for $\kappa >0$. Its imaginary parts at those two arguments remove the unknown vertical-side normal derivative. With $G_j^s(\kappa )={\int }_0^{\infty }\sin (\kappa x)g_j(x)dx$ and $S_j(\kappa )={\int }_0^{\infty }\sin (\kappa x)q_y(x,j)dx$, the result isThe lower outward normal derivative is $-q_y$, so its Dirichlet-to-Neumann map coefficient is $-S_0$, whereas the upper coefficient is $S_l$.

// Target: analysis.bigb

For $q_t=q_{xx}+\alpha q_x$, the exponential change of unknown gives $u_t=u_{xx}-{\alpha }^{2}u/4$. Its Fourier sine transform satisfies a closed scalar evolution with a boundary forcing term. The initial transform uses $e^{\alpha x/2}{q}_0(x)$; ordinary integrable-transform arguments require sufficient decay of this weighted datum. For $q_0=e^{-ax}$ this requires $a>\alpha /2$. In contrast, the unweighted sine transform of $q$ couples to its cosine transform through the drift term, so it does not directly diagonalize this half-line problem.

For a primitive Dirichlet character modulo $N>1$ of parity $a\in \{0,1\}$, put ${\Theta }_{\chi }(\tau )={\sum }_n{n}^a\chi (n)e^{\pi i{n}^2\tau }$, extending the character by zero on nonunits. The square root is analytic with positive value when $-i\tau$ is positive real. Apply Poisson summation separately in each residue class modulo $N$, and use the finite Fourier transform of a primitive Dirichlet character. The Fourier transforms of $e^{\pi i\tau x^2}$ and $x{e}^{\pi i\tau x^2}$ are $(-i\tau {)}^{-1/2}{e}^{-\pi i{t}^2/\tau }$ and $(t/\tau )(-i\tau {)}^{-1/2}{e}^{-\pi i{t}^2/\tau }$. These give the formula, including its odd-character phase.

Integer row vectors forming a square unimodular matrix are a basis of the corresponding free abelian group: the adjugate formula makes the inverse integral. The subgroup spanned by any subset of that basis is therefore a direct summand, complemented by the remaining basis vectors. In contrast, a proper finite-index subgroup of a free abelian group cannot be a direct summand: its nonzero finite quotient would have to embed as a subgroup of a torsion-free group.

For $f:{\mathbb{P}}_k^1\to \mathrm{Spec}k$, let $D$ consist of $0$ and infinity. Its ideal sheaf is $\mathcal{O}(-2)$, giving an exact coherent sequence $0\to \mathcal{O}(-2)\to \mathcal{O}\to {\mathcal{O}}_D\to 0$. The map on global sections is the diagonal $k\to k\oplus k$, which is not surjective. Since direct image to the point is global sections, projectivity does not make direct image right exact.

For a continuous map $f:X\to Y$, the derived direct image $R{f}_{\ast }$ is obtained by applying $f_{\ast }$ to an injective resolution. Composition with derived global sections gives $R\Gamma (Y,R{f}_{\ast }E)\cong R\Gamma (X,E)$.

Use ${ϵ}_{123}=1$ and orientation $d\tau \wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$. A $\tau$-independent gauge curvature has mixed component ${\mathcal{F}}_{i\tau }=D_i\Phi$. If $B_i=D_i\Phi$, its two-form is ${\sum }_{\mathrm{cyclic}}{B}_i(d{x}^j\wedge d{x}^k-d\tau \wedge d{x}^i)$. The Hodge star operator interchanges the two displayed basis forms, so $\ast \mathcal{F}=-\mathcal{F}$. With orientation $d{x}^1\wedge d{x}^2\wedge d{x}^3\wedge d\tau$ the same lift is self-dual; changing the sign of its $d\tau$ component instead recovers anti-self-duality. Orientation must therefore accompany any stated sign in this reduction.

