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The boundary of the radius-$R$ ball has area $2{\pi }^{n/2}{R}^{n-1}/\Gamma (n/2)$.

The radius-$R$ ball in ${\mathbb{R}}^n$ has volume ${\pi }^{n/2}{R}^n/\Gamma (n/2+1)$.

Spherical coordinates $(r,\theta ,\varphi )$ in ${\mathbb{R}}^3$ use

A Hermitian metric on a holomorphic vector bundle $E\to X$ is a smoothly varying positive-definite Hermitian form $h_x$ on every complex fiber $E_x$.

Such a metric makes the underlying smooth bundle a Hermitian vector bundle.

A holomorphic line bundle is a complex line bundle with holomorphic local trivializations and holomorphic nonvanishing transition functions.

A crystal basis is the combinatorial limit of a basis of a quantum-group representation. Its colored directed graph records the actions of the Kashiwara lowering operators.

For a dominant integral weight $\lambda$, the irreducible character is

For a module with finite-dimensional weight spaces, its formal character is $\mathrm{ch}V={\sum }_{\mu }(\dim V_{\mu })e^{\mu }$ in the completed group algebra of the weight lattice.

For a Borel subalgebra $\mathfrak{b}=\mathfrak{t}\oplus {\mathfrak{n}}^{+}$ and a weight $\lambda$, the Verma module iswhere ${\mathfrak{n}}^{+}$ kills ${\mathbb{C}}_{\lambda }$ and $h\in \mathfrak{t}$ acts by $\lambda (h)$.

A vector $v$ has weight $\mu \in {\mathfrak{t}}^{\ast }$ when $hv=\mu (h)v$ for every $h$ in the Cartan subalgebra.

A Weight is the eigenvalue functional; a weight vector is a nonzero vector carrying it.

If $\mu$ and $\lambda$ are weights of a finite-dimensional representation, $\mu$ is dominant, and $\mu ⪯\lambda$, thenSuccessively subtract a simple root ${\alpha }_i$ for which $⟨\lambda ,{\alpha }_i^{\vee }⟩>0$ and use injectivity of the corresponding ${\mathfrak{sl}}_2$ lowering operator.

If $\lambda ={\sum }_i{m}_i{\omega }_i$ is dominant integral, the tensor product ${⨂}_{i}V({\omega }_i{)}^{\otimes m_i}$ contains $V(\lambda )$ as the irreducible summand generated by the tensor product of highest-weight vectors.

For Hermitian operators $A$ and $B$, one writes $A\le B$ when $B-A$ is positive semidefinite. This partial order is also called the semidefinite order.

The spectral theorem for normal operators gives a unique decompositionwhere $A_{+},A_{-}\ge 0$ and their supports are orthogonal. They are called the positive and negative parts of $A$. If $\mathrm{Tr}A=0$, then $\mathrm{Tr}{A}_{+}=\mathrm{Tr}{A}_{-}=\Vert A{\Vert }_1/2$.

At the upright spinning state, $p_{\varphi }=p_{\psi }=\lambda$. The reduced potential is locally minimized at zero inclination whenso sufficiently large axial angular momentum stabilizes the gravitationally inverted top.

For $u_t=D{u}_{xx}$ and $u(x,0)=1_{[-a,a]}(x)$, convolution with the heat kernel gives

For simple random walk on ${\mathbb{Z}}^d$ with $d\ge 3$,Consequently the probability of ever hitting $x$ from the origin has the same order.

The dual geometric representation acts on $V^{\ast }$ byThe open fundamental chamber consists of the functionals positive on every simple basis vector.

The Coxeter Gram matrix is the Gram matrix $G=(-2\cos (\pi /m_{ij}){)}_{i,j\in I}$ of the distinguished basis in the Geometric representation of a Coxeter group.

For a connected finite-dimensional Riemannian manifold, completeness of the Riemannian distance, geodesic completeness, and compactness of every closed bounded subset are equivalent. These conditions imply that any two points are joined by a minimizing geodesic. Pairwise existence of minimizing geodesics alone does not imply completeness: an open Euclidean ball is incomplete, yet the line segment between any two of its points stays in the ball and minimizes length.