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The total variation of a function $f:[a,b]\to \mathbb{R}$ measures its accumulated rises and falls, rather than just its change between the endpoints. For each partition of an interval $\Pi =(a=t_0<\cdots <t_n=b)$, add the absolute values of its sampled increments. Then take the supremum over all such partitions of an interval:The function has bounded variation when $V_a^b(f)<\infty$. This definition uses its value at every point of the interval, including both endpoints.^[ref]

For a nondecreasing function, all increments are nonnegative and the sum telescopes to $f(b)-f(a)$. More generally, the triangle inequality gives $|f(b)-f(a)|\le V_a^b(f)$. Equality can fail dramatically when the function changes direction: an object can travel a long distance and finish near its starting position. Here we measure movement of the function's value; arc length of its function graph also includes horizontal movement.

Every Riemann-integrable function on a compact interval can be approximated in $L^1$ by continuous functions. Consequently their integrals converge uniformly over all subintervals.

For a tagged partition $a=x_0<\cdots <x_n=b$ with $t_i\in [x_{i-1},x_i]$, the corresponding Riemann sum isIts limits over increasingly fine partitions define the Riemann integral.

In dimension greater than two, if sectional curvature is pointwise independent of the two-plane, the contracted Bianchi identity forces that curvature to be constant on each connected component.

The positive-definite specialization is Schur's lemma.

The Ricci tensor and scalar satisfy ${\nabla }^{\alpha }{R}_{\alpha \beta }=\frac{1}{2}{\nabla }_{\beta }R$.

The algebraic symmetries of the Riemann curvature tensor leave only one independent curvature component in two dimensions and forceConsequently the Einstein tensor vanishes identically.

A Drude response has a single relaxation pole, $G_R(\omega )={\sigma }_0/(1-i\omega {\tau }_0)$. Causality places the pole in the lower half plane when ${\tau }_0>0$.

The spectral function records the weighted energy differences connected by an operator. With the convention $G_R=-i\theta ⟨[A,A]⟩$, it is proportional to $-\mathrm{Im}{G}_R$.

In the circular restricted three-body problem the two massive bodies move on circular orbits about their barycentre. The problem is autonomous in their uniformly rotating frame.

The sperm number compares filament length with an elastohydrodynamic penetration length. For periodic forcing at angular frequency $\omega$, the usual convention isIts fourth power is a viscous-to-bending ratio on the full filament length. In steady cross-flow problems a different convention also occurs: $\mathrm{Sp}={\zeta }_{\perp }{L}^{3}U/A$, where $U$ is flow speed and $A$ the filament bending modulus. The defining formula distinguishes these conventions.

If a stationary Gaussian process has covariance spectral density $\hat{K}$, its reproducing-kernel Hilbert space has normon the functions for which this integral is finite.

The kernel trick evaluates inner products of feature map outputs through $k(x,z)=⟨\varphi (x),\varphi (z)⟩$ without explicitly constructing those vectors. An algorithm expressed entirely through these inner products can therefore operate on nonlinear derived coordinates using only evaluations of a positive-semidefinite kernel on the original inputs. A classifier linear in the derived coordinates can have a nonlinear boundary in the original coordinates.

The polynomial kernel gives a finite, fully visible example: $k(x,z)=(x^{T}z{)}^2$ corresponds to $\varphi (x)=(x_1^2,\sqrt{2}{x}_1{x}_2,x_2^2{)}^T$. The XOR separation by a product feature below makes the interaction coordinate $x_1{x}_2$ useful, while dual prediction with a kernel shows how to predict without calculating it explicitly.

An arbitrary score called a similarity need not be a positive-semidefinite kernel. Symmetry alone, even with nonnegative entries, is insufficient: the symmetric matrixhas $a^{T}Sa=-2$ for $a=(1,-1{)}^T$, so it cannot be a Gram matrix of real inner products. A positive-semidefinite kernel requires a positive semidefinite matrix for every finite choice of inputs, not just the observed training data.^[ref] The existing Moore-Aronszajn theorem connects this requirement to a Reproducing kernel Hilbert space; this example needs only finite linear algebra.

For feature vectors $\varphi (x_i)$, the empirical uncentered kernel covariance operator is

Every positive-semidefinite kernel $k$ has a Hilbert-space feature map $\varphi$ satisfying $k(x,y)=⟨\varphi (x),\varphi (y)⟩$, and determines a unique reproducing-kernel Hilbert space.

The representer theorem reduces regularized optimization over a reproducing-kernel Hilbert space to the span of kernel sections at the observed points.

For $m,n\ge 0$,The identity follows by multiplying the weight characters and comparing their nested weight strings.

On $V_n$, the central element $-I$ acts as $(-1{)}^n$. It therefore acts trivially on $V_n\otimes V_n$, although it need not act trivially on the tensor square of a representation containing irreducibles of both parities.

The standard invariant alternating form identifies $V_1$ with its dual. Its symmetric powers identify every $V_n$ with $V_n^{\ast }$, and complete reducibility then gives $V\cong V^{\ast }$ for every finite-dimensional complex $S{U}_2$-representation.

Every finite-dimensional complex $S{U}_2$-representation is a direct sum of the pairwise nonisomorphic irreducibles $V_n={\mathrm{Sym}}^n({\mathbb{C}}^2)$.