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A vacuum manifold is the space of field values minimizing the potential in the specified homogeneous classical description. For a global symmetry group $G$ acting transitively on one orbit of minima, a chosen vacuum with stabilizer $H$ gives that orbit as $G/H$.

The orbit need not be connected: a real scalar field with two symmetry-related minima provides a disconnected example. Restricting to the identity component requires the corresponding connected subgroup and stabilizer, rather than automatically identifying every $G/H$ with one connected component. Continuous vacuum directions motivate Goldstone modes, while nontrivial surrounding maps can classify topological defects. In a gauge theory, some minimum-field values are gauge-equivalent, so the distinction between field-space minima and physical vacuum labels must also be stated.

A second-rank tensor invariant under every member of the orthogonal group on ${\mathbb{R}}^n$ is a scalar multiple of the Kronecker delta ${\delta }_{ij}$. For a radial weight $f(|x|)$ with absolutely integrable second moments,Reflections force off-diagonal entries to vanish, and coordinate exchanges make all diagonal entries equal. Taking the matrix trace determines their common value. The same proof applies to a ball or a spherical shell centered at the origin.

For $n\ge 3$, invariance under all proper rotations already forces this second-rank form. In two dimensions, proper rotations alone also preserve an antisymmetric tensor; full orthogonal invariance, or the symmetry of $x_i{x}_j$, excludes that extra term. Higher moments use all possible pairings of Kronecker deltas, as in radial tensor moments in three dimensions.

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

Mantle convection is buoyancy-driven motion in a rocky planetary mantle. Heating and cooling create density differences; chemical and mineral-phase differences can also affect buoyancy. Although the mantle is mostly solid, deformation by material creep permits slow flow over geological time.

A simple model balances viscous resistance with buoyancy and transports heat by both flow and thermal conduction. Its Rayleigh number compares those effects. Temperature-dependent mantle viscosity and the strength of the lithosphere help determine whether the surface moves as plates or remains a stagnant lid. Seismic structure, surface deformation and heat flux constrain different parts of this model.

For an integer $n>1$, the largest prime factor $P^{+}(n)$ is the greatest prime number dividing $n$. For example, $P^{+}(72)=3$ and $P^{+}(70)=7$: multiplicity affects the size of $n$, but does not repeat a factor in this maximum.

In notation involving smooth numbers, one often extends the definition by $P^{+}(1)=1$. With that convention and $y\ge 2$, a positive integer is $y$-smooth exactly when $P^{+}(n)\le y$. The number zero is excluded, since every prime number divides zero and there is no largest one.

For a finite set $S$ of prime numbers, an S-smooth number is a positive integer all of whose prime factors belong to $S$. It has the unique form ${\prod }_{p\in S}{p}^{e_p}$ with nonnegative integer exponents. The empty product includes one, so if $S$ is empty, one is the only S-smooth number.

For $S=\{2,5\}$, the number $200=2^3{5}^2$ qualifies, whereas $30$ does not because of its factor three. Restricting to positive integers is part of the convention here; the related B-smooth integer notation explicitly permits a negative sign. A numerical smooth number bound $y$ corresponds to taking $S$ to be all prime numbers at most $y$.

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 idealizes a thin shear region as a surface across which tangential velocity jumps while normal velocity remains continuous. Vorticity is then concentrated on the surface in the zero-thickness description. Its sheet strength is related to the tangential velocity difference, rather than to an ordinary finite vorticity value evaluated inside a resolved layer.

A shear layer between two streams motivates the approximation when its thickness is small relative to the wavelength being studied. The ideal model can support Kelvin-Helmholtz instability, but its very short-wave behavior can differ strongly from that of a finite-thickness viscous layer. The surface's kinematic boundary condition and the appropriate dynamic stress balance must be supplied as well as the velocity jump. It is therefore a useful scale-dependent interface model, not a literal claim that a physical fluid has an infinitely sharp resolved velocity profile.

An ordinal records the order type of a well-order. In the von Neumann representation, it is a transitive set well-ordered by membership. The first examples are $0=\varnothing$, $1=\{0\}$, $2=\{0,1\}$, and $\omega =\{0,1,2,\dots \}$.

Every well-order is isomorphic to exactly one ordinal. Ordinal order distinguishes arrangements with the same cardinality: $\omega$ and $\omega +1$ are both countably infinite, but only the latter has a greatest element. Transfinite induction and transfinite recursion extend finite reasoning through these order types. Their ordinal addition generally fails commutativity because it records concatenation order.

A cardinal number records the size of a set up to bijection. Two sets have the same cardinality exactly when such a bijection exists. Finite cardinals are $0,1,2,\dots$; the least infinite one is ${\aleph }_0$, the size of the natural numbers.

With the axiom of choice, every set can be well-ordered, and its cardinal is represented by the least ordinal in its bijection class, an initial ordinal. Infinite size does not behave like finite counting: the even natural numbers and all natural numbers have equal cardinality via $n\mapsto 2n$. This notion is distinct from the order type recorded by an ordinal.

For functions $f:A\to C$ and $g:B\to C$, the fiber product of sets isIt retains only pairs whose images agree. Its two projections commute with the maps to $C$, and any other compatible pair of maps factors through it uniquely.

For example, let $f,g$ send an integer to its parity class. Their fiber product consists of all integer pairs with the same parity. If $C$ is a singleton, every pair qualifies and the construction reduces to the Cartesian product. This elementary model explains the matching-data condition in more general pullbacks.

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.