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For an element $a$ of a complex Banach algebra,

Mantle convection is the slow buoyancy-driven motion of a rocky planetary mantle. On geological timescales the solid mantle behaves as a highly viscous Newtonian fluid to first approximation.

The largest prime factor $P^{+}(n)$ of an integer $n>1$ is the greatest prime number dividing $n$.

For a finite set $S$ of primes, an integer is $S$-smooth when every prime dividing it belongs to $S$. Bounding the largest prime factor is equivalent to restricting the integer to be $S$-smooth for a suitable finite $S$.

A sine-Gordon breather is a spatially localized solution that is periodic in time and can be interpreted through a complex-conjugate pair of discrete scattering eigenvalues.

This is a breather of the Sine-Gordon equation; the general breather concept also occurs in other nonlinear systems.

For every $l>0$,solves $u_{XT}=\sin u$. The scaling $(X,T)\mapsto (e^{s}X,e^{-s}T)$ transforms $l$ by multiplication with $e^{-s}$.

Quantum order finding Fourier-transforms a superposition of arguments having the same modular-exponentiation value, producing peaks near multiples of the reciprocal period.

If $r$ is even, $a^r\equiv 1(\mathrm{mod}N)$, and $a^{r/2}\not\equiv -1(\mathrm{mod}N)$, then $\gcd (a^{r/2}-1,N)$ is a nontrivial factor unless the order was not minimal.

A vortex sheet is a surface across which tangential velocity is discontinuous while normal velocity remains continuous.

An ordinal is a transitive set well-ordered by membership and represents the order type of a well-order.

In the presence of choice, a cardinal number may be represented by the least ordinal in its bijection class.

The fiber product of maps f:A to B and g:A-prime to B consists of pairs with equal images in B.

The disjoint union tags the elements of its summands, for example $X\bigsqcup Y=(X\times \{0\})\cup (Y\times \{1\})$.

The Cartesian product $X\times Y$ is the set of ordered pairs $(x,y)$ with $x\in X$ and $y\in Y$.

The image of a function asks which outputs a chosen set of inputs reaches. A preimage asks which inputs reach a chosen set of outputs. Both constructions apply to every function, including one that loses information by sending several inputs to the same output. Their difference is useful whenever a transformation must be followed forward or a condition on its result must be traced back.

Consider the function $f:A\to B$ withEvery input has exactly one output, but zero has two inputs and three has none. The full image of a function is $f(A)=\{0,1,2\}$, a proper subset of the codomain $B$. Selecting inputs $\{a,c\}$ gives image $\{0,1\}$. Selecting outputs $\{0,2\}$ instead gives preimage $\{a,b,d\}$.

The function is not injective, because $f(a)=f(b)$, and is not surjective onto $B$, because three is absent from its image. Restricting only the codomain to $\{0,1,2\}$ makes the same rule surjective; it remains noninjective. This separates the output set actually reached from the target set specified in the definition. The definitions of images of a function and preimages, including the distinction from an inverse function, are given in Richard Hammack, Book of Proof, §12.6: Image and Preimage.

In ZF, if $X\times X\cong X$ for every infinite $X$, apply this to $X\bigsqcup h(X)$. The product-sum comparison lemma applied to $X$ and its Hartogs ordinal gives either an impossible injection $h(X)\to X$ or a surjection $h(X)\to X$. In the latter case, ordering each $x\in X$ by its least ordinal preimage well-orders $X$. The converse is the well-orderable-cardinal identity ${\kappa }^2=\kappa$.

For every set $X$ there is a least ordinal $h(X)$ that does not inject into $X$.

The least ordinal supplied by the theorem is the Hartogs number.

Every family of nonempty sets has a choice function. In ZF this is equivalent to the well-ordering theorem.

For each formula $\phi$, the axiom schema of specification forms the subsetof any set $x$. When interpreting the schema inside a class model, the formula must be evaluated internally.

The cumulative hierarchy is defined by $V_0=\varnothing$, $V_{\alpha +1}=\mathcal{P}(V_{\alpha })$, and $V_{\lambda }={\bigcup }_{\alpha <\lambda }{V}_{\alpha }$ at limit ordinals.