A vector field $X$ is projectable through a smooth map between manifolds $f:M\to N$ if there is a smooth vector field $Y$ on $N$ with $d{f}_p{X}_p=Y_{f(p)}$ for every $p$. Agreement on fibres is necessary. For a surjective submersion it is also sufficient: smooth local sections of the submersion express $Y$ locally as $df(X)$ and prove its smoothness. An arbitrary non-surjective map may instead give a field only along its image, with extension to $N$ requiring additional choices.

The chain rule shows that $\alpha {\varphi }_t{\alpha }^{-1}$ is the local flow of the pushforward of a vector field ${\alpha }_{\ast }X$. If this equals $X$, uniqueness of integral curves of a vector field gives commutation. Conversely, differentiate the commuting identity at $t=0$ to obtain $d{\alpha }_p{X}_p=X_{\alpha (p)}$. All identities hold on their common domains; the vector field need not be complete.

A vector field along a smooth map between manifolds $f:M\to N$ assigns $Y_p\in T_{f(p)}N$ smoothly to each $p$. It is a section of the pullback tangent bundle, not necessarily a vector field on $N$. Along a curve $\gamma$, the velocity $\dot{\gamma }$ is such a section.

For a morphism of ringed spaces, the pullback of a sheaf of modules is left adjoint to the direct image sheaf functor. The unit sends a local section $h$ to $1\otimes h$ after inverse image. On the structure sheaf it becomes the given ring homomorphism ${\mathcal{O}}_Y\to f_{\ast }{\mathcal{O}}_X$.

The restriction meant here is the pullback of a sheaf of modules to the closed subvariety, including tensoring with its structure sheaf. On an affine chart, tensoring $M$ with $A/I$ gives $M/IM$. This distinction matters because restricting only the underlying topological sheaf would retain the ambient ring action.

The pullback of a covector is the dual linear map to the differential of a smooth map. In coordinates its components are $({\varphi }^{\ast }\eta {)}_i=({\partial }_i{y}^{\alpha }){\eta }_{\alpha }$. The same rectangular Jacobian defines vector pushforward, with its other index contracted. No invertibility is needed.

For an embedded submanifold $i:M\hookrightarrow N$, the ambient affine connection induces a pullback connection on $i^{\ast }TN=TN{|}_M$. Extend a local section off $M$ and differentiate in a direction tangent to $M$. If two extensions agree on $M$, their difference has zero coefficients on $M$, whose tangential derivatives are zero. Thus the restricted derivative is independent of the extension. Its value may have a normal component, so obtaining a connection on $TM$ requires a further projection.

An unbiased likelihood estimator has expectation equal to the likelihood function at each parameter. A pseudo-marginal algorithm requires this estimator to be nonnegative. Importance sampling provides $\hat{L}=M^{-1}{\sum }_j\ell (F_j)p_{\theta }(F_j)/g_{\theta }(F_j)$ for independent random variables $F_j\sim g_{\theta }$, provided the proposal covers the target support. Relative variance affects the mixing time even though unbiasedness guarantees the correct marginal target.

An involutive Metropolis proposal uses a deterministic bijection $S$ satisfying $S^2=I$. For a volume-preserving $S$ and target probability density function $\rho$, accept $S(z)$ with probability $\min (1,\rho (S(z))/\rho (z))$. The identity $\rho (z)\alpha (z)=\min (\rho (z),\rho (S(z)))$ and a change of variables under $S$ prove detailed balance. A non-unit Jacobian determinant must be included when volume is not preserved.

On a unit interval with gauge parameters vanishing at its ends, $e=s+\dot{ϵ}$ separates the worldline einbein into its invariant average $s$ and a canonical gauge transformation. The Jacobian for nonconstant modes is $\det {\partial }_t$ between the appropriate boundary-condition spaces. On a circle its prime excludes the constant gauge-parameter zero mode. This Faddeev-Popov determinant can be represented by anticommuting $b,c$ fields with an action proportional to $\int b\dot{c}dt$.

