A first-order formula is ${\Pi }_1^{\mathrm{ZF}}$ if ZF proves it equivalent, with the same free variables, to a universal unbounded quantifier block followed by a bounded formula in set theory. Bounded quantifiers do not increase its level in the Lévy hierarchy. Such properties are downward absolute between appropriate transitive models of ZF.

A first-order formula is syntactically ${\Sigma }_1$ when it is an unbounded existential quantifier block followed by a bounded formula in set theory. Equivalence provable in a specified theory gives the corresponding modulo-theory notion. Existential witnesses show upward absoluteness between transitive models containing the parameters.

A syntactically ${\Pi }_1$ first-order formula has an unbounded universal quantifier block followed by a bounded formula in set theory. Universal quantification makes it downward absolute between transitive models. A Pi-one formula modulo ZF may have another syntax but has such an equivalent provable in ZF.

A first-order formula is ${\Delta }_1$ modulo ZFC if that theory proves it equivalent, with the same free variables, both to a Sigma-one formula in set theory and to a universal unbounded quantifier block with a bounded formula in set theory as matrix. The theory used for provable equivalence matters: modulo ZF is a stronger requirement.

In unit binary separation and mean motion, displacements obey $\ddot{X}-2\dot{Y}=(1+2B)X$ and $\ddot{Y}+2\dot{X}=(1-B)Y$, where $B={\mu }_1/r_1^3+{\mu }_2/r_2^3$ at $L_2$. The characteristic polynomial is ${\lambda }^4+(2-B){\lambda }^2+(1+2B)(1-B)$. The small-secondary limit has $B\to 4$.

Use the derivative-index-first velocity gradient $G_{ij}={\partial }_i{u}_j$ and $\omega =G-G^T$. For an objective time derivative the tensor transformation $A^{\ast }=QA{Q}^T$ under $x^{\ast }=Q(t)x+c(t)$ must imply $\mathcal{D}{A}^{\ast }/\mathcal{D}t=Q(\mathcal{D}A/\mathcal{D}t)Q^T$. If $R=\dot{Q}{Q}^T$, then $G^{\ast }=QG{Q}^T-R$ andConsequently the extra terms cancel in $D_{t}A+\frac{1}{2}(\omega A-A\omega )$, proving covariance. With the component-index-first gradient $L_{ij}={\partial }_j{u}_i$, the same physical rate is $D_{t}A-WA+AW$, with spin tensor $W=(L-L^T)/2=-\omega /2$. Changing the gradient convention without changing the commutator sign destroys objectivity.

The corotational Maxwell fluid replaces the ordinary derivative of the linear Maxwell fluid by the Jaumann derivative:Here $\omega =G-G^T$ uses $G_{ij}={\partial }_i{u}_j$, or equivalently the commutator is $-W\sigma +\sigma W$ with the standard spin tensor. The stress has relaxation time $\lambda$ and small-rate viscosity $\mu$. The commutator makes the constitutive equation nonlinear while maintaining an objective time derivative.

Elliptic curves isogenous over a finite field have the same number of rational points: the isogeny intertwines Frobenius, and multiplicativity of degree gives equal degrees for $1-\pi$. Their groups need not be isomorphic. Over ${\mathbb{F}}_7$, $y^2=x^3-x$ and $y^2=x^3+4x$ are two-isogenous and have groups $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}$ and $\mathbb{Z}/8\mathbb{Z}$, respectively.

At a material velocity jump with continuous background mass density, vertical displacement $\eta$ and fluid pressure are continuous, while $w=ikU\eta$ need not be. For a stationary component, $p=U^2{\eta }_z$ and the signed upward-wave impedance is $Z=U^{2}m$. Reflection and vertical-displacement transmission are $R=(Z_1-Z_2)/(Z_1+Z_2)$ and $T=2Z_1/(Z_1+Z_2)$.

For $A\in \mathcal{U}$ and an idempotent ultrafilter, put $A^{\ast }=\{n\in A:A-n\in \mathcal{U}\}$. Then $A^{\ast }\in \mathcal{U}$, and $A^{\ast }-n\in \mathcal{U}$ for every $n\in A^{\ast }$. For the second assertion, apply idempotence to $A-n$ and intersect its resulting good-translation set with $A-n$. This allows each new finite-sums generator to be chosen from finitely many translation constraints, proving the Idempotent-ultrafilter proof of Hindman's theorem.

