A map between affine varieties is a morphism when its coordinate functions are restrictions of ambient polynomials. Pullback identifies these morphisms contravariantly with homomorphisms of their coordinate rings.

A classical variety is affine if finitely many global regular functions generate the unit ideal and their nonvanishing opens are affine. Localization of global sections on a principal open identifies their coordinate rings with $A_{f_i}$, where $A=\Gamma (X,{\mathcal{O}}_X)$. Choose a finite-type subalgebra of $A$ containing the functions, their unit-ideal coefficients and numerators of generators of every $A_{f_i}$. Its affine variety has the same principal affine charts, whose isomorphisms glue globally. This avoids assuming finite generation of $A$ before affineness is known.

Writing $u=e^{-\beta x/2}v$ transforms $u_t=u_{xx}+\beta u_x$ into $v_t=v_{xx}-{\beta }^{2}v/4$. A prescribed coordinate derivative $u_x=g$ at an endpoint becomes $v_x-(\beta /2)v=e^{\beta x/2}g$ there. Thus a Neumann boundary condition becomes a Robin boundary condition. The signs are coordinate-derivative signs at both endpoints; outward normal derivatives require a sign change at the left endpoint.

On $z>0$, consider $p_t+a{p}_z=D(z{p}_z{)}_z$ with $a,D>0$, zero scalar flux $ap-Dz{p}_z$ at the endpoints for $t>0$, and unit initial impulse at the origin. Put $r=a/D$ and $\eta =z/(Dt)$. The normalized similarity solution isThe ansatz $p=(Dt{)}^{-1}f(\eta )$ gives $[\eta f^{\prime }+(\eta -r)f{]}^{\prime }=0$. Endpoint decay sets this integrated constant to zero and hence $f\propto {\eta }^r{e}^{-\eta }$; the gamma function normalizes its integral to one. Its scale is $Dt$, so it converges weakly to a unit impulse as $t\downarrow 0$. Its gamma distribution shape is $1+r$, its maximum occurs at $z=at$, and its expected value is $(a+D)t$. For width proportional to $z$, the associated horizontally averaged plume concentration is proportional to $z^{r-1}{e}^{-z/(Dt)}$: its interior maximum is $(a-D)t$ when $a>D$, its supremum occurs at the origin when $a=D$, and it is singular there when $0<a<D$. The physical finite source regularizes this ideal-origin behaviour.

For the advection-diffusion equation $u_t=u_{xx}+2a{u}_x$, the multiplication $w=e^{ax}u$ gives $w_t=w_{xx}-a^{2}w$. A Dirichlet boundary condition remains a prescribed value, multiplied by $e^{ax}$ at the endpoint. This real exponential conjugation removes the first derivative without changing a bounded spatial interval.

Running phase dynamics means that an unwrapped phase keeps winding rather than approaching a locked value. For the deterministic Adler phase equation, $\omega >ϵ>0$ makes $\dot{\theta }=\omega -ϵ\sin \theta$ positive everywhere. There are no equilibrium points, and one full winding takesFor the integral, shift the angle so that the denominator is $\omega -ϵ\cos \varphi$ and use $u=\tan (\varphi /2)$. Splitting at the tangent's pole combines the two pieces intowhich has the displayed value. Thus the average winding speed is $2\pi /T=\sqrt{{\omega }^2-{ϵ}^2}$.

If $\omega =ϵ+\delta$ with small positive $\delta$, the average speed is asymptotic to $\sqrt{2ϵ\delta }$. Motion spends a long time near the vanished locking equilibria before completing each winding. The tilted washboard potential is monotone in this regime; noise-driven barrier hopping and deterministic running describe different mechanisms.

For $0<\omega <ϵ$, set $\alpha =\arcsin (\omega /ϵ)$ and $\kappa =\sqrt{{ϵ}^2-{\omega }^2}$. The neighboring saddles of the tilted washboard potential have barriers $\Delta V_{+}=2\kappa -\omega (\pi -2\alpha )$ and $\Delta V_{-}=2\kappa +\omega (\pi +2\alpha )$ above the minimum at $\alpha$. Their difference is $2\pi \omega$; both have curvature $-\kappa$, so forward and backward Kramers escape rates have equal leading prefactors.

For a classical risk model with adjustment coefficient $R>0$, the ultimate ruin probability from capital $u\ge 0$ satisfies $\psi (u)\le e^{-Ru}$. Stop the exponential continuous-time martingale at ruin or a finite horizon, bound its value on the ruin event, and then increase the horizon.

For positive claims with finite nonzero expected value $\mu$, the function $(M_X(r)-1)/r$ is continuous and strictly increasing on the positive finite-transform domain, starting at $\mu$. If $M_X$ diverges at a finite upper endpoint, or is finite for all positive arguments, the secant slope tends to infinity. In the latter case use $M_X(r)\ge \mathbb{P}(X\ge a)e^{ar}$ for some $a>0$ of positive tail probability. Every target $(1+\theta )\mu$ with positive relative safety loading therefore has exactly one positive root.

