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Approximate $h$ uniformly by finite sums $\sum f_j(x)g_j(y)$; their integral operators have finite-dimensional range and converge in operator norm, so $T_h$ is compact. If $h(x,y)=0$ for $x\le y$, iterated integration over an ordered simplex gives $\Vert T_h^n\Vert \le \Vert h{\Vert }_{\infty }^n/n!$. Thus $r(T_h)=0$, and compact spectral theory gives $\sigma (T_h)=\{0\}$.

Not for an arbitrary signed $L^1$ kernel. One can place very tall, very narrow spikes of $g$ at separated locations, with summable masses and total integral one, and matching still narrower spikes of an $L^1$ function accumulating at a Lebesgue point. Along a selected subsequence of dilations, a kernel spike lands on the matching function spike and contributes a fixed amount, while the total widths make the point a Lebesgue point. Thus compact support (or a suitable integrable radial majorant) cannot simply be dropped.

On disjoint events $A_n$ with $P(A_n)=c/n^2$, set $X_n=n{1}_{A_n}$. Then $X_n\to 0$ almost surely and $E{X}_n^2=c$, but any common dominator has integral at least ${\sum }_{n}nP(A_n)=c\sum 1/n=\infty$. Thus no integrable dominating variable exists.

With phase $\varphi (z)=(1+i)z-z^2/2$, the original ray is a contour integral. At the endpoint $z=-1$, the steepest direction solves $\mathrm{Im}(\varphi (z)-\varphi (-1))=0$ with decreasing real part; locally it is the direction for which ${\varphi }^{\prime }(-1)(z+1)$ is negative real.

Stirling and replacing the magnetization sum by an integral gives the stated form with $P(m)=-\beta (Bm+3J{m}^2+2K{m}^3)$, up to subexponential constants.

The second Adams operation $\chi \mapsto [g\mapsto \chi (g^2)]$ lies in the integral character ring, so $\tilde{\chi }=\sum n_i{\chi }_i$ with integers $n_i$. Norm one forces exactly one coefficient to be $\pm 1$; evaluation at the identity makes the sign positive. Squaring is bijective on conjugacy classes as well, so distinct irreducibles remain distinct. Thus the ${\tilde{\chi }}_i$ permute the irreducible characters.

A Poisson point process of intensity $\lambda (x)$ has independent counts on disjoint Borel sets, with $\Pi (A)\sim \mathrm{Poisson}({\int }_A\lambda (x)dx)$ whenever the integral is finite.

Multiplying two asymptotic eigenfunctions by $q$ reduces the integral, after the phase coordinate $s={\int }_a^x\sqrt{q}$, to a constant multiple of ${\int }_0^L\sin (n\pi s/L)\sin (m\pi s/L)ds=0$.

The identity theorem says analytic functions agreeing on a set with an interior accumulation point agree throughout a connected domain. Analytic continuation is an analytic function on a larger connected domain agreeing on a nonempty overlap. Factor the denominator as $(t+i)(t-z)$. For $\Im z<0$, contour closure below giveswith a removable value at $-i$; this entire expression is the continuation. For $\Im z>0$, the displayed real-axis integral encloses only $-i$ and equals $2\pi i{e}^{-5}/(z+i)$, so it is not that continuation. Finally $1/n\to 0$ only at a boundary point of $D$, so the zeros do not determine $G$; for example $G(z)=\sin (\pi /z)$ is nonzero and has them all.

Use $E=m{c}^2+p^2/(2m)+\cdots$ and replace the Fermi denominator by its Boltzmann form. The Gaussian integral gives

A probability density is a nonnegative function whose integral over an event gives its probability.

Let ${\delta }_t=\sqrt{p_t}-\sqrt{p}=(t/2)g\sqrt{p}+r_t$, with $\Vert r_t{\Vert }_2=o(|t|)$. Since $p_t-p=2\sqrt{p}{\delta }_t+{\delta }_t^2$, the Cauchy-Schwarz inequality gives $\Vert p_t-p-tpg{\Vert }_1\le 2\Vert r_t{\Vert }_2+\Vert {\delta }_t{\Vert }_2^2=o(|t|)$. Thus differentiability in quadratic mean suffices to differentiate bounded density integrals, even when the density itself has no useful pointwise derivative.

A bounded continuous factor $k_1$ and an integrable factor $k_2$ give a bounded velocity-integral gain on bounded continuous functions, with bound $\Vert Kf{\Vert }_{\infty }\le \Vert k_1{\Vert }_{\infty }\Vert k_2{\Vert }_1\Vert f{\Vert }_{\infty }$. The dominated convergence theorem proves joint continuity of the velocity integral. Combined with damped transport, the factorial bound for a Volterra iterate proves existence on arbitrary finite time slabs.

A perturbation can have negligible Hellinger distance but a large integral of its density's fourth power. At the uniform density on $[0,1]$, let $b(s)=6s(1-s)$ on $[0,1]$ and zero elsewhere, and let $f_t(u)=1-t^4+t^{-2}b(u/t^6)$. Then $\int f_t=1$ and $\int (\sqrt{f_t}-1{)}^2\le 2t^4=o(t^2)$, so the score function is zero. However $\int f_t^4\ge 72/(35t^2)$. Thus the density fourth-power functional is not continuous along this differentiable-in-quadratic-mean path, despite a bounded baseline and individually bounded nearby densities.

A measurable function is square-integrable when the integral of its squared absolute value is finite. Its equivalence class belongs to the L2 space; on a probability space, this means a finite second moment.

A Stäckel potential allows additive separation of the Hamilton-Jacobi equation in an orthogonal coordinate system. In axisymmetric prolate spheroidal coordinates, the relative-potential form is $\psi =[G(\lambda )-G(\mu )]/(\lambda -\mu )$. It gives energy, axial angular momentum and an independent third separation integral. Here $G$ is a one-variable generating function, not the gravitational constant. Generic bounded orbits are regular invariant-torus motions; resonance and coordinate degeneracy require separate treatment.

A statistical functional assigns a target value to each probability measure in a statistical model. Examples include an expected value, a quantile, and an integral of a power of a probability density function. Its behavior along statistical paths determines whether first-order influence-function representers are available.

A regular phase-mixed distribution $F(H,p_{\varphi },I)$ in an axisymmetric Stäckel potential is even separately in $v_{\lambda }$ and $v_{\mu }$, because these integrals depend on those components only through squares. Odd integration then gives all three mixed second moments zero, even if the $p_{\varphi }$ dependence produces azimuthal streaming. Bare stationarity alone is insufficient in degenerate resonant potentials with additional integrals. For example the isotropic harmonic Hamiltonian has the extra conserved $K=v_x{v}_y+{\omega }^{2}xy$; a positive Gaussian $\exp (-aH+ϵK)$ with $a>|ϵ|>0$ is a steady tracer and has nonzero $⟨v_x{v}_y⟩$. Its harmonic gravitational potential is a Stäckel potential in these spheroidal coordinates. This tracer need not itself source that harmonic gravitational potential. The alignment claim thus needs the phase-mixed three-integral hypothesis, not merely the weak form of Jeans theorem.

For $M>0$ with integral one, the Cauchy-Schwarz inequality makes the mass functional bounded on the weighted Hilbert space. The vector $M$ has norm one, so the velocity-reset projection is the orthogonal projection onto its span. Consequently $L=\Pi -I$ is bounded and self-adjoint. This Hilbert geometry differs from the norm-one positive reset projection on phase-space $L^1$.