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The graph of $f:A\to B$ is the relation represented by $(1_A,f):A\hookrightarrow A\times B$. Its converse is right adjoint to it in the pointwise order on relations.

A rubblising collision shatters a body but does not supply enough energy to disperse its fragments against self-gravity, leaving a gravitationally bound rubble pile.

An object $A$ of a Cartesian closed category is tiny when exponentiation $(-{)}^A$ itself has a right adjoint.

The extra adjunction defining a tiny object should not be confused with the filtered-colimit condition for a compact object (mathematics).

The radial density is proportional to $r^2{e}^{-r^{2n}/(V^{2n/3}{k}_{B}T)}$, whose mode is

The canonical map of an even-degree hyperelliptic curve isso it identifies the two points $(x,y)$ and $(x,-y)$ of a general fibre and is not an embedding.

For $g\ge 2$, a basis of the space of regular differentials defines the canonical map $X\to {\mathbb{P}}^{g-1}$. Equivalently, after choosing a differential with divisor $K$, one may use a basis of $L(K)$.

A smooth plane curve of degree $d$ has genus

For the affine coordinate $t$ on ${\mathbb{P}}^1$, put $s=t^{-1}$ at infinity. Since $dt=-s^{-2}ds$,so $K_{{\mathbb{P}}^1}\sim -2\infty$. More generally,because $L(d\infty )$ consists of polynomials of degree at most $d$ when $d\ge 0$ and only the zero function when $d<0$.

When $u$ and $v$ have diffusivities $D$ and $1$, respectively, and the homogeneous Brusselator equilibrium is stable, a Turing instability occurs precisely whenAt onset,

The spatially homogeneous Brusselator has its unique positive equilibrium at $(r,s/r)$. Its Jacobian there has determinant $r^2$ and trace $s-1-r^2$. If $s-1>r^2$, the equilibrium is a repeller inside the Brusselator trapping region; the Poincaré-Bendixson theorem then supplies a periodic orbit.

For the spatially homogeneous Brusselator with parameters $r,s>0$, set $\alpha =r/(1+s)$. The quadrilateralis a trapping region. The four inward-pointing tests use $\dot{x}\ge 0$ on $x=\alpha$, $\dot{y}\ge 0$ on $y=0$, $\dot{y}\le 0$ on $y=s/\alpha$, and $\dot{x}+\dot{y}=r-x\le 0$ on the sloping edge.

For one-dimensional representations $\theta ,\phi$ of the upper-triangular subgroup $B\le S{L}_2({\mathbb{F}}_p)$,where ${\theta }^w(t)=\theta (w^{-1}tw)$.

BRST cohomology identifies physical states and observables with BRST-closed objects modulo BRST-exact ones. A change of gauge-fixing fermion changes the action by a BRST-exact term and therefore leaves BRST-cohomology classes unchanged when the measure has no BRST anomaly.

A gauge-fixing fermion is a Grassmann-odd functional whose BRST transformation $s\Psi$ supplies the gauge-fixing and ghost terms. Nilpotence gives $s(s\Psi )=0$ immediately.

For standard Brownian motion $W$ and $f\in C_b^2(\mathbb{R})$, the processis a true martingale. The Itô formula writes it as ${\int }_0^t{f}^{\prime }(W_s)d{W}_s$, whose bounded integrand has finite expected integrated square on every finite horizon, by the Itô isometry. For a general $C^2$ function, the same expression is a local martingale; additional integrability is needed for a true martingale. Integrability of $f(W_t)$ alone does not justify removing localization. For example, $f(x)=x^2$ gives $W_t^2-W_0^2-t$, a true martingale by finite Gaussian moments even though $f$ is unbounded.

For $a,b>0$ and the convention $\sup \varnothing =0$,

For $a,y>0$,

For $a,b>0$,This follows from the Brownian reflection principle weighted by the Cameron-Martin theorem for a linear drift.

For a suitable twice differentiable function $u$ and an integrable stopping time $T$,

It is the Dynkin's formula with the Brownian motion generator $\frac{1}{2}\Delta$.

A body-centered cubic lattice consists of the points of a cubic lattice together with their body centers. Its reciprocal lattice is face-centered cubic.