An algebraic variety whose irreducible components all have the same dimension of an algebraic set. The components of a generic fibre in the fiber dimension theorem have this property even when that fibre is reducible.

For a diffusion process with diffusion generator $L$, the Itô formula makes $\phi (X_t)-\phi (X_0)-{\int }_0^{t}L\phi (X_s)ds$ a local martingale. Under stopping and integrability conditions making it a true martingale, taking expectations gives the displayed identity. For Brownian motion, $L=\frac{1}{2}\Delta$, recovering the usual Brownian formula.

After diffusion arrest of Gaussian packet compression, define the logarithmic area increase $A_t=\frac{1}{2}\log [\det J(t)/\det J(0)]$. It is nonnegative. Assume $A_t/t$ has a large deviation principle with convex rate function $\mathcal{I}$, minimized at $\lambda >0$. The exact amplitude relation $f(t)=f(0)e^{-A_t}$ and the Laplace principle for probability measures give the displayed moment decay rate. The rate function refers to area, so rare width fluctuations are included rather than discarded without justification. For differentiable strictly convex $\mathcal{I}$, the interior minimizer satisfies ${\mathcal{I}}^{\prime }(s_{\mu })=-\mu$. If ${\mu }_{\ast }=-{\mathcal{I}}^{\prime }(0)$ is finite, then ${\gamma }_{\mu }=\mathcal{I}(0)$ for $\mu \ge {\mu }_{\ast }$: rare nearly unstretched packets dominate high moments. Below the threshold, $d({\gamma }_{\mu }/\mu )/d\mu =-\mathcal{I}(s_{\mu })/{\mu }^2<0$. These statements concern a positive Gaussian packet with flow-only averaging; a homogeneous signed random initial scalar requires a different averaging kernel. Arbitrary random $C(t)$ alone does not imply this statistical model.

The Rayleigh criterion calls two equal, mutually incoherent optical point sources just resolved when one circular-pupil Airy disk's centre coincides with the other's first zero. It is a specified discrimination convention, distinct from full width at half maximum and from precision in estimating one source's position.

For this circular optical pupil, the Rayleigh criterion uses angular separation $s_{\mathrm{R}}={\theta }_1\simeq 1.22\lambda /D$, or $s_{\mathrm{R}}=r_1\simeq 1.22\lambda f/D$ in the paraxial approximation to the focal plane. The point spread functions still overlap.^[ref]

For the Airy disk design example with $D=5.00\mathrm{mm}$, $f=100\mathrm{mm}$ and $\lambda =550\mathrm{nm}$, the angular convention gives $s_{\mathrm{R}}=1.34164\times 10^{-4}\mathrm{rad}$, or about $27.67$ arcseconds, and the image-plane separation is $13.4164\mu \mathrm{m}$. Two equally bright, mutually incoherent optical point sources at separations $0.7$, $1$ and $1.5$ times this scale would have image separations $9.391$, $13.416$ and $20.125\mu \mathrm{m}$ respectively. These calculated cases are compared in incoherent two-source Airy profile.

The Rayleigh criterion is useful because it ties an easily recognized two-peak convention to a predictable optical scale. It does not assert that all information vanishes below that separation. Inferring two unknown positions also depends on photon shot noise, background, unequal brightness and uncertainty in the point spread function; locating a single isolated source is a different task. A theoretical analysis shows that alternative optical measurements can estimate the separation of two incoherent optical point sources below this scale.^[ref]

The mutual optical coherence condition is essential. In incoherent imaging, time-averaged light intensities add. In coherent imaging, wave amplitudes add before squaring, leaving a wave interference term controlled by relative wave phase. Two holes illuminated by the same coherent laser therefore do not implement the equal-incoherent-source convention merely because their individual patterns resemble Airy disks. The coherent single-aperture laboratory test establishes the response of each aperture; it does not establish the pair's mutual incoherence.