For positive $k$ take the $k$th tensor power of the hyperplane line bundle; for negative $k$ take the corresponding power of its dual; for $k=0$ take the trivial line bundle. On the complex projective line, global sections for $k\ge 0$ correspond to polynomials of degree at most $k$ and form a space of dimension $k+1$.

For $1<a\le 2$, the radial function $f(x)=(1+|x|{)}^{-a}$ belongs to $L^2({\mathbb{R}}^2)$ and has finite distributional total variation seminorm on a domain, but is not in $L^1({\mathbb{R}}^2)$. Cutting it off outside radius $T$ gives bounded-variation space functions tending to $f$ in $L^2$, with variation tending to that of $f$: the added jump costs $2\pi T(1+T{)}^{-a}\to 0$. Thus the penalty equal to TV on $BV\cap L^2$ and infinity elsewhere is not sequentially lower semicontinuous in $L^2$. The closed homogeneous bounded-variation space extension avoids this domain issue.

On a complex vector space with linear coordinates, the Euler vector field is $E={\sum }_j{z}_j{\partial }_{z_j}$. It generates scalar dilations, and the Euler homogeneous function theorem says $E(f)=kf$ for a homogeneous function of degree $k$. Contracting a top form with $E$ gives a horizontal form for the quotient by nonzero scalings; descent also requires the appropriate scaling invariance.

For a holomorphic vector bundle $E$, $\mathcal{O}(E)(U)$ is its space of holomorphic sections on $U$, with their restrictions. It is the kernel of the bundle Dolbeault operator on smooth sections.

Dualize the fibres of a holomorphic vector bundle. If its transitions are $g_{ij}$, the dual transitions are $g_{ij}^{-T}$, which are holomorphic and preserve the evaluation pairing. These define the natural holomorphic structure on the smooth dual bundle.

A prime chain of length $s$ in a Noetherian local ring forces its maximal-ideal length function to be at least $c{t}^s$ for large $t$. Quotient by the initial prime to reduce to a domain, choose a nonzero $a$ in the next prime, and use the Artin-Rees lemma to obtain $H_R(t)\ge H_R(t-c_0)+H_{R/(a)}(t)$. Induction on chain length and summing about $t/(2c_0)$ terms prove the bound. This yields $\dim R\le d(R)$ without assuming the Krull height theorem.

For an operator $O_H=U^{†}{O}_{S}U$ in the Heisenberg picture, differentiating the evolution operators gives $d{O}_H/dt=i[H_H,O_H]+U^{†}(\partial O_S/\partial t)U$ in units $\hslash =1$. With no explicit operator time dependence, the second term vanishes. Applying this commutator identity to the canonical momentum and real scalar field gives their operator field equations.

A generalized Heegaard diagram permits compression bodies and a nonempty negative boundary. Equivalently, a suitable handlebody with two-handles attached along only part of a compressing system can retain torus boundary components. Disk crossings supply generators and attaching-curve words supply relators for its fundamental group.

Winds in a weakly ionized exoplanet atmosphere can drive electric currents through a planetary magnetic field. Their Ohmic heating is a proposed power source for hot-Jupiter radius inflation. A model must determine the electrical conductivity, the complete current paths and the depth at which the power is deposited.^[ref]

In a nonrelativistic, isotropically conducting fluid, let $\mathbf{v}$ be its velocity, $\mathbf{E}$ the laboratory electric field, $\mathbf{B}$ the magnetic field, $\mathbf{J}$ the current density and $\sigma >0$ the electrical conductivity. The simple resistive Ohm's law isThe irreversible heat production per unit volume and time is thereforeThe laboratory electrical power $\mathbf{J}\cdot \mathbf{E}$ alone also contains mechanical work by the Lorentz force:Thus identifying $\mathbf{J}\cdot \mathbf{E}$ directly with resistive heat requires an appropriate rest frame or a vanishing mechanical-work term. The total deposited power is $L_{\mathrm{Ohm}}=\int q_{\mathrm{Ohm}}dV$. A large value confined to optically thin upper layers need not maintain the deep interior's entropy; magnetic braking and current closure also limit what a prescribed wind can supply.

The cellwise error of the Haar projection estimator in Gaussian white noise is $\sqrt{2^J/n}$ times a standard normal variable. Its supremum norm is therefore exactly that factor times the Gaussian maximum over $2^J$ cells. The Gaussian maximum bound without independence gives expected error at most $\sqrt{2^J(2J+2)\log 2/n}$. This concerns the stochastic error about the projection; approximation bias must be added when estimating the full drift.