If each claim includes an independent expense $A$, replace its payment law by the convolution of independent random variables $X+A$. Its moment-generating function is $M_X(r)M_A(r)$ and its expected value is $\mathbb{E}X+\mathbb{E}A$. Keeping the relative safety loading $\theta$ fixed therefore changes the premium rate as well. The new coefficient solves $M_X(R)M_A(R)-1=(1+\theta )(\mathbb{E}X+\mathbb{E}A)R$, within the common finite-transform domain.

A functor $F:\mathcal{X}\to \mathcal{A}$ preserving the possibly large colimits of the projections $U_A:(F\downarrow A)\to \mathcal{X}$ has a right adjoint exactly when all those colimits exist. The universal cocone makes its vertex a terminal object of $(F\downarrow A)$, hence a representation of $\mathcal{A}(F-,A)$.

For a functor $U:\mathcal{C}\to \mathcal{D}$ between locally small categories, with $\mathcal{C}$ a complete category, $U$ has a left adjoint if and only if it preserves small categorical limits and satisfies the solution-set condition. The dual statement uses cocompleteness, small-colimit preservation and weakly terminal solution sets to characterize existence of a right adjoint.

The left Kan extension of $Q:\mathcal{C}\to \mathcal{E}$ along $F:\mathcal{C}\to \mathcal{D}$ is universal among functors on $\mathcal{D}$ equipped with a transformation from $Q$ to their restriction along $F$. When it exists, it is left adjoint to precomposition. For small indexing categories and cocomplete $\mathcal{E}$, it is computed by comma-category colimits.

A process $H$ is progressively measurable when, for every $T$, its restriction to $[0,T]\times \Omega$ is measurable for $\mathcal{B}([0,T])\otimes {\mathcal{F}}_T$. Continuous adapted processes are progressively measurable, and a Borel function of such a process remains so. A progressively measurable square-integrable Brownian integrand has an equivalent predictable integrand for the Itô integral, up to equality for $dtd\mathbb{P}$.

For an adapted process with right-continuous paths and a stopping time $T$, $X_t^T=X_{t\wedge T}$ is adapted. At fixed $t$, round $t\wedge T$ upward on finite partitions of $[0,t]$. Every approximating value is ${\mathcal{F}}_t$-measurable, and right continuity gives the limit. Completeness handles a null exceptional set if right continuity holds only almost surely. A martingale assumption and boundedness of $T$ are unnecessary.

The subgradient of $\max \{0,1-a_{i}x\}$ is $-{\lambda }_i{a}_i$, with ${\lambda }_i=1$ below the margin, ${\lambda }_i=0$ beyond it and ${\lambda }_i\in [0,1]$ at equality. For quadratic regularization the one-dimensional optimality condition is $\alpha nx={\sum }_i{\lambda }_i{a}_i$. The optimizer can be unique even if the weights at equality are not.

With fixed positive quadratic regularization, if all reference hinge margins are nonzero, the one-dimensional minimizer has local formula $x(a)=(\alpha n{)}^{-1}{\sum }_{i\in I}{a}_i$. The active hinge set is verified using continuity of this explicit candidate, then strong convexity identifies it as the optimizer. Its exact Euclidean Lipschitz modulus is $\sqrt{|I|}/(\alpha n)$.

A pinned semi-infinite membrane scatters an incoming guided mode into reflected guided modes and outgoing sound. For unit incident displacement the scattered endpoint value is $-1$. Its Half-range Fourier transform therefore has nonzero endpoint terms even though the total displacement vanishes. The Wiener-Hopf equation kernel is $2/(m{\omega }^2-T{k}^2)+\gamma /({\rho }_0{\omega }^2)$ with the outgoing branch ${\gamma }^2=k^2-{\omega }^2/c_0^2$.

For a low-Mach number source with stress $O({\rho }_0{U}^2)$ and advective time $\ell /U$, the three-dimensional compact acoustic quadrupole has density amplitude $O({\rho }_0{M}^4\ell /r)$, where $M=U/c_0$. The power depends on these source-time and stress hypotheses, not only on the quadrupole label.

The kinematic jerk is the time derivative of acceleration. For a Kepler orbit with relative position $\mathbf{r}$, relative velocity $\mathbf{v}$, $r=|\mathbf{r}|$, $\dot{r}=\mathbf{r}\cdot \mathbf{v}/r$ and total mass $M$, differentiation of Newton's law of universal gravitation gives $\mathbf{j}=-GM(\mathbf{v}-3\dot{r}\mathbf{r}/r)/r^3$.