Perturb the Couette flow between walls moving at +-U by a small pressure gradient -G. The mean shear rate remains U/h, and the stress variation across the gap is -Gy. Linearizing the monotone generalized Newtonian fluid law divides this variation by differential viscosity, giving the displayed velocity and flux per unit width. Require $Gh\ll (U/h){\eta }_d(U/h)$ and a smooth constitutive law on the perturbed rates. For a power-law fluid, the denominator is $nk(U/h{)}^{n-1}$.

Linearizing the differential pom-pom model about its relaxed state leaves no first-order stretch perturbation in incompressible flow. Its anisotropic orientation obeys $\dot{a}+a/{\tau }_1=A_1$, producing the displayed relaxation modulus. In a slow-flow expansion, $A=I+{\tau }_1{A}_1+{\tau }_1^2(2A_1^2-A_2)+O(3)$. Trace normalization and the first stretch correction affect only isotropic stress at this order, giving the displayed second-order fluid coefficients. Linear small-amplitude response allows arbitrary frequency, whereas this local second-order expansion also requires slow time dependence.

For extension rate $s>0$ and $x=s{\tau }_1<1/2$, the steady conformation tensor is $\mathrm{diag}((1-2x{)}^{-1},(1+x{)}^{-1},(1+x{)}^{-1})$. Its normalized orientation has axial component $(1+x)/[3(1-x)]$ and transverse components $(1-2x)/[3(1-x)]$. Stretch is finite only if $s^2{\tau }_1{\tau }_2<1-x$, when ${\lambda }_{\infty }=(1-x)/(1-x-s^2{\tau }_1{\tau }_2)$ and the displayed extensional viscosity follows. Above $x=1/2$, the unnormalized axial tensor grows exponentially but $B\to \mathrm{diag}(1,0,0)$; then stretch remains bounded only if $s{\tau }_2<1$. Orientation normalization and stretch relaxation therefore impose different thresholds.

For a real-valued smooth function on a smooth manifold, its differential of a smooth map is a differential one-form: $d{f}_p$ maps a tangent vector to the directional derivative of the function. In local coordinates, $df={\sum }_i({\partial }_{i}f)d{x}^i$. The exterior derivative extends this operation from functions to higher-degree differential forms.

Take $X$ with continuous uniform distribution on $(0,1)$ and set $Y=X^2$. This is an increasing invertible transformation on that interval. Its cumulative distribution function is $F_Y(y)=\sqrt{y}$, givingAlthough the probability density diverges near zero, it is integrable. The logarithmic stretch is integrable too:The inverse function $X=\sqrt{Y}$ recovers every input in the interval. The changed differential entropy records a changed coordinate volume, rather than a failure of recovery.

// Target: probability-and-statistics.bigb

The Fresnel equations give reflection and transmission amplitudes for both transverse-electric and transverse-magnetic polarization of an electromagnetic wave at a planar interface. They follow by applying interface conditions for electromagnetic fields to incident, reflected and refracted plane electromagnetic waves.

For a computably enumerable first-order theory $T$ extending Robinson arithmetic, choose the standard arithmetic arithmetical provability predicate ${\mathrm{Prov}}_T$. The diagonal lemma, carried out in the weak base theory, gives a sentence $G$ withIf $T$ is consistent, $T$ cannot prove $G$: an actual finite proof would be verifiable in the weak base, so $T$ would prove both its provability and its unprovability. There is therefore no actual proof of $G$, and the displayed equivalence, true in the standard natural numbers, shows that this canonical $G$ is true there.

If $T$ is omega-consistent, it cannot prove $\neg G$ either. Such a proof would imply the existence of a proof code for $G$, while each standard numeral can be individually verified not to be such a code. Ordinary consistency suffices for unprovability of $G$; the original argument for unprovability of its negation uses the stronger hypothesis. Requiring the fixed-point equivalence in a sound weak base is important: an arbitrary equivalence provable only in an unsound extension need not yield a true sentence.