On a compact metric space, every sequence of Borel probability measures has a subsequence with weak convergence of probability measures to a Borel probability measure. To see this, choose a countable uniformly dense subset of continuous functions, extract a diagonal subsequence of their bounded integrals, and extend the resulting positive normalized functional to all continuous functions. The Riesz representation theorem supplies its Borel probability measure. This is useful for constructing an invariant measure from orbit empirical measures.

A weakly closed uniformly tight family of Borel probability measures on a Polish space is weakly compact. Embed the underlying space into a compact metric closure, take subsequential pushforward limits, and use the closed-set Portmanteau theorem on the common tightness compact sets. The limit retains full mass in the original space; inverse continuity of the probability pushforward embedding theorem then returns the weak limit.

For the prismatic triangular-channel shallow water equations, $c^2=G\varphi h/2$. The characteristic speeds are $u,u\pm c$. Along $dx/dt=u$, $d\varphi /dt=-2w\varphi /h$. Along $dx/dt=u\pm c$, the hyperbolic system has compatibility relationThe concentration differential cannot generally be discarded to obtain Riemann invariants $u\pm 4c$. Positive depth and concentration are required for this nondegenerate form.

For a power-law matter spectrum, smoothed variance scales as $\sigma \propto M^{-\beta }$ with $\beta =(n+3)/6$. Differentiating the Press-Schechter collapsed fraction gives a halo mass function proportional to $M^{\beta -2}\exp [-(M/M_{h,\mathrm{ch}}{)}^{2\beta }]$. A constant stellar-to-halo mass ratio preserves both exponents and rescales characteristic mass and normalization.

Defining characteristic mass by an exponential factor $\exp [-(M/M_{h,\mathrm{ch}}{)}^{2\beta }]$ fixes its peak height to $\nu =\sqrt{2}$, rather than one. Hence $\sigma (M_{h,\mathrm{ch}},z)={\delta }_{\mathrm{sc}}/\sqrt{2}$ and the variance at other masses follows the same power-law scaling. The precise characteristic-mass convention determines this factor.

For a fixed narrow mass bin with time-independent smoothing slope, the Press-Schechter halo mass function has time dependence $n\propto \nu e^{-{\nu }^2/2}$, with halo peak height $\nu \propto D^{-1}$. Differentiating gives the displayed rate. It is positive only for $\nu >1$; the bin can lose objects to higher masses later. Cumulative counts require integrating the differential abundance, and a moving mass boundary adds a selection term.

A sieve on $U$ is a subfunctor of the representable functor $yU$, equivalently a family of arrows into $U$ closed under precomposition. For $f:V\to U$, the pullback sieve consists of arrows $g$ into $V$ for which $fg$ belongs to the original sieve.

A Grothendieck topology assigns covering sieves to each object. The maximal sieve covers; pulling back a covering sieve gives a covering sieve; and a sieve locally covering along all arrows of a covering sieve itself covers. It defines the sheaves on a site. Families of arrows generate sieves and can be used to generate the topology.

For the positive-sign covariant gauge-fixing inverse, $p^a{G}_{ab}=\xi p_b/p^2$. It is not identically zero for fixed $\xi \ne 0$ and is a nonzero vector at a nonzero non-null four-momentum. Null momenta require the chosen pole prescription, and individual zero components are possible. The covariant Landau gauge limit suppresses this longitudinal part. The Ward identity removes longitudinal contributions from physical conserved-current amplitudes.

The Ringel lemma on bricks makes every indecomposable a brick module; positive definiteness and the Ringel form then force $q_Q=1$ and rigidity. Conversely, a maximal-orbit representation at a positive root cannot split: nonzero cross extensions increase orbit dimension, while vanishing cross extensions make the quadratic form of a split sum at least two. Open dense orbits give uniqueness. A compact bound on the integer roots gives finite representation type.