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A boson has a symmetric complete multiparticle state under exchange of identical particles. The symmetry includes internal degrees of freedom as well as position. Under the locality and positivity hypotheses of relativistic quantum field theory in three spatial dimensions, the Spin-statistics theorem relates this exchange behavior to integer spin.

Symmetry permits several bosons to occupy one single-particle mode. Creation into an already occupied mode has the bosonic square-root occupation factor, contributing to stimulated processes and Bose-gas behavior. Photons and helium-4 atoms are examples at different descriptive levels. Symmetry does not force every preparation to exhibit a particular amount of bunching or to form a condensate: measured correlations also depend on the state, interactions and detector resolution.

If $R$ is Noetherian, $I$ is an ideal, and $N\subseteq M$ are finitely generated modules, then some $k$ satisfiesfor every $n\ge k$.

An $I$-filtration $(M_n)$ is stable when $I{M}_n=M_{n+1}$ for all sufficiently large $n$. Every stable $I$-filtration is equivalent to the $I$-adic filtration.

The $x$-adic filtration is the filtration by powers of the principal ideal $(x)$.

Special parameter choices can identify a hypergeometric solution with an elementary solution of the same second-order differential equation by matching its local normalization.

The axis of a hyperbolic tree isometry $g$ is its unique invariant line. The isometry acts on this line by translation through its translation length.

If $s>n/2+\alpha$ with $0<\alpha \le 1$, then Fourier inversion, $|e^{it}-1|\le C_{\alpha }|t{|}^{\alpha }$, and Cauchy-Schwarz giveThe weighted frequency integral converges precisely because $s-\alpha >n/2$.

On a bounded regular domain, a bounded subset of $C^{k,\alpha }$ is relatively compact in $C^{l,\beta }$ whenever $k+\alpha >l+\beta$. This is an Arzelà-Ascoli theorem argument applied successively to the derivatives.

On a ball of radius $R$, lower derivative norms of $u\in C^{l,\alpha }$ are bounded by an arbitrarily small multiple of the top Hölder seminorm plus a multiple of $\Vert u{\Vert }_{C^0}$. One endpoint form is

The fiber class $F$ is the divisor class of a fiber of ${\mathbb{F}}_k\to {\mathbb{P}}^1$. Its complete linear system gives the ruling.

The negative section $S\subset {\mathbb{F}}_k$ has self-intersection $S^2=-k$. Together with a fiber $F$ of the ruling, it generates the Picard group and satisfies $F^2=0$ and $S\cdot F=1$.

For a Hilbertian basis $(e_i)$,

For a chosen additive function such as module length, the Poincare series of a graded module is the generating function of that function on its homogeneous components. In this usage it is a Hilbert series.

Unitary gauge uses a gauge transformation to remove the Goldstone fields from the Higgs multiplet, leaving the physical radial Higgs excitation and massive vector fields.

A critical point of a real function with continuous second partial derivatives is nondegenerate when its Hessian matrix is an invertible matrix there, equivalently when none of its eigenvalues is zero. A positive-definite matrix gives a strict local minimum, a negative definite Hessian gives a strict local maximum, and opposite-sign eigenvalues give a saddle point of a scalar function. A nondegenerate critical point need not be an extremum: $x^2-y^2$ has Hessian matrix $\mathrm{diag}(2,-2)$, which is an invertible matrix.

Such critical points are the local building blocks of Morse theory.

A Toeplitz matrix is constant along each diagonal, so its entries have the form $T_{nm}=t_{n-m}$.

A set $x$ is reasonable when every member of $\mathrm{TC}(\{x\})$ is countable. In ZFC this is equivalent to $x$ being hereditarily countable.

On $\mathbb{C}[x]$, the operators $d/dx$, multiplication by $x$, and the identity satisfy $[d/dx,x]=1$. Sending $a,b,c$ to these operators gives a faithful irreducible infinite-dimensional representation of the Heisenberg Lie algebra.

This is a finite-group realization of an Iwahori-Hecke algebra. For a finite group $G$ with a BN-pair, the Iwahori-Hecke algebra over $k$ is the endomorphism algebra of the permutation module $k[G/B]$, up to the conventional opposite algebra. Its standard basis is indexed by the Bruhat double cosets.

Fixing the cyclic momenta $p_{\psi }=\lambda$ and $p_{\varphi }=\mu$ leaves the canonical pair $(\theta ,p_{\theta })$ and a one-degree-of-freedom Hamiltonian with effective potential