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On a positively oriented smooth simple closed contour, set $\mathcal{C}f=(\pi i{)}^{-1}\mathrm{PV}{\int }_{\Gamma }f(t)/(t-z)dt$. The Sokhotski–Plemelj theorem gives the sum and difference of the two Cauchy boundary values. Changing the sign of the exterior branch converts the equation $\mathcal{C}f=g$ into a normalized additive Riemann-Hilbert problem with jump $g$. Applying the boundary formula again yields $f=\mathcal{C}g$, proving the involution on a suitable smooth or Hölder function class.

For $u\in H_0^1(0,1)$, write $u(x)=(1-x){\int }_0^x{u}^{\prime }-x{\int }_x^1{u}^{\prime }$. The squared $L^2$ norm of this piecewise constant test kernel is $x(1-x)$, so the Cauchy-Schwarz inequality gives the displayed pointwise estimate. Integrating against a nonnegative weight $w$ gives $\int w|u{|}^2\le [\int wx(1-x)]\int |u^{\prime }{|}^2$. In particular $w=x^2$ gives the constant $1/20$, useful for controlling a negative quadratic potential by derivative energy. The estimate is sharp at each fixed evaluation point, whose equality profile is a tent function, although its integrated constant need not be sharp.

For a positive integer-valued random variable with finite mean $\mu$, the size-biased distribution has probabilities $k\mathbb{P}(Y=k)/\mu$. They sum to one by the definition of expected value. Multiplying the differentiated probability generating function by $s/\mu$ gives the displayed identity for $0\le s<1$, with the endpoint interpreted by its limit.

For the positive-support geometric distribution with success probability $p$, size bias gives $\mathbb{P}(\tilde{Y}=k)=k{p}^2(1-p{)}^{k-1}$. The sum of two independent copies has $k$ positive-integer decompositions at value $k+1$, so its shifted probability is exactly this expression. Its probability generating function is $p^{2}s/(1-(1-p)s{)}^2$, also obtained by the size-biased probability generating function identity.

Reweight probabilities proportionally to the nonnegative value being observed, then divide by its positive finite expected value. This produces a size-biased distribution. Values twice as large receive twice the original relative weight; this is a change in the sampling law rather than a deterministic multiplication of the sampled value.

For a finite-dimensional commutative complex algebra, a simple module is a cyclic quotient of the algebra and hence finite-dimensional. Every action map is a module endomorphism, since the algebra is commutative. The complex Schur's lemma makes each action scalar, so every vector subspace is a submodule; simplicity forces dimension one. The finite-dimensional hypothesis cannot simply be deleted: $\mathbb{C}(t)$ acting on itself is a simple module of infinite complex dimension.

A Hahn decomposition of a signed measure partitions its space into measurable sets $P,N$ such that every measurable subset of $P$ has nonnegative measure and every measurable subset of $N$ has nonpositive measure. For finite total variation measure, set ${\mu }^{+}=(|\mu |+\mu )/2$ and choose a ${\mu }^{+}$-null set $N$ maximizing $|\mu |(N)$: a countable union of approximating null sets attains the supremum. If some subset of its complement has negative signed mass, successively remove disjoint subsets with positive mass at least half the supremum still available. Their masses tend to zero because their sum is bounded by total variation. The remainder has no positive-measure subset, has negative mass, and is ${\mu }^{+}$-null, contradicting maximality after adjoining it to $N$. On $P$ one has $|\mu |=\mu$ and on $N$ one has $|\mu |=-\mu$. Consequently $d{\mu }^{+}/d|\mu |=1_P$, an explicit Radon-Nikodym derivative.

A viscous shock is a smooth travelling transition produced when positive diffusion resolves a shock of a conservation law. For $u_t+q(u{)}_x=ϵ[D(u)u_x{]}_x$ with $D>0$, an inner coordinate $X=(x-ct)/ϵ$ gives $D(U)U^{\prime }=q(U)-q(u_{-})-c(U-u_{-})$. The endpoint conditions impose the Rankine-Hugoniot speed. If $q$ is strictly concave, the right side is positive between its roots, so the admissible profile increases from its rear state to its front state. This supplies the entropy selection directly from a regularizing mechanism.

The Shapley value is a linear map on real characteristic functions with zero value at the empty coalition. In each player permutation, the marginal contribution to $av+bw$ is $a$ times the marginal to $v$ plus $b$ times the marginal to $w$. Taking their expectations proves the formula for arbitrary real $a,b$. This directly proves real homogeneity as well as cooperative-game additivity.

To prove equality of two sets, show each set inclusion separately. If both inclusions hold, every element belongs to one set exactly when it belongs to the other; the axiom of extensionality gives equality. Each direction may use different hypotheses. For example, when $x\in A$, the condition $x\notin A\setminus B$ implies $x\in B$; without $x\in A$, that inference would be invalid.

Two disjoint compact subsets of a Hausdorff space have disjoint open neighbourhoods. For every $p\in P$, separation of a point and a compact set gives disjoint open sets $G_p\ni p$ and $H_p\supseteq Q$. Choose finitely many $G_p$ covering $P$, then take their union for $U$ and the matching finite intersection of the $H_p$ for $V$. The compactness of both subsets is used, once in the point-versus-compact set result and once in the outer finite selection. The ambient Hausdorff space need not be compact or a normal space.

Let $(u_n,v_n)\to (0,0)$ with both coordinates nonzero, $r_n=|u_n|/4$, $s_n=|v_n|/4$, and $\varphi (t)=\max (1-|t|,0)$. The displayed function equals one at every $(u_n,v_n)$ and zero on both coordinate axes. Each coordinate line off the corresponding axis meets only finitely many supports, so its restriction is a finite maximum of continuous functions. On the axes it is zero. Hence $B$ is separately continuous and disproves convergence of this sequence in any topology making every separately continuous function continuous. Isotropic radial bumps need not have this property when one coordinate approaches zero much faster than the other.

A finite-dimensional complex semisimple Lie algebra which is an ideal in a finite-dimensional Lie algebra has the displayed direct complement. The restriction of the Killing form of $M$ to $L$ equals the Killing form of $L$: the induced adjoint maps of elements of $L$ on $M/L$ vanish. This restriction is nondegenerate by the Cartan criterion for semisimplicity. Invariance makes its orthogonal complement an ideal. Nondegeneracy on $L$ gives a direct vector-space sum with that complement, and the bracket of the two ideals lies in their zero intersection, so they commute. The ambient algebra need not itself be semisimple.

Let $V$ be a finite-dimensional complex vector space and let $L$ be a Lie subalgebra of its traceless endomorphisms acting irreducibly. Its solvable radical $R$ is simultaneously triangularizable by Lie's theorem. The resulting finite set of diagonal characters is invariant under conjugation by $e^{tx}$ for $x\in L$, because this conjugation preserves the ideal $R$ and gives an equivalent representation. Each character varies continuously with $t$ in a finite set, hence is constant; differentiation gives $\lambda ([x,r])=0$.

A nonzero common $R$-eigenspace exists by Lie's theorem. The displayed vanishing makes it invariant under all of $L$: $r(xv)=\lambda (r)xv+\lambda ([r,x])v=\lambda (r)xv$. Irreducibility therefore makes it all of $V$. Thus $R$ acts by scalars, and the zero-trace condition forces those scalars to vanish. The matrix representation is faithful, so $R=0$. The traceless condition matters: adjoining scalar matrices to an irreducible semisimple algebra gives a reductive algebra with nonzero centre.

Let a semisimple complex Lie algebra $L={⨁}_i{L}_i$ act faithfully and irreducibly on a space $V$ of prime dimension. Complete reducibility for $L_1$ splits $V$ into isotypic components. The commuting other ideals preserve these components, so irreducibility leaves just one, of the form $W_1\otimes {\mathrm{Hom}}_{L_1}(W_1,V)$. Repeating with the other factors gives $V\cong {⨂}_i{W}_i$, with each $W_i$ an irreducible module for its simple ideal.

Faithfulness makes every factor nontrivial. A nonabelian simple Lie algebra has only the trivial one-dimensional module, so $\dim W_i\ge 2$. Their product is the prime $\dim V$, forcing just one simple ideal. Hence $L$ is simple. This is a representation-dimension obstruction, independent of the classification of root systems.

A point interaction concentrates a coefficient of a linear differential equation at a single interior point. On an interval $0<x<\ell$, with $0<c<\ell$, consider $L_{\alpha }y=y^{\prime \prime }+\alpha y(c){\delta }_c$. The domain consists of continuous functions that are twice differentiable away from $c$, with the required one-sided derivatives. Multiplication of the Dirac delta by $y$ means the unambiguous point value $y(c){\delta }_c$. It does not mean an undefined product of a delta with an arbitrary discontinuous function.

The equation is interpreted using distributional derivatives. Smooth forcing gives an ordinary equation on each side, with transmission conditions for a delta coefficient at the interaction. A delta potential in the one-dimensional Schrödinger equation is a physical example after choosing the corresponding sign and constants.

The third derivative is the derivative of the second derivative. On the interior of a cubic polynomial piece it is constant. A cubic spline can have different constants on adjacent pieces, so its third derivative need not be continuous across a spline knot.

The antiplane field $w=b\arg (x+iy)/(2\pi )$ has a smooth local gradient on the punctured plane and satisfies the differential strain compatibility condition there. Its circulation around the defect axis is nevertheless $b\ne 0$, excluding a globally single-valued displacement on that domain. A radial cut permits a branch with an axial jump $b$ between the two cut faces; joining faces with that mismatch is the Volterra construction of a screw dislocation with Burgers vector parallel to the axis. The stress has a singular core where linear elasticity requires microscopic or plastic regularization. This example separates a local differential integrability test from global closure around a hole.

On a bounded Lipschitz domain, the Poisson equation $-\Delta u=f$ with Robin boundary condition ${\partial }_{n}u+\beta u=0$ has the displayed weak formulation in $H^1(\Omega )$ when $\beta$ is a bounded boundary coefficient. Boundary integrals use Sobolev traces. It follows from Green's first identity, requiring only first weak derivatives of $u$. If $\beta$ is uniformly positive, testing the difference of two solutions against itself proves uniqueness: its gradient and its boundary Sobolev trace vanish. The Poincare inequality with a boundary trace also makes this a coercive bilinear form.