A natural bijection between set-valued functors is a natural transformation whose component functions are bijections. Their inverse functions are automatically natural, so it is a natural isomorphism.

The first-order string action isThe multipliers impose the two Virasoro constraints. The canonical equations give a spatial momentum flux $J_m=e{T}^2{X}_m^{\prime }+u{P}_m$.

For a nondegenerate induced worldsheet metric $g_{ab}={\partial }_{a}X\cdot {\partial }_{b}X$, variation gives ${\partial }_a(\sqrt{-g}{g}^{ab}{\partial }_b{X}^m)=0$. In conformal gauge this is the two-dimensional wave equation, supplemented by the Virasoro constraints.

A circular closed string contracts and re-expands with $X^0=Rt$ and $X^1+i{X}^2=R\cos t{e}^{i\sigma }$. Its proper length is $2\pi R|\cos t|$, while its energy remains $2\pi TR$. At collapse the induced worldsheet metric degenerates, but the Nambu-Goto phase-space action supplies a regular continuation.

At an interior point where the regression function is differentiable, a symmetric compactly supported regression kernel makes the first-order bias of an estimator vanish asymptotically. In regular fixed-design nonparametric regression with $h\to 0$ and $nh\to \infty$, the local constant estimator then has squared bias of an estimator $o(h^2)$ and variance $R(K)/(nh)+o((nh{)}^{-1})$ for a unit-integral regression kernel. The differentiability remainder is pointwise and need not hold uniformly over a function class.

For two nearly edge-on orbits with small orbital inclination difference $\Delta I$ and small longitude of ascending node difference $\Delta \Omega$, expanding the dot product of their normals gives $I_m^2=(\Delta I{)}^2+(\Delta \Omega {)}^2+O({\delta }^4)$, with angles in radians and all offsets from edge-on of order $\delta$. Thus equality of observed sky-plane inclinations alone does not imply aligned orbital planes.

For a finite multitemperature blackbody disk, the low-frequency Rayleigh-Jeans law gives $F_{\nu }\propto {\nu }^2\int rT(r)dr$. If $T\propto r^{-3/4}$ away from the inner edge, outer annuli dominate because the weight per logarithmic radial interval grows as $r^{5/4}$. A measure-zero zero-temperature inner edge does not invalidate the leading law when $rT(r)$ is integrable.

For a blackbody disk with $T\propto r^{-3/4}$, the substitution $x=h\nu /(k_{B}T)$ in $F_{\nu }\propto {\nu }^3\int r(e^x-1{)}^{-1}dr$ produces $F_{\nu }\propto {\nu }^{1/3}\int x^{5/3}(e^x-1{)}^{-1}dx$. When the inner and outer temperatures make the limits approximately zero and infinity, that convergent integral is independent of frequency. The law applies between the outer-edge turnover and the hot inner-disk cutoff.

For a complementary-slackness optimum ${\mu }^{\ast }$, defineinterpreting a zero-reference summand as ${\mu }_j/{\kappa }_j$. This is a nonnegative generalized reverse-relative-entropy expression for positive resource prices. Along multiplicative resource-price dynamics, with $p=A^T\mu$ and $g=Ax-C$,Its sublevel bounds keep all route prices away from zero and permit the LaSalle invariance principle, including optima with some zero individual prices.

For a real measurable function $m$ on a sigma-finite measure space, multiplication on $D(M_m)=\{f\in L^2:mf\in L^2\}$ is an unbounded self-adjoint operator. Testing its adjoint relation on $\{|m|\le n\}$ identifies the same maximal domain. Outside the essential range, multiplication by $(m-z{)}^{-1}$ is a bounded resolvent. Inside it, normalized indicators of finite-positive-measure sets where $m$ is close to $z$ are approximate eigenvectors, so the inverse cannot be bounded.

For $q_i=2{\alpha }^i$, $r_i=1/2+2^{-i-1}$, a uniform power gain for elliptic solutions bounds the successive norms by factors $[DR{2}^{i+2}{]}^{{\alpha }^{-i}}$. Their infinite product isIt is finite because both $\sum {\alpha }^{-i}$ and $\sum i{\alpha }^{-i}$ converge. Uniform control of the $L^{q_i}$ norms on $B_{1/2}$ then controls the essential supremum, by comparing with the measure of a set where the function exceeds the proposed bound. A constant independent of $q$ left outside the $2/q$ power at every step would instead give a divergent product.

A morphism of schemes into an affine scheme is uniquely determined by its homomorphism on global sections. Conversely, a ring homomorphism $A\to \Gamma (X,{\mathcal{O}}_X)$ restricts on every affine open subscheme of $X$ to a map into $\mathrm{Spec}A$; equality on affine covers of overlaps glues these maps. No quasi-compactness or affineness assumption on $X$ is needed.

For morphisms of schemes $f:X\to Y$ and $g:Y\to Z$, the graph is the displayed map. It is the pullback of the diagonal morphism $Y\to Y{\times }_{Z}Y$ along $(f\circ p_X,p_Y)$. Thus a separated morphism $g$ makes ${\Gamma }_f$ a closed immersion. If also $g\circ f$ is a closed immersion, the projection $X{\times }_{Z}Y\to Y$ is a closed immersion by base change of a morphism of schemes, and its composite with ${\Gamma }_f$ proves that $f$ is a closed immersion.

For an order-$k$ B-spline with $k\ge 2$, its Greville abscissa is the arithmetic mean of its $k-1$ interior knots. These are precisely the coefficients reproducing the linear function in the monomial B-spline coefficients formula. The irregular plural is Greville abscissae.

A monoid homomorphism $h:M\to N$ preserves multiplication and identity: $h(xy)=h(x)h(y)$ and $h(1_M)=1_N$. It therefore preserves every finite ordered product, including the empty product. The monoids with these maps form the category $\mathbf{Mon}$.

A connected three-sheeted cover has transitive monodromy. If it also has a simple branch point, its monodromy contains a transposition and must be the full symmetric group on three letters. A local triple branch supplies a three-cycle explicitly.

With unit tension and $X^0=t$, $X^1=\sigma$, the Virasoro constraints give $P_1=-\Pi {\Phi }^{\prime }$ and $P_0=-\sqrt{(1+{\Phi }^{\prime 2})(1+{\Pi }^2)}$. The reduced energy density is $\mathcal{H}=-P_0$. On the chiral branch $\Pi ={\Phi }^{\prime }$, the Hamiltonian field equation becomes $\dot{\Phi }={\Phi }^{\prime }$.

For $X=(t,\sigma ,k(\sigma +t))$, the vertical coordinate velocity is $k$, but its normal projection has magnitude $|k|/\sqrt{1+k^2}<1$. Tangential motion merely changes string labels. The induced worldsheet metric has determinant $-1$ even when constant-label curves become spacelike for $|k|>1$.

For the nested partial unary operation category tower, each one-step forgetful functor is a monadic adjunction right adjoint: it reflects isomorphisms and creates its split coequalizers. The Beck monadicity theorem identifies the first Eilenberg-Moore category with the next category in the tower. A long composite has that same first monad because free extension of nested partial unary operations leaves no new top fixed points. Repeating the comparison forgets one fewer operation at each step. The remaining forgetful functor is not full until no operation remains to forget, giving monadic length $m-n$.

For $Z$ with a standard normal distribution and $a\ge 0$, split the Gaussian integral at zero and complete the square to obtain $\mathbb{E}{e}^{a|Z|}=2e^{a^2/2}\Phi (a)\le 2e^{a^2/2}$, where $\Phi$ is the standard normal distribution function. This converts a bound on an absolute Gaussian random variable into a two-sided Gaussian tail bound using Markov